Limited Downside Risk in Portfolio Selection among U.S. and Pacific Basin Equities

Transcription

1 October 2000 Limited Downside Risk in Portfolio Selection among U.S. and Pacific Basin Equities Dennis W. JANSEN Texas A&M University and P.E.R.C. Abstract In this paper we demonstrate safety first portfolio selection using extreme value theory. We show that Roy s safety first criterion can be improved on by exploiting the fat tail property of asset returns. Using daily data for a set of international stock indices for the period 1986-May 2000, we calculate the so-called tail indexes, which are accurate measures of the fat-tailedness of the stock return distributions, and use these to calculate minimum threshold return levels given very low exceedence probabilities for investors. This example is but one way that the theory of extremes can be utilized in economics and finance.. *I thank the International Journal of Economics for the invitation, and the Texas ARP and the Private Enterprise Research Center at Texas A&M University for research support. I also acknowledge an intellectual debt to my collaborators on other applications of extreme value methods to economics and finance, Casper de Vries and Kees Koedijk. Dennis W. Jansen Department of Economics 4228 TAMU Texas A&M University College Station, TX phone d-jansen@tamu.edu

2 I. INTRODUCTION Portfolio selection usually attempts to balance between expected return and risk through diversification. The common choice as an empirical measure of risk is the standard deviation of returns. But we show that when investors desire limited downside risk the portfolio selection process may differ somewhat from the traditional mean-variance formulation. This paper also provides an example of portfolio selection with limited downside risk, using portfolios of U.S. and various pacific basin nation stock indices, including Hong Kong, Japan, Korea, and Taiwan. Why might we be interested in limited donwside risk? First, even if agents are endowed with standard concave utility functions such that to a first order approximation they would be mean-variance optimizers, there may be constraints that elicit asymmetric treatment of upside potential and downside risk. For instance, regulators require commercial banks to report a single number, the so-called Value at Risk (VaR), which gives the expected loss on their trading portfolio if the lowest 1% quantile return would materialize. Capital adequacy is judged on the basis of the size of this expected loss. If inadequate, the bank has to lower its exposure or increase its capital. Risk management therefore has an incentive to steer clear from this lower boundary. Likewise, pension funds are often required by law to structure their investment portfolio such that the risk of underfunding is kept low. Second, agents may well treat losses and gains asymmetrically, see e.g. Kahneman, Knetsch and Thaler (1990). They argue that agents exhibit loss aversion, and if so there is good reason to consider models that try to incorporate this feature. Modeling limited downside risk has often been identified with the safety first criterion developed by Roy (1952) 1. Applications of Roy s safety-first criterion were traditionally regarded as uninteresting for two reasons. First, the safety first criterion was originally stated in terms of securing a minimal return level with a high probability, and the criterion was made operational through the use of the Chebyshev bound. But for minimal return levels below the riskless rate, the portfolio problem under the safety first criterion is degenerate; see e.g. Levy and Sarnat (1972). Arzac and Bawa (1977) moreover, noted that the original criterion fails to order risky assets which are unambiguously ordered by the principle of absolute preference. But Arzac and Bawa considered a lexicographic form of the 1 Bernstein (1992) has an interesting account of Roy s work on safety-first at the time of Markowitz s work on mean-variance.

3 2 safety first principle, in which the investor is only concerned about safety when the failure probability is above a critical level. Otherwise investors will maximize expected return. With this feature, and by allowing borrowing and lending, Arzac and Bawa were able to resolve the theoretical shortcomings of the original criterion. The second concern with the safety-first choice criterion has been the calculation of the failure probability. If the distribution of returns is unknown, as is almost always the case in practice, then there is need to calculate the failure probability without relying on a distributional assumption. The literature has traditionally proceeded by using the Chebyshev bound. But this bound, while robust, it highly inaccurate. Our task, then, is to demonstrate that we can improve on the Chebyshev bound for limited downside risk portfolio selection. We employ a semi-parametric method for modelling the tail behavior of an (unknown) distribution, a method taken from extreme value theory in statistics. The setup of this paper is as follows. In section 2 we outline the basics of safety-first portfolio selection, and section 3 outlines some material from statistical extreme value theory. In section 4 we use the latter theory to estimate tail indices for portfolios of stocks, and in section 5 these estimates for tail fatness are used to construct portfolios with limited downside risk. Section 6 contains our conclusion. II. SAFETY FIRST THEORY AND EXTREME VALUE THEORY Roy s (1952) safety first criterion was extended by Arzac and Bawa (1977) to allow lexicographic ordering and borrowing and lending. Here we briefly summarize Arzac and Bawa s analysis. Investors (lexicographic) preference ordering is represented as (B,:), where B=1 if the probability P of a large negative return is less than a specified critical value *, P<*, and otherwise B=1-P. The critical loss value * will be called the risk probability, RP. For a given B, the investor then maximizes the expected return, :. We can state the portfolio problem as:

