Health Care
Health Care
Health Care
5-1-2010
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by
Brian C. Payne
A DISSERTATION
Major: Business
Lincoln, Nebraska
May, 2010
The first essay investigates whether health care is a priced factor in asset returns.
Specifically, in the search for empirical relationships between macroeconomic factors
and asset returns, health care appears to be a significant US economic force receiving less
attention than others such as (aggregate) inflation, production, or consumption measures.
We use the medical care component of the Consumer Price Index to measure medical
inflation shocks as a candidate macroeconomic factor whose riskiness the market
rewards. Incorporating multiple model specifications during the period 1967-2009, we
find this inflationary component to be a relatively robust source of priced risk in US stock
returns.
The second essay demonstrates how a genetic algorithm (GA) technique with
standard parameters and the appropriate fitness function can generate five-asset portfolios
that effectively hedge macroeconomic risks, including health care cost inflation.
Investigating 40 macroeconomic series-year combinations, the GA generates 36 (11)
hedging portfolios that are weakly (unambiguously) preferred to unmitigated risk
exposure in an out-of-sample analysis between 2005 and 2008. This same technique can
create parsimonious mimicking or tracking portfolios for investable assets such as mutual
funds and exchange-traded funds (ETFs), particularly in the down market of 2008.
iv
Dedication
I dedicate this work, as with all my efforts, to my wife, Melissa, and our children. You
are simultaneously my greatest heroes and biggest fans.
v
Acknowledgements
Thank you to my committee, Dr. John M. Geppert (Chair), Dr. Donna M.
Dudney, Dr. Gordon V. Karels, and Dr. David B. Smith, for your time, energy, and
assistance. In particular, completing this research would have been literally impossible
without you, Dr. Geppert. In 17 years with the Air Force, I have met many, many
patriots, public servants, and teachers; your commitment to this country, your
community, and your students places you in the upper tail of the distribution. You
H[HPSOLI\WKHZRUGWHDFKHUDQG I am forever indebted to you for mentoring me.
Dr. Jennings and Lieutenant Colonel (Retired) Fraser, you initiated my interest in
this topic by allowing me to participate in your research. Thank you.
Mom and Dad, you taught me work ethic and perseverance and have endlessly
supported me. I strive to become to my children a IUDFWLRQRIZKDW\RXve been to me.
You know how much you have impressed upon my life. Thank you.
Bob, Rita, and Mary, these past three years never would have occurred without
you. You gave us a reason for being here and have supported our family for the duration.
Thank you.
Uncle John, you uniquely know the instrumental, no critical, role you have played
in this chapter of my life. I will forever remember it and hope to pay it forward someday.
Thank you.
Jeff and Jirka, through this experience, in my vernacular, we have become
brothers-in-arms. Whether it was for personal or professional reasons, I have leaned on
you both many times; you have never wavered. I will reciprocate anytime. Thank you.
vi
To all of the other graduate students, before and after, I appreciate the individual
roles you have played in my experience as well as the larger role you have played for the
success of this department and program. You include: Chris, Jeff, Jin, Stoyu, Lee,
Rashiqa, Yi, Daria, and Thuy.
Lastly, but most importantly, I thank God. You are ultimately the source of all
good things, and I am not worthy of the grace, guidance, and countless Blessings you
have provided to me during this journey.
vii
Grant Information
I would like to recognize and thank the U.S. Air Force and the U.S. Air Force Academy
Department of Management for respectively funding and sponsoring this research. The
views expressed here are solely mine and do not necessarily represent the views of these
sponsors.
viii
Table of Contents
Essay #1: Health Care as a Priced Factor in Asset Returns
Introduction
Literature Review
Hypothesis Development
12
Methodology
13
13
20
25
Results
Fama-MacBeth (1973) Two-Pass Method
29
29
40
Discussion
42
Conclusion
45
References
47
Appendix A: Figures
49
Appendix B: Tables
58
92
ix
Essay #2: Using Genetic Algorithms for Hedging Health Care Costs,
Managing Macroeconomic Risk, and Tracking Investments
94
Introduction
94
Literature Review
95
Data
98
98
105
Results
110
118
118
125
Conclusions
133
References
136
Appendix A: Figures
137
Appendix B: Tables
152
167
x
List of Figures and Tables
Essay #1: Health Care as a Priced Factor in Asset Returns
Figure 1: Consumer Price Index (CPI) Measures, January 1967-August
2009
50
51
52
53
54
55
56
59
60
61
63
64
66
69
Table 8: Fama-MacBeth (1973) Results for Priced Factors, January 1967August 2009
70
76
xi
January 1967-August 2009
Table 10: Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for
Priced Factors-Flannery & Protopapadakis (2002) Macroeconomic Factors,
January 1967- August 2009
80
84
88
Essay #2: Using Genetic Algorithms for Hedging Health Care Costs, Managing
Macroeconomic Risk, and Tracking Investments
Figure 1: Monthly Consumer Price Index (CPI) Measures, January 1967August 2009
138
139
140
141
142
143
144
147
148
150
Table 1: Fitness Level for Asset Portfolios across Generations using Proof
of Concept Returns, January 1967-August 2009
153
xii
Table 2: Fitness Level for Asset Portfolios over Generations using Actual
Target (Medical Inflation) and Investable Assets (303 Stocks from CRSP,
Long Government Bond, Short Government Bond, and Treasury Bill),
During the Test and Validation Periods, January 1967-December 2007
154
Table 3: Portfolio Hedging Effectiveness against Medical Inflation, Outof-Sample Period, January 2008-December 2008
155
156
157
158
163
165
1
Essay #1: Health Care as a Priced Factor in Asset Returns
Introduction
Health care represents a significantand growingportion of the US economy.
Nationally, in 2009 health care spending is expected to reach $2.5 trillion, which
represents over 17 percent of the gross domestic product (GDP). In other words,
currently one of every six dollars spent in this country is health care related. This
percentage has doubled over the past 30 years, and the Congressional Budget Office
predicts it will double again over the next 25 years.1 The impact of such health care costs
can be catastrophic to individuals, as documented by Himmelstein et al (2009). They
find 62 percent of personal bankruptcies filed in 2007 were linked to medical expenses
even though nearly 80 percent of those filing for bankruptcy had health insurance. This
rate of medical-cost-induced personal bankruptcies has increased by almost 50 percent
since 2001.
Extrapolating this result from individual agents to the firms that compose the US
financial market would indicate that firms with greater exposure to health care expenses
should face higher risk of financial distress. One software CFO summarizes the risk of
health care costs VXFFLQFWO\ZKHQKHVWDWHV+HDOWK-care costs are increasing faster than
SUHWW\PXFKDQ\RWKHUFRPSRQHQWRIRXUFRVWVWUXFWXUHWKH\KDYHWKHSRWHQWLDOWRFURZG
RXWLQYHVWPHQWLQRWKHUDUHDVDQGXOWLPDWHO\PDNHXVOHVVFRPSHWLWLYH2 According to
portfolio theory, unless this risk is diversifiable across firms, investors should price this
firm-specific risk and demand greater firm-specific returns for bearing it. The purpose of
1
http://www.forbes.com/2009/07/02/health-care-costs-opinions-columnists-reform.html
K^<ZW^,-Care Reform is High, but Some CEOs Take
sCFO December 2009, pp. 38-43. This magazine is published by a subsidiary of The
Economist.
this research is to determine the impact of health care costs on asset prices in the US
market and to assess the degree to which medical costs are a priced risk in the US
economy.
http://ehbs.kff.org/
recruitment and UHWHQWLRQEHQHILW4 Consequently firms face medical care costs they
cannot control yet must absorb due to their need to attract and keep high-quality labor.
This trend appears valid for publicly-traded (i.e., larger) firms despite recent popular
press reports that unaffordable premium increases for small businesses will likely
decrease the amount they cover.5 Thus it appears escalating medical care costs represent
a reasonable candidate as a systematic risk to stockholders who hold publicly-traded
firms.
Literature Review
For decades now research has sought to establish an empirical connection
between macroeconomic events and stock price movements that theoretically ought to
exist. This study extends the prior efforts that have documented a contemporaneous
relationship between certain macroeconomic factors and returns. As we describe in more
detail later, our focus on medical inflation augments previous findings regarding
aggregate inflation. Studies show aggregate inflation surprises6 and returns tend to be
negatively-related over time (see, for example, Chen, Roll, and Ross (1986), Flannery
and Protopapadakis (2002) and Hong, Torous, and Valkanov (2007)). Further, Chen,
RROODQG5RVVSURYLGHZHDNHYLGHQFHWKDWLQIODWLRQLVDSULFHGULVNIDFWRUZKLOH
the others do not investigate this result.
Before summarizing the foundation of literature upon which this study will build,
it is important to understand the difference between a factor explaining stock returns and
the factor serving as a priced source of risk. While a macroeconomic factor, such as
aggregate inflation, might exhibit high covariance with particular stock returns (i.e., have
DKLJKEHWDLQDWLPHVHries regression), this relationship does not say whether the
market views this factor as a risky one worthy of return premia. In order for the factor to
EHFRQVLGHUHGULVN\LWLVQHFHVVDU\IRUDQ\VHFXULW\VEHWDWRFRUUHODWHZLWKWKH
VHFXULW\VH[FHVV returns in the cross-section. The Fama-MacBeth (1973) two-pass
procedure represents the classical way to determine whether a factor is priced by the
PDUNHW:HZLOOGHVFULEHWKLVSURFHGXUHLQPRUHGHWDLOXQGHUWKH0HWKRGROJ\VHFWLRQ
The seminal study on the relationship between macroeconomic data and stock
returns, Chen, Roll, and Ross (1986), investigates monthly stock returns between 1958
and 1984 with a goal of bridging the gap that existed between the theoretical idea that
macroeconomic events drive stock prices at some level and the fact that nobody had
found empirical evidence of such a connection. Specifically, the authors study whether
industrial production, inflation (both expected and unexpected), a term risk premium
(difference between return on long government bond and short Treasury bill), and a
default risk premium (difference between return on portfolio of Bbb rated bonds and
short Treasury bill) explain expected stock returns over time. They admit these
macroeconomic series are by no means exhaustive in their inclusion. In briefly
addressing other theoretical predictions and as a robustness test, the authors augment
their model with the market risk premium, a measure of consumption, and an oil price
index (PPI for crude), ultimately concluding that the former has a negligible effect, and
neither of the latter factors are priced.
Methodologically, CRR (1986) form twenty size-based portfolios whose returns
are used as the dependent variables in their models, since using portfolios helps to
mitigate errors-in-variables problems. They then implement a Fama-MacBeth (1973)
two-pass methodology to assess whether the aforementioned macroeconomic factors are
priced. While the inclusion of the market returneither value- or equal-weighted
performs well in the first-pass time series regressions, as we expect from the Sharpe
(1964) and Lintner (1965) Capital Asset Pricing Model (CAPM), this factor is not priced
in the presence of certain macroeconomic factors once the second-pass cross-sectional
regressions are completed. The authors indicate that relatively smooth macroeconomic
measures will inherently fail to explain a substantial amount of the variance in noisy
stock returns. As a result, none of their models depict the coefficient of determination
PHDVXUHLH5-VTXDUHGIRUDVVHVVPHQW:HDQWLFLSDWHVLPLODUO\XQLPSUHVVLYH5squared values for our first-pass regressions that include only macroeconomic factors.
The consensus of this research is that industrial production, changes in the market
risk premium, yield curve twists, and measures of unanticipated inflation and expected
inflation changes are all significant in explaining expected stock returns. The effect of
these variables on stock returns is robust to the inclusion of the market return factor (per
CAPM) as well as to the inclusion of consumption and oil robustness variables. In sum,
this study represents a hallmark effort in tying together the theory and empirical
representation of macroeconomic events influencing stock returns.
In contrast to the study of macroeconomic factors, Fama and French (1993) study
and find evidence of a parsimonious factor model that explains the variation in both stock
and bond returns. Specifically, they contend that the following five factors explain the
returns: excess market return (value-weighted market return minus one-month Treasury
bill), SMB (Small-minus-Big, calculated by subtracting the return of the decile of the
largest stocksby market capitalizationfrom the decile of smallest stocks), HML
(High-minus-Low, calculated by subtracting the return of the stock decile having the
lowest book-to-market equity ratio from the decile with the highest book-to-market
equity ratio), DEF (default risk premium, calculated by subtracting the long government
bond from a Baa-and-below portfolio of similar duration corporate bonds), and TERM
(term risk premium, calculated by subtracting the one-month Treasury from the long
government bond).
These authors investigate monthly returns from 1963 to 1991 for 32 different
portfolios of returns, which include 25 stock portfolios and 7 bond portfolios. They form
the stock portfolios by intersecting the quintiles of size and book-to-market equity ratio.
Their bond portfolios include two government portfolios, short- and long-term, and five
corporate portfolios ranging in grade from Aaa to low-grade (or junk) bonds.
While their model is admittedly atheoretical and strictly empirically-founded, the
Fama and French (1993) results are econometrically impressive. Their model parameters
are highly statistically significant, and the coefficient of determination values are
extremely large across the 32 portfolios. They do not price these particular factors in the
traditional Fama-MacBeth (1973) two-pass manner, since they indicate adding bonds to
the cross-VHFWLRQDOUHJUHVVLRQVZRXOGEHGLIILFXOWEHFDXVHVL]HDQGERRN-to-market
HTXLW\KDYHQRREYLRXVPHDQLQJIRUJRYHUQPHQWDQGFRUSRUDWHERQGV,QVWHDGRI
pricing their factors, the authors test for their cross-sectional effectiveness in the market
by jointly-testing whether the intercept terms for all 32 portfolios are zero using the
Gibbons, Ross, and Shanken (1989) methodology. While this test does not unequivocally
support their modelmainly due to the small/low book-to-market portfolio having a nonzero intercepttheir results indicate the factors explain stock and bond returns rather
well. Finally, they perform a variety of robustness tests, including examining the January
effect and bisecting the sample, and find the results tend to hold.
Since theory indicates macroeconomic factors should influence stock returns by
serving as nondiversifiable risk factors (Ross (1976)), Flannery and Protopapadakis
(2002) investigate the impact of 17 macroeconomic series on daily stock return mean and
conditional volatility for the period between 1980 and 1996. Ultimately the authors
confirm that inflation (CPI), the Producer Price Index (PPI), and a monetary aggregate
(M1 and M2) influence stock returns, as previous research has indicated. Additonally,
they make the novel discoveries that balance of trade (BOT), employment, and housing
VWDUWVH[SODLQVWRFNUHWXUQVFRQGLWLRQDOYRODWLOLW\7KH\GHWHUPLQHQHZVIRUWKHVHVL[
variables is also associated with higher trading volume, an expected empirical result.
Meanwhile, they fail to find influences from Industrial Production or GNP, as previous
research has documented.
Their data set of macroeconomic series is arguably the most comprehensive to
date, and the authors utilize a cRQYLQFLQJPHDQVWRPHDVXUHWKHVXUSULVHFRPSRQHQWRI
the measures. Their method is important, because it is the surprise, or unexpected
component, of macroeconomic data that should theoretically induce stock price changes,
or returns.7 The authors measure surprise by using data from MMS International (now a
VXEVLGLDU\RI6WDQGDUG 3RRUVRQDQDO\VWVH[SHFWDWLRQVRIPDFURHFRQRPLFGDWDYDOXHV
for a given date. By comparing these expectations to the actual announced value, the
authors quantify the surprise component. In using daily returns and volatility, the authors
argue they can quantify most precisely the effect the news has on the market.
To mitigate criticism, Flannery and Protopapadakis (2002) employ various
techniques. To avoid allegations of model misspecification, they include a host of
conditioning variables, including: lagged market return, lagged risk-free rate, lagged junk
bond premium (AAA-BAA returns), lagged term risk premium, lagged dividend-to-price
ratio, lagged firm size value, and a host of timing controls to account for post-holiday
returns and the January effect. Addtionally, they forestall the econometric problem of
heteroskedasiticity by employing a generalized autoregressive conditional
heteroskedasticity (GARCH) model to investigate returns.
We aim to augment and extend Flannery and Protopapadakis (2002) since (1)
their study looks at aggregate inflation (versus medical inflation) as one of the
macroeconomic series and (2) their study identifies priced factor candidates, but they
never determine whether these candidates are priced. While we also confirm the negative
relationship between contemporaneous inflation (and its surprise), we also investigate
LQIODWLRQVVXE-component related to medical costs. Finally, whereas these authors
determine which macroeconomic factors explain stock returns (and volatility) over time
(i.e., they complete the Fama-MacBeth first-pass), our study investigates whether any
Theory tells us factors proxy for the stochastic discount factor (SDF), which is a ratio of the present and
expected future marginal utilities of consumption. Under the permanent income hypothesis, consumption
is a random walk, which induces prices that necessarily deviate from expected levels and generate returns.
candidate factors we discover are indeed priced in equilibrium. That is, we complete
both the first- and second-pass for the relevant factor candidates.
While prior studies have investigated contemporaneous macroeconomic
YDULDEOHVDVVRFLDWLRQZLWKVWRFNUHWXUQV+RQJHWDOLQYHVWLJDWHWKHLQIRUPDWLon
diffusion theory by determining that certain industry returns lead the broad market
returns. These authors determine that portfolios for retail, services, commercial real
estate, metal, and petroleum forecast the stock market, in some cases by up to two
months. Their finding is generally robust to the eight-largest non-US stock markets.
Additionally, they relate their results to economic theory by discovering that industries
that forecast the market also generally forecast two macroeconomic series (Industrial
Production Growth and the Stock and Watson (1989) coincident index of economic
activity) that explain returns.
Using monthly returns from 1946-2002, Hong et al (2007) investigate the ability
of the Fama-French 38 industry sectors to explain broad market returns. Their intent is to
test the information diffusion hypothesis (see Merton (1987) and Stein (1999)), which
assumes that news travels slowly across markets and due to limited informationprocessing capacity, implying investors might not pay attention to or extract information
from asset prices of industries they do not pay close attention to. Excluding five
industries for missing data and generating a commercial real estate industry portfolio, the
authors ultimately determine 14 of the 34 industries lead the market by one month. These
industries are: commercial real estate, mines, apparel, print, petroleum, leather, metal,
transportation, utilities, retail, money or financial, services, non-metallic minerals, and
television. They interpret this finding as evidence that information diffuses less-than-
10
instantly across industry sectors to have an effect on the aggregate market and that
information takes on the order of two months to be incorporated from industries into the
broad market index. With respect to international data, the authors study returns for
Japan, Canada, Australia, UK, Netherlands, Switzerland, France, and Germany for the
period 1973-2002 and find the results hold up remarkably well.
These authors also control for similar factors as the other studies, specifically,
lagged values for: excess market return, inflation, default spread (BAA-AAA), market
GLYLGHQG\LHOGDQGPDUNHWYRODWLOLW\1RWDEO\WKHVHDXWKRUVKLJKOLJKWWKDWIURPWKH
literature on stock market predictabilLW\WKDWEHLQJDEOHWRSUHGLFWQH[WPRQWKVUHWXUQLV
already quite an achievement, as it is notoriously difficult to predict the market at long
KRUL]RQV
The gaps in this research we intend to fill are that health care is not an explicit US
sector these authors studied, so its leading ability in the US is not clear. Notably, the
other international stock markets studied have a health care sector, and its leading effect
is unfortunately indeterminate based on the presented results. Additionally, our health
care measure is a macroeconomic series versus a composition of returns series, so we are
bridging a gap between leading indicators and macroeconomic factors that is not
addressed in previous literature.
Another study, Lamont (2001), presents a purely atheoretical model to estimate,
or track, non-investable macroeconomic series over time. The author uses 13 base assets
and their lagged returns to track these macroeconomic series. The base asset series
include four bond portfolios, eight industry-sorted stock portfolios, and the market
portfolio for the stock market. The key macroeconomic series estimated include:
11
12
Hypothesis Development
Asset pricing theory (see Cochrane (2005) or Campbell, Lo, and Mackinlay
JHQHUDWHVUHODWLRQVKLSVEHWZHHQH[SHFWHGVHFXULW\UHWXUQVDQGLQGLYLGXDOV
consumption as well as their investment opportunity set. That is, changes in security
SULFHVJHQHUDWHUHWXUQVDQGWKHVHFKDQJHVDUHLQGXFHGE\HYHQWVWKDWDOWHULQGLYLGXDOV
ability to consume or their opportunities to invest. The most natural and intuitive events
that alter these items are macroeconomic phenomena. For instance, higher-than-expected
LQIODWLRQDIIHFWVLQYHVWRUVDELOLW\WRFRQVXPHZKLFKVKRXOGLQWXUQDIIHFWWKHGHPDQGIRU
stocks and influence their prices. As another macroeconomic example, a spike in
unemployment would likely affHFWLQGLYLGXDOVDJJUHJDWHFRQVXPSWLRQDQGWKHUHIRUH
WKHRUHWLFDOO\DIIHFWVWRFNUHWXUQV,WVWKLVWKHRUHWLFDOUHODWLRQVKLSEHWZHHQ
macroeconomic phenomena and stock returns that forms the basis for this study.
Specifically, this study analyzes the macroeconomic phenomenon of medical
inflation and its effect on security returns. Based on the relevance of medical care costs
to the US economy, the fact that all firms with human capital appear to be exposed to this
13
cost, and the devastating effect these costs have had on some individuals (see
Introduction for these details), this study proposes that medical care costs affect financial
markets in a material manner, since they affect the riskiness of firm cashflows to the
extent firms are exposed to medical inflation. Specifically, we hypothesize that the
PDUNHWSULFHVILUPVH[SRVXUHWRPHGLFDOFDUHFRVWVDVDVRXUFHRIV\VWHPDWLFULVN7KXV
those securities whose returns covary positively with medical inflation should earn excess
returns. We anticipate medical inflation surprises price negatively since those assets that
covary positively with medical inflation shocks serve as hedging instruments against
unanticipated spikes in health care costs.
Methodology
Is medical inflation different from aggregate inflation?
We first establish that although the Medical Care Consumer Price Index (CPI) is a
component of aggregate CPI, its (unexpected) behavior is sufficiently different from the
aggregate measure to warrant its consideration as a separate priced factor. Figure 1 plots
the monthly time series from January 1967 to August 2009 of CPI for All Urban
Consumers (CPIAUCSL), which we call aggregate CPI for brevity, and some of its major
components. For future reference, Appendix A includes all figures, and Appendix B
includes all tables. All levels in this study are seasonally-adjusted whenever possible.
Specifically, series include Medical Care CPI (CPIMEDSL), Housing CPI (CPIHOSSL),
Food and Beverage CPI (CPIUFDSL), and Transportation CPI (CFDTRNSL). These
components currently represent almost 80 percent of the aggregate CPI, with the Medical
component representing 6.39 percent, Housing 43.42 percent, Food 15.76 percent, and
14
Transportation 15.31 percent.8 While the composition of aggregate CPI has certainly
changed over time, these composition changes are not material for our purposes since our
focus is on inflation in the medical component over time.
Our data sample begins in January 1967 since it represents the time when Medical
CPI begins a trend of over 40 years of month-to-month variation. While data exist
beginning in 1947, data for two key factors, DEF and TERM, only becomes available
beginning in April 1953 when the 10-year government bond data originated. More
importantly, Medical Care CPI exhibits 108 months of zero changes in those first 238
months of measurement, which is almost 50 percent of the time and clearly deviant
behavior considering the more recent pattern of consistent upward monthly variation.
The sample ends in August 2009, for a total of 512 months of data.
While a visual analysis of all panels of Figure 1 shows that medical CPI levels
clearly diverges from the other displayed CPI components beginning in approximately
1985, statistical tests of the relation between aggregate and medical CPI further confirms
this observation. Dickey-Fuller tests for unit roots indicate all subject CPI series are
integrated of order 1, or I(1), for all time periods shown, yet none of them are
cointegrated either pairwise or as a group. Therefore, to work with stationary series,
from here we will use the monthly percent changes in these series. We use the term
LQIODWLRQWRUHIHUWRSHUFHQWFKDQJHVLQDSDUWLFXODU&3,PHDVXUHHJPRQWKO\Pedical
inflation represents the monthly percent change in medical CPI). To calculate per period
percent changes we difference the natural log of the levels. For internal consistency, we
also convert any discrete returns to their continuous values throughout this study. All
8
For data availability and relevance reasons, we omit data on Apparel (3.69 percent), Recreation (5.74
percent), Education and Communication (6.30 percent), and Other goods and services (3.39 percent).
15
monthly and quarterly percent changes are stationary according to Dickey-Fuller tests,
with the exceptions coming in annual sampling where Food and Transportation inflation
maintains a unit root.
16
represented, and thus the sum of all cells equals our 512-month dataset. Shaded cells
differ from a UDQGRPvalue by occurring with extremely high (bold and shaded) or low
(underlined and shaded) frequency. We use = 0.10 on each side to determine the nonrandom pairings.
Figure 2 indicates that monthly medical inflation is positively related to monthly
aggregate inflation, especially at the extreme values (quartiles 1,1 and 4,4), which makes
sense since (a) medical inflation is a component of aggregate inflation, and even with its
low weighting within aggregate inflation, extreme values will magnify its affect on
aggregate inflation and (b) it is also positively-relatedalbeit to a lesser extentto
Housing inflation, Food inflation, and Transportation inflation, which are obviously also
components of the aggregate inflation measure. The bold (underlined) cells indicate
where the cell values are above (below) the 90th percentile values for that cell across the
one-thousand bootstrapped iterations.
This positive relationship is also shown using Table 1, which shows the various
monthly inflation correlations on the upper triangle. Variances (covariances) are on the
diagonal (lower triangle). For example, Panel A indicates a correlation of 0.405 between
aggregate inflation and medical inflation. Monthly medical inflation is less correlated
with aggregate inflation than it is with housing inflation. While the correlation with food
and transportation inflation is still positive, these values are lower yet. As one would
expect, the correlations all grow monotonically as the measurement frequency decreases,
or time horizon increases, to quarterly and annual data. Interestingly, across all sampling
frequencies, medical inflation has the lowest variance, with the variance measures located
on the diagonal. Coupled with the time series level plots in Figure 1, this result indicates
17
medical inflation appears to be moving upward at a relatively constant rate relative to the
other inflation measures. On the contrary, transportation inflation exhibits much higher
volatility than the other measures across all sampling frequencies.
