Sharpe Ratios and Alphas in Continuous Time

Transcription

1 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL. 39, NO. 1, MARCH 2004 COPYRIGHT 2004, SCHOOL OF BUSINESS ADMINISTRATION, UNIVERSITY OF WASHINGTON, SEATTLE, WA Share Ratios and Alhas in Continuous Time Lars Tyge Nielsen and Maria Vassalou Λ Abstract This aer rooses modified versions of the Share ratio and Jensen s alha, which are aroriate in a simle continuous-time model. Both are derived from otimal ortfolio selection. The modified Share ratio equals the ordinary Share ratio lus half of the volatility of the fund. The modified alha also differs from the ordinary alha by a second-moment adjustment. The modified and the ordinary Share ratios may rank funds differently. In articular, if two funds have the same ordinary Share ratio, then the one with the higher volatility will rank higher according to the modified Share ratio. This is justified by the underlying dynamic ortfolio theory. Unlike their discrete-time versions, the continuous-time erformance measures take into account that it is otimal for investors to change the fractions of their wealth held in the fund vs. the riskless asset over time. I. Introduction This aer rooses and analyzes modified versions of the Share ratio and Jensen s alha, which are derived from otimal ortfolio selection in a simle continuous-time model. The ordinary Share ratio was roosed by Share (1966) and is much used by ractitioners. It is the ratio between exected or average excess return and risk, where risk is measured as standard deviation of return. According to static mean-variance ortfolio theory, if investors face an exclusive choice among a number of funds, then they can unambiguously rank them on the basis of their Share ratios. A fund with a higher Share ratio will enable all investors to achieve a higher exected utility. The modified or instantaneous Share ratio is effectively the same as the discrete Share ratio, excet that the rates of return over finite time intervals are relaced by instantaneous rates of return. We show that if investors face an exclusive choice among a number of funds, each of which has a constant instantaneous Share ratio and if they are able to dynamically reallocate wealth between their chosen fund and a money market Λ Nielsen, lars.nielsen@morganstanley.com, Morgan Stanley, 750 7th Avenue, New York, NY 10019, and Vassalou, maria.vassalou@columbia.edu, Graduate School of Business, Columbia University, 3022 Broadway, New York, NY The authors thank Zhiwu Chen (associate editor and referee) and esecially Stehen J. Brown (the editor) for many useful comments. We are resonsible for any errors. The views, thoughts and oinions exressed in this aer are those of the authors and should not in any way be attributed to Morgan Stanley or to Lars Tyge Nielsen as a reresentative, officer, or emloyee of Morgan Stanley. 103

2 104 Journal of Financial and Quantitative Analysis account, then they can unambiguously rank the funds on the basis of their instantaneous Share ratios. A fund with a higher instantaneous Share ratio will enable all investors to achieve a higher exected utility. The assumtion of constant instantaneous Share ratios is obviously restrictive, but it does allow the volatilities and exected excess returns of the funds to change stochastically over time. As long as a fund invests in an underlying ortfolio that has a constant instantaneous Share ratio, it may well engage in a dynamic strategy with resect to the fraction of asset value invested in the ortfolio and the fraction invested in the riskless asset, or the degree of leverage emloyed. If the underlying ortfolio has constant volatility, then the fund may also engage in a strategy that involves buying and selling contingent claims such as ut and call otions on the ortfolio. Even though our linkage of exected utility maximization and the instantaneous Share ratio allows for dynamic strategies, it does not contradict the concerns about gaming of the Share ratio exressed, for examle, in Goetzman et al. (2002). That aer alies the Share ratio to zero-investment strategies and to a fairly general model of returns distributions, where maximizing the Share ratio is not necessarily in the best interest of the fund s investors. By contrast, we assume a ositive investment, and we limit the returns distributions under consideration to those for which we can establish a theoretical foundation for the instantaneous Share ratio based on exected utility maximization. In the case where the value of a fund has constant volatility, we can comare the instantaneous Share ratio with a discrete Share ratio calculated from continuously comounded rates of return. It turns out that the instantaneous Share ratio equals the discrete Share ratio lus half of the volatility of the fund. Unlike the assumtion of a constant instantaneous Share ratio, the assumtion of a constant volatility rules out various dynamic strategies and otion strategies on the art of the fund manager. Moreover, under the joint assumtion of constant volatility and a constant instantaneous Share ratio, the excess returns are i.i.d., and the standard estimation theory for the Share ratio derived by Lo (2002) alies. Lo shows that care must be taken in estimating the Share ratio when returns are not i.i.d., such as, for examle, when they are autocorrelated. The relative size of the volatility adjustment to the Share ratio does not deend on whether returns are exressed er day, er month, or er year. The same is true of the ranking of ortfolios roduced by the instantaneous Share