port performance manipulation

8/7/2019 port performance manipulation

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Yale School of Management, PO Box 208200, New Haven, CT 06520-8200 Brown University, Department of Economics Box B, Providence, RI 02912

This paper has benefited from the comments of an anonymous referee and numerous colleagues including theparticipants at the Five Star Conference hosted by New York University, the Hedge Fund Conference hosted byBorsa Italiana, and the Berkeley Program in Finance.

Portfolio Performance Manipulation andManipulation-Proof Performance Measures

William Goetzmann, Jonathan Ingersoll,

Matthew Spiegel, Ivo Welch

March 15, 2006


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Portfolio Performance Manipulation and

Manipulation-Proof Performance Measures

Fund managers naturally claim that they can provide superior performance. Investorsmust somehow verify these claims and discriminate among managers. In 1966, William Sharpe,using mean-variance theory, introduced the Sharpe ratio to do this in a quantifiable fashion.Together with its close analogues, the information ratio, the squared Sharpe ratio and M-squared,the Sharpe ratio is now widely used to rank investment managers and to evaluate theattractiveness of investment strategies in general. Other measures such as Jensens alpha and theHenriksson-Merton market timing measure, although they evaluate other aspects of performanceare also based on the same or similar theories.

If investors use performance measures to reward money managers, then money managershave an obvious incentive to manipulate their performance scores. Our paper contributes in twoways to this literature. First, it points out the vulnerability of traditional measures to a number ofsimple dynamic manipulation strategies. Second, it offers a formal definition of the propertiesthat a manipulation-proof measure should have and derives such a measure.

For a measure to be manipulation proof, it must not reward information free trading. Inthis regard the existing set of measures suffer from two weaknesses. First, most were designedto be used in a world where asset and hence portfolio returns have nice distributions like thenormal or lognormal. But, this ignores the fact that managers can potentially use derivatives (ortrading strategies) to radically twist return distributions. Hedge fund returns, in particular, havedistributions that can deviate substantially from normality, and the hedge fund industry is one inwhich performance measures like the Sharpe ratio are most commonly employed. Second,exact performance measures can be calculated only in theoretical studies (e.g., Leland (1999)and Ferson and Siegel (2001)); in general, they must be estimated. Since standard statisticaltechniques are designed for independent and identically distributed variables, it is possible tomanipulate the existing measures by using dynamic strategies resulting in portfolios whose

returns are not so distributed.We show here that portfolios using dynamic strategies investing in index puts and calls

can be used to manipulate the common portfolio measures. This is true even when only veryliquid options are used and when transactions costs are extremely large. Numerical results arepresented for the Sharpe ratio, Jensens alpha, Treynor ratio, Appraisal ratio, generalized alpha,Sortino ratio (1991), Van der Meer, Plantinga and Forsey (1999) ratio, and the timing measuresof Henriksson and Merton (1991), and Treynor and Mazuy (1966). In every case, themanipulations presented generate statistically and economically significant better portfolioscores even though all of the simulated trades can be carried out by an uninformed investor.

The paper describes general strategies for manipulating a performance measure. (1)Static manipulation of the underlying distribution to influence the measure. Such strategies can

enhance scores even if the evaluator calculates the measure without any estimation errorwhatsoever. (2) Dynamic manipulation that induces time variation into the return distribution inorder to influence measures that assume stationarity. This can enhance a portfolios score even ifthe measure is calculated without any estimation error but without regard to the (unobservable)time variation in the portfolios return distribution. (3) Dynamic manipulation strategies thatfocus on inducing estimation error. Since measures have to be estimated with real world data,there are strategies that can induce positive biases in the resulting values. As a simple example,imagine an evaluator who estimates a funds Sharpe ratio by calculating its excess returns


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standard deviation and average excess return over a 36 month period using monthly data.Further assume the fund manager simply wishes to maximize the expected value of thecalculated Sharpe ratio. A simple strategy that accomplishes this has the fund sell an out of themoney option in the first month, while investing the remaining funds in the risk free asset. If theoption expires worthless the portfolio is then invested wholly in the risk free asset for the

remaining 35 months. Whenever this happens the portfolio has a zero standard deviation, apositive excess return, and thus an infinite Sharpe ratio. Since there is a strictly positiveprobability that the option will expire worthless the expected value of the calculated Sharpe ratiomust be infinity. Worse yet, as the paper shows even with goals other than maximizing theexpected calculated performance value, and concern for other moments of the test statistic amanager can deliberately induce large economically significant measurement errors to hisadvantage. Also notice that increasing the frequency with which returns are observed does notshield the set of existing measures from this type of dynamic manipulation.

If the set of current measures are vulnerable to manipulation the question naturally arisesas to whether a manipulation proof measure exists. To answer this question, one first needs todetermine what manipulation proof means. This paper defines a manipulation proof

performance measure (MPPM) as one that has three properties:

1. The scores value should not depend upon the portfolios dollar value.2. An uninformed investor cannot expect to enhance his estimatedscore by deviatingfrom the benchmark portfolio. At the same time informed investors should be able toproduce higher scoring portfolios and can always do so by taking advantage of arbitrageopportunities.3. The measure should be consistent with standard financial market equilibriumconditions.

It turns out that these three requirements are enough to pin down the measure. The first simplyimplies that returns are sufficient statistics rather than dollar gains or losses. The second andthird, however, provide a great deal more structure. As the earlier discussion indicates there areseveral ways to manipulate a performance measure and the second condition implies that aMPPM must be immune to all of them. In particular that means the manager cannot expect tobenefit by trying to alter the scores estimation based upon observable data. To accomplish thisgoal the score must be (1) increasing in returns (to recognize arbitrage opportunities), (2)concave (to avoid increasing the score via simple leverage or adding unpriced risk), (3) timeseparable to prevent dynamic manipulation of the estimated statistic, and (4) have a power form

to be consistent with an economic equilibrium. The resulting measure ( ) suggested here is

11

1

1 1 n (1 ) (1 ) .(1 )

T

ft ft t t

r r xt T

=

+ + +

(1)

The coefficient can be thought of as the evaluators risk aversion and should be selected tomake holding the benchmark optimal for an uninformed manager. The variable tis the lengthof time between observations while Tequals the total number of observations. These twovariables (tand T) simply help normalize the measures final calculation. The portfolios returnat time tisxt, and the risk free assets return is rft. Note the measure is easy to calculate and hasan intuitive interpretation as the average per period welfare of a power utility investor in the


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portfolio over the time period in question. As an example, consider a fund with returns of.1,.05, .17, and .02. Assume the risk free rate is .01. In this case T=4, rft is .01, and if equals 2

then equals 1.036. For equal to 3, equals 1.109.Our paper was originally motivated by the question of whether the existing set of

performance measures is sufficiently manipulation proof for practical use. This is particularly

relevant in the presence of transactions costs, which may offset whatever performance gains amanager might hope to generate from trading for the purpose of manipulating the measure.Therefore, our paper explores how difficult it is to meaningfully game the existing measures.For the seven measures examined here four ratios (Sharpe (1966), Sortino (1991), Leland(1999), and Van der Meer, Plantinga, and Forsey (1999)), and three regression intercepts (theCAPM alpha, Treynor and Mazuy (1966), and Henriksson and Merton (1981)) the answer isnot encouraging. Simple dynamic strategies that only relever the portfolio each measurementperiod or buy (very liquid) at the money options can produce seemingly spectacular results, evenin the presence of very high transactions costs. For example, consider the Henriksson andMerton (1981) measure. A simple options trading scheme in the presence of transactions costsequal to 20% produces very good results. The final regression statistics report that the

portfolio has returns that are superior to the markets nearly 65% of the time, and (using a 5%critical value) statistically significantly better 9% of the time. Obviously, lower and morerealistic transactions costs only make matters worse. The other measures analyzed here aresimilarly susceptible to having their estimated values gamed.

Our results have a number of implications for investment management. Hedge funds andother alternative investment vehicles have broad latitude to invest in a range of instruments,including derivatives. Mitchell and Pulvino (2001) document that merger arbitrage, a commonhedge fund strategy, generates returns that resemble a short put-short call payoff. Recentresearch by Agarawal and Naik (2001) shows that hedge fund managers in general follow anumber of different styles that are nonlinear in the returns to relevant indices. In a mannersimilar to Henriksson and Merton, Agarawal and Naik use option-like payoffs as regressors to

capture these non-linearities. In fact, option-like payoffs are inherent in the compensation-structure of the typical hedge fund contract. Goetzmann, Ingersoll and Ross (2001) show that thehigh water mark contract, the most common type in the hedge fund industry, effectively leavesthe investor short 20% of a call option. The call is at-the-money each time it is reset by apayment and out-of-the money otherwise.

The paper is structured as follows. Section 1 discusses various ways in which the Sharperatio can be manipulated. Section 2 discusses the manipulation of a number of other measuresincluding those using reward to variability (Section 2A) and the Henriksson-Merton andTreynor-Mazuy market timing measures (Section 2B). Section 3 derives the manipulation proofperformance measure, and Section 4 concludes.