4 3 (1) (2) (3) where V j denotes the initial market value of asset j, X j the final market value of asset j, W the initial wealth of the investor, b the amount of borrowing (with negative amounts indicating lending), r the risk free gross rate of return, and ( j the amount of risky asset j in the portfolio. Note that a safety first investor must specify both the disaster level of wealth, s, and the maximal acceptable probability of this disaster, *, which we refer to as the risk level. The above problem can be restated in terms of gross returns R, R=X/V, by rewriting the safety first condition as (4) It is useful in this case to define a value q * (R) that such that there is an * percent chance of returns less than or equal to this value. (5) The negative of this quantile q * can be considered as a measure of the Value at Risk, or VaR, as that term is used in current practice. Note that the safety first criterion is violated whenever

5 4 (6) A safety first investor will exhibit risk aversion if the critical wealth level s is smaller than his secure final wealth Wr, an assumption that seem reasonable in application. For example, s could be stated as a fraction of wealth, say.8w or.9w, in which case Wr most certainly exceeds s. In this case, a safety first investor will decline a fair risk that violates the safety first criterion in favor of pure lending at the risk free rate r. Risk-averse safety first investors will always buy some part of a divisible favorable risk, the maximum amount satisfying (W+b)q * (R) - br = s, or (7) excess returns: With favorable assets available, the portfolio problem can be rewritten in a more familiar form as the maximization of the max : = Wr + (W+b)( G R-r) (8) ( j,b where the maximization is carried out among those portfolios that satisfy the safety first criterion; that is, among those portfolios that have a probability of 1-* or greater of maintaining a value in excess of s. By combining (7) and (8) this problem reduces to max : = Wr - (s-wr)( G R-r)/(r-q * (R)) (9) ( j,b This implies that the risk averse safety first investor can first maximize the ratio of the risk premium to the return opportunity loss that he is willing to incur with probability *, i.e. max ( G R-r)/(r-q * (R) (10) ( j

6 5 and then the investor can pick the scale of the risky part of his portfolio. A natural question to ask is when and how the portfolio selection under safety first will differ from such traditional methods as mean-variance selection procedures. The parameter * and the portfolio distribution determines the VaR, q * (R). The VaR is a measure of risk that may be preferred to the second moment because it is only based on large negative returns and it includes situations in which the second moment is not sufficient to measure the probability of a large negative shock. There are some cases in which the safety first problem can be restated as an equivalent mean-variance problem. Arzac and Bawa themselves noted that if q * (R) could be written as R G - g(*)h((,1), where 1 are the parameters of the distribution, then safety first reverts to mean variance. This happens, for example, when the distribution of returns is normal, student t, or stable with common characteristic exponent in the interval (1,2). It also happens when q * (R) is derived using Chebyshev's inequality. Thus if returns belong to certain classes of distributions, there are similarities between safety first and mean variance portfolio selection. In both cases h((,1) equals the dispersion of the portfolio. But we have little reason to believe that all asset returns are generated by, say, student-t distributions with the same degrees of freedom. If the distribution of the returns is known, then the two-part optimization can be relatively straightforward. But in practice the return distribution is almost never known, so q * (R) has to be estimated. Roy (1952) initially proposed using the Chebyshev inequality, and the subsequent literature has followed up on this suggestion. But we will show an alternative procedure that yields much sharper bounds. III. EXTREME VALUE THEORY AND ESTIMATION OF EXCEEDENCE VALUES 2 Consider a stationary sequence X 1,X 2,... of i.i.d. random variables with distribution function F(). We are interested in finding the probability that the maximum M n = max(x 1,...,X n ) of the first n random variables is below a certain value x. This probability is given by P(M n <x) = F n (x). Extreme value theory studies the limiting distribution of the order statistic M n. It can be shown that the 2 This material draws heavily from Jansen, Koedijk, and de Vries (forthcoming).

