Part Two Chapter 2 - Risk and Return
Part Two Chapter 2 - Risk and Return
Part Two Chapter 2 - Risk and Return
5
Let’s introduce now the concepts concerning the returns on investments.
First of all, we define the holding period return (HPR) as the total return on an
investment, including all income. It is a major return calculation: ending value
of investment divided by beginning value of investment, over a stated period of
time. The holding period return can also be computed as follows: ending price
minus beginning price plus intermediate cash-flows, all divided by the
beginning price over a stated period of time (one year).
The holding period return has to be greater than or at least 1.00. For HPR
= 1, there is no gain or loss for the investor. When HPR > 1.00, the investment is
profitable and there is an increase in someone’s wealth. Of course, with a HPR <
1.00, the investor is losing money. The time period during which an investor
owns an asset is called holding period.
Usually, the evaluation of an investment return is expressed in annual
percentage terms. This indicator is called holding period yield (HPY).
HPY = HPR – 1
In the previous example:
HPY = 1.15 – 1 = 0.15 or 15 percent.
So far, we have considered that the holding period = 1.
If n > 1, we compute first the annual holding period return (AHPR).
AHPR = HPR1/n
Suppose an individual invests today $700 and this investment is worth
$925 two years from now.
HPR = $925 / $ 700 = 1.321
AHPR = 1.3211/2 = 1.149
Annual holding period yield (AHPY) is
AHPY = AHPR – 1 , so AHPY = 1.15 – 1 = 0.15 or 15 percent.
6
Now we have to proceed further and introduce the computation of the
mean rate of return for a single investment over some period of time, and for a
portfolio of investments for one year or more.
We will consider first the case of a single investment.
There are two classical methods of computing the mean rate of return:
- The arithmetic mean (AM)
∑ HPY
AM = , where ∑ HPY = the sum of annual HPYs
n
n = the number of years
- The geometric mean (GM)
GM = ∏1/n – 1 , where ∏ = the product of annual HPRs
= HPR1⋅ HPR2⋅ … ⋅ HPRn
Suppose we make an investment in an asset and the economic results for
three years are as follows:
Beginning Ending
Year HPR HPY
Value Value
1 100 120 1.20 0.20
2 120 150 1.25 0.25
3 150 129 0.86 - 0.14
(0.20) + (0.25) + (-0.14)
AM = = 0.1033 or 10.33 percent
3
GM = (1.20⋅ 1.25⋅ 0.86)1/3 – 1 = 1.291/3 – 1 = 1.09 – 1 = 0.09 or 9 percent
If the rates of return vary over the years, the geometric mean is lower than
the arithmetic mean.
We also notice that if we compound 9 percent for three years we get 1.29,
which is the actual ending value: 129/100.
That is why we may define the geometric mean as the measure of the
long-term mean rate of return expressed as the compound annual rate of return.
We may express the geometric mean as the n-th root of product of the annual
HPR for n years minus 1.
7
The arithmetic mean has the disadvantage of an average method, and
sometimes does not measure efficiently an investment. Let’s consider the
following example:
Beginning Ending
Year HPR HPY
Value Value
1 100 50 0.5 - 0.5
2 50 125 2.5 1.5
3 125 100 0.8 - 0.2
We may compute the weighted HPY, by determining first the portfolio’s HPR:
8
and HPY = HPR – 1 = 1.19047 – 1 = 0.19047 or 19 percent
Suppose that the nominal rate of return on U.S. Treasury Bills was 10
percent and the rate of inflation for the same time period was 5 percent. It results
a real risk-free rate of
1.10 / 1.05 – 1 = 4.76 percent.
The risk-free investment assumes the investor is certain of the future cash
flows and their timing. Still, most investments are subject to risk – the
9
uncertainty that an investor will earn its expected rate of return. Risk may also
be defined as the possible variation associated with the expected return
measured by the standard deviation. Consequently, the investors will expect an
additional return for making a risky investment rather than a safe one is called
the risk premium and it will increase the required rate of return over the nominal
risk-free rate.
The most considered sources of risk are: business risk, financial or
leverage risk, liquidity risk, exchange rate risk, and country risk.