1. Manipulation of the Sharpe Ratio

Although the Sharpe ratio is known to be subject to manipulation, it remains one of if notthe most widely used and cited performance measures. It, therefore, makes a good example tointroduce performance manipulation in more detail. Other measures are examined in the nextsection.


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1A. Static Analysis

A number of papers have shown that by altering the distribution function governingreturns the statistical mean and variance can be manipulated to increase the Sharpe ratio to somedegree. Ferson and Siegel (2001) look at a fairly general case (that includes potentially private

information) while Lhabitant (2000) shows how using just a couple of options can generateseemingly impressive values. Since the focus of this paper is on dynamic strategies and thedevelopment of a MPPM to prevent all sources of manipulation the only static result presentedhere is the characterization of the return distribution that maximizes the Sharpe ratio. Readersinterested in additional details should consult one of the above cited articles.

The Appendix shows that Sharpe ratio maximizing portfolio in a complete market is

characterized by a state i return in excess of the interest rate ( SRi ) of

MSR

MSR 2

MSR

11 .i ii

p px x

= +

S(2)

Here1/ 2

2 1MSR i ip p S is the maximum possible Sharpe ratio, and pi and ip are the trueand risk-neutral probabilities of state i. Since the Sharpe ratio is invariant to leverage, we are

free to specify the portfolios mean excess return, MSRx , at any level desired.

While the MSR has a number of interesting properties it is also true that one way tomitigate its impact on a funds apparent performance is to sample returns more frequently. Asone does so the percentage difference between the market portfolios Sharpe ratio and that of theMSR goes to zero. However, as will be seen below frequent sampling does not help whenmanagers use dynamic manipulation strategies and thus the paper now turns to this issue.

1B. Dynamic Analysis

Calculated values of the Sharpe ratio and virtually all other performance measures usestatistics based on the assumption that the reported returns are independent and identicallydistributed. While this may be a good description of typical portfolio returns in an efficientmarket, it clearly can be violated if a portfolios holdings are varied dynamically depending onits performance.

Consider a money manager who has been lucky or unlucky with an average realizedreturn high or low relative to the portfolios realized variance. The return distribution in thefuture will likely not be similar to the experience. Consequently, to maximize the Sharpe ratio,the portfolio can be modified to take into account the difference in the distributions between the

realized and future returns just as if the returns distribution was not ex ante identically distributedeach period. This dynamic manipulation makes the past and future returns dependent whencomputed by an unconditional measure.

To illustrate, suppose the manager has thus far achieved an historical average excess

return of hx with a standard deviation of h . The portfolios average excess return and standard

deviation in the future are denoted by x and . Then its measured Sharpe ratio over the entire

period will be will be


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2 2 2 2 2

2 2 2 2 2

(1 )

( ) (1 )( ) [ (1 ) ]

(1 )

(1 1/ ) (1 ) (1 1/ ) [ (1 ) ]

h f

h h f f h f

h f

h h f f h f

x x

x x x x

x x

x x x

+ =

+ + + +

+ =

+ + + +

S

S S

(3)

where is the fraction of the total time period that has already passed, and h and f are the pastand future Sharpe ratios.

By inspection, the overall Sharpe ratio is maximized by holding in the future a portfolio

that maximizes the future Sharpe ratio,MSRf =S S . However, regardless of what Sharpe ratio

can be achieved in the future, leverage is no longer irrelevant as it was in the static case.Maximizing the Sharpe ratio in (3), we see that the optimal leverage gives a target mean excessreturn of

2 2(1 ) (1 ) for 0

for 0.h h f h

f

h

x x

x x

+ + >

=

S S

(4)

If the manager has been lucky in the past and achieved a higher than anticipated Sharpe

ratio, h f>S S , then the portfolio should be targeted in the future at a lower mean excess return

(and lower variance) than it has realized. This allows the past good fortune to weigh more

heavily in the overall measure.1

Conversely, if the manager has been unlucky and ,h f

+ + =

+

S S S S S

S SS

S SS

(5)

Figure 1 plots the overall Sharpe ratio as a function of the realized historical Sharpe ratio forhistories of different durations. The overall Sharpe ratio can, on average, be maintained above

1 For a fixed Sharpe ratio, the overall mean is linear in the future leverage while the overall standard deviation isconvex. The proportional changes in the mean and standard deviation with respect to future leverage are equal whenthe future and historical Sharpe ratios are equal. Therefore, when the historical Sharpe ratio is less than the futureSharpe ratio, increasing leverage increases the overall mean at a faster rate than the standard deviation and viceversa.

2 Because the optimal future leverage is infinite when 0,hx the past Sharpe ratio does not affect the overallSharpe ratio, and the value of the overall Sharpe ratio for 0h


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(is forced below) the theoretical maximum, MSRS , whenever the past performance has been good

(bad). The realized Sharpe ratio, of course, has the most impact when the past history is long (1).

[Insert Figure 1 here]

As can be seen in the graph or from equation (5), the over-all Sharpe ratio is increasing andconvex in the historical Sharpe ratio. Therefore, dynamic-Sharpe-ratio-maximizing strategiesshould, on average, be able to produce a Sharpe ratio higher than SMSR.

3Simulations show that dynamic manipulation of the Sharpe ratio can have a substantial

effect. In a lognormal market with a risk premium of 12%, a manager re-levering his portfolioafter thirty months of a five-year evaluation period can achieve a Sharpe ratio of 0.714. This is18% higher than the static MSR (and 13% higher than the standard biased estimate of the same).

In addition, dynamic manipulation works in exactly the same way for portfolios that have notbeen Sharpe optimized each period. Any portfolio with a Sharpe ratio ofSf can be levered asshown in (4) to achieve a higher average Sharpe ratio than this on average. For example, in thesame lognormal market, if the market is relevered after 30 months, its Sharpe ratio improves on

average from 0.597 to 0.672, an increase of 13%.In addition, we have assumed here that a single measurement period is shorter than the

time between rebalancings. If the portfolio manager can alter his portfolio within a singlemeasurement period, then the distribution of a single return can be affected. For example, if aportfolio increases sharply in value during the first part of a measurement period, switching to aless aggressive position may make the ultimate measured return smaller and keep the Sharperatio high. Goetzmann, Ingersoll, and Ivkovic (2000) analyze the problem of performancemeasurement when the investment rebalancing period is shorter than the measurement period.

Particular circumstances may permit additional opportunities to dynamically manipulatethe Sharpe ratio of a portfolio. Smoothing returns over time will leave the portfolios meanreturn unchanged but decrease its variance so smoothing returns will increase the Sharpe ratio.

Funds with illiquid assets whose prices are only reported occasionally may benefit here. Hedgefunds or other portfolios with high-water mark performance fees will also benefit. A standardperformance contract calls for expensing the performance fee monthly but paying it onlyannually, but unpaid fees can be lost based on poor later fund performance. This contractualprovision moves recorded returns from periods of good performance to periods of poorperformance and smoothes the reported returns.

1C. Jensens Alpha and Related Measures

Several other performance measures are related to the Sharpe ratio. The most familiar of

these is Jensens alpha which is the intercept of the market model regression, t mt t x x= + + .If the CAPM holds, this intercept term should be zero when the regression is expressed in excessreturn form.

Jensens alpha measures the marginal impact to the Sharpe ratio of adding a small

3 Brown, et al (2004) provide evidence that some Australian money managers engage in a pattern of tradingconsistent with the behavior described here.


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amount of an asset to the market.4 Consequently, any portfolio with a Sharpe ratio in excess ofthe markets must have a positive alpha. In particular, the alphas of the maximal Sharpe ratioportfolios described above are5

2 2

MSR MSR mkt MSR (1 ) 0 .x = >S S (6)

Clearly, alpha is subject to severe manipulation. A MSRP can be created with any desiredleverage, so its alpha can in theory be made as large as desired by levering.

The Treynor (1965) ratio and the Treynor appraisal ratio were introduced in part to avoidthe leverage problem inherent in alpha. The Treynor measure is the ratio of alpha to beta6 whilethe appraisal measure is the ratio of alpha to residual standard deviation. Like the Sharpemeasure, and unlike alpha, the two Treynor measures are unaffected by leverage.

Both Treynor measures indicate superior performance for the MSRP. The MSRPs twoTreynor ratios are

7

( )2 2MSRMSR MSR mkt mktMSR

2 2MSRMSR MSR mkt

MSR MSR mkt

1 0

0 .Var[ ]

x

x x

= >

= >

T S S

A S S

(7)

Any MSRP has the largest possible Treynor appraisal ratio, but Treynor ratios in excess of theMSRPs can be achieved by forming portfolios with positive alphas and betas close to zero.

