Portfolio Performance Manipulation and Manipulation-Proof Performance Measures

Transcription

1 Portfolo Performance Manpulaton and Manpulaton-Proof Performance Measures Wllam Goetzmann, Jonathan Ingersoll, Matthew Spegel, Ivo Welch March 5, 006 Yale School of Management, PO Box 0800, New Haven, CT Brown Unversty, Department of Economcs Box B, Provdence, RI 09 Ths paper has benefted from the comments of an anonymous referee and numerous colleagues ncludng the partcpants at the Fve Star Conference hosted by New York Unversty, the Hedge Fund Conference hosted by Borsa Italana, and the Berkeley Program n Fnance.

2 Portfolo Performance Manpulaton and Manpulaton-Proof Performance Measures Abstract Over the years numerous portfolo performance measures have been proposed. In general they are desgned to capture some partcular enhancement that mght result from actve management. However, f a prncpal uses a measure to judge an agent, then the agent has an ncentve to game the measure. Our paper shows that such gamng can have a substantal mpact on a number of popular measures even n the presence of extremely hgh transactons costs. The queston then arses as to whether or not there exsts a measure that cannot be gamed? As ths paper shows there are condtons under whch such a measure exsts and fully characterzes t. Ths manpulaton-proof measure looks lke the average of a power utlty functon, calculated over the return hstory. The case for usng our alternatve rankng metrc s partcularly compellng n the hedge fund ndustry, n whch the use of dervatves s unconstraned and manager compensaton tself nduces a non-lnear payoff and thus encourages gamng.

3 Portfolo Performance Manpulaton and Manpulaton-Proof Performance Measures Fund managers naturally clam that they can provde superor performance. Investors must somehow verfy these clams and dscrmnate among managers. In 966, Wllam Sharpe, usng mean-varance theory, ntroduced the Sharpe rato to do ths n a quantfable fashon. Together wth ts close analogues, the nformaton rato, the squared Sharpe rato and M-squared, the Sharpe rato s now wdely used to rank nvestment managers and to evaluate the attractveness of nvestment strateges n general. Other measures such as Jensen s alpha and the Henrksson-Merton market tmng measure, although they evaluate other aspects of performance are also based on the same or smlar theores. If nvestors use performance measures to reward money managers, then money managers have an obvous ncentve to manpulate ther performance scores. Our paper contrbutes n two ways to ths lterature. Frst, t ponts out the vulnerablty of tradtonal measures to a number of smple dynamc manpulaton strateges. Second, t offers a formal defnton of the propertes that a manpulaton-proof measure should have and derves such a measure. For a measure to be manpulaton proof, t must not reward nformaton free tradng. In ths regard the exstng set of measures suffer from two weaknesses. Frst, most were desgned to be used n a world where asset and hence portfolo returns have nce dstrbutons lke the normal or lognormal. But, ths gnores the fact that managers can potentally use dervatves (or tradng strateges) to radcally twst return dstrbutons. Hedge fund returns, n partcular, have dstrbutons that can devate substantally from normalty, and the hedge fund ndustry s one n whch performance measures lke the Sharpe rato are most commonly employed. Second, exact performance measures can be calculated only n theoretcal studes (e.g., Leland (999) and Ferson and Segel (00)); n general, they must be estmated. Snce standard statstcal technques are desgned for ndependent and dentcally dstrbuted varables, t s possble to manpulate the exstng measures by usng dynamc strateges resultng n portfolos whose returns are not so dstrbuted. We show here that portfolos usng dynamc strateges nvestng n ndex puts and calls can be used to manpulate the common portfolo measures. Ths s true even when only very lqud optons are used and when transactons costs are extremely large. Numercal results are presented for the Sharpe rato, Jensen s alpha, Treynor rato, Apprasal rato, generalzed alpha, Sortno rato (99), Van der Meer, Plantnga and Forsey (999) rato, and the tmng measures of Henrksson and Merton (99), and Treynor and Mazuy (966). In every case, the manpulatons presented generate statstcally and economcally sgnfcant better portfolo scores even though all of the smulated trades can be carred out by an unnformed nvestor. The paper descrbes general strateges for manpulatng a performance measure. () Statc manpulaton of the underlyng dstrbuton to nfluence the measure. Such strateges can enhance scores even f the evaluator calculates the measure wthout any estmaton error whatsoever. () Dynamc manpulaton that nduces tme varaton nto the return dstrbuton n order to nfluence measures that assume statonarty. Ths can enhance a portfolo s score even f the measure s calculated wthout any estmaton error but wthout regard to the (unobservable) tme varaton n the portfolo s return dstrbuton. (3) Dynamc manpulaton strateges that focus on nducng estmaton error. Snce measures have to be estmated wth real world data, there are strateges that can nduce postve bases n the resultng values. As a smple example, magne an evaluator who estmates a fund s Sharpe rato by calculatng ts excess return s

4 standard devaton and average excess return over a 36 month perod usng monthly data. Further assume the fund manager smply wshes to maxmze the expected value of the calculated Sharpe rato. A smple strategy that accomplshes ths has the fund sell an out of the money opton n the frst month, whle nvestng the remanng funds n the rsk free asset. If the opton expres worthless the portfolo s then nvested wholly n the rsk free asset for the remanng 35 months. Whenever ths happens the portfolo has a zero standard devaton, a postve excess return, and thus an nfnte Sharpe rato. Snce there s a strctly postve probablty that the opton wll expre worthless the expected value of the calculated Sharpe rato must be nfnty. Worse yet, as the paper shows even wth goals other than maxmzng the expected calculated performance value, and concern for other moments of the test statstc a manager can delberately nduce large economcally sgnfcant measurement errors to hs advantage. Also notce that ncreasng the uency wth whch returns are observed does not sheld the set of exstng measures from ths type of dynamc manpulaton. If the set of current measures are vulnerable to manpulaton the queston naturally arses as to whether a manpulaton proof measure exsts. To answer ths queston, one frst needs to determne what manpulaton proof means. Ths paper defnes a manpulaton proof performance measure (MPPM) as one that has three propertes:. The score s value should not depend upon the portfolo s dollar value.. An unnformed nvestor cannot expect to enhance hs estmated score by devatng from the benchmark portfolo. At the same tme nformed nvestors should be able to produce hgher scorng portfolos and can always do so by takng advantage of arbtrage opportuntes. 3. The measure should be consstent wth standard fnancal market equlbrum condtons. It turns out that these three requrements are enough to pn down the measure. The frst smply mples that returns are suffcent statstcs rather than dollar gans or losses. The second and thrd, however, provde a great deal more structure. As the earler dscusson ndcates there are several ways to manpulate a performance measure and the second condton mples that a MPPM must be mmune to all of them. In partcular that means the manager cannot expect to beneft by tryng to alter the score s estmaton based upon observable data. To accomplsh ths goal the score must be () ncreasng n returns (to recognze arbtrage opportuntes), () concave (to avod ncreasng the score va smple leverage or addng unprced rsk), (3) tme separable to prevent dynamc manpulaton of the estmated statstc, and (4) have a power form to be consstent wth an economc equlbrum. The resultng measure ( ˆΘ ) suggested here s T ˆ ρ Θ n ( rft ) ( rft xt ). ( ) t T () ρ t= The coeffcent ρ can be thought of as the evaluator s rsk averson and should be selected to make holdng the benchmark optmal for an unnformed manager. The varable t s the length of tme between observatons whle T equals the total number of observatons. These two varables ( t and T) smply help normalze the measure s fnal calculaton. The portfolo s return at tme t s x t, and the rsk free asset s return s r ft. Note the measure s easy to calculate and has an ntutve nterpretaton as the average per perod welfare of a power utlty nvestor n the

