a dynamic model of active portfolio

8/3/2019 A Dynamic Model of Active Portfolio

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A dynamic model of active portfoliomanagement with benchmark orientation q

Yonggan Zhao *

School of Business Administration and RBC Center for Risk Management, Faculty of Management,Dalhousie University, 6100 University Avenue, Suite 2010, Halifax, NS, Canada B3H 3J5

Available online 18 April 2007

Abstract

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factormodel similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset

pricing model. Journal of Economic Theory, 13, 342360] and that portfolio managers adopt a meantracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 1322], we develop a dynamic model of activeportfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike thecase of constant proportional portfolios for standard utility maximization, our optimal portfoliopolicy is state dependent, being a function of time to investment horizon, the return on the bench-mark portfolio, and the return on the investment portfolio. We dene a dynamic performance mea-sure which relates portfolios return to its risk sensitivity. Abnormal returns at each point in time arequantied as the difference between the realized and the model-tted returns. Risk sensitivity is esti-mated through a dynamic matching that minimizes the total tted error of portfolio returns. Forillustration, we analyze eight representative mutual funds in the U.S. market and show how thismodel can be used in practice. 2007 Elsevier B.V. All rights reserved.

0378-4266/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jbankn.2007.04.007

q The author is very grateful to Zhiping Chen, Gunter Dufey, Iraj Fooladi, Greg Hebb, Leonard C. MacLean,Kuan Xu, and William T. Ziemba for their insightful comments and support. Comments and suggestions fromtwo anonymous referees are greatly appreciated. Financial support from the Canada Research Chairs program isalso acknowledged.

*

Tel.: +1 902 494 6972; fax: +1 902 494 1503.E-mail address: [email protected]

Available online at www.sciencedirect.com

Journal of Banking & Finance 31 (2007) 33363356www.elsevier.com/locate/jbf
mailto:[email protected]:[email protected]


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JEL classication: B23; C51; C61; G11; G23

Keywords: Active portfolio management; Benchmark portfolio; Growth optimum portfolio; Risk sensitivity;Mutual fund performance evaluation

1. Introduction

Professional fund managers are frequently judged by their ex post excess returns relativeto a prescribed benchmark. Most money managers adopt an optimal strategy that maxi-mizes an expected excess return adjusted by the tracking error relative to the benchmark;see, Roll (1992). This is a sensible investment approach because fund sponsors wiselyexpect their investment portfolios to maintain a performance level that is close to a desiredbenchmark.

We analyze an optimal dynamic portfolio and asset allocation policy for investors whoare concerned about the performances of their portfolios relative to that of a given bench-mark. Maximizing the expected utility of the excess return over a chosen benchmark issometimes referred to as active portfolio management, while passive portfolio manage-ment just establishes a portfolio that possibly tracks the chosen benchmark; see, Roll(1992) and Sharpe (1964) . There are many professional and institutional investors who fol-low this benchmark-oriented procedure. For example, many equity mutual funds take theS&P 500 Index as a benchmark and try to beat it. Some bond funds try to exceed the per-formance of Lehman Brothers Bond Index. For an analysis of active portfolio manage-

ment in a static setting, see Grinold and Kahn (2000) .In the standard utility maximization with constant relative risk aversion (CRRA), theoptimal policies are all constant proportion portfolio allocation strategies. The portfoliois continuously rebalanced so as to always keep a constant proportion of the total fundvalue in the various asset classes, regardless of the level of the fund. Although such policieshave a variety of optimality properties for the ordinary portfolio problem and are used inasset allocation practice (see, Perold and Sharpe (1988) and Black and Perold (1992) ),some investors are reluctant to use constant proportion strategies in the belief that theirexpectations suggest that varying weights would be more protable. By maximizing theprobability that the investment fund achieves a certain performance goal before falling

below a predetermined shortfall relative to the benchmark, Browne (2000) relates the opti-mal portfolio policy to a state variable, the ratio of the level of investment portfolio to thebenchmark portfolio , which leads to an analytical solution in a complete market setting.