8

In addition, since the Sharpe ratio is subject to dynamic manipulation, alpha and both

4 If asset i is combined with the market into a portfolio mkt(1 )iwx w x+ , then ( ) mkt0 .iww = = S 5

The beta of the maximal-Sharpe-ratio portfolio,xMSR, is

2

MSR mkt MSR mkt MSR mkt MSR MSR mktMSR 2

mkt MSR mkt MSR mkt mkt MSR

Cov[ , ] Cov[ , ] Var[ ] Var[ ]

Var[ ] Var[ ] Var[ ] Var[ ]

m

m

x x x x x x x x

x x x x x x = = =

S

S

where Smkt is the Sharpe ratio of the market. The third equality follows since the MSRP is mean-variance efficient

by definition so mkt MSR MSR mkt MSR Cov[ , ] Var[ ] x x x x x= .

6 Treynors (1965) original definition of his measure was / ;fr however, / is now the commonly accepted

definition.

7 The Treynor ratio can be computed from the MSRPs in equation (6) and its in footnote 5. To compute theTreynor appraisal ratio, we use and

2

MSR MSR mkt MSR MSR MSR mkt MSR mkt

22 2 2 2 2

2 MSR MSR mkt mkt MSR mktMSR MSR mkt 2 2 2 2 2

mktMSR MSR mkt MSR MSR

Var[ ] Var[ ] 2 Cov[ , ] Var[ ]

Var[ ] Var[ ] 1 .

x x x x x x

x x x xx x

x

= +

= = =

S S

S S S S S

8 A portfolio that shorts MSRdollars in the market for every dollar in the MSRP will have a beta of zero and apositive alpha. Therefore, its Treynor ratio will be +.


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Treynor measures are as well and can be increased above these statically achieved values. Themanipulation is illustrated in Figure 2. If returns have been above (below) average, thendecreasing (increasing) the leverage in the future, will generate a market line for the portfoliowith a positive alpha. Not surprisingly, this is very similar to the dynamic manipulation thatproduces a superior Sharpe ratio leverage is decreased (increased) after good (bad) returns.


Alpha-like measures can also be computed from models other than the CAPM. Underquite general conditions, the generalized alpha of an asset or portfolio in a single-period modelis9

genCov[ (1 ), ]

where .Cov[ (1 ), ]

f m

p p p m p

f m m

u r x x x B x B

u r x x

+ + = =

+ +

(8)

Bp is a generalized measure of systematic risk, rfis the per-period (not-annualized) interest rate,

and u() is the utility function of the representative investor holding the market. The systematicrisk coefficient,B, can be estimated by regressingx onxm using u() as an instrumental variable.Ingersoll (1987) and Leland (1999) suggest using a power utility function where

n[ (1 )] n(1 )(1 ) (1 ) with .

Var[ n(1 )]

f m f

f m f m

f m

r x ru r x r x

r x

+ + + + + = + + =+ +

E

(9)

This generalized alpha gives a correct measure of mispricing assuming that the represen-tative utility function is correctly matched to the market portfolio; that is, if the market portfoliodoes maximize the utility function employed. But this statement, tautological as it is, onlyapplies to single-period or static manipulation. Even if the utility is correctly matched to the

market, strategically rebalancing the portfolio over time can give an apparent positive alpha dueto the deviation between the average and a properly conditioned expectation. The basictechnique is the same decrease (increase) leverage after good (bad) returns.

Beyond the single-factor CAPM, there are several alpha-like performance measures inuse. These models augment the single factor CAPM model with additional risk factors such asthe Fama-French factors. However, within the simulations discussed in the next sub-section, no

such factors exist. The market returns are generated in an environment where (8) with gen = 0 isthe correct way to price. This single factor model should do at least as well in the simulatedenvironment as any model with additional, but within the simulation, unpriced, factors. Sim-ilarly, if a multi-factor model were simulated, a multi-dimensional rebalancing of the portfolioshould produce positive alphas. For this reason, the Chen and Knez (1996) measure and similar

measures have not been examined.

1D. Manipulating the Sharpe Ratio and Alpha with Transactions Costs

In practice, a manager can change a portfolios characteristics far more frequently thanonce during the typical measurement period. On the other hand, the costs of transacting may

9 See Ingersoll (1987) for the derivation of this general measure of systematic risk.


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eliminate the apparent advantage of many manipulation strategies. As we show below even veryhigh transaction costs cannot prevent managers from manipulating the Sharpe ratio or otherperformance measures.

Determining the optimal dynamic manipulation strategies for all of the popular perfor-mance measures is beyond the scope of our paper. The optimal manipulation strategy depends

on the size of the transaction costs, the complete set of returns to date within the evaluationperiod, the distribution of future returns, and the number of periods remaining in the evaluationperiod. However, for our purposes, an optimal strategy is not necessary. We only want toestablish whether reasonably simple trading strategies can distort the existing measures even inthe presence of transactions costs, and if so, by how much.

[Insert Table I here]

Table I shows the Sharpe ratio performance of a dynamically rebalanced portfolio. Theportfolio is always invested 100% in the market with leverage; no derivatives are used to alterthe distribution. The leverage starts at one. After the first year, the portfolio is levered at the

beginning of each month so that the target mean is given by equation (4) though the leverage isalso constrained to be between 0.5 and 1.5 so the portfolio will not be too extreme in nature. Theleverage is achieved by buying or selling a synthetic forward contract consisting of a longposition in calls and a short position in puts that are at-the-money in present value i.e., the strike

price per dollar invested in the market is ert. Trades in these two options are assessed a round-

trip transaction cost of 0%, 10%, or 20%. The simulation consisted of 10,000 repetitions.On average the dynamically-manipulated portfolios Sharpe ratio is 13% higher than the

markets and 7% higher than the MSRPs. In the 10,000 trials, the manipulated portfolio had aSharpe ratio higher than the markets 82.6% of the time in the absence of transactions costs.

10

Even with a 20% round-trip transactions cost, the dynamically-manipulated portfolio still beatthe market almost three-quarters of the time. It should be mentioned here that comparing these

winning percentages to 50% is a conservative test of how good the manipulated Sharpe ratio is.The CAPM predicts that any portfolios Sharpe cannot be bigger than the markets not that itwill be bigger or smaller with some measurement error.

The Sharpe-manipulated portfolio beats the market frequently, but is it substantiallybetter statistically; that is, is the difference in the Sharpe ratios significant? In practice thisquestion is seldom asked because Sharpe ratios are at least as difficult to estimate precisely as aremean returns.

11However, differences between two Sharpe ratios can be more precisely estimated

10 The frequency with which one portfolio beats another must be interpreted with caution when the winningportfolio holds derivatives. To illustrate, consider two portfolios that are almost identical. The only difference is thesecond portfolio sells derivative assets that have a positive payoff only rarely, like deep out-of-the-money options.The proceeds are invested in bonds. Whenever, the payoff event for the derivative does not occur, then the second

portfolios returns will be uniformly greater than the first portfolios returns. This will lead to a better outcomefor the second portfolio by almost any performance measure. Since the payoff event can be made as rare as desiredwith derivative contracts, any portfolio, even an ex ante optimal one, can be beaten often. In our simulations hereonly the leverage is changed so they are not subject to this problem

11Lo (2002) has shown that the asymptotic standard error of the Sharpe ratio of a portfolio with iid returns is2 1/ 2[(1 ) / ]T+S so the coefficient of determination is 1 2 1/ 2[(1 ) / ] .T +S S The coefficient of determination for the

mean return is 1/ 2 1 1 1/ 2 .T x T = S The former is larger by a factor of 2 1/ 2(1 )+S so the same difference inpercentage terms is less significant for a Sharpe ratio than it is for an average return.


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difference is that the excess returns are calculated relative to a risky benchmark portfolio ratherthan the risk-free rate. Ifx andxb are the excess returns on the portfolio and the benchmark, theinformation ratio is

infomation .Var( )

b

b

x x

x

=

S

(10)

Since excess returns are the returns on zero net cost (or arbitrage) portfolios, they can becombined simply by adding them together without any weighting. Clearly the arbitrage portfolio

that is a combination of the excess returns and bx has an information ratio with respect to thebenchmark, b , numerically equal to the Sharpe ratio of alone; therefore, the information ratiois subject to the same manipulation as the Sharpe ratio. In particular, adding the MSRP excessreturns to the market portfolio will achieve the highest possible information ratio relative to themarket.

One criticism commonly leveled against the Sharpe ratio is that very high returns arepenalized because they increase the standard deviation more than the average. This is the reasonthe MSRP has bounded returns. To overcome this problem, it has been suggested to measure

risk using only bad returns. In particular, Sortino (1991) and others have measured risk as theroot-mean-square deviation below some minimum acceptable return. Van der Meer, Plantingaand Forsey (1999) have further suggested that the reward in the numerator should only countgood returns. Sortinos downside-risk and van der Meer, Plantinga and Forseys upside-potential Sharpe-like measures are

( ) ( )1/ 2 1/ 2

2 2

[ ] [Max( ,0)].