5 portfolo over the tme perod n queston. As an example, consder a fund wth returns of.,.05,.7, and.0. Assume the rsk free rate s.0. In ths case T=4, r ft s.0, and f ρ equals then ˆΘ equals.036. For ρ equal to 3, ˆΘ equals.09. Our paper was orgnally motvated by the queston of whether the exstng set of performance measures s suffcently manpulaton proof for practcal use. Ths s partcularly relevant n the presence of transactons costs, whch may offset whatever performance gans a manager mght hope to generate from tradng for the purpose of manpulatng the measure. Therefore, our paper explores how dffcult t s to meanngfully game the exstng measures. For the seven measures examned here four ratos (Sharpe (966), Sortno (99), Leland (999), and Van der Meer, Plantnga, and Forsey (999)), and three regresson ntercepts (the CAPM alpha, Treynor and Mazuy (966), and Henrksson and Merton (98)) the answer s not encouragng. Smple dynamc strateges that only relever the portfolo each measurement perod or buy (very lqud) at the money optons can produce seemngly spectacular results, even n the presence of very hgh transactons costs. For example, consder the Henrksson and Merton (98) measure. A smple optons tradng scheme n the presence of transactons costs equal to 0% produces very good results. The fnal regresson statstcs report that the portfolo has returns that are superor to the market s nearly 65% of the tme, and (usng a 5% crtcal value) statstcally sgnfcantly better 9% of the tme. Obvously, lower and more realstc transactons costs only make matters worse. The other measures analyzed here are smlarly susceptble to havng ther estmated values gamed. Our results have a number of mplcatons for nvestment management. Hedge funds and other alternatve nvestment vehcles have broad lattude to nvest n a range of nstruments, ncludng dervatves. Mtchell and Pulvno (00) document that merger arbtrage, a common hedge fund strategy, generates returns that resemble a short put-short call payoff. Recent research by Agarawal and Nak (00) shows that hedge fund managers n general follow a number of dfferent styles that are nonlnear n the returns to relevant ndces. In a manner smlar to Henrksson and Merton, Agarawal and Nak use opton-lke payoffs as regressors to capture these non-lneartes. In fact, opton-lke payoffs are nherent n the compensatonstructure of the typcal hedge fund contract. Goetzmann, Ingersoll and Ross (00) show that the hgh water mark contract, the most common type n the hedge fund ndustry, effectvely leaves the nvestor short 0% of a call opton. The call s at-the-money each tme t s reset by a payment and out-of-the money otherwse. The paper s structured as follows. Secton dscusses varous ways n whch the Sharpe rato can be manpulated. Secton dscusses the manpulaton of a number of other measures ncludng those usng reward to varablty (Secton A) and the Henrksson-Merton and Treynor-Mazuy market tmng measures (Secton B). Secton 3 derves the manpulaton proof performance measure, and Secton 4 concludes.. Manpulaton of the Sharpe Rato Although the Sharpe rato s known to be subject to manpulaton, t remans one of f not the most wdely used and cted performance measures. It, therefore, makes a good example to ntroduce performance manpulaton n more detal. Other measures are examned n the next secton. 3

6 A. Statc Analyss A number of papers have shown that by alterng the dstrbuton functon governng returns the statstcal mean and varance can be manpulated to ncrease the Sharpe rato to some degree. Ferson and Segel (00) look at a farly general case (that ncludes potentally prvate nformaton) whle Lhabtant (000) shows how usng just a couple of optons can generate seemngly mpressve values. Snce the focus of ths paper s on dynamc strateges and the development of a MPPM to prevent all sources of manpulaton the only statc result presented here s the characterzaton of the return dstrbuton that maxmzes the Sharpe rato. Readers nterested n addtonal detals should consult one of the above cted artcles. The Appendx shows that Sharpe rato maxmzng portfolo n a complete market s characterzed by a state return n excess of the nterest rate ( x MSR ) of x MSR pˆ p = xmsr +. SMSR () / Here S ˆ MSR p p s the maxmum possble Sharpe rato, and p and p ˆ are the true and rsk-neutral probabltes of state. Snce the Sharpe rato s nvarant to leverage, we are free to specfy the portfolo s mean excess return, x MSR, at any level desred. Whle the MSR has a number of nterestng propertes t s also true that one way to mtgate ts mpact on a fund s apparent performance s to sample returns more uently. As one does so the percentage dfference between the market portfolo s Sharpe rato and that of the MSR goes to zero. However, as wll be seen below uent samplng does not help when managers use dynamc manpulaton strateges and thus the paper now turns to ths ssue. B. Dynamc Analyss Calculated values of the Sharpe rato and vrtually all other performance measures use statstcs based on the assumpton that the reported returns are ndependent and dentcally dstrbuted. Whle ths may be a good descrpton of typcal portfolo returns n an effcent market, t clearly can be volated f a portfolo s holdngs are vared dynamcally dependng on ts performance. Consder a money manager who has been lucky or unlucky wth an average realzed return hgh or low relatve to the portfolo s realzed varance. The return dstrbuton n the future wll lkely not be smlar to the experence. Consequently, to maxmze the Sharpe rato, the portfolo can be modfed to take nto account the dfference n the dstrbutons between the realzed and future returns just as f the returns dstrbuton was not ex ante dentcally dstrbuted each perod. Ths dynamc manpulaton makes the past and future returns dependent when computed by an uncondtonal measure. To llustrate, suppose the manager has thus far acheved an hstorcal average excess return of x h wth a standard devaton of σ h. The portfolo s average excess return and standard devaton n the future are denoted by x f and σ f. Then ts measured Sharpe rato over the entre perod wll be wll be 4