ratio. The fact that the instantaneous Share ratio differs from the discrete Share ratio by half of the volatility of the fund imlies that the discrete and instantaneous Share ratios may well roduce different rankings of funds. The instantaneous Share ratio does not enalize fund managers as much for taking risks as the discrete ratio does. In articular, if two funds have the same discrete Share ratio but different volatilities, then the fund with higher volatility will be the better erformer. The intuition behind this result is that the static one-eriod theory on which the discrete Share ratio is based overestimates the riskiness of high volatility funds, because it does not take into account the investors ability to change the fraction of their wealth allocated to the fund over time. The instantaneous Share ratio does not reward the fund manager for taking risks without regard to the exected rate of return. If he increases the volatility

3 Nielsen and Vassalou 105 of the fund, then he has to raise the instantaneous excess rate of return at least roortionally to kee the same instantaneous Share ratio, and he has to raise the instantaneous excess return more than roortionally to increase the instantaneous Share ratio. If either the ordinary or the instantaneous Share ratio is used to make statements about whether funds on average have a higher Share ratio than a benchmark, then it is subject to survival bias. This issue ertains equally to ordinary and to instantaneous Share ratios. However, if the Share ratios are used to rank funds, then survival bias should not be an issue. Jensen s alha was roosed by Jensen (1968), (1969) and is used by both ractitioners and academics. To construct a version of Jensen s alha that is aroriate in continuous time, we need to interret it in terms of otimal ortfolio choice. If an investor identifies a fund that has a ositive alha, then what exactly does that tell him about how to maximize his exected utility? The literature seems to have been silent on this oint, although the following answer is not surrising. Suose the investor initially holds a combination of the riskless asset and an index ortfolio. He considers whether to tilt his ortfolio holdings towards an actively managed fund by investing a small roortion of his wealth in it. He should do so only if it raises his exected utility and, hence, only if it raises the Share ratio of his overall ortfolio. We show that Jensen s alha is roortional to the first derivative of the overall Share ratio with resect to the roortion invested in the active fund. Hence, a ositive alha means that the investor can increase his exected utility by investing at least a small amount in the fund. This relation between Jensen s alha and the Share ratio holds in a dynamic model as well as in a static model. In a dynamic model, the relevant version of alha is the instantaneous alha. It is effectively the same as the discrete alha, excet that in calculating it, the rates of return over finite time intervals are relaced by instantaneous rates of return. We show that the instantaneous alha is equal to the discrete alha lus half the variance of the ortfolio minus half the covariance of the ortfolio with the benchmark. The rest of the study is organized as follows. Section II shows how the instantaneous Share ratio can be used as a erformance criterion. Section III derives the relation between the instantaneous and the discrete Share ratio. Section IV derives the exlicit relation between Jensen s alha and the Share ratio. Section V discusses the instantaneous Jensen s alha and its relation to the discrete Jensen s alha. We conclude in Section VI. II. The Instantaneous Share Ratio The discrete Share ratio of a ortfolio is the ratio between the exected excess rate of return and the volatility, S =(Er r f )= var(r ), where r is the rate of return on the ortfolio, and r f is the riskless rate. In this section, we define the instantaneous Share ratio and show that it unambiguously ranks funds in a continuous-time setting for exected utility maximizing investors who invest in one fund and in the riskless asset. For a general introduction to continuous-time finance models, see Nielsen (1999). There is an instantaneously riskless asset with value M(t) = M(0)

4 106 Journal of Financial and Quantitative Analysis rdsg, where r is the instantaneously riskless interest rate. The value rocess F of a ortfolio or fund, with dividends reinvested, is assumed to be a ositive Itô rocess with differential df=f = μ dt + ff dw, where μ is the instantaneous exected rate of return and ff is the K-dimensional row vector of instantaneous relative disersion coefficients. The volatility of the fund will be ffff >. It is assumed to be ositive. The instantaneous Share ratio of the fund, denoted by S inst, is defined as R t exf 0 S inst =(μ r)= ffff >. We will assume that the investor chooses one fund and then slits his wealth between this fund and the money market account. The way in which he slits it may change over time in resonse to new information. In other words, he imlements a ortfolio strategy, which in this context is a onedimensional rocess q. The interretation is that he uts the fraction q of his wealth in the fund and 1 q in the money market account. If q > 1, then he uses leverage. The resulting wealth rocess V has dynamics dv=v =(q(μ r)+r) dt + qff dw. We assume a finite time horizon [0; T]. The investor chooses q so as to maximize his exected utility of final wealth V(T). Proosition 1 is the theoretical foundation for using the instantaneous Share ratio for