[Min ( ,0)] [Min ( ,0)]

x x x

x x x x

= =

D U

E E

E E

(11)

The minimal acceptable excess return, ,x is commonly chosen to be zero.12

While these two measures do avoid the problem inherent in the Sharpe ratio of penalizingvery good outcomes, they do this too wellso that the highest possible returns are sought in lieuof all others. As shown in the Appendix, the Sortino downside-risk and VPF upside-potentialmaximizing portfolios are very similar. They both hold the MSRP, an extra very large invest-ment in the state security for the state with the highest market return (the lowest likelihood

ratio / / I I i ip p p p ), and bonds.

12 The minimum acceptable return is sometimes set to the portfolios average return for the Sortino measure. Thismakes the denominator the semi-standard deviation. The average return should not be used like this for the VPFmeasure as the ratio is then invariant to a uniform upward shift in the entire probability distribution.


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2MDR 2 MSR MDR 2MSR

MSR2 2 2

MDR MDR MDR

1

MUP 2 2 MSR 2 MUP

MSR MSR 2

MUP

22

MDR 2

1 (1 ) 1 (1 )

(1 )(1 ) (1 ) 1

where 1

I I I i i I

I I I

I I Ii I i I

I I I

I

I

p p px x x x i I x

p p p

p p px x p x x i I x

p p p

p

p

= + + < =

= + < =

= +

SS D

D D D

U S SU

D

2 12 2 2

MUP 21 1

/ 2 and / .

I II I

i i i i

i iI I

p p p p p p

p p

= =

= U

(12)

DMDRand UMUP are the largest possible Sortino downside risk and VPF upside potential ratios.

Figure 3 illustrates the maximal Sortino and VPF ratio portfolios. The one-month returnon the market has a twenty-period binomial distribution that approximates a lognormal market

with a continuously-compounded risk premium of 12% and a logarithmic volatility of =20%.

13There are twenty periods over the one month and twenty-one final states. Both port-

folios achieve a very high return in the best market outcome state and have a negative excessreturns in all but this best state.14 The annualized Sortino and VPF ratios of the market are 1.00and 2.86. The scores for the optimized Sortino ratio portfolio are over twice as high at 2.83 and6.04. The same is true for the optimized VPF portfolio; it scores 2.76 and 6.17.


Clearly the maximal Sortino and VPF ratio portfolios are far from optimal, and even acursory examination of the period-by period returns would likely attract very negative attention.For example, both of these portfolios have Sharpe ratios that are less than a tenth that of themarket. Conversely the MSRP has a Sortino and VPF ratio just a bit below that of the market.

The maximal-ratio strategies are also risky in a sampling sense. The best market state islikely to have only a small probability of occurring, and should it fail to occur during theevaluation period, the sample averages and therefore the ratios would be negative even in a bull

market. On the other hand, should no below- outcomes occur, both ratios would be infinite. Of

course, the maximal-ratio portfolios need not be employed to manipulate the measures. Thepayoffs in Figure 2 indicate that a high ratio can probably be achieved merely by purchasingcalls the further out-of-the-money the better. For example, in our simulation environment,simply investing the entire portfolio in at-the-money market index calls each month has a Sortinoratio of 1.20 and a VPF ratio of 3.68 substantially higher than the markets. Using ten percentout-of-the-money calls increases these ratios to 1.79 and 5.10.

As with the Sharpe ratio, dynamic manipulation can also increase the measure or increasethe measure further when used along with static manipulations. One simple scheme can often

achieve an infinite ratio when the minimal acceptable excess return is zero (or negative). If thefirst excess return is positive (which should be true more than one-half of the time), then holding

13 A binomial market environment is used as an illustration here because in a lognormal market or any market with acontinuous distribution, the maximal Sortino and VPF portfolios have an infinitesimal negative excess return exceptin the highest return state. In this state, which occurs with zero probability in a market with a continuum of states,the return is infinite.

14 For any choice of ,x both portfolios have returns less than x in all but the best state.


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13

the risk-free asset for every other period will give a positive numerator and a denominator ofzero resulting in an infinite measure.

To illustrate the dynamic manipulation possibilities of these two measures even with veryrestricted portfolios, we employ simulations similar to those used for the Sharpe ratio. If thehistorical returns up through time t0 have an average excess return of h and a Sortino measure

ofDh, then the portfolios measured Sortino ratio over the entire period will be

0 0

2 2 2 2 2 2

(1 ) (1 )

Max ( ,0) (1 ) Max ( ,0) (1 )

h f h f

t t h h f f t t t t

x x x x

x x x x < >

+ + = =

+ + D

D D(13)

where the subscriptfdenotes future performance. As with the Sharpe ratio, the overall Sortinoratio is maximized by maximizing the Sortino ratio in the future and selecting the leverage sothat

2 2 for 0

for 0.h f h h

f

h

x xx

x >=

D D (14)

Table II shows the Sortino and VPF ratio performance of a dynamically rebalancedportfolio under the same conditions used for the Sharpe ratio. Again no derivative assets wereused to statically manipulate the ratios. The market is held each month, and only the leverage ischanged, each month after the 12th, as described in (14). The manipulated portfolios producestatistics that are superior to simply holding the market portfolio for both modified measures aswell as the Sharpe ratio.

15The Sortino ratio is higher than the markets more than 82% of the

time even with 20% round-trip transactions costs. It is significantly higher at the 5% confidencelevel almost 14% of the time.

[Insert Table II here]

The Sharpe and VPF ratios are also higher nearly three-quarters of the time for thismanipulation. The same manipulation produces good results for all three ratios because each ofthe ratios has the same general dynamic manipulation rule decrease leverage after good luck(after the historical score has been high).

The best dynamic manipulation VPF ratio is almost the same. The target in the future isproportional to the upside realization in the past and the ratio of the squares of the expected and

realized VPF measures; i.e., [Max( ,0)]fx =E 2 2Avg[Max( ,0)]h f hx U U . The simulation results

for the VPF measure manipulation are not presented here as they were very similar. Understand-ably, this manipulation produces somewhat higher average VPF ratios and beats the marketsVPF ratio more often while producing somewhat lower Sortino ratios.

2B. Henriksson-Merton and Treynor-Mazuy Timing Measures

15 The simulations also show that for a base lognormal distribution, the sample Sortino and VPF ratios are almostcertainly biased (like the Sharpe ratio for the MSRP and unlike (approximately) the Sharpe ratio for the market).The true Sortino and VPF ratios for the market are 1.002 and 2.844. The sample averages were 1.117 and 2.972with standard errors of 0.00935 and 0.00831. So the sample averages were higher than the true values by more than12 and 15 standard errors, respectively.


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14

The timing measures of Treynor and Mazuy (1966) and Henriksson and Merton (1981)are bivariate regressions that use an extra market factor to capture managers timing abilities(rather than to capture an additional source of risk). The regressions used are

0 1 2t t t t x m w= + + + (15)

where tx and tm are the excess returns on the portfolio and the market and tw is the variable usedto capture timing ability. The timing variables are tw Max( ,0)tm and

2

t tw m for HM andTM, respectively. The HM model is consistent with a manager who changes his portfolioleverage between two values depending on whether he forecasts that the market return willexceed the risk-free rate or not. As shown by Admati, Bhattacharya, Pfleiderer, and Ross (1986),the TM model is consistent with a manager whose target beta varies linearly with his marketforecast.

Unlike the measures we have already considered, performance here is characterized by

two numbers. In each case a positive 2 is an indication of market timing ability and a positive 0is considered a sign of superior stock selection. However, as shown by Jagannathan and

Korajczyk (1986), simple trading strategies including or mimicking options can produce positive2s and negative 0s or vice versa.

16 An unequivocal demonstration of timing ability must,therefore, satisfy a joint test on the two coefficients.

Merton (1981) has shown that the total contribution of the managers timing andselectivity in the HM model is 0e

rt + 2P(1, t, ert) per dollar invested. P(S, , K) is the

value of a -period put option on the market with a strike price ofK. The intuition for thisconclusion is that a market timer who could forecast perfectly whether or not the market return

would exceed the interest rate and who adjusted his portfolio to be fully invested in the market or bonds could essentially provide such a put for free. In addition his selectivity ability wouldprovide the present value of0 per dollar invested. The total contribution is the amount by whichthe value of the protective put exceeds its average cost measured by the lowered present value

of the extra average return.Similarly, under the TM model, the total contribution of the managers investment ability

per dollar invested is given by the same formula where the value of a derivative contract thatpays the square of the markets excess return replaces the puts value. This value is derived in

Appendix D. The total contributed value, V, of the money managers contribution to timing and

selectivity is

2

HM 0 2

TM 0 2

(1, , )

( 1) .

r t r t

r t r t t

e t e

e e e

= +

= +

V P

V(16)

Unfortunately with a complete market of derivative assets, the TM and HM measures can

be manipulated to any degree desired; that is, any values for the s can be achieved with manydifferent portfolios. The minimum-variance portfolio that achieves any particular (positive)

target values for0 and 2 will be long in the MSRP and the timing put struck at-the-money in

16 For example, buying (writing) calls produces a positive (negative) 2 and a negative (positive) 0. The MSRP alsohas a negative 2 and a positive 0. It should be noted that due to the 2 term in these regressions, 0 and 1 are notthat standard Jensens alpha and CAPM beta.