7 γ xh + ( γ) xf S = γ ( x +σ ) + ( γ )( x +σ ) [ γ x + ( γ ) x ] h h f f h f γ xh + ( γ) xf = γ x ( + / S ) + ( γ ) x ( + / S ) [ γ x + ( γ ) x ] h h f f h f (3) where γ s the fracton of the total tme perod that has already passed, and h and f are the past and future Sharpe ratos. By nspecton, the overall Sharpe rato s maxmzed by holdng n the future a portfolo that maxmzes the future Sharpe rato, S f = S MSR. However, regardless of what Sharpe rato can be acheved n the future, leverage s no longer rrelevant as t was n the statc case. Maxmzng the Sharpe rato n (3), we see that the optmal leverage gves a target mean excess return of xh( h ) ( f ) for xh 0 x + S + S > f = for xh 0. (4) If the manager has been lucky n the past and acheved a hgher than antcpated Sharpe rato, Sh > S f, then the portfolo should be targeted n the future at a lower mean excess return (and lower varance) than t has realzed. Ths allows the past good fortune to wegh more heavly n the overall measure. Conversely, f the manager has been unlucky and Sh < S f, then n the future, the portfolo should be targeted at a hgher mean excess return than that so far realzed. In the extreme, f the average excess return realzed has been negatve, then the manager should use as much leverage as possble (deally nfnte) to mnmze the mpact of the poor hstory. The Sharpe rato that can be acheved over the entre perod s S S +γ S + ( γ) S S = γ S MSR +γsmsr MSR h h MSR + ( γ ) Sh +γsmsr for S > 0 h for S 0. h (5) Fgure plots the overall Sharpe rato as a functon of the realzed hstorcal Sharpe rato for hstores of dfferent duratons. The overall Sharpe rato can, on average, be mantaned above For a fxed Sharpe rato, the overall mean s lnear n the future leverage whle the overall standard devaton s convex. The proportonal changes n the mean and standard devaton wth respect to future leverage are equal when the future and hstorcal Sharpe ratos are equal. Therefore, when the hstorcal Sharpe rato s less than the future Sharpe rato, ncreasng leverage ncreases the overall mean at a faster rate than the standard devaton and vce versa. Because the optmal future leverage s nfnte when xh 0, the past Sharpe rato does not affect the overall Sharpe rato, and the value of the overall Sharpe rato for S 0 s the same as for S 0. h < h = 5

8 (s forced below) the theoretcal maxmum, S MSR, whenever the past performance has been good (bad). The realzed Sharpe rato, of course, has the most mpact when the past hstory s long (γ ). [Insert Fgure here] As can be seen n the graph or from equaton (5), the over-all Sharpe rato s ncreasng and convex n the hstorcal Sharpe rato. Therefore, dynamc-sharpe-rato-maxmzng strateges should, on average, be able to produce a Sharpe rato hgher than S MSR. 3 Smulatons show that dynamc manpulaton of the Sharpe rato can have a substantal effect. In a lognormal market wth a rsk premum of %, a manager re-leverng hs portfolo after thrty months of a fve-year evaluaton perod can acheve a Sharpe rato of Ths s 8% hgher than the statc MSR (and 3% hgher than the standard based estmate of the same). In addton, dynamc manpulaton works n exactly the same way for portfolos that have not been Sharpe optmzed each perod. Any portfolo wth a Sharpe rato of S f can be levered as shown n (4) to acheve a hgher average Sharpe rato than ths on average. For example, n the same lognormal market, f the market s relevered after 30 months, ts Sharpe rato mproves on average from to 0.67, an ncrease of 3%. In addton, we have assumed here that a sngle measurement perod s shorter than the tme between rebalancngs. If the portfolo manager can alter hs portfolo wthn a sngle measurement perod, then the dstrbuton of a sngle return can be affected. For example, f a portfolo ncreases sharply n value durng the frst part of a measurement perod, swtchng to a less aggressve poston may make the ultmate measured return smaller and keep the Sharpe rato hgh. Goetzmann, Ingersoll, and Ivkovc (000) analyze the problem of performance measurement when the nvestment rebalancng perod s shorter than the measurement perod. Partcular crcumstances may permt addtonal opportuntes to dynamcally manpulate the Sharpe rato of a portfolo. Smoothng returns over tme wll leave the portfolo s mean return unchanged but decrease ts varance so smoothng returns wll ncrease the Sharpe rato. Funds wth llqud assets whose prces are only reported occasonally may beneft here. Hedge funds or other portfolos wth hgh-water mark performance fees wll also beneft. A standard performance contract calls for expensng the performance fee monthly but payng t only annually, but unpad fees can be lost based on poor later fund performance. Ths contractual provson moves recorded returns from perods of good performance to perods of poor performance and smoothes the reported returns. C. Jensen s Alpha and Related Measures Several other performance measures are related to the Sharpe rato. The most famlar of these s Jensen s alpha whch s the ntercept of the market model regresson, xt =α+β xmt +ε t. If the CAPM holds, ths ntercept term should be zero when the regresson s expressed n excess return form. Jensen s alpha measures the margnal mpact to the Sharpe rato of addng a small 3 Brown, et al (004) provde evdence that some Australan money managers engage n a pattern of tradng consstent wth the behavor descrbed here. 6