7 6 distribution function F n (x) of M n converges when suitably normalized and for large n to a limiting distribution G(x), where G(x) is one of three asymptotic distributions. Thus, the tail behavior of distributions can be characterized by something analogous to the central limit theorem, in that various distributions behave similarly in the tails, at least in the limit. Moreover, even though there are three possible limiting distributions, one holds for distributions with finite support and can be excluded on a priori grounds when looking at stock returns 3. As for the remaining two, one is characterized by the existence of all moments, while the second is characterized by the absence of higher moments. Because stock returns are fat tailed we consider the limiting distribution G(x) which is characterized by a lack of some higher moments. This limiting distribution was given in equation (12) above, and rewritten here for a suitably normalized x as: G(x) = 0, if x<0 (13) = exp(-x) -1/( = exp(-x) -", if x>0. Here (>0, and " is known as the tail index. An introductory reference to this result is Mood, Graybill and Boes (1974), and Leadbetter, Lindgren and Rootzen (1983) provide a more comprehensive treatment. Leadbetter et al. also demonstrate that this result holds if the assumption of independence is violated, provided the dependence is not too strong. What type of distributions are included in G(x)? The student t with finite degrees of freedom, the stable distribution, and the ARCH process are all included. For the student t, the tail index " is the degrees of freedom. Note that the normal distribution is a thin tailed distribution with all moments existing, and hence the tail behavior of the normal distribution is not characterized by the G(x) given above. Instead, the normal distribution would belong to the other limit distribution valid for distributions of unbounded random variables that have all moments existing. 3 With returns measured as log differences of index levels, both negative and positive returns are in principle unbounded.

8 7 In our application, we assume we are dealing with a distribution that converges to the distribution G(x) given above. That is, we assume we are dealing with a distribution that is fat tailed and regularly varying at infinity. We estimate ", which provides information on the tail shape of the limiting distribution and also indicates the number of moments that exist. This is interesting in itself, as it provides a way to answer the question of how many finite moments exist in the distribution of stock returns without committing to a specific distribution. Second, the estimate of " allows us to calculate exceedence values, values of x that will only be exceeded with a stated probability. This is especially valuable because we can calculate exceedence values for probabilities that are much lower than 1/n of our sample!. That is, because we are relying on the limit distribution in the tail, we can sensibly extrapolate the tail far outside the sample experience to make statements about the exceedence values corresponding to very small probabilities. Or, turning the problem around, we can state a large value x, even one far outside the sample experience, and calculate the probability of its occurrence. To estimate the tail index " we use Hill's (1975) moment estimator. We first calculate the order statistics X (n),x (n-1),...,x (1) for our random variables, where X (n) > X (n-1) > X (n-2), etc. Then the Hill estimator is: (14) where m is the number of upper order statistics included and n is again the total number of observations. This estimator may be biased in small samples, depending on the underlying distribution, but Mason (1982) proves that under some regularity conditions the estimator is consistent for (. Furthermore, Goldie and Smith (1987) show that (1/" - 1/")m.5 is asymptotically normal N(0,1/" 2 ). Danielsson et al (1994) provide monte carlo evidence of the size of the bias in estimates of ". Note that the Hill estimator can be applied to either tail of a distribution merely by calculating order statistics from the opposite tail, either by ordering from lowest to highest or by multiplying the original data by -1 and applying the above procedure directly. We can test for the equivalence of " in the two tails and if desirable we can combine the tail observations (by taking absolute values) to estimate a common ". One issue in using the Hill estimator is the choice of order statistics, m. It is known that m should be chosen so that m(n) goes to infinity with n, but that m/n remains finite. Recently Danielsson and de Vries (1997) and Danielsson, de Haan, Peng, and de Vries

9 8 (1999) have provided methods for choosing the number of order statistics to use in estimation by a procedure that builds on the original Hall (1990) bootstrap procedure by taking advantage of different rates of convergence of alternative estimators for the tail index. After calculating " we can turn to estimating the quantiles q p. Based on a corollary to Dekkers et al. (1989) given in the appendix of de Haan et al. (1994), we have: (15) This formula can be used to estimate quantiles q p that will only be exceeded with probability p. Alternatively, it can also be rearranged to estimate the probabilities p corresponding to a given quantile q p. For the quantile estimation we also have that (16) is asymptotically N(0, 1/" 2 ). IV. DATA ANALYSIS - Tail Estimates for Stocks Turning now to the analysis of the data, we consider the problem of choosing between investing in a mutual fund of international stocks. We take the viewpoint of a U.S. investor, and employ daily data from January 1986 through the end of May or beginning of June Our sample consists of 3762 daily observations on stock index returns from the United States, Hong Kong, Japan, Korea, and Taiwan. The data is described in more detail in Table 1. Table 2 presents summary statistics for our sample of stock returns. This table includes summary statistics on the index returns in both local currency terms and in U.S. dollar terms, although we will later focus on the U.S. dollar returns. The stock returns