The business risk of the firm’s investment opportunities reflects the
uncertainty due to the nature of a firm’s business and it expresses the risk of the
real assets held by the firm. The investors assume a financial risk when the firm
issues debt in order to finance the real investments. Liquidity risk is the inability
of financial intermediaries to meet short-term financial claims. In other words,
this is the uncertainty due to the ability to sell or buy investment in a secondary
market. The exchange risk is the loss or gain in value due to a change in foreign
exchange rate. The country risk or political risk is the possibility that unexpected
events within a host country will influence a client firm’s or government’s
ability to repay a loan.
The investor will have to choose between a risky investment or a safe and
certain investment. Most investors are risk-averse i.e. they demand high safety
and require a high degree of return for increases in risk. If we take into account
the case of the perfect certainty, there will be only one possible return and the
probability of receiving that return will be one. But the real world is an uncertain
one and we do not deal with certain returns but expected returns. As a
consequence, the expected return from an investment E (Ri) will be computed as
follows:
Expected return = ∑i (probability of return)⋅(possible return)
E (Ri) = ∑i (Pi)⋅(Ri)
10
As we see, the expected return is the average return based on the
probabilities of the individual returns. The measures of risk for an investment
are the variance of rates of return and standard deviation of rates of return.
These statistical measures give the possibility to compare risk and return directly.
Variance is a measure of variability equal to the mean squared deviation
from the expected value.
Variance ( σ2 ) = ∑ (Probability) ⋅ (Possible Return – Expected Return)2 =
= ∑ (Pi)[Ri – E(Ri)]2 = (1/n)⋅ ∑ [Ri – E(Ri)]2
If we consider the case of a perfect certainty, the variance will be zero,
expressing the fact that in this particular case there is no variance, no deviation
from the expected values, no risk.
Example:
σ2 = ∑ ( 1 ) ⋅ ( 0.10 – 0.10 ) = 0
Now, suppose the following example:
Probability of possible return (Pi) Possible return (Ri) (Pi)⋅(Ri)
0.1 0.25 0.025
0.1 -0.25 -0.025
0.8 0.15 0.12
∑ = 0.12
The expected return E(Ri) = 0.12 or 12 percent.
The dispersion of the distribution is computed in the table below:
Probability (Pi) Return (Ri) Ri – E(Ri) [Ri – E(Ri)]2 Pi ⋅[Ri – E(Ri)]2
0.1 0.25 0.13 0.0169 0.00169
0.1 -0.25 -0.37 0.1369 0.01369
0.8 0.15 0.03 0.0009 0.00072
∑ = 0.0161
σ2 = 0.0161
The variance in this case is σ2 = 0.0161. The larger the variance for an
expected rate of return, the greater the dispersion of expected returns and the
greater the risk for the investment.
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The standard deviation, which is also a measure of variability, is equal to
the square root of the variance:
σ = (0.0161)1/2 = 0.12688
As a conclusion for our example, the distribution has an expected value of
12 percent and a standard deviation of 12.688 percent.
Variance and standard deviation are absolute measures of dispersion. In
order to compare the dispersion of two series it is important to use a relative
measure of dispersion, which is the coefficient of variation. This indicator
reflects the risk per unit of return and is computed as a ratio between the
standard deviation and the expected value. A larger value of this coefficient
means greater dispersion relative to the arithmetic mean of the series, greater
risk per unit of return.
Standard deviation of return
Coefficient of variation =
Expected rate of return
12
Rate of
return High Risk
Average
Risk Security Market Line
Low Risk ∝
Risk
Figure 1 – The relationship between risk and return
13
The change in the risk premium determines a change in the slope of the
market line Rm’. If we plot this change on a graph, we can show that if the
market risk premium increases, there will be an increase in the slope of the
market line, too.
E(R)
SML’
P’
) ∝’
Rm’
SML
P
Rm ) ∝’
RFR
rm Risk
- changes in one of the variables that affects all investments i.e. changes in
capital market conditions such as ease or tightness of money or changes in the
expected rate of inflation. These changes will affect all investments and will
determine a parallel shift of SML.
E(R)
SML’
SML
RFR’
RFR
Risk
14
Chapter 3 – Diversification
60
40
U.S. Stocks
20 International Stocks
11.7
1 10 20 30 40 50 60
Number of stocks
15
The risk reduction benefits of international diversification would therefore
be even greater to UK investors and other investors from smaller markets than
the US with less domestic diversification potential.
