1、Lionel MartelliniRisk and Asset Management Research Center,EDHEC Graduate School of Businesslionel.martelliniedhec.eduConfrence Gestion Alternative2 Avril 2019The Alpha and Omega of Hedge Fund Performance EvaluationJoint work with Nol Amenc(EDHEC)and Susan Curtis(USC)Outline IntroductionStandard CAP
2、M ModelAdjusting CAPM for the Presence of Stale PricesPayoff Distribution ModelMulti-Factor Models Implicit Factor ModelExplicit Factor ModelExplicit Index ModelPeer BenchmarkingComparative Performance AnalysisImpact on Attributes on Funds PerformanceConclusionIntroductionPapers on MF and HF Perform
3、ance There is ample evidence that portfolio managers following traditional active strategies on average under-perform passive investment strategies Examples are:Jensen(1968),Sharpe(1966),Treynor(1966),Grinblatt and Titman(1992),Hendricks,Patel and Zeckhauser(1993),Elton,Gruber,Das and Hlavka(1993),B
4、rown and Goeztman(2019),Malkiel(2019),Elton,Gruber and Blake(2019),or Carhart(2019),among many othersRecently,many papers have focused on hedge fund performance evaluationExamples are:Ackermann,McEnally,and Ravenscraft(2019),Amin and Kat(2019),Agarwal and Naik(2000a,2000b),Brown,Goetzmann and Ibbots
5、on(2019),Edwards and Caglayan(2019),Fung and Hsieh(2019,2019a,2019b),Gatev,Goetzmann and Rouwenhorst(2019),Liang(2000),Lhabitant(2019),Lo(2019),Mitchell and Pulvino(2019),Schneeweis and Spurgin(2019,2000)Because these studies are based on a variety of models for risk-adjustment,and also differ in te
6、rms of data used and time period under consideration,they yield very contrasted resultsThe present paper can be viewed as an attempt to provide an unified picture of hedge fund managers to generate superior performanceTo alleviate the concern of model risk on the results of performance measurement,w
7、e consider an almost exhaustive set of pricing models that can be used for assessing the risk-adjusted performance of hedge fund managers.IntroductionModelsWhile we find significantly positive alphas for a sub-set of hedge funds across all possible models,our main finding is perhaps that the dispers
8、ion of alphas across models is very largeHedge funds appear to have significantly positive alphas on average when normal returns are measured by an explicit factor model,even when multiple factors serving as proxies for credit or liquidity risks are accounted forHowever,hedge funds on average do not
9、 have significantly positive alphas once the entire distribution is considered or implicit factors are includedOn the other hand,all pairs of models have probabilities of agreement greater than.50In other words,while different models strongly disagree on the absolute risk-adjusted performance of hed
10、ge funds,they largely agree on their relative performance in the sense that they tend to rank order the funds in the same wayIntroductionPreview of the ResultsOur analysis is conducted on a proprietary data base of 1,500 individual hedge fund managers(MAR-CISDM data base)We use the 581 hedge funds i
11、n the MAR database that have performance data as early as January 2019The data base contains monthly returns and alsoFund size(asset under management)Fund type(MAR classification system)Fund age(defined as the length of time in operation prior to the beginning of our study)Location(US versus non US)
12、Incentive feesManagement feesMinimum purchase amountIntroductionDataStandard CAPM ModelNormal and Abnormal ReturnsFactor models allow us to decompose managers (excess)returns intoNormal returns(risk premium)Abnormal returns(investment opportunity)Statistical noise(illusion)Normal returns are generat
13、ed as a fair reward for the risk(s)taken by fund managersAbnormal returns are generated managers unique ability to“beat the market”in a risk-adjusted sense,generated through superior access to information or better ability to process commonly available informationNeed some model to understand what a
14、“normal”return is;benchmark model is the CAPM(Sharpe(1964)Standard CAPM Model ResultsThe average alpha across all funds is significantly positiveThe majority of hedge funds have positive alphas,and about a third are statistically significantVery few funds have significantly negative alphasStatisticV
15、alue under CAPMAlpha(average fund)5.83%Std.Err.Alpha(average fund)2.85%p-value(for average alpha not 0)0.045St.Dev.Alpha(across funds)10.02%of funds with alpha significantly031.3%of funds with alpha significantly00.7%Standard CAPM Model Distribution of CAPM AlphasDistribution of CAPM Alphas0%2%4%6%8
16、%10%12%14%16%-15-13-11-9-7-5-3-1135791113151719212325Alpha Value(annual%,midpoint of bin)PercentofFundsStandard CAPM Model Distribution of CAPM BetasDistribution of CAPM Betas0%5%10%15%20%25%-1-0.8-0.6-0.4-0.200.20.40.60.811.21.41.61.82Beta Value(midpoint of range)PercentofFundsAdjusting CAPM for St
17、ale Prices The Presence of Stale PricesIt has been documented(Asness,Krail and Liew(2019)that a fair number of hedge funds hold illiquid securitiesFor monthly reporting purposes,they typically price these securities using either the last available traded price or estimates of current market pricesSu
