Building a low-carbon economy: The inaugural report of the UK Committee on Climate Change

Essays on Banking and Portfolio Choice Bo Larsson D I S S ER TA T IO N S IN E C ONO M IC S 2 0 0 5 : 4 © Copyright by Bo Larsson, 2005. All rights reserved. Department of Economics Stockholm University ISBN 91-7155-123-9 Printed by Akademitryck AB, Valdermarsvik, Sweden, 2005. To Malin v Contents Acknowledgements Introduction I II vii 3 Banking and Optimal Reserves in an Equilibrium Model 11 Optimal Rebalancing of Portfolio Weights under Time-varying Return Volatility 39 III Can Parameter Uncertainty Help Solve the Home Bias Puzzle? 73 vi vii Acknowledgments What a long strange trip this has been, as The Grateful Dead ones wrote. I don’t think anyone that knew me in my early years could have predicted that I would write a thesis one day. Being 15 and having to choose which path of education to take, the choice was easy: only nerds and geeks took science to prepare for University. Better to take the combined Business and Language program where all the girls were and get a job after High school. But when I spent a year in Cazenovia NY being surrounded by friends and family that were going (or had been) to University I think that I started to reconsider. Therefore I want to thank them who sowed the seed of higher education in my stubborn mind, cheers family and friends in Caz. I also want to thank my boss when working at Handelsbanken, Pernilla Berggren, who is the person that encouraged me to apply to the Ph.D. program. I think that three ingredients are essential in order to success in the Ph.D. program: character, friends and theoretical skills. My commanding officers during the recruit and UN armed forces respectively, Per Dahlbom and Henrik Lettius, have helped me build my character by being exceptionally good role models (in case someone think I’m more of a character than having any, Per and Henrik aren’t to be blamed), which have help me through rough times. When it comes to friends there are “drängarna” in Strängnäs whom I haven’t had time to see so much the last couple of years. My High school friends “E3-sacks” whom I don’t see very often but when we do, it rocks! Of course I mostly hang with other Ph.D. students and colleagues. The year I started on the program we were twelve new candidates that have had great fun over the years. In fact it seemed that the theories of Engle could be applied also on calendars, after immensely long periods without weekends, all of a sudden there could be arbitrarily many Saturdays in a row, often associated with the taking of exams. Being such good friends didn’t only mean a lot of party but also good support when times were hard, especially during the first year. Every year I also get to see fantastic views and have great adventures with Mats and Jesper when we go hiking, though Micke call it suffering, up north. When I’m particular fortunate my advisor in sports Lennart takes me to see Speedway featuring Tony Rickardsson and company, the gladiators of today. To see them race around and hit the boards in 80 mph really makes you realize that not being able to solve a model is not that big of a deal. Since Mackan said that one should drive MC during Ph.D. studies, I want to thank Putte and Nyqvist & Monell Trafikskola, Åkerman family, Anders, Kent and Dad for helping me get my MC drivers license. My family also has a big part in this thesis; they are probably responsible for my interest for social sciences and economics. As long as I can remember there have always been discussions about politics and economics at home, especially my grandfather Vallis has been an inspiration through his strong engagement for current viii affairs. My family has also been good practical support, mom do my taxes, dad has hooked me up with people in Banking to discuss my models with, my brother fix technical stuff and my sister have filled my