While monthly medical inflation is contemporaneously positively related to
aggregate inflation and its major components, contemporaneous medical and aggregate
inflation are not ultimately our primary variables of concern. Since theory argues the
market imputes any expected information into returns, it is the surprises in these variables
that should truly impact stock returns. Therefore it is the correlation between these
surprises, or shocks, that matter most. While we explicitly address this relationship next,
to foreshadow, the respective correlations are much lower among surprises irrespective of
the measurement method we use.
VAR decomposition
Given the inherent and exhibited contemporaneous relationship between medical
inflation and aggregate inflation coupled with our inability to create a compelling
rationale for one exogenously determining the other, we believe each to have an
autoregressive component as well as an explanatory component that includes lagged
values of the other inflation factor. Additionally, since we are ultimately interested in
WKHVHLQIODWLRQIDFWRUVDIIHFWRQWKHVWRFNPDUNHWone method to get surprises involves
using a vector autoregression model (VAR) that incorporates these three variables.
Specifically, we estimate the VAR model in equations (1) to (3) to determine how similar
or different the relationships between medical and aggregate inflation are with respect to
the market.
18
= 0 + 1, + 2, + 3, +
(1)
= 0 + 1, + 2, + 3, +
= 0 + 1, + 2, + 3, +
(2)
(3)
d<&
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
19
month. On the other hand, it does not have a relationship with medical inflation. Instead,
both one- and two-month lagged medical and aggregate inflation explain current medical
inflation. Medical inflation leads aggregate inflation by two months, while aggregate
inflation is DXWRFRUUHODWHGDWRQHPRQWKFRQWUROOLQJIRUWKHRWKHUYDULDEOHVLQIOXHQFHV
Finally, only the one-month lagged market risk premium has any explanatory power for
the current market excess return.
Impulse response
The plots in Figure 3 show the univariate impulse response functions among these
variables. Specifically, these plots demonstrate the 18-month effect of a one standard
deviation shock to each variable along with the asymptotic standard error confidence
bands. Plots (1,2) and (2,1) are of primary interest, as they indicate that a univariate
shock to medical or aggregate inflation has a two- to three-month upward effect on its
counterpart. But this upward effect reverses and eventually dies out over the ensuing 15
months. Thus a shock to medical (aggregate) inflation has no permanent effect on
aggregate (medical) inflation according to the results from this model. So despite the
relationship between aggregate and medical inflation that occurs by-construction, this
result provides some of the strongest evidence to this point that medical and aggregate
inflation can be considered different macroeconomic phenomena.
Variance decomposition
While the impulse response functions demonstrate the results of a unilateral shock
to a particular variable on another variable, for example, the timing and magnitude of a
20
Is Medical Inflation Different from Rm-Rf, SMB, HML, MOM, DEF, and TERM?
To explore whether medical inflation is a priced risk factor in stock returns, it is
also worthwhile ensuring it does not simply proxy for factors already known to perform
well in explaining stock returns. Specifically, Fama and French (1993) demonstrate the
efficacy of their well-known SMB and HML factors to explain the time series of stock
returns. Additionally, they echo the CRR (1986) findings that DEF and TERM have
21
important relationships with stock returns. While CRR (1986) find DEF and TERM are
priced risk factors in stock returns, Fama and French (1993) find these two factors help
generate a more unifying model to explain time series returns of both stocks and bonds,
not solely stocks as approached in CRR (1986). Finally, we look at the relationship
between medical inflation and Momentum (MOM), initiated by Jegadeesh and Titman
(1993) and implemented by Carhart (1996 and 1997) to help explain stock returns.
For this section, we collect SMB, HML, and MOM values IURP.HQ)UHQFKV
data library. We calculate the default risk premium, DEF, as the difference between the
monthly 0RRG\VVHDVRQHG%DDFRUSRUDWHERQG\LHOGand the 10-year Treasury constant
maturity rate. TERM represents the term risk premium, or by-month difference between
the 10-year Treasury constant maturity rate and the three-month Treasury bill. For these
measures we obtain the Baa portfolio, long government bond, and three-month Treasuries
after January 1995 from the Federal Reserve Statistical Release H.15, Selected Interest
Rates. We are gratefully acknowledge Jeffrey Pontiff for providing pre-1995 data on the
three-month Treasuries.
Joint distribution
Analogous to the bootstrapping method employed above, Figure 4 shows the
quartile joint distributions between medical inflation and various factors. The only real
potential for a relationship between medical inflation and known factors occurs with the
term risk premium (TERM). While a clear negative pattern fails to emerge, the
abundance of abnormally high and low contemporaneous relationships (10 of 16 cells) is
concerning. Although the relationship is not as pronounced as we anticipated, this
22
indication of some relationship is not surprising given that TERM effectively represents
the difference between the limits of the yield curve, which many interpret as an indicator
of future inflation. To the extent the market interprets current high inflation as
unsustainable going forward, one would expect the somewhat negative relationship we
observe. A consistent result occurs in the final grid, which indicates that medical
LQIODWLRQVnegative relationship with TERM is similar to the relationship between
aggregate inflation and TERM. This result becomes evident by comparing the bottom
two results in Figure 4.
Figure 4 indicates the relationship of SMB, HML, MOM, and DEF with medical
inflation is virtually no different than a random pairing of the monthly values. While
DEF shows some slight evidence of a positive relationship with medical inflation, a
confounding result is that DEF is also high when medical inflation is low an abnormal
amount of time (see sector 1,4).
The correlations in Table 4, which again contain the correlations in the upper
triangle and covariances on the diagonal and lower triangle, show medical inflation is not
very correlated with any of the other previously-demonstrated factors using monthly
sampling. Again, the absolute values of the correlations generally grow monotonically as
the sampling frequency decreases. Medical inflation once again has the lowest variance
of all factors studied in this section. The other notable facts from Table 4 are that once
again medical inflation is much less volatile than the other measures, and also the excess
market return, MKTRF, has a variance is generally larger than the other factors. Thus
even though medical inflation might be related to other factors, as CRR (1986) highlight,
we must consider the relative volatility of various macroeconomic series. Specifically, at
23
the monthly frequency, the next least volatile factor, DEF, has a variance (0.548), which
is over seven times that of medical inflation (0.073). The other factors have variance
values greater than medical inflation by two orders of magnitude. Thus if the visual,
bootstrapping simulation, and correlation results are not enough to separate medical
inflation as its own factor, then its relative smoothness over time should suffice. This
smoothness could prove detrimental, as prior authors have pointed out the consequent
FKDOOHQJHRIVXFKUHODWLYHO\VPRRWKPDFURHFRQRPLFVHULHVKDYLQg much explanatory
power considering the highly-varying nature of asset returns.
VAR analysis
Akin to our earlier VAR analysis that includes inflation and the market risk
premium, in this analysis we expand the VAR model to include the factors considered in
this section that may explain and influence medical inflation. In addition to aggregate
inflation and the market risk premium, we augment the VAR with SMB, HML, MOM,
DEF, and TERM. Thus our system is represented by the following:
= 1 +
24
clarify the above description, equation (4) illustrates the first equation (of 8) represented
by the entire VAR system.
= 0 + 1, + 2,
+ 3, + 4, + 5, + 6, + 7,
+ 8, + ( 4)
Using the Schwarz criterion to determine the appropriate lag length of 1, Table 5 depicts
the VAR parameter estimates for this specification.
These results indicate that besides medical inflation itself, only aggregate inflation
leads medical inflation by one month, controlling for the other lagged factors. We
anticipated this result based on the earlier three-variable VAR, which showed the
relationship to aggregate inflation. Lagged medical inflation does not have explanatory
power for any of these other factors except for aggregate inflation and TERM, both of
which it significantly leads by one month.
The variance decomposition reported in Table 6 shows the major source of
variance over time for medical inflation stems from its own volatility (88.7 percent). The
next major source of variance is aggregate inflation (9.1 percent), followed by MKTRF,
HML, and TERM, which change slightly depending on the variable ordering, however,
not substantially enough to warrant further comment. While the low self-values for DEF
and TERM might initially cause concern, these values are order-sensitive and change
dramatically when moved forward within the system. In general for all these variables,
25
order matters, and shocks to their own values persist over time. In untabulated results,
the volatility in the market risk premium plays the major secondary role for these factors.
26
section), and as a result, the Fisher (1930) relationship between aggregate inflation and
interest rates does not necessarily hold for its components, such as medical inflation. The
advantage of its straightforwardness is evident. For completeness, Figure 5 shows the
plot of actual aggregate inflation (CPIAUCMO) and expected aggregate inflation
(EXPINF) using this Kalman Filter method. Expected aggregate inflation (EXPINF) is
clearly a smoothed version of the more volatile actual inflation (CPIAUCMO) series.
The other readily-available alternative we analyze is to estimate the VAR model
along the lines of those we have created in earlier sections and use the residual from
equations (1) and (4), , as the unexpected component of medical inflation.
27
WKHVLPLODULW\RIPHGLFDOLQIODWLRQVbehavior WRDJJUHJDWHLQIODWLRQVXVLQJWKLVWHFKQLTXH
encourages us this method is not entirely inappropriate.
VAR estimate
Using the earlier VAR models, which we will refer to as the Inflation VAR (3
variables: CPIMEDMO, CPIAUCMO, and MKTRF) and the Factor VAR (8 variables:
CPIMEDMO, CPIAUCMO, MKTRF, SMB, HML, MOM, DEF, and TERM), we create
two series of unexpected medical inflation using the residuals from the first equation in
each VAR specification. Figure 7 shows, respectively, a time series plot of the expected
medical inflation values from the Inflation VAR (EXPMED3VAR) and Factor VAR
(EXPMED8VAR) juxtaposed with the actual medical inflation (CPIMEDMO) time
series. These results indicate the expected medical inflation tracks actual medical
inflation quite closely, with the exception of a few deviations in the early-1980s and late1990s.
Table 7 summarizes the relationships between our various measures of
unexpected medical inflation and other key variables of interest by quantifying the
correlations between these various measures and some of the key variables we are
concerned might demonstrate redundancy with medical inflation.
Whereas previous results indicate a possibility that medical inflation might simply
pick up the effect of aggregate inflation or perhaps even the TERM risk premium, the
correlations between our various measures for medical inflation surprises and these
variables are nearly zero per Table 7. The reason for the slight difference between some
of these results and Table 1 is the loss of observations with the lagged terms in the VARs.
28
These results provide more compelling evidence that surprises in medical inflation, which
might affect stock returns, do not simply reflect previously-documented factors that have
been shown to explain returns. Again, our measures of medical inflation surprises are the
Kalman Filter series (UNEXPMEDKALMAN), the Inflation VAR residual series
(UNEXPMED3VAR), and the Factor VAR residual series (UNEXPMED8VAR). While
aggregate inflation earlier proved to be our chief concern for redundancy with a positive
correlation of 0.404, the correlations between the medical inflation surprises generated by
the Kalman Filter and VAR models are much lower, ranging from 0.009 to 0.119.
Further, none of the medical inflation shock measures are correlated with aggregate
inflation shocks (correlations from -0.024 to 0.003). Recall that CRR (1986) show
aggregate inflation is (mildly) priced in returns. Additionally, the relationships between
medical inflation and TERM are effectively zero (ranging from -0.002 to 0.003). Finally,
it is encouraging that all measures of unexpected medical inflation are highly-positively
correlated with one another (between 0.853 and 0.892). In other words, the medical
inflation surprises appear quite insensitive to the mechanism used to generate them, and
they are not correlated with other factors that prior research has explored.
Given these results in Table 7, we proceed using only the unexpected medical
inflation from the state space model (UNEXPMEDKALMAN) time series. This time
series is most nearly orthogonal to both aggregate inflation and TERM, which to this
point have been the factors of greatest concern in terms of redundancy.
29
Results
Fama-MacBeth (1973) Two-Pass Method
Having established unexpected medical inflation as a distinct factor from others
that have shown an association with US stock returns in past studies, the next step is to
determine whether it is priced in equilibrium. To do so, we employ the Fama-MacBeth
(1973) two-pass method of time series and cross-sectional regression models. To
minimize the errors-in-variables problems with individual stock returns, we use portfolios
to measure returns. Specifically, we use the 25 Fama-French portfolio returns, which are
formed by independently-sorting all NYSE, AMEX, and NASDAQ stocks available on
CRSP into quintiles based on size and book-to-market ratio.10 In the first pass, we use 60
PRQWKVRIWLPHVeries returns to generate betas according to equation (5).
, = 0 + , , +
(5)
=1
:HJUDWHIXOO\DFNQRZOHGJH.HQ)UHQFKVGDWDOLEUDU\IRUSURYLGLQJWKHVHUHWXUQV
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
30
, = 0 + , , +
(6)
=1
This analysis occurs on a rolling basis, beginning with 60-month beta estimation
from January 1967 to December 1971, followed by cross-sectional regressions from
January to December 1972. We then increment the beta estimation period by 12 months,
re-calculate the betas using equation (5), and re-price the factors using equation (6). With
data beginning in January 1967 and ending in August 2009, we ultimately have just under
38 yHDUVZRUWKRIPRQWKO\ULVNSULFHV ), or 452 monthly observations. Calculating the
statistics on these risk price series allows us to determine whether the price of risk for the
respective factor differs significantly from zero. In other words, for each unit of factor
ULVNWKHDVVRFLDWHGFRHIILFLHQWTXDQWLILHVWKHH[Wra return required for bearing such risk.
We determine whether medical inflation is a priced risk factor using a variety of
specifications. First and most simply, we include medical inflation expected and
unexpected components as the only factors and then add aggregate inflation components.
Next, we augment the CRR (1986) model specifications to include medical inflation
components. Additionally, we extend the more recent results of Flannery and
Protopapadakis (2002), explicitly pricing those macroeconomic series they indicate
perform well in a first-pass scenario and further augmenting these measures with medical
inflation. Finally, in a sort of hybrid macroeconomic-characteristics model, we price the
Fama and French (1993) five factors, include momentum, and augment these factors with
31
medical inflation. Generally speaking, it appears medical inflation in some form loads as
a priced risk factor across the model frameworks.
Two-pass method with unexpected medical inflation (and unexpected aggregate inflation)
Table 8, Panel A contains the results for the first set of second-pass regressions.
Specifically, this table shows the second-pass results for 6 different model specifications
having completed first-pass regressions for the period January 1967 to August 2009, or
512 months.
7RJHWDEDVHOLQHRQRXUPRGHOVperformance for the subject period, the first
specification includes only the market excess return (i.e., simple single-factor CAPM),
both for the first-pass time series regressions and the second-pass. To provide an
example of the first-pass results from equation (5), we present Table 8, Panel B. It shows
first-pass results across the entire time period (512 months), while the rolling regression
methodology used in Table 8, Panel A runs the first pass time series regression for sixty
months at a time. The results in Panel B, columns 4 through 6 indicate that the market
risk premium explains each of the portfolio returns well across the entire time series (see
the high t-statistics and coefficients of determination).
Turning back to Table 8, Panel A, the second column shows that in our period of
study, the market risk premium factor is indeed priced in equilibrium during this period.
While contrary to theory, we find a well-known empirical result that covariance with the
market statistically has a negative return premium, which is opposite of that predicted by
the Sharpe-Lintner Capital Asset Pricing Model. This result indicates the market risk
premium is negative, or that this sample has a negatively-sloped Security Market Line.
32
As pointed out by Ahn, Gadarowski, and Perez (2009), this result could simply be an
artifact of the relatively constant beta values (range from 0.81 to 1.45) across the 25
portfolios. Unfortunately such a negative-sloping Securities Market Line (SML) is a
commonly-found but theoretically-discouraging result.
The first novel two-pass test we complete involves only expected and unexpected
medical inflation as the possible risk factors. Table 8, Panel C, shows results from the
associated first-pass regressions, which are analogous in format to those for the market
factor shown in Table 8a.
From the first-pass regressions, it is evident that while the market risk premium is
contemporaneously highly-SRVLWLYHO\FRUUHODWHGZLWKHDFKSRUWIROLRVUHWXUQVWKHVDPH
cannot be said of the contemporaneous relationship between medical inflation (expected
RUXQH[SHFWHGDQGWKHSRUWIROLRVUHWXUQV from the first-pass regressions. Like the
market risk premium, which is statistically priced (Table 8, Panel A, Column 2), the
results for medical inflation indicate the contemporaneous expected and unexpected
medical inflation are both priced at conventional statistical levels (Table 8, Panel A,
Column 3). Additionally, results for unexpected medical inflation (Column 4) are robust
to the replacement of expected medical inflation with its first difference series, the analog
to the change in expected aggregate inflation used in the CRR (1986) study. The fourth
specification (Column 5) in Table 8, Panel A, which substitutes aggregate inflation
expected and unexpected components for medical inflation, demonstrates that neither
expected nor unexpected aggregate inflation are priced when used alone in the model.
Finally, and perhaps most notably, mHGLFDOLQIODWLRQVORDGLQJDVDSULFHGULVNIDFWRULV
robust to including aggregate inflation components as factors in the fifth specification
33
(Columns 7 and 8). While the economic price of medical inflation decreases
substantially when including aggregate inflation as a factor (from 0.10 to 0.05),
unexpected medical inflation is nonetheless priced at conventional significance levels
whether one uses the time series of expected medical and aggregate inflation or their first
differences. Since Dickey-Fuller tests confirm that expected medical inflation
(EXPMED) is a stationary time series, we proceed using it in future specifications versus
the change in it.
While concerns about multicollinearity clearly arise when including both
aggregate and medical inflation in the same model, the robustness of mHGLFDOLQIODWLRQV
pricing both alone and despite the potential multicollinearity is encouraging. And we
know the relationship between unexpected medical inflation, the variable of primary
interest, is effectively uncorrelated with the other inflation measures. Overall, these
models that incorporate medical inflation in relatively simple specifications entice us to
explore more comprehensive specifications for evidence of medical inflation as a priced
risk factor.
Although it is encouraging that the medical inflation factor prices in equilibrium
during this period of study, one surprise result is that this factor prices positively. As
with aggregate inflation, it seems most plausible that as stock (or portfolio) returns
covary positively with medical inflation, then these assets would represent hedges for the
EDGVWDWHRIKLJKPHGLFDOLQIODWLRQ$VDUHVXOWLQYHVWRUVORRNLQJWRprotect their
wealth against such a bad state would bid up the prices of these assets and consequently
reduce returns to these assets. Thus one would predict that a high covariance between
returns and medical inflation shocks would lead to lower expected returns. In other
34
words, we anticipate the medical inflation factor we have constructed should price
negatively in equilibrium.
To determine if this unexpected result occurs consistently over time, we now
subdivide the period of study into two sub-periods. The first period runs from January
1967 to December 1984; the second period from January 1985 to August 2009. Given
the 60-month beta formation period at the beginning of our sample, these sub-periods
FRQWDLQDQGPRQWKVRIUHWXUQGDWDUHVSHFWLYHO\7KH rationale for subdividing
the entire period at this date comes from Figure 1. The mid-1980s appears to be the time
when medical inflation begins diverging from the other components of medical inflation
and represents a reasonable basis for partitioning the sample.11
Table 8, Panel D contains the results of the subdivided sample in columns 3 and 4
for the specification 6 in Table 8, Panel A. For our basic specifications, although medical
inflation surprises are priced positively across the entire sample period, this result only
occurs because of their positive pricing in the early sub-period. In the more recent
period, medical shocks price negatively and significantly, as we would predict.
Additionally, while expected medical inflation prices (positively) in the earlier period, it
fails to price in the latter period. Unfortunately the positively-priced medical inflation
result in the early period remains a mystery despite multiple robustness tests and
explorations, as we describe later.
11
The pattern described in the following sub-period results also generally holds when we arbitrarily
subdivide the sample into its 4 decades (1970s, 80s, 90s, and 2000s).
35
36
37
38
loading for the entire period is driven strictly by the highly-positive pricing during the
period when medical inflation and other inflation components exhibited similar trends in
their levels (see Figure 1).
The next, and final, analysis involves augmenting three model specifications
based on the well-known Fama-French factors with medical inflation. Table 11, Panel A
depicts the results on these second-pass regressions. Specifications 1, 4, and 7 (columns
2, 5, and 8) are the baseline specifications for a 3-factor, 5-factor, and 6-factor (i.e., 5factor plus Momentum) analysis. Each respective subsequent specification (i.e., 2, 5, and
8) includes expected and unexpected medical inflation. The final specifications in each
progression, 3, 6, and 9, replace disaggregated medical inflation with the composite
value.
Regarding our variables of interest, contrary to our prior results, in no case is
unexpected medical inflation priced when we incorporate it across the entire period.
Furthermore, the composite medical inflation value is not priced in these specifications.
Expected medical inflation is positively priced, albeit at a substantially lower economic
value than we have seen in the prior tables. These results clearly do not support our
hypothesis, per se, and as before, it is concerning that expected medical inflation carries
some return premia here. As for the other factors, MKTRF (negative) and HML
(positive) price in all specifications, TERM (positive) prices in all specifications save
one, and MOM (negative) fails to price. SMB (positive) never prices, and DEF
(negative) only prices at conventional levels in the final specification.
Once again, the results support our hypothesis when we subdivide the sample.
Table 11, Panel B summarizes the subdivided second-pass of the final specification from
39
Table 11, Panel A, which includes all factors. While medical inflation surprise does not
price across the entire period, the early sub-period again drives this insignificant result.
In the more recent sub-period, medical inflation surprises price negatively, consistent
with the results in our prior specifications. Additionally, the level of concern diminishes
about expected medical inflation pricing across the entire time period. The subdivided
results indicate expected medical inflation fails to price in the last 25 years. Finally, in
more recent times, these data indicate MKTRF (negative), HML (positive), DEF
(negative), and TERM (positive) all price significantly.
In an attempt to explain the counterintuitive finding that medical inflation prices
positively from 1972 to 1984, we consider a couple possibilities. Given the Jensen,
Mercer, and Johnson (1996) finding regarding the relationship between asset returns and
WKH)HGHUDO5HVHUYHVPRQHWDU\ policy stance, we posit that perhaps medical inflation
pricing is conditioned upon the same monetary policy phenomenon. We divide the
sample into contractionary and expansionary monetary policy periods according to the
method described in Jensen, Mercer, and Johnson (1996), using their data augmented
with data from the Federal Reserve Statistical Release H.15, Selected Interest Rates for
more recent months. We then explore medical inflation pricing using two methods. In
the first method, we simply test the means of medical inflation factor prices (i.e., from
equation (6)) for any difference across the two monetary policy environments. While the
evidence is not statistically convincing, perhaps this area is one for further future
exploration since we find contractionary periods tend to load higher than expansionary
periods with p-values ranging from 0.110 to 0.436 depending on the specification (results
not tabulated).
40
In the next method, we test for a time-series difference in the relationship between
asset returns and medical inflation surprise by interacting the medical inflation surprise
with an indicator variable for whether each month occurs during a contractionary or
expansionary timeframe. This method involves running the time series regressions of
equation (5) for the whole period with an additional interaction term. Again, these
untabulated results show no significant difference between the covariance between
medical inflation shocks and asset returns associated with different monetary policy
periods.
41
use this ordering for the ensuing 12 months. We create quintiles of these stocks ordered
by medical inflation news beta and calculate an equally-weighted monthly average of
firm returns each quintile. Differencing the average returns of the high-beta quintile and
low-beta quintile forms our MedHML factor. We roll the beta estimation period forward
by 12 months and repeat the process. In the end we generate 444 MedHML returns for
the period ranging from January 1972 to December 2008. Across the whole period there
exists no statistical difference between the average returns of the high- and low-beta
quintiles (i.e., MedHML is statistically no different than zero), which is not problematic
(see Cochrane (2005)).
Given this MedHML series, we incorporate it as a factor and proceed with the
Fama-MacBeth (1973) two-pass procedure using equations (5) and (6) to determine
whether it is priced. We present the first-pass results for the entire period in Table 12,
Panel A. Contrary to the less-than-impressive first-pass results in Table 8, Panel C, these
results indicate that across the entire time period MedHML covaries significantly with the
Fama-French 25 portfolio returns for 14 of the 25 portfolios at conventional significance
levels. Additionally, the covariance is positive for small firms and then changes sign as
firms grow in size. All else equal, small firms correlate better over time with medical
inflation than do larger firms. We do not discern a pattern with the book-to-market
measure.
The second-pass results are located in Table 12, Panel B. The result found in
earlier specifications persists here. The medical inflation surprise factorin this case
MedHMLprices negatively in the most recent time period (see column 4). Because we
ORVHDQH[WUDPRQWKVRIGDWDPRYLQJIURPWKHSUHYLRXVVSHFLILFDWLRQVWRWKLVRQHWKH
42
more recent sub-period is substantially larger than the earlier sub-period, and thus this
negative pricing in the recent sub-period appears to dominate across the whole timeframe
(column 2). Interpreted, this negative price on the MedHML zero investment portfolio
indicates that as the difference between the medical inflation surprise betas (or returns)
grows, expected returns decrease. In the end the market prices a risk factor formed by
simultaneously taking a long (short) position in stocks exhibiting high (low) covariance
with medical inflation surprises similarly to how it prices the macroeconomic medical
inflation measure. Thus it appears that our finding of the market pricing the risk
associated with health care costs is robust to creating a firm characteristics-based factor
in addition to the macroeconomic factor analyzed earlier.
Discussion
While it appears at the aggregate level the risk to firms of health-care related costs
are priced, one might wonder whether it would be more appropriate to partition the
sample of firms into those who are large enough to self-insure versus those who purchase
coverage from external agencies. Such an indicator variable would capture the
incremental relationship between returns and self-insuring firms, but we believe the firm
(i.e., portfolio) betas already capture these differences. Whether a firm self-insures or
opts for an externally-managed plan, it will ultimately bear the cost of medical care for
insured employees. Perhaps a delay in this cost recognition could occur if an external
insurance company has to recoup an unexpected rise in costs (i.e., higher medical
inflation) in a future period, but the firm will ultimately bear the expense. To the extent
the market incorporates this information, it should be imputed into returns and allow the
43
ILUPVEHWDVWRFDSWXUHWKHHIIHFWV6HSDUDWHO\EXWUHODWHGLWLVQRWLPPHGLDWHO\FOHDUWKDW
that by-firm self insurance data is available.