9 amount of an asset to the market. 4 Consequently, any portfolo wth a Sharpe rato n excess of the market s must have a postve alpha. In partcular, the alphas of the maxmal Sharpe rato portfolos descrbed above are 5 α = x ( S S ) > 0. (6) MSR MSR mkt MSR Clearly, alpha s subject to severe manpulaton. A MSRP can be created wth any desred leverage, so ts alpha can n theory be made as large as desred by leverng. The Treynor (965) rato and the Treynor apprasal rato were ntroduced n part to avod the leverage problem nherent n alpha. The Treynor measure s the rato of alpha to beta 6 whle the apprasal measure s the rato of alpha to resdual standard devaton. Lke the Sharpe measure, and unlke alpha, the two Treynor measures are unaffected by leverage. Both Treynor measures ndcate superor performance for the MSRP. The MSRP s two Treynor ratos are 7 αmsr T ( S S ) = x > 0 MSR MSR mkt mkt βmsr αmsr AMSR = SMSR Smkt > 0. Var[ x β x ] MSR MSR mkt (7) Any MSRP has the largest possble Treynor apprasal rato, but Treynor ratos n excess of the MSRP s can be acheved by formng portfolos wth postve alphas and betas close to zero. 8 In addton, snce the Sharpe rato s subject to dynamc manpulaton, alpha and both 4 If asset s combned wth the market nto a portfolo wx ( w) xmkt 5 The beta of the maxmal-sharpe-rato portfolo, xmsr, s +, then ( S ) =α σmkt w w = 0. Cov[ x, x ] Cov[ x, x ] Var[ x ] x Var[ x ] x β = = = MSR mkt MSR mkt MSR mktm MSR MSR mkt MSR Var[ xmkt ] Var[ xmsr ] Var[ xmkt ] xmsr Var[ xmkt ] xmkt m SMSR S where S mkt s the Sharpe rato of the market. The thrd equalty follows snce the MSRP s mean-varance effcent by defnton so xmkt = xmsr Cov[ xmsr, xmkt ] Var[ xmsr ]. 6 Treynor s (965) orgnal defnton of hs measure was rf α/ β; however, αβ / s now the commonly accepted defnton. 7 The Treynor rato can be computed from the MSRP s α n equaton (6) and ts β n footnote 5. To compute the Treynor apprasal rato, we use α and Var[ x β x ] = Var[ x ] β Cov[ x, x ] +β Var[ x ] MSR MSR mkt MSR MSR MSR mkt MSR mkt xmsr xmsr Smkt xmkt xmsr Smkt MSR MSR xmkt SMSR xmkt SMSR Smkt SMSR SMSR = Var[ x ] β Var[ ] = =. 8 A portfolo that shorts β MSR dollars n the market for every dollar n the MSRP wll have a beta of zero and a postve alpha. Therefore, ts Treynor rato wll be +. 7

10 Treynor measures are as well and can be ncreased above these statcally acheved values. The manpulaton s llustrated n Fgure. If returns have been above (below) average, then decreasng (ncreasng) the leverage n the future, wll generate a market lne for the portfolo wth a postve alpha. Not surprsngly, ths s very smlar to the dynamc manpulaton that produces a superor Sharpe rato leverage s decreased (ncreased) after good (bad) returns. [Insert Fgure here] Alpha-lke measures can also be computed from models other than the CAPM. Under qute general condtons, the generalzed alpha of an asset or portfolo n a sngle-perod model s 9 Cov[ u ( + r ), ] gen f + x m x α p = xp Bpxm where Bp =. (8) Cov[ u ( + r + x ), x ] f m m B p s a generalzed measure of systematc rsk, r f s the per-perod (not-annualzed) nterest rate, and u( ) s the utlty functon of the representatve nvestor holdng the market. The systematc rsk coeffcent, B, can be estmated by regressng x on x m usng u ( ) as an nstrumental varable. Ingersoll (987) and Leland (999) suggest usng a power utlty functon where n[ E( + rf + x m)] n( + rf ) ρ u ( + rf + xm) = ( + rf + xm) wth ρ=. Var[ n( + r + x )] f m (9) Ths generalzed alpha gves a correct measure of msprcng assumng that the representatve utlty functon s correctly matched to the market portfolo; that s, f the market portfolo does maxmze the utlty functon employed. But ths statement, tautologcal as t s, only apples to sngle-perod or statc manpulaton. Even f the utlty s correctly matched to the market, strategcally rebalancng the portfolo over tme can gve an apparent postve alpha due to the devaton between the average and a properly condtoned expectaton. The basc technque s the same decrease (ncrease) leverage after good (bad) returns. Beyond the sngle-factor CAPM, there are several alpha-lke performance measures n use. These models augment the sngle factor CAPM model wth addtonal rsk factors such as the Fama-French factors. However, wthn the smulatons dscussed n the next sub-secton, no such factors exst. The market returns are generated n an envronment where (8) wth α gen = 0 s the correct way to prce. Ths sngle factor model should do at least as well n the smulated envronment as any model wth addtonal, but wthn the smulaton, unprced, factors. Smlarly, f a mult-factor model were smulated, a mult-dmensonal rebalancng of the portfolo should produce postve alphas. For ths reason, the Chen and Knez (996) measure and smlar measures have not been examned. D. Manpulatng the Sharpe Rato and Alpha wth Transactons Costs In practce, a manager can change a portfolo s characterstcs far more uently than once durng the typcal measurement perod. On the other hand, the costs of transactng may 9 See Ingersoll (987) for the dervaton of ths general measure of systematc rsk. 8

11 elmnate the apparent advantage of many manpulaton strateges. As we show below even very hgh transacton costs cannot prevent managers from manpulatng the Sharpe rato or other performance measures. Determnng the optmal dynamc manpulaton strateges for all of the popular performance measures s beyond the scope of our paper. The optmal manpulaton strategy depends on the sze of the transacton costs, the complete set of returns to date wthn the evaluaton perod, the dstrbuton of future returns, and the number of perods remanng n the evaluaton perod. However, for our purposes, an optmal strategy s not necessary. We only want to establsh whether reasonably smple tradng strateges can dstort the exstng measures even n the presence of transactons costs, and f so, by how much. [Insert Table I here] Table I shows the Sharpe rato performance of a dynamcally rebalanced portfolo. The portfolo s always nvested 00% n the market wth leverage; no dervatves are used to alter the dstrbuton. The leverage starts at one. After the frst year, the portfolo s levered at the begnnng of each month so that the target mean s gven by equaton (4) though the leverage s also constraned to be between 0.5 and.5 so the portfolo wll not be too extreme n nature. The leverage s acheved by buyng or sellng a synthetc forward contract consstng of a long poston n calls and a short poston n puts that are at-the-money n present value.e., the strke prce per dollar nvested n the market s e r t. Trades n these two optons are assessed a roundtrp transacton cost of 0%, 0%, or 0%. The smulaton conssted of 0,000 repettons. On average the dynamcally-manpulated portfolo s Sharpe rato s 3% hgher than the market s and 7% hgher than the MSRP s. In the 0,000 trals, the manpulated portfolo had a Sharpe rato hgher than the market s 8.6% of the tme n the absence of transactons costs. 0 Even wth a 0% round-trp transactons cost, the dynamcally-manpulated portfolo stll beat the market almost three-quarters of the tme. It should be mentoned here that comparng these wnnng percentages to 50% s a conservatve test of how good the manpulated Sharpe rato s. The CAPM predcts that any portfolo s Sharpe cannot be bgger than the market s not that t wll be bgger or smaller wth some measurement error. The Sharpe-manpulated portfolo beats the market uently, but s t substantally better statstcally; that s, s the dfference n the Sharpe ratos sgnfcant? In practce ths queston s seldom asked because Sharpe ratos are at least as dffcult to estmate precsely as are mean returns. However, dfferences between two Sharpe ratos can be more precsely estmated 0 The uency wth whch one portfolo beats another must be nterpreted wth cauton when the wnnng portfolo holds dervatves. To llustrate, consder two portfolos that are almost dentcal. The only dfference s the second portfolo sells dervatve assets that have a postve payoff only rarely, lke deep out-of-the-money optons. The proceeds are nvested n bonds. Whenever, the payoff event for the dervatve does not occur, then the second portfolo s returns wll be unformly greater than the frst portfolo s returns. Ths wll lead to a better outcome for the second portfolo by almost any performance measure. Snce the payoff event can be made as rare as desred wth dervatve contracts, any portfolo, even an ex ante optmal one, can be beaten often. In our smulatons here only the leverage s changed so they are not subject to ths problem Lo (00) has shown that the asymptotc standard error of the Sharpe rato of a portfolo wth d returns s / / [( +S ) / T ] so the coeffcent of determnaton s S [( + S ) / T ]. The coeffcent of determnaton for the / mean return s T x T / / σ =S. The former s larger by a factor of ( +S ) so the same dfference n percentage terms s less sgnfcant for a Sharpe rato than t s for an average return. 9