Managers of actively managed mutual funds are interested in shifting the investmentpolicy with changes of returns on both their investment portfolios and the benchmarkportfolio from time to time. Academic researchers dene this market activity as markettiming; see, Becker et al. (1999), Coggin et al. (1993), Ferson and Warther (1996) , andTreynor and Mazuy (1966) . This paper addresses this issue for a general incomplete mar-ket where the investor is allowed to invest in a large number of stocks which may includeall the individual components of equity indices. In a static setting, all efficient portfolioscan be obtained from the market portfolio by using leverage, assuming normally distrib-uted returns or quadratic utilities. However, as shown here, this is generally not true in adynamic setting except for the very special case that the market portfolio is equivalent to aleveraged growth optimum portfolio.

Y. Zhao / Journal of Banking & Finance 31 (2007) 33363356 3337


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Consistent with the standard risk/return tradeoff, the objective of the optimizationmodel is to maximize the expected differential returns on the investment portfolio andthe benchmark adjusted by its quadratic variation over the investment horizon. Thisobjective is intuitive and easily understood; it has a simple model structure and has beenused widely in the practice of portfolio management. Based on this setting, we derive ananalytical solution assuming that the model parameters are time varying. The optimalportfolio is a linear combination of the riskless asset, the growth optimal portfolio andthe benchmark portfolio.

Existing models have more or less ignored the activities in shifting the portfolio weightsduring the investment horizon. Dynamic optimal portfolios require portfolio managers toreformulate their portfolios given new observations. Especially for those mutual fundswhose performances are tied to a selected benchmark, updating of a portfolio over timeappears to be even more appropriate.

A very common question often arises as how to dene managers portfolio strategies,without observing their risk sensitivity. Our rst step is to infer the risk sensitivity of port-folio managers by utilizing the outcome of the portfolio returns over time. Estimation of risk sensitivity is based on dynamic matching. In other words, portfolios abnormalreturns are measured as the difference between the actual outcomes of the portfolio returnand the models implied return. The estimated risk sensitivity is then dened as the min-imizer of the total deviation of portfolio returns from the models implied returns. Subse-quently, the performance measure is dened as the expected value of the discountedabnormal returns over the investment horizon. Foster and Stutzer (2002) examined perfor-mance and risk aversion of funds with benchmarks using a large deviations approach.

The paper proceeds as follows. Section 2 discusses the setting of the nancial marketmodel for asset prices and the formulation of the problem to be studied. Section 3 presentsthe solutions for the optimal investment policies and the optimal portfolio returns overtime. Section 4 discusses a dynamic performance measure with risk sensitivity estimatedthrough dynamic matching. Section 5 presents the empirical analysis of the model andits implication for portfolio management. Section 6 concludes.

2. Market setting and model formulation

In this section, we discuss the setting of the nancial market and the formulation of the

optimization model. We extend the factor structure of Ross (1976) to a continuous timeframework with time varying factors.

2.1. The nancial market

We consider a nancial market consisting of m + 1 primary assets: m risky securities,S 1 , . . ., S m , and a (riskless) cash bond, S 0 . The investor may invest in the cash bond andthe risky securities. The cash bond earns a continuously compounded rate r(t) such that

dS 0t

S 0t r t dt ; 1

where r(t) is locally riskless. The probabilistic setting for the risky assets is as follows: nindependent Brownian motions, Z 1 (t), . . ., Z n(t), characterize the market uncertainty forall risky securities. These factors are not observable. The uncertainty is partially revealed

3338 Y. Zhao / Journal of Banking & Finance 31 (2007) 33363356


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through the securities prices. In addition, all risky securities have their own risky compo-nents driven by independent Brownian motions, W 1 (t), . . ., W m(t). Both Z i s and W i s aredened on a ltered probability space, X; f F t g; F ; P , where F t is the P -augmentationof the natural ltration (see Duffie, 2000, for a brief review of the relevant terminology ).Thus, we assume that these n + m Brownian motions generate the prices of the m riskysecurities. Specically, we assume that the risky security prices follow a multi-variate dif-fusion process,

dS it S it

r t hit dt Xn

j1r ij t dZ jt cit dW it ; for i 1; . . . ; m; 2

where hi (t) is the instantaneous risk premium of security i and r ij (t) stands for the instan-taneous volatility of security i with the change on the risk factor Z j (t), and ci (t) is S i s idio-syncratic risk . For computational simplicity, those parameters are required to be at mosttime varying. 1 Eqs. (1) and (2) are consistent with asset pricing models in continuous timeas in Merton (1971), Browne (2000), and Gomez and Zapatero (2003) . The above settingfor the asset pricing model equips us with an explicit modeling of systematic and idiosyn-cratic risks, as in the Arbitrage Pricing Theory of Ross (1976) in discrete time.