18、ch non-synchronous return data can lead to understated estimates of actual market exposure,and therefore to mismeasurement of hedge fund risk-adjusted performanceIn that context,some adjustment must be performed to account for the presence of stale pricesAdjusting CAPM for Stale Prices Model and Res
19、ultsModel:run regressions of returns on both contemporaneous and lagged market returnsResults:the number of funds with alpha values significantly greater than zero has been cut in halfri,trf,tik 0KikrM,t krf,t ki,tStatisticValue under CAPMValue under Lagged CAPMAlpha(average fund)5.83%2.14%Std.Err.A
20、lpha(average fund)2.85%3.21%p-value(average fund alpha not 0)0.0450.51%of funds with alpha significantly031.3%16.9%of funds with alpha significantly00.7%2.8%Adjusting CAPM for Stale Prices Model and ResultsCAPM has more funds with alphas near 10%,and lagged CAPM model has more funds with alphas betw
21、een-10%and 0Comparison of Alphas under CAPM and Lagged CAPM0%2%4%6%8%10%12%14%16%-23-21-19-17-15-13-11-9-7-5-3-11357911131517192123Alpha Values(bin midpoints)PercentofFundsCAPMLagged CAPMPayoff Distribution Function ApproachNon-Linear Exposure to Standard Asset ClassesHedge fund returns exhibit non-
22、linear option-like exposures to standard asset classes becauseThey can use derivativesThey follow dynamic trading strategiesFurthermore,the explicit sharing of the upside profits under the form of incentive fees implies that post-fee returns have option-like element even if pre-fee returns do notIn
23、this context,mean-variance CAPM based performance measures will fail to account for non trivial preferences about skewness and kurtosisThere exists a method allowing an investor to account for the whole distribution of returnsPayoff Distribution Function ApproachMethodologyMethodology introduced by
24、Dybvig(see Dybvig(1988a,1988b),applied to hedge fund performance evaluation by Amin and Kat(2019)First step:recover the cumulative probability distribution of the monthly hedge fund payoffs as well as the S&P 500 from the available data set assuming$100 are invested at the beginning of the period A
25、normal distribution is assumed for the S&P 500(i.e.,we only need to estimate the mean and standard deviation of the monthly return on the S&P 500 over the period),but not for the hedge fundsSecond step:generate payoff functions for each hedge fundA payoff function is a function f that maps the retur
26、n distribution of the S&P 500 into a relevant return distribution for the hedge fundPayoff Distribution Function Approach Methodology(cont)Third step:we use a discrete version of a geometric Brownian motion as a model for the underlying S&P price process generate 20,000 end-of-month valueFrom these
27、20,000 values,we generate 20,000 corresponding payoffs for each hedge fund,average them,and discount them back to the present to obtain a fair price for the payoffThis“price”thus obtained can be thought of the minimum initial amount that needs to be invested in a dynamic strategy involving the S&P a
28、nd cash to generate the hedge fund payoff functionIf the price thus obtained is higher than 100,this means that more than$100 needs to be invested in S&P to generate a random terminal payoff comparable to the one obtained from investing a mere$100 in the hedge fund.We therefore take this as evidence
29、 of superior performance.On the other hand,if the price obtained is lower than$100,we conclude that one may achieve a payoff comparable to that of the hedge fund for a lower initial amount.The percentage difference is computed as a relative measure of efficiency lossPayoff Distribution Function Appr
30、oach Cumulative Probability DistributionsThe slope for the average hedge fund is much steeper than for the S&P 500,indicating a much narrow distribution of returnsPayoff Distribution Function Approach Performance of High-and Low-Rated Funds The low-rated fund has a wide distribution of returnsTop ra
31、ted funds were found to be of two types:high volatility funds with exceptionally high returns,and low volatility fundsComparison of Hedge Funds with S&P 500(Real Data)0.00.20.40.60.81.0859095100105110115120Final Fund Value(Initial=100)CumulativeProbabilityDistributionS&PTop High Risk FundTop Low Ris
32、k FundWorst FundPayoff Distribution Function Approach Distribution of Efficiency Gain or LossDistribution of Hedge Fund Efficiency0%10%20%30%40%50%60%3%Efficiency Gain or LossPercentofHedgeFundsPayoff Distribution Function Approach Performance of Hedge FundsOn average,hedge funds do not outperform t
33、he marketHowever,the statistics are influenced by a few funds with large negative efficiencies.Over half of the funds have positive efficiency measuresNote that in the implementation of PDPM,we must make an assumption about the volatility of the S&P 500Here,we used a16%volatility as measured during