spare time with the opportunity to ride magnificent horses. I think I was well prepared for graduate studies by Björn Hansson who was my undergraduate thesis advisor and also gave a really challenging course. He has also helped me with data to this thesis. During the graduate program I have really been inspired by Timo Teräsvirta, both through courses and discussions of problems. Taking courses by John Campbell and Sune Karlsson has also been rewarding, both essay two and three were started as term papers in their respective courses. I also want to thank Dan both as friend and co-author. The final touch on the language in the thesis has been supplied by Christina, which resulted in many improvements. When it comes to actually finishing this thesis I cannot thank my advisor Hans Wijkander enough. He has both been like a coach in creative writing and a demanding listener to assure that my thinking is straight and also improving my modeling. Through his suggestions I have really improved my skills in writing scientific papers. Last I want to thank Malin for putting up with me through my ups, and downs. Listening to endless explanations of integrals and probability functions for which you probably couldn’t care less. For proof reading my articles, and maybe the most important thing living with me! For those of you that I have kicked in the shins, tackled and/or managed to shoot in the face during lunch football and ball-hockey, remember that I have the perfect excuse being Rose Mary’s Baby. . . Stockholm, August 2005 Bo Larsson Funding from the department of Economics at Stockholm University, Sparbankernas forskningsstiftelse, Siamon and Wallenbergs Jubileumsfond is gratefully acknowledged. Introduction Introduction Throughout time, economists have often encountered empirical problems and/or behavior that cannot easily be explained with existing models and theory. As a social scientist, it is both challenging and inspiring to address problems where standard models have previously failed. Although this thesis might seem quite divided as the three essays are in rather different fields, their connection is, in fact, that they all discuss problems where little is known about the cause of economic agents actions. In the last paper, coauthored with Dan Nyberg, we discuss one of the more famous economic puzzles, termed “the home-bias puzzle”. The first two papers discuss questions of great importance within banking and finance that have received relatively little attention from researchers. In chapter one, I address the question of why banks in reality are so “overcapitalized” in relation to regulatory requirements. Since banks profit from more leverage, they ought to seek as low reserves as possible. To strengthen international capital markets, banking is now regulated to ensure that sound banking is practiced, which puts a bound on how low the reserves can be. Most developed countries have adopted the reserve requirements of the Bank of International Settlements (BIS), which regulates tier one capital, i.e. short-term assets and cash, and tier two capital, i.e. banks’ long-term debt, BIS (2003). The reserve requirements stipulate that the ratio of tier one and tier two capital to a bank’s risk-adjusted loan portfolio should be at least eight percent, and that the considered tier two capital cannot exceed the tier one capital. This effectively means that the tier one ratio should be at least four percent. However, banks typically have a much larger tier one capital; in Sweden the market leading banks have as high tier one ratios as 7-8 percent. Jackson et. al. (1999) show that internationally, for G-10 banks, the average reserve ratio is 11.2 %. Reserves in a bank cannot be directly compared to cash holdings in a regular company, but some similarities exist. For both banks and regular firms, cash holdings raise the endurance, i.e. the bank/firm can withstand