Let x i (t) be the proportion of the investors wealth invested in the risky security i at timet, for i = 1, . . ., m, with the remainder, 1 Pmi1 xit , invested in the cash bond. Assumethat x 1 (t), . . ., xm(t) are admissible and F t is an adapted control processes, that is, x i (t)is a non-anticipative function that satises the condition of bounded variation

R T

0

Pmi

1 x2i t dt < 1 for an investment horizon T < 1 . Instead of writing the investors

fund value, we consider the gross return which is equal to the investment portfolio valuedivided by the initial portfolio value at any point in time. Let R (t) be the portfolios grossreturn at time t. The portfolio return follows:

d Rt Rt

1 Xm

i1 xit dS 0t S 0t X

m

i1 xit

dS it S it

r t

X

m

i1 xit hit dt

X

m

i1

X

n

j1 xit r ij t dZ jt xit cit dW it " #; 3

with initial value R(0) being equal to 1. To have a better exposition, we adopt matrix nota-tion. Denote

r t r ij t ;

ht h1t ; . . . ; hmt > ;ct diagc1t ; . . . ; cmt ;

1

The methodology developed in this paper can be extended to the cases in which parameters depend on thestate variables. For example, the short term rate process r(t) is usually considered random as in Vasicek (1977),Cox et al. (1985), and Heath et al. (1992) . However, the computational complexity would be extreme andanalytical solutions would be almost impossible. From the results developed later in this paper, these parameterscan be relaxed to satisfy the condition that the market price for the aggregated risk depends only on time.



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xt x1t ; . . . ; xmt > ;

Z t Z 1t ; . . . ; Z nt > ;

W t W 1t ; ; W mt > :

Then r (t) is an m n matrix and c(t) is a diagonal matrix of size m m. h(t) and x(t) arem-dimensional column vectors. Z (t) and W (t) are n-dimensional and m-dimensionalBrownian motions, respectively. Thus, Eq. (3) can be rewritten as

d Rt Rt

r t xt > ht dt xt > r t dZ t xt > ct dW t : 4Note that we have adopted an incomplete market asset pricing model as in He and

Pearson (1991a,b) . To eliminate redundant assets, we assume that [ r c ] is of full row rank,that is, rr > + cc> is an invertible matrix.

Although no other restrictions are imposed on the choices of admissible portfolios, suchas short sales, continuous trading, and transactions costs, the following usual conditionsfor admissibility of a portfolio policy x(t) must be satised for the stochastic integrability:

i E R T 0 j xt > l t jdt h i< 1 ;ii E R T 0 xt > r t r t > ct ct > xt dt h i< 1 ;

8>:5

where E stands for the expectation operator. Let C T be the set of all m-dimensionalstochastic processes that satisfy (5) and the condition of bounded variation

Z T

0 xt > xt dt < 1 for some T < 1 :

Hence, C T consists of admissible portfolios.

2.2. Benchmark portfolio and the growth optimum

Institutional money managers are typically judged by the return on their actively man-aged funds relative to the return of a preset benchmark. As indicated earlier, the focus of this paper is to determine an investment strategy that is optimal relative to the perfor-mance of a benchmark. The benchmark might be an equity fund (S&P 500 Index), a bondfund (Lehman Brothers Bond Fund), or a balanced fund (mix of stocks and bonds),depending on the fund properties and investors motivation. In other applications, abenchmark portfolio could be a stream of liabilities, such as a pension fund. We assumethat the selected benchmark carries only the market risk. That is, the benchmark grossreturn follows the stochastic process

d M t M t

qt dt bt > dZ t ; 6

where q(t) and b(t) are the instantaneous rate of return and the volatility vector associatedwith the latent factors, respectively. This specication is in line with the Ross (1976) modeland consistent with existing linear asset pricing models. The major difference of this modelfrom the existing asset pricing models is that this model does not rely on a set of predeter-mined factors which are desired to price all assets in the economy. Going back to thedynamics of the nancial instruments, we could interpret that each of the individual



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instruments has a component of risk that is driven by the same latent factors (the Brown-ian motion Z (t)), in addition to its instrumental specic risk (the Brownian motionW (t)). Ferson et al. (1993) conducted general tests of latent variable models. As theypointed out, latent variable models assume that expected returns vary over time as func-tions of a small number of risk premiums, which are common across assets. Similar totheir model, our model does not assume a functional form for the conditional expectedreturns and therefore avoids biased inference from a misspecied model.