erformance measurement in a dynamic framework. It says that an investor who slits his wealth between a money market account and a fund can obtain a higher exected utility the higher is the instantaneous Share ratio of the fund that he chooses, rovided that the interest rate varies in a deterministic manner and that the instantaneous Share ratios of the funds under consideration are constant. It follows that if the investor is choosing one and only one among a number of funds, each of which has a constant instantaneous Share ratio, then he will refer the one that has the highest instantaneous Share ratio. In this sense, the instantaneous Share ratio can be used to rank funds in the dynamic framework exactly like the discrete Share ratio in a static model. Proosition 1. Suose the interest rate r is deterministic. Consider two funds whose rice rocesses F 1 and F 2 have differentials df i =F i = μq i dt + ff i dw for i = 1; 2. Suose the instantaneous Share ratios S inst;i =(μ i r)= ff i ff i >, i=1; 2, are ositive constants. Given the investor s utility function, the maximum exected utility he can obtain from a ortfolio strategy that involves only the fund F 1 and is adated to F is strictly larger than the maximum exected utility he can obtain from a ortfolio strategy that involves only the fund F 2 and is adated to F, if and only if S inst;1 > S inst;2. The roof of Proosition 1 is in the Aendix. It is not entirely simle for several reasons: i) the Wiener rocess W is otentially high dimensional, which allows the two funds to be less than erfectly instantaneously correlated, ii) the investor s trading strategy may in rincile be contingent on much more information than just observing the value of the fund he is trading, and iii) the relative disersion vector ff i of the fund may be stochastically time varying. Proosition 1 assumes that the instantaneous q Share ratios are constant, but it does not assume that the volatilities ff i ff i > or the excess returns μ i r are constant. This has two imortant imlications. First, so long as a fund invests in an underlying ortfolio that has constant instantaneous Share ratio, it may well

5 Nielsen and Vassalou 107 engage in a dynamic strategy with resect to the fraction of asset value invested in the ortfolio and the fraction invested in the riskless asset. These fractions may be stochastically time varying and may involve leverage. For examle, the fund may follow a strategy of doubling u, or increasing its bets when it suffers losses. Such strategies do not affect the instantaneous Share ratio, so the fund will still have a constant instantaneous Share ratio. Second, the fund may engage in a strategy that involves buying and selling contingent claims on the underlying ortfolio, at least if the latter has constant volatility. If the ortfolio has constant volatility, then it conforms to the dynamics underlying the Black-Scholes model, and the excess return and volatility of a contingent claim equals the claim s elasticity times the excess return and volatility, resectively, of the ortfolio (see Nielsen (1999), Chater 6, Section 6.2). Hence, the instantaneous Share ratio of the claim equals the instantaneous Share ratio of the underlying ortfolio. The fund will have stochastically time-varying excess return and volatility, but since all the contingent claims are erfectly correlated with the underlying ortfolio and have the same constant instantaneous Share ratio, the fund also has that same constant instantaneous Share ratio. III. Discrete and Instantaneous Share Ratios In this section, we derive a relation between the instantaneous Share ratio and the discrete Share ratio. We need some manageable assumtion about the volatility to calculate the ordinary Share ratio. The simlest assumtion that will do is constant volatility. Hence, we assume that the volatility ff and the exected instantaneous excess rate of return μ r of the fund are constant. This imlies that the instantaneous Share ratio of the fund will be constant. Like in Proosition 1, we assume that the interest rate r is deterministic. Although the exected instantaneous excess return μ r is constant, the exected instantaneous return μ itself may not be constant. The continuously comounded rate of return r f on the money market account R t+fi t over the time interval [t; t + fi ] is r f =lnm(t + fi ) ln M(t) = rds. It is deterministic. Since we are now considering only one fund at a time, we can assume that the Wiener rocess is one-dimensional and that ff is one-dimensional. The mean of the continuously comounded rate of return r on the ortfolio over the time interval [t; t + fi ] is Er = r f + mfi, where m = μ r 1 /2ff 2. The variance is ff 2 fi, and the standard deviation is ff fi. The discrete Share ratio is the ratio between the mean and standard deviation of excess rates of return over a discrete eriod. If the rates of return are exressed as continuously comounded rates er eriod of length fi, then the discrete Share ratio is (Er r f )= var(r )=(mfi)=(ff fi )=S fi, where S = m=ff is the discrete Share ratio based on continuously comounded annualized rates. By substituting the definition of m into the definition of the instantaneous Share ratio, we find the relation between the discrete Share ratio and the instantaneous Share ratio, S inst =(μ r)=ff=(m+ 1 /2ff 2 )=ff=s+ 1 /2ff. So, the instantaneous Share ratio differs from the discrete Share ratio by a bias equal to ff=2. This bias of course comes from the difference of ff 2 =2 between the instantaneous mean excess return μ r and the discrete mean excess return m.