10 9 vary from a low of about 1.9% per year on the Japanese index in local currency, to a high of 19.9% per year on the Taiwanese index in dollar terms. The average U.S. index return was about 15.2% per year over this period. The standard deviation of stock returns varies somewhat across countries, with the lowest being on the Japanese index in local currency terms, and the highest being on the Taiwanese index in dollar terms. Looking only at dollar returns, the Japanese index had a slightly lower standard deviation than the U.S. index, and Taiwan of course had the largest. Table 2 also reports information on the excess kurtosis and skewness of the distribution of returns, and on the range of values during this time period. We do not report any tests for normality, but the Jarque-Bera test strongly rejects normality. The kurtosis in stock returns for some of the countries, especially Hong Kong and the U.S., gives strong evidence of the fat-tailed property so often noted in the distribution of returns. Clearly there is little reason to think that these stock returns are normally distributed.. We next estimated the tail indices for our various return series. In Table 3 we report estimates of the tail index for several methods of choosing the number of order statistics. The first grouping of results uses the subsample bootstrap method suggested by Danielsson, de Haan, Peng, and de Vries. There is a wide range of values for m, the number of order statistics included in the estimation. In terms of the tail index, values range from a low of for Hong Kong to a high of for Korea. This indicates that the estimated tail thickness is highest for Hong Kong, lowest for Korea. We also report estimated two-standard error confidence bands for estimates of the tail index. The second and third grouping of results in Table 3 reports estimates of the tail index when we arbitrarily fix the number of order statistics at a small fraction of the sample, either 0.25% or 0.5%, respectively. For these choices for m, the tail estimates vary a bit from those reported in the top grouping, although overall the estimates are not hugely different across the three sets of estimates. Only in the case of Hong Kong and the U.S. does an estimate reported in the second and third set of estimates not lie within the two standard error confidence band reported for the first group of estimators. An interesting feature of Table 3 is that it provides information on the existence of the second moment of stock returns. It the tail index is greater than two, then the second moment exists, while if the tail index is less than two then the second moment does

11 10 not exist. For our first group of estimators we find that all our estimates of the tail index are more than two standard deviations from two, although this is not alway true of the second and third grouping of estimates. In Table 4 we have in the rightmost three columns our calculations of the exceedence values corresponding to various probabilities. The first of these columns is headed p=1/4000, to indicate that this calculation applies when the probability is such that we would expect about one occurrence in our sample. Looking at the first row, Hong Kong, we see that the estimated exceedence value in the lower tail is -.25 or a -25% return, with a confidence interval of -24% to -26%. Thus we estimate that there is a 1 in 4000 chance that we will experience a daily return of -25% or less. For Japan, Korea, and Taiwan, we estimate the exceedence value for a 1 in 4000 probability as basically in the -11% to -13% range, which is similar to their in-sample experience. In fact, the two countries whose exceedence levels are not in good accord with their in-sample experience are Hong Kong (-26% exceedence estimate versus -40%) and the U.S. (-11% exceedence estimate versus -22%). In the second grouping of estimates in Table 4, we report estimated exceedence values when we arbitrarily set the number of order statistics to nine, which is 0.25% of the sample size. In this case both Hong Kong and the U.S. have exceedence estimates that are closer to their in-sample experience, although still neither has an exceedence estimate that equals the in-sample number or even falls within a two-standard error distance from the in-sample value. One interpretation is that these countries experienced in-sample realizations of returns during the period that are much rarer than their occurrence would otherwise indicate. In fact, it is only when we specify a probabily of 1 in 8000 (or one in 30 years) that our confidence bands for both Hong Kong and the U.S. begin to include realizations like that which occured once in each country in our sample. In any case, it seems clear that, in terms of the chance of a one-day large negative return, both Korea and Taiwan are safer stock markets for a U.S. investor than are Hong Kong and, perhaps, Japan. V. DATA ANALYSIS - Portfolio Selection and Exceedence Estimates