larger losses. Accordingly, one explanation for carrying large reserves could be risk reduction over time. Ingvar Kamprad, the successful founder of IKEA, stated that having large amounts of capital is the best way of ensure long-term growth, even though analysts frequently complain about inefficient use of capital. IKEA is thus independent of capital markets when new projects are slow starters, thereby allowing it to pursue 3 4 Introduction long-term investments despite possibly very poor short-term outcomes.1 However, in this article, I study another possibility. How does systematic risk affect optimal banking strategies? In many earlier capital market articles, it is often assumed that systematic risk is absent, e.g. Diamond (1984) or Williamson (1986). But the financial literature has supplied massive support for the presence of systematic risk, and path breaking models such as both CAPM by Sharpe (1964) and APT by Ross (1976) build on the separation between idiosyncratic and systematic risk. By introducing systematic risk in the banking model by Williamson (1986), I can show that there could be insurance motives for carrying reserves, independent of the banking regulations. In fact, banks have always carried reserves independent of regulations. With systematic risk, banks have a positive probability of default, which results in expected auditing costs for depositors. By holding cash reserves, banks can reduce this expected auditing cost. Auditing costs are unproductive and therefore, the expected return on lending, to be split between banks and depositors, can be increased with the use of reserves. Swedish regional banks in general carry more than twice the reserves of market leading national banks. If conjecturing that small regional banks are less diversified since they cannot pool regional risk, this might be the reason why they carry more reserves than large banks. I show that more systematic risk in my model indeed optimally leads to larger reserve ratios. In chapter two, optimal long-term investment strategies given different return generating processes are analyzed. For a long time, the common investment advice has been to suggest a larger fraction of stocks in a portfolio if the investor has a long investment horizon. This advice has been given by both professionals as well as the popular press. A striking example of this is also the marketing of generation tailored mutual funds in the new Swedish pension system, where costumers are supposed to pick their investments based on birth year. The reason for this investment behavior to be preferred is that stocks are less “risky” in the long run, where risk is often measured as standard deviation; see, for example, Siegel (1994). Campbell and Viceira (2002) point out that if return is independent and identically distributed, i.i.d., risk measured as standard deviation is inversely related to the square root of the investment horizon. If there is to be any reduction in risk with longer investment horizons, there must be some sort of predictability in returns, i.e. they cannot be i.i.d. I restrict the analysis to wealth alone, i.e. investors do not have any labor income. Campbell, Cocco, Gomes and Maenhout (2001) show that if investors have labor income, this could lead to a lower demand for stocks as investors approach their retirement. However, this effect is unrelated to the relative riskiness of stocks, but is 1 Discussion in the documentary “Mannen som ville möblera om världen” on Swedish national television, SVT2 July 10, 2005. Introduction 5 a substitution effect between the risk-free asset and labor income. The marketing of “generation funds” in Sweden emphasized the old mantra of lower “risk” in stocks for long horizons. Several authors have investigated portfolio choice decisions for long-term investors. But it seems that no one has considered the case where returns are distributed