Reward-to-Risk Ratios of Fund of Hedge Funds

Transcription

1 Reward-o-Risk Raios of Fund of Hedge Funds YIGIT ATILGAN Assisan Professor of Finance, Sabanci Universiy TURAN G. BALI Dean s Research Professor of Finance, Georgeown Universiy K. OZGUR DEMIRTAS Associae Professor of Finance, Baruch College, CUNY, and Sabanci Universiy ABSTRACT This chaper examines wheher he fund of hedge fund porfolios dominae he U.S. equiy and bond markes based on alernaive measures of reward-o-risk raios. Sandard deviaion is used o measure oal risk and boh nonparameric and parameric value-arisk is used o measure downside risk when he reward-o-risk raios are consruced. We find ha he fund of funds index has higher reward-o-risk raios compared o several sock and bond marke indices. This resul is especially srong when he risk measures are calculaed from he mos recen year s daa and is robus as he measuremen window is exended o four years.

2 . INTRODUCTION Invesors base heir porfolio asse allocaion decisions on he ineracions beween risks and reurns of available financial securiies. The assumpion of risk aversion implies ha securiies wih greaer risk should demand greaer reurn. Alhough he rade-off beween risk and reurn is well-esablished in financial economics, he abiliy o generae higher expeced reurns per uni risk can vary from one securiy o anoher. This chaper compares various reward-o-risk raios for he Fund of Hedge Fund (FoHF) index wih hose of several bond and sock marke indices. Tradiional risk measures used in porfolio performance measuremen assume ha reurns are normally disribued and herefore he sandard deviaion of he empirical reurn disribuion is a good esimae of risk only if he underlying reurn disribuion is close o normal. The firs measure of reward-o-risk ha we use is he Sharpe raio (966) which is equal o he raio of he mean excess reurn of a porfolio o is sandard deviaion. The Sharpe raio is he mos common measure of how well he reurn of a porfolio compensaes he invesor for he risk aken. However, a common criicism is ha i is oo broad since i includes he oal risk of a porfolio in is denominaor. Anoher poenial issue regarding he calculaion of Sharpe raios for he FoHF index is he non-normaliy of hedge fund reurn disribuions. The hedge fund lieraure provides evidence ha disribuions of hedge fund reurns end o deviae from normaliy. Malkiel and Saha (005) repor ha he disribuion of hedge

3 fund reurns generally have high kurosis and negaive skewness. The documened deviaion from normaliy can be raced o he unique invesmen sraegies ha hedge funds follow. Fung and Hsieh (997) observe ha hedge fund managers are flexible o choose among a diverse se of asse classes and hey can use dynamic rading sraegies ha involve shor sales, leverage and derivaives. Such sraegies have he poenial o induce opion-like payous and exposure o ail evens for hedge funds. In a follow-up sudy, Fung and Hsieh (00) focus on hedge funds ha use rend-following sraegies. They consruc several rend-following facors ha can replicae key feaures of hedge fund reurns such as skewness and posiive reurns during exreme marke movemens. Michell and Pulvino (00) invesigae merger arbirage sraegies and conjecure ha reurns o risk arbirage are relaed o marke reurns in a nonlinear way. Their resuls indicae ha merger risk arbirage is similar o wriing uncovered index pu opions. Agarwal and Naik (004) find ha nonlinear payoff srucures exis for a wide range of hedge fund sraegies including equiy-oriened posiions. They sae ha ignoring he downside risk of hedge funds can resul in significanly higher losses during large marke downurns. Brown, Gregoriou and Pascalau (009) look a he diversificaion effec of invesing in FoHFs and find ha he magniude of skewness is an increasing funcion of diversificaion offered by FoHFs. Their finding suggess ha downside risk exposure may no be diversifiable. Finally, Bali, Gokcan and Liang (007) and Liang and Park (007) provide direc evidence ha downside risk measures such as value-a-risk, expeced shorfall and ail risk can explain he cross-secion of hedge fund reurns. 3