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15

present value terms.17 It might be short or long in the market and bonds depending on thetargeted mean excess return and market beta.


Figure 4 illustrates the payoffs on the minimum-variance zero-beta portfolios with atiming target of2 = 0.1 and a selectivity target of0 = 10 basis points per month. Theenvironment is the same as before a lognormal market with a risk premium of 12% per yearand a volatility of 20%. The graph displays approximately the middle 95% of the returndistribution. The timing returns fluctuate from local extremes near plus and minus 20% over the

market range from 7% to +7%. In practice, of course, only scattered points on the curveswould be seen, and the portfolio would probably be described as somewhat volatile butotherwise normal.

Reexamining Figure 2 we see that the alpha-manipulating strategy that levers down aftergood returns and levers up after poor returns has a general convex shape. This strategy willlikely produce a false positive timing performance. Simulations verify that this is true; however,

while 2 and the total contribution as measured in (16) are positive, simply changing leveragealso yields a (false) negative selection ability. Both 0 and 2 can be made positive if a callposition is written against the portfolio since this produces an extra return when the market has asmall return.

[Insert Table III here]

Table III shows the performance of a manipulated portfolio that holds the market andwrites a ten percent out-of-the-money call on 2% of its holdings; that is, the portfolio shares only98% of the market returns above 10% in any month. After one year, the leverage is changedevery month to equal 0.5 or 1.5 if the fraction of months that the markets return was less thanaverage was more or less than half. The resulting portfolio produces a total contributed value of

around 98 to 156 basis points per year depending on transactions costs as measured by either HMor TM. Our simulations show that the total contributed value was positive about 65% to 70% ofthe time and significantly so (at the 5% level) 9% to 14% of the time. The results are not asstrong as for the other measures, but this is not surprising. The HM and TM measures are timingmeasures specifically designed to for timing ability and should be able to better reject ourdynamic manipulations than the other measures discussed so far.

The components of performance, selectivity and timing, do not have as strong an indivi-dual showing because the simulation was designed to produce over all good performance and thetwo measures are negatively correlated. Increasing the out-of-the-money call sales, for example,

would increase 0 and decrease 2. However, each component if positive more than half of thetime and significantly so more often than chance would allow.

3. Manipulation Proof Performance Measures

In the preceding sections of this paper, we have established that popular measures of

17 See section D of the Appendix for a proof. For the Treynor-Mazuy measure the portfolio needs to be long in anexotic derivative contract paying the square of the market price rather than put options.


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16

performance are susceptible to manipulation even when transaction costs are high. We haveshown that manipulation requires no superior information and little technical sophistication.Simple schemes like trading puts and calls or simply altering a portfolios leverage can oftencreate performance that looks superior. Furthermore, even if money managers are not seeking tomanipulate their performance numbers, competitive evolution may favor those managers whose

strategies happen to look like our manipulation strategies if these measures are widely used. Wehave shown that these problems can be of concern even when evaluating well-diversified port-folios of equities, but increasingly money is being invested in hedge funds and other venueswhose return distributions differ substantially from those historically found in all-equity funds.For such funds even more problems may arise.

Such problems could be avoided with a manipulation-free measure. But what does itmean for a measure to be manipulation-free? What exactly should it encourage a manager to door not do? Intuitively, if a manager has no private information and markets are efficient, thenholding some benchmark portfolio, possibly levered, should maximize the measures expectedvalue. The benchmark might be the market, but in some contexts other benchmarks could beappropriate. Static manipulation is the tilting of the portfolio away from the (levered)

benchmark even when there is no informational reason to do so. Dynamic manipulation isaltering the portfolio over time based on performance rather than new information. Our goal isto characterize a measure that punishes uninformed manipulation of both types.

Formally a performance measure is a function of the portfolios probability distributionacross the outcome states. That is, ifr is the vector of returns across all the possible outcomes,

then the performance measure is a real-valued function, ( ), r of those returns. 18 In practice, ofcourse, we do not know the true distribution, and the estimated performance measure must be afunction of the returns realized over time. We will denote the estimated performance measure as

( )1 , .T

t t tr s

= 19 Each time period and its return, rt, can, but need not, be identified with the state,

st, that occurred. For example, estimating the Sharpe ratio requires knowing only the returns

while to estimate Jensens alpha we need to know the contemporaneous market return for eachreturn.One property that any manipulation-proof performance measure (MPPM) must have is

that it recognizes arbitrage opportunities as good. If the portfolio-return distribution r1dominates the portfolio return distribution r2, then the performance measure must rank it

higher, 1 2( ) ( ). > r r This same property should hold for the estimated performance measure,though the dominance in this case may only be apparent as not all states may have been realized.We already know that the Sharpe ratio fails this simple test if the returns in one or more states

18 In principle, a performance measure could be pair or more of numbers (like 0 and 2 in the TM and HM models ormean and variance). In general, of course, such performance measures would not provide a complete ranking of

portfolios. In one sense multi-valued performance measures would be harder to manipulate since they wouldprovide more information. In another more practical sense, multi-valued performance measures would be easier to

manipulate since without a complete ranking there would be more portfolios that the market benchmark did not beat.

We confine our attention to single-valued performance measures like the Sharpe ratio.

19 Using the time series of returns in the estimated performance measure to represent the true performancemeasure assumes that the realizations of portfolio returns (and states) are sufficiently independent and identicallydistributed over time so that the frequencies are will be equal to the ex ante probabilities in a sufficiently largesample..


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17

are increased sufficiently, the ratio will decrease even though the return distribution unequivo-cally improves based on the observed evidence.

However, this property while necessary is not sufficient to make a measure manipulation-proof. To have a performance measure that is proof against dynamic manipulation, the functionmust have a strong independence property such that the altering of some components does not

affect the relative ranking based on other components. A performance measure cannot bedynamically manipulated only if the relative rankings of different futures are unaffected by theparticular history that has occurred.20 Not that this does notsay that the history does not affectrankings only that two different futures can be compared independently of the past.

This independence property is known from utility theory where it is called (strong) utilityindependence or strong utility separation. (See e.g., Debreu (1960).) Furthermore, a ranking(i.e., ordinal) function has this independence property if and only if it has an additive represen-tation. That is, the only possible MPPMs can be expressed as

( )11

1 , ( , ) .T

T

t t t t t tt

r s r sT= =

=

(17)

The function () is any increasing function and is irrelevant for ranking; therefore, it can be taketo be the identify function except when otherwise more convenient. We chose to write theargument as an average rather than a simple sum because most performance measures are based

on some kind of average. Each function tmust be increasing if is to satisfy the arbitrage isgood criterion.

Generally, each time period is treated identically, apart from any state dependence, so

that all tare the same.21 In this case, any performance measure free from dynamic manipulation

must be equivalent to a simple average of some, possibly state-dependent, function of returns.Some simple performance measures that cannot be dynamically manipulated are the average

return, (rt) = rt; the average market-adjusted return, (rt,st) = rtmt(st); and the geometricaverage return, (rt) = n (1 )tr+ .

Any function will create a performance measure, , that is proof against dynamicmanipulation. To prevent static manipulation, must be strictly increasing, as previously noted,and concave in the returns. If is not concave, then levering the portfolio by borrowing willincrease the measure as will simply adding unpriced risk. Concavity is a natural property todemand since it is also necessary (and sufficient) if we want to rate as better the less risky (in aRothschild-Stiglitz sense) of two portfolios with the same average return.

This analysis shows that manipulation-proof evaluations of portfolios are computed asthe expectation, or in practice the average, of some function of returns. The natural set of perfor-mance measure functions thus is the same as the set of common utility functions, in that both are

time-separable, increasing and concave. The only remaining question is what functions can be

20Precisely, the measure is proof against dynamic manipulation only if ( ) ( )0 01 11 1 , ,T T

t t t t r r r r

+ + > implies

( ) ( )1 1 1 1 , ,T T

t t t t r r r r

+ + > for any history

1tr

. This independence property must also hold for all

conditioning subsets.21 An identical treatment of each periods return is not a requirement for the measure to be manipulation proof. For

example, the time-dependent average, t(rt) =T t

tr can be used. This puts extra emphasis on recent returns.


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18

used.

Any measure as defined in (17) with tan increasing, concave function of the return isa manipulation-proof performance measure. But if we wish the measure to be consistent with a

market in equilibrium, must be further restricted to have a power form. There are two justifi-cations for this conclusion.