Another item of possible concern is whether medical inflation represents only the
costs of medical care but is independent of the actual expenditures firms incur to
purchase the care, which is a more accurate indicator of the risk posed by unexpected cost
escalation in this area. As discussed, earlier, medical care inflation represents the cost
index for a basket of medical care commodities and services. This component represents
6.39 percent of the aggregate inflation measure quantified by the Consumer Price Index
(CPI). On the other hand, expenditures on medical care are generally tracked as a
fraction of GDP. According to the Congressional Budget Office, spending on health care
has grown from approximately 5 percent of GDP in 1960 to 17 percent today.12 So while
it is conceivable that costs increase but expenditures do not, which would argue against
the use of medical inflation as a proxy for firm exposure to health care cost growth, the
growth in medical care costs as a fraction of GDP indicates that expenditures are rising in
addition to costs. Addtionally, results from the Kaiser Family Foundation 2009 Survey
of Employer Health Benefits indicates that firms have increased their payment of medical
benefits by 132 percent between 1999 and 2009. In the same period, the cost of medical
care as measured by medical CPI has risen by 52 percent. Thus it appears using this costEDVHGPHDVXUHRIILUPVH[SRVXUHWRPHGLFDOFDUHLQIODWLRQFRXOGEHPRUHFRQVHUYDWLYH
than looking DWILUPVDFWXDOH[SHQGLWXUHV
Perhaps the cost versus expenditure question loses some relevance when we
consider the medical inflation surprise beta could account for it. Once again, the medical
12
http://www.cbo.gov/ftpdocs/87xx/doc8758/MainText.3.1.shtml
44
inflation surprise beta captures the by-firm (i.e., portfolio) relationship between firm
returns and medical inflation shocks and should account for the unique effect of price
changes versus expenditure changes. For instance, if medical care costs (i.e., inflation)
rise 10 percent but a firm implements a wellness program designed to decrease medical
care needs for employees that exacts a 10 percent decrease in medical care services used,
then these effects offset each other. Firm earnings experience no medical care induced
shock, leading to no return difference due to this shock and exacting a zero beta (i.e.,
covariance) between the return and medical care shock. One might argue that firms who
pay health insurance premiums are relatively-penalized since they pay these premiums
regardless of medical care services used, but we assume the insurers act in an actuariallyneutral fashion and adjust premia in the subsequent time periods to reflect cost and
expenditure changes. In other words, if there is a positive shock to medical inflation
(cost) and health care usage (expenditures), then for an externally-insured firm the insurer
will adjust future premiums to offset the unexpected loss. Conversely, these insurers will
also adjust to offset an unexpected gain. This assumption makes the cost versus
expenditure question moot, because ultimately firm earnings bear the brunt of any
changes in medical care costs.
Finally, a clear contemporary policy question involves financing expanded health
care in this country. Since the results of this study indicate investors in firms that are
more exposed to medical inflation demand lower returns due to the hedging effect of such
DVVHWVDJRYHUQPHQWLQWHUYHQWLRQWKDWUHGXFHVILUPVULVNRIPHGLFDOLQIODWLRQZRXOG
drive down prices of hedging assets and drive up prices of their risky counterparts. While
45
this wealth transfer from hedge-portfolio holders to those holding risky assets would
occur, it is not apparent the government needs to correct it with a redistribution scheme.
Conclusion
Health care represents a major component of the US economy. In the past
quarter-century its costs have risen much faster than price in the balance of the economy
as measured by the Consumer Price Index. Despite these escalating costs, firms have
shown resilience in their commitment to provide medical care as an employee benefit.
To the extent firms are exposed to medical costs differently (i.e., more or less employees,
better or worse health care plans, more or less leverage when negotiating rates with
providers), their ultimate cash flows change as this component of their cost structure
changes. To the extent the investors cannot diversify away this risk across firms, they
will demand excess returns for bearing the risk of escalating medical expenses that firms
evidently will not or cannot trim.
This study is an attempt to determine whether the market does indeed consider
medical care costs a source of undiversifiable and hence priced risk. By separating the
medical inflation component from the other basket of goods that composes aggregate
inflation, one can generate a macroeconomic factor to test this question. Looking at
monthly returns for the time period between January 1967 and August 2009, we lean on
earlier factor models, specifically those generated by CRR (1986) and Fama and French
(1993) as a baseline. Additionally, we incorporate the novel findings regarding
macroeconomic factors from Flannery and Protopapadakis (2002). We augment these
models to include an expected and unexpected medical inflation component, which we
46
generate based on Fama and Gibbons (1982) and Ansley (1980). At the top-level our
findings generally support the contention that (unexpected) medical inflation does
represent a priced risk factor, particularly in the last 25 years. Work still needs to be
done for the earlier period of this study, 1967 to 1984, to determine why medical inflation
surprises price in the opposite direction as one would anticipate.
Given these findings that suggest medical inflation surprises are priced in the
market recently, they are by definition non-diversifiable. Since many large entities are
liable for current and future medical care costs and must decide where to invest today to
offset these future liabilities, our results indicate these entities cannot simply invest in the
market and expect to fully fund health care expenses. Furthermore, Jennings, Fraser, and
Payne (2009) highlight that more targeted and seemingly natural hedging investments
such as health care mutual funds are not effective instruments to offset medical inflation.
Faced with non-diversifiable medical care cost risk that is not naturally hedged, future
work could determine what investable assets would serve as a mechanism to best hedge
the risks associated with unanticipated health care cost changes.
47
References
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49
Appendix A
Figures
50
Figure 1
Panel A: Monthly Consumer Price Index (CPI) Levels
51
Figure 2
Monthly Joint Distributions of Inflation Components, Sorted into Quartiles
CPIMEDMO vs. CPIAUCMO
CPIHOSMO
1
2
3
4
1
54
34
24
13
2
34
45
29
16
3
22
28
43
33
CPIMEDMO vs.
4
17
15
31
64
1
2
3
4
1
42
39
23
20
1
32
33
33
25
2
40
44
26
17
3
30
33
32
31
4
25
12
36
53
2
52
38
28
9
3
23
37
37
26
4
10
8
39
71
CPIMEDMO vs.
1
2
3
4
1
42
37
23
25
2
35
29
38
21
3
22
23
45
34
4
28
33
21
46
These 4x4 grids represent the results of a 1000-iteration bootstrapping simulation in which the
first time series, monthly medical inflation (CPIMEDMO), values are placed into quartiles. Next,
the values for the time series listed second (e.g., CPIAUCMO in the first example) are also placed
into quartiles. The numbers in each quartile intersection represents the number of times the
quartile values match for the period January 1967 to August 2009. For example, monthly
medical inflation (CPIMEDMO) and monthly aggregate inflation (CPIAUCMO) are both in their
lowest quartile range (i.e., 1,1) 54 times during the period of study. To determine whether this
value is statistically higher (or lower) than it would be if the series were independent, we
randomly reorder the second time series 1000 times. Generally, at the 10 percent level, the values
for all quartile combinations fall between 27 and 32 for these random pairings. The bold font
(underlined font) and shaded cells represent quartile intersections that are statistically higher
(lower) than these 10 percent cutoffs. Variables are defined in Figure 1.
52
Figure 3
Impulse Response Functions for Three-Variable Vector Autoregression (VAR) Model
Response to Cholesky One S.D. Innovations 2 S.E.
Response of CPIMEDMO to CPIMEDMO
.24
.24
.24
.20
.20
.20
.16
.16
.16
.12
.12
.12
.08
.08
.08
.04
.04
.04
.00
.00
.00
-.04
-.04
-.04
10
12
14
16
18
10
12
14
16
18
.30
.25
.25
.25
.20
.20
.20
.15
.15
.15
.10
.10
.10
.05
.05
.05
.00
.00
.00
-.05
-.05
-.05
10
12
14
16
18
10
12
14
16
18
-1
2
10
12
14
16
18
10
12
14
16
18
10
12
14
16
18
-1
.30
-1
2
10
12
14
16
18
10
12
14
16
These results depict the 18-month response of medical inflation (CPIMEDMO), aggregate
inflation (CPIAUCMO), and WKHPDUNHWVUHWXUQH[FHVVRIWKHULVNIUHHUDWHMKTRF) in the
following VAR model to a one-standard deviation shock to itself and the other variables in the
system for the period January 1967 to August 2009.
= 0 + 1, + 2, + 3, +
= 0 + 1, + 2, + 3, +
= 0 + 1, + 2, + 3, +
Note i = 1, 2 for a two-lag model using the Schwarz criterion for lag length. Implicit causal
ordering is shown above (CPIMEDMO, CPIAUCMO, and MKTRF), with the results insensitive
to changes in this ordering. These response functions use the Cholesky decomposition to
orthogonalize the residuals, and dashed lines represent the two-standard-deviation confidence
bands.
18
53
Figure 4
Monthly Joint Distributions of Potential Risk Factors, Sorted into Quartiles
CPIMEDMO vs. SMB
1
2
3
4
1
42
30
32
23
2
28
31
37
31
3
24
30
30
39
1
2
3
4
1
34
29
31
30
1
31
36
32
26
2
20
32
37
36
3
36
27
34
28
1
2
3
4
2
48
23
19
34
3
32
27
42
25
3
30
30
27
38
4
33
32
35
27
1
2
3
4
1
27
29
28
34
2
30
31
34
31
2
35
33
26
35
3
22
37
49
19
4
43
23
24
38
1
2
3
4
1
32
20
16
56
2
28
33
29
34
3
26
32
43
25
4
39
39
38
12
These 4x4 grids represent the results of a 1000-iteration bootstrapping simulation in which the
first time series, monthly medical inflation (CPIMEDMO), values are placed into quartiles. Next,
the values for the time series listed second (e.g., SMB in the first example) are also placed into
quartiles. The numbers in each quartile intersection represents the number of times the quartile
values match for the period January 1967 to August 2009. For example, monthly medical
inflation (CPIMEDMO) and the Fama-French (1992) SMB factor are both in their lowest quartile
range (i.e., 1,1) 42 times during the period of study. To determine whether this value is
statistically higher (or lower) than it would be if the series were independent, we randomly
reorder the second time series 1000 times. Generally, at the 10 percent level, the values for all
quartile combinations fall between 27 and 32 for these random pairings. The pink (green)
shading represent quartile intersections that are statistically higher (lower) than these 10 percent
cutoffs. Variables include the Fama and French (1993) factors SMB, HML, DEF, and TERM;
the Carhart (1997) momentum (MOM) factor; and aggregate inflation (CPIAUCMO).
Time series plot of actual monthly aggregate inflation (CPIAUCMO) versus the expected monthly aggregate inflation (EXPINF). The expected
monthly inflation series is calculated based on the Fisher (1930) relationship between inflation and interest rates according to a Kalman Filter
methodology developed by Ansley (1980) as described in Fama and Gibbons (1989). This procedure permits the constant within a regression
model to change dynamically based on past data WUHQGVLQDQHIIRUWWRGLVFHUQWKHGLIIHUHQFHEHWZHHQWKHWUXHVLJQDODQGQRLVH associated with
the signal. ,QWKLVDQDO\VLVWKHH[SHFWHGLQIODWLRQFRPSRQHQWUHSUHVHQWVWKHVLJQDODQGWKHXQH[SHFWHGFRPSRQHQWLVWKHQRLVH
Figure 5
Plot of Actual (CPIAUCMO) vs. Expected (EXMEDINF) Aggregate Inflation using Kalman Filter Model (January 1967-August 2009)
54
Time series plot of actual monthly medical inflation (CPIMEDMO) versus the expected monthly medical inflation (EXPMEDINF). The expected
monthly inflation series is calculated based on the Fisher (1930) relationship between inflation and interest rates according to a Kalman Filter
methodology developed by Ansley (1980) as described in Fama and Gibbons (1989). This procedure permits the constant within a regression
PRGHOWRFKDQJHG\QDPLFDOO\EDVHGRQSDVWGDWDWUHQGVLQDQHIIRUWWRGLVFHUQWKHGLIIHUHQFHEHWZHHQWKHWUXHVLJQDODQGQRLVHDVVRFLDWHGZLWK
the signal. In this analysis the expecteGLQIODWLRQFRPSRQHQWUHSUHVHQWVWKHVLJQDODQGWKHXQH[SHFWHGFRPSRQHQWLVWKHQRLVH
Figure 6
Plot of Actual (CPIMEDMO) vs. Expected (EXMEDINF) Medical Inflation using Kalman Filter Model (January 1967-August 2009)
55
Time series plots of actual monthly medical inflation (CPIMEDMO) and two different measures of the expected component of monthly medical
inflation. EXPMED3VAR is the estimated value of CPIMEDMO from a twice-lagged three-variable vector autoregression (VAR) including
Panel B: Acutal Medical Inflation and Expected Medical Inflation from an Eight-Variable Vector Autoregression Model
(January 1967-August 2009)
Panel A: Acutal Medical Inflation and Expected Medical Inflation from a Three-Variable Vector Autoregression Model
(January 1967-August 2009)
Figure 7
56
monthly medical inflation, aggregate inflation, and the excess return on the value-weighted market above the risk-free rate of a one-month
Treasury bill (MKTRF). Pearson correlation between these series is 0.688. EXPMED8VAR includes the aforementioned three variables
augmented with (once-lagged) SMB, HML, DEF, TERM, and MOM factors. Pearson correlation between these series is 0.590.
57
58
Appendix B
Tables
59
Table 1
Panel A: Monthly Covariance/Correlation Table
CPIMEDMO
CPIAUCMO
CPIHOSMO
CPIUFDMO
CPITRNMO
CPIMEDMO
0.073
0.036
0.041
0.014
0.040
CPIAUCMO
0.405
0.107
0.081
0.082
0.253
CPIHOSMO
0.452
0.741
0.112
0.048
0.104
CPIUFDMO
0.109
0.525
0.302
0.227
0.047
CPITRNMO
0.141
0.736
0.296
0.095
1.103
Table 1 shows the Pearson correlations among inflation variables at various sampling frequencies
in the upper triangles. Covariances are shown on the diagonal and lower triangles. Variable
definitions follow.
CPIMEDxx: Medical Component
CPIAUCxx: Aggregate Inflation
CPIHOSxx: Housing Component
CPIUFDxx: Food Component
CPITRNxx: Transportation Component
60
Table 2
Vector Autoregression (VAR) Estimates
(January 1967-August 2009)
Included observations: 509 after adjustments
Standard errors in ( ) & t-statistics in [ ]
CPIMEDMO
CPIAUCMO
MKTRF
0.235
0.065
-0.351
(0.040)
(0.051)
(0.944)
[ 5.802]
[ 1.273]
[-0.372]
0.401
0.166
-0.619
(0.040)
(0.050)
(0.931)
[ 10.060]
[ 3.298]
[-0.665]
0.0809
0.518
-0.556
(0.036)
(0.045)
(0.830)
[ 2.276]
[ 11.565]
[-0.671]
0.107
0.065
0.410
(0.036)
(0.046)
(0.843)
[ 2.965]
[ 1.429]
[ 0.487]
0.002
0.006
0.096
(0.002)
(0.002)
(0.045)
[ 0.841]
[ 2.621]
[ 2.142]
-0.001
0.000
-0.048
(0.002)
(0.002)
(0.045)
[-0.493]
[-0.041]
[-1.056]
0.117
0.033
0.920
(0.021)
(0.027)
(0.492)
[ 5.563]
[ 1.240]
[ 1.869]
R-squared
0.473
0.425
0.016
Adj. R-squared
0.467
0.418
0.004
CPIMEDMO(t-1)
CPIMEDMO(t-2)
CPIAUCMO(t-1)
CPIAUCMO(t-2)
MKTRF(t-1)
MKTRF(t-2)
Constant
Table 2 depicts the VAR parameters for the period January 1967 to August 2009 for the
following model, which includes medical inflation (CPIMEDMO), aggregate inflation
(CPIAUCMODQGWKHPDUNHWVUHWXUQH[FHVVRIWKHULVNIUHHUDWHMKTRF).
= 0 + 1, + 2, + 3, +
= 0 + 1, + 2, + 3, +
= 0 + 1, + 2, + 3, +
Note i = 1, 2 for a two-lag model using the Schwarz criterion for lag length.
61
Table 3
Variance Decomposition
(January 1967-August 2009)
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
62
16
17
18
0.33
0.33
0.33
11.58
11.63
11.67
87.06
87.01
86.97
1.36
1.36
1.36
63
Table 4
Panel A: Monthly Covariance/Correlation Table
CPIMEDMO MKTRF
CPIMEDMO
0.073
-0.039
MKTRF
-0.049
21.405
SMB
0.027
4.427
HML
0.027
-4.771
MOM
0.037
-2.953
DEF
-0.016
0.175
TERM
-0.027
0.105
SMB
0.031
0.298
10.342
-2.487
-0.387
0.166
0.247
HML
0.033
-0.340
-0.255
9.193
-2.207
-0.094
-0.074
MOM
0.030
-0.142
-0.027
-0.162
20.147
-0.523
-0.137
DEF
-0.081
0.051
0.070
-0.042
-0.158
0.548
0.287
TERM
-0.080
0.018
0.061
-0.019
-0.024
0.306
1.607
DEFQ
-0.098
0.039
0.140
-0.073
-0.251
4.723
2.710
TERMQ
-0.091
0.057
0.099
0.037
-0.081
0.345
13.096
DEFA
-0.083
-0.189
0.168
0.079
-0.042
49.665
31.242
TERMA
-0.092
0.185
0.293
0.134
-0.257
0.330
180.672
CPIMEDQ
0.462
-0.352
0.145
0.211
0.312
-0.145
-0.224
MKTRFQ
-0.067
59.047
17.652
-11.133
-7.155
0.651
1.577
SMBQ
0.039
0.421
29.757
-7.066
-0.610
1.659
1.947
HMLQ
0.060
-0.278
-0.249
27.084
-5.903
-0.828
0.692
MOMQ
0.066
-0.134
-0.016
-0.164
48.026
-3.774
-2.024
CPIMEDA MKTRFA
6.244
-0.103
-4.651
328.455
7.093
56.794
2.503
-71.086
1.401
-20.567
-1.457
-24.137
-3.087
45.176
SMBA
0.222
0.246
162.802
15.697
-46.742
15.135
50.295
HMLA
0.069
-0.272
0.085
208.432
-80.212
8.041
26.077
MOMA
0.040
-0.081
-0.262
-0.398
195.279
-4.184
-48.338
Table 4 shows the Pearson correlation among various factors at various sampling frequencies in
the upper triangles. Covariances are shown on the diagonal and lower triangles. Variable
definitions follow.
CPIMEDxx: Medical Component of inflation
MKTRFxx: Value-weighted market return in excess of the risk-free rate
SMBxx: SMB factor as described by Fama and French (1993)
HMLxx: HML factor as described by Fama and French (1993)
MOMxx: Momentum factor as described by Carhart (1997)
DEFxx: Default risk premium as described by Chen, Roll, and Ross (CRR) (1986)
TERMxx: Term risk premium as described by CRR (1986)
64
Table 5
Vector Autoregression (VAR) Estimates
(January 1967-August 2009)
Included observations: 509 after adjustments
Standard errors in ( ) & t-statistics in [ ]
CPIMEDMO
(t-1)
CPIAUCMO
(t-1)
MKTRF(t-1)
SMB(t-1)
HML(t-1)
DEF(t-1)
TERM(t-1)
MOM(t-1)
Constant
R-squared
Adj. Rsquared
CPIMEDMO
CPIAUCMO
MKTRF
SMB
HML
DEF
TERM
MOM
0.447
-0.040
[ 11.273]
0.174
-0.044
[ 3.909]
-0.630
-0.831
[-0.758]
0.212
-0.564
[ 0.375]
0.446
-0.537
[ 0.831]
0.037
-0.034
[ 1.080]
0.195
-0.084
[ 2.321]
0.719
-0.788
[ 0.912]
0.199
-0.036
[ 5.560]
0.002
-0.002
[ 0.829]
0.004
-0.003
[ 1.148]
0.006
-0.004
[ 1.573]
0.009
-0.015
[ 0.632]
-0.002
-0.008
[-0.240]
0.000
-0.002
[ 0.090]
0.191
-0.040
[ 4.775]
0.348
0.502
-0.040
[ 12.530]
0.004
-0.003
[ 1.467]
0.006
-0.004
[ 1.782]
-0.003
-0.004
[-0.817]
-0.056
-0.016
[-3.417]
-0.026
-0.009
[-2.747]
-0.002
-0.003
[-0.910]
0.252
-0.045
[ 5.614]
0.438
-0.219
-0.750
[-0.291]
0.068
-0.050
[ 1.359]
0.088
-0.068
[ 1.293]
-0.026
-0.076
[-0.343]
0.221
-0.307
[ 0.719]
0.021
-0.174
[ 0.122]
-0.004
-0.048
[-0.077]
0.276
-0.839
[ 0.329]
0.018
0.262
-0.509
[ 0.515]
0.158
-0.034
[ 4.673]
-0.023
-0.046
[-0.498]
0.016
-0.051
[ 0.316]
0.433
-0.208
[ 2.081]
0.039
-0.118
[ 0.327]
-0.027
-0.033
[-0.839]
-0.982
-0.569
[-1.726]
0.062
0.557
-0.484
[ 1.151]
0.067
-0.032
[ 2.100]
0.019
-0.044
[ 0.438]
0.185
-0.049
[ 3.800]
-0.181
-0.198
[-0.915]
0.137
-0.113
[ 1.217]
-0.023
-0.031
[-0.732]
0.048
-0.542
[ 0.088]
0.045
-0.027
-0.031
[-0.858]
-0.014
-0.002
[-7.009]
-0.006
-0.003
[-2.010]
-0.007
-0.003
[-2.382]
0.973
-0.013
[ 76.957]
-0.009
-0.007
[-1.187]
0.007
-0.002
[ 3.428]
0.068
-0.035
[ 1.969]
0.935
-0.051
-0.076
[-0.671]
0.018
-0.005
[ 3.571]
0.014
-0.007
[ 1.972]
0.027
-0.008
[ 3.548]
0.171
-0.031
[ 5.531]
0.888
-0.018
[ 50.462]
-0.002
-0.005
[-0.342]
-0.261
-0.085
[-3.086]
0.866
-0.410
-0.711
[-0.577]
-0.165
-0.047
[-3.497]
0.113
-0.064
[ 1.758]
-0.145
-0.072
[-2.028]
-1.139
-0.291
[-3.915]
0.105
-0.165
[ 0.636]
0.004
-0.046
[ 0.097]
2.760
-0.795
[ 3.470]
0.062
0.338
0.429
0.002
0.047
0.030
0.934
0.864
0.047
Table 5 depicts the VAR parameters for the period January 1967 to August 2009 for the
following model.
= 1 +
65
66
Table 6
Variance Decomposition
(January 1967-August 2009)
Variance Decomposition of CPIMEDMO:
Period
S.E.
CPIMEDMO CPIAUCMO MKTRF
1 0.221
100.000
0.000
0.000
2 0.250
95.776
3.657
0.068
3 0.262
92.665
6.387
0.276
4 0.267
90.976
7.845
0.422
5 0.269
90.128
8.555
0.498
6 0.271
89.696
8.889
0.535
7 0.271
89.462
9.046
0.555
8 0.271
89.321
9.119
0.567
9 0.271
89.226
9.152
0.575
10 0.272
89.153
9.167
0.581
11 0.272
89.091
9.172
0.587
12 0.272
89.035
9.173
0.593
13 0.272
88.982
9.172
0.599
14 0.272
88.930
9.169
0.605
15 0.272
88.878
9.165
0.612
16 0.272
88.827
9.161
0.619
17 0.272
88.776
9.157
0.626
18 0.272
88.726
9.152
0.633
SMB
0.000
0.100
0.173
0.212
0.230
0.237
0.241
0.242
0.243
0.243
0.244
0.244
0.244
0.244
0.244
0.244
0.245
0.245
HML
0.000
0.389
0.466
0.474
0.472
0.470
0.469
0.468
0.468
0.468
0.467
0.467
0.467
0.467
0.467
0.467
0.467
0.467
DEF
0.000
0.007
0.011
0.012
0.012
0.013
0.015
0.022
0.033
0.049
0.071
0.097
0.128
0.163
0.201
0.242
0.283
0.326
TERM
0.000
0.001
0.013
0.043
0.086
0.137
0.188
0.235
0.276
0.311
0.338
0.360
0.377
0.389
0.398
0.405
0.409
0.412
MOM
0.000
0.001
0.009
0.016
0.020
0.023
0.024
0.026
0.027
0.028
0.029
0.031
0.032
0.033
0.034
0.036
0.037
0.038
SMB
0.000
0.544
0.652
0.693
0.709
0.716
0.719
0.720
0.721
0.721
0.721
0.721
0.721
0.722
HML
0.000
0.038
0.035
0.036
0.039
0.042
0.044
0.046
0.049
0.051
0.054
0.056
0.059
0.063
DEF
0.000
0.041
0.109
0.214
0.362
0.557
0.796
1.077
1.392
1.736
2.101
2.480
2.867
3.255
TERM
0.000
0.145
0.387
0.660
0.925
1.161
1.357
1.514
1.634
1.722
1.784
1.825
1.851
1.865
MOM
0.000
0.122
0.163
0.185
0.202
0.216
0.229
0.242
0.254
0.266
0.277
0.289
0.301
0.312
67
15
16
17
18
0.318
0.319
0.320
0.321
8.364
8.333
8.302
8.271
82.262
81.843
81.440
81.055
2.754
2.814
2.873
2.930
0.722
0.723
0.724
0.725
0.066
0.069
0.073
0.076
3.639
4.015
4.380
4.731
1.870
1.870
1.866
1.860
0.323
0.333
0.343
0.353
SMB
0.000
0.354
HML
0.000
0.025
DEF
0.000
0.176
TERM
0.000
0.003
MOM
0.000
0.007
SMB
91.665
86.264
HML
0.000
0.057
DEF
0.000
0.777
TERM
0.000
0.008
MOM
0.000
0.146
SMB
3.581
3.460
HML
84.276
83.385
DEF
0.000
0.174
TERM
0.000
0.156
MOM
0.000
0.117
SMB
0.281
1.131
HML
0.415
0.894
DEF TERM
97.323 0.000
79.235 0.940
MOM
0.000
2.008
SMB
0.071
0.386
HML
2.851
0.744
DEF TERM
16.687 79.903
22.237 68.144
MOM
0.000
0.582
SMB
0.360
1.304
HML
4.435
5.083
DEF
0.005
2.012
MOM
93.283
87.893
TERM
0.022
0.109
Cholesky Ordering: CPIMEDMO CPIAUCMO MKTRF SMB HML DEF TERM MOM
Table 6 depicts the variance decomposition for the following VAR system for the period January
1967 to August 2009.