according to the popular GARCH-models first suggested by Nobel Laureate Robert Engle (1982). One reason for the popularity of GARCH models is that they capture the fact that return volatility clusters over time; see, for example, Mandelbrot (1963). Since these models are frequently used in practice, I investigate in what optimal portfolio rules for multi-period investors they result. My approach is to first analyze the implications for long-term portfolio choice when only risk, measured as conditional variance, is time-varying. Earlier analyses of similar return structures have shown that for investors with infinite horizons, shocks to volatility affect portfolio choice such that there are some horizon effects; see Chacko and Viceira (2000). In contrast to that result, I find there to be no effects on portfolio weights when the investment horizon is altered. Naturally, shocks to volatility affect the optimal weights of the risky asset, but these have an equal effect on all investors, given the same level of risk aversion. Merton (1973) showed that expected returns should be related to the level of risk through the coefficient of risk aversion. The intuition is that if you are risk averse, more risk should be compensated by higher expected returns. Moreover, the more risk averse you are, all else equal, you will require more compensation in terms of expected return to hold risky assets. Therefore, I also perform analyses where expected returns on risky assets depend on conditional volatility, which is my measure of risk. When returns are predictable, the optimal portfolio weights do change when the investment horizon is altered. These effects are small, however, and only present for about thirty periods, i.e. 2-3 years with monthly data. When there is a positive relationship between risk and expected return, investing in stocks is a hedge against low returns. This is because a lower than average return today raises the expected return tomorrow. In the final chapter, I and my coauthor Dan Nyberg address international portfolio choice. A well-known puzzle in international finance is the equity home bias: in contrast to the prescriptions of standard portfolio theory, international diversification is not used as a means for decreasing the risk of a portfolio, e.g. French and Poterba (1991) and Lewis (1999). In this essay, we illustrate a mechanism where the exchange rate estimation risk causes an equity home bias. Estimation risk is introduced into a standard mean-variance portfolio framework using return time-series of different lengths. We take the perspective of a domestic investor considering a foreign stock-market index investment. The investor cares about the local-currency return of the foreign investment as the investor’s consump- 6 Introduction tion basket is denominated in local currency and observes historical data on stock index returns and the return on the exchange rate. The returns are assumed to be generated from a multivariate normal distribution and the investor uses a stochastic model to forecast future returns on the stock indices and the exchange rates. The return history of the exchange rate is argued to be shorter than the available time series of equity index returns due to e.g. exchange rate shifts. A change in the exchange rate regime implies that the past time series on local currency returns of a foreign investment can no longer be expected to be informative about the future, and the sample needs to be truncated. To deal with this issue, we use a framework devised by Stambaugh (1997) to analyze investments whose histories differ in length. Stambaugh derives maximum likelihood and Bayesian predictive distribution meanvariance estimators of the combined sample. If the investors form expectations of future returns based on the ex post sample moments, then the present estimation risk is ignored. On the other hand, if investors take the