Since the growth optimum portfolio plays an important role later, we present a brief review of this specialized portfolio. It is well known that maximizing the expected log util-ity of wealth is equivalent to maximizing the geometric growth rate. In the setting of Mer-ton (1971), the vector of the growth optimum portfolio weights is

gt r t r t > ct ct >

1

ht ; 7

where r (t), c(t), h(t) and r(t) are general stochastic processes with minor restrictions forstochastic integrability. Furthermore, for any utility function, that is, hyperbolic absoluterisk aversion (HARA), the optimal portfolio can be decomposed into two parts. The rstpart is the growth optimum, and the second part is a hedging component for the stochasticinvestment opportunities.

Denote G (t) as the return of the growth optimum portfolio at time t. The dynamic of G (t) is

dG t G t

r t gt > ht dt gt > r t dZ t gt > ct dW t : 8The optimal expected growth rate, dened as the average of the optimal value function

over the investment horizon, E 1T ln G T , is equal toE 1T Z T 0 r t 12 ht > r t r t > ct ct > 1ht dt !:

If r (t), c(t), h(t) and r(t) are constant over time, the optimal growth rate is

r 12

h> rr > cc> 1h:

A well-known result of the growth optimum is that, in the long run, it can beat anyportfolio with probability 1. This produces a strong motivation for investing in the growth

optimum portfolio when the investment horizon is long term.

2.3. Active portfolio management

Asset or portfolio managers are hired to ensure the achievement of the nancial goals of their clients. A portfolio management contract is basically a principal-agent relationship,whereby the principal (clients) establishes a set of guidelines (the investment mandate) forthe management of his or her assets, which portfolio managers (agents) are to observe.This not only involves rules and guidelines for the management of assets, but also requiresthe establishment of adequate incentives for the agent in order that the agent will managethe assets properly and in accordance with clients wishes.

Actively managed portfolios focus on the tradeoff between reward (excess return) andrisk, relative to a benchmark. Roll (1992) discussed this approach, using the tracking errormeanvariance framework. He pointed out that the tracking error mean variance optimal



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portfolio is not necessarily meanvariance efficient due to an implicit constraint that tiesthe portfolios performance to a selected benchmark. The objective of a portfolio manageris basically to generate a cash ow that matches the desired cash ow of each client as clo-sely as possible. The objective is not necessarily constrained to maximizing the expectedportfolio return on a risk adjusted basis. Instead, the portfolio policy may be associatedwith a performance against a benchmark portfolio that has been chosen by the investmentclients. Assuming an ability for selecting securities, portfolio managers are expected tohave better performance than the chosen benchmark portfolio. Since the risk of an activelymanaged portfolio is partially characterized by the performance of the selected bench-mark, it may be more desirable to focus on the squared portfolio return in excess of thebenchmark return. For example, if the excess return of a portfolio is constant and equalto a negative value, the risk in the tracking error mean variance analysis would be quan-tied as zero, which is certainly unacceptable. There is no reason for any investor to prefersuch a portfolio with a sure loss to any one that may have larger standard deviation. Toavoid such a pitfall, we use the squared tracking error as the risk measure.

With k as the sensitivity toward risk, the investors optimal portfolio model can beformulated as

max xt 2C T

E RT M T 12

kE f RT M T 2g

subject tod Rt Rt

r t xt > ht dt xt > r t dZ t xt > ct dW t d M t M t

qt dt bt > dZ t :

9

The active portfolio management problem is formulated as a standard stochastic controlproblem. For clarity and conciseness, the time argument for a stochastic process is omittedwhenever possible.