4 Downside risk is a funcion of he higher order momens of a reurn disribuion and even wihou he exisence of nonlinear payoffs, higher order momens such as skewness and kurosis have been found o play an imporan role in asse pricing. The mean-variance porfolio heory of Markowiz (95) has been exended by Ardii (967) and Kraus and Lizenberger (976) o incorporae he effec of skewness. These sudies presen hreemomen asse pricing models wih invesors ha hold concave preferences and prefer posiive skewness. The main implicaion of hese models is ha asses ha increase a porfolio s skewness are more desirable and should command lower expeced reurns. Harvey and Siddique (000) exend hese uncondiional pricing models and incorporae condiional co-skewness. Again, he implicaion is ha risk-averse invesors prefer posiively skewed asses o negaively skewed asses. As far as he fourh-momen is concerned, Dimar (00) builds on he heoreical works of Kimball (993) and Pra and Zeckhauser (987) and finds preference for lower kurosis. Asse disribuions wih lower probabiliy mass in heir ails are preferred and herefore asses ha increase a porfolio s kurosis are less desirable and should command higher expeced reurns. Downside risk increases wih kurosis and decreases wih skewness (Cornish and Fisher (937)). Given he imporance of hese reurn momens for asse pricing and he prevalence of downside risk in hedge fund reurns, we place special emphasis on he concep of downside risk in our reward-o-risk analysis. To invesigae how much expeced reurn each index commands per uni of downside risk, we use boh a nonparameric and parameric measure of value-a-risk in he consrucion of he alernaive reward-o-risk raios. For he nonparameric VaRSharpe raio, he 4

5 denominaor is he absolue value of he minimum index reurn over various pas sample windows. For he parameric reward-o-downside risk measure (PVaRSharpe), he denominaor is based on he lower ail of Hansen s (994) skewed -densiy. The resuls indicae ha he FoHF index ouperforms he bond and sock marke indices based on radiional Sharpe raios on average. Alhough he Sharpe raios decrease for every index as he sampling window for he calculaion of sandard deviaion is exended and his decline is mos pronounced for he FoHF index, i has he highes Sharpe raio regardless of he sampling window. When we ake downside risk ino accoun hrough nonparameric and parameric value-a-risk, he resuls are similar. The FoHF index has higher downside risk-adjused Sharpe raios compared o all bond and sock marke indices and his resul is especially srong a shorer sampling windows for value-a-risk measuremen. The chaper is organized as follows. Secion discusses he mehodology for calculaing he reward-o-risk raios. Secion 3 explains he daa and presens he summary saisics. Secion 4 discusses he empirical resuls. Secion 5 concludes.. METHODOLOGY We esimae hree reward-o-risk raios ha differ from each oher based on he risk measure used in he denominaor. The firs of hese raios is he sandard Sharpe raio: 5

6 Sharpe i, Ri, R f, = () SDev i, where R i, denoes he monh reurn on he fund of funds, bond or sock marke index i and R f is he risk-free rae as measured by he -monh Treasury bill reurn. The sandard deviaion for index i is compued using he squared deviaions of monhly reurns from heir means. For each monh and index i, pas k monhs are used o compue he sandard deviaion where k akes he alernaive values of, 4, 36 or 48. Specifically, SDev i, = k k ( R j Ri ) i, j= 0 () In order o ake downside risk ino accoun, we firs use a nonparameric measure of value-a-risk which measures how much he value of a porfolio could decline in a fairly exreme oucome. In our analysis, we use he minimum index reurns observed during pas k monhs of daily daa where k again akes he alernaive values of, 4, 36 or 48. These original value-a-risk measures are muliplied by - before he consrucion of he reward-o-risk raios so ha higher magniudes of he measure correspond o greaer downside risk. Afer we calculae nonparameric value-a-risk measures each monh using rolling windows, Sharpe raios ha incorporae hese nonparameric value-a-risk esimaes are compued. Specifically, VaRSharpe is defined as: VaRSharpe i, Ri, R f = (3) VaR i, 6

7 7 where VaR i, is he nonparameric value a risk. Finally, for he parameric measure of value-a-risk, we use he skewed -densiy, which accouns for skewness and excess kurosis in he daa. Hansen (994) inroduces a generalizaion of he Suden -disribuion where asymmeries may occur, while mainaining he assumpion of a zero mean and uni variance. This skewed (ST) densiy is given by: < + + = + + b a z a bz v bc b a z a bz v bc v z f v v / if / if ),,, ; ( λ λ λ σ µ (4) where σ µ = R z is he sandardized excess marke reurn and he consans a, b, and c are given by = 4 v v c a λ 3 a b + = λ, Γ + Γ = ) ( v v v c π (5) The parameric approach o calculaing value-a-risk is based on he lower ail of he ST disribuion. Specifically, we esimae he parameers of he ST densiy (µ, σ, υ, λ) using he pas, 4, 36 or 48 monhs of reurn daa and hen find he corresponding percenile of he esimaed disribuion. Assuming ha ) (, z f R v λ = follows an ST densiy, parameric value-a-risk is he soluion o

