A COMPARISON AMONG PERFORMANCE MEASURES IN PORTFOLIO THEORY

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1 A COMPARISON AMONG PERFORMANCE MEASURES IN PORFOLIO HEORY Sergio Ortobelli * Almira Biglova ** Stoyan Stoyanov *** Svetlozar Rachev **** Frank Fabozzi * University of Bergamo Italy ** University of Karlsruhe Germany *** FinAnalytica Inc. and University of Sofia Bulgaria **** University of Karlsruhe Germany and University of California Santa Barbara Yale University Connecticut Abstract: his aer examines some erformance measures to be considered as an alternative of the Share Ratio. More secifically we analyze allocation roblems taking into consideration ortfolio selection models based on different erformance ratios. For each allocation roblem we comare the maximum exected utility observing all the ortfolio selection aroaches roosed here. We also discuss an ex-ost multi-eriod ortfolio selection analysis in order to describe and comare the samle ath of the final wealth rocesses. Coyright 005 IFAC Keywords: Performance ratios heavy tails ortfolio choice risk measure exected utility efficient frontier.. INRODUCION More than thirty five years ago Share (966 introduced the so called Share Ratio a erformance measure for mutual funds that is justified by the classic Markowitz mean-variance analysis. Leaving behind the assumtion of normality in return distributions the classic Share erformance measure has become a uestionable tool for ranking ortfolio choices. As a matter of fact the many shortcomings and ambiguous results of the emirical and theoretical mean-variance analysis reresent the main reason and justification for the creation of alternative erformance measures such as those roosed in the last decade (see Goetzmann Siegel Svetlozar Rachev's research was suorted by grants from Division of Mathematical Life and Physical Sciences College of Letters and Science University of California Santa Barbara and the Deutschen Forschungsgemeinschaft. Sergio Ortobelli's research has been artially suorted under Murst 40% 60% and CNR- MIUR-Legge 95/95. Ingersoll Welch (003 Farinelli ibiletti (003 Dowd (00 Sortino (000 Pedersen and Satchell (00. In the sirit of these recent researches we want to consider more general risk-reward ratios best suited to comare skewed and heavy tailed return distributions with resect to a benchmark. In the sirit of these recent researches we want to consider more general risk-reward ratios best suited to comare skewed and heavy tailed return distributions with resect to a benchmark. In view of this consideration we introduce and discuss several erformance measures. In articular we comare the classic Share ratio with other ratios roosed in literature: minimax ratio (Young (998 Stable ratio (Ortobelli Rachev Schwartz (003 MAD ratio (Konno and Yamazaki (99 Farinelli-ibiletti ratio (Farinelliibiletti (003 Sortino-Satchell ratio (Sortino (000 Pedersen Satchell (00 Cologtye ratios (Giacometti Ortobelli (00 VaR and CVaR ratios (Favre and Galeano (00 and Martin Rachev Siboulet (003 Rachev-tye ratios (see

2 Biglova Ortobelli Rachev Stoyanov (004 and Ortobelli Rachev Biglova Stoyanov and Fabozzi (004. First we roose an ex-ante static comarison among ortfolio selection models based on different erformance measures. In articular we comare the exected utility of investors when the market ortfolio is comuted by maximizing a given erformance measure. We analyze two allocation roblems for investors with different risk aversion coefficients. We determine the efficient frontiers generated by linear combinations of the market ortfolio and the riskless asset. Each investor characterized by his/her utility function will refer the ortfolio which maximizes his/her exected utility on the efficient frontier. Hence the ortfolios obtained with this methodology reresent the otimal investors' choices in each distinct case. Second we roose an ex-ost dynamic analysis as another aroach to erformance comarison. Here we comare the final wealth of investors who maximize the exected logarithmic utility function under several ortfolio selection models. Section introduces the erformance ratios. Section 3 rooses a comarison among the different models. In the last section we briefly summarize the results.. PERFORMANCE RAIOS his section describes the different erformance ratios examined in ortfolio selection roblems. Particularly we analyze the roblem of otimal allocation among n assets: n of those assets are risky with returns (continuously comounded z = [ z... z n ]' and the (nth asset is risk-free with return z 0. No short selling is allowed i.e. the ortfolio risky weights xi [0] for every...n n and the riskless weight = xi are eual to or greater than zero. Assume that all ortfolios are uniuely determined by the mean and by a risk measure consistent with some stochastic dominance order. he investor will choose an otimal ortfolio which is the linear combination of the riskless asset and an otimal risky ortfolio. he otimal risky ortfolio is given by the ortfolio that maximizes the erformance ratio. hus for any erformance measure H we comute a "market ortfolio" x M ' z that is the solution of the following otimization roblem max ( x n xi = ( xi 0 For different erformance measures we obtain different otimal ortfolios. herefore the market ortfolio comosition xm ' = [ xm... xm n]' found for each erformance measure H is based on a diverse risk ercetion. In articular we consider the following erformance measures a Share ratio (see Share (966 (994 SD( x ' z where SD(xJz is the standard deviation of the ortfolio xjz. b Minimax ratio (see Young (998 MM ( x ' z where MM ( x ' z = z0 min x ' zt and zt is the vector t of returns at time t. c Stable ratio (see Ortobelli Rachev Schwartz (003 where = xqx ' Q = [ i j ] is the disersion matrix of the vector z that we assume to be K-stable sub-gaussian distributed. he elements of Q are defined for every [K ( / ( ij= A f( zi zj < > i j i j ; where ( ( j f z z = E z z E z A( = ; z i = z i µ i ; < > j = z sgn( z z ; the otimal is j j determined as in Lamantia Ortobelli Rachev (004 and K is the index of stability that we assume being the mean of indexes of stability. d MAD ratio (see Konno and Yamazaki (99 N where = Exz ( ' and z N k = oints out the k-th observation of vector z. e Gini ratio (see Yitzhaki (98: n where = xi( zit zik ( t> k f Farinelli-ibiletti ratio (See Farinelli-ibiletti 003 (( ' (( ' E x z t E x z t where E( ( x' z t = ( x' z t ( ' xz t ( max( t 0 / = and