However, risk reduction can be improved by concentrating on poorly
correlated stock markets. Table 1 gives the correlation coefficients between
major stock markets for the period 1981 to 1990. It also gives the average
annual returns and standard deviations of returns for all these markets. The
returns, calculated quarterly, allow for changes in exchange rates and therefore
include exchange risk. Notice how the correlation coefficients vary between a
high of 0.77 for the Swiss and German stock markets and a low of 0.14 between
Austria and Japan. These correlation coefficients between stock markets are in
general lower than are correlation coefficients between shares in the same
market; hence the risk reduction benefits of international diversification. By
using a quadratic programming computer software package and imputing return,
risk, and correlation coefficient data for all these markets, a lower risk portfolio
than that achieved by naïve diversification can be derived.
Of course, there is little point in trying to reduce risk to a minimum if
returns are substantially reduced as well. If, say, returns on all overseas
investments were substantially less than those available to Mr. Jon Doe on the
UK stock market, it might be that the benefit of reduced risk obtained from
international diversification was more than offset by reduced returns.
However, this is not the case as we can see from Table 1, which also
shows the average annual sterling returns that could have been obtained on the
18 different stock markets during the period January 1981 to September 1990.
Nine of those outperformed the UK stock market over that period, with the
Japanese stock market outperforming the UK market by 3% per year over 10
years. If Mr. Doe had chosen the right overseas stock markets in which to invest,
not only could he have reduced risk by diversification, but he could have
significantly increased his achieved sterling returns.
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Equities Return Volatility Correlation – GBP
Ratio
% % US JP UK DM CH F ND I SW ES B DK NW OE CN AS SING HK
USA 16.3 21.6 0.8 *
Japan 21.8 22.9 1.0 0.43 *
UK 18.8 20.4 0.9 0.61 0.42 *
Germany 20.7 23.0 0.9 0.48 0.36 0.43 *
Switzerland 14.9 18.5 0.8 0.64 0.45 0.55 0.77 *
France 20.0 25.1 0.8 0.49 0.42 0.42 0.62 0.60 *
Netherlands 22.2 20.1 1.1 0.72 0.47 0.61 0.70 0.74 0.58 *
Italy 16.0 26.8 0.6 0.40 0.40 0.38 0.50 0.47 0.54 0.49 *
Sweden 26.9 24.5 1.1 0.53 0.46 0.52 0.49 0.56 0.36 0.53 0.47 *
Spain 21.5 26.1 0.8 0.47 0.46 0.43 0.41 0.48 0.49 0.45 0.43 0.46 *
Belgium 28.4 21.3 1.3 0.56 0.46 0.47 0.62 0.62 0.63 0.63 0.47 0.42 0.47 *
Denmark 20.8 20.4 1.0 0.49 0.35 0.28 0.48 0.51 0.40 0.50 0.33 0.33 0.29 0.44 *
Norway 17.9 27.5 0.7 0.56 0.30 0.56 0.46 0.54 0.48 0.61 0.27 0.48 0.38 0.51 0.38 *
Austria 21.7 23.7 0.9 0.28 0.14 0.21 0.60 0.46 0.39 0.39 0.36 0.35 0.28 0.35 0.19 0.29 *
Canada 10.4 21.8 0.5 0.83 0.39 0.63 0.39 0.60 0.42 0.67 0.43 0.50 0.41 0.45 0.46 0.51 0.23 *
Australia 9.4 31.5 0.3 0.53 0.29 0.57 0.32 0.46 0.34 0.44 0.26 0.43 0.44 0.37 0.29 0.51 0.15 0.64 *
Singapore 8.6 32.6 0.3 0.67 0.37 0.52 0.40 0.50 0.60 0.54 0.25 0.50 0.40 0.47 0.42 0.57 0.29 0.59 0.59 *
Hong Kong 11.0 39.1 0.3 0.48 0.31 0.55 0.40 0.44 0.31 0.54 0.33 0.48 0.46 0.37 0.24 0.50 0.26 0.49 0.53 0.51 *
World – GDPppp 19.3 17.4 1.1
Table 1 – Returns, risk and correlation coefficients of major stock market (sterling terms)
(Source: UBS Phillips & Drew, World Markets Review, Table 3.1 ( c ) for data from January 1981 to September 1990)
17
Suppose Mr. Doe were trying to decide in September 1990 in which
overseas markets it was optimal to invest. He would be concerned with expected
returns and risks of markets rather than the historical figures given in Table 1. If
he used historical figures, he would doubtless have put a substantial amount of
his overseas portfolio into Japanese shares, which offered, during the 1980s,
relatively high returns, risk on a level with that of the UK market, and relatively
low correlation with the other stock markets. However, past performance is no
guide to the future and the Japanese market in fact crashed in December 1990
and fell almost continuously for the next two years.