8 ΓST ( Φ) f v λ ( z dz = Φ (6), ) where Γ ST (Φ) is he value-a-risk hreshold based on he ST densiy wih a loss probabiliy of Φ. Sharpe raios ha incorporae parameric value-a-risk are defined as: PVaRSharpe i, Ri, R f = (7) PVaR i, 3. DATA AND DESCRIPTIVE STATISTICS We gaher he daa for he FoHF index reurns from he Hedge Fund Research (HFR) daabase. The daabase repors monhly index values for various hedge fund sraegies beginning from January 990 and he sample period used in he following analysis exends unil December 0. HFR indices are broken down ino four main sraegies, each wih muliple sub-sraegies. These sraegies include equiy hedge (equiy marke neural, quaniaive direcional, shor-bias, ec.), even driven (disressed / resrucuring, merger arbirage, ec.), macro (commodiy, currency, ec.) and relaive value (converible arbirage, fixed-income corporae, ec.). HFR also repors a Fund of Funds Composie index which includes over 650 consiuen funds. FoHFs inves wih muliple managers hrough funds or managed accouns and heir main benefi is designing a diversified porfolio of managers o reduce he risk of invesing wih an individual manager. Fund of Funds Composie index is an equally-weighed index and i is commonly used by hedge fund managers as a performance benchmark. A fund needs o repor monhly gross reurns and reurns ne of all fees o be included in he index. Moreover, he asses need o be repored in US dollars and he fund needs o have a leas $50 million under 8

9 managemen or have been acively rading for a leas welve monhs. Funds are included in he composie index he monh afer heir addiion o he daabase. We also collec daa for various bond and sock marke indices for comparison purposes. Specifically, we collec price daa for indices ha rack Treasury bonds wih mauriies of 5, 0, 0 and 30 years. For equiies, we focus on he S&P 500 index and he NYSE/AMEX/NASDAQ index wih disribuions. All he daa for he bond and sock marke indices come from he Cener for Research in Securiy Prices (CRSP). The yield for he -monh Treasury bill which is used o proxy for he risk-free rae is downloaded from Kenneh French s online daa library. Table repors he descripive saisics for all indices. A comparison of means shows ha he NYSE/AMEX/NASDAQ index has he highes monhly reurn (0.78%), however he S&P 500 index has no generaed as high an average reurn (0.58%). This difference can be explained by he greaer reurns generaed by small socks hisorically. The mean reurns on he bond indices increase by ime o mauriy wih he 5-year bond index delivering 0.56% per monh and he 30-year bond index delivering 0.73% per monh. In erms of means, he FoHF index sis somewhere in he middle in his picure wih a monhly mean reurn of 0.6%. The medians ell a similar sory wih he bigges difference being ha boh sock marke indices have generaed higher median reurns han all oher indices. NYSE/AMEX/NASDAQ index had a median reurn of.34% over he sample period whereas S&P 500 index had a median reurn of.0%. Again, he 9

10 median reurns for he bond indices increase by ime o mauriy and vary from 0.58% o 0.89%. The FoHF index sill posiions iself in he middle wih a median reurn of 0.77%. Table. Descripive Saisics for Fund of Hedge Funds, Bond and Equiy Indices This able presens descripive saisics for he reurns of various fund of hedge funds, bond and equiy indices in he US. The four bond marke indices are based on 5-year, 0-year, 0-year and 30-year mauriy Treasury bonds. The wo equiy indices are he S&P 500 index and he NYSE/AMEX/NASDAQ Composie index. The descripive saisics ha are presened in he able are he mean, sandard deviaion, minimum, 5 h percenile, median, 75 h percenile, maximum, skewness and kurosis. Mean S Dev Min 5h Median 75h Max Skew Kurosis Fund of Funds year bond year bond year bond year bond S&P NYSE/AMEX/ NASDAQ Wih respec o he sandard deviaions, we find ha he sock marke indices are generally more volaile compared o he bond marke indices. NYSE/AMEX/NASDAQ and S&P 500 indices have monhly sandard deviaions of 4.55% and 4.39%, respecively. The sandard deviaions of he bond marke indices increase from.8% for he 5-year bond index o.90% for he 30-year bond index. This finding is in line wih he higher ineres rae sensiiviies associaed wih bonds of longer duraions. The FoHF index has he second lowes sandard deviaion which is equal o.7%. The paerns for sandard deviaions also manifes hemselves when we look a he maximum and minimum reurns. The highes (lowes) maximum (minimum) reurns belong o he equiy indices and he bond indices wih longer imes o mauriy. For example, here has been a monh in which he NYSE/AMEX/NASDAQ index has gained 0