3 / (( ' = ( ' E x z t x z t ( ' x z t ( max( t x ' z 0 =. We use t = z0 t = z 0 / = = n i and = / n i = and also we use = / and = = n n i. i = g Sortino-Satchell ratio (see Sortino F. (000 Pedersen C. Satchell S. E. (00 ( t where ( t = ( t x' z either with k = / ( t = E( ( t = ( t. We suose t = z 0 /. h Colog ratio (see Giacometti Ortobelli (00 VARxz ' where VARxz ' is the variance of x z. i Cologdsr ratio (see Giacometti Ortobelli (00 ( t where ( ( t = x' z t and t = z 0 /. k = j VaR ratio (see Favre and Galeano (00 VaR99% ( x ' z z0 where VaR99% ( x ' z is the oosite of % uantile imlicitly defined by Pxz ( ' < VaR99% ( = 0.0. k CVaR ratio (Martin Rachev Siboulet (003 Favre and Galeano (00 CVaR99% ( x ' z z0 where CVaR99% Exz ( ' / VaR99% (. l Rachev ratio (Biglova Ortobelli Rachev Stoyanov (004 and Ortobelli Rachev Biglova Stoyanov and Fabozzi (004 CVaR ( z0 x ' z CVaR ( x ' z z0 for some oortune K and M belonging to [0]. m Rachev generalized ratio (Biglova Ortobelli Rachev Stoyanov (004 and Ortobelli Rachev Biglova Stoyanov and Fabozzi (004 EL ( z0 x ' z EL ( x ' z z0 where EL ( X = E ( max( X 0 / X VaR. ( 3. AN EMPIRICAL COMPARISON In this section we roose two distinct tyes of comarison: an ex-ante comarison where we consider several ex-ante utility maximizers and an ex-ost comarison where we analyze the samle aths of final wealth obtained with the different aroaches. 3. An ex-ante comarison Once we determine the otimal market ortfolios we can comare the efficiency of alternative erformance measures from the oint of view of different decision making rocesses. In articular assuming that no short sales are allowed we examine the issue of otimal allocation among the riskless return LIBOR and n asset returns. In articular we consider daily returns in the eriod he first analysis aroximates the exected utility of the following utility functions ( r ur ( = with < ur ( = log( r cr ur ( = e withc> 0 where r = z0 ( x' M z and N is the allocation in the riskless asset. In order to comare the various models we use the algorithm roosed by Giacometti and Ortobelli (004. Considering i.i.d. observations z( i of the vector z( i = z( i z( i... zn( i ' the main stes of our comarison are the following Ste. Fit the efficient frontiers corresonding to the different market ortfolio xm = [ x M... xn M ]'. Ste. Choose a utility function u with a given coefficient of risk aversion. Ste. 3 Calculate for every efficient frontier max u( z0 ( xm ' z( i Ste. 4 Reeat stes and 3 for every utility function and for every risk aversion coefficient c and P. Finally we obtain tables which aroximate the maximum exected utility for the multilicative factor. We imlicitly assume the aroximation u( z0 ( xm ' z( i. Eu ( ( z0 ( xm ' z( i We know that too large or too small risk aversion coefficients imly that investors choose resectively either the riskless or the market ortfolio. herefore in order to obtain significant results we calibrate risk aversion coefficients so that the ortfolios which maximize the exected utility are otimal ortfolios in the segment of the efficient frontier considered. As a conseuence of this analysis it follows that the Rachev-tye ratios Farinelli-ibiletti ratio and the