In practice, therefore, investors using a portfolio theory model to
determine their optimal international equity portfolio use their own forecasts for
expected returns (bearing in mind their view of future exchange rates
movements) but will generally use historical estimates of market volatilities and
correlation coefficients, since it is more difficult to have an intuitive feel for
these. They, then, as for the domestic equivalent, are offered by their computer
model a range of portfolios on the efficient frontier, each with its own risk and
return levels. Investors can choose whichever portfolio most suits their risk
preference.
Notice that investors from different countries looking at market returns,
risks and correlation coefficients for the same historical period will be shown
different numbers, reflecting the use of a different base currency. Returns for US
investors will be calculated in dollars and returns for German investors in
deutschemarks and will be different from those given in Table 1. This will
naturally lead to different opportunity sets, efficient frontiers and optimal
portfolios for different nationality investors. Also, investors within the same
country may have different return expectations for markets or different tax
positions, again leading to variations in optimal portfolios between investors.
19
Capital Asset Pricing Model
20
There is at present no real agreement on the form that an international
CAPM would take (e.g. there are problems in defining a risk-free interest rate in
international CAPM context), but this view of the capital markets as integrated
or segmented does give an understanding of how international diversification
reduces risk. If markets were fully integrated, and investors had identical
expectations and investment preferences (that is, were passive investors as
opposed to the more active investors inputting expected returns into the portfolio
theory model), a world CAPM would lead, as does the domestic CAPM, to the
result that all investors should hold the same market portfolio, in other words,
hold a proportion of all the stock markets in the world in amounts reflecting the
market’s relative importance. In this case, if investors chose to restrict
themselves to a purely domestic portfolio, they would be bearing diversifiable
risk for which they would not be rewarded.
On the other hand, if markets were segmented, international investment
would also offer a reduction in risk since risk, which was systematic in a
national context, would become diversifiable in an international context.
Anyhow, markets are segmented if there are impediments preventing investors
from including non-domestic securities in their portfolios, so only those
investors lucky enough to be able to circumvent the restrictions would be able to
benefit from international diversification if markets were segmented.
There is some evidence that markets are partially segmented although the
tax, legal and informational barriers are gradually being removed. More and
more investors are able, if they wish, to invest in international portfolios or
index funds, designed to represent an international market portfolio. Thus, if
they are passive investors as assumed by the CAPM, they can in practice
apportion their assets between, say Treasury bills in their domestic currency and
a world index fund.
Table 2 shows the relative sizes of the major stock markets expressed as
percentages of the total world stock market valued at the end of 1991. In Table
21
2, the US represents the largest single stock market, followed by Japan, the UK,
France and Germany. If these five markets are added up, they represent 85.3%
of the total global stock market capitalisation.
% of total
US 42.8
Japan 25.4
UK 10.5
Germany 3.3
France 3.3
Switzerland 2.0
Canada 1.8
Other 10.9
100.0
Table 2 – Relative importance of major stock markets (beg 1993)
Integrated CAPM says that all investors, regardless of which country and
currency they are based in, should put the majority of their money into these five
stock markets. The optimal portfolio under CAPM will include countries
according to the size of their stock markets. This is in contrast to an optimal
portfolio under a portfolio theory model approach, which will choose countries
according to their expected returns, risks and the correlation coefficients
between countries.