estimation risk into account when calculating the next period’s expected return and covariances, Bayesian predictive mean-variance estimators should be used. The impact of estimation risk on an optimal portfolio is illustrated with data from Sweden and the United States. Our results show the introduced estimation risk to mainly be associated with the exchange rate, and that explicitly accounting for the estimation risk causes the domestic investor to increase the fraction of domestic assets. While the introduction of exchange rate estimation risk is not powerful enough to explain the entire home bias observed in data, the results of this paper illustrate a potentially important mechanism that is often overlooked in discussions of international portfolio diversification. My articles do not fully reveal the underlying mechanisms of economic agents behavior in the problems I study. However, I feel that the articles do shed some new light on these problems of great practical importance. Introduction 7 References BIS, (2003), “The New Basel Capital Accord,” Basel Commitee on Banking Supervision, http://www.bis.org/bcbs/cp3full.pdf Campbell, John, and Luis Viceira, (2002), “Strategic Asset Allocation,” Oxford University Press, Oxford. University. Campbell, John, Joao Cocco, Francisco Gomes, and Pascal Maenhout, (2001), “Investing Retirement Wealth: A Life-Cycle Model,” in John Campbell and Martin Feldstein eds. Risk Aspects of Investment-Based Social Security Reform, University of Chicago Press, Chicago. Chacko, George, and Luis Viceira, (2000), “Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets,” working paper Harvard University. Diamond, D. W., (1984), “Financial Intermediation and Delegated Monitoring,” Review of Economic Studies, 51:3, 393-414. Engle, Robert, (1982), “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50/4: 9871006. French, K., and Poterba, J. (1991), “Investor Diversification and International Equity Markets,” American Economic Review, 81, 222-26. Jackson, P., et. al. (1999), “Capital Requirements and Bank Behaviour: the Impact of the Basle Accord,” Basle Committee on Banking Supervision Working Papers, no. 1. Lewis, K. (1999), “Trying to Explain the Home Bias in Equities and Consumption,” Journal of Economic Literature, 37, 571-608. Mandelbrot, Benoit, (1963), “The Variation of Certain Speculative Prices,” Journal of Business, 36/4: 394-419. Merton, Robert, (1973), “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41: 867-87. Williamson, Stephen D., (1986), “Costly Monitoring, Financial Intermediation, and Equilibrium Credit Rationing,” Journal of Monetary Economics, 18, 159-79. Sharpe, William F., (1964), “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal of Finance, 19:3, 425-42. Siegel, Jeremy, (1994), “Stocks for the Long Run,” McGraw-Hill, New York, NY. Stambaugh, R. (1997), “Analyzing Investments whose Histories Differ in Length,” Journal of Financial Economics, 45:3, 285-331. Williamson, Stephen D., (1986), “Costly Monitoring, Financial Intermediation, and Equilibrium Credit Rationing,” Journal of Monetary Economics, 18, 159-79. Essay I ! " #$ % & '($ ) .&/ *&+++, - && 0$ 1 1 2 3 4 4 5 1 6 % % 6 6 && 78 1 % 9 : &0 ; < = *&+>#, ? 2 7 . @ 1 *&++>, *&+'+, 4 ? A 2 8 ? A 4 7 ? A : *&+'+, = *&+(B (' , 7 *&+(#, . 4 8 ; % *&+(>, 7 *&+(B (' , 7 ? A 7 5 ! @ (' , *&+(B &+(B " C ; 2 6 8 D 8 7 *&+(B, 8 < E 5 4 1 7 ? 4 *&+(B (' , 8 < &F ; 8 % 6 4 G ; 7 *&+(' , % 8 A 8 *&+(' +& 0//0, 7 % )H1 1 ? 1 I1A ? 3 8 7 % *&+(B, J *&++', 2 7 *&+(&, . *&+(+, ? 6 *&++', 3 8 G < H *&+'B, 6 ?