4 Cologdsr ratio models often show a suerior erformance with resect to the classic Share ratio. In contrast the other aroaches do not seem to diverge significantly from the mean-variance one even though we observe that the otimal ortfolio weights are significantly different. his result imlicitly suorts the hyothesis that Rachev-tye ratios Farinelli-ibiletti ratio and the Cologdsr ratio cature the distributional behavior of the data (tyically the comonent of risk due to heavy tails better than the classic mean-variance model. 3. An ex-ost multi-eriod comarison Let us suose that investors with logarithmic utility function invest their wealth in the market assuming that the market ortfolio is determined by maximizing one of the above erformance measures. hen the investors will choose a convex combination between the riskless and the market ortfolio. We want to comare the samle ath and the final wealth obtained from the several aroaches. hus everyday and for each erformance measure H we have to solve two otimization roblems: the first in order to determine the market ortfolio and the second to determine the otimal exected utility on the efficient frontier. In articular everyday we calibrate the ortfolio using the last 50 observations. herefore without considering transaction costs and taxes we first determine the market ortfolio solving the otimization roblem (. hus after k eriods we get the market ortfolio comosition xm and the investors will choose the ortfolio that maximizes their exected utility given 49 k ( i by max log( z0 ( xm ' z( i 50 k ( i where z0 is the corresonding i-th observation of the LIBOR. In this way we get the otimal investment N [0] in the riskless and the vector comosition of risk assets ( x M after k eriods. herefore the final wealth at the k-th ste is given by (50 k Wk = Wk ( z0 ( xm ' z(50 k. his ex-ost multi-eriod analysis generally confirms the results of the revious analysis. In articular we observe that Rachev-tye ratios resent the best erformances during all the eriod considered. 4. CONCLUSIONS his aer rooses and comares alternative ortfolio selection models. In the first art we describe several erformance measures. Secifically we justify the imortance of some new ortfolio choice models because they consider the fundamental financial imact of the tail distribution. As it follows from the revious considerations the erformance ratios introduced here can be theoretically imroved and emirically tested. However a more general theoretical and emirical analysis with further discussion and comarisons will be the subject of future research. he emirical comarison confirms that the classic Share ratio resents less forecast abilities than other erformance measures roosed in literature. he Rachev-tye ratios resent better erformance for most decision makers among the alternative models roosed. In addition these erformance measures reveal a high degree of efficiency for large ortfolios. hus it is reasonable to believe that imlementing these ortfolio selection models for online calculation is a realistic issue. REFERENCES Artzner P. F. Delbaen J-M. Heath and D. Eber (999 Coherent Measures of Risk. Math. Finance Biglova A. Ortobelli S. Rachev S. Stoyanov S. (004. Different aroaches to risk estimation in ortfolio theory Journal of Portfolio Management Dowd K. (00 Share thinking Risk. Risk Management for investor secial reort June S-S4. Favre L. Galeano (00 Mean modified Value at Risk otimization with hedge funds Journal of Alternative Investment 5 Fall. Farinelli S. ibiletti L. (003 Share thinking with asymmetrical references. echnical Reort University of orino. Giacometti R. Ortobelli S. (004 Risk measures for asset allocation models Chater 6 in the Volume (eds. Szegò Risk measures for the st century Giacometti R. Ortobelli S. (00 A comarison among three disersion measures. echnical Reort University of Bergamo. Goetzmann W. Ingersoll J. Siegel M. Welch I. (003 Sharening Share Ratios. echnical Reort Yale School of Management. Konno H. Yamazaki H. (99 Mean-absolute deviation ortfolio otimization model and its alication to okyo stock market. Management Science Lamantia F. Ortobelli S. Rachev S.. (004 Value at Risk with stable distributed returns. echnical Reort University of Karlsruhe. Markowitz H. (959 Portfolio selection; efficient diversification of investment. New York: Wiley. Martin D. Rachev S. Siboulet F Phi-alha otimal ortfolios and Extreme Risk Management Wilmott Magazine of Finance November Ogryczak W. and Ruszczynski A. (998 On stochastic dominance and mean-semi-deviation models. Rutcor Research Reort Ortobelli S I. Huber S. Rachev and E. Schwartz (00 Portfolio choice theory with non- Gaussian distributed returns. Handbook of Heavy ailed Distributions in Finance. North Holland Handbooks of Finance Chater 4 (Series Editor W.. Ziemba.

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