= 1 +
68
0.889
0.039
0.038
0.000
0.004
0.001
0.039
0.027
0.029
0.031
0.000
0.001
0.000
0.030
0.853
0.892
0.048
-0.001
0.009
-0.001
0.048
UNEXPMED8VAR
0.003
0.003
-0.024
0.046
0.051
-0.003
0.001
UNEXPINF
0.009
0.065
0.119
0.730
0.107
-0.122
0.036
CPIAUCMO
-0.001
0.003
-0.002
-0.010
-0.296
1.603
-0.028
TERM
0.678
0.726
0.807
0.009
0.404
-0.083
0.074
CPIMEDMO
Table 7 shows the Pearson correlation among various factors at various sampling frequencies in the upper triangles. Covariances are shown on the
diagonal and lower triangles. Variable definitions follow.
CPIMEDMO
TERM
CPIAUCMO
UNEXPINF
UNEXPMED8VAR
UNEXPMED3VAR
UNEXPMEDKALMAN
UNEXPMED3VAR
UNEXPMEDKALMAN
Table 7
Covariance/Correlation Table for Various Measures of Unexpected Medical Inflation and Other Possible Proxies
(January 1967-August 2009)
69
70
Table 8
Panel A
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors
(January 1967-August 2009)
Constant
MKTRF
Parameter
Parameter
Parameter
Parameter
Parameter
Parameter
Parameter
P-value
P-value
P-value
P-value
P-value
P-value
P-value
1.092
0.802
0.673
0.723
0.481
0.770
0.390
0.000
0.001
0.009
0.001
0.031
0.001
0.075
-0.577
0.097
EXPMED
0.032
0.046
0.035
0.004
EXPMED
UNEXPMED
0.009
0.038
0.648
0.028
0.099
0.087
0.048
0.047
0.002
0.006
0.050
0.077
EXPINF
0.009
-0.010
0.714
0.607
EXPINF
UNEXPINF
N=
452
452
452
-0.010
-0.006
0.187
0.323
-0.007
0.025
-0.028
0.008
0.779
0.314
0.246
0.736
452
452
452
452
Table 8, Panel A depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Shading indicates a coefficient significant at the 10% level (or p-value < 0.10).
MKTRF is the market return net of the risk-IUHHUDWHWDNHQIURP.HQ)UHQFKV'DWD/LEUDU\
EXPMED (UNEXPMED) is the expected (unexpected) component of medical inflation as
determined by a state space model described in Fama and Gibbons (1982). (;30(D is the
first-differenced series (i.e., time t minus time t-1) of medical inflation. Definitions for aggregate
inflation (EXPINF, EXPINF, and UNEXPINF) are analogous to those of medical inflation.
7KHVHGDWDVHULHVRFFXUVEHWZHHQ-DQXDU\DQG$XJXVWIRUPRQWKVZRUWKRIGDWD
The second-pass results in 452 data points due to the initial 60-month beta formation period.
71
Table 8
Panel B
First-Pass Regression Results
(January 1967-August 2009)
Constant
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
-0.449
0.254
0.341
0.556
0.630
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
-0.228
0.147
0.437
0.493
0.506
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
-0.181
0.208
0.298
0.405
0.604
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
-0.029
0.017
0.216
0.355
0.362
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
-0.046
0.081
0.031
0.135
0.243
T-Stat
MKTRF
Size (Small)
-2.022
1.445
1.318
1.230
2.131
1.084
3.590
1.005
3.694
1.060
Size 2
-1.357
1.414
1.092
1.167
3.485
1.043
3.933
0.993
3.262
1.086
Size 3
-1.300
1.348
2.021
1.108
2.841
0.985
3.651
0.922
4.313
1.004
Size 4
-0.279
1.234
0.199
1.082
2.188
1.012
3.444
0.936
2.706
1.021
Size (Big)
-0.597
0.991
1.073
0.935
0.340
0.871
1.237
0.814
1.743
0.847
T-Stat
Adjusted
R-Squared
30.243
29.681
31.543
30.208
28.908
0.641
0.633
0.660
0.641
0.620
39.214
40.482
38.753
36.887
32.570
0.750
0.762
0.746
0.727
0.675
45.075
50.195
43.660
38.651
33.359
0.799
0.831
0.789
0.745
0.685
54.308
57.438
47.723
42.254
35.492
0.852
0.866
0.817
0.777
0.711
59.417
57.850
43.827
34.722
28.262
0.874
0.868
0.790
0.702
0.610
Table 8, Panel B shows results for the Fama-MacBeth (1973) first-pass regression for the
following model.
72
, = 0 + , , +
=1
where , is the month t excess return on portfolio p, p t represents time, k represents
the number of factors, and represents a factor used to explain returns. , represents a
portfolio-specific parameter estimated in the model and is calculated as the covariance of the
factor and portfolio return normalized by the variance of the factor (i.e., , = (, , )/
)) (. For this specification, the portfolios p are the Fama and French size- and book-tomarket sorted quintiles, t represents the 512 months spanning from January 1967 to August 2009,
and the only factor is the market return in excess of the risk-free rate (MKTRF). Shading
indicates parameters that are significant at conventional (90 percent) level.
73
Table 8
Panel C
First-Pass Regression Results
(January 1967-August 2009)
UNEXPMED
Constant
T-Stat
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
0.767
1.439
1.087
1.172
1.170
0.617
1.354
1.186
1.361
1.262
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
1.218
0.912
1.722
0.978
0.787
1.081
0.990
2.076
1.223
0.866
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
1.488
1.165
1.226
0.926
1.129
1.432
1.389
1.604
1.259
1.349
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
1.590
1.206
0.963
1.152
0.365
1.722
1.501
1.246
1.573
0.437
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
1.600
1.131
0.704
0.233
1.231
2.187
1.629
1.036
0.347
1.648
EXPMED T-Stat
Size (Small)
-1.314
-0.568
-1.420
-0.718
-0.673
-0.394
-0.483
-0.301
-0.283
-0.164
Size 2
-1.757
-0.839
-0.620
-0.361
-1.739
-1.127
-0.222
-0.149
0.247
0.146
Size 3
-2.241
-1.159
-1.033
-0.662
-1.067
-0.750
-0.322
-0.236
-0.268
-0.172
Size 4
-2.210
-1.287
-1.469
-0.982
-0.680
-0.472
-0.842
-0.618
0.767
0.494
Size (Big)
-2.447
-1.798
-1.323
-1.024
-0.625
-0.495
0.442
0.354
-1.288
-0.927
KALMAN
T-Stat
Adjusted
R-Squared
-0.659
-1.132
-0.702
-0.571
-0.032
-0.295
-0.594
-0.427
-0.369
-0.019
-0.003
-0.002
-0.003
-0.004
-0.004
-1.182
-1.252
-0.639
-1.341
-0.946
-0.585
-0.757
-0.429
-0.934
-0.581
-0.002
-0.003
-0.001
-0.002
-0.003
-0.877
-0.842
-1.383
-1.435
-0.513
-0.470
-0.560
-1.008
-1.088
-0.341
-0.001
-0.002
-0.001
-0.002
-0.004
-1.476
-1.612
-1.345
-1.348
-1.363
-0.891
-1.118
-0.969
-1.025
-0.909
0.001
0.000
-0.002
-0.001
-0.002
-0.970
-0.667
-0.370
-1.237
-0.387
-0.739
-0.535
-0.304
-1.027
-0.289
0.003
-0.001
-0.003
-0.002
-0.002
Table 8, Panel C shows results for the Fama-MacBeth (1973) first-pass regression for the
following model.
74
, = 0 + , , +
=1
where , is the month t excess return on portfolio p, p t represents time, k represents
the number of factors, and represents a factor used to explain returns. , represents a
portfolio-specific parameter estimated in the model and is calculated as the covariance of the
factor and portfolio return normalized by the variance of the factor (i.e., , = (, , )/
)) (. For this specification, the portfolios p are the Fama and French size- and book-tomarket sorted quintiles, t represents the 512 months spanning from January 1967 to August 2009,
and the factors k are expected and unexpected medical inflation (EXPMED and
UNEXPMEDKALMAN, respectively). Shading indicates parameters that are significant at
conventional (90 percent) level.
75
Table 8
Panel D
Variable
Constant
EXPMED
UNEXPMED
EXPINF
UNEXPINF
N=
Jan 72-Aug 09
Jan 72-Dec 84
Jan 85-Aug 09
Parameter
Parameter
Parameter
P-Value
0.770
P-Value
0.377
P-Value
0.977
0.001
0.318
0.001
0.046
0.100
0.018
0.004
0.008
0.206
0.048
0.207
-0.035
0.050
0.001
0.031
-0.010
-0.085
0.030
0.607
0.067
0.073
-0.028
-0.133
0.027
0.246
0.003
0.355
452
156
296
Table 8, Panel D depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
(UNEXPMED) is the expected (unexpected) component of medical inflation as determined by a
state space model described in Fama and Gibbons (1982). Definitions for aggregate inflation
(EXPINF and UNEXPINF) are analogous to those of medical inflation. These data series occurs
EHWZHHQ-DQXDU\DQG$XJXVWIRUPRQWKVZRUWKRIGDWD7KHVHFRQG-pass results
for the first specification containts 452 data points due to the initial 60-month beta formation
period. The second and third specifications depict results when the sample entire sample is split
into two time periods.
76
Table 9
Panel A
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Chen, Roll,
and Ross (1986) Mactoreconomic Factors
(January 1967-August 2009)
Constant
Parameter
Parameter
Parameter
Parameter
Parameter
P-value
0.802
P-value
0.769
P-value
1.391
P-value
0.865
P-value
1.435
0.002
0.001
0.000
0.001
0.000
0.033
0.019
0.019
0.169
0.054
0.052
0.028
0.021
0.078
0.065
0.014
0.026
EXPMED
UNEXPMED
CPIMED
MKTRF
INDPRO
EXPINF
UNEXPINF
DEF
TERM
N=
-1.033
-1.078
0.000
0.000
-0.154
-0.121
0.008
-0.102
-0.025
0.139
0.201
0.930
0.310
0.787
-0.003
-0.006
-0.007
-0.001
-0.004
0.612
0.181
0.155
0.791
0.463
0.011
-0.008
-0.023
0.011
-0.013
0.606
0.678
0.312
0.565
0.554
-0.129
-0.083
-0.075
-0.106
-0.070
0.043
0.171
0.188
0.093
0.241
0.080
0.239
0.351
0.084
0.182
0.585
0.090
0.007
0.549
0.147
452
452
452
452
452
Table 9, Panel A depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
(UNEXPMED) is the expected (unexpected) component of medical inflation as determined by a
state space model described in Fama and Gibbons (1982). MKTRF is the market return net of the
risk-free rate (taken from Ken FrenchV'DWD/LEUDU\,1'352LVWKHPRQWKO\FKDQJHLQ
Industrial Production. (;3INF is the first-differenced series (i.e., time t minus time t-1) of
aggregate inflation. UNEXPINF is analagous to UNEXPMEDINF but for aggregate inflation.
DEF is the difference between the 10-year Treasury bond and a portfolio of Baa corporate bonds.
77
TERM is the difference between the 10-year Treasury bond and the 90-day Treasury bill. These
GDWDVHULHVRFFXUVEHWZHHQ-DQXDU\DQG$XJXVWIRUPRQWKVZRUWKRIGDta. The
second-pass results in 452 data points due to the initial 60-month beta formation period.
78
Table 9
Panel B
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Chen, Roll,
and Ross (1986) Mactoreconomic Factors, Divided Sample
(January 1967-August 2009)
Table 9, Panel A, Column 3
(Specification 2)
Jan 72Jan 72Jan 85Aug 09
Dec 84
Aug 09
Constant
EXPMED
UNEXPMED
Parameter
Parameter
Parameter
Parameter
Parameter
Parameter
P-Value
P-Value
P-Value
P-Value
P-Value
P-Value
0.769
0.107
1.118
1.391
0.778
1.715
0.001
0.794
0.000
0.000
0.080
0.000
0.033
0.059
0.019
0.019
0.033
0.011
0.019
0.055
0.170
0.169
0.275
0.406
0.054
0.216
-0.032
0.052
0.206
-0.029
0.028
0.001
0.045
0.021
0.000
0.072
-1.033
-0.672
-1.222
0.000
0.198
0.001
MKTRF
INDPRO
EXPINF
UNEXPINF
DEF
TERM
N=
-0.121
-0.360
0.005
0.008
0.006
0.009
0.201
0.113
0.955
0.930
0.976
0.922
-0.006
-0.006
-0.006
-0.007
-0.007
-0.007
0.181
0.485
0.247
0.155
0.442
0.225
-0.008
-0.048
0.012
-0.023
-0.080
0.008
0.678
0.190
0.606
0.312
0.057
0.771
-0.083
0.033
-0.143
-0.075
0.027
-0.128
0.171
0.801
0.020
0.188
0.810
0.042
0.239
-0.301
0.523
0.351
-0.024
0.549
0.090
0.194
0.003
0.007
0.906
0.001
452
156
296
452
156
296
Table 9, Panel B depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
(UNEXPMED) is the expected (unexpected) component of medical inflation as determined by a
state space model described in Fama and Gibbons (1982). MKTRF is the market return net of the
risk-free rate (taken from Ken FrenchV'DWD/LEUDU\,1'352LVWKHPRQWKO\FKDQJHLQ
Industrial Production. (;3INF is the first-differenced series (i.e., time t minus time t-1) of
aggregate inflation. UNEXPINF is analagous to UNEXPMEDINF but for aggregate inflation.
DEF is the difference between the 10-year Treasury bond and a portfolio of Baa corporate bonds.
TERM is the difference between the 10-year Treasury bond and the 90-day Treasury bill. These
79
GDWDVHULHVRFFXUVEHWZHHQ-DQXDU\DQG$XJXVWIRUPRQWKVZRUWKRIGDta. The
second-pass results in 452 data points due to the initial 60-month beta formation period. The
second, third, fifth, and sixth specifications depict results when the sample entire sample is split
into two time periods.
80
Table 10
Panel A
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Flannery &
Protopapadakis (2002) Mactoreconomic Factors
(January 1967- August 2009)
Constant
Parameter
Parameter
Parameter
Parameter
Parameter
P-Value
P-Value
P-Value
P-Value
P-Value
0.592
0.732
1.035
1.246
1.152
0.012
0.002
0.000
0.000
0.000
0.036
0.039
0.008
0.005
0.044
0.056
0.042
0.010
EXPMED
UNEXPMED
CPIMED
0.083
0.077
0.011
0.009
0.040
EXPINF
0.011
EXPINF
UNEXPINF
PPIAgg
PPICrude
M1
M2
HOUST
0.002
-0.003
-0.001
0.003
0.765
0.614
0.793
0.624
0.014
-0.003
0.007
-0.007
-0.009
0.541
0.908
0.764
0.769
0.708
0.123
0.105
0.118
0.062
0.052
0.118
0.243
0.166
0.473
0.575
-0.089
0.015
0.414
0.794
0.969
0.271
-0.081
-0.149
-0.076
-0.035
-0.022
0.162
0.016
0.187
0.535
0.703
-0.127
-0.156
-0.095
-0.088
-0.079
0.002
0.000
0.010
0.010
0.024
-2.234
-1.906
-1.721
-1.977
-1.238
0.014
0.034
0.060
0.031
0.166
-0.716
-0.906
-0.797
0.011
0.002
0.008
452
452
452
MKTRF
N=
452
452
Table 10, Panel A depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
81
82
Table 10
Panel B
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Flannery &
Protopapadakis (2002) Macroeconomic Factors, Divided Sample
(January 1967- August 2009)
Table 10, Panel A, Specification 4
Constant
EXPMED
UNEXPMED
Jan 72-Aug 09
Jan 72-Dec 84
Jan 85-Aug 09
Parameter
Parameter
Parameter
P-Value
P-Value
P-Value
1.246
0.914
1.420
0.000
0.054
0.000
0.036
0.080
0.013
0.008
0.012
0.287
0.044
0.184
-0.030
0.042
0.001
0.035
-0.001
-0.003
0.000
0.793
0.741
0.966
-0.007
-0.069
0.026
0.769
0.090
0.358
0.062
-0.178
0.189
0.473
0.308
0.048
-0.035
0.007
-0.057
0.535
0.927
0.456
-0.088
-0.112
-0.075
0.010
0.081
0.060
-1.977
-0.805
-2.594
0.031
0.625
0.018
-0.906
-0.794
-0.966
0.002
0.136
0.005
452
156
296
CPIMED
EXPINF
EXPINF
UNEXPINF
PPI (Agg)
PPI (Crude)
M1
M2
HOUST
MKTRF
N=
83
Table 10, Panel B depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
(UNEXPMED) is the expected (unexpected) component of medical inflation as determined by a
state space model described in Fama and Gibbons (1982). CPIMED is aggregate medical
inflation, or the sum of expected and unexpected medical inflation. (;3,1)LVWKHILUVWdifferenced series (i.e., time t minus time t-1) of aggregate inflation. UNEXPINF is analagous to
UNEXPMEDINF for aggregate inflation. PPIAgg (PPICrude) is the monthly change in Producer
Price Index for all commodities (crude materials). M1 (M2) is the monthly change in M1 (M2)
Money Stock, seasonally adjusted. HOUST is the monthly change in total new housing starts.
The prior 5 series come from the St. Louis Federal Reserve Federal Reserve Economic Database
(FRED). MKTRF is the market return net of the risk-free rate IURP.HQ)UHQFKV'DWD/LEUDU\
These data series occur between January 1967 and $XJXVWIRUPRQWKVZRUWKRIGDWD
The second-pass results in 452 data points due to the initial 60-month beta formation period. The
second and third specifications depict results when the sample entire sample is split into two time
periods.
MOM
TERM
DEF
HML
SMB
MKTRF
CPIMED
UNEXPMED
EXPMED
Constant
-0.815
0.002
0.083
0.578
0.499
0.001
-0.852
0.001
0.072
0.633
0.523
0.001
-1.018
0.001
0.521
0.601
0.079
0.002
0.001
-0.061
0.246
0.299
0.013
-0.038
0.492
0.210
0.074
0.506
0.545
0.091
0.001
0.001
0.512
0.599
0.080
0.000
-0.893
0.036
0.253
0.424
-0.045
0.001
0.510
0.606
0.077
0.001
-0.925
0.954
0.975
-0.784
0.001
0.000
1.234
P-Value
Parameter
0.001
0.952
0.827
0.000
0.001
0.000
1.218
0.004
0.000
1.310
P-Value
0.006
0.000
0.000
1.096
P-Value
Parameter
0.022
1.159
1.154
P-Value
Parameter
0.032
P-Value
P-Value
Parameter
0.027
Parameter
Parameter
0.021
-0.395
-0.259
0.292
0.151
-0.078
0.001
0.488
0.511
0.097
0.001
-0.930
0.918
0.002
0.002
0.039
0.000
1.261
P-Value
Parameter
0.119
0.191
0.241
-0.065
0.001
0.514
0.585
0.082
0.000
-1.007
0.000
1.317
P-Value
Parameter
Table 11
Panel A
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Statistical Factors
(January 1967-August 2009)
-0.258
0.062
0.236
-0.064
-0.111
0.001
0.508
0.595
0.079
0.001
-0.949
0.866
0.004
0.000
1.266
P-Value
Parameter
84
452
452
452
452
452
452
452
452
0.193
452
0.371
Table 11, Panel A depicts results from the second-pass of the Fama-MacBeth (1973) rolling regression procedure to assess priced risk factors in
stock returns for multiple model specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market portfolios.
Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED (UNEXPMED) is the expected (unexpected) component of
medical inflation as determined by a state space model described in Fama and Gibbons (1982). CPIMED is aggregate medical inflation, or the
sum of expected and unexpected medical inflation. MKTRF is the market return net of the risk-free rate. SMB is calculated by subtracting the
return of the decile of the largest stocksby market capitalizationfrom the decile of smallest stocks. HML is calculated by subtracting the
return of the stock decile having the lowest book-to-market equity ration from the decile with the highest book-to-market ratio. See Fama and
French (1993) for additional details regarding MKTRF, SMB, and HML. DEF is the difference between the 10-year Treasury bond and a portfolio
of Baa corporate bonds. TERM is the difference between the 10-year Treasury bond and the 90-day Treasury bill. MOM is a momentum factor
found by subtracting the returns of a stock portfolio having the lowest recent returns from a portfolio having the highest recent returns. MKTRF,
60%+0/DQG020FRPHIURP.HQ)UHQFKV'DWD/LEUDU\These data series occurs between January 1967 and August 2009, for 512 months
worth of data. The second-pass results in 452 data points due to the initial 60-month beta formation period.
N=
0.358
85
86
Table 11
Panel B
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors-Statistical
Factors, Divided Sample
(January 1967-August 2009)
Table 11, Panel A, Specification 8
Constant
EXPMED
UNEXPMED
MKTRF
SMB
HML
DEF
TERM
MOM
N=
Jan 72-Aug 09
Jan 72-Dec 84
Jan 85-Aug 09
Parameter
Parameter
Parameter
P-Value
P-Value
P-Value
1.261
0.627
1.595
0.000
0.230
0.000
0.039
0.078
0.018
0.002
0.003
0.152
0.002
0.079
-0.039
0.918
0.105
0.008
-0.930
-0.538
-1.136
0.001
0.325
0.001
0.097
0.258
0.013
0.511
0.277
0.946
0.488
0.798
0.324
0.001
0.001
0.085
-0.078
0.056
-0.149
0.151
0.566
0.021
0.292
-0.076
0.486
0.021
0.720
0.002
-0.395
-0.263
-0.464
0.193
0.498
0.264
452
156
296
Table 11, Panel B depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. EXPMED
(UNEXPMED) is the expected (unexpected) component of medical inflation as determined by a
state space model described in Fama and Gibbons (1982). MKTRF is the market return net of the
risk-free rate. SMB is calculated by subtracting the return of the decile of the largest stocksby
market capitalizationfrom the decile of smallest stocks. HML is calculated by subtracting the
return of the stock decile having the lowest book-to-market equity ration from the decile with the
highest book-to-market ratio. See Fama and French (1993) for additional details regarding
87
MKTRF, SMB, and HML. DEF is the difference between the 10-year Treasury bond and a
portfolio of Baa corporate bonds. TERM is the difference between the 10-year Treasury bond
and the 90-day Treasury bill. MOM is a momentum factor found by subtracting the returns of a
stock portfolio having the lowest recent returns from a portfolio having the highest recent returns.
0.75)60%+0/DQG020FRPHIURP.HQ)UHQFKV Data Library. These data series
RFFXUVEHWZHHQ-DQXDU\DQG$XJXVWIRUPRQWKVZRUWKRIGDWD7KHVHFRQG-pass
results in 452 data points due to the initial 60-month beta formation period. The second and third
specifications depict results when the sample entire sample is split into two time periods.
T-Stat
4.07
1.78
0.62
1.79
2.26
-0.81
-3.76
-1.98
-3.06
-2.75
-0.84
-1.42
-2.66
-2.00
-0.88
-0.18
MedHML
16.49
5.22
1.49
4.52
5.95
-2.40
-10.25
-5.02
-7.68
-7.21
-2.40
-4.41
-7.84
-5.73
-3.02
-0.48
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
B/M (Low)
1.09
1.09
1.02
1.03
1.09
1.11
1.10
0.99
1.00
1.06
1.16
1.01
0.90
0.94
0.99
1.13
MKTRF
63.73
49.21
54.63
53.45
53.91
59.76
64.72
60.83
60.76
59.79
60.40
58.74
54.72
60.42
52.17
43.06
T-Stat
0.40
0.51
0.36
0.42
0.51
0.74
0.87
0.71
0.75
0.87
1.00
1.05
0.97
1.05
1.27
1.31
SMB
17.67
17.16
14.68
16.43
19.10
30.15
38.61
32.97
34.03
36.84
39.02
45.86
44.66
50.68
50.39
37.38
HML
-0.40
Size 4
0.80
0.65
0.50
0.25
-0.44
Size 3
0.79
0.58
0.44
0.18
-0.37
Size 2
0.67
0.44
0.30
0.06
-0.30
Size (Small)
T-Stat
-15.51
23.67
23.13
17.24
8.10
-15.66
30.35
23.22
17.37
6.74
-12.56
25.69
17.81
12.85
2.12
-7.60
T-Stat
0.00
0.45
0.22
0.29
0.26
-0.20
-0.06
0.05
0.20
0.15
-0.12
-0.22
0.24
0.17
-0.15
-0.38
DEF
0.01
3.17
1.82
2.32
1.98
-1.69
-0.58
0.45
1.92
1.36
-0.93
-2.00
2.25
1.68
-1.21
-2.22
T-Stat
-0.03
-0.06
0.00
-0.05
0.00
0.03
0.03
0.00
0.04
-0.10
-0.05
0.12
0.01
0.00
-0.02
-0.10
TERM
Table 12
Panel A
Fama-MacBeth (1973) First-Pass Time Series Results for Priced Factors
(January 1967-December 2008)
-0.58
-0.89
-0.03
-0.78
-0.05
0.46
0.49
0.00
0.77
-1.80
-0.75
2.14
0.18
-0.03
-0.28
-1.15
T-Stat
-0.01
-0.05
-0.01
-0.02
-0.02
-0.05
-0.01
-0.01
0.00
-0.02
-0.04
-0.03
0.02
0.03
0.01
-0.03
MOM
-0.48
-2.30
-0.72
-0.86
-1.00
-2.53
-0.48
-0.42
-0.18
-1.23
-2.07
-1.78
1.24
1.73
0.29
-0.99
T-Stat
0.95
0.88
0.89
0.89
0.91
0.95
0.94
0.93
0.93
0.94
0.95
0.94
0.94
0.95
0.95
0.93
Adj
R-sq
88
-1.86
-3.03
-0.22
-1.34
-2.67
-1.31
-0.48
1.08
-6.12
-8.80
-0.79
-2.99
-7.45
-4.09
-1.25
4.39
B/M 3
B/M 4
B/M (High)
B/M (Low)
B/M 2
B/M 3
B/M 4
B/M (High)
1.05
1.00
1.01
1.03
0.97
1.15
1.05
1.13
1.13
39.38
58.18
49.38
56.76
66.51
50.13
55.62
52.76
54.02
-0.11
-0.21
-0.23
-0.21
-0.27
0.19
0.20
0.17
0.23
-3.22
-9.26
-8.40
-8.68
-13.89
6.15
8.00
6.12
8.40
0.77
0.62
0.31
0.13
-0.37
Size (Big)
0.79
0.62
0.52
0.28
18.96
23.65
9.90
4.62
-16.87
22.63
21.41
15.88
8.67
-0.07
0.02
0.01
0.20
-0.09
-0.05
0.15
0.15
0.32
-0.39
0.19
0.05
1.71
-0.96
-0.33
1.21
1.09
2.34
-0.03
-0.12
0.00
-0.01
0.06
-0.02
-0.12
-0.07
-0.09
-0.30
-2.14
0.03
-0.21
1.39
-0.21
-2.03
-1.08
-1.29
-0.04
-0.06
0.02
0.01
-0.02
-0.04
-0.02
-0.05
-0.03
-1.53
-3.53
1.11
0.29
-1.29
-1.71
-1.23
-2.23
-1.32
0.79
0.89
0.85
0.89
0.94
0.87
0.89
0.88
0.89
where , is the month t excess return on portfolio p, p t represents time, k represents the number of factors, and represents a factor
used to explain returns. , represents a portfolio-specific parameter estimated in the model and is calculated as the covariance of the factor and
portfolio return normalized by the variance of the factor (i.e., , = (, , )/)) (. For this specification, the portfolios p are the Fama
and French size- and book-to-market sorted quintiles, t represents the 504 months spanning from January 1967 to December 2008, and the factors
k are described as follows. MedHML is the return formed by subtracting the returns of a portfolio having a low beta with medical inflation from
the returns of a portfolio having a high beta with medical inflation. These portfolios are formed by sorting individual stocks into quintiles based
on their medical inflation beta. MKTRF is the market return net of the risk-free rate. SMB is calculated by subtracting the return of the decile of
the largest stocksby market capitalizationfrom the decile of smallest stocks. HML is calculated by subtracting the return of the stock decile
having the lowest book-to-market equity ration from the decile with the highest book-to-market ratio. See Fama and French (1993) for additional
details regarding MKTRF, SMB, and HML. DEF is the difference between the 10-year Treasury bond and a portfolio of Baa corporate bonds.