12 11 In order to illustrate safety first portfolio selection, we proceed to construct hypothetical portfolios consisting of linear combinations of our stock indices. We consider four portfolios, each with the U.S. index and the index of one of the other four countries. For each pair, we vary the fraction of non-u.s.stocks from 0% to 100% by steps of size 10%. This gives us 11 portfolios, and for each we calculate the exceedence probabilities and the safety-first portfolio selection criteria. Table 5 reports the portfolio selection information for the U.S. - Japanese portfolio. Here we calculated the exceedence estimates based on our estimates of the tail index reported in Table 4 when we used m=9. Since convolutions of random variables should exhibit the tail behavior of the distribution with the thickest tail, we use the tail index from the thickest -tailed distribution in the portfolio for all cases in which the portfolio was not at 100% in any one index. The top half of Table 5 reports our results when we set the probability at 1/4000 when calculating the exceedence levels, while the bottom half of the table reports our results when we set the probability at 1/8000. Based on estimates of " and estimated exceedence values for our various portfolio combinations (which are listed in Table 5), and based on calculations of average returns for our various portfolio combinations, we can calculate the safety-first porfolio selection criterion. Recall that a safety first investor must specify the probability * that wealth will not decline to the critical level or lower. In our application, we are taking this * to be either 1/4000 or 1/8000, the values used in calculating the exceedence levels. Other choices for * would involve recalculating the exceedence levels. The first stage of the safety-first portfolio choice problem is to choose the portfolio composition that maximizes ( R-r)/(rq * (R)). We calculate this quantity for two values of r, the risk free rate. We use r =0 and r= The later value is a 3.5% annual G risk free rate of return. It turns out that the choice of risk-free rate is important. Using r=0, we find that the optimal portfolio for a safety-first U.S. investor is to hold 30% Japanese stocks and 70% U.S. stocks. Using r= , the optimal portfolio is to hold 10% Japanese stocks and 90% U.S. stocks. To see why this matters, consider a concrete example. Assume that the safety-first investor also specifies that s, the safety first wealth level, is equal to.80w, or s =.80W. With this knowledge, we can calculate the amount borrowed or lent at the risk free rate from (W+b)q * (R) - br = s. For r=1 and the optimal portfolio of 30% Japanese stocks, we have that b = (.80-q d (R))W/(q * (R)-r)

13 12 = W/( ) = W. If we normalize wealth to unity, W = 1, we have borrowing of an additional $.9212 to invest in a portfolio consisting of 30% Japanese stocks and 70% U.S. stocks. The mean return on this portfolio is (11.9%/year), so this investor earns a mean return of ( x ) = (24.1%/year). If disaster strikes, his wealth ends up as ( x.8959) , or.80, just as the safety first criterion demands. What is wrong with investing 100% in U.S. stocks? Clearly the mean return is higher than the mixed portfolio, (14.4%/year). But the safety first criterion severely limits the amount of wealth our investor can place in such a risky investment. With s =.8W (and r=1) we calculate that our investor will borrow b = (.80-q d (R))W/(q * (R)-r) = ( )W/( ) =.5186W, i.e. he will borrow.5186 for every dollar of wealth. The expected return on this portfolio is ( x ) = (22.6%/year), less than the return on the portfolio containing 30% Japanese stocks. If disaster strikes, the 100% stock portfolio will decline in value to ( x.8683) =.80, just as required by the safety first constraint. The conclusion, then, is that a leveraged portfolio containing 30% Japanese stocks and 70% U.S. stocks generates a superior rate of return given the constraint of holding the risk of a gross return of.80 to a probability. Tables 6, 7, and 8 report similar exercises for portfolios of U.S. stocks and Hong Kong, Korean, and Taiwanese stocks, respectively. For Hong Kong stocks, the optimal portfolio for a U.S. investor is either 20% Hong Kong stocks (when the probability of disaster is.00025) or 0% (when the probability of disaster is It appears that Hong Kong stocks are too risky, even though the returns on Hong Kong stocks have exeeded the average returns on U.S. stocks during the sample period. The rapid increase in the exceedence levels as the portfolio mix increases in favor of Hong Kong stocks is too rapid to be compensated for by the slower increase in the average rate of return. In Table 7, the optimal mix of U.S. and Korean stocks is an mix regardless of the two probability measures considered. In fact, the larger value for the tail index for the Korean stock index leads to a spike in the safety-first criterion when the portfolio mix turns to 100% Korean stock, because the Korean stock index returns have a much thinner tail than the U.S. index returns, leading to