Since utility is defined over wealth or consumption, while performance measures aredefined over returns, an MPPM that treats returns identically is consistent with expected utilitytheory only if the utility of returns is independent of wealth. This is true amongst additivelyseparable utility functions only for power utility. Additionally as shown by Rubinstein (1976)and He and Leland (1993), if the market portfolios rate of return is independent and identicallydistributed over time and markets are perfect, then the representative investor must have powerutility.

The MPPM measure we choose here is

11

1

1 1 n (1 ) (1 ) .(1 )

T

ft ft t

t

r r xt T

=

+ + + (18)

where rftandxtare the per-period (not annualized) interest rate and the excess return on theportfolio over period t. The function () is similar to a power utility function with relative riskaversion of.22 We have transformed the measure (using ) so that can be interpreted as theannualized continuously-compounded excess return certainty equivalent of the portfolio. That is,a risk-free portfolio earning exp[ n (1 ) ]ftr t+ + each period would also have a measuredperformance of .

Finally, we want to associate the MPPM with some benchmark portfolio. This wouldtypically be some market index. In the absence of any private information, we want the MPPMto score the chosen benchmark highly. If the benchmark portfolio has a lognormal distribution,

then the parameter should be selected so that

n (1 ) n(1 ).

Var n(1 )

f b f

f b

r x r

r x

+ + + = + +

E

(19)

Historically this number is about 2 to 4 for the CRSP value-weighted market portfolio dependingon the time period and frequency of data used. We have used a relative risk aversion of 3 in oursimulations.

Table IV shows the performance for our manipulated portfolios in our previous simula-tions. The MPPMs are determined for three different risk aversions, 2, 3, and 4.

23The

22 This measure is notthe same as using power utility of terminal wealth. Power utility of terminal wealth would be1 1(1 ) (1 )

ft t r x + + or a monotonic transformation of this. Utility of terminal wealth is a valid MPPM but is

not a useful risk-adjusted measure since for any particular realization of returns, it always rates as best whichever

portfolio achieved the largest total accumulation regardless of the risk aversion assumed.

23 The range of relative risk aversions in the market might be much wider than 2 to 4; however, this range is meantto capture the range of risk aversions that would be relevant for the construction of mutual funds or other welldiversified portfolios. Relative risk aversions of 2 to 4 correspond to portfolios that would range from leveraged

positions of 1.5 down to 0.75. This should encompass most portfolios that would be ranked.


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19

simulation was constructed with a market portfolio in which the risk aversion was 3, so thenumbers in that column are correct. In practice we would not know the correct risk aversion touse so the other MPPMs are given to check the sensitivity.

[Insert Table IV here]

The computed MPPMs show correctly that the manipulated portfolios are not as good asthe market.24 For example, the Sharpe-manipulated portfolio had a Sharpe ratio exceeding themarkets 82.6% of the time and was significantly larger 20.4% of the time at the 5% confidencelevel. On the other hand, the MPPM measure shows that this portfolio beat the market only46.3% of the time and was significantly (5% confidence) better only 0.4% of the time. Con-versely it significantly underperformed the market 9.6% of the time as measured by the MPPMbut only 1.4% of the time as measured by the Sharpe ratio. The average performance was 96basis points below the market on a certainty equivalent basis.

The other portfolios show similar results. Their underperformances range from 87 to 126basis points and they never beat the market more than 50% of the time. They significantly

underperform the market more than 9% of the time and outperform it from 0.4% to 2.4% of thetime. The results are similar for MPPMs computed assuming risk aversions of 2 or 4.Of course, as well as penalizing poor performance, we would like an MPPM to recognize

good performance when it occurs. Tables V and VI show the performance of money managerswho try to provide stock selection and market timing, respectively.

Table V compares the annualized Sharpe ratio, alpha, and MPPM for informed anduninformed money manager managers. The informed money manager can provide an annualalpha of 1% at the cost of holding a portfolio with undiversified risk. The three panels showportfolios with annual residual risks of 20%, 2%, and 0.2%.25 These are portfolios of approxi-mately one, one hundred, and ten thousand component stocks. The uniformed manager holdssimilar under-diversified portfolios but has an alpha of zero. The table shows the average,standard deviation, and frequencies for the difference between the portfolios and marketsSharpe ratios and MPPMs and for the portfolios alpha.

The portfolio alphas are just as expected. They average 1% and 0% for the informed anduninformed managers. For the uninformed manager, they are significantly positive or negativejust about the predicted 5%. For the informed manager they are significantly positive (negative)more (less) than 5% of the time regardless of the diversification again as expected since alphadoes not penalize for under-diversification.

For 10,000 stock portfolios all three of the measures show that the informed managersportfolio is better than the market and the uniformed managers portfolio is essentially identicalto the market. For one hundred stock portfolios, the MPPM and Sharpe have almost the sameresults though the MPPM does marginally better at recognizing both informed and uninformed

24 Only the zero-transactions-cost cases are reported. With transactions costs, the MPPM scores would be evenlower.

25 The logarithmic market return over a period of length tis 2 212

[ ( )] ( ) ,t t + + + + where and are standard normal variables. The markets expected rate of return and log variance per unit time are and 2,respectively. The informed trader provides an of 1% at a cost of a residual variation of of 20%, 2%, or 0.2%.The uninformed trader has the same residual variation with an of zero.


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20

managers. For one stock portfolios, the MPPM does substantially better at illustrating that the1% alpha does not properly compensate for the lack of diversification.

Table VI compares the MPPM and Henriksson-Merton and Treynor-Mazuy timingmeasures for informed an uninformed market timer. The informed market timer knows that atime varying mean explains a portion of the observed market variance. In the two panels, the

informed trader information explains 0.1% and 1% of the markets variation, and he adjusts hisleverage optimally.26

The uninformed trader believes he has similar quality information, butdoes not so he adjusts his leverage to the same degree, but randomly.

The table shows the average, standard deviation, and frequencies for the timing

coefficients, 2, and the total contributed value (as given in equation (16)) and the differencebetween the portfolios and markets MPPMs. The HM and TM models are designed torecognize informed timing, but not to penalize uninformed timing. Thus, they show resultsconsistent with the null hypothesis for the uniformed timer just as they would for a manager nottrying to time at all. The MPPM, on the other hand, shows that uninformed timer is definitelyhurting the portfolios performance, and that the uninformed timer who thinks he has betterinformation, hurts the portfolio more. The HM and TM measures recognize the informed trader

more frequently because, unlike the MPPM, they do not penalize the portfolios performance forthe induced lack of complete intertemporal diversification.

4. Conclusion

Portfolio performance measurement has been a topic of interest within both thepractitioner and academic communities for a long time. However, a reasonable concern amongthose that use a particular measure is whether or not the manager being evaluated might react byattempting to manipulate it. Several articles have noted that even when the evaluator knows themoments of the return distribution it is still possible to use informationless trades to boost the

expected Sharpe ratio.What has not been recognized is that because any performance measure must be

estimated a fund manager can potentially game the estimation procedure as well. As this paperhas shown, a manager that seeks to manipulate many of the more popular measures can indeedproduce very impressive performance statistics. For example, a simple rebalancing strategy canyield a market beating Jensens alpha 86% of the time even when transactions costs are set at anunrealistically high 20% of each dollar traded. But, these statistics are fraudulent in that they areproduced entirely without private information. Indeed, most investors would probably prefer tohold the market index rather than the portfolios that manipulate the various measures tested inthis paper.

If the current set of measures can be manipulated, can one then find one that cannot be

26 The logarithmic market return over a period of length tis 2 212

( ) (1 ) , s t t + + where is a standardnormal variable. The markets unconditional expected rate of return and logarithmic variance per unit time are and 2, respectively. The information about the changing mean is in the signal, ,s which is normally distributed withmean zero and variance 2 2 t where is the fraction of variation known to the informed trader. In the simula-tions, is set to 0.1% and 1%. The optimal market holding conditional on a signal,s, is equal to the conditional risk

premium divided by the relative risk aversion times the conditional variance, 2 2( ) (1 ) .s r + Since theunconditional risk premium is equal to the relative risk aversion times the unconditional variance, the optimal

leverage conditional on a signals is 2 2(1 / ) (1 )s+ .


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21

and is simultaneously of some value to investors? This paper has shown that indeed such ameasure exists. Manipulation-proof performance measures can be completely characterized (upto a monotonic transformation) as the weighted average of a utility-like function. Not only is itmanipulation proof but it is no more difficult to calculate than the Sharpe ratio, and considerablyeasier to calculate than many of the other performance measures that have been proposed in the

past to remedy the Sharpe ratios shortcomings.


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Appendix

A. The Maximal-Sharpe-Ratio Portfolio in a Complete Market

Consider a portfolio with excess returnxi in state i. The probability of state i ispi. The

Sharpe ratio of this portfolio is

( )22

i i

i i i i

p x

p x p x

=

S (A1)

This ratio is invariant to scaling so when maximizing it, we can fix the expected excess return atany positive value, 0,i ip x x = > with no loss of generality. Then maximizing the Sharpe ratio isequivalent to minimizing the mean squared return subject to an expected return of with a costof zero.