12 f the underlyng returns are correlated as would generally be true and s certanly true n our example. Statstcal tests of portfolo returns generally assume that the returns are ndependent and dentcally dstrbuted over tme. Such s not the case here for the manpulated portfolo whch makes dervng the dstrbuton of the dfference n the Sharpe ratos a dffcult task. Furthermore, there s lttle pont n dong so as our smulaton s a constructed example, and the statstc we derve would be applcable only to that case. Fortunately we need not do so. Our smulatons gve us a sample dstrbuton of the dfferences, so we can easly determne how often the dfference s k standard devatons above or below zero. For example, wth no transactons costs, the dfference between the manpulated Sharpe rato and the market Sharpe rato was more than.65 standard devatons above zero 0.4% of the tme. It was never more than.65 standard devatons below zero. Were the dfference normally dstrbuted, each of these should have occurred 5% of the tme. Of course, the dfferences between the Sharpe ratos s not normal, but ths s stll evdence that a properly constructed test would conclude that the dfference was sgnfcant more often than chance would prescrbe f the true expected dfference were zero. Table I gves the percentages of tmes that the Sharpe rato dfference exceeds.65 standard devatons. Usng Jensen s alpha or one of the Treynor measures to evaluate performance gves even better results. The dynamc portfolo has an average annualzed alpha over % and the alpha s postve more than 9% of the tme n the absence of transactons costs. Even wth 0% round-trp costs, the average alpha s stll.6% and s postve over 85% of the tme. The generalzed alpha shows only slghtly weaker performance for the manpulated portfolo. In the smulatons we had the luxury of knowng the correct utlty functon, but superor performance as measured by the generalzed alpha s nsenstve to rsk averson assumed. Smlar superor performance was found for rsk aversons throughout the range to 4. It s not surprsng that the alphas can be manpulated more easly than the Sharpe ratos. The CAPM null hypothess s that α = 0. Wth measurement error postve and negatve devatons are approxmately equally lkely. However, the CAPM null hypothess on the Sharpe rato s that t s less than the market s. The manpulaton on the Sharpe rato has to frst make up ths dfference before beatng t.. Manpulatng Other Measures A. Reward-to-Varablty Measures Many other performance measures have been proposed over the years to correct perceved flaws n the Sharpe rato or to extend or modfy ts measurement. Some of these measures use a benchmark portfolo usually some market ndex. All of them are subject to the same type of manpulaton that can be used on the Sharpe rato. In ths and the next sectons we examne many of the more popular alternatves. Modglan and Modglan's (997) M-squared score s smply a restatement of the Sharpe rato. The M-squared measure s the expected excess return that would be earned on a portfolo f t were levered so that ts standard devaton was equal to that on the benchmark. Clearly maxmzng the Sharpe rato also maxmzes the M-squared measure relatve to any benchmark. Therefore, t s also subject to the same manpulaton. Sharpe s nformaton rato s another smple varaton on the orgnal Sharpe rato. The 0

13 dfference s that the excess returns are calculated relatve to a rsky benchmark portfolo rather than the rsk-free rate. If x and x b are the excess returns on the portfolo and the benchmark, the nformaton rato s x xb Snfomaton =. (0) Var( x x ) Snce excess returns are the returns on zero net cost (or arbtrage) portfolos, they can be combned smply by addng them together wthout any weghtng. Clearly the arbtrage portfolo that s a combnaton of the excess returns x and x b has an nformaton rato wth respect to the benchmark, x b, numercally equal to the Sharpe rato of x alone; therefore, the nformaton rato s subject to the same manpulaton as the Sharpe rato. In partcular, addng the MSRP excess returns to the market portfolo wll acheve the hghest possble nformaton rato relatve to the market. One crtcsm commonly leveled aganst the Sharpe rato s that very hgh returns are penalzed because they ncrease the standard devaton more than the average. Ths s the reason the MSRP has bounded returns. To overcome ths problem, t has been suggested to measure rsk usng only bad returns. In partcular, Sortno (99) and others have measured rsk as the root-mean-square devaton below some mnmum acceptable return. Van der Meer, Plantnga and Forsey (999) have further suggested that the reward n the numerator should only count good returns. Sortno s downsde-rsk and van der Meer, Plantnga and Forsey s upsdepotental Sharpe-lke measures are b E[ x ] E[Max( x x,0)] D = U = ( E[Mn ( x x,0)] ) ( E[Mn ( x x,0)] ) / /. () The mnmal acceptable excess return, x, s commonly chosen to be zero. Whle these two measures do avod the problem nherent n the Sharpe rato of penalzng very good outcomes, they do ths too well so that the hghest possble returns are sought n leu of all others. As shown n the Appendx, the Sortno downsde-rsk and VPF upsde-potental maxmzng portfolos are very smlar. They both hold the MSRP, an extra very large nvestment n the state securty for the state wth the hghest market return (the lowest lkelhood rato pˆ / p pˆ / p ), and bonds. I I The mnmum acceptable return s sometmes set to the portfolo s average return for the Sortno measure. Ths makes the denomnator the sem-standard devaton. The average return should not be used lke ths for the VPF measure as the rato s then nvarant to a unform upward shft n the entre probablty dstrbuton.