6 108 Journal of Financial and Quantitative Analysis It is imortant to recognize that i) while the discrete and instantaneous Share ratios do deend on whether returns are exressed er day, er month, or er year, the ranking of ortfolios that they roduce does not, ii) the relative size of the bias does not deend on whether returns are exressed er day, er month, or er year, and iii) when the Share ratios are estimated from data, the imortance of the bias is indeendent of the frequency of the data. To make oints i) and ii), we calculate the instantaneous and discrete Share ratios for returns exressed er eriod of length fi, and then we exress the bias as a fraction of the discrete Share ratio. Observe that the definition of the instantaneous Share ratio as S inst =(μ r)=ff is based on instantaneous returns er eriod of length one, say one year. The instantaneous Share ratio corresonding to rates of return er time eriod of length fi is (μfi rfi )=(ff fi )=S inst fi = S fi + 1 /2ff fi. It is clear that the rankings of funds roduced by S inst fi and S fi are indeendent of fi, which was oint i). The size of the bias is S inst fi S fi =1 /2ff fi, which of course goes to zero as the length fi of the time interval goes to zero. However, exressed as a fraction of the discrete Share ratio S fi, the bias is (S inst fi S fi )=(S fi )=(Sinst S)=S, which is indeendent of fi. This was oint ii). The relative bias can also be written as (S inst fi S fi )=(S fi )=(μ r m)=m=(μfi rfi mfi )=(mfi ). It equals the difference between the instantaneous and the discrete exected excess return er eriod of length fi, exressed as a fraction of the discrete exected excess return. Finally, iii) if the Share ratios are estimated from data, then the quality of the estimate will of course be better the more data is used and, in articular, the higher the frequency of the data. However, the true underlying values of the ratios are unaffected, rovided that they are exressed in terms of returns er eriod of a fixed length, such as a year, indeendently of the samling frequency. When estimating the instantaneous Share ratio, we have to take into account the fact that while the arameters μ and ff refer to instantaneous returns, we can actually only observe returns over discrete-time eriods such as days, weeks, months, or years. The equation, S inst =(m + 1 /2ff 2 )=ff, has the virtue of exressing the instantaneous Share ratio in terms of discrete-time moments of the rates of return, since m is the exectation of the annualized discrete-time rate of return and ff is the standard deviation. The fact that the instantaneous Share ratio equals the discrete Share ratio lus half of the volatility of the fund imlies that the ranking of funds based on the discrete and the instantaneous Share ratios may well be different. In articular, if two funds have the same discrete Share ratio but one has higher volatility than the other, then they will be ranked as equal by the discrete Share ratio while the one with higher volatility will be ranked higher by the instantaneous Share ratio. In other words, given the mean annualized excess rate of return m, the instantaneous ratio enalizes the fund manager less than does the discrete ratio for taking risk in the form of volatility. The fund with higher volatility will enable the investor to achieve a higher exected utility. The intuition behind this result is that the static one-eriod theory on which the discrete Share ratio is based overestimates the riskiness of high volatility funds, because it does not take into account the investors ability to change the fraction of their wealth allocated to the fund over time. Take as an examle an

7 Nielsen and Vassalou 109 investor who wants to hold 50% of his wealth in the fund and 50% in the riskless asset, and whose investment horizon T is one year. In the static framework, he initially invests half of his money in the fund and half in the riskless asset, and then he waits for a year to see what haens. However, already after a month, the value of the fund may have gone u so that he actually holds 60% in the fund and only 40% in the riskless asset. During the course of the year, this situation may be further exacerbated. By contrast, in the dynamic framework, the investor will immediately react to the increase in the value of the fund by selling some of it and investing the roceeds in the riskless asset, so that he always holds exactly 50% in each. This lowers the overall riskiness of his strategy. The difference is reflected in the modification of the Share ratio. The fact that the instantaneous Share ratio enalizes the fund manager less for taking risk does not mean that it rewards him for taking risks without regard to the exected rate of return. If he increases the volatility of the fund, then he has to raise the instantaneous excess rate of return μ r at least roortionally to kee the same instantaneous Share ratio. To imrove its instantaneous Share ratio, and thereby imrove its relative ranking, the fund has to increase its instantaneous excess return more than roortionally to any increase in its volatility. IV. Jensen s Alha There are various versions of Jensen s alha, corresonding to different asset ricing models. Here we will only discuss the