In light of Roll (1992), the objective function in (9) is dened as the risk adjusted excessreturn over the benchmark. The riskness of the investors portfolio is gauged by theparameter k. A smaller value of k indicates a riskier portfolio, while a large value of kdrives the portfolio closer to the benchmark. For example, if the value of k approacheszero, the optimal investment portfolio will be identical to the benchmark or the closest rep-lication portfolio of the benchmark in the metric of mean-square error. One thing whichdifferentiates our model from the Roll (1992) model is that the meanvariance analysis isnot generally equivalent to a quadratic utilit y maximization in the presence of an extrastate variable and incomplete market setting. 2

If the capital market is complete, the set of Arrow-Debreu state prices is unique and thenotion of a stochastic discount factor and abnormal returns can be well dened. However,in an incomplete market setting, one may have to assume homogeneous belief so that the

2 However, if the market is complete, the continuous time version of the meanvariance analysis is equivalent tothe standard quadratic utility maximization. Let n be the unique state price density process and R be the target

excess expected return. Then, the relation betweenk

andR

is given by1k

RE n2 E n

V n

where E and V represent the mean and variance operators.



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market is effectively complete. The homogeneous belief may not be realistic in an econ-omy with delegated portfolio management. The main drive for delegated portfolio man-agement is basically information asymmetry. We assume that only the representativeactive portfolio manager has information and his clients do not. In this paper, we adoptan incomplete market setting in the hope that our model formulation is more general. Thesystematic and idiosyncratic risks are the main objects of the study which can be moreeffectively explored with an incomplete market setting as specied in (2). The incompletemarket setting allows for the case where the benchmark is not replicable and then, relativeto the benchmark, individual investment elements will carry both the idiosyncratic riskand the systematic risk to the investors. On the other hand, the benchmark portfoliowould be redundant for a complete market setting (the model would become a mere exten-sion of Rolls model to the continuous time analogue).

3. Solution to the dynamic investment model

3.1. Optimal portfolio weights

For the objective function considered in this paper, the relevant state variables are thereturns of the investment portfolio and the benchmark portfolio, namely R(t) and M (t).The outcome from this approach reveals how the optimal performance function and theassociated optimal control polices can be obtained as the solution to a standard stochasticcontrol problem. The portfolios gross return process, R (t), is controlled by x(t), while M (t)is a given process that describes the evolution of the benchmark portfolios gross return. We

seek such a solution, x(t), that is a function of t, R(t) and M (t). Denote the optimal perfor-mance function starting from time t to the end of the investment horizon T as

J R; M ; t max x2 C T

E RT M T 12

k RT M T 2jF t !: 10That is, J (R , M , t) is the conditional expectation of the risk adjusted performance of theportfolio over the benchmark, given the current information F t . Assume J is differentiablewith respect to t and twice continuously differentiable with respect to R and M . DenotingJ . and J .. as the rst order and the second order derivatives, respectively, the Hamilton JacobiBellman optimality equation is

J t sup x2 C T

r x> h RJ R q MJ M 12 x> rr > cc> xR2 J RR x> rb RMJ RM 12

b> b M 2 J MM & ' 0;11with boundary condition J R; M ; T R M 12 k R M

2. Assume J is strictly concavein R . Solving the embedded maximization problem in (11) reveals the following theorem.

Theorem 1. Suppose J is differentiable with respect to t and conti nuously twice differentiablewith respect to R and M. If J is strictly concave with respect to R, 3 then the optimal portfolioweight vector, x * , in the risky securities, is a linear combination of the growth optimum

portfolio and the replicating portfolio of the benchmark at any point in time, namely,

3 This assumption will be veried when we obtain the optimal value function by solving the optimalityequation.



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x J R

RJ RRg

MJ RM RJ RR

p ; 12

where g = ( rr > + cc> ) 1 h and p = ( rr > + cc> ) 1 rb are the growth optimum portfolio and the benchmark replication portfolio weights in the risky securities, respectively.

It can be observed in (12) that the weight in the growth optimum is equal to the inverseof the relative risk aversion level in the standard utility maximization. The weight in thebenchmark replication portfolio can be interpreted as the tradeoff between the optimalportfolio and the benchmark portfolio. Thus, the optimal portfolio weights have a simplestructure consisting of three components. The rst component is related to the growthoptimum portfolio, the second component describes how the optimal portfolio tradesoff with the benchmark portfolio, and the third component is the risk free asset. Asexpected, the optimal portfolio weight invested in the benchmark portfolio changes withthe return on the benchmark portfolio. This simple structure leads us to conclude that athree fund separation holds for the specied investment objective. Thus, all actively man-aged portfolios can be generated from three funds, namely the cash bond, the growth opti-mum and the (replicating) benchmark portfolio.