14 MDR x p MSR MSR MDR ˆ I S pi x p I x = ( + SMSR ) x ( ) I xi ˆ + < = ˆ D DMDR pi DMDR pi DMDR pi I MDR = + pˆ I MUP pˆ MSR MUP ˆ I x pi( p ) I x ( ˆ = U pi) S MSR x x( MSR ) I xi p + S < = ˆ () I pi piu MUP where D p p p / and /. I I I I pˆ ˆ p U MUP = p p = pˆ ˆ I pi = D MDR and U MUP are the largest possble Sortno downsde rsk and VPF upsde potental ratos. Fgure 3 llustrates the maxmal Sortno and VPF rato portfolos. The one-month return on the market has a twenty-perod bnomal dstrbuton that approxmates a lognormal market wth a contnuously-compounded rsk premum of % and a logarthmc volatlty of σ = 0%. 3 There are twenty perods over the one month and twenty-one fnal states. Both portfolos acheve a very hgh return n the best market outcome state and have a negatve excess returns n all but ths best state. 4 The annualzed Sortno and VPF ratos of the market are.00 and.86. The scores for the optmzed Sortno rato portfolo are over twce as hgh at.83 and The same s true for the optmzed VPF portfolo; t scores.76 and 6.7. [Insert Fgure 3 here] Clearly the maxmal Sortno and VPF rato portfolos are far from optmal, and even a cursory examnaton of the perod-by perod returns would lkely attract very negatve attenton. For example, both of these portfolos have Sharpe ratos that are less than a tenth that of the market. Conversely the MSRP has a Sortno and VPF rato just a bt below that of the market. The maxmal-rato strateges are also rsky n a samplng sense. The best market state s lkely to have only a small probablty of occurrng, and should t fal to occur durng the evaluaton perod, the sample averages and therefore the ratos would be negatve even n a bull market. On the other hand, should no below- x outcomes occur, both ratos would be nfnte. Of course, the maxmal-rato portfolos need not be employed to manpulate the measures. The payoffs n Fgure ndcate that a hgh rato can probably be acheved merely by purchasng calls the further out-of-the-money the better. For example, n our smulaton envronment, smply nvestng the entre portfolo n at-the-money market ndex calls each month has a Sortno rato of.0 and a VPF rato of 3.68 substantally hgher than the market s. Usng ten percent out-of-the-money calls ncreases these ratos to.79 and 5.0. As wth the Sharpe rato, dynamc manpulaton can also ncrease the measure or ncrease the measure further when used along wth statc manpulatons. One smple scheme can often acheve an nfnte rato when the mnmal acceptable excess return s zero (or negatve). If the frst excess return s postve (whch should be true more than one-half of the tme), then holdng 3 A bnomal market envronment s used as an llustraton here because n a lognormal market or any market wth a contnuous dstrbuton, the maxmal Sortno and VPF portfolos have an nfntesmal negatve excess return except n the hghest return state. In ths state, whch occurs wth zero probablty n a market wth a contnuum of states, the return s nfnte. 4 For any choce of x, both portfolos have returns less than x n all but the best state.

15 the rsk-free asset for every other perod wll gve a postve numerator and a denomnator of zero resultng n an nfnte measure. To llustrate the dynamc manpulaton possbltes of these two measures even wth very restrcted portfolos, we employ smulatons smlar to those used for the Sharpe rato. If the hstorcal returns up through tme t 0 have an average excess return of x h and a Sortno measure of D h, then the portfolo s measured Sortno rato over the entre perod wll be γ xh + ( γ) xf γ xh + ( γ) xf D = = γ Max ( x,0) + ( γ) Max ( x,0) γ x D + ( γ) x D t< t t t h h f f 0 t> t0 (3) where the subscrpt f denotes future performance. As wth the Sharpe rato, the overall Sortno rato s maxmzed by maxmzng the Sortno rato n the future and selectng the leverage so that xh f h for xh 0 x D D > f = (4) for xh 0. Table II shows the Sortno and VPF rato performance of a dynamcally rebalanced portfolo under the same condtons used for the Sharpe rato. Agan no dervatve assets were used to statcally manpulate the ratos. The market s held each month, and only the leverage s changed, each month after the th, as descrbed n (4). The manpulated portfolos produce statstcs that are superor to smply holdng the market portfolo for both modfed measures as well as the Sharpe rato. 5 The Sortno rato s hgher than the market s more than 8% of the tme even wth 0% round-trp transactons costs. It s sgnfcantly hgher at the 5% confdence level almost 4% of the tme. [Insert Table II here] The Sharpe and VPF ratos are also hgher nearly three-quarters of the tme for ths manpulaton. The same manpulaton produces good results for all three ratos because each of the ratos has the same general dynamc manpulaton rule decrease leverage after good luck (after the hstorcal score has been hgh). The best dynamc manpulaton VPF rato s almost the same. The target n the future s proportonal to the upsde realzaton n the past and the rato of the squares of the expected and realzed VPF measures;.e., E[Max( x f,0)] = Avg[Max( x h,0)] U f U h. The smulaton results for the VPF measure manpulaton are not presented here as they were very smlar. Understandably, ths manpulaton produces somewhat hgher average VPF ratos and beats the market s VPF rato more often whle producng somewhat lower Sortno ratos. B. Henrksson-Merton and Treynor-Mazuy Tmng Measures 5 The smulatons also show that for a base lognormal dstrbuton, the sample Sortno and VPF ratos are almost certanly based (lke the Sharpe rato for the MSRP and unlke (approxmately) the Sharpe rato for the market). The true Sortno and VPF ratos for the market are.00 and.844. The sample averages were.7 and.97 wth standard errors of and So the sample averages were hgher than the true values by more than and 5 standard errors, respectvely. 3