original Jensen s alha, which corresonds to the mean-variance CAPM, and its continuous-time modification. The usual interretation of alha is that it is a risk-adjusted erformance measure that adjusts exected or average returns for beta risk. However, this interretation does not exlicitly relate alha to otimal ortfolio choice or say recisely what an investor should do if he identifies one or more funds with ositive alha. This section gives an interretation of Jensen s alha in terms of ortfolio otimization and exlains the relation between Jensen s alha and the Share ratio. The rates of return in the formulas to follow can be interreted either as rates of return over discrete-time eriods, as will be aroriate in a static model, or as instantaneous rates of return, for use in a dynamic model. The exectations, variances, and covariances should be interreted accordingly. Jensen s alha of a ortfolio, relative to an index or benchmark x, is defined as ff = Er r f fi(er x r f ), where r f is the riskless rate, r and r x are the rates of return on the ortfolio and on the index x, and fi = cov(r ; r x )= var(r x ) is the beta of the ortfolio with resect to the index. If indeed the index x is efficient, then the true alha of every security and every ortfolio will be zero, although an estimated alha may be different from zero because of estimation error. However, alha can be calculated and given a recise interretation in terms of ortfolio otimization even if the index is not efficient. Suose the investor initially holds a combination of the riskless asset and an index ortfolio tracking the index x, in roortions 1 ν and ν. He now considers whether to tilt his ortfolio a little bit in the direction of the fund. In other

8 110 Journal of Financial and Quantitative Analysis words, he considers taking a small fraction ffl of his wealth and investing it in the ortfolio, while reducing the fractions held in the riskless asset and the index to (1 ffl)(1 ν) and (1 ffl)ν, resectively. Let S(ffl) denote the Share ratio (or instantaneous Share ratio) of the new ortfolio. Proosition 2. The derivative of S(ffl) with resect to ffl, evaluated at ffl = 0, is S 0 (0) =ff=(ν var(rx )). The roof of Proosition 2 is in the Aendix. Proosition 2 leads to the following interretation of alha. If ff>0, then an investor who basically invests in the index or in a combination of the index and the riskless asset can increase his Share ratio and hence his exected utility by investing a small ositive amount in the fund. Of course, if ff<0, then he can achieve the same effect by short-selling the fund, if this is ossible. When ffl varies, the standard deviation and mean of the investor s entire ortfolio traces out a hyerbolic curve, which is in fact the risky ortfolio frontier generated by two assets, the initial ortfolio and the fund. We illustrate this frontier in Figure 1, where ff>0. The frontier should not be confused with the usual frontier constructed from all available securities. r FIGURE 1 The Frontier Generated by the Index and the Fund ε+ ( 1 ε) y y r f Frontier of all risky assets Combinations of y and σ When ffl = 0, we are at the oint y. As ffl increases, we move u along the uer branch of the small hyerbola. The Share ratio S(ffl) is initially increasing and then decreasing. It has a maximum oint ffl >0, which reresents the otimal fraction of wealth to take out of the initial ortfolio and ut into the fund. It corresonds to the oint ffl + (1 ffl)y in Figure 1. It is alternatively ossible that S(ffl) does not have a maximum but is increasing for all ffl>0. This occurs if the riskless rate is at or above the exected rate of return on the minimum variance ortfolio formed from the index and the fund, which corresonds to the to-oint of the small hyerbolic curve in the figure. This resembles the situation where the riskless rate is at or above the global

9 Nielsen and Vassalou 111 minimum variance ortfolio formed from the risky securities, as analyzed, for examle, in Huang and Litzenberger (1988). Observe that alha alone does not say how much the investor should otimally invest in the fund. In other words, we cannot calculate ffl, the otimal value of ffl, knowing only the value of alha. The idiosyncratic variance of the fund also matters. If the investor uts too large a fraction of his wealth into the fund, then the idiosyncratic risk may result in a lower Share ratio and a lower exected utility. The analysis above alies not only in a static model but also in a continuoustime model, when the rates of return over a discrete-time interval are relaced by instantaneous rates of return. The Share ratio will be relaced by an instantaneous Share ratio, and alha will be relaced by an instantaneous alha, which we shall define in the following section. V. The Instantaneous Alha In this section, we define the instantaneous alha and derive a relation between the instantaneous and the discrete alhas. Let F x be the value of the index fund with dividends reinvested, and let F be the value of the other fund with dividends reinvested. Assume that they follow the rocesses F x (t)=f R t x (0)exf 0 (μ x 1 /2ffx 2 )ds + R t 0 ff x dz x and F (t)=f R t (0) exf 0 (μ 1 /2ff 2 )ds + R t 0 ff dz g, where μ x and μ are the instantaneous exected