TERM is the difference between the 10-year Treasury bond and the 90-day Treasury bill. MOM is a momentum factor found by subtracting the
returns of a stock portfolio having the lowest recent returns from a portfolio having the highest recent returns. Shading indicates parameters that
are significant at conventional (90 percent) level.
=1
, = 0 + , , +
Table 12, Panel A shows results for the Fama-MacBeth (1973) first-pass regression for the following model.
-2.48
-8.00
B/M 2
89
90
Table 12
Panel B
Fama-MacBeth (1973) Second-Pass Cross-Sectional Results for Priced Factors,
with Divided Sample
(January 1967-December 2008)
CONSTANT
MedHML
MKTRF
SMB
HML
DEF
TERM
MOM
N=
Jan 77-Dec 08
Parameter
P-value
1.442
0.000
-0.006
0.039
-1.021
0.002
0.082
0.642
0.539
0.001
0.108
0.101
0.103
0.466
1.393
0.000
384
Jan 77-Dec 84
Parameter
P-value
0.124
0.801
-0.006
0.180
-0.225
0.757
0.270
0.449
0.709
0.015
0.121
0.461
0.047
0.865
0.569
0.295
96
Jan 85-Dec 08
Parameter
P-value
1.882
0.000
-0.006
0.096
-1.286
0.000
0.019
0.926
0.483
0.015
0.103
0.133
0.122
0.460
1.667
0.000
288
Table 9, Panel B depicts results from the second-pass of the Fama-MacBeth (1973) rolling
regression procedure to assess priced risk factors in stock returns for multiple model
specifications. Test assets are the 25 Fama-French quintile-sorted size and book-to-market
portfolios. Coefficients significant at the 10% level (or p-value < 0.10) are shaded. MedHML is
the return formed by subtracting the returns of a portfolio having a low beta with medical
inflation from the returns of a portfolio having a high beta with medical inflation. These
portfolios are formed by sorting individual stocks into quintiles based on their medical inflation
beta. MKTRF is the market return net of the risk-free rate. SMB is calculated by subtracting the
return of the decile of the largest stocksby market capitalizationfrom the decile of smallest
stocks. HML is calculated by subtracting the return of the stock decile having the lowest bookto-market equity ration from the decile with the highest book-to-market ratio. See Fama and
French (1993) for additional details regarding MKTRF, SMB, and HML. DEF is the difference
between the 10-year Treasury bond and a portfolio of Baa corporate bonds. TERM is the
difference between the 10-year Treasury bond and the 90-day Treasury bill. MOM is a
91
momentum factor found by subtracting the returns of a stock portfolio having the lowest recent
returns from a portfolio having the highest recent returns. MKTRF, SMB, HML, and MOM
FRPHIURP.HQ)UHQFKV'DWD/LEUDU\These data series occur between January 1967 and
December 2008, for 504 PRQWKVZRUWKRIGDWDPRQWKVRIGDWDDUHXVHGWRJHQHUDWHWKH
medical inflation surprise beta coefficient to create deciles, leaving 444 months for the two-pass
method. The second-pass in this case results in 384 data points due to the initial 60-month beta
formation period. The second and third specifications depict results when the entire sample is
split into two time periods.
92
Appendix C:
Using State Space Models to Disentangle Expected and Unexpected Inflation
The Kalman filter model is a state-space representation where model parameters are
continually updated to reflect new information. The model establishes dynamic parameter values
and facilitates finite-sample forecasts. This Appendix presents a brief summary of the topic,
which follows Hamilton (1994a), to which the reader is referred for a more extensive treatment.
The generic state-space representation of the dynamics of an observed variable y
associated with an unobserved variable is given by the following system.
t+1 = Ft + vt+1 (1)
yt = A xt + H t + wt (2)
ZKHUH)$DQG+DUHPDWULFHVDQGxt is a vector of exogenous variables, which could include
lagged values of y if uncorrelated with and w at all leads and lags. Equation (1) represents the
state or transition equation; equation (2) the observation or measurement equation. The two error
series v and w are shocks to the respective transition process and measurement equation,
respectively, and represent white noise. Assumptions in this model are that the error terms are
uncorrelated at all lags, x is uncorrelated with all lags of DQGZDQGWKHV\VWHPVREVHUYDWLRQV
are a finite series with the first unobservable state uncorrelated with all subsequent shock (i.e.,
v and w) values.
Given this general setup, Fama and Gibbons (1982) investigate the unobservable ex ante
real interest rate, , which is a function of the nominal interest rate (i.e., lagged Treasury bill), i,
inflation, I, and the average ex ante real interest rate, r, according to Fisher (1930) and
represented by equation (3).
t = it E(I)t r (3)
where E(*) is the expectations operator. Assuming the expected real interest rate follows an
AR(1) process, the state equation becomes (4) below.
t+1 = t + vt+1 (4)
Since we have observations on the ex post real interest rate, which is the nominal interest rate
minus actual inflation, we can write the measurement equation as follows.
it It = it EIt + EIt It (5)
or substituting from (3),
it It = r + t + EIt It = r + t + ut (6)
where ut equals EIt It , or the negative value of unexpected inflation. This term represents
error in the inflation forecast. If these inflation forecasts are made in optimal fashion, then these
errors, ut , should be uncorrelated with their lags and with the ex ante real interest rate, t . Thus
the conditions of equation (2) are met, and one can see how the general Kalman filter model in
93
equations (1) and (2) apply when we let F = , yt = it It , = r, and H = 1 to get the
following system.
t+1 = t + vt+1
it It = r + t + ut = r + t + EIt It
Using the Dynamic Linear Model (DLM) capability in the RATS computer program and setting
initial values of 0.1 for the coefficient on the nominal interest rate (i.e., lagged Treasury bill), i,
and 1.0 for the variances of the error terms v and u, we iteratively solve this system to extract the
time-varying constant parameter. Doing so in turn allows us to separate aggregate inflation into
its unexpected and expected components.
94
Essay #2: Using Genetic Algorithms for Hedging Health Care Costs, Managing
Macroeconomic Risk, and Tracking Investments
Introduction
The intuition behind a natural hedge is straightforward enough, and can be easily
illustrated through example. If an individual is concerned about wealth decreases from
rising gasoline and heating oil prices, then he or she could offset the wealth decreases by
investing in oil companies (with fixed production factor costs). Likewise, one would
think it is possible to offset health care costs by investing in health care-related firms.
Empirically, however, this natural hedge for health care does not exist according to the
analysis in Jennings, Fraser, and Payne (2009). By investigating the correlation between
various investable health care mutual funds and health care inflation, they find these
funds do a poor job of hedging health care costs that have outpaced general inflation
since the mid-1980s. Since hedging such non-investable macroeconomic factors has
substantial practical relevance for health care and beyond, the purpose of this paper is to
present an implementable technique to form a hedging or risk management strategy.
Narrowly, this research demonstrates the ability of a genetic algorithm (GA)
methodology to identify a portfolio of assets that offsets the risk posed by monthly
medical inflation. More broadly, the GA procedure implemented here could be used to
find risk-managing portfolios for virtually any non-investable time series, representing a
significant risk management tool. Further, the technique translates directly into creating
SRUWIROLRVWKDWPLPLFLQYHVWDEOHDVVHWV7KLVSDSHULOOXVWUDWHVWKHWRROVIOH[LELOLW\E\
demonstrating its ability to find asset portfolios that track mutual funds and exchangetraded funds (ETFs).
95
Literature Review
We provide a brief summary of GAs here, referring the interested reader to Bauer
(1994) for a more detailed description and Holland (1975) for the mathematical proofs
behind the methods. A genetic algorithm (GA) is an iterative computational method
based on an analogy with Darwinian natural selection and mutation. As with any
optimization method, a GA begins with choosing an objective function. In this
application, the first objective is to minimize the variance of a hedged portfolio consisting
RIDVKRUWSRVLWLRQLQPHGLFDOLQIODWLRQDQGDORQJSRVLWLRQLQDSRUWIROLRRILQYHVWDEOH
assets. Following the analogy of natural selection, a particular candidate solution is
knoZQDVDQLQGLYLGXDOZKLOHWKHH[WHQWWRZKLFKDSDUWLFXODUFDQGLGDWHVROXWLRQPHHWV
WKLVREMHFWLYHLVNQRZQDVWKHILWQHVVRIWKHLQGLYLGXDO7KHPHFKDQLFDOGHWDLOVRIWKLV
SDSHUV*$DSSOLFDWLRQDUHJLYHQLQWKH0HWKRGVVHFWLRQ+ROODQGSURYLdes the
eloquent math demonstrating the efficiency of this problem-solving methodology.
The contribution of this effort consists of using the GA method to create
parsimonious economic tracking or risk management portfolios. The Bauer (1994)
presentation focuses on using GAs to generate trading strategies based on certain rules,
but to our knowledge nowhere has anyone (publicly) discussed implementing them to
generate portfolios that offset the risk posed by uninvestable macroeconomic series. On
the other hand, Lamont (2001) describes a model that uses 13 stock and bond
portfolioswhile controlling for other lagged variablesto predict future
macroeconomic time series. His method is exclusively regression-based and assumes
investment in all 13 stock and bond portfolios, or essentially the market, to track each of
the 7 macroeconomic series he investigates. While (aggregate) inflation is among the
96
PDFURHFRQRPLFVHULHVKHORRNVDWKHGRHVQRWDVVHVVWKHVHDVVHWVDELOLW\WRSUHGLFW
the medical component of inflation. While medical inflation (CPIMEDSL) has outpaced
aggregate inflation (CPIAUCSL) since the early-1980s (see Figure 1), we nonetheless
suspect his main conclusions would hold for medical inflation, too. For future reference,
Appendix A includes all figures, and Appendix B includes all tables. Our contribution
EH\RQG/DPRQWVHIIRUWVLVWRXVHDFRPSOHWHO\GLIIHUHQWWRROWRILQGPRUH
parsimonious, flexible, and perhaps better-performing, set of investable assets to track a
non-investable (e.g., macroeconomic) series.
GAs lend themselves to the medical cost hedging problem presented in Jennings,
Fraser, and Payne (2009) for a variety of reasons, especially when compared to using a
straightforward multiple regression approach. First, multiple regressions typically satisfy
a single objective function: minimize the mean-squared error between the estimated and
actual data. While this might be the appropriate objective function for a GA to solve, it is
flexible enough to solve other objective functions as well. For instance, perhaps one
desires a portfolio prohibiting short-selling. One can implement a rule for positive asset
weights and then implement a GA to solve this problem. Additionally, since the GA is
not a hill-climbing algorithm, it is capable of handling non-linear as well as discontinuous
objective functions. In all, as evidenced with the example in this paper, the iterative
nature of the GA allows for much more flexibility and control over the desired objective
function than a multiple regression.
Besides their adaptability, GAs are computationally efficient. For the purpose of
this research, there exist literally tens of thousands of possible investable assets
worldwide that one could use to find the best hedge against medical inflation in this
97
country. Unfortunately the lack of data and degrees of freedom make it impossible to run
a multiple regression using all these assets in a single model. In this instance, since the
sample includes monthly medical inflation values between January 1967 and August
2009, there are only 512 data points. Using too many assets as independent variables
quickly consumes degrees of freedom. Further, if one were to explore combinations of
sub-samples of the assets, an exhaustive search of the combinations could take months or
years to complete.13 Obviously the hedging strategy in such a case could become
obsolete by the time it is discovered. Besides computational considerations, another
advantage of GAs relates to parsimony and user-defined objectives. For example,
transactions costs play a role in any applied investment decisions. By stipulating the
number and population of traded assets, users have greater control over these transactions
costs. Alternatively, using an exhaustive multiple regression solution could encourage
the investor to trade every asset in the model depending on the weights assigned to each
asset.
As Bauer (1994) highlights, there certainly exists a major caveat to GAs. Because
of their iterative nature and the sensitivity to the user-defined inputs (e.g., initial
population, breeding population, amount of gene crossover, mutation frequency and
method, etc.), there exists a real probability that a GA will not find the single optimal
solution. However, as he demonstrates, it will generally find near-optimal solutions, and
in some cases the optimal solution. It is possible the GA will converge too quickly on a
sub-optimal solution. As he further articulates, however, in a practical sense achieving
13
As a simple example, if one wanted to select 5 assets from a population of 300 assets, there exist over
19.6 billion unique combinations. If a computer processes 1000 multiple regressions per second, then an
exhaustive analysis would take over 7.5 months to complete. The processing time further increases with
portfolio size and the asset population.
98
the time-consuming optimal solution can become less-preferred than the quicklydeveloped acceptable one. Thankfully certain researchers, such as De Jong (1975) have
quite effectively established appropriate input values and quantified the sensitivity to
changes in them. Given the unique problem in this paper, the GA methodology generally
adheres to convention and uses these recommended input values, but we also provide
rationale for any deviations.
Data
Before describing the GA methodology employed, it is useful to outline data
sources briefly. Medical inflation comes courtesy of the Federal Reserve Economic
Database (FRED) hosted online by the Federal Reserve Bank in St. Louis. The security
returns as well as the long and short government bonds and Treasury bill rates come from
the Center for Research in Security Prices (CRSP).
99
not track the target precisely (i.e., > 0 for most or all periods t). We discuss this
auxiliary goal later when using the GA in a real-world application.
1
where =
,(
time t.
,QWKLVSURRIRIFRQFHSWRSWLPL]DWLRQWKHFKRLFHYDULDEOHVWKDWDFFRPSOLVKWKH
objective are a set of five assets chosen from a universe of 3,000 assets. These 3,000
assets are a random subsample of over 16,000 stocks from CRSP that have at least 60
months of returns between 1967 and 2009. In the vernacular of a GA, the universe of
assets LVNQRZQDVWKHSRSXODWLRQDQGWKHFKRLFHYDULDEOHVDUHNQRZQDVJHQHV$
VHWRIJHQHVWKHQFRPSRVHVWKHLQGLYLGXDO,QWKLVFDVHWKHUHH[LVWILYHDVVHWVJHQHV
that compose each portfolio (individual). The goal of the GA is then to form a hedge
portfolio by choosing a subset of five assets (i.e., genes) out of the universe of 3,000
assets (i.e., population) that has the minimum error variance over the investment horizon.
,Q*$WHUPVWKHJRDOLVWRILQGDQLQGLYLGXDOWKDWKDVDVHWRIJHQHV that generates
WKHKLJKHVWILWQHVVDPRQJWKHSRSXODWLRQRILQGLYLGXDOV
To illustrate the method, we must first define and quantify how well a potential
solution meets the objective function. Doing so means choosing 5 assets: i, j, k ,l and m,
from 3,000 assets without replacement and equally weighting their returns to form a
hedge asset with a return series given by equation (2).
, = 0.2 + 0.2 + 0.2 + 0.2 + 0.2
( 2)
100
In this equation, represents the return on a particular asset at time t, and the 0.2
coefficients represent the equal-weighting of each asset.
Next we construct a target portfolio according to equation (3),
, = 1 + 2 + 3 + 4 + 5 ( 3)
14
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
101
heart of the joint distribution of all the tracking assets. If the GA technique works as
anticipated, we should expect five assets to track the target quite well in this proof of
concept.
Having a target series along with a sample of possible assets to include in a
KHGJLQJSRUWIROLRZHWKHQFUHDWHSDUHQWSRUWIROLRVRUFDQGLGDWHVROXWLRQVZKLFK
are random combinations of five assets. This set is defined as the initial population (see
Figure 2). Each candidate in turn has a quantifiable fitness level according to the
objective function described in equation (1). Establishing the initial parent portfolios and
their respective fitness levels initializes the GA algorithm and leads to subsequent
iterations, whose goal is to improve the best solution.
The GA improves on the population of initial candidate solutions by creating new
candidates as partial combinations of existing population members. The production of
QHZFDQGLGDWHVROXWLRQVLVNQRZQLQWKH*$OLWHUDWXUHDVDEUHHGLQJ7KHSURFHVV
begins by first ordering the initial population according to its fitness level as shown in
Figure 3, Panel A. The popuODWLRQLVWKHQGLYLGHGLQWRWZRJURXSVFDOOHGWKHEUHHGLQJ
SRSXODWLRQDQGQRQ-EUHHGLQJSRSXODWLRQ)RUWKLVVWXG\ZHUHWDLQWKHEHVW-fit 10percent members as the breeding population and the remaining 90-percent as nonbreeding members. New candidate solutions come only from the breeding population,
and a new candidate is formed by randomly pairing two members of this breeding
population. For example, using the results in Figure 3, Panel A, the first new candidates
could be formed from candidates1000 and 2. The next pair considered might be 1 and
721. Keeping with the natural selection analogy, the pre-existing or current two
FDQGLGDWHVROXWLRQVDUHFDOOHGSDUHQWV7KHSDUHQWVDUHHDFKGHILQHGE\WKHLUJHQH
102
sequence (candidate 1000 by assets 2, 6, 28, 7 and 11 and candidate 2 by assets 23, 9, 7,
29 and 31). A breeding produces two new candidate solutions, or offspring, as follows.
We randomly generate an integer between 1 and 3. This integer, called the crossover
number, indicates how many genes (counted from the left) from the two parents to swap
LQJHQHUDWLQJWZRQHZFDQGLGDWHVROXWLRQVFDOOHGRIIVSULQJ)RUH[DPSOHIRUWKHILUVW
pairing, if the crossover number is 2, then the two new offspring would be as shown in
Figure 3, Panel B. In the second pairing (see Panel C), if the crossover number is 3,
Offspring 3 and 4 would be as shown. This process continues until enough pairings have
occurred for the breeding population to replace the non-breeding group. Using our
parameters, since we have 1,000 members in the population, 100 serve as the breeding
population, leaving 900 to be replaced through the breeding process. Thus we perform
450 pairings to generate the 900 replacement members.
The next step is to evaluate the fitness level of each offspring using the process
described for the initial population (see equation (1)). Finally, we modify the existing
population by determining whether the offspring have better fitness levels than their
parents. If a particular offspring has a fitness measure better than one of its parents, then
we replace one member of the non-breeding population with the superior offspring.
Doing so guarantees that the offspring has higher fitness than the candidate solution it
replaces because of the original sorting of the population by fitness and division into
breeding and non-breeding groups. If an offspring has an inferior fitness level compared
to its parents, then the offspring does not go into the population but instead becomes
replaced with a new candidate whose assets are chosen at random. This scenario
UHSUHVHQWVUDQGRPPXWDWLRQLQWKHSRSXODWLRQVJHQHSRRO$VZLWKWKHRWKHUPHPEHUV
103
we calculate the fitness level of this new candidate and replace one member of the nonbreeding population with this new candidate. Doing so at worst weakens the nonbreeding population but at best creates a mutation that will move into the breeding
population for the next generation. After repeating this process for all the offspring, we
again sort the population of candidate solutions by fitness level. This entire series of
FDOFXODWLRQVLVNQRZQDVFUHDWLQJDJHQHUDWLRQ7KHEUHHGLQJSRSXODWLRQWRSpercent most fit) after each generation will weakly dominate both the initial and prior
population.
The GA approach quickly converges to (near) optimal solutions by two
PHFKDQLVPV)LUVWWKHDOJRULWKPH[SORLWVWKHIDFWWKDWDJURXSRIJRRGVROXWLRQVZLOO
generally contain similar features; in our case this group of assets forms a good hedging
portfolio for a target series. By selectively swapping combinations of assets among
candidates that are by themselves good solutions, better candidates emerge. The second
IHDWXUHD*$H[SORLWVLVDFRQFHSWFDOOHGLPSOLFLWSDUDOOHOSURFHVVLQJ,QRXU
3,000
application, we have 5 assets to choose out of a population of 3,000 assets, or
5
possible hedges. If, after several generations, the algorithm determines that asset 20 does
not contribute to overall fitness, then the algorithm has effectively eliminated all
2,999
candidates have implicitly
4
been eliminated from consideration.15 The combination of these two features allows the
GA to cover a vast number of candidate solutions very rapidly.
Running the proof of concept for multiple generations provides encouraging
results. Table 1 shows the generation-by-generation results of the GA for 10 generations.
15
There would be 2,999 other assets to choose from once asset 20 was eliminated and 4 assets to choose.
104
The generation-by-generation population minimum, mean, and maximum fitness levels
vary quite substantially. Again, we calculate these fitness levels using Equation (1), and
our goal is to minimize the fitness value. The final column shows the mean fitness level
for only the breeding population by generation, or the top 100-performing portfolios in
this construction. As expected, the minimum fitness values decrease (Column 2), or
improve, over generations until a certain point at which convergence likely occurs. In
this instance the minimum value appears to converge in the fifth generation. Notably, the
mean and maximum (i.e., worst) fit of the population fluctuate over time, which occurs
because we allow substantial mutation from generation to generation. We notice two
things about the mean fit of the breeding population (Column 5). First, its fitness value
steadily improves (i.e., decreases) asymptotically over time. Secondly, the breeding
population has converged to the minimum value shown in Column 2 by the eighth
JHQHUDWLRQ7KXVWKHEHVWVROXWLRQKDVEUHGRXWDOORWKHUSRVVLELOLWLHVDQGWKHRQO\
possible further improvement would come from mutations in future generations.
Figure 4, Panel A shows a graphical representation of the monthly returns for the
hedge portfolio for the best solution (RETURNHEDGE), which is an equally-weighted
portfolio of assets indexed by numbers 2745, 2432, 2011, 1086, and 2151 (of our 3,000
original assets), and the target return series (TARGET) between January 1967 and
August 2009. Again, equation (2) shows how we calculate the notional target series. The
correlation between these two series exceeds 0.96. Because Panel A makes it difficult to
discern the difference between the two series, Figure 4, Panel B captures a scatter plot of
these values. These results are encouraging, since the corresponding values cluster
around a line through the origin with slope equal to one.
105
Clearly this proof of concept is artificialalbeit realisticsince all assets are not
available for investment during the entire period. This scenario contains look-ahead bias
since we estimate the coefficients using all information available in the sample.
Realistically, at any given point in time, we could only estimate the relevant betas using
past information instead of across the entire time period as shown here. Additionally, this
proof of concept does not allow for dynamic factor models that might provide additional
insight. We incorporate a form of conditioning factor model where portfolio weights
FKDQJHZLWKWKHVWDWH-of-the-ZRUOGLQWKHIXWXUHUHDO-world applications. With these
considerations in mind and with the belief the GA works as intended, we now turn to outof-sample testing using an actual target and investable assets.
106
funding standpoint. We subsequently provide a more tangible explanation of this
argument.
In this real-world exploration, the choice variables that accomplish the objective
are a set of five assets chosen from a universe of 306 assets. The 306 assets include 303
stocks pulled from CRSP that have returns as of January 1967 and December 2008,
which is the period for which we have relevant medical inflation data. Additionally,
these stocks have uninterrupted return data for at least 100 months before 2005, which is
the earliest out-of-sample period we consider for the macroeconomic series. We also
include three government bond monthly return series: the 10-year bond, one-year note,
and 30-GD\ELOO&OHDUO\DOORIWKHVHDVVHWVDUHWUXO\LQYHVWDEOHOHDGLQJXSWRWKHRXW-ofsample periods in a practical sense, unlike those shown in the proof of concept scenarios.
Further, the results shown are but a starting point considering the recent work of Brandt
and Santa-Clara (2006). These authors show that applying conditioning and timing
methods to asset returns can expand the asset universe essentially without bound. While
we consider conditioned returns for our mutual fund and exchange-traded fund (ETF)
applications, we do not incorporate timing methods here.