16 The tmng measures of Treynor and Mazuy (966) and Henrksson and Merton (98) are bvarate regressons that use an extra market factor to capture managers tmng abltes (rather than to capture an addtonal source of rsk). The regressons used are x =γ +γ m +γ w +ε (5) t 0 t t t where x t and m t are the excess returns on the portfolo and the market and w t s the varable used to capture tmng ablty. The tmng varables are w t Max( m t,0) and w t m t for HM and TM, respectvely. The HM model s consstent wth a manager who changes hs portfolo leverage between two values dependng on whether he forecasts that the market return wll exceed the rsk-free rate or not. As shown by Admat, Bhattacharya, Pflederer, and Ross (986), the TM model s consstent wth a manager whose target beta vares lnearly wth hs market forecast. Unlke the measures we have already consdered, performance here s characterzed by two numbers. In each case a postve γ s an ndcaton of market tmng ablty and a postve γ 0 s consdered a sgn of superor stock selecton. However, as shown by Jagannathan and Korajczyk (986), smple tradng strateges ncludng or mmckng optons can produce postve γ s and negatve γ 0 s or vce versa. 6 An unequvocal demonstraton of tmng ablty must, therefore, satsfy a jont test on the two coeffcents. Merton (98) has shown that the total contrbuton of the manager s tmng and selectvty n the HM model s γ 0 e r t + γ P(, t, e r t ) per dollar nvested. P(S, τ, K) s the value of a τ-perod put opton on the market wth a strke prce of K. The ntuton for ths concluson s that a market tmer who could forecast perfectly whether or not the market return would exceed the nterest rate and who adjusted hs portfolo to be fully nvested n the market or bonds could essentally provde such a put for free. In addton hs selectvty ablty would provde the present value of γ 0 per dollar nvested. The total contrbuton s the amount by whch the value of the protectve put exceeds ts average cost measured by the lowered present value of the extra average return. Smlarly, under the TM model, the total contrbuton of the manager s nvestment ablty per dollar nvested s gven by the same formula where the value of a dervatve contract that pays the square of the market s excess return replaces the put s value. Ths value s derved n Appendx D. The total contrbuted value, V, of the money manager s contrbuton to tmng and selectvty s r t r t VHM =γ 0e +γp (, t, e ) (6) r t r t σ t V =γ e +γ e ( e ). TM 0 Unfortunately wth a complete market of dervatve assets, the TM and HM measures can be manpulated to any degree desred; that s, any values for the γ s can be acheved wth many dfferent portfolos. The mnmum-varance portfolo that acheves any partcular (postve) target values for γ 0 and γ wll be long n the MSRP and the tmng put struck at-the-money n 6 For example, buyng (wrtng) calls produces a postve (negatve) γ and a negatve (postve) γ 0. The MSRP also has a negatve γ and a postve γ 0. It should be noted that due to the γ term n these regressons, γ 0 and γ are not that standard Jensen s alpha and CAPM beta. 4

17 present value terms. 7 It mght be short or long n the market and bonds dependng on the targeted mean excess return and market beta. [Insert Fgure 4 here] Fgure 4 llustrates the payoffs on the mnmum-varance zero-beta portfolos wth a tmng target of γ = 0. and a selectvty target of γ 0 = 0 bass ponts per month. The envronment s the same as before a lognormal market wth a rsk premum of % per year and a volatlty of 0%. The graph dsplays approxmately the mddle 95% of the return dstrbuton. The tmng returns fluctuate from local extremes near plus and mnus 0% over the market range from 7% to +7%. In practce, of course, only scattered ponts on the curves would be seen, and the portfolo would probably be descrbed as somewhat volatle but otherwse normal. Reexamnng Fgure we see that the alpha-manpulatng strategy that levers down after good returns and levers up after poor returns has a general convex shape. Ths strategy wll lkely produce a false postve tmng performance. Smulatons verfy that ths s true; however, whle γ and the total contrbuton as measured n (6) are postve, smply changng leverage also yelds a (false) negatve selecton ablty. Both γ 0 and γ can be made postve f a call poston s wrtten aganst the portfolo snce ths produces an extra return when the market has a small return. [Insert Table III here] Table III shows the performance of a manpulated portfolo that holds the market and wrtes a ten percent out-of-the-money call on % of ts holdngs; that s, the portfolo shares only 98% of the market returns above 0% n any month. After one year, the leverage s changed every month to equal 0.5 or.5 f the fracton of months that the market s return was less than average was more or less than half. The resultng portfolo produces a total contrbuted value of around 98 to 56 bass ponts per year dependng on transactons costs as measured by ether HM or TM. Our smulatons show that the total contrbuted value was postve about 65% to 70% of the tme and sgnfcantly so (at the 5% level) 9% to 4% of the tme. The results are not as strong as for the other measures, but ths s not surprsng. The HM and TM measures are tmng measures specfcally desgned to for tmng ablty and should be able to better reject our dynamc manpulatons than the other measures dscussed so far. The components of performance, selectvty and tmng, do not have as strong an ndvdual showng because the smulaton was desgned to produce over all good performance and the two measures are negatvely correlated. Increasng the out-of-the-money call sales, for example, would ncrease γ 0 and decrease γ. However, each component f postve more than half of the tme and sgnfcantly so more often than chance would allow. 3. Manpulaton Proof Performance Measures In the precedng sectons of ths paper, we have establshed that popular measures of 7 See secton D of the Appendx for a proof. For the Treynor-Mazuy measure the portfolo needs to be long n an exotc dervatve contract payng the square of the market prce rather than put optons. 5

18 performance are susceptble to manpulaton even when transacton costs are hgh. We have shown that manpulaton requres no superor nformaton and lttle techncal sophstcaton. Smple schemes lke tradng puts and calls or smply alterng a portfolo s leverage can often create performance that looks superor. Furthermore, even f money managers are not seekng to manpulate ther performance numbers, compettve evoluton may favor those managers whose strateges happen to look lke our manpulaton strateges f these measures are wdely used. We have shown that these problems can be of concern even when evaluatng well-dversfed portfolos of equtes, but ncreasngly money s beng nvested n hedge funds and other venues whose return dstrbutons dffer substantally from those hstorcally found n all-equty funds. For such funds even more problems may arse. Such problems could be avoded wth a manpulaton-free measure. But what does t mean for a measure to be manpulaton-free? What exactly should t encourage a manager to do or not do? Intutvely, f a manager has no prvate nformaton and markets are effcent, then holdng some benchmark portfolo, possbly levered, should maxmze the measure s expected value. The benchmark mght be the market, but n some contexts other benchmarks could be approprate. Statc manpulaton s the tltng of the portfolo away from the (levered) benchmark even when there s no nformatonal reason to do so. Dynamc manpulaton s alterng the portfolo over tme based on performance rather than new nformaton. Our goal s to characterze a measure that punshes unnformed manpulaton of both types. Formally a performance measure s a functon of the portfolo s probablty dstrbuton across the outcome states. That s, f r s the vector of returns across all the possble outcomes, then the performance measure s a real-valued functon, Θ(), r of those returns. 8 In practce, of course, we do not know the true dstrbuton, and the estmated performance measure must be a functon of the returns realzed over tme. We wll denote the estmated performance measure as ˆ (, T Θ r ). t s 9 t Each tme perod and ts return, r t= t, can, but need not, be dentfed wth the state, s t, that occurred. For example, estmatng the Sharpe rato requres knowng only the returns whle to estmate Jensen s alpha we need to know the contemporaneous market return for each return. One property that any manpulaton-proof performance measure (MPPM) must have s that t recognzes arbtrage opportuntes as good. If the portfolo-return dstrbuton r domnates the portfolo return dstrbuton r, then the performance measure must rank t hgher, Θ ( r) >Θ( r ). Ths same property should hold for the estmated performance measure, though the domnance n ths case may only be apparent as not all states may have been realzed. We already know that the Sharpe rato fals ths smple test f the returns n one or more states 8 In prncple, a performance measure could be par or more of numbers (lke γ 0 and γ n the TM and HM models or mean and varance). In general, of course, such performance measures would not provde a complete rankng of portfolos. In one sense mult-valued performance measures would be harder to manpulate snce they would provde more nformaton. In another more practcal sense, mult-valued performance measures would be easer to manpulate snce wthout a complete rankng there would be more portfolos that the market benchmark dd not beat. We confne our attenton to sngle-valued performance measures lke the Sharpe rato. 9 Usng the tme seres of returns n the estmated performance measure ˆΘ to represent the true performance measure assumes that the realzatons of portfolo returns (and states) are suffcently ndependent and dentcally dstrbuted over tme so that the uences are wll be equal to the ex ante probabltes n a suffcently large sample.. 6