Form the Lagrangian27

( ) ( )212 0i i i i i ip x x p x p x= + + L (A2)

The first-order conditions for a minimum are

0 0 0i i i i i i i ii

p x p p x p x p xx

= = + = = = = L L L

(A3)

The second-order condition for an interior minimum is also met.Solving the first equation in (A3) gives the MSRP excess return in state i as

/ .i i i x p p = + (A4)

To determine the Lagrange multipliers, multiply (A4) bypi and ip and sum over states. Recog-

nizing that 1,i ip p= = gives

2 2

0 / /

i i i i

i i i i i i i

x p x p p

p x p p p p p

= = + = +

= = + = +

(A5)

These equations can be solved to determine the multipliers values

2 / 1i i

xx

p p

= =

(A6)

27The zero-net-wealth budget constraint is expressed here using the risk-neutral probabilities in place of the state

prices. Since the state price is ,rT ie p a portfolio with a risk-neutral expected excess return of zero has a zero cost.


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23

So the maximal-Sharpe-ratio portfolio is

( / 1) .i i i x x p p = + (A7)

The variance of the maximal-Sharpe ratio portfolio and the square of its Sharpe ratio are

2 2 2 2 2 2

12 2 2 2

22 2

MSR 2

( ) ( / 1) / 2

/ 1 / 1

/ 1.

x i i i i i i i i

i i i i

i i

x

p x x p p p p p p p

p p x p p

xp p

= = = +

= =

= =

S

(A8)

The maximal Sharp ratio portfolio is therefore

1/ 22 2

MSR (1 / ) with / 1i i i MSR i i x x x p p p p

= + = S S (A9)

The sum in (A9) can be expressed as 2 2 2 ( / ) [ / ]i i ip p p p p= E , so the square of the maximalSharpe ratio is one less than the expectation of the square of the realized probability likelihoodratio.

B. Sortino (Downside Risk) and Van der Meer, Plantinga, and Forsey (Upside

Potential) Measures

The Sortinos (1991) downside risk measure and Van der Meer, Plantinga and Forseys(1999) upside potential measure both compute the risk term in the denominator by considering

only excess returns below some minimum acceptable return,x . The upside potential measure

also only counts returns above this in the numerator. The two measures are

( ) ( )1/ 2 1/ 2

2 2

[Max( ,0)][ ].

[Min ( ,0)] [Min ( ,0)]

i

i i

x xx

x x x x

= =

D U

EE

E E

(B1)

The usual choice forx is zero. In this case both ratios are invariant to leverage like the Sharpe

ratio.28

This is the choice we use in this paper.

28 The measures are also invariant to leverage ifx is set proportional to the average return. If it is equal to the mean

return, the denominator of each measure is the semi-standard deviation, and this is sometimes done for the Sortino

ratio. This is a poor choice for the VPF upside measure, though, as is then invariant to a uniform upward shift in

the entire portfolio distribution.


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The Sortino Downside-Risk Measure

For 0,x = the Sortino measure can be maximized exactly like the Sharpe ratio by fixingthe mean excess return and minimizing the risk measure in the denominator. Forming theLagranian

( ) ( )212 Min ( ,0) 0i i i i i i p x x p x p x= + + L (B2)

and differentiating gives the first-order conditions as

0 Min( ,0) 0 0i i i i i i i ii

x p p x p x p xx

= = + = = = =

L L L

(B3)

The second-order condition for an interior minimum is also met.Note that the first condition in (B3) is independent ofxi whenxi > 0; therefore, it can be

satisfied only if just a single state has a positive excess return (or two or more states with the

same likelihood ratio have the same positive excess return). The state with a positive excessreturn must be that state with the smallest likelihood ratio, / .i ip Since the likelihood ratio is

inversely proportional to the marginal utility in a state, the single state with the positive excessreturn is the state with the highest market return denoted as stateI. Using the first order

conditions to solve for the Lagrange multipliers and the return in the highest return state, I ,

gives1 1

1 1 1

1 1 12 2

1 1 1 1

(1 ) (1 )

0 / (1 ) /

0 .

I I I

i i I i i i I I I I

i i i

I I I I

i i I I i i i I I I i i

i i i i

I I

x p x p x p p p x p p

p x p x p p p p x p p p

p p

= = =

= = = =

= = + + = + +

= = + + = + +

=

(B4)

Solving these three equations for, , and MDR ,Ix and substituting back into the firstequation in (B3) gives the down-side risk maximizing portfolio

MDR MDR 2

12

( )

where 2 ( ) .

i I I i i I I I i i I

I I I I i i

x p p p p i I x p p p p p

x p p p p p p

= < =

(B5)

The square of the minimized downside risk in the denominator is


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25

1 1 1 1 12 2 2 2 2 2

1 1 1 1 1

2 212 2 2 2 2 2

2 21 1

2

[Min ( ,0)] ( / ) 2 /

(1 ) 2 (1 ) / 2 /

2

I I I I I

i i i i i i i i i

i i i i i

I I I I I I

I I i i i i

i i I I I I

I I I i

I I I

x p x p p p p p p p

p p p p p p p p p p

p p p p

p p px p

p p p

= = = = =

= =

= = + = + +

= + = +

= +

E

12

1

/ .I

i

i

p

=

(B6)

So the maximized squared Sortino downside risk measure is

222 2

MDR 2 21

1 / 2 . [Min ( , 0)]

II I

i i

iI I

p pxp p

p p== = + D

E (B7)

The Van der Meer, Plantinga, and Forsey (Upside Potential) Measures

Maximizing the upside potential measure is very similar. Again if the minimal accep-table return, ,x is zero the VPF upside-potential measure is invariant to leverage so to maximizeit, we can fix the numerator, [Max( ,0)],ixE and minimize the square of the denominator,

2[Min ( ,0)].xE We form the Lagrangian and optimize

( )21

Min ( ,0) Max( ,0) 02

for 00

for 0 .

i i i i i i

i i i i

i i ii

p x N p x p x

p x p x

p p xx

+ +

L

L(B8)

As with the maximized Sortino measure only one state can have a positive excess return, and itmust be the state with the lowest likelihood ratio and the highest market return.

Using the first order conditions to solve for the Lagrange multiplier and the return in the

highest return state, MUPI , gives

1 11 1 1

MUP MUP 2 2 2

1 1 1 1

0 / / / .

I I I I I

i i I I i i I I i i i i

i i i iI

pp x p x p p p x p p N p p

p

= = = =

= = + = =

(B9)

The denominator of the squared VPF ratio is

121 1 12 2 2 2 2 2

21 1 1

[Min ( ,0)] ( / ) / ,

I I I I

i i i i i i i

i i iI

px p x p p p N p p

p

= = =

= = =

E (B10)

and the squared ratio itself is


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26

22 12 2

2 21

/ .[Min ( , 0)]

II

i i

iI

pNp p

x p

=

= = UE

(B11)

For comparison purposes, it is better to express results in terms of the meanexcess return rather than the numerator using

1 1

21 1 1

( / ) (1 ) (1 ) .

I I I I

i i I I i i i i i I I

i i i I

pNx p x p x p x N p p p N p N p

p

= = =

= = + = + = + = U

(B12)

So the maximal upside potential portfolio is

11

MUP 2 MUP

2

(1 ) ( / ) (1 ) / 1 .

I I

i I I I i i I

I I

p pxx x p p p p p i I x

p p

= < =

UU

(B13)

Comparing the maximal Sortino and VPF portfolios to the maximal Sharpe portfolio, wesee that for all but for the highest return state the excess returns are linearly related

2MDR 2 MSR MSR

MSR2 2

MDR MDR

1

MUP 2 2 MSR 2

MSR MSR

1 (1 ) for

(1 ) (1 ) for .

I Ii i

I I

Ii I i

I

p px x x i I

p p

p p x x i I

p

= + +


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27

The Sharpe ratio of the market is

( )2

2

( )

mkt 1/ 2 1/ 2 1/ 22

[ ] 1.