The CAPM model assumes that the investors is passive, accepts the
market consensus on stock market values and returns, and wishes to achieve an
average return on a fully diversified portfolio. The portfolio theory model
assumes that the investor is more active, with views different from the
consensus. He will therefore expect to achieve a better return for risk ratio than
the investor in a global index fund by ‘overweight’ in those stock markets which
he thinks will outperform and ‘underweight’ in those which he considers
undervalued.
Although there is less theoretical support for an international CAPM than
for its domestic counterpart, international index funds, which offer investors the
opportunity to invest in countries in proportion to their stock market and bond
22
market values have had some success. This has been helped by the practical
difficulties which investors face when attempting to select likely outperforming
stocks in foreign stock markets, where they have to deal through unknown
intermediaries and are not as at ease with the local accounting methods and
types of equity analysis employed.
Such index funds are available to individual investors through the medium
of unit trusts, which adopt a more active approach, attempting to outperform
their index fund counterparts. Also, it must not be forgotten that investing in the
UK stock market does expose investors such as Mr. Doe to exchange risk and to
some element of international diversification, due to the large element of
revenues and costs incurred overseas by UK-based companies.
Finally, for example most UK investors consider sterling to be their base
currency. Whatever the time horizon of their investment strategy, their future
liabilities are likely to be expressed in sterling terms. Both the portfolio theory
and CAPM approaches to international investment should lead UK-based
investors to hold a majority of non-sterling-denominated bonds or equities in
portfolios that are optimal in terms of expected returns per unit of risk. This type
of asset structure would expose the investors to significant exchange risks
although investors who do not wish to bear the exchange risk inherent in
overseas investment can, to a large extent, hedge this away.
At any rate, it is still the case that UK and other investors tend to keep a
majority of their investments in securities denominated in their base currency,
applying integrated CAPM only to the funds which are not dedicated to
domestic investments or adjusting the portfolio theory model to include a
predetermined percentage of domestic security markets.
23
Chapter 4 – Eurodollar Markets
24
International markets are a free market response to domestic regulation. A
transaction is said to occur in the international market when the currency
employed is outside the control of the issuing monetary authority. A dollar-
denominated loan to an American company in London, a German mark loan to a
Japanese company in Singapore and a French franc bond issue by a Dutch
company underwritten in Frankfurt are all examples of international market
transactions.
In each instance, the transaction occurs in a locale that is beyond the
direct regulatory reach of the issuing monetary authority. Thus, the US Federal
Reserve has trouble regulating banking activities in London even when the
activities involve American companies and are denominated in dollars, just as
the Bundesbank has difficulty regulating German mark activities in Singapore.
International financial markets got their start in London shortly after
World War II, and were originally limited to dollar transactions in Europe;
hence the name Eurodollar. And while the markets have since grown well
beyond dollar transactions in Europe to become truly global, they are still
known generically as Eurodollar markets.
Euromarket activity has burgeoned in recent years because the markets
provide access to large pools of capital, denominated in a number of currencies,
at very competitive prices. Moreover, the absence of regulation drives issue
costs and reporting requirements to an absolute minimum.
Two important reasons Eurodollar markets can offer lower cost financing
than domestic markets are the absence of reserve requirements on Eurodollar
deposits and the ability to issue bonds in what is known as bearer form. In the
United States and many other domestic markets, banks must abide by reserve
requirements stipulating that they place a portion of each deposit in a special,
noninterest-bearing account at the central bank. Because these reserves tie up
resources without yielding a return, domestic loans must carry a higher interest
rate than Eurodollar loans to yield the same profit.
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The chief appeal of bearer bonds is that they make it easier for investors
to avoid paying taxes on interest income. The company issuing bearer bonds
never knows its owners and simply makes interest and principal payments to
whoever presents the appropriate coupon at the appropriate time. In contrast, the
issuer of a registered security maintains records of the owner and the payments
made. Because bearer securities facilitate tax avoidance, they are illegal in the
US. Their use in Eurodollar markets means that Eurodollar bonds can carry
lower coupon rates than comparable, domestic bonds and still yield the same
aftertax returns.
The ability of Eurodollar markets to draw business away from domestic
markets has sharply accelerated the deregulation of domestic financial markets.
As long as companies and investors can avoid onerous domestic regulations by
simply migrating to Eurodollar markets, regulators face a Hobson’s choice: they
can either remove the offending regulations, or watch Eurodollar markets grow
at the expense of domestic markets.