19 are ncreased suffcently, the rato wll decrease even though the return dstrbuton unequvocally mproves based on the observed evdence. However, ths property whle necessary s not suffcent to make a measure manpulatonproof. To have a performance measure that s proof aganst dynamc manpulaton, the functon must have a strong ndependence property such that the alterng of some components does not affect the relatve rankng based on other components. A performance measure cannot be dynamcally manpulated only f the relatve rankngs of dfferent futures are unaffected by the partcular hstory that has occurred. 0 Not that ths does not say that the hstory does not affect rankngs only that two dfferent futures can be compared ndependently of the past. Ths ndependence property s known from utlty theory where t s called (strong) utlty ndependence or strong utlty separaton. (See e.g., Debreu (960).) Furthermore, a rankng (.e., ordnal) functon has ths ndependence property f and only f t has an addtve representaton. That s, the only possble MPPMs can be expressed as T ( r s t= ) T ˆ Θ t, t = ϒ θt( rt, st). T (7) t= The functon ϒ( ) s any ncreasng functon and s rrelevant for rankng; therefore, t can be take to be the dentfy functon except when otherwse more convenent. We chose to wrte the argument as an average rather than a smple sum because most performance measures are based on some knd of average. Each functon θ t must be ncreasng f ˆΘ s to satsfy the arbtrage s good crteron. Generally, each tme perod s treated dentcally, apart from any state dependence, so that all θ t are the same. In ths case, any performance measure free from dynamc manpulaton must be equvalent to a smple average of some, possbly state-dependent, functon of returns. Some smple performance measures that cannot be dynamcally manpulated are the average return, θ(r t ) = r t; the average market-adjusted return, θ(r t, s t ) = r t m t (s t ); and the geometrc average return, θ(r t ) = n ( + r t ). Any functon θ wll create a performance measure, Θ, that s proof aganst dynamc manpulaton. To prevent statc manpulaton, θ must be strctly ncreasng, as prevously noted, and concave n the returns. If θ s not concave, then leverng the portfolo by borrowng wll ncrease the measure as wll smply addng unprced rsk. Concavty s a natural property to demand snce t s also necessary (and suffcent) f we want to rate as better the less rsky (n a Rothschld-Stgltz sense) of two portfolos wth the same average return. Ths analyss shows that manpulaton-proof evaluatons of portfolos are computed as the expectaton, or n practce the average, of some functon of returns. The natural set of performance measure functons thus s the same as the set of common utlty functons, n that both are tme-separable, ncreasng and concave. The only remanng queston s what functons can be 0 0 τ T 0 τ T Precsely, the measure s proof aganst dynamc manpulaton only f ˆ ( r ˆ t, r t ) ( rt, r τ+ t τ+ ) τ T τ T ( r, ) ˆ t r t ( rt, r τ+ t τ+ ) Θ ˆ >Θ for any hstory condtonng subsets. Θ >Θ mples r τ. Ths ndependence property must also hold for all t An dentcal treatment of each perod s return s not a requrement for the measure to be manpulaton proof. For T example, the tme-dependent average, θ t (r t ) = δ can be used. Ths puts extra emphass on recent returns. t r t 7

20 used. Any measure Θ as defned n (7) wth θ t an ncreasng, concave functon of the return s a manpulaton-proof performance measure. But f we wsh the measure to be consstent wth a market n equlbrum, θ must be further restrcted to have a power form. There are two justfcatons for ths concluson. Snce utlty s defned over wealth or consumpton, whle performance measures are defned over returns, an MPPM that treats returns dentcally s consstent wth expected utlty theory only f the utlty of returns s ndependent of wealth. Ths s true amongst addtvely separable utlty functons only for power utlty. Addtonally as shown by Rubnsten (976) and He and Leland (993), f the market portfolo s rate of return s ndependent and dentcally dstrbuted over tme and markets are perfect, then the representatve nvestor must have power utlty. The MPPM measure we choose here s T ˆ ρ Θ n ( rft ) ( rft xt ). ( ) t T (8) ρ t= where r ft and x t are the per-perod (not annualzed) nterest rate and the excess return on the portfolo over perod t. The functon θ( ) s smlar to a power utlty functon wth relatve rsk averson of ρ. We have transformed the measure (usng ϒ) so that ˆΘ can be nterpreted as the annualzed contnuously-compounded excess return certanty equvalent of the portfolo. That s, a rsk-free portfolo earnng exp[ n ( + r ) ˆ ft +Θ t] each perod would also have a measured performance of Θ ˆ. Fnally, we want to assocate the MPPM wth some benchmark portfolo. Ths would typcally be some market ndex. In the absence of any prvate nformaton, we want the MPPM to score the chosen benchmark hghly. If the benchmark portfolo has a lognormal dstrbuton, then the parameter ρ should be selected so that n E( + rf + x b) n( + rf) ρ=. Var n( + rf + x b) (9) Hstorcally ths number s about to 4 for the CRSP value-weghted market portfolo dependng on the tme perod and uency of data used. We have used a relatve rsk averson of 3 n our smulatons. Table IV shows the performance for our manpulated portfolos n our prevous smulatons. The MPPMs are determned for three dfferent rsk aversons,, 3, and 4. 3 The Ths measure s not the same as usng power utlty of termnal wealth. Power utlty of termnal wealth would be ρ ( ρ ) ( + rft + xt ) or a monotonc transformaton of ths. Utlty of termnal wealth s a vald MPPM but s not a useful rsk-adjusted measure snce for any partcular realzaton of returns, t always rates as best whchever portfolo acheved the largest total accumulaton regardless of the rsk averson assumed. 3 The range of relatve rsk aversons n the market mght be much wder than to 4; however, ths range s meant to capture the range of rsk aversons that would be relevant for the constructon of mutual funds or other well dversfed portfolos. Relatve rsk aversons of to 4 correspond to portfolos that would range from leveraged postons of.5 down to Ths should encompass most portfolos that would be ranked. 8