( 1)Var[ ] ( 1)

r t t r t r t

tt t

b e e e e

eb e e

= = =

SE

(C2)

The square of the downside risk in the denominator of the Sortino and VPF ratios is29

( ) ( )

2

22 n 2 n n 2

0 0

2 2 23 1 1(2 ) ( ) 22 2 2

[Min ( ,0)] ( ) 2 ( )

( ) ( ) ( )2 .

r t r t e er t b r t b r t b r t

t r t r t

b e e e p b db e e e e p b db

r t r t r t e e e

t t t

+ +

= = +

+ = +

E (C3)

The upside potential in the numerator of the VPF measures is

( )n2 21 1

2 2

[Max( ,0)] ( )

( ) ( )

r tr t b r t

e

t r t

b e e e dF b

r t r t e e

t t

=

+ =

E

(C4)

For the maximal-ratio portfolios, analysis similar to that in Appendix A yields a maximalsquared Sharpe ratio of

2 2 2 2

MSR ( )/ ( ) 1 [ ( )/ ( )] 1p b p b db p b p b= = S E (C5)

for a portfolio with an excess return of *( ) [ ( )/ ( ) 1]x b x p b p b= + where 2MSR/x = S as before.To determine the maximal Sharpe ratio portfolio, we need the likelihood ratio

( ) ( )

2

2 22 21 1

2 2

2 2

( ) /

2

n n ( )exp

( ) 2 2

exp 1 .2

r

b r t b t p b

p b t t

r rb t

= +

+ =

(C6)

The square of the maximal Sharpe ratio is

29 For a normal random variable,z, with mean z and variance v2, the upper and lower partial exponential momentsare

( ) ( )2 2

2 2 2 21 12 20

( ) exp ( ) exp .K

z z

K

K z v K K z ve dF z z v e dF z z v

v v

+ + = + = +


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28

22 2 2 2( ) /

MSR 2 ( )/ ( ) 1 exp 1 ( ) 1.r

r p b p b r t b

+ = = S E E (C7)

Because n b is normally distributed with mean 212

( ) t and variance 2 ,t n b is also

normally distributed with21

2[ n ] ( )b t

= E and variance2 2

Var[ n ]b t

= . Using themoment generating function for a normal random variable,30 the maximal Sharpe ratio is

( )

22 21

MSR 22 2 2

22 2

2 2

2( )exp 1 ( ) exp 2 ( ) 1

( ) ( )exp 2 1 exp 1.

r r rr t t t

r rt r r t

+ = +

= + + =

S

(C8)

The maximal-Sortino ratio and maximal-VPF ratio portfolios will be degenerate with aninfinite holding of the security for the highest return state and an infinitesimal holding of allother state securities. Both maximized ratios will be infinite as well. However, if we restrictattention to well-formed portfolios then the ratios can be computed.

For example, consider a call option with a strike price ofHwritten on one dollar invested

in the market. The payoff on this portfolio is C(b) = Max(bH, 0)

The Sharpe, Sortino, and VPF ratios of the portfolio are

0

2 2

0 0

2 2

0 0

[ ( )]

[ ( )] [ ( )]

[ ( )] [Max( ( ) ,0)]

[Max ( ( ),0)] [Max ( ( ),0)]

r t

r t r t

r t r t

C b C e

C b C b

C b C e C b C e

C e C b C e P b

=

= =

S

D U

E

E E

E E

E E

(C9)

where C0 is the cost of the option. The expectations in the Sharpe ratio are

22 2 (2 ) 2

21

2

[ ( )] ( ) ( ) ( ) ( )

[ ( )] ( ) ( ) ( ) 2 ( ) ( )

n ( )where .

t

H HH

t t

H H H H

K K K

C b b H p b db e h H h

C b b H p b db e h He h H h

K th h h t t

+ +

= =

= = +

+ +

E

E

(C10)

The square of the risk measure in the denominators of the Sortino and VPF ratios is

30 The moment generating function for a normal random variable,z, with mean z and variance 2z is ( )s 2 21

2[exp( )] exp( )z sz sz s= + E .


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29

0 0

0

2

0 0

22 2 2

0 0 00 0

2 2 2

0 0

(2 )0

Max( ,0) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2( ) ( ) ( ) ( ) (

r t r t

r t

r t

C e H H C e H r t r t r t

H

r t r t

H H C e H

r t t t H HC e H C e

C e b H p b db C e p b db C e b H p b db

C e h C e H h h

C e H e h h e h h

+ +

+

+ ++

= + +

= + +

+ +

) .r t H+ +

(C11)

The numerator for the Sortino ratio is the same as for the Sharpe ratio. For the VPF ratio, it is

0 000 0( ) ( ) ( ) ( ) ( ) .r t r t r t

r t t r t

H C e H C eH C eb H C e p b db e h H C e h

+ ++ = + (C12)

D. Timing Measures: Henriksson-Merton and Treynor-Mazuy

The Henriksson-Merton (HM) and Treynor-Mazuy (TM) measures are based on theregression

0 1 2t t t t x m w= + + + (D1)

where tx and tm are the excess returns on the portfolio and the market and tw is a variable used

to capture timing ability. The variables used by HM and TM are Max( ,0)t tw m and2 ,t tw m

respectively.

To achieve a good score on these measures we want both 0 and 2 as large as possible.For an OLS regression, the coefficients will be

2 2

1 2 0 1 22 2 2 2 2 2. xm w xw mw xw m xm mw

m w mw m w mw

m w

= = =

(D2)

The parameters, 2 2, , , , and ,m w mwm w are fixed by the environment so the regressioncoefficients are determined by the portfolios expected return and its covariances with the marketand the timing variable, w. Since the expected return and the covariances are linear in the

individual state returns, any values for , , and xm xwx and hence the regression coefficients canbe achieved with four or more assets.

31

With more than four assets, we have extra degrees of freedom. It makes sense to find the

minimum-variance portfolio that achieves a given HM or TM score. Therefore, we fix xm, xw,andx (which fixes 0, 1, and 2) and minimize the portfolios variance. The optimizationproblem is

31 If the CAPM holds, then the mean excess return would be proportional to its covariance with the market for all

assets, and we would not be able to set them independently for a portfolio. As we have seen, the CAPM cannot hold

in a market with sufficient number of derivative assets.


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30

212

Min

subject to ( ) ( ) 0

i i

xm i i i xw i i i i i i i

p x

p x m m p x w w x p x p x = = = =

(D3)

where the last constraint fixes the cost of the portfolios excess returns to zero. Forming the

Lagranian and minimizing gives

211 22

3 4

1 2 3 4

1 2 3 4

( ) ( )

0

0 ( ) ( )

( ) ( ) / .

i i xm i i i xw i i i

i i i i

i i i i i i i i

i

i i i i i

p x p x m m p x w w

x p x p x

p x p m m p w w p px

x m m w w p p

+ +

+ +

= =

= + + +

L

L(D4)

The optimizing portfolio in a complete market can be split into four parts. It will hold bonds, 1

units in the market, 4 in the MSRP, and 2 units in a contract that pays w. For HM, the lastwill be a put option that is at-the-money in present value terms. For TM, it will be a derivativecontract with paying off the squared excess market return.

The Lagrange multipliers that determine these positions are the solutions to the con-straints32

[ ]1 2 3 4 3 4

2

1 2

3 4

21 2 4

2

1 2

3 4

( ) ( ) /

( ) ( ) ( )( )

( ) ( )

( ) ( )( ) ( )

( ) (

i i i i i i i

xm i i i i i i i i

i i i i

m mw

xw i i i i i i i i

i i i i

x p x p m m w w q p

p m m x p m m p m m w w

p m m p m m

m

p w w x p m m w w p w w

p w w p w

= = + + + = +

= = +

+ +

= +

= = +

+ +

[ ]

2

1 2 4

1 2 3 4

2

1 2 3 4

)

( )

0 ( ) ( ) /

( ) /

mw w i i

i s i i i i i

i i i i

w

p w w

p x p m m w w p p

m p w w p p

= + +

= = + + +

= + + +

(D5)

Solving for the Lagrange multipliers we have

32 Recall that the risk-neutral expectation of the excess market return m, i ip m is zero. This is not true ofw as it isnot the value of a marketed asset. The sum ( )i i p w w is the difference between the risk-neutral and true expec-tations ofw, [ ] [ ].w w E E This is the negative of the forward risk-premium.


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31

( )4 1 22 2 2 2 2

MSR MSR 1 2 2 2 2 2 2

MSR MSR

2 2 2 2 2

MSR MSR 2 2 2 2

MSR

/ 1 ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(where

i i i i i

xm w w xw mw w mw w w

m w w mw mw w mw w w

xm mw w xw m m w mw

m w

x x q p m m w

m x m

m m m

m m x m

= + + +

+ =

+ +

=

S S

S S

S S

S2 2 2

MSR

3 42 2 2 2 2

4 2 2 2 2 2 2

MSR MSR

) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

[ ] [ ].

w mw mw w mw w w

xm w mw w xw m w mw m w mw

m w w mw mw w mw w w

w

m m mx

m m x

m m m

w w

+ =

+ + =

+

S

S S

E E

(D6)

The coefficient 2 is a measure of the managers timing ability while 0 measures stock selectionability.

Merton (1981) has shown that in the context of the HM model, the total contribution of

the managers timing and selectivity ability per dollar invested over a time interval tis

0 2 (1, ; )r t r t e t e + P where P(S, ;K) is the value of a put option with strike price Kand time

to maturity written on the market. The manager provides an extra average return of0. Thefirst term is the present value of this. (It is discounted at the risk-free rate since the extra returnis not correlated with the market.) In addition, through timing the manager provides the fraction

2 of a protective put. The total contribution is the amount by which the

port performance manipulation

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