14 13 a lower chance of a large negative one-day return. But this is not sufficient to overcome the values of diversifying, as the exceedence quantile is actually lowest for the optimal mix of stocks. In Table 8, the relatively thin tail of the Taiwanese stock market together with its higher average return over the sample period makes a portfolio of 100% Taiwanese stock the optimal choice for a U.S. investor! The second best choice is 40% Taiwanese and 60% U.S. stock. Again this result depends on the use of the Taiwanese tail index to calculate the exceedence values for the 100% Taiwanese portfolio while using the U.S. tail index to calculate the exceedence values for the mixed portfolios. So it only holds when an investor is willing to plunge into Taiwanese stocks. VI. CONCLUSION. In this paper we operationalize safety first portfolio selection using extreme value theory. We show that the conventional safety first criterion as developed by Roy can be successfully improved upon by exploiting the fat tail property of asset returns. Using daily data for stocks for the period 1986-May 2000, we calculate the so-called tail indexes, which are accurate measures of the fattailedness of the stock return distributions, and minimum threshold return levels given very low exceedence probabilities for investors. Finally, we point out that the portfolio selection problem for safety first investors is a very useful application of the theory of extremes, and one that may have very large practical applications. As an example, we illustrate how a safety-first investor might build a portfolio from two indexed mutual funds, one a fund of US equities and one a fund of foriegn equities. The extension to constructing portfolios among many financial instruments is straightforward but tedious.

15 14 REFERENCES Arzac, Enrique R. and Vijay S. Bawa, "Portfolio Choice and Equilibrium in Capital Markets with Safety-First Investors, Journal of Financial Economics, 4, 1977, Bernstein, Peter. Capital Ideas, (Free Press, New York), Danielsson, Jon, Dennis W. Jansen, and Casper G. de Vries, "The Method of Moments Ratio Estimator for the Tail Shape Parameter," Communications in Statistics, Theory and Methods, 25-4, Danielsson, Jon, L. de Haan, L. Peng, and Casper G. de Vries, Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation, manuscript, Erasmus University, August Danielsson, Jon and Casper G. de Vries, Beyond the Sample: Extreme Quantile and Probability Estimation, manuscript, Tinbergen Institute, December Dekkers, Arnold L.M., J.H.J. Einmahl and Laurens de Haan, "A Moment Estimator for the Index of an Extreme Value Distribution," The Annals of Statistics, 1989, Goldie, Charles M. and Richard L. Smith, "Slow Variation with Remainder: Theory and Applications," Quarterly Journal of Mathematics, 38, 1987, de Haan, Laurens, "Fighting the Arch-Enemy with Mathematics," Statistica Neerlandica, 1990, de Haan, Laurens, Dennis W. Jansen, Kees Koedijk, and Casper G. de Vries, "Safety First Portfolio Selection, Extreme Value Theory and Long Run Asset Risks." In Galombos, ed., Proceedings from a Conference on Extreme Value Theory and Applications, Kluwar Press, 1994, Hall, Peter, "Using the Bootstrap to Estimate Mean Squared Error and Select Smoothing Parameter in Nonparametric Problems," Journal of Multivariate Analysis, 1990, Hill, Bruce M., "A Simple General Approach to Inference About the Tail of a Distribution," The Annuals of Statistics, 1975, Jansen, Dennis and Casper G. de Vries, "On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective," The Review of Economics and Statistics, 1991, Jansen, Dennis, Kees G. Koedijk, and Casper G. de Vries, Portfolio Selection with Limited Downside Risk, Journal of Empirical Finance, 2000, forthcoming. Koedijk, Kees G., M.M.A. Schafgans and Casper G. de Vries, "The Tail Index of Exchange Rate Returns," Journal of International Economics, 1990,