11 .53% in value and he 30-year bond index has gained 7.4% in value. Similarly, here has been a monh during which he NYSE/AMEX/NASDAQ index has los 8.46% of is value and he 30-year bond index has los 4.74% of is value. The exreme reurns for he FoHF index are milder wih a minimum monhly reurn of -7.47% and a maximum monhly reurn of 6.85%. This finding is consisen wih he diversificaion effecs inheren in fund of funds sraegies as argued in Fung and Hsieh (000). Finally, we compare he higher order momens of he indices. The FoHF index has he second mos negaive skewness saisic (-0.67) afer he NYSE/AMEX/NASDAQ index (-0.68). The oher sock marke index, S&P 500, also has negaive skewness (-0.56). This is consisen wih earlier findings in he lieraure ha he ails of he hedge fund and equiy reurn disribuions are longer on he lef side compared o he righ side. The negaive skewness associaed wih hese indices was also foreshadowed by heir higher medians compared o he means. For he bond marke indices, he skewness saisic increases wih ime o mauriy. The 5-year bond index has a skewness saisic of -0.8 whereas he 30-year bond index disribuion is posiively skewed wih a saisic of 0.9. The kurosis of he FoHF index is again subsanial and equal o 6.7. In oher words, he FoHF reurn disribuion has more mass on is ails compared o he normal disribuion and hus, is lepokuric. Kurosis again increases wih ime o mauriy for he bond marke indices from 3.3 o The kurosis for sock marke indices lie somewhere in he middle among he bond marke indices wih a kurosis saisic for he NYSE/AMEX/NASDAQ (S&P 500) index equal o 4.00 (4.3).

12 4. EMPIRICAL RESULTS Table presens he radiional Sharpe raios ha incorporae he sandard deviaion of a porfolio in is denominaor. We calculae hese monhly Sharpe raios in a rolling window fashion and use differen sampling windows o calculae he sandard deviaions. The lengh of he sampling windows ranges from o 48 monhs. We presen boh he ime-series mean and he sandard deviaions of he reward-o-risk raios for all indices. Table. Sandard Deviaion-Based Sharpe Raios for Fund of Hedge Funds, Bond and Equiy Indices This able presens he sandard deviaion-based Sharpe raios for various fund of hedge funds, bond and equiy indices in he US. The four bond marke indices are based on 5-year, 0-year, 0-year and 30-year mauriy Treasury bonds. The wo equiy indices are he S&P 500 index and he NYSE/AMEX/NASDAQ Composie index. The numeraor of he sandard deviaion-based Sharpe raio is equal o he monhly reurn of he index minus he risk-free rae. The denominaor is equal o he sandard deviaion of monhly reurns over he pas, 4, 36 or 48 monhs. Each row repors he means of each raio and he sandard deviaions are presened in parenheses. Sharpe Sharpe4 Sharpe36 Sharpe48 Fund of Funds (0.58) (0.366) 0.69 (0.363) (0.57) 5-year bond 0.3 (0.3709) 0.7 (0.83) (0.338) (0.857) 0-year bond 0.80 (0.378) (0.080) (0.59) 0.56 (0.3) 0-year bond (0.304) (0.639) 0.74 (0.04) (0.08) 30-year bond (0.307) 0.5 (0.576) 0.3 (0.9) 0.79 (0.0744) SP (0.360) 0.9 (0.47) 0.04 (0.988) (0.79) NYSE/AMEX/ NASDAQ 0.34 (0.3703) (0.443) (0.967) (0.656) When he sandard deviaion is calculaed from he mos recen year s daa, he FoHF index generaes he highes excess reurn per uni risk. The Sharpe raio for FoHF is equal o 0.35 which implies ha he index demands exra 35 basis poins of expeced reurn per % increase in sandard deviaion. The comparison beween he bond and sock