<H1 0 7 *&+(' , *&+B#, &# 2 - 4 < 4 F 7 6 # ? D 8 8 & < 8 8 - 8 2 ; 8 8 ? ! " 6 & 2 2 G -4 @ 7 *&+(B, 8 -4 6 4 7 2 E - &> d Pro fit _ RD , r D es Loan Rich (B) D, Re s er v De po sit (1 ) Consumers (C) Loan R, r _ R, r _ an Lo BANK Entrepreneurs (E) 6 &5 . = 2 7 *&+(B, % 6 & < 4 7 4 @ 4 4 &B 4 8 4 C 4 4 *&, *&, K 4 4 2 *0, *0, 7 *&+(B (' , 8 4 2 4 *0, = 8 4 2 *0, *F, *F, 8 4 *#, *#, 4 - &' 4 *#, - - ! " 3 &+(( ++ &0 FB F( & F ? F 2 4 6 0 F> 7 Expected Utility for Lender and Entrepreneur with Direct Lending Expected Utility for Lender and Entrepreneur with Direct Lending 1.4 1.4 Entrepreneur 1.2 Entrepreneur 1.2 Optimal contract 1 Optimal contract Expected utility Expected utility 1 0.8 0.6 0.4 0.8 0.6 0.4 Lender Lender 0.2 0.2 0 0.2 0.4 6 0.6 0.8 1 1.2 Repayment 05 . 1.4 1.6 1.8 0 0.2 2 0.4 0.6 0.8 1 1.2 Repayment 1.4 1.6 4 4 & F 8 8 & &/( 0 4 & & /F>' & /F#B 1.8 2 &( 8 4 4 8 ( 7 8 *0, *#, G 5 8 E ; 1 = - 4 L M < 8 8 5 < 8 0 & 6 @ *0, 0 * /'0F, 8 6 0 4 4 ; * /F>', *& &/(, 4 - - 8 @ - 6 4 8 4 1 8 @ 8 8 7 *&+(B, ? *&+((, ? = 3 *0///, % 6 7 *&+(B, 2 @ &+ 4 < -4 5 6 ! 8 " 8 - < 6 - - D - 2 4 4 8 8 = - 5 8 7 *>, 7 *>, *B, ; 6 7 *', 3 4 8J - *&++(, *(, 0/ *(, % % - - - *+, *+, C - *+, 6 *+, *&/, *&/, # $ % $ # 7 $ 7 *&+(B 8 (' , 8 A *&+(' +& 0//0, J *0///, . = *&&, *&&, 0& 6 4 - *&0, *&&, ! 4 " !# *&F, *&0, *&#, *&#, *&>, 2 $ " *&>, 7 *&#, 5 " *&B, *&&, 5 *&', 4 *&0, *&B, 00 Matching SHB Losses to a Continuous Distribution 180 Credit Losses SHB 160 3.00% 140 2.00% 120 1.50% 100 f(x) Loss 2.50% 1.00% 80 0.50% 60 0.00% 19 78 19 * 80 19 82 19 84 19 86 19 88 19 90 19 92 19 94 19 96 19 98 20 00 20 02 40 20 Year 0 6 F5 0 2 0.01 0.02 0.03 0.04 0.05 0.06 Loss fractions 0.07 0.08 0.09 0.1 % *&(, 2 ! " - M 2 M @ 2 - 8 K *&B, 2 *&(, $ *&(, 4 4 * % , 6 ? = 8 % F 8 *0//#, &$ F 8 0F & 2 2 A 4 @ *0//0, ; 4 < % % *&+, ? *&+, J 4 2 &// & 4 *0///, - ' - 6 E *0/, & ' ' *0/, 2 *0/, *0&, & & ' E *0&, ' < 8 4 - 4 ! " 4 - % 0 '# 0> % 8 < 2 *&+++, 0# ( @ - *00, ) *00, ) * - *0F, * < 4 *0F, 4 *00, 4 3 ) *&++#, *00, *0#, * ) * *0#, ( 4 @ - ( ; 7 7 5 - < - *&+(B, 0> ( ) " *&, *0, 4 2 4 4 *0', 2 , + + - + *0&, + 2 *0>, ( + *0>, 2 $ - *&(, - + ; + 4 2 + *0>, , + .$ . .$ + , + + +# + +# *0B, + . & 8 *0', + , + + *0', 4 % 4 4 2 *0(, +, + + / /- *0(, 0& 0B 0 7 *&+(B (' , 2 4 4 2 K < 4 G @ 4 ? + - *&++', * 2 0 2 0 6 1 K + - , + + 0 *0+, 0 + *0+, +, + 0 7 + - /- *&+(B (' , @ 4 8 2 4 0 # 6 +, + + - /- * , + 0 , + 0 0 - 0 *F&, /, - 4 4 *F/, C 4 0' + , + + 0 0, / *F0, 4 *F/, 2 4 2 + - *F/, + 4 4 6 - *F/, / 0 2 4 : 0& = 4 - /0 $ % 8 / 2 4 + *F/, - *F/, 2 - ; *F/, *F&, 4 ( - % " 0 D 4 2 4 *FF, *FF, 4 4 2 4 % 0( Expected Utility for Banks and Entrepreneurs with no Systematic Risk 1.4 Expected Utility for Banks and Entrepreneurs with Systematic Risk Entrepreneur Entrepreneur 1.2 1.2 Optimal contract Optimal contract 1 Expected utility Expected utility 1 0.8 0.6 0.4 0.8 0.6 0.4 Bank Lender 0.2 0 6 0.2 0 0.5 1 Repayment 1.5 0 2 0.5 1 Repayment #5 1.5 7 8 & &+0> '#+' 4 & /(F& ; F 4 0> $ 4 & /B0B & &F(+ 7 +### 4 *F#, $ ( ) *F#, 0 7 *F#, 7 8 *&+(B (' , 4 2 *&+(B, @ 8 ; F ; 8 D E 6 # @ 6 0 0+ 7 6 # 4 6 # 7 & &+0> 0> $ 4 '#+' ( F& $ 8 4 +$ *F >' $, * +###, B 0B $ 4 & &F(+ & &F(+ 4 4 *B 0B $, D H 50% rho.01 45% rho .1 40% rho .3 rho .5 Probability 35% 30% 25% 20% 15% 10% 5% 27% 25% 24% 22% 20% 18% 16% 15% 13% 9% 11% 7% 6% 4% 2% 0% 0% Loss Fraction 6 >5 H 8 /& > >$ F ; & G 4 4 > 4 F/ Expected utility per unit of reserves for banks 1.25 Expected utility per unit of reserves 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 6 B5 . 0.03 0.21 0.7 1.7 3.41 Default exposure in percent 4 4 7 4 2 < 4 //0 0$ < 4 &> > $ & &B#B 8 4 6 > (( 7 6 B : 4 4 0$ ! &> >$ " 4 & &B#B G &B $ F& ρ =.2 Expected utility to banks when 1.18 1.14 1.175 1.13 1.12 Expected utility Expected utility 1.17 1.165 1.16 1.11 1.1 1.09 1.155 1.15 1.08 1.145 1.07 1.14 6 ρ =.5 Expected utility to banks, 1.15 1.185 0.01 0.07 Default exposure in percent 0.31 1.06 0.92 0 0.002 0.007 Default exposure in percent 0.018 0.036 '5 4 0 0> $ > &F ' $ &' +($ ; 4 &F +' $ G &+ 0 $ 8 B$ D 0 $ 4 - 7 F0 3 C 7 4 2 F0 4 6 ' 4 > 7 7 0 6 ' 4 0 &F ' $ 0> $ 7 &' +( $ - < < - + 4 & < 8 ; 8 7 < ; 2 - - 7 4 2 - FF 4 < M @ 4 ; 4 ; 1 4 % J *&++', < 2 - 2 E 8 G 4 4 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$ ; "# ? /222 %8 " < 7 IJ IJ E( IJ IJ 7 8 # ! + ! ( G ) . ( ! C " + ( E ( &*# " # + 59 %+ ( .) 