16 Leadbetter, M.R., Georg Lindgren and Holger Rootzen, Extremes and Related Properties of Random Sequences and Processes, (Springer-Verlag, Berlin), Levy, H. and M. Sarnat, "Safety First--An Expected Utility Principle," Journal of Financial and Quantitative Analysis, 1972, Longin, Francois, "The Asymptotic Distribution of Extreme Stock Market Returns, Journal of Business, 1996, Loretan, Mico and P.C.B. Phillips, "Testing the Covariance Stationarity of Heavy-Tailed Time Series: An Overview of the Theory With Applications to Several Financial Datasets," Journal of Empirical Finance, 1, 1994, Markowitz, H.M., Portfolio Selection, (Wiley, New York), Mason, David M., "Laws of Large Numbers for Sums of Extreme Values," Annals of Probability, 1982, Mood, A.M., T.A. Graybill and D.C. Boes, Introduction to the Theory of Statistics, (McGraw Hill, New York), Pagan, A.R., G.W. Schwert, "Alternative Models for Conditional Stock Volatility," Journal of Econometrics, 1990, Roy, A.D., "Safety First and the Holding of Assets," Econometrica, 1952,

17 Table 1 List of Data Used Country Equity Index Other Data begin date end date Hong Kong Hang-Seng Index exchange rate January 02/1986 June 05/2000 Japan Nikkei 255 Stock Index exchange rate January 02/1986 June 05/2000 Korea Taiwan KOSPI (Korea SE Composite) Taiwan SE Weighted Index exchange rate January 02/1986 May 31/2000 exchange rate January 02/1986 June 05/2000 United States S&P January 02/1986 June 05/2000 Data source in all cases is the DataStream electronic database

18 Table 2: Summary Statistics Mean Return (x1000) Std. Deviation (x10) Kurtosis Skewness Minimum Maximum Number of Observations HangSeng HangSeng ($) Kospi Kospi ($) Nikkei Nikkei255 ($) Taiwan Taiwan ($) US-SP

19 Table 3 Calculation of the Tail Index Parameter k* ( " X n-(k*+1) ( (+/- 2 s.e.) " (+/- 2 s.e.) Hong Kong , , Japan , , Korea , , Taiwan , , U.S , , Hong Kong , , Japan , , Korea , , Taiwan , , U.S , , Hong Kong , , Japan , , Korea , , Taiwan , , U.S , , 4.547

20 Table 4 k " X n-k Q p (quantile estimates for given probabilities) p = 1/4000 p = 1/6000 p = 1/8000 Hong Kong (-.236,-.259) (-.277,-.302) (-.310,-.336) Japan (-.119,-.140) (-.132,-.155) (-.143,-.167) Korea (-.100,-.136) (-.102,-.147) (-.103,-.156) Taiwan (-.100,-.128) (-.105,-.137) (-.109,-.143) U.S (-.103,-.111) (-.117,-.126) (-.128,-.138) Hong Kong (-.206,-.335) (-.253,-.406) (-.295,-.464) Japan (-.108,-.172) (-.121,-.197) (-.132,-.216) Korea (-.103,-.149) (-.109,-.163) (-.114,-.174) Taiwan (-.096,-.136) (-.101,-.147) (-.105,-.156) U.S (-.100,-.163) (-.119,-.193) (-.135,-.217)

21 Table 5: Portfolio - Percent Japan Portfolio Selection: U.S. and Japan Q p (p=.00025) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% % % % % % % % ** % % ** 0% Portfolio - Percent Japan Q p (p= ) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% % % % % % % % ** % % **

22 0% Table 6 Portfolio - Percent Hong Kong Portfolio Selection: U.S. and Hong Kong Q p (p=.00025) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% % % % % % % % % **.2953** 10% % Portfolio - Percent Hong Kong Q p (p= ) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% % % % % % % %

23 20% % % **.2189** Table 7 Portfolio - Percent Korea Portfolio Selection: U.S. and Korea Q p (p=.00025) Mean Return in Sample (x1000) r=0; (values x 100) 100% % % % % % % % % **.3590** 10% % r= ; (values x 100) Portfolio - Percent Hong Kong Q p (p= ) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% % % % % % %

24 30% % **.2689** 10% % Table 8 Portfolio - Percent Taiwan Portfolio Selection: U.S. and Taiwan Q p (p=.00025) Mean Return in Sample (x1000) r=0; (values x 100) 100% **.4893** 90% % % % % % **.4152** 30% % % % r= ; (values x 100) Portfolio - Percent Taiwan Q p (p= ) Mean Return in Sample (x1000) r=0; (values x 100) r= ; (values x 100) 100% **.4360** 90% % % % %

25 40% **.3110** 30% % % %