13 marke indices does no presen any clear paern. Alhough he NYSE/AMEX/NASDAQ index has a superior Sharpe raio (0.3) compared o all he bond indices, he S&P index lags behind mos of he bond indices wih a Sharpe raio of There is a declining paern for he bond indices wih Sharpe raios of 0. for he 5-year bond index and 0.40 for he 30-year bond index. Anoher poin o noe is ha he FoHF index also has he highes variaion in Sharpe raios. We observe his paern for he oher raios as well. Exending he sampling window for calculaing sandard deviaions o 4 monhs does no dramaically aler he resuls. The Sharpe raio of he FoHF index declines o 0.98 from 0.35, bu i is sill he index ha generaes he highes excess reurn per uni risk. Noe ha he reducion in he Sharpe raio is mechanical due o he posiive relaion beween sandard deviaion and ime horizon and his reducion is encounered for all indices. The NYSE/AMEX/NASDAQ index has a greaer Sharpe raio (0.90) compared o all bond indices excep he 5-year bond index (0.3). On he oher hand, he S&P 500 index has a smaller Sharpe raio (0.9) compared o all bond indices excep he 30-year bond index (0.5). For sampling windows of 36 and 48 monhs, he resuls are similar excep ha S&P 500 now has he lowes Sharpe raios and he 0-year bond index begins o ouperform he NYSE/AMEX/NASDAQ index. Mos imporanly, he FoHF index has he highes Sharpe raio regardless of he sampling window for he sandard deviaion. One final poin is ha he decrease in he Sharpe raios as he sampling window increases is sharper for he FoHF index compared o he oher indices. For he -monh window, 3

14 he Sharpe raio of he FoHF index exceeds is closes follower by 0.0 (0.35 vs. 0.3) whereas he difference is reduced o (0.36 vs. 0.04) for he 48-monh window. These resuls collecively sugges ha he FoHF index generaes a higher excess reurn per uni risk when risk is measured by sandard deviaion. However, here is enough evidence in he lieraure o believe ha he sandard deviaion is an incomplee measure of risk for hedge fund reurns whose disribuion deviaes from normaliy. This is also evidenced by he negaively skewed and lepokuric behavior of he FoHF index reurns in Table. Therefore, o ake he nonlineariies hedge fund reurns ino accoun, we calculae alernaive Sharpe raios based on nonparameric and parameric value-a-risk. Table 3 presens Sharpe raios ha are based on nonparameric value-a-risk. These VaRSharpe raios scale expeced excess reurns by he absolue value of he minimum reurn of a porfolio during a recen sample window where he lengh of he window varies beween and 48 monhs. When we focus on VaRSharpe, we find ha he FoHF index generaes he highes excess reurn per uni downside risk. The raio for he FoHF index is equal o.07 and exceeds hose of he oher indices muliple-fold. We again noe ha he ime-series sandard deviaion of he VaRSharpe measure is he greaes for he FoHF index. In oher words, alhough he FoHF index easily ouperforms he oher indices based on his paricular meric, his ouperformance seems o be variable hrough ime. VaRSharpe for he NYSE/AMEX/NASDAQ index is equal o 0.54) and greaer han hose of all bond marke indices excep he 5-year bond index which has a VaRSharpe of We observe ha he downside risk-adjused Sharpe raio has a 4