34、the time period of the dataWhen we repeated the analysis with a higher volatility of 20%,the average hedge fund has a slightly higher efficiency and is no longer significantly different from zeroPDPMMean Efficiency(annual rate)-0.92%St.Dev.Efficiency10.66%of funds with efficiency058%Multi-Factor Mod
35、elsOther Sources of RiskHedge funds are typically exposed to a variety of risk sources including volatility risks,credit or default risks,liquidity risks,etc.,on top of standard market risksIf one uses CAPM while the“true”model is a multi-factor model,then estimated alpha will be higher than true al
36、phaModern portfolio theory and practice is based upon multi-factor models(Merton(1973),Ross(1976)The return on asset or find i is Rit=mi+bi1F1t+.+biKFKt+eit Fkt is factor k at date t(k=1,K)eit is the asset specific return bik measures the sensitivity of Ri to factor k,(k=1,K)Multi-Factor Models Four
37、 Types of Factor Models Implicit factor models Factors:principal components,i.e.,uncorrelated linear combinations of asset returns Explicit factor models macro factorsFactors(Chen,Roll,Ross(1986):inflation rate,growth in industrial production,spread long-short treasuries,spread high-low grade corpor
38、ate interest rate Explicit factor models micro“factors”Factors(actually attributes):size,country,industry,etc.Explicit factor model index“factors”Factors are stock and bond market indicesImplicit Factor Model The ModelUse principal component analysis to extract statistical factors(linear combination
39、s of returns)Challenge is determine the optimal#of factors(K)Select K by applying results from the theory of random matricesCompare the properties of an empirical covariance matrix to a null hypothesis purely random matrix Distribution function for eigenvalues under the null hypothesisHere,we regard
40、 as statistical noise all factors associated with an eigenvalue lower than lambda min TNTNTNTNNTf21212minmaxminmaxImplicit Factor Model The Results The mean alpha is less than zero under this model This suggests that there are factors influencing hedge fund performance that are captured in the Impli
41、cit Factor Model but not captured in CAPMStatisticValue under Implicit Factor ModelMean Alpha(annual rate)-1.04%Std.Dev.Alpha12.59%p-value(for mean alpha not 0)0.047%of funds with alpha044%Explicit Factor Model The ModelWe test an explicit macro factor model where we use financial variables to proxy
42、 deeper economic effectsThe following factors are usedUS equity risk is proxied by the return on the S&P 500 index,and world equity risk is proxied by the return on the MSCI World Index ex USEquity volatility risk,proxied by using the changes in the average of intra-month values of the VIXFixed-inco
43、me level risk is proxied by the 3 months T-Bill rate Slope risk or term premium risk is proxied by monthly differences between the yield on 3 months Treasuries and 10-year TreasuriesCurrency risk is proxied by changes in the level of an exchange volume-weighted index of currencies versus US dollarCo
44、mmodity risk is proxied by changes in the level of a volume-weighted index of commodity pricesCredit risk is proxied by changes in the monthly observations of the difference between the yield on long term Baa bonds and the yield on long term AAA bondsLiquidity risk is proxied by changes in the month
45、ly market volume on then NYSEExplicit Factor Model The ResultsSome of the above variables do not appear to command a premiumThe following variables seems to be rewarded risk factors:S&P 500,MSCI ex-US,oil price return,change in credit spread,change in VIXThis is the 5-factor model that we use Statis
46、ticValue under 5-Factor Macro ModelAlpha(average fund)7.25%Std.Err.Alpha(average fund)2.35%p-value(for average alpha not 0)0.01Std.Dev.Alpha(across funds)9.81%of funds with alpha significantly039%of funds with alpha significantly01%Explicit Factor Model The Analysis The average alpha is higher than
47、under the CAPM This is primarily due to the inclusion of the MSCI ex-US indexMany of the funds in our database are global funds,with a higher beta on the MSCI index than on the S&P 500.Since the MSCI index underperformed the S&P 500 during this time period,the inclusion of the MSCI factor makes the
48、alpha values higher for these funds Thus,in fact,these funds are outperforming the world index but are underrated in the CAPM model since their performance is compared to the S&P 500The inclusion of other factors(oil prices,change in credit spread,change in VIX)tends to lower the alphas(compared to
49、a 2-variable model using S&P500 and MSCI only),but does not erase the gains made by including the MSCIThus,we conclude that when all of these significant factors are included,the average hedge fund still has a positive alpha Style analysisSharpe(1992):equity styles are as important as asset classesE
50、xamples:growth/value,small cap/large cap,etc.Model:Rit=wi1F1t+wi2F2t+.wikFkt+eit Rit=(net of fees)excess return on a given portfolio or fundFkt=excess return on index j for the period twik=style weight(add up to one)eit=error term Divide the fund return into two partsStyle:wi1F1t+wi2F2t+.wikFkt(part