The IET is an example. When first imposed, the tax had the desired effect
of restricting foreign companies’ access to dollar financing. Over time, however,
borrowers found they could avoid the tax by simply going to the Eurodollar
markets. The longer-run effect of the IET, therefore, was to shift business away
from the US without greatly affecting the total volume of dollar financing. An
awowed goal in repealing the IET was to make US markets more competitive
with Eurodollar markets.
Not all regulations are bad, of course. Regulatory oversight of financial
markets and the willingness of governments to combat panics have greatly
stabilized markets and economies for over 50 years. The ongoing question is
whether the deregulatory pressures created by the Eurodollar markets are
improving efficiency by stripping away unwarranted restraints, or whether they
are in dangerously destabilizing the world economy.
26
Chapter 5 – Efficient Markets
27
Market efficiency is a description of how prices in competitive markets
respond to new information. The arrival of new information to a competitive
market can be likened to the arrival of a lamb chop to a school of flesh-eating
piranha, where investors are – plausibly enough – the piranha. The instant the
lamb chop hits the water, there is turmoil as the fish devour the meat. Very soon,
the meat is gone, leaving only the worthless bone behind, and the water returns
to normal.
Similarly, when new information reaches a competitive market, there is
much turmoil as investors buy and sell securities in response to the news,
causing prices to change. Once prices adjust, all that is left of the information is
the worthless bone. No amount of gnawing on the bone will yield any more
meat, and no further study of old information will yield any more valuable
intelligence.
An efficient market, then, is one in which prices adjust rapidly to new
information and in which current prices fully reflect available information about
the assets traded. “Fully reflect” means that investors rapidly pounce on new
information, analyze it, revise their expectations, and buy or sell securities
accordingly. They continue to buy or sell securities until price changes eliminate
the incentive for further trades. In such an environment, current prices reflect the
cumulative judgement of investors. They fully reflect available information.
The degree of efficiency displayed by a particular market depends on the
speed with which prices adjust to news and the type of news to which prices
respond. It is common to speak of three levels of informational efficiency.
A market is weak-form efficient if current prices fully reflect all
information about past prices.
A market is semistrong-form efficient if current prices fully reflect all
publicly available information.
A market is strong-form efficient if current prices fully reflect all
information public or private.
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Extensive tests of many financial markets suggest that with limited
exceptions, most financial markets are semistrong-form efficient, but not strong-
form efficient. This statement needs to be qualified in two respects. First, there
is the issue of perspective. The above statement applies to the typical investor,
subject to brokerage fees and without special information-gathering equipment.
It does not apply to market makers. Second, it is impossible to test every
possible type and combination of public information for efficiency. All we can
say is that the most plausible types of information tested with the most
sophisticated techniques available indicate efficiency. This does not preclude the
possibility that a market may be inefficient with respect to some as yet untested
information source.
But what are the implications of efficiency? If financial markets are semi-
strong form efficient, the following statements are true:
- publicly available information is not helpful in forecasting future
prices.
- in the absence of private information, the best forecast of future price
is current price, perhaps adjusted for a long-run trend.
- without private information, a company cannot improve the terms on
which it sells securities by trying to select the optimal time to sell.
Individuals without private information have two choices. They can
reconcile themselves to efficiency and quit trying to forecast security prices, or
they can attempt to make the market inefficient from their perspective. This
involves becoming a market insider by acquiring the best available information-
gathering system in hopes of learning about events before others. A
A variation – usually illegal – is to seek inside information. Advance
knowledge that Carl Ichan will attempt to acquire TWA, for example, would
undoubtedly be useful in forecasting TWA’s future stock price. A third strategy
used by some is to purchase the forecasts of prestigious consulting firms, the
29
chief virtue of which appears to be that there will be someone to blame if things
go wrong.
As the above comments suggest, market efficiency is a subtle and
provocative notion with a number of important implications for investors as well
as companies. The treatment of the topic has been brief here, but sufficiently
long enough to suggest that unless executives have inside information or
superior information-gathering and analysis systems, there may be little to be
gained from trying to forecast prices in financial markets. This conclusion
applies to many markets in which companies participate, including those for
government and corporate securities, foreign currencies and commodities.
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