15 declining paern for he bond marke indices as he ime o mauriy increases and he 30- year bond index has a VaRSharpe of The S&P 500 index has a similar performance wih a VaRSharpe of To summarize, he FoHF index is he superior performer based on VaRSharpe and neiher he bond nor he sock marke indices clearly dominae each oher. Table 3. Nonparameric Value a Risk-Based Sharpe Raios for Fund of Hedge Funds, Bond and Equiy Indices This able presens he nonparameric value a risk-based Sharpe raios for various fund of hedge funds, bond and equiy indices in he US. The four bond marke indices are based on 5-year, 0-year, 0-year and 30-year mauriy Treasury bonds. The wo equiy indices are he S&P 500 index and he NYSE/AMEX/NASDAQ Composie index. The numeraor of he nonparameric value a risk -based Sharpe raio is equal o he monhly reurn of he index minus he risk-free rae. The denominaor is equal o he minimum monhly index reurn over he pas, 4, 36 or 48 monhs. Each row repors he means of each raio and he sandard deviaions are presened in parenheses. VaRSharpe VaRSharpe4 VaRSharpe36 VaRSharpe48 Fund of Funds.073 (5.854) (3.9006) (0.793) (0.67) 5-year bond (0.69) (0.003) 0.60 (0.35) 0.04 (0.0973) 0-year bond 0.87 (0.4049) 0.8 (0.40) (0.0974) (0.06) 0-year bond (0.8470) (0.09) (0.0676) (0.045) 30-year bond (.4856) (0.037) (0.063) (0.0364) SP (0.678) (0.398) (0.050) (0.085) NYSE/AMEX/ NASDAQ (0.457) 0.06 (0.473) (0.4) (0.0874) When we exend he sampling window o calculae nonparameric value-a-risk, he VaRSharpe raios again decline mechanically. The reason is ha he absolue value of he minimum reurn during he las 48 monhs has o be equal o or greaer han ha during he las monhs. Analyzing he longer horizon VaRSharpe raios makes some paerns apparen. Firs, he FoHF index coninues o be he bes performer regardless of he sampling window. Second, he 5-year bond index coninues o have he highes VaRSharpe raio afer he FoHF index and for he 36-monh and 48-monh horizons, he 5

16 0-year bond index also ouperforms he NYSE/AMEX/NASDAQ index. Third, he S&P 500 index coninues o have he lowes excess reurn per uni downside risk afer he 30- year bond index. Finally, similar o he resuls from he radiional Sharpe raio analysis, he margin by which he VaRSharpe raio of he FoHF index exceeds hose of he oher indices declines as he sampling window increases. For example, VaRSharpe of he FoHF index is seven imes as much as ha of he 5-year bond index which is closes follower. However, as he sampling window is exended o 48 monhs, he difference beween he VaRSharpe raios decreases subsanially. This is due o he fac ha he reducion in he VarSharpe raios is much seeper for he FoHF index compared o he oher indices. VaRSharpe48 measures for he FoHF and he 5-year bond indices are equal o 0.3 and 0.0, respecively. Nex, we invesigae he reward-o-risk raios ha have parameric value-a-risk based on Hansen s (994) skewed -densiy in heir denominaors. Table 4 presens he resuls. The inference from he analysis of PVaRSharpe raios corroboraes he findings from Table 3. When we focus on he -monh sampling horizon for he consrucion of he parameric downside risk measure, we find ha he FoHF index again has he highes reward-o-risk raio wih a PVaRSharpe of.04. One can also see ha he 0-year bond index also performs well for his meric wih a PVaRSharpe of The sock marke indices, namely he NYSE/AMEX/NASDAQ and S&P 500 indices have PVaRSharpe raios of 0.67 and 0.3, respecively. These values are lower han hose of all bond marke indices wih he excepion of he 30-year bond index. The exension of he sampling window again reduces he reward-o-risk raios for all indices. The FoHF index coninues 6

17 o be he bes performer regardless of he lengh of he sampling window. However, as observed for he radiional and nonparameric value-a-risk based Sharpe raios, he decline in he PVaRSharpe raio is seeper han he oher indices. For example, he raio of PVaRSharpe of he FoHF index o ha of he 5-year bond index is more han 4 when he -monh sampling window is used whereas for he 48-monh sampling window, he FoHF and 5-year bond indices have PVaRSharpe raios of 0.37 and 0.08, respecively. A closer look a he resuls reveals ha he bond marke indices generally ouperform he sock marke indices and here is a downward rend in he reward-o-risk raios among he bond marke indices especially for longer sampling windows. Table 4. Parameric Value a Risk -Based Sharpe Raios for Fund of Hedge Funds, Bond and Equiy Indices This able presens he parameric value a risk-based Sharpe raios for various fund of hedge funds, bond and equiy indices in he US. The four bond marke indices are based on 5-year, 0-year, 0-year and 30- year mauriy Treasury bonds. The wo equiy indices are he S&P 500 index and he NYSE/AMEX/NASDAQ Composie index. The numeraor of he parameric value a risk-based Sharpe raio is equal o he monhly reurn of he index minus he risk-free rae. The denominaor is equal o he firs percenile of Hansen s (994) skewed -densiy esimaed using he monhly reurns from over he pas, 4, 36 or 48 monhs. Each row repors he means of each raio and he sandard deviaions are presened in parenheses. PVaRSharpe PVaRSharpe4 PVaRSharpe36 PVaRSharpe48 Fund of Funds.037 (4.3839) (.7954) 0.64 (0.487) (0.79) 5-year bond (0.7375) 0.36 (0.7) 0.68 (0.65) 0.08 (0.0956) 0-year bond (7.68) (0.) (0.096) (0.0597) 0-year bond 0.64 (.4548) (0.848) (0.0739) (0.04) 30-year bond (0.904) (0.3949) (0.0755) (0.0409) SP (0.936) 0.07 (0.8) (0.094) (0.0769) NYSE/AMEX/ NASDAQ (0.965) (0.53) (0.0993) (0.0799) Figures and presen plos of radiional and nonparameric value-a-risk based Sharpe raios, respecively. For hese figures, we choose only one bond marke and one sock marke index o show he relaive performance of he FoHF index o keep he exposiion 7