rates of return, ff x and ff are the instantaneous volatilities or standard deviations of the rates of return, and Z 1 and Z 2 are two otentially correlated standard Wiener rocesses with correlation coefficient ρ. The instantaneous alha of the fund, denoted by ff inst, is defined as ff inst = μ r fi inst (μ x r), where fi inst =(ff ff x ρ)=ffx 2 =(ff ρ)=ff x. The instantaneous alha is effectively the discrete alha with the rates of return over finite time intervals relaced by instantaneous rates of return. For the urose of deriving a relation between the instantaneous and the discrete alha, assume that the interest rate varies in a deterministic manner, and that the correlation ρ, the volatilities ff and ff x, and the instantaneous exected excess rates of return μ r and μ x r are constant. The means of the continuously comounded rates of return r x and r on x and over the time interval [t; t + fi ] are Er = r f + m fi and Er x = r f + m x fi, where m x = μ x r 1 /2ffx 2 and m = μ r 1 /2ff 2. The variances and the covariance are var(r )=ff 2 fi,var(r x)=ffx 2 fi, and cov(r ; r x )=ff ff x ρfi. If the rates of return are exressed er eriod of length fi, then the discrete Jensen s alha is m fi ((cov(r ; r x ))=(var(r x )))m x fi = m fi ((ff ff x ρfi )=(ffx 2 fi ))m xfi =(m fim x )fi = fffi, where fi = cov(r ; r x )=var(r x )=(ff ff x ρfi )=(ffx 2 fi )=(ff ff x ρ)=ffx 2 = fi inst and ff = m fim x is the discrete Jensen s alha based on annualized returns. By substituting the definitions of m and m x into the definition of the instantaneous alha, we find the relation between the discrete alha and the instantaneous alha, ff inst = ff + 1 /2(ff 2 ff ff x ρ). It follows that the instantaneous Jensen s alha differs from the discrete Jensen s alha by a bias equal to (ff 2 ff ff x ρ)=2. Similar to the Share ratios, it is imortant to recognize that i) while the discrete and instantaneous alhas do deend on whether returns are exressed er

10 112 Journal of Financial and Quantitative Analysis day, er month, or er year, the ranking of ortfolios that they roduce does not, ii) the relative size of the bias does not deend on whether returns are exressed er day, er month, or er year, and iii) if the alhas are estimated from data, then the imortance of the bias is indeendent of the frequency of the data. To make oints i) and ii), we calculate the instantaneous alha for returns exressed er eriod of length fi, and then we exress the bias as a fraction of the discrete alha. Observe that the definition of the instantaneous alha as ff inst = μ r fi inst (μ x r) is based on instantaneous returns er eriod of length one, say one year. The instantaneous alha corresonding to rates of return er time eriod of length fi is (μ r)fi fi inst (μ x r)fi =ff inst fi. It is clear that the rankings of funds roduced by ff inst fi and fffi are indeendent of fi. This illustrates oint ii). The size of the bias is ff inst fi fffi = 1 /2(ff 2 ff ff x ρ)fi, which goes to zero as the length fi of the time interval goes to zero. However, exressed as a fraction of the discrete alha, the bias is (ff inst fi fffi )=(fffi )=(ff inst ff)=ff, which is indeendent of fi. This demonstrates oint ii). Finally, iii) if the alhas are estimated from data, then the same arguments made for the instantaneous and discrete Share ratios aly. Similar to the case of the instantaneous Share ratio, the equation, ff inst = ff + 1 /2(ff 2 ff ff x ρ), exresses the instantaneous alha in terms of discrete-time moments of the rates of return. This is useful when estimating it from data. VI. Conclusions This aer has roosed modifications of the Share ratio and Jensen s alha, which are consistent with exected utility maximization in a continuous-time model. Secifically, the modifications take into account the fact that investors may change the slit of their wealth between the fund and the riskless asset over time. The theory assumes that the Share ratios are constant, but it allows for stochastically time-varying volatilities, which could arise, for examle, from a dynamic leverage strategy or from a strategy of buying contingent claims such as uts and calls on an underlying ortfolio. The instantaneous Share ratio does not necessarily deliver the same ranking of funds as its discrete version. In fact, in the secial case of constant volatility, we related these two versions and found that the instantaneous Share ratio enalizes a fund less for taking risk than does the discrete ratio. We derived a recise interretation of Jensen s alha in terms of otimal ortfolio choice by relating it to the Share ratio. Secifically, a ositive alha of a fund means that an investor who initially holds a benchmark index fund can imrove his Share ratio by diverting a small fraction of his wealth into the fund. The modified erformance evaluation criteria roosed in this aer have been derived under the simlest ossible assumtions. There is scoe to exlore the modifications to the theory required when the Share ratios change over time. There is also scoe to exlore the estimation of all the erformance measures, in a way that would be consistent with theory, when the volatilities and exected excess returns of the funds are not constant. Such extensions go beyond the boundaries of this aer.