Broadly speaking, the problem at hand can be characterized as follows. Starting
today, you have a known liability (assuming medical prices remain constant) over the
QH[W\HDUUHVXOWLQJIURPPHGLFDOH[SHQVHV:HDUHDVVXPLQJQRTXDQWLW\ULVNLQWKLV
medical expense. That is, we assume the level of medical services remains constant, and
the only uncertainty comes from medical price changes. Given that medical expenses are
likely to increase in an uncertain fashion (see Figure 1), you need to invest in a set of
assets that will increase in value at or above the rate of medical inflation, and at the same
107
time result in a combined net asset position that has volatility lower than if the medical
liability were hedged with risk-IUHHLQYHVWPHQWV6XFKULVN-IUHHLQYHVWPHQWVHJ
JRYHUQPHQWERQGVRU&'VEDVLFDOO\UHSUHVHQWWKHEDVHOLQHKHGJLQJVWUDWHJ\VLQFHLWLV
unlikely an exposed entity would let available funds sit idle and bear no interest.
Panels A through C in Figure 5 depict relevant scenarios. With return on the yaxis and time on the x-axis, Short Target represents a short position in the series (e.g.,
medical inflation) you wish to hedge. In Panel A, the ideal Hedging Portfolio would
mirror the Short Target series exactly, leading to the ideal Hedged Portfolio, which in this
example has zero variance. To the extent this Hedged Portfolio lies above or below the
zero return line, it is possible to hedge the target using more or less funds than the known
liability. Specifically, the Hedged Portfolio in Panel A indicates a hedging portfolio with
a mean periodic return higher than that of the target series, which means you could invest
less than the current known liability and still offset the future liability exactly. Thus
Panel A is an ideal scenario: a hedged position that is less risky than a natural short
position along with a fully-funded (in fact over-funded) liability. Panel B depicts a
scenario where a constant return asset such as a T-Bill serves as the Hedging Portfolio.
Clearly the Hedged Portfolio exhibits higher return variance than in Panel A, and in this
case the variance ratio of the hedged portfolio to the target series equals one since the
variance of the Hedged Portfolio equals the variance of the Short Target. Although we
show a situation where the constant return of the Hedging Portfolio is greater than the
absolute value of the Short Target return, if this constant return series were less than the
absolute value of the Short Target return, then we would have a situation where the
liability would be underfunded over time. Thus it is important to consider both the
108
variance ratio and return levels when considerLQJWKHKHGJLQJSRUWIROLRVHIIHFWLYHQHVV
from a risk management standpoint. Finally, Panel C depicts a more real-world scenario
where the Hedging Portfolio correlates with the Target with an absolute value between
zero and one. In this case, the Hedging Portfolio relationship with the Short Target series
creates a Hedged Portfolio with higher variance than the Short Target position. However,
the liability is always fully-funded since the returns to the Hedged Portfolio are nonnegative. It is straightforward to envision the alternative scenario whereby the Hedged
Portfolio variance is lower than the Short Target with negative returns.
Returning to our GA, the objective in this case it to choose five assets, (i, j, k , l,
and m) from 306 assets without replacement and weight them to form a hedging portfolio
with return according to equation (4),
= 0 + 1 + 2 + 3 + 4 + 5 (4)
where , represents the return on a particular asset and LP are the indices that
indicate one of the 306 assets. 1 . . 5 are hedge ratios determined by OLS regression.
We next define the hedged position, or hedged portfolio, as in equation (5).
= ( 5)
Here, serves as the medical inflation target we are trying to hedge and
is the return series of the hedging portfolio which consists of five investable
assets and is shown fully in equation (4). Since we are naturally short the medical
inflation position in that we must pay medical care costs each time period, we take a long
position in to offset the movements in , which will ideally
stabilize and minimize the risk of escalating health care costs. is analogous to the
series in equation (1).
109
,QHYDOXDWLQJWKHKHGJLQJSRUWIROLRVILWQHVVZHDJDLQUDQGRPO\JHQHUDWH
candidate hedge portfolios consisting RIILYHJHQHVRUDVVHWVDQGGHILQHWKLVVHWDVWKH
initial population. To evaluate the performance out-of-sample, we divide the data into
three regions as shown in Figure 6. For our baseline analysis, these periods include the
Test period (January 1967 to December 2006), Validation period (January to December
2007), and the Out-of-Sample period (January to December 2008). For robustness
purposes we also explore calendar years 2005-2007 as out-of-sample periods. We then
regress medical inflation ( ) on these assets according to Equation (6) using
only WKHGDWDLQWKH7HVWSHULRG(TXDWLRQVKRZVDQH[DPSOHIRUPRIWKLVUHJUHVVLRQ
using assets 1, 7, 25, 36, and 41.
= 0 + 1 1 + 2 7 + 3 25 + 4 36 + 5 41 (6)
From this regression come estimates for the hedge ratios 0 . 5 . Using these
parameters, it is possible to calculate the time series for RYHUWKH7HVW
SHULRGSHUHTXDWLRQDQGFRQWLQXHXVLQJWKHKHGJHUDWLRVWKURXJKWKH9DOLGDWLRQ
SHULRG6LQFHWKH9DOLGDWLRQSHULRGLVRXWVLGHWKHUDQJHXVHGWRHVWLPDWHWKH
SDUDPHWHUVLWDFFRXQWVIRUSHUIRUPDQFHRIWKHPRGHOIRUPHGGXULQJWKH7HVWSHULRGLQ
a pre-out-of-sample manner and reduces the criticism of overfitting the model based on
past known information. Next we calculate the mean and variance of , ,
and IRUERWKWKH7HVWDQG9DOLGDWLRQSHULRGV7KHILWQHVVOHYHOLQWKLV
application, which we seek to minimize, consists of a weighted measure that accounts for
these attributes as shown in equation (7),
2 ,
2
= 1 ,
+ 2
2 ,
(7)
110
2
where ,
is the variance of return series x during period y. Using the two terms
incorporates the variance of the hedged portfolio in both the Test and Validation periods.
Specifically, the first term simply measures the variance of the hedged portfolio, H,
weighted by factor 1 . The second term expresses the idea that it is desirable for the
variance of the hedged portfolio in the Validation period to remain consistent with or
lower that it is in the Test period. We consider it a stability measure and weight its
importance by 2 . To emphasize, in this construct lower fitness values are preferred.
Initially we set (1 , 2 ) to (1, 1) but also complete sensitivity checks using other
weight vector values. We repeat regression (6) and calculate the fitness using equation
(7) for all 1,000 candidate solutions in the initial population. Having initialized the
candidate solution population, we use the iterative GA procedure, or breeding, as
described in the proof of concept above to evolve an improved solution over multiple
generations. The major differences between this real-world application and our earlier
proof of concept involve the set of possible hedging assets and the fitness level measure.
The breeding process and its user-defined inputs (e.g., number of genes, number of
parents, breeding population, mutation procedure, and crossover rate) remain consistent
with the proof of concept.
Results
Table 2 shows the generation-by-generation results of a GA run with the fitness
weights (1 , 2 ) set to (1, 1). The generation-by-generation population minimum, mean,
and maximum fitness measures vary as anticipated. The minimum decreases over time,
the maximum fluctuates randomly given our allowance for mutations, and the mean also
111
fluctuates, but to a lesser degree. Again, we calculate these fitness measures using
Equation (7). The final column shows the mean fitness measure for only the breeding
population, which are the top 100-performing portfolios in each generation. As expected,
again the minimum fitness values decrease (Column 2), or improve, over generations
until convergence in the tenth generation. In this case, we again notice two things about
the mean fit of the breeding population (Column 5). While its fitness value steadily
improves (i.e., decreases) asymptotically over time, the breeding population has not
converged to the minimum value shown in Column 2. Thus it is theoretically possible to
obtain even better performance if we were to allow for additional generations, and the
possibility of mutations always makes this possibility hold. However, improvement
seems improbable given the best existing solution in most cases is a unique one. This
PRVWILWPHPEHULVRIWHQMXVWRQHPHPEHULQD-member breeding population, and
sinFHZHDOORZSDLULQJVWRRFFXUUDQGRPO\DVRSSRVHGWRIRUFLQJWKLVPRVWILWVROXWLRQ
WREUHHGHDFKJHQHUDWLRQLVQRPRUHOLNHO\WREUHHGWKDQDQ\RIWKHRWKHUEUHHGLQJ
members.
Performing a brief sensitivity analysis by varying (1 , 2 ) leads to Table 3.
Clearly there are many more possibilities than those shown, but the point is merely to
demonstrate the potential for the GA to generate nice solutions. Column 1 shows varying
weight values, while Column 2 depicts the ratio of the variance of the hedged portfolio to
the variance of actual medical inflation for the out-of-sample evaluation. Columns 3
through 7 depict the assets that compose the best hedge portfolio as determined by the
GA; the Asset Key table at bottom shows the associated CRSP tickers for these assets.
Appendix C lists all asset-ticker-industry combinations for the GA-generated portfolios
112
for the rest of this study. Column 8 shows the generation at which the minimum fitness
level converges for each respective weight combination. In the analysis to date, we
constrain the process to end after 10 generations. The final column shows the mean outof-sample excess monthly return by investing in the hedging portfolio. For instance, by
investing in stocks 274, 79, 199, 33, and 214,16 (i.e., weight vector (1,1)) and
simultaneously remaining naturally short the medical inflation measure, one would earn
an average return of 0.316 percent per month on hedged portfolio between January and
December 2008, indicating the liability is fully-funded as described earlier. And since
the mean fitness level of the breeding population has not yet converged to the minimum
value in virtually any of these cases (not shown), again it is theoretically possible that an
even better solution exists than shown here. Of course, mutations could always improve
the population even after convergence.
For this relatively small sample of weight values, Column 2 of Table 3 (which
quantifies the variance ratio between the hedge portfolio and medical inflation) indicates
that the GA-determined hedge portfolio eliminates up to 49 percent (i.e., ratio equals
0.51) of the variance in medical inflation for the out-of-sample period. That is, the
variance of simultaneously having a short position in medical inflation and a long
position in the five GA-selected stocks and/or bonds generates a portfolio variance less
than one-half of what it would be simply remaining exposed to the medical inflation
VHULHV,QWKLVEHVWFDVHWKHSRUWIROLRFRQVLVWVRIIRXUVWRFNV17 and the 30-day Treasury
Bill, which is represented by return series 306. Figure 7, Panels A through C depict the
16
The major industries of the stocks that correspond to these index numbers are Processed & Packaged
Goods, Entertainment, Specialty Retail, Security & Protection Services, and Industrial Equipment.
17
The stock industries include Accident & Health Insurance, Restaurants, Rental & Leasing Services, and
Domestic Telecommunication.
113
out-of-VDPSOHUHWXUQVHULHVDQGDVVHWVUHODWLYHZHLJKWVZLWKLQWKHKHGJLQJSRUWIROLR
respectively. Clearly for the weight vector (1,0) portfolio, a long position in the T-Bill
dominates the position, with an investor holding very small positions in the other 4 assets,
ZKLFKDUHVWRFNV1RWDEO\KRZHYHUHYHQWKHVHVPDOOSRVLWLRQVDIIHFWWKHSRUWIROLRV
overall return series non-trivially. This effect becomes evident by comparing this
SRUWIROLRVYDULDQFHUDWLRRIWRDUDWLRRIZKHQXVLQJRQO\WKH7-bill as a
hedging portfolio. We discuss this T-bill situation in more detail below. The other two
portfolios consist solely of a set of common stocks. Clearly the performance is sensitive
WRWKHZHLJKWYDOXHVDQGHYHQLQWKHZRUVWFDVHGHSLFWHGLHWKHYDULDQFHRI
the hedged portfolio is still below unity at 0.97. While the variance ratios are appealing,
perhaps as relevant from a broader risk management perspective is that in each hedged
portfolio the monthly returns on average dominate the rise in medical inflation, and in
every case these hedging portfolio returns are greater every month (see Figure 7 and its
subsequent explanation for the graphical evidence). Thus while the portfolio might not
always track the target of medical inflation precisely, investing in a GA-selected portfolio
of five common stocks and government bonds appears to offset the rise in health care
costs in the out-of-sample period studied here, while simultaneously lowering the liable
HQWLW\VULVNH[SRVXUH
Knowing the sensitivity to the weights, we must select a set of weights for further
DQDO\VLVDQGDSSOLFDWLRQ:KLOHDSSHDUVWREHWKHEHVWUHVXOWIrom Table 3, we are
concerned this set of weights fails to account for the Validation period results and the
associated performance stability we desire and described earlier. The weight vector (1,1)
is also appealing, since its hedged portfolio reduces the risk of exposure to medical care
114
costs by approximately 14 percent. However, since the weight vector (0,1) also provides
convincing results (i.e., variance ratio is less than one and the mean monthly return of the
hedged portfolio is positive), it seems to present the most conservative yet still effective
case. Therefore we select the vector (0,1) for analysis from here forward. We label this
ZHLJKWYHFWRUWKHJRRGFDVHDVRSSRVHGWRWKHEHWWHURUEHVWFDVHgoing
forward. Making this selection provides us with an effective hedge for medical inflation
since the variance ratio is less than one, allows us to account for both the Test and
9DOLGDWLRQSHULRGGDWDDQGKRSHIXOO\IRUHVWDOOVDOOHJDWLRQVWKDWZHDUHFKHUU\-SLFNLQJ
for the best subsequent results.
To expand a bit on this weight set using Figure 7, it is visually difficult to discern
that the hedge portfolio, H, has lower variance than medical inflation, since the ratio is
close to unity. However, it is clear the monthly returns for the hedging portfolio for the
JRRGFDVH5(7+('*(3257RXWSDFHPHGLFDOLQIODWLRQ,QRWKHUZRUGVLQYHVWLQJLQ
this hedging portfolio of assets 81, 76, 84, 229, and 11718 FRYHUVDILUPVPHGLFDO
inflation exposure and on average provides a mean monthly excess return of 0.328
percent (or roughly 4 percent annually) during the out-of-sample period over medical
inflation.
While we introduce the significance of excess mean return in the discussion
surrounding Figure 5, Table 4 quantifies what this excess return means for the entity
seeking to fund a future liability for the year 2008. Assuming an entity is setting aside $1
million to fund the anticipated medical care expense it will incur sometime during 2008,
Column 2 shows monthly medical inflation, with Column 3 showing what the $1 million
18
The stock industries include Chemicals, Business Equipment, Electric Utilities, Manufacturing, and Steel
& Iron. Appendix A lists the stock index numbers, tickers, and industries.
115
liability would cost if incurred at the end of the associated month. Column 4 shows the
monthly hedging portfolio return using assets 81, 76, 84, 229, and 117 (see Figure 7,
Panel B), and Column 5 translates these returns into dollar values if the $1 million setaside is invested in the hedging portfolio. Column 6 shows how much of the original $1
million would have been needed to offset the liability exactly, with Column 7 (8)
showing the excess initial funds (as a percentage) at the beginning of the year if the
whole liability were to occur at the end of the associated month. The ranJHRIH[FHVV
funds at the outset is between 0.09 percent and 3.93 percent in this example. Again, this
overfunding occurs due to the hedging portfolio returns outpacing medical inflation every
month of 2008, and the GA finds an investment portfolio that creates less risk for the
exposed entity than doing nothing and remaining exposed to medical inflation. And
while a portfolio of all stocks like the one with weight vector (0,1) perhaps creates more
risk than simply investing in T-bills, using T-bills requires investing all of the liable
funds in this single asset. A shock to the one-to-one relationship between medical
inflation and T-bills could be devastating for the liable entity, whereas a portfolio of
multiple (e.g., five) assets could more easily absorb a shock to the relationship between
one of its assets and the medical inflation target, not to mention the inferior performance
of the T-bill as a lone hedging asset discussed in the subsequent paragraph. Additionally,
while the all-stock portfolio with weight (0,1) is inferior from a volatility standpoint, it is
possible that the excess mean return compensates for this loss.
Figure 8, Panel A, depicts the potential portfolios in mean-standard deviation
space. Clearly all three of the hedged positions dominate a short position in medical
LQIODWLRQ0HG,QIZLWKERWKDKLJKHUPHDQDQGORZHUVWDQGDUGGHYLDWLRQLQWKHLURXW-
116
of-sample returns, and it appears the hedging portfolio with weights (0,1) is slightly
inferior to (1,1), again confirming it as the conservative choice going forward. Finally,
since the question naturally occurs, untabulated analysis shows simply taking a long
position in T-bills provides a slightly inferior variance ratio of 0.995 but also means an
average return deficit of 0.22 percent per month in 2008. In other words, simply
investing in T-bills is (slightly) more risky than the GA-generated hedging portfolio
while also leaving the entity underfunded for the year.
Another natural question that arises concerns the effectiveness of the S&P 500 as
a hedge for medical inflation. Intuitively, since medical inflation has generally been
positive since the 1970s (Figure 1) and the S&P 500 index has also tended to grow over
time, one might think these two trends might correlate well. Contrary to this intuition,
investing in the S&P 500 would serve as a relatively poor hedge for medical inflation
when considering the GA-generated portfolios. Taking a long position in the S&P 500
does not serve as an effective hedge for medical inflation in terms of its variance ratio.
Respectively, for calendar years 2005, 2006, 2007, and 2008, the variance ratios of the
hedged portfolio (i.e. long S&P 500; short medical inflation) to the medical inflation
target series are 271.04 , 232.25, 380.67, and 4,203.12. The respective correlations
between the S&P 500 monthly return series and monthly medical inflation of 0.21, -0.40,
-0.13, and -0.19 for 2005 to 2008 also provide weak support for using the S&P 500 to
hedge medical inflation. Finally, using the S&P 500 not only makes the hedged portfolio
more risky but also fails to manage risk from the standpoint of funding the medical
liability. In 2005, 2007, and 2008, the average monthly returns of a hedged portfolio
117
consisting of a long position in the S&P 500 are negative (-0.11, -0.13, and -4.27 percent,
respectively).
5HWXUQLQJWRWKHJRRGSRUWIROLRUHVXOWLQJIURPWKH*$LWLVIXUWKHUSRVVLEOHWR
generate the parameters in equation (4) to see the out-of-sample hedge ratios for assets
that form the hedging portfolio. Table 4 shows these results for the Test period according
to equation (7). As an aside, the relative weights in Figure 7, Panel B come from
normalizing these regression parameters (i.e., dividing each individual parameter by the
sum).
= 1 + 2 81 + 3 76 + 4 84 + 5 229 + 6
117 +
(7)
In this model, the labels refer to the monthly security returns for stock x as
described earlier. Table 4 indicates hedge ratios on the individual assets are statistically
insignificant at conventional levels. However, this statistical insignificance during the Test
Period becomes less relevant when we consider that our ultimate concern is the out-ofsample performance.
Two natural concerns with implementing this method are transactions costs and
the ability to take short positions in the required assets. Since in this application the
hedging portfolio is formed at the beginning of the out-of-sample calendar year and
frozen for the rest of the year, transactions costs occur once. Therefore, since portfolio
turnover is almost nonexistent, we do not account for transactions costs in our
macroeconomic examples. However, in foreshadowing, the mutual fund and ETF
extensions in this paper exhibit greater turnover and do account for such transaction
costs. Regarding short sale constraints, for this version of the paper, we proceed in the
spirit of Brandt and Santa-Clara (2006), who also exhibit optimal portfolios consisting of
118
short positions in both stocks and bonds. At an applied level, clearly short sales must be
dealt with by the practitioner on an asset-by-asset basis. Potential future work involves
altering the fitness function to avoid any short positions.
Finally, to put these findings into graphical perspective and relate them to past
work, Figure 8, Panel B compares the possible medical inflation hedging scenarios in the
mean/standard deviation space. Knowing the hedged position for the all GA-generated
cases dominates the natural short position in both the first and second moments from
3DQHO$3DQHO%SUHVHQWVDUHODWLYHFRPSDULVRQRIWKH*$KHGJLQJSRUWIROLRV
performance to the health care mutual funds in the spirit of Jennings, Fraser, and Payne
(2009). Clearly the GA hedging portfolio performs superior to the intuitive natural
hedges represented by the health care-related mutual funds offered by Eaton Vance,
Vanguard, and Fidelity (ETHSX, VGHCX, and FSPHX, respectively).
119
to investable assets such as stock indexes, mutual funds, or exchange-traded funds
(ETFs). As the following results show, while the GA performance does not universally
reduce the risk as measured by portfolio variance, it gains appeal when considering the
mean return provided to offset this (sometimes very slight) additional risk. Finally, the
GA-developed portfolios perform quite well out-of-VDPSOHZKHQPLPLFNLQJRWKHr
investable assets such as mutual funds and ETFs.
The prior results surrounding medical inflation invite the criticism of look-ahead
bias. While up to this point we have chosen a fitness measure weight vector
retrospectively based on out-of-sample performance that has already occurred, to
strengthen the argument for applying this GA procedure requires selecting a fitness
measure a priori without the foresight of which (1 , 2 ) weight vector is best. As
discussed and decided upon earlier, we run all future results with and only with (1 , 2 )
HTXDOWRZKLFKZHFDOOWKHJRRGFDVHDQGRQHFRXOGDUJXHLVFRQVHUYDWLYHVLQFHLW
GRHVQRWSURYLGHWKHEHVWRXW-of-sample hedging portfolio for medical inflation.
Making this decision in advance represents the situation we would face in a real-world
application.
We select macroeconomic series based on practical relevance as well as data
availability. Not all target series are available beginning in January 1967; we note the
available dates for each series parenthetically in future descriptions. Additionally, we
think the results presented could be conservative for a reason beyond the fitness weight
vector. While some target series begin in, say, the mid-1980s, we do not expand the
investable stocks available to hedge the particular series to those that have returns
beginning in the mid-1980s and continuing until the present. So while we constrain the
120
number of available assets in the hedging portfolios at one level, on another level we
allow them to expand. As Brandt and Santa-Clara (2006) highlight, we could include
conditional assets to expand the set available almost infinitely. To foreshadow, we do
increase the available assets in the mutual fund and ETF analyses by incorporating a
conditioning variable (see Cochrane (1996)), but doing so comes with a price, as altering
the portfolio more often than annually increases transaction costs. We account for these
costs in this analysis.
One rationale for using this GA technique to hedge other macroeconomic series
could be for insurers to hedge the payouts to claimants, particularly for homeowners
policies. Since various macroeconomic series might best capture these liabilities, we
look at a host of possibilities, including the housing component of inflation (January 1967
to December 2008), the Case-Shiller 10-City Composite Housing Price Index (January
1987 to December 2008), and the Producer Price Index for Residential Construction
(June 1986 to December 2008). The first and last series come from FRED, and the Case6KLOOHU,QGH[GDWDFRPHVIURPWKH6WDQGDUGDQG3RRUVZHEVLWH19
From a transportation and energy perspective, there could be multiple uses for an
investment portfolio to hedge associated liabilities such as fuel prices. For this reason,
we apply the GA to the energy component of inflation (January 1967 to December 2008),
transportation inflation (January 1967 to December 2008), spot oil price for a barrel of
West Texas Intermediate (January 1967 to December 2008), spot price for a gallon of
New York Harbor kerosene-type jet fuel (April 1990 to December 2008), and monthly
price per gallon for diesel fuel (March 1994 to December 2008). FRED provides data on
19
http://www.standardandpoors.com/indices/sp-case-shiller-home-price-indices/en/us/?indexId=spusacashpidff--p-us----
121
the inflation measures, Dow Jones provides information on the spot oil prices, and the
Department of Energy is the source for both jet and diesel fuel prices.
Last, but certainly not least, two macroeconomic series of general interest include
aggregate inflation and changes in the S&P 500 index. Bodie (1976) investigated the
question of hedging aggregate inflation, so this question is not new, and our GAgenerated results support his conclusion that shorting common stocks is an effective
hedge for inflation. However, the GA in this paper tells us exactly which five stocks do
an effective job. While we anticipate the GA will find a set of five stocks that will
generally track the S&P 500 index, we do not believe it will be easy for such a portfolio
to have lower variance given the Statman (1987) finding that well-diversified portfolios
require 30 to 40 stocks as opposed to the five we select using this GA. Finally, from a
time series econometrics standpoint, since all of these data series mentioned are captured
in levels, we difference the natural log values between periods to convert them into
returns to correspond with the return values for the hedge portfolios as well as move from
non-stationary to stationary time series.
7KHUHVXOWVRIWKH*$VSHUIRUPDQFHIRUWKHVHYDULRXVPDFURHFRQRPLFVHULHVDUH
VKRZQLQ7DEOH3DQHOV$WKURXJK'RI7DEOHGHSLFWFDOHQGDU\HDUVZRUWKRIRXWof-sample results. The Panels are in reverse chronological order so we can view more
recent out-of-sample results first. We use the calendar years 2005 to 2008 to capture
recent results across a range of economic climates, with the earliest 2 years (2005-2006)
showing evidence of economic growth as measured by the S&P 500, followed by a
UHODWLYHO\IODW\HDULQDQGHQGLQJZLWKWKHREYLRXVO\FKDOOHQJLQJHFRQRP\LQ the
year 2008.
122
Overall, these results show that selecting a set of five investable assets to hedge
various macroeconomic phenomena is imperfect, which supports the Chen, Roll, and
Ross (1986) finding that the first-pass Fama-MacBeth (1973) regressions of
macroeconomic factors on stock portfolios do not perform very well, but it is not
impossible. The best variance ratios occur with the S&P 500, the only series having
ratios of less than unity across all time periods. It has variance ratios ranging from 0.62
to 0.87, which indicates the GA-identified hedged positions reduce variance by 13 to 38
percent. Additionally, the S&P hedged portfolio provides positive mean monthly returns
for the most recent two years studied, 2007 and 2008, indicating that perhaps the GA
technique is most effective in flat or downward-moving markets. Globally, the worst
variance ratio is the 2008 Case-Shiller 10-City Composite Index, where the GAgenerated hedged portfolio is 29 percent more volatile than the Case-Shiller Index.
Looking at medical inflation, the GA portfolios perform well in our estimation.
In both 2007 and 2008, the variance ratios of the hedged portfolio is lower than 1.0, and
the worst variance ratio occurs in 2006 at 1.07. Nevertheless, in every case the entity
embracing the hedging portfolio is fully funded out-of-sample, with monthly (annual)
excess returns ranging from 0.13 (1.58) percent in 2007 to 0.33 (4.01) percent in 2008.