Theorem 1 is based on the condition that the HJB equation (11) is solvable. Substitut-ing the optimal weights x * in (11), the optimality equation is

J t rRJ R q MJ M 12

b> b M 2 J MM 1

2 J RR J Rh MJ RM rb >

rr > cc> 1 J Rh MJ RM rb 0: 13

3.2. Solution to the optimality equation

We provide a general solution to (13) with the required boundary condition

J R; M ; T R M 12

k R M 2: 14

Theorem 2. The solution to the optimality Eq. (13) with boundary condition (14) is given as

J R; M ; t ct b1t R b2t M 12

a11t R2 a12t RM 12

a22t M 2; 15

where

ct 12k R T t h> geR T

ur 32h

> g d sdu;

b1t eR T t r h> g d s;b2t eR T t qdt 1 R T t h> g peR

T

ur 32h

> gh> p d sdu ;a11t keR T t r 12h> g d s;a12t ke

R T

t r qh> g pd s;

a22t keR T

t q 12b

> b d s 1 12 R T t h rb > g peR T

ur q32h

> g2h> p12b> b d sdu :

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 16



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The proof of Theorem 2 can be found in Appendix . Since a 11 < 0, the condition that theperformance function J is strictly concave for the embedded maximization in (11) auto-matically holds. Therefore, the necessary conditions are sufficient for the embeddedmaximization.

Having obtained the optimal performance function, we now explicitly derive the opti-mal portfolio weights at each point in time. As a result of the direct calculation using The-orems 1 and 2 , the formula for the corresponding optimal portfolio weights for given risksensitivity and other model parameters follows.

Theorem 3. The optimal weight vector in the risky assets at each point in time is

x 1

k Re

R T

t 12h

> gd s M R

e

R T

t sd s 1 g M R e

R T

t sd sp : 17

where s q 12 h> g h> p .

The proof of Theorem 3 is given in Appendix .

3.3. Interpretation of the optimal investment portfolio

Similar to Browne (2000), our actively managed portfolio is a dynamic linear combina-tion of weights in the growth optimum and the benchmark. However, the percentages allo-cated to the growth optimal portfolio and the benchmark must depend independently on

the two state variables as opposed to Browne (2000) model, in which allocation dependson a single state variable, the ratio of returns on the actively managed portfolio and thebenchmark. Our model is more general than Browne (2000) because it provides investorswith more exible allocation strategies. It is implied from examining the optimal dynamicportfolio weights that three fund separation holds for our model. However, the allocationsin the three funds are dynamically changing over time.

The optimal weight in the benchmark portfolio dynamically depends on the ratio, M R , of the return on the benchmark to the return on the actively managed portfolio, modied byan exponential factor e

R T

t sd s, which is jointly determined by the risk premiums of the

benchmark replicating portfolio and the growth optimum. If R T

t sd s

>0, the proportioninvested in the benchmark is greater than the ratio of the returns on the benchmark

and the actively managed portfolio. Similarly, if R T t sd s < 0, the proportion invested inthe benchmark is less than the ratio of the returns on the benchmark and the actively man-aged portfolio. It is also implied that, if the benchmark outperforms the fund, the fundallocates more to the benchmark portfolio than to the growth optimum. This portfoliotrading activity involves market timing. Much research has devoted to measuring portfoliomanagers timing abilities; see e.g. Treynor and Mazuy (1966), and Henriksson and Merton(1981).

On examining the dynamic portfolio weights in (17), we can easily reach the following

straightforward corollaries.

Corollary 1. The optimal portfolio weight tilts more toward the benchmark as the value of therisk parameter k increases.



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The intuition is that when the risk sensitivity parameter is larger, the manager is morerisk averse, therefore, his portfolio tilts more toward the benchmark-replicating portfolio.

Corollary 2. For a xed risk sensitivity parameter k, the optimal portfolio weight tilts toward the benchmark, if the active portfolio under-performs the benchmark.