11 VII. Aendix: Formalities and Proofs Nielsen and Vassalou 113 The investors information structure is reresented by a filtration (F t) t2[0;t ] on an underlying robability sace (Ω;F; P). The interretation is that F t is the information set available to the investors at time t. Random fluctuations in securities rices are driven by a K-dimensional rocess W, which is a K-dimensional Wiener rocess with resect to the filtration. Portfolio strategies, and the instantaneous means and disersions of value rocesses, are assumed to be measurable and adated to the filtration. The roof of Proosition 1 relies on the following lemma. Lemma 1. Suose the interest rate r is deterministic. Let B be a one-dimensional standard Brownian motion, and let F B be the augmented filtration generated by B. Consider two funds whose rice rocesses F and ˆF have differentials (df)=f = μ dt + ff dw and (d ˆF)= ˆF =(s 2 + r) dt + sdb, where s =(μ r)= ffff > is assumed to be constant. Given the investor s utility function, the maximum exected utility he can obtain from a ortfolio strategy that involves only fund F and is adated to F is the same as the maximum exected utility he can obtain from a ortfolio strategy that involves only fund ˆF and is adated to F B. Proof. Set ffi = s= ffff > =(μ r)=(ffff > ) and = R ffiff, then > = s 2. Define a onedimensional standard Brownian motion C by C(t) = t 0 (1=s) dw and let FC be the augmented filtration generated by C. It follows from the results in Nielsen and Vassalou (1997) that the otimal ortfolio strategy when trading fund F has the form q = affi, where a is a rocess that is adated to F C. A strategy of this form gives the following dynamics of wealth, (dv)=v =(q(μ r) + r) dt + qff dw =(affis ffff > + r) dt + affiff dw =(as 2 + r) dt + a dw =(as 2 + r) dt + as dc. Consider fund F ffi, which arises from trading fund F using ortfolio strategy ffi. It has dynamics (df ffi )=F ffi =(s 2 + r) dt + sdc. The wealth dynamics arising from trading fund F using ortfolio strategy affi is (dv)=v =(as 2 + r) dt + as dc, which is the same as the wealth dynamics arising from trading fund F ffi using ortfolio strategy a. Hence, the maximum exected utility from trading fund F using ortfolio strategies that are adated to F is identical to the maximum exected utility from trading fund F ffi using ortfolio strategies that are adated to F C. The latter is obviously identical to the maximum exected utility from trading fund ˆF using ortfolio strategies that are adated to F B. Proof of Proosition 1. Let B be a one-dimensional standard Brownian motion, and let F B be the augmented filtration generated by B. For each s, consider a fund whose rice rocesses ˆF[s] have differential (d ˆF[s])= ˆF[s] =(s 2 + r) dt + sdb. According to Lemma 1, given the investor s utility function, the maximum exected utility he can obtain from a ortfolio strategy that involves only fund F i and is adated to F is the same as the maximum exected utility he can obtain from a ortfolio strategy that involves only fund ˆF[S inst;i] and is adated to F B. Therefore, what we need to show is that if S inst;1 > S inst;2, then the maximum exected utility the investor can obtain from trading in fund ˆF[S inst;1] with a ortfolio strategy that is adated to F B is strictly larger than the maximum exected utility he can obtain from trading in fund ˆF[S inst;2] with