From a mean-standard deviation perspective, the hedging portfolios in 2007 and 2008
provide an unambiguously better position for the exposed entity. The Pearson
correlations between medical inflation and the hedging portfolio presented in Column 5
provide mixed results; the positive values in 2007 and 2008 are desirable, but the
negative correlations in 2005 and 2006 are concerning. The 2005 and 2006 results
indicate at worst the entity must seriously consider the tradeoff between risk and return,
123
since with slightly more risk it can clearly fund the health care liability. In no case is the
GA-generated hedging portfolio unambiguously worse (i.e., higher variance and lower
return).
The housing-related series tend to provide mixed results over time. As a whole,
the variance ratios are better during the economic growth years of 2005 and 2006 with 2
of the 3 series having ratios below 1.0 in each year (housing inflation and Case-Shiller in
2005; Case-Shiller and PPI residential construction in 2006). However, from the
perspective of fully-funding the liability, the latter years of 2007 and 2008 provide better
risk management portfolios with positive mean monthly returns to the hedge portfolios
for both housing inflation and the Case-Shiller Index. The target-hedging portfolio
correlations are positive for PPI residential construction in all years except 2008, but no
clearly apparent trends emerge for the other series over time or based on the economic
environment.
The energy and transportation area includes five series that provide mixed results
in terms of hedging effectiveness. Approximately one-half of the series-year
combinations show reduced risk from taking the hedged position, since 11 of the 20
combinations exhibit variance ratios under 1.0, and exactly 10 of the 20 series-year
combinations indicate the hedged portfolio fully funds the liability. While in the difficult
economic environment of 2008 all hedged portfolios are slightly more volatile than the
target with variance ratios ranging from 1.01 (energy inflation, transportation inflation,
and diesel fuel PPG) to 1.13 (jet fuel spot price), all of the liabilities are overfunded based
on the positive mean monthly returns to the hedged portfolios. At the opposite extreme,
in 2005, all variance ratios are less than one, ranging from 0.86 (jet fuel spot price) to
124
0.99 (energy inflation and diesel fuel PPG), but the hedged portfolios exhibit negative
mean monthly returns in all but one case (transportation inflation). Additionally, as a
group the positive correlations between the target and hedging portfolios are more
appealing for the year 2005 than for 2008. In the more flat economic period of 2007, a
hedged position would reduce risk in four of five cases, with transportation inflation
representing the exception, but the hedging portfolio would underfund the liability in
every case. During the economic growth year of 2006, hedging performance is
noteworthy for both the oil spot price and transportation inflation, as both exhibit riskreducing hedged portfolios that would fully fund the associated liability. All things
conVLGHUHGLWDSSHDUVWKH*$KHGJLQJSRUWIROLRVHIIHFWLYHQHVVLQDGGUHVVLQJ
energy/transportation-related risk is sensitive to the economic environment, with the
slightly more effective results appearing to come during difficult economic environments
like the one that occurred in calendar year 2008.
Finally, with the other two series of interest, aggregate inflation and the S&P 500
return series, hedging performance generally appears solid across all periods. For
aggregate inflation, the hedged portfolio fully funds the liability in every year studied.
Both 2008 and 2005 are years where the benefit of the hedging portfolio is unambiguous,
since besides having mean excess returns, in these years taking the long position in the
hedging portfolio reduces the volatility of the return series. While the correlations could
certainly be higher, with the maximum (minimum) correlation of 0.23 (0.02) occurring in
2005 (2006), it is encouraging they are non-negative for all out-of-sample years. For the
S&P 500, as previously mentioned, it is the only macroeconomic series studied where the
hedged portfolio exhibits less volatility than the target in every year. Again, the concern
125
is certainly the underfunding that occurs out-of-sample during the economic growth years
of 2005 and 2006. This concern is somewhat mitigated by the correlations between the
target and the five-stock hedging portfolio, which range from 0.38 in 2007 to 0.70 in
2008. Oddly enough, of all the macroeconomic series investigated here, the only two
containing a bond in the GA-generated hedging portfolio are the energy inflation in 2006
and the S&P 500 in 2005. Untabulated results show this short (one-year) bond composes
over 99 percent of these portfolios, with the energy inflation hedge consisting of a short
position in the bond and the S&P 500 consisting of a long position.
Other Investable Assets: Mutual Funds and Exchange Traded Funds (ETFs)
The recent S&P 500 results encourage us to expand our analysis further regarding
the GA performance for investable assets. In 2008, the five stocks found by the GA
create a hedged portfolio that is approximately 40 percent less risky than the S&P 500
while exhibiting a mean out-of-sample monthly (annual) return of 3.38 percent (49.10
percent). While one might be concerned this performance occurs in 2008, a notoriously
poor market, we are encouraged the GA finds good solutions exactly when investors need
them. Again, these GA-generated hedging portfolios consist of stocks listed on the major
US exchanges, not more speculative assets such as derivatives, which would surely have
higher expected returns but also likely higher risk. If an industry such as insurance were
to invest according to these results, regulators would more likely raise issues if the
hedging assets for medical inflation consist of corn or soybean futures than common
stocks.
126
From an investment standpoint, the S&P results indicate the potential for the GA
to find appealing investment tracking portfolios. Specifically, the S&P results force us to
investigate whether the GA can find a portfolio of five stocks that track a mutual fund or
ETF with perhaps either less return variance and/or higher mean returns out-of-sample.
To perform an initial exploration of this question, we randomly select ten mutual fund
series with varying return history lengths and run the GA using these funds as respective
targets. As with the macroeconomic series, we keep the weight vector (1 , 2 ) equal to
(0,1) to avoid any look-ahead bias. These funds include those with ticker symbols
ACMVX (April 2004 to December 2008), BRGIX (December 1998 to December 2008),
EXOSX (July 2002 to December 2008), EXTAX (April 1998 to December 2008),
FBALX (January 1987 to December 2008), FCNTX (May 1989 to December 2008),
FDVLX (July 1989 to December 2008), JMCVX (September 1998 to December 2008),
SPHIX (November 1994 to December 2008), and WAGTX (January 2001 to December
2008). We list the funds and their associated Morningstar style boxes in Figure 9.20
The major difference between the mutual fund and prior macroeconomic series
analysis involves the use of conditional weighting on our hedging assets. Cochrane
(1996) demonstrates the relevance of using returns that are conditioned on some state
variable, and Brandt and Santa-Clara (2006) discuss how doing so greatly increases the
effective assets under consideration. As we discuss later, doing so comes with the
transactions costs imposed by frequent rebalancing.
In this case we condition the hedging asset weights on the predicted value of the
mutual fund under consideration. To summarize the problem, the objective in this case is
20
127
to choose five assets, (i, j, k, l, and m) from 306 assets without replacement and weight
them to form a hedging portfolio according to equation (8).
= 0 + 1 + 2 + 3 + 4 + 5
+1 + 2 + 3 + 4 + 5
(8)
where , represents the return on a particular asset and LP are the indices that
indicate one of the 306 assets. 1 . . 5 and 1 5 are hedge ratios determined by OLS
regression. is a conditioning variable, which is the best predicted mutual fund
UHWXUQZHFDQJHWIURPDVWHSZLVHUHJUHVVLRQDFURVVWKH7HVWSHULRGWKDWLQFOXGHV
PRQWKVODJJHGYDOXHVRIWKHPXWXDOIXQGWKHPDUNHWUHWXUQQHWRIWKHULVN-free rate, the
S&P 500 return, inflation, the long government bond return, short government bond
return, and 30-day Treasury Bill returns. In other words, the conditioning variable is
formed only using past information that is readily available to predict the future fund
return. Notationally, the stepwise regressions include the aforementioned variables at
time t-NZKHUHN:KLOHGDWDPLQLQJLVFOHDUO\DFULWLFLVPRIWKLVVSHFLILF
technique, the practical goal of this step is to generate the best forecast for the respective
mutual fund using all relevant information available. The resulting conditioning variable
and estimation equation varies by fund, but for example, the conditioning variable for
fund ACMVX looks like equation (9),
= 0 + 1 3 + 2 3 + 3 102 (9)
=
where represents the fund return for month t, is the market return
minus the risk-free rate, and 10 is the 10-year government bond return for month t.
This conditioning variable allows the hedge ratios on the individual assets to change with
128
the state-of-the-world as suggested in Cochrane (1996). As before, we next define the
hedged position, or hedged portfolio, as in equation (10).
= ( 10)
129
funds. Thus the out-of-sample investments in the hedging portfolios for these two funds
are fixed for the entire out-of-sample period, just as they are for the prior macroeconomic
target series.
To emphasize the point that the GA-generated solutions are unambiguously
better, we look at adjusting the hedging portfolios for risk using both the Sharpe ratio and
-HQVHQVDOSKD Because it is practically difficult to take a short position in a mutual fund
per se without simply shorting each stock in the fund, our risk-adjusted comparisons
juxtapose the GA-generated hedging portfolio with the target mutual fund. Columns 5
and 6 of Table 7 quantify the Sharpe ratios for both the hedging portfolio and the target.
The shaded values indicate which series has the higher Sharpe ratio, and in every case the
GA-generated portfolios offer greater returns out-of-sample for each unit of risk (as
measured by standard deviation). We emphasize this point using Figure 9, Panel A,
which depicts the out-of-sample alternatives for an investor: (1) either invest in ACMVX
or one of the other 9 funds or (2) invest in the GA-generated five-stock portfolio
DOWHUQDWLYHRUFRXQWHUSDUWODEHOHG$$&09;RU$[[[[[ZKHUH[[[[[
represents the ticker for one of the other nine funds. The arrows on this plot originate at
the target fund and proceed to the GA-generated five-asset portfolio. For example, the
circled points indicate alternative (1) above, which is to invest in ACMVX, connected
with alternative (2), which is to invest in the GA-JHQHUDWHGPLPLFNLQJILYH-asset
portfolio, or (A) ACMVX. Since most investors with typical mean-variance preferences
would desire higher return and lower risk, it is clear (A) ACMVX presents both for this
out-of-VDPSOHSHULRG%H\RQGWKLVVSHFLILFH[DPSOHLQDOOEXWRQHFDVHWKHQRUWKZHVW
direction of the arrows unambiguously indicates the GA-generated portfolio provides
130
GRPLQDQWILUVWDQGVHFRQGPRPHQWV7KHORQHQRUWKHDVWDUURZLHIXQG:$*7;
presents a scenario where an investor must make a judgment call based on risk
preferences over the mean and standard deviation of returns. Notably, if one were able to
short the fund and go long the GA-generated portfolio, the results would become stronger
than depicted.
As mentioned before, one clear consideration when opting for the GA-generated
five-asset portfolio is transaction costs. While the investing policy (possibly) changes
annually, the conditional nature of the hedging portfolio means its composition likely
changes monthly based on updated predictions about the target series according to
equation (11), which is simply equation (8) re-arranged to group terms associated with
each asset i, j, k, l, and m.
= 0 + 1 + 1 + (2 + 2 ) + (3
+ 3 ) + 4 + 4 + (5 + 5 )
The weight placed on each asset varies with the state-of-the-world, , or the
predicted value of the fund described with equation (9). As a result, each of the five
securities in the hedging portfolio is likely traded monthly to establish the optimal
position going forward. Thus the transactions costs will impact the return to the hedging
portfolio. Assuming that each stock is traded once per month, each year will see 60
WUDGHV,QWKHZRUVWFDVHRI)&17;ZKLFKVKRZVDQDYHUDJHH[FHVVUHWXUQWRWKH
hedged portfolio of 2.48 percent per month without transaction costs, the effective annual
return is 34.17 percent. Roughly calculated, dividing this excess annual return by the 60
trades indicates the hedging portfolio still proves more appealing than the actual fund (in
the second moment) if trades cost less than 0.57 percent on average.
131
5HVHDUFKRQWUDQVDFWLRQFRVWLQGLFDWHVWKDWODUJHWUDGHUVVXFKDVLQ)LGHOLW\V
agreement with Lehman Brothers costs can be as low as approximately 2 to 2.5 cents per
share.21 This same book excerpt indicates an overall rate for sales traders, block traders,
program traders, and algorithm trading for transaction costs are approximately 3.3 cents
per share. According to another source, in fourth quarter 2004, NYSE (NASDAQ)
transactions cost the average investment manager 0.26% (0.35%) per trade.22 To remain
conservative in our analysis, we assume the GA identifies only NASDAQ stocks and that
each of the five stocks identified are traded every month. Thus we assume a monthly loss
of approximately 1.75% due to transaction costs. Continuing the ACMVX example
shown above, the circled points in Table 10, Panel B show the monthly effect in the
return-risk space of monthly transactions costs of 1.75 percent. The return to the GAgenerated portfolio, (A) ACMVX, decreases by 1.75 percent compared to the return
shown in Panel A. Even though (A) ACMVX exhibits a negative mean monthly return,
an investor with typical mean-variance preferences would still prefer this GA-generated
portfolio to the ACMVX fund return-risk combination. Overall, Figure 9, Panel B
demonstrates that even with this conservative assumption toward transaction costs, the
GA hedging portfolios remain unambiguously better out-of-sample than the target funds
in 9 of 10 cases for the calendar year 2008.
%HVLGHVWKH6KDUSH5DWLR&ROXPQVDQGVKRZWKHFRPSDULVRQRI-HQVHQV
DOSKDXVLQJDVLQJOHIDFWRULH&$30PRGHO,QRIWKHIXQGVWKH-HQVHQVDOSKD
is absolutely higher for the GA-generated portfolio than for the target fund out-of-sample.
21
http://www.automatedtrader.net/online-exclusive/631/transaction-cost-research
22
http://www.capco.com/files/pdf/71/02_SERVICES/06_Market%20impact%20Transaction%20cost%20ana
lysis%20and%20the%20financial%20markets%20%28Opinion%29.pdf
132
However, in only 2 of these cases (i.e., ACMVX and FBALX) is the alpha significantly
different than zero. Our caveat with this measure is that we do not set up the fitness
IXQFWLRQWRPD[LPL]HWKLV-HQVHQVDOSKDPHDVXUHVRWKHVHUHVXOWVDUe but a positive
externality of the fitness function we define in equation (7). Certainly altering the fitness
function to incorporate alpha is a possibility when applying the GA technique, and we
leave it for future research. Overall, it appears from this sample of mutual funds that
using the GA allows one to mimic various mutual funds quite well and possibly
outperform them out-of-sample in the first two statistical moments.
Finally, since mutual funds exhibit a short-selling constraint, we move to
Exchange-Traded Funds, or ETFs, which do allow short positions. This time we use 5
randomly-selected ETFs as the target series for this GA technique. Table 8 and its Panels
A through E, along with Figure 10 Panels A and B, depict the results for these ETFs that
are analogous to the mutual fund results provided earlier. The ETFs sampled are EWC
(April 1996 to December 2008), DIA (January 1998 to December 2008), IWD (May
2000 to December 2008), IXN (November 2001 to December 2008), and IJT (August
2000 to December 2008). As with the mutual funds results, the GA finds quite effective
mimicking, or hedging, portfolios for these random ETFs. In every case the hedged
portfolio of going long the five-asset portfolio and shorting the ETF reduces the out-ofsample variance relative to simply going short the ETF. This reduction ranges from
approximately 17 (EWC) to 85 percent (IXN). Additionally, in every case shown there
exists an average positive excess return to holding the hedged portfolio. This monthly
(annual) return premium ranges from 1.74 (22.92) to 4.39 (67.51) percent in our out-ofsample period. The correlations are again positive in all ETF cases. The Sharpe ratios
133
for the hedging portfolios outpace those for the target ETFs in every case, and the
JenVHQVDOSKDPHDVXUHVDUHDJDLQPL[HG)LJXUH3DQHO$3DQHO%VKRZVWKH
performance of the ETF and GA-generated hedging portfolio in mean/standard deviation
space without (with) transaction costs as conservatively quantified earlier. In all five
cases the GA set of assets is preferred, even when we include the assumed transaction
costs as discussed with mutual funds. In summary, the GA again generates effective outof-sample hedging portfolios.
To reiterate the caveat discussed earlier, while the results shown so far appear to
support the GA as a viable technique to create hedging portfolios, we have started the
Test Period in the case of each target, whether macroeconomic series, mutual fund, or
ETF, to maximize the amount of time series information possible. There is certainly
merit to using a shorter time horizon for estimating asset weights than beginning in 1967
since using data from four decades ago inherently assumes relationship stability that has
existed over that timeframe will continue for another 12 months. It is almost a certainty
that using a shorter Test Period will change the results, but we cannot currently speculate
as to the direction. Therefore, we leave this data collection and incorporation effort to a
future generation of this research.
Conclusions
Historically it seems the effort to tie asset pricing theory and empirical
phenomena has focused on using macroeconomic data to explain asset returns (e.g.,
Chen, Roll, and Ross (1986)). However, there exist important risk-reduction reasons for
using inveVWDEOHDVVHWVWRH[SODLQRUPRUHDSSURSULDWHO\KHGJHQRQ-investable
134
macroeconomic series. In just one example, as Jennings, Fraser, and Payne (2009)
highlight, many large entities could benefit from reducing their future exposure to
medical care costs. From an insurance standpoint, there exist an almost boundless
number of non-investable risky phenomena that one might wish to protect against by
investing in particular assets. For example, property insurers might wish to manage the
risk imposed by construction costs, or mass carriers might wish to offset their
transportation or energy consumption risk. The Genetic Algorithm (GA) technique
presented in this paper provides a viable alternative for addressing such problems.
The purpose of this paper is to demonstrate that the Holland (1975) Genetic
Algorithm (GA) process can find such hedging portfolios based on user-defined
parameters in a relatively efficient manner. In the case of medical inflation, across the
out-of-sample time period from 2005 to 2008, these five-asset GA-generated portfolios
appear to perform much better than the current investable mutual funds at managing the
risk of escalating health care expenses. Using these GA-generated hedging portfolios at
worst approximates the same risk (i.e., variance) level as exposure to medical inflation
itself, but they also provide superior returns in the out-of-sample months investigated,
UHVXOWLQJLQDQHQWLW\VDELOLW\WRIXOO\IXQGWKHDVVRFLDWHGOLDELOLWLHVE\LQYHVWLQJLQWKHVH
portfolios.
The results for nine other macroeconomic series presented here, while mixed
overall, generally show a weakly-preferred situation. The GA-generated portfolios at
least provide an opportunity for entities to tradeoff their desired levels of risk and return.
It is rare that the hedged position formed by holding a long position in the GA portfolio
against a natural short position in the macroeconomic series presents an unambiguously
135
worse scenario for the hedging entity. In only 4 of the 40 series-year combinations (PPI
residential construction in 2008, transportation inflation in 2007, diesel fuel in 2006, and
Case-Shiller 10-City Composite Index in 2005) are the hedged portfolio variance higher
than the target series and the entity underfunded in the out-of-sample year. On the other
hand, for almost three times as many (11) of the series-year combinations, the GArecommended hedged position unambiguously improves a hedging entities situation by
fully funding the liability with less out-of-sample risk. The remaining 25 series-year
combinations allow the liable party to tradeoff risk and return when deciding whether to
manage its risk using the GA portfolio.
The GA technique described in this paper provides a mechanism to find
commonly-traded asset combinations to address exposure to non-traded risk. One
extension of this research could involve including additional assets in the hedging
portfolios (e.g., 10 stocks) with the hope of decreasing the inherent unsystematic risk in
the existing portfolios. And the results for tracking investable assets, while less robust in
this presentation, are encouraging and present another avenue for future research.
136
References
Bauer, Richard J., 1994, Genetic Algorithms and Investment Strategies (John Wiley &
Sons, NY).
Bodie, Zvi, 1976, Common stocks as a hedge against inflation, Journal of Finance 31,
459-470.
Brandt, Michael W., and Pedro Santa-Clara, 2006, Dynamic portfolio selection by
augmenting the asset space, Journal of Finance 61, 2187-2218.
Carhart, Mark M., 1997, On persistence in mutual fund performance, Journal of Finance
52, 57-82.
Cochrane, John H., 1996, A cross sectional test of an investment based asset pricing
model, Journal of Political Economy 104, 572-621.
De Jong, K., 1975, An analysis of the behavior of a class of genetic adaptive systems,
PhD dissertation, University of Michigan.
Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on
stocks and bonds, Journal of Financial Economics 33, 3-56.
Flannery, Mark J. and Aris A. Protopapadakis, 2002, Macroeconomic factors do
influence aggregate stock returns, Review of Financial Studies 15, 751-82.
Holland, John H., 1975, Adaptation in Natural and Artificial Systems (University of
Michigan, MI).
Jennings, William W., Steven Fraser, and Brian C. Payne, 2009, Do health care
investments hedge health care liabilities? Journal of Investing 18, 69-74.
Jensen, Michael C., 1968, The performance of mutual funds in the period 1945-1964,
Journal of Finance 23, 389-416.
Lamont, Owen, 2001, Economic tracking portfolios, Journal of Econometrics 105, 16184.
Sharpe, William F., 1966, Mutual fund performance, Journal of Business 39, 119-38.
Statman, Meir, 1987, How many stocks make a diversified portfolio? Journal of
Financial and Quantitative Analysis 22, 353-63.
137
Appendix A
Figures
138
Figure 1
Monthly Consumer Price Index (CPI) Measures
January 1967-August 2009
139
Figure 2
Initial Population for Genetic Algorithm (GA)
Candidate Solution #
1
2
3
*
*
*
721
*
*
1000
Fitness Level
1.02
0.8
2
*
*
*
1.05
*
*
0.25
We choose 1,000 possible asset combinations to form hedge portfolios. These are referred to as
parents or candidate solutions. Each candidate solution includes the return series from 5 assets
out of a universe of 3,000 assets. For example, candidate 1 forms a hedge portfolio using assets
1,7, 25, 36, and 41. The hedge is estimated by equally-weighting these assets into a portfolio
according to the equation below,
, = 0.2 + 0.2 + 0.2 + 0.2 + 0.2
where i = 1, j = 7, k = 25, l = 36, and m = 41.
The degree of hedging effectiveness is then calculated and UHFRUGHGDVWKHFDQGLGDWHVILWQHVV
level. Fitness levels indicate the extent to which the objective function (see equation (1)) is
optimized.
140
Figure 3
GA Pairing Example
Panel A
Ranking by Fitness
Candidate Solution #
1000
2
1
721
3
*
Ret(i)
2
23
1
22
7
*
Ret(j)
6
9
7
15
3
*
Ret(k)
28
7
25
14
2
*
Ret(l)
7
29
36
9
19
*
Ret(m)
11
31
41
38
15
*
Fitness Level
0.25
0.8
1.02
1.05
2
*
The initial candidate population is ranked according to fitness. The most fit candidates
are called the breeding population.
Panel B
Pairing 1: Crossover Equal to 2
Candidate Solution #
1000
2
Offspring 1
Offspring 2
Ret(i)
2
23
23
2
Ret(j)
6
9
9
6
Ret(k)
28
7
28
7
Ret(l)
7
29
7
29
Ret(m)
11
31
11
31
Fitness Level
0.25
0.8
Candidate 1000 and 2 constitute the first pairing based on their best fitness. They produce two
offspring or alternative solutions by the process of crossover. With a crossover of 2, Ret(i) and
Ret(j) from candidate 1000 and 2 are swapped while the remaining three assets remain the same.
Panel C
Pairing 2: Crossover Equal to 3
Candidate Solution #
1
721
Offspring 3
Offspring 4
Ret(i)
1
22
22
1
Ret(j)
7
15
15
7
Ret(k)
25
14
14
25
Ret(l)
36
9
36
9
Ret(m)
41
38
41
38
Fitness Level
1.02
1.05
Candidate 1 and 721 constitute the second pairing based on their fitness. They produce two
offspring or alternative solutions by the process of crossover. With a crossover of 3, Ret(i) ,
Ret(j) and Ret(k) from candidate 1 and 721 are swapped while the remaining two assets remain
the same.
141
Figure 4
GA Proof of Concept Performance
Panel A
0RQWKO\5HWXUQRI7DUJHW6HULHV7DUJHWYHUVXV%HVW+HGJLQJ3RUWIROLR5HWXUQKHGJH
Chosen by the GA
January 1967-August 2009
Panel B
6FDWWHU3ORWRI7DUJHW6HULHV7DUJHWYHUVXV+HGJLQJ3RUWIROLR5HWXUQKHGJH
January 1967-August 2009
Correlation = 0.96
142
Figure 5
Hedging Portfolio Examples
Panel A: Ideal Hedging Portfolio
143
Figure 6
Division of the Data and Evaluation of Fitness
2
,
2
,
2
,
is the variance of series x during period y. x consists of the hedging portfolio consisting of
five assets, the target series (i.e., medical inflation in this case), or the hedged portfolio, H, which
represents a long position in the hedging portfolio and short position in the target series (i.e.,
medical inflation). y consists of the Test, Validation, or Out-of-Sample period as shown and
described above. (1 , 2 ) represent the subjectively-assigned weights for each respective term.