18 clean. To be conservaive, we focus on he 5-year bond and NYSE/AMEX/NASDAQ indices which have proved o be he bond and sock marke indices ha have performed he bes over he sample period. Moreover, we presen he graphs for he reward-o-risk measures ha use sandard deviaion and nonparameric value-a-risk calculaed from a 48-monh sampling window since he superior performance of he FoHF index becomes less pronounced as he sampling window is exended. Figure. Sandard Deviaion-Based Sharpe Raios year bond NYSE/AMEX/NASDAQ FoHF This figure plos he sandard deviaion-based Sharpe raios for he fund of hedge funds, 5-year bond and NYSE/AMEX/NASDAQ indices beween January 994 and December 0. The numeraor of he sandard deviaion-based Sharpe raio is equal o he monhly reurn of he index minus he risk-free rae. The denominaor is equal o he sandard deviaion of monhly reurns over he pas 48 monhs. The figures show ha he FoHF index had a superior performance a he beginning of he sample period based on boh reward-o-risk merics, bu he Sharpe raios dropped o he level of he NYSE/AMEX/NASDAQ index by 996. We see ha he superior performance of he FoHF index is no uniform hrough ime. This observaion is consisen wih he large volailiy associaed wih he reward-o-risk raios of he FoHF 8

19 index uncovered in he earlier analysis. There have been periods in which eiher he 5- year bond or he NYSE/AMEX/NASDAQ or boh have ouperformed he FoHF index. One such period is he period afer he recen global financial crisis and i can clearly be seen ha he reward-o-risk raios ook a downward urn in he second half of 008. During his period, he performance of he sock marke has also been dismal and he 5- year bond index has generaed higher reurns per uni risk. Boh figures also capure he sock marke crash of he early las decade afer he inerne bubble burs as evidenced by he seep decline in he reward-o-risk raios of he NYSE/AMEX/NASDAQ afer 000. Figure. Nonparameric Value a Risk-Based Sharpe Raios year bond NYSE/AMEX/NASDAQ FoHF This figure plos he parameric value a risk-based Sharpe raios for he fund of hedge funds, 5-year bond and NYSE/AMEX/NASDAQ indices beween January 994 and December 0. The numeraor of he parameric value a risk-based Sharpe raio is equal o he monhly reurn of he index minus he risk-free rae. The denominaor is equal o he firs percenile of Hansen s (994) skewed -densiy esimaed using he monhly reurns from over he pas 48 monhs. 9

20 CONCLUSION We invesigae wheher he fund of hedge fund porfolios ouperform various bond and sock marke indices in erms of being able o generae higher reurns per uni risk. Due o he poenial non-normaliy associaed wih hedge fund reurns, we give special emphasis o he concep of downside risk in our analysis. Consequenly, apar from he radiional Sharpe raio, we also consruc reward-o-risk raios ha use non-parameric or parameric measures of value-a-risk in heir denominaor for various indices. Our main finding is ha he FoHF index has superior reward-o-risk raios compared o all bond and sock marke indices. Alhough his superior performance is more pronounced when he risk measures are calculaed using daa from he las monhs, he abiliy of he FoHF index o generae higher reurns per uni risk is robus regardless of he sampling window. We also find ha he documened ouperformance is no a phenomenon ha has been observed consisenly hrough ime and here have been periods in which he FoHF index has lagged behind he oher indices. 0

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