a ortfolio strategy that is adated to F B. Let a be the otimal ortfolio strategy when he trades fund ˆF[S inst;2]. Since this fund has constant instantaneous mean and disersion, it is known from Merton (1971) that a has the form, a = fl(μ 2 r)=(ff 2ff > 2 )=fl(sinst;2= ffff > ), where fl>0 is the relative risk tolerance of the investor s utility function. Hence, a > 0. If the investor uses the ortfolio strategy a to trade the fund ˆF[S inst;i], then his wealth dynamics is (dv i)=v i =(as 2 inst;i +r) dt+ as inst;i db. The logarithm of his final wealth will be ln V i(t) =lnv(0) + R t 0 (as2 inst;i + r 1 /2a 2 ) dt + R t 0 asinst;i db. Hence, R ln V1(T) ln V2(T)= t 0 a(s1 inst;i Sinst;i 2 ) dt > 0. Therefore, the investor obtains a strictly higher exected utility by using a to trade fund ˆF[S inst;1] than by using it to trade the fund ˆF[S inst;2]. Proof of Proosition 2. Let y denote the initial ortfolio. Its rate of return is r y =(1 ν)r f + νr x. The exected excess rate of return and variance of the new ortfolio will be

12 114 Journal of Financial and Quantitative Analysis Er y r f + ffl(er Er y) and (1 ffl) 2 var(r y) + ffl 2 var(r ) +2ffl(1 ffl) cov(r y; r ), resectively, and the Share ratio (or instantaneous Share ratio) will be S(ffl) = Er y r f + ffl(er Er y) : (1 ffl) 2 var(r y) + ffl 2 var(r ) +2ffl(1 ffl) cov(r y; r ) Set E(ffl)=Er y r f + ffl(er Er y), v(ffl)=(1 ffl) 2 var(r y)+ffl 2 var(r )+2ffl(1 ffl) cov(r y; r ), and ff(ffl) = v(ffl), Then S(ffl) = Er y r f + ffl(er Er y) (1 ffl) 2 var(r y) + ffl 2 var(r ) +2ffl(1 ffl) cov(r y; r ) = E(ffl) ff(ffl) : To calculate S 0 (0), first observe the following, E(0) =Er Er y, v(0) =var(r y), ff(0) = var(ry), E 0 (ffl)=er Er y, v 0 (ffl)= 2(1 ffl) var(r y) +2ffl var(r ) +2(1 2ffl) cov(r y; r ), v 0 (0) = 2var(r y) + 2 cov(r y; r ), ff 0 (ffl) = 1 /2(v 0 (ffl))=(ff(ffl)), and ff 0 1 v 0 (0) (0) = 2 ff(0) 1 2var(r y) + 2 cov(r y; r ) = 2 var(ry) = var(ry) +cov(ry; r) : var(ry) Moreover, Er r f +(cov(r ; r y)= var(r y))(er y r f )=Er r f +(ν cov(r ; r x)=ν 2 var(r x))(νer x +(1 ν)r f r f )=Er r f + (1=ν)fiν(Er x r f )=Er r f + fi(er x r f )=ff. Now, S 0 (0) = E0 (0)ff(0) E(0)ff 0 (0) v(0) " 1 = (Er Er y) var(r y) = = 1 var(ry) 1 ν var(rx) var(ry) (Er y r f )» Er Er y + (Er y r f ) (Er y r f ) [Er r f + fi(er y r f )] = # var(ry) +cov(r; ry) var(ry) cov(r; ry) var(r y) ff : ν var(rx) References Goetzman, W.; J. E. Ingersoll; M. I. Siegel; and I. Welch. Sharening Share Ratios. Working Paer 02-08, Yale IFC (February 2002). Huang, C., and R. Litzenberger. Foundations for Financial Economics. Amsterdam: North-Holland (1988). Jensen, M. C. The Performance of Mutual Funds in the Period Journal of Finance, 23 (1968), Risk, the Pricing of Caital Assets, and the Evaluation of Investment Portfolios. Journal of Business, 42 (1969), Lo, A. W. The Statistics of Share Ratios. Financial Analysts Journal, 58 (2002), Merton, R. C. Otimum Consumtion and Portfolio Rules in a Continuous Time Model. Journal of Economic Theory, 3 (1971), Nielsen, L. T. Pricing and Hedging of Derivative Securities. Oxford, U.K.: Oxford Univ. Press (1999). Nielsen, L. T., and M. Vassalou. Portfolio Selection with Randomly Time-Varying Moments: The Role of the Instantaneous Caital Market Line. htt:// Working Paer, Columbia Univ. (1997). Share, W. F. Mutual Fund Performance. Journal of Business, 34 (1966),