59" %8 (.* 4 =! #/0 %A $ /22/ %A (( ? ! () ? ) + ! C (4 . / % $ 1 %+ (( ? (0 < - ) 4 &*# " # ( ) C (4 . * " 0 /$ / 7 (( 5 %+ ! C 4 ( G ) 92 2 9 %= + ( !+ , 4 ( F .* " / 0/ /1$ 51 %+ ( 8 G ! ! ! C .* "/ 5 $ 9 A + ( ! &*# " # + ! C * " 2/0 $ $ G ! 3#( (( 8 # ( !C 3#( (( +", $01 1 / 1 59 %? " " - ) A (( A ? ) .7 4 ? $ A A (( ? " " - ( "# .7 4 ? #/0 ) E # . $0 / 12 1/ Essay II !!! " $ $ # $ $ %&&'( $ $ $ $ % )!!( $ $ $ * + $ , $ - . $ - . $ * & * / ' " $ 7 ; $ / - 0 "3 ' 0 7 : < 7 " $ $ $ . $ ,1 4 2 5 8 6 6 ' 9 ; $. -' $ ' * " " /'=> + 7 $ & $ & / $ * ?@ $ &&' $ * A! $ $ B $ $ $ +!!C $ 5 +! $ $ , 1 @!C 5!C $ * C $ $ $ $ $ / 1 * $ D ; $ $" %+@E@( %+@E@( ' $ * $ 0 $ % ( ' %+@)+ $ $ $ )?( , - . $ * $ , $ $ D " 1 2 ' F $ %+@GG( H % F = 6 F %+@@ ( 0 $ = ; %+@@E( ( $ 6 : * %+@@@( $ $ $ $ $ H 0 $ * $' $ %+@E@( 0 $ % !!!( A+ 2 H %+@)E( $ $ $ $ : $6 $ # % !!!( $ $ $ $ 6 $ %+@@@( 6 : 0 : % !!!( $ $ * $ $ $ $ < I/<60 $ I/<60 7 $ 1 " 8 7 & * # 0 % !!+( % !!+( I/<60 I/<60 2 $ $ $ % !!+( $ J 6 I $ 6 ' % !!+( $ I/<60 $ $ $ $ $ * $ H %+@@?( I9< I/<60 / , $ J %+@GE( I/<60 I 9 > J %+@@+( 8I/<60 $ ' $ $8 2 < %+@G)( I/<60 ' ' $ $ $ I/<60 < $ J $ %+@)?( 0 I/<60 %+@)?( $ D 7 $ A " ' %+@E@( ' $ : % !!!( < * %+@)?( 6 $ %+@E@( $ $ $ H < $ % !!!( $ I/<60 $ ! @@C . $ 7 $ $ 0 , I/<60 $ = $ ?$ $ 7 B K $ $ $ $ $ " " $ ? H % !!!( " @A % " A* 7 ;" ( 0 %+@@ ( " / " $ $ 6<"& +@5A +@+@ +@@@ $ '$ " 5 E 6 $ $ 6/&' A? $ $ $ $ $ $ $ * $ , D 7 $ $ $ % L!( $ $ 7 M $ $ ?! * % L+( $ @ $ $ * / L @ $ * $ $ $ $ * $ * $ J B %+( % ( %+( $ * $ $ > $ $ $ B %?( %?( $ $ $ AA $ " ? $ * $ $ %?( %A( %A( H B % !!!( :/< K , + !!! !!! 7 F %+@@!( * $ $ - $ $ $ ? H J $ $ $ ." $ M$ $ # ? $ $ $ $ $ %+( B * H J $ H %5( $ $ / %+@)+( 0 / * %E( " $ H $ % !!!( ' & %+@G5( %+@E@( %E( ? H %+@5)( A5 = $ K * - $ . # ' & %+@G5( / K $ 5 E , $ K %)( " $ H %5( $H 7 * $ % ( * B %G( ; $ %E( $ %5( %@( %@( * 7 %@( %+!( H$ $ % ( %+!( %++( > $ %+ ( AE H $ * H$ 1 * %5( H %+?( ; %+?( $ $ $ %A( 0 $ * B # %+?( $ $ H$ $ % !!!( K $ A! 1 H 5 # $ K 8 $ $ $ ! @@C $ $ 0 ' " 1 A 1 $ * $ %?( $ 7 $ = $ ! E @@C 7 $ K ! @@C ?!C $ $ $ A ' K $ A) ! " 4 7 I/<60 I/<60% $ * ( H $ I/<60% J %+@)E( $ $ $ ( I/<60 I/<60 $ $' %+@)?( $ !# 7 $ M$ $ $ I/<60 $ " $ / B %+A( 0 8 %+@G ( $ /<60% ( %+A( $ $ $ I/<60% H $ 2 ( $ %+@GE(5 $ %+5( %+E( 7 %+5( I/<60 $ 5 * /<60 / $ $ $ $ $ %+@GE( H AG %+5( %+E( %+A( %+5( $ $ $ * E I/<60 $ % K $(B %+)( %+G( %+@( / I/<60% ( $ * $ ) $ > %+@@+( I/<60% $ 0 N ( N $ $ $ $ $ > I/<60 $ $ !# $ % $ > J %+@@+( 8I/<60 $ H$ $ % !( 6 $ % !( I/<60 $ I/<60 E ) = $ > $ H $ $ %+5( A@ $ / * H %+@)E( $ $ $ O %+@@?( < $ 1 G I/<60 %+@@?( I > 9 < J 8I/<60 $ $ I/<60 $ I/<60 $ I I9< I/<60 $ $ H$ $ + ! $ $ % +( ! I9< I/<60 $ I/<60 % ( I/<60 # $ $ $ I/<60 ! $ 8I/<60 $ $ ! !#! ' & %+@)?( K ' J # $ $ $ $ $ 8 < %+@G)( $ /<60 ' I/<60 ' % !!!( $ G > %+@@+( I 2 $ 1 $ 8 I8 H 5! # $ " % ?( % ?( 6 %+E( % !( % +( > $ %+@@+( $ N $ $ " H * $ $ M$ $ !#' / ( $ $ $ I/<60 %+@@?( I/<60 * I/<60 * # $ * I/<60 % ?( %+5( I/<60 # # $ * % A( >1 I/<60 ) % E( $ % A( % 5( # # % A( $ %+5( $ $ $ $ / $ & $ $ % E( 1 $ % 5( 5+ S caled high frequency volatility vs True volatility 3 scaled true 2.5 2 1.5 1 0.5 0 0 7 +B 1000 2000 3000 4000 5000 6000 I/<60%+ +( $ I/<60%+ +( $ % )( $ $ * %+@( % )( $ % G( * $ 8 % G( $ H$ 1 % G( $ $ I/<60 %+)( $ $ + $ $ $ $ 7 $ %+G( / $ I/<60%+ +( $ 5 I/<60 $ $ $ >1 %+@@?( 7 H % !!!( $ 6 : % !! ( $ $H $ $ $ $ $ I/<60 0 $0 P %+@@@( $ $ * $ , H % !!!( $ $ # $ * 7 %+@@+( % ?( % +( 0 % @( %?!( " % % @( " % % %?!( $ %?+( & H $ ' $ % %?+( $ I9< I/<60 ' H " A 5? $ $ $ $ $ 0 $ $ ' $ * $ * 7 $ 6<"& * 7 %+@@ ( ( * 7 * $ +@5 @A % " " +@+@ @@ 0 %+@@ ( " $7 $ + ;" $ - $ $ . < $ 1 $ 1 $ 7 / " %+@@ ( %+@+@ G!( 6 " H 6 6 $ * H 6 * %+@G! @@( $ $ ' * I $ @ " 9 > @ Q %+@G5( 5A ' 2 $H + $ , $ ,- ,' %+@@ ( $. & */ ;" , , ,, / 6<"& $ $ $ $ / " 0 7 $ " $ " 6 H $ @@ $ ;" * +@G! " $ $ % 6 H " $( B * ) % ' ! !!G+? ! !!