As the ratio, M R , becomes large, the weight in the benchmark-replicating portfolio pincreases. However, the changing direction of the weight in the growth optimum dependsalso on the other parameters such as the coefficient of g.

The above two corollaries roughly give the changing direction through the state vari-ables, but the magnitude of changes depends on the parameter values and the outcomesof the state variables over time. For a numerical examination of changing portfolioweights, see Figs. 1 and 2.

4. Portfolio performance evaluation

Empirical research has shown that the majority of mutual funds have consistentlyunder-performed relevant benchmarks. Does this indicate that managers have inefficientlyallocated funds? As Gruber (1996) argued, management ability and transaction costs maybe the driving factors that are not included in the measurement of mutual fund perfor-mance. The traditional measurement for mutual fund performance is closely related tothe performance of the market portfolio, whose return may not include transaction costsfor trading.

4.1. Measuring abnormal returns

Given the model (9) and the investors risk sensitivity k, we can explicitly calculate theoptimal investment policy over time using Eq. (17). If a manager is reluctant to acquirenew information at a point in time, he is expected to act as an efficient agent with an opti-mal portfolio investment policy. On the other hand, if the manager is willing to obtain newinformation for a cost, his/her portfolio policy will denitely differ from that of the unin-formed one. Hence, the difference between the actual outcome of the portfolios return andthe model-tted (efficient portfolio) return should measure a managers abnormal perfor-mance for a given degree of information. Let R k be the efficient portfolio return withparameter k. The abnormal performance at time t is dened as b Rt Rkt , where b Rt is the actual outcome of the portfolios return at time t.One issue that has not been directly addressed in the area of investment management ishow to infer a managers risk sensitivity k. The regression based models, such as theCAPM, are not proper for dynamic investment management, since the assumption forthose models is that all returns are independent and identically distributed for all periods.In the CAPM based models, risk sensitivity is characterized by the beta coefficient, and themanagers abnormal performance is quantied as the intercept from regressing the fundon a benchmark portfolios return. For dynamic portfolio management, managers are

required to constantly shift their portfolio weights from securities to securities with newobservations. So, a static regression model is not proper for characterizing managers riskattitudes in a dynamic nancial world. Ferson and Schadt (1996) discussed measurementof fund strategy and performance in changing economic conditions.



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4.2. A measure for performance evaluation

We estimate the risk sensitivity parameter k by dynamically matching the actual port-folio return to its expectation from the model under a given measure, k k. The estimated kminimizes the total difference of the actual returns and the model expected returns,

k arg mink b R Rkn o: 18We call k the implied risk sensitivity of a portfolio. For our empirical analysis, k k is

chosen as the usual l 2 measure:

Z T 0 b Rt Rkt 2dt ; 19

where

b Rt and R k (t) are the realizations of the actual returns and the model tted returns

at time t . The quantity (19) is the tting error .If there is a perfect t with the impled k, the manager has demonstrated an efficient per-formance as if he/she were uninformed. Similarly, if there are tting errors in the impliedk, it is inferred that the manager possesses some skills (either good or bad). However, themanagers performance depends on the actual outcome of asset returns. Based on theabove discussion, we dene the performance index , P k , as the total abnormal performancefor the entire period,

P k

b R Z T 0

b Rt Rkt dt : 20

P k b

R

> 0 indicates that the manager has outperformed the model suggested portfolio,which has a performance index P k = 0. Similarly, P k b R< 0 indicates that the managerhas under-performed the model suggested returns. We now dene the criterion for rankingportfolio performances.Denition 1. Portfolio 1 with a realized return b R and risk sensitivity k is superior toportfolio 2 with a realized return b R

and the risk sensitivity k* , if

P k b RP P k b R:

The above categorization provides us with two hierarchies for portfolio performanceranking. The rst ranking criterion focuses on the portfolios total abnormal performance,ignoring the effect of individual risk sensitivity. The second criterion considers the level of risk sensitivity with which rankings are determined among portfolios with similar risk per-spectives. If our intention is to compare portfolios performances with similar risk sensi-tivities, then we should use the latter approach for the purpose of ranking funds.

5. An event study

Data for the period from September 2001 to December 2005 were collected from Data-stream and the Fama and French web site. We used the twelve industry portfolios char-acterized in the Fama and French web site as our investment instruments. Eight USmutual funds were selected from Datastream for our study.