144
Figure 7
Genetic Algorithm Portfolio Performance for Out-of-Sample Period
January 2008-December 2008
Industry Membership
Processed & Packaged Goods
Entertainment
Specialty Retail
Security & Protection Services
Industrial Equipment
Relative Weight
-1.12
-0.38
0.39
-0.11
0.21
145
Panel B: Weight Vector (0,1)
Industry Membership
Chemicals
Business Equipment
Electric Utilities
Manufacturing
Steel & Iron
Relative Weight
-0.11
-0.02
-0.53
0.02
-0.36
146
Panel C: Weight Vector (1,0)
Industry Membership
T-Bill
Accident & Health Insurance
Restaurants
Rental & Leasing Services
Telecommunications
Relative Weight
1.00
0.00
0.00
0.00
0.00
147
Figure 8
Medical Inflation and Potential Hedging Portfolio Return Characteristics in MeanStandard Deviation Space for Out-of-Sample Period
January 2008-December 2008
Panel A: GA Portfolios Relative to Each Other
Monthly Return
0.4
0.3
0.2
Med Inf
0.1
0.0
0.000
-0.1
0.050
0.100
0.150
-0.2
-0.3
Monthly Return
1.0
0.5
0.0
-0.5 0.0
Med Inf
2.0
4.0
6.0
8.0
-1.0
-1.5
-2.0
VGHCX
-2.5
FSPHX
-3.0
ETHSX
-3.5
Definitions:
Med Inf: medical inflation series
(H) Med Inf (x,y): hedged portfolio (RetPortfolio Medical Inflation) using (w1,w2) = (x,y)
ETHSX: Eaton Vance Worldwide Health Science A
VGHCX: Vanguard Health Care
FSPHX: Fidelity Select Health Care
148
Figure 9
Mutual Fund and Potential Hedging Portfolio Return Characteristics
Mean-Standard Deviation Space
Out-of-Sample Period
January 2008-December 2008
Panel A: No Transaction Costs
149
ACMVX: American Century Mid Cap
Value Inv
Unavailable
150
Figure 10
Exchange-Traded Fund (ETF) and Potential Hedging Portfolio Return Characteristics
Mean-Standard Deviation Space
Out-of-Sample Period
January 2008-December 2008
Panel A: No Transaction Costs
151
EWC: iShares MSCI Canada Index
152
Appendix B
Tables
153
Table 1
Fitness Level for Asset Portfolios across Generations using Proof of Concept Returns
January 1967-August 2009
Generation
1
2
3
4
5
6
7
8
9
10
Minimum
Fitness Level
7.61
7.00
5.29
4.68
4.28
4.28
4.28
4.28
4.28
4.28
Population
Mean
Fitness Level
40.34
36.09
37.36
38.27
36.60
37.98
41.47
56.43
54.81
56.56
Breeding Sample
Maximum
Fitness Level
309.31
379.93
302.12
358.80
480.70
423.33
234.77
359.41
338.54
349.44
This table shows how the GA population evolves over many generations. Fitness level is
the quantified measure for the variance of a time series called , which is the difference
between a target series and portfolio of five assets seeking to hedge the target series. The
objective function is shown below.
where = , ,
, ( , ) is the return of the target series (hedge portfolio series) at time
t.
In this proof of concept example, the hedging assets are simulated investable assets, with
representative factor loadings. To generate the simulated assets, we estimate betas of a fivefactor model for each asset from a random sample of 3,000 assets from CRSP, pulled from the
DVVHWVZLWKDWOHDVWPRQWKVUHWXUQVEHWZHHQ-DQXDU\DQG'HFHPEHU7KH
five factors include the market risk premium, SMB, HML, default risk premium (DEF), and term
risk premium (TERM). The target series is a simulated asset composed of the median factor beta
values from the 3,000 sampled assets.
In the table, minimum fitness level is for the portfolio of five assets among the entire
breeding population, which has 1,000 members, that has the lowest, or best, fitness measure. The
mean (maximum) fitness level is the population mean (maximum). In this example, the breeding
population consists of 10 percent of the population, or 100 members, and the final column shows
the mean fitness level for this breeding sub-population.
154
Table 2
Fitness Level for Asset Portfolios over Generations using Actual Target (Medical Inflation)
and Investable Assets (303 Stocks from CRSP, Long Government Bond, Short Government
Bond, and Treasury Bill) During the Test and Validation Periods
January 1967-December 2007
Weight =
(1,1)
Generation
1
2
3
4
5
6
7
8
9
10
Breeding Sample
Population
Minimum
Fitness Level
0.271
0.250
0.246
0.242
0.242
0.240
0.233
0.233
0.233
0.233
Mean
Fitness Level
0.339
0.338
0.332
0.333
0.339
0.343
0.339
0.339
0.337
0.347
Maximum
Fitness Level
0.504
0.504
0.536
0.506
0.575
0.609
0.533
0.521
0.500
0.516
This table shows how the GA population evolves over many generations. Fitness
level is the quantified by the following.
=
2
1 ,
2
,
+ 2
2
,
2
is the variance (mean) of series x during period y. x consists of the hedging portfolio
,
consisting of five assets, the target series (i.e., medical inflation in this case), or the hedged
portfolio, H, which represents a long position in the hedging portfolio and short position in the
target series (i.e., medical inflation). y consists of the Test, Validation, or Out-of-Sample period
as shown and described above. (1 , 2 ) represent the subjectively-assigned weights for each
respective term and equal to (1,1) here.
The investable assets in this example consist of 303 stocks from CRSP that have returns
from January 1967 to December 2008 and three government bond/bill return series (i.e., 10-year
bond, one-year bond, and 30-day bill).
In the table, minimum fitness level is for the portfolio of five assets among the entire
breeding population, which has 1,000 members, that has the lowest, or best, fitness measure. The
mean (maximum) fitness level is the population mean (maximum). In this example, the breeding
population consists of 10 percent of the population, or 100 members, and the final column shows
the mean fitness level for this breeding sub-population.
155
Table 3
Portfolio Hedging Effectiveness against Medical Inflation
Out-of-Sample Period
January 2008-December 2008
Weights
(w1, w2)
(1,1)
(0,1)
(1,0)
Variance Ratio
,
,
0.866
0.971
0.513
Assets
(Return Series Stock Number)
274
81
306
79
76
297
199
84
288
33
229
225
Generation
Converged
Mean Excess
Monthly Return
of Hedging
Portfolio vs.
Medical
Inflation
9
7
3
0.316
0.328
0.105
214
117
32
Column 1 shows the weights implemented in the fitness measure shown in equation (7). Column
2 depicts the ratio of the variance of the hedged portfolio, H, to the variance of medical inflation
for the Out-of-Sample period (January 2008-December 2008). Column 3 shows the assets that
compose the hedging portfolio. Each number, 1 to 306, represents an index referring to a return
series of an traded stock or government bond who has returns in CRSP from January 1967 to
'HFHPEHU*HQHUDWLRQ&RQYHUJHGVKRZVWKHJHQHUDWLRQRIWKHJHQHWLFDOJRULWKPZKHUH
the best solution appears to have stabilized. The final Column quantifies the mean monthly
excess return to the hedging portfolio over the mean monthly medical inflation, or Ret(Hedge)
CPIMEDMO, for the Out-of-Sample period.
Assets
Stock Number
274
79
199
33
214
81
76
84
229
117
306
297
288
225
32
Industry Membership
Processed & Packaged Goods
Entertainment
Specialty Retail
Security & Protection Services
Industrial Equipment
Chemicals
Business Equipment
Electric Utilities
Manufacturing
Steel & Iron
T-Bill
Accident & Health Insurance
Restaurants
Rental & Leasing Services
Telecommunications
156
Table 4
Hedging Portfolio Performance in Funding $1M Health Care Liability
January 2008-December 2008
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Medical
Inflation
(%)
0.48
0.12
0.22
0.15
0.14
0.26
0.10
0.21
0.27
0.15
0.24
0.27
Health Care
Costs
($M)
1.005
1.006
1.008
1.010
1.011
1.014
1.015
1.017
1.019
1.021
1.024
1.026
Portfolio
Return
(%)
0.57
0.53
0.53
0.53
0.45
0.56
0.54
0.51
0.60
0.66
0.57
0.58
Portfolio
Assets
($M)
1.006
1.011
1.016
1.022
1.026
1.032
1.038
1.043
1.049
1.056
1.062
1.068
Initial Investment
Required to Offset Cost
($M)
0.999
0.995
0.992
0.988
0.985
0.982
0.978
0.975
0.972
0.967
0.964
0.961
Excess at
Time 0
(%)
0.09
0.50
0.81
1.19
1.50
1.80
2.23
2.51
2.84
3.32
3.64
3.93
This table presents the funding performance of an investment in the medical inflation hedging
portfolio consisting of assets 81, 76, 84, 229, and 117. Assuming an entity is setting aside $1
million to fund the anticipated medical care expense it will incur sometime during 2008, Column
2 shows monthly medical inflation, with Column 3 showing what the $1 million liability would
cost if incurred at the end of the associated month. Column 4 shows the monthly hedging
portfolio return, and Column 5 translates these returns into dollar values if the $1 million setaside is invested in the hedging portfolio. Column 6 shows how much of the original $1 million
would have been needed to offset the liability exactly, with Column 7 showing the excess initial
funds as a percentage at the beginning of the year if the whole liability were to occur at the end of
the associated month.
157
Table 5
Hedge Ratios for *RRG Genetic Algorithm Portfolio for Test Period
January 1967-December 2006
Parameter P-Value
0.537
0.000
-0.00037
0.850
-0.00006
0.970
-0.00171
0.453
0.00007
0.938
-0.00118
0.556
0.003
420
Constant
Ret(81)
Ret(76)
Ret(84)
Ret(229)
Ret(117)
R-Squared
N
Industry Membership
Chemicals
Business Equipment
Electric Utilities
Manufacturing
Steel & Iron
0.97
1.01
1.29
1.16
1.01
1.01
1.02
1.13
1.01
Housing Inflation
Case-Shiller 10-City Composite
PPI Residential Construction
Energy Inflation
Transportation Inflation
Oil Spot Price*
Jet Fuel Spot Price
Diesel Fuel PPG
0.15
0.70
0.10
-0.19
-0.46
Aggregate Inflation
0.99
0.41
5.05
S&P 500
0.62
3.38
49.10
Notes:
Weights (0,1)
*Best solution in 10th generation; convergence not guaranteed
2.95
33.80
-0.12
(Annual) ,
4.01
0.17
0.10
0.01
-0.19
-0.38
0.03
0.24
1.76
7.40
6.54
3.79
0.24
2.46
-0.01
(Monthly)
0.33
2.95
23.23
135.56
113.92
56.28
Macroeconomic (Target)
Series
Medical Inflation
253
41
84
288
51
263
204
84
194
26
1
81
Panel A
Validation Period: January-December 2007
Out-of-Sample Period: January-December 2008
Test Period: Varies-December 2006
16
44
90
229
137
184
199
90
46
188
2
76
198
11
192
192
185
174
37
192
192
291
3
84
Assets
Table 6
MACROECONOMIC SERIES
266
20
145
149
231
172
45
145
108
87
4
229
192
189
82
49
287
219
229
82
156
192
5
117
1/67 - 12/08
1/67 - 12/08
1/67 - 12/08
1/67 - 12/08
1/67 - 12/08
4/90 - 12/08
3/94 - 12/08
1/67 - 12/08
1/67 - 12/08
6/86 - 12/08
1/67 - 12/08
158
This table summarizes the hedging performance for a portfolio of five assets identified by the Genetic Algorithm (GA) technique using the fitness
function in equation (6) for the out-of-sample period December to January 2008. Column 1 shows the target macroeconomic series. Column 2
shows the variance ratio of the hedged portfolio, H, which consists of a long position in the five-asset hedging portfolio and a short position in the
target macroeconomic series. Column 3 (4) shows the average monthly (annualized) return of the hedged portfolio, H. Column 5 shows the
Pearson correlation between the target macroeconomic series and the five-asset hedging portfolio. Columns 6-10 identify the investable assets that
from the portfolio by their author-generated index numbers. Please see Appendix C for a list of the industries associated with these assets. The
final column shows the relevant time period for the target macroeconomic series. Future Panels show the same results for Out-of-Sample periods
2007, 2006, and 2005.
159
1.03
1.04
0.99
0.94
1.01
0.99
0.86
0.98
Housing Inflation*
Case-Shiller 10-City Composite*
PPI Residential Construction
Energy Inflation
Transportation Inflation
Oil Spot Price*
Jet Fuel Spot Price
Diesel Fuel PPG
-0.99
-0.25
-2.50
-1.72
-1.48
0.16
1.43
-0.42
(Monthly)
0.13
Aggregate Inflation
1.02
0.10
S&P 500
0.87
0.37
Notes:
Weights (0,1)
*Best solution in 10th generation; convergence not guaranteed
0.90
1.16
4.59
-11.27
-3.02
-26.16
-18.81
-16.34
1.99
18.53
-4.96
0.09
0.38
0.24
0.08
0.11
0.37
0.16
-0.03
-0.54
0.21
(Annual) ,
1.58
0.34
229
125
146
288
233
72
10
196
254
192
1
229
Table 6
Panel B
Validation Period: January-December 2006
Out-of-Sample Period: January-December 2007
Test Period: Varies-December 2005
287
84
288
229
254
203
229
192
186
153
2
87
200
219
51
195
266
3
89
247
53
138
3
13
Assets
192
249
192
192
149
229
287
21
215
175
4
272
15
117
15
15
51
25
105
30
190
56
5
251
1/67 - 12/07
1/67 - 12/07
1/67 - 12/07
1/67 - 12/07
1/67 - 12/07
4/90 - 12/07
3/94 - 12/07
1/67 - 12/07
1/67 - 12/07
6/86 - 12/07
1/67 - 12/07
160
1.13
0.88
0.98
1.04
0.93
0.96
1.01
1.13
Housing Inflation
Case-Shiller 10-City Composite*
PPI Residential Construction*
Energy Inflation
Transportation Inflation
Oil Spot Price*
Jet Fuel Spot Price
Diesel Fuel PPG
0.24
0.20
0.15
0.74
-0.30
0.15
0.53
-0.28
(Monthly)
0.24
Aggregate Inflation
1.03
0.18
S&P 500
0.86
-0.84
Notes:
Weights (0,1)
*Best solution in 10th generation; convergence not guaranteed
1.07
2.21
-9.58
2.93
2.47
1.76
9.26
-3.57
1.83
6.60
-3.28
0.02
0.47
-0.04
0.58
0.28
0.05
-0.34
-0.53
0.35
0.25
(Annual) ,
2.95
-0.22
192
122
304
288
72
229
229
216
187
113
1
229
Table 6
Panel C
Validation Period: January-December 2005
Out-of-Sample Period: January-December 2006
Test Period: Varies-December 2004
287
83
288
229
111
70
137
132
289
227
2
137
246
124
51
287
51
66
161
229
138
3
3
161
Assets
24
219
229
192
253
114
192
243
40
192
4
192
229
266
237
15
230
198
263
194
275
171
5
263
1/67 - 12/06
1/67 - 12/06
1/67 - 12/06
1/67 - 12/06
1/67 - 12/06
4/90 - 12/06
3/94 - 12/06
1/67 - 12/06
1/67 - 12/06
6/86 - 12/06
1/67 - 12/06
161
0.99
1.00
0.96
0.99
0.96
0.89
0.86
0.99
Housing Inflation
Case-Shiller 10-City Composite
PPI Residential Construction
Energy Inflation
Transportation Inflation
Oil Spot Price
Jet Fuel Spot Price
Diesel Fuel PPG*
-1.18
0.10
-1.92
-1.42
-1.08
0.10
-0.77
-0.36
(Monthly)
0.20
Aggregate Inflation
0.97
0.13
S&P 500
0.70
-0.10
Notes:
Weights (0,1)
*Best solution in 10th generation; convergence not guaranteed
1.02
1.56
-1.20
-13.27
1.15
-20.73
-15.77
-12.24
1.22
-8.83
-4.23
0.23
0.62
0.12
0.26
0.33
0.53
0.10
0.09
-0.02
0.34
(Annual) ,
2.39
-0.13
229
228
51
288
51
229
258
197
287
139
1
157
Table 6
Panel D
Validation Period: January-December 2004
Out-of-Sample Period: January-December 2005
Test Period: Varies-December 2003
88
229
288
229
229
64
82
86
15
143
2
199
287
304
178
178
38
174
123
192
249
192
3
229
Assets
104
297
192
97
184
145
229
46
46
34
4
253
162
267
287
192
45
255
245
257
48
233
5
277
1/67 - 12/05
1/67 - 12/05
1/67 - 12/05
1/67 - 12/05
1/67 - 12/05
4/90 - 12/05
3/94 - 12/05
1/67 - 12/05
1/67 - 12/05
6/86 - 12/05
1/67 - 12/05
162
30.305
111.102
53.485
38.070
34.167
179.319
37.093
-0.549
0.247
-0.013
-0.134
-0.519
0.376
0.004
0.274
-0.559
-0.556
-0.514
-0.560
-0.600
-0.517
-0.383
-0.333
Sharpe
Target
-0.144
2.015
0.995
1.139**
-0.298
5.055
0.581
3.019**
Portfolio
1.502
Target
0.777
82
76
236
153
120
8
77
3
82
20
188
89
303
219
127
43
153
53
156
154
257
179
135
190
53
120
168
105
176
157
Assets
273
269
301
151
231
160
250
124
194
219
277
118
36
238
219
249
291
140
256
304
4/04 - 12/08
12/98 12/08
7/02 - 12/08
4/98 - 12/08
1/87 - 12/08
5/89 - 12/08
7/89 - 12/08
9/98 - 12/08
11/94 12/08
1/01 - 12/08
Target
Beg End
(Mo/Yr)
This table summarizes the hedging performance for a portfolio of five assets identified by the Genetic Algorithm (GA) technique using the fitness
function in equation (6) for the out-of-sample period December to January 2008. Column 1 shows the target mutual fund symbol. Column 2
shows the variance ratio of the hedged portfolio, H, which consists of a long position in the five-asset hedging portfolio and a short position in the
0.280
0.577
2.230
6.424
3.635
2.725
2.480
8.937
2.664
51.425
Sharpe
Portfolio
SPHIX
0.945
2.680
37.349
0.481
-0.399
0.441
0.857
WAGTX
1.061
4.611
71.767
-0.135
-0.620
2.979
-0.637
Notes: Weights (0,1)
*,** indicate significance at the 0.10 and 0.05 level, respectively
+
Best solution in 10th generation; convergence not guaranteed
#
Future values are unpredictable, so portfolio consists of unconditioned assets
0.519
1.135
0.636
0.423
0.638
1.477
0.795
BRGIX+#
EXOSX
EXTAX#
FBALX
FCNTX
FDVLX
JMCVX+
3.518
(Annual)
0.805
0.266
0.676
0.804
0.643
0.248
0.520
0.424
ACMVX
(Monthly)
0.089
-0.021
0.252
0.025
-0.289
0.565
1.2575*
Fund
Symbol
Monthly Out-of-Sample Results for Ten Random Mutual Funds Using Annually-Established Policy
Validation Period: January-December 2007
Out-of-Sample Period: January-December 2008
Test Period Varies by Fund
Table 7
MUTUAL FUNDS
163
Fund Symbol
ACMVX
BRGIX
EXOSX
EXTAX
FBALX
FCNTX
FDVLX
JMCVX
SPHIX
WAGTX
Fund Name
American Century Mid Cap Value
Bridges Investment
Manning & Napier Overseas
Manning & Napier Tax Managed A
Fidelity Balanced
Fidelity Contrafund
Fidelity Value
Janus Perkins Mid Cap Value T
Fidelity High Income
Wasatch Global Science & Technology
target mutual fund series. Column 3 (4) shows the average monthly (annualized) return of the hedged portfolio, H. Column 5 (6) shows the
Sharpe Ratio for the five-DVVHWKHGJLQJSRUWIROLRWDUJHWPXWXDOIXQG&ROXPQVKRZVWKH-HQVHQVDOSKDIUom the single-factor CAPM for
the five-asset hedging portfolio (target mutual fund). Column 9 shows the Pearson correlation between the target mutual fund series and the fiveasset hedging portfolio. Columns 10-14 identify the investable assets that from the portfolio by their author-generated index numbers. Please see
Appendix C for a list of the industries associated with these assets. The final column shows the relevant time series for the target mutual fund
series. Below is the legend showing the stocks associated with the index numbers above.
164
Portfolio
0.356
1.384*
0.932
-0.342
-0.319
Target
0.923
0.599
0.566
0.817
0.709
102
245
158
149
247
32
71
Assets
4/96 - 12/08
1/98 - 12/08
5/00 - 12/08
11/01 12/08
8/00 - 12/08
Target
Beg End
(Mo/Yr)
This table summarizes the hedging performance for a portfolio of five assets identified by the Genetic Algorithm (GA) technique using the fitness
function in equation (6) for the out-of-sample period December to January 2008. Column 1 shows the target ETF symbol. Column 2 shows the
variance ratio of the hedged portfolio, H, which consists of a long position in the five-asset hedging portfolio and a short position in the target ETF
series. Column 3 (4) shows the average monthly (annualized) return of the hedged portfolio, H. Column 5 (6) shows the Sharpe Ratio for the
five-DVVHWKHGJLQJSRUWIROLRWDUJHW(7)&ROXPQVKRZVWKH-HQVHQVDOSKDIURPWKHVLQJOH-factor CAPM for the five-asset hedging
portfolio (target ETF). Column 9 shows the Pearson correlation between the target ETF series and the five-asset hedging portfolio. Columns 1014 identify the investable assets that from the portfolio by their author-generated index numbers. Please see Appendix C for a list of the industries
associated with these assets. The final column shows the relevant time series for the target ETF series. Below is the legend showing the stocks
associated with the index numbers above.
-0.281
-0.407
-0.263
Sharpe
Target
1.109
-0.119
67.508
26.908
35.286
Sharpe
Portfolio
IXN
0.149
1.735
22.924
-0.357
-0.519
+
IJT
0.658
2.006
26.914
-0.355
-0.403
Notes:
Weights (0,1)
*indicates significance at the 0.10 level
+
Best solution in 10th generation; convergence not guaranteed
4.393
2.006
2.550
(Annual)
0.309
0.300
1.017
0.834
0.390
0.508
EWC
DIA+
IWD
(Monthly)
-0.448
-0.607
-0.616
Fund
Symbol
Monthly Out-of-Sample Results for Five Random ETFs Using Annually-Established Policy
Validation Period: January-December 2007
Out-of-Sample Period: January-December 2008
Test Period Varies by Fund
Table 8
Exchange Traded Funds (ETFs)
165
ETF Symbol
EWC
DIA
IWD
IXN
IJT
ETF Name
iShares MSCI Canada Index
SPDR Dow Jones Industrial Average
iShares Russell 1000 Value Index
iShares S&P Global Technology
iShares S&P SmallCap 600 Growth
166
167
Appendix C
Stock Numbers, Tickers, and Industries
Number
Ticker
Industry
Number
Ticker
Industry
SUN
81
DOW
Chemicals
ABL
82
DPL
Diversified Utilities
ABT
Pharmaceuticals
83
DTE
Electric Utilities
AEE
Diversified Utilities
84
DUK
Electric Utilities
10
AIP
Paper Products
86
EDE
Electric Utilities
11
AIT
Industrial Equipment
87
EGN
Gas Utilities
13
ALK
Airlines
88
EIX
Electric Utilities
15
AMR
Airlines
89
EK
Photo Equipment
16
AP
Industrial Machinery
90
EML
Tools
20
ASA
Financial
93
ESP
Electronics
21
ASH
Chemicals
97
Auto Manufacturing
24
AVP
Personal Products
102
FMC
Chemicals
25
AVT
Electronics
103
FO
Home Furnishings
30
BAX
Medical Supplies
104
FOE
Chemicals
32
BCE
Telecommunications
105
FPL
Electric Utilities
34
BDK
108
GAM
Financial
36
BFA
Beverage
111
GD
Aerospace/Defense
37
BFB
Beverage
113
GIS
Food
38
BGG
Industrial Machinery
114
GLW
Communication Equip
40
BMY
Pharmaceuticals
117
GNI
41
BP
118
GR
Aerospace/Defense
42
BRN
120
GV
Heavy Construction
43
BWS
Textiles
122
GY
Conglomerate
44
CAS
Materials Wholesale
123
HAL
45
CAT
Heavy Equipment
124
HE
Electric Utilities
46
CBE
Conglomerate
127
HL
Silver Mining
48
CEG
Electric Utilities
128
HNZ
Food
49
CEM
Chemicals
132
HPQ
Computer Systems
51
CFS
Management Services
135
HSC
53
CHG
Diversified Utilities
137
HUBA
Electronic Equpiment
56
CMC
138
HUBB
Electronic Equpiment
64
CR
Conglomerate
139
IBM
Computer Systems
66
CSC
IT Services
140
IDA
Electric Utilities
70
CUB
Science Instruments
143
IMO
71
CUO
Building Materials
145
IR
Industrial Equipment
72
CVR
Auto Manufacturing
146
IRF
Semiconductors
76
DBD
Business Equipment
149
JCP
Department Store
77
DD
Chemicals
151
Food
168
Number
Ticker
Industry
Number
Ticker
Industry
153
KO
Beverage
228
ROL
Business Services
154
KR
Grocery
229
RONC
Manufacturing
156
Property/Casualty Ins
230
RRD
Business Services
157
LDR
Research Services
231
RSH
Electronics Retail
158
LG
Gas Utilities
233
Wireless Comm
160
LMT
Aerospace/Defense
234
SCG
Utilities
161
LUK
Lumber Production
236
SCX
Manufacturing
162
LZ
Chemicals
237
SEB
Meat Products
168
MDP
Publishing
238
SGP
Healthcare
171
MHP
Publishing
241
SJM
Food
172
MMM
Conglomerate
243
SLB
174
MOGB
Aerospace/Defense
245
SLI
Electronics
175
MOT
Telecommunications
247
SNR
Electronics
176
MRK
Pharmaceuticals
249
SO
Electric Utilities
177
MRO
250
SPA
Electronics
178
MSB
Financial-Land
251
STL
Banking
179
MUR
253
SWK
184
NEM
Mining
254
SXI
Industrial Equipment
185
NEU
Chemicals
255
SYNL
186
NFG
Gas Utilities
256
TE
Electric Utilities
187
NGA
Industrial Equipment
257
TEG
Diversified Utilities
188
NI
Diversified Utilities
258
TJX
Department Store
190
NOC
Aerospace/Defense
263
TPL
Financial-Land
192
NR
266
TSTY
Food
194
NVE
Diversified Utilities
269
TXT
Conglomerate
195
NXY
272
UIS
IT Services
196
OGE
Electric Utilities
273
UL
Food
197
OKE
Gas Utilities
275
UST
Tobacco
198
OLN
Synthetics
277
UVV
Tobacco
199
OMX
Business Equipment
287
WEDC
Semiconductors
200
OXM
Textiles
288
WEN
Fast Food
202
PAS
Food
289
WEYS
Textiles
203
PBI
Business Equipment
291
WGL
Gas Utilities
204
PBY
Auto Parts
297
WSC
212
PG
Personal Products
301
XEL
Electric Utilities
215
PKE
Circuit Boards
303
ZAP
Holding Company
216
PKI
Medical Supplies
304
Long Bond
Long Bond
219
POM
Electric Utilities
305
Short Bond
Short Bond
226
ROG
306
T-Bill
T-Bill
227
ROH
Chemicals