++) ! !!?+E ! !!A)) ! !A@+ ! !!+! ! !AE! ! !! + % R; '# 1 & & +C 0 I/<60%+ +( + @5 @@ ;" " $ G $ " ;" " $ / ? I/<60 >1 H %+@@+( 55 H I/<60 $ " . 1% 02) 13 * $ 0 1% 02 ' " *1% 02 " H , %+@@ ( Q ' & * $ 8 B I/<60B 8I/<60B I/<60 ( () I9< I/<60 ( () 8I/<60 ( () R $ / $ $ ? 7 I/<60 $ I9< I/<60 I9< I/<60 * " 7 $ $ $ $ ;" ;" $ 5E $ ;" +! $ @+ 8I/<60 , $ $ = " $ $ $ I9< I/<60 0 * $ * $ '# " , $ ;" * $ 7 $ 7 $ * ;" H % !!!( ! !!? " " $ # 8 ;" I/<60 ' ' %+@)?( < " K ? AE A! I$ " 6 +5 ;" / AE $ % !!5( : K ? $ I9< I/<60 ' $ $ $ 8I/<60 ' * $ I/<60 $ $ $++ " I/<60 ' J $ / $ $ +! ++ = %2 $ $ & !% +@@@( IS<60 ! !!+( $ $ $ 5) ! 1% 02 ") 13 8 $ %+@@ ( Q * " * 1% 02 ") ' *1% 02 " " H ' , & $ 8 8 $ & " B I/<60B 8I/<60B I/<60 ' ( ( () I9< I/<60 ' () 8I/<60 ' ( ( ( ( " I/<60 ' I9< I/<60 ' ;" K 8 $I $ 2 < 9 %+@G)( " < * %+@@?( ;" 7 5G $ - 7 . * 8I/<60 ' $ $ /<%+( $ $ * A! $ $ * $ 8I/<60 ' 0 8I/<60 ' ; + E!! I/<60 ' I9< I/<60 ' 5 $ $ * * ? $ $ $ H $ $ 0 $ 1 H$ A! * K " $ $ - . @! @GC ! !!!5 ! !!E ; # $ 7 $ $ $ $ H 7 % $ : $ %+?( @@ C $ L+ ( A! 5@ $ H K $ % !!!( + !!! !!! # , $ $ K 5! !!! $ $ = $ + 1 Optimal allocation to stocks when initial chock is varied, with EGARCH(1,1), A=5 100 h =.0010 t ht=.0011 ht=.0012 Precentage allocation to stocks 95 ht=.0013 ht=.0014 90 h =.0015 t 85 80 75 70 7 0 5 10 15 20 25 Investment horizon in months B 30 35 $ & $ $ ;" -# 0 " " , $ $ $ + / $ $ 0 $ %+@@)( I/<60 E! 7 8I/<60%+ +( $ & ;" * " $ $ ' $ %+@E@( " %+@E@( 7 $ !!+5 !!+A $ AC !!++ , $ $ !!+ $ ' %+@)?( $ )C D 7 * % !!!( : $ $ $ 6 $ * $ $ * $ $ F # $ ' $ $ $ 0 $ $ $ $ $ 1 $ $ $ -# K H N $ " , $ $ $ $ $ $ $ $ 7 $ * I/<60 ' $ $ $ " ;" 7 $ ? * I9< I/<60 ' $ E+ 5C ' $ I/<60 ' $ $ 0 $ " * $ @$ $ Optimal allocation to stocks with RRA, A=6 90 80 % alloc ated to s toc k s GJR-GARCH-M 70 60 GARCH-M 50 40 30 0 7 5 10 15 20 25 30 35 Investment horizon in months ?B I ' 40 I/<60 ' ;" $ !!+G I/<60 ' $ I9< I/<60 ' +@5 @A I9< I/<60 ' ) !! $ $ I9< I/<60 ' 7 7 ? % ?$ ( $ $ , $ %+ C( I/<60 ' $ 45 $ I9< I/<60 ' $ $ $ $ $ $ H * $ B $ - E . " 1 * 0 * 6 6 -#! ' % !!+( $ $ $ I9< I/<60 ' $ K $ $ $ $ B I9< I/<60 ' " % % $ $ ( 7 ( $ $ ! @5 A / $ , !A $ $ * $ / !G $ $ $ $ $ $ $ $ $ $ $ $ 0 - $ . $ , $ $ $ $ $ $ $ $ $ $ K % * $ I/<60 $ K I9< I/<60%+ +( 7 ? @+G5 $ $ $ ( + $ K K $ * ++E?E " $ ++E?E E? The effect of varying persistence and asymmetry in GJR-GARCH-M 52 % allocated to stocks 50 48 46 44 42 40 38 0.8 0.7 35 30 25 0.6 20 15 0.5 10 0.4 β, β+γ=0.95 7 AB 5 0 Investment horizon in months ? $ I9< I/<60 ' A@ " $ @5 I/<60 0 $ ;" $ - . $ + $!+ $ $ 7 5 " $ $ * $ -1 , K * . * K E $ $ $ 6 K : %+@@@( EA Optimal allocation to stocks with negative mean effect in GJR-GARCH(1,1)-M 100 95 % allocation to stocks 90 γ = - 0.11636 85 80 75 70 γ = 0.11636 65 60 55 50 0 5 7 5B " % ? @+G5( 10 15 20 Investment horizon in months $ $ 25 30 % ! ++E?E() $ " % ? @+G5( 7 E K $ @ ? 1 $ E?C I9< I/<60%+ +( ' E 7 E K 7 ? $ $ * %+@@@( / * : 6 K @ 7 * * @ K * $ E $ '$ $ $ $ $ $ $ 8 E5 The effect of varying A, The Coefficient of Relative Risk Aversion 80 78 % allocated to stocks 76 74 72 70 68 66 64 62 9 8 7 40 6 30 5 20 4 10 3 2 Relative risk aversion parameter A 7 EB Investment horizon in months ? K I/<60 ' / 7 ;" <</ " $ I9< $ $ * $ $ $ 7 $ $ 7 4 $ $ $ 0 0 $ $ $ $ $ $ . 0 I/<60 $ $ * $ " $" %+@E@( EE ' %+@E@( 0 $ $ , I/<60 ' $ $ $ $ $ * I/<60 $ $ % ( $ / * $ 1 $ +N C +C $ + C $ '$ $ $ $ $ $ 7 $ $ $ $ K $ $ $ K $ $ $ $ $ 0 $ $ $ $ , $ K ' $ K $ $ $ 7 $$ / $ $ , $ E , / $ K $ K K @ $ E) $ 6 $ $ $ $ $ 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" MM ?< 3 : ) < ! < 6 < # = << ( * & () :. : : / : * %3 6 + B 5 ) % 6 + MM 2 ) ! ( 4 * ) %+ 4 6 , -!. :/ 3 N 0 5+ '& * & ! 0 6 + 5 ) ) / . ! ( ) . " : '' 2 " 7 ! 5 =5 , . !: / (6 3 21 ? 0 3 2 * Essay III Can Parameter Uncertainty Help Solve the Home Bias Puzzle? Published in Ekonomia volume 6 issue 2, 2003 http://ideas.repec.org/a/ekn/ekonom/v6y2003i2p187-209.html