First, we applied general factor analysis to decompose the riskiness of the twelve indus-try portfolios into factors. Then, we decided the optimal number of factors through a risk



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minimization model on the ratio of the benchmark and the portfolio specic risk. To beconsistent with the literature, we used the S&P 500 index as the benchmark portfoliofor our study.

After determining the optimal number of risk factors, we estimated the risk sensitivityparameters for the selected mutual funds using our suggested model in Section 3. We pro-vided a comparison between the performance using our model and the actual realizations.We will discuss how our model can be used for measuring portfolio performance in adynamic environment as opposed to the Sharpe Ratio.

5.1. Factor analysis for risk characterization

We assume a continuous time model similar to the factor structure of Ross (1976) forthe dynamics of the asset prices. Sharpe (1992) presents a twelve asset model for the intro-duction of his style analysis for investment. Grinblatt and Titman (1983) discussed theessence of factor pricing in a nite economy. However, we do not conne ourselves indeciding the candidate factors to avoid selection biases. Instead, we employ a statisticalfactor analysis to determine the optimal number of latent factors. Ferson et al. (1993) pro-vided general tests for the latent variable models. This methodology for asset pricing pur-poses has been robustly tested.

Our analysis is carried out on a daily basis. Instead of simply dictating the number of factors, we use the signal to noise ratio criterion for determining the optimal number of factors. This criterion is detailed below.

It is expected that the specic variances generated from factor analysis and the bench-

mark residues from the regression model are diminishing with the number of factors.Thus, we look at the ratio of the variance of the regression residues of the benchmarkand the average specic variance of the industrial portfolios. The intuition is that we can-not over-specify the number of factors, because the ratio of residue to the specic variancebecomes arbitrarily large. On the other hand, we cannot under-specify the number of fac-tors, otherwise the residue for the benchmark portfolio tends to be large, which increasesthe ratio of residues. So, we expect to obtain the correct number of factors by thiscriterion.

We rst apply factor analysis to the twelve industry portfolios for a range of factors andobtain the factor loadings, the portfolio specic risk, and the factor scores. Then we

regress the benchmark returns on the obtained factor scores to get the benchmark specicrisk. Since our assumption is that the benchmark carries factor risk only, we nally deter-mine the optimal number of factors by minimizing the ratio of the benchmark and theindustry specic risk. In expressions, let Z be the factor scores. The residue for the regres-sion of the benchmark return, rb, on Z is written as

b r b q b> Z :

The optimal number of factors minimizes the signal to noise ratio

Variance of b1n

Pni1c2i

;

where n is the number of industrial portfolios and c2i is the specic variance for industryportfolio i from the factor analysis.



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5.2. Empirical results

A model is practical only if it recovers reality and has a strong performance. We presenta numerical example to illustrate how our model can be used in practice.

Following the above discussion, we obtain the benchmark residues and the averageportfolio specic variance by varying the number of factors (the risk free rate is discretion-ally specied as 0.75% per annum throughout our numerical example, to reect a lowinterest rate period from September 2001 to December 2005).

From Table 1 , we see that the optimal number of factors is 3 for this study using thesignal to noise criterion. Table 2 provides the estimated parameters with the three factormodel. This is consistent with the famous Fama and French three factor model.

Table 1Residues with varying number of factors

Number of factors 1 2 3 4 5

Benchmark specic variance 0.0010 0.0010 0.0006 0.0006 0.0005Average portfolio specic variance 0.0137 0.0122 0.0114 0.0099 0.0078Ratio of specic variances 0.0698 0.0823 0.0568 0.0610 0.0688

Table 2Parameter estimation for the twelve industry portfolios and the S&P 500 index

Portfolio Mean return ( l ) Factor loading ( r ) Portfolio specic ( c)

NoDur 0.0248 0.0409 0.0973 0.0311 0.0718Durbl 0.0059 0.1539 0.1219 0.0936 0.1348Manuf 0.0414 0.1079 0.0929 0.1275 0.0136Enrgy 0.0596 0.0599 0.0984 0.0787 0.1706Chems 0.0348 0.0551 0.1208 0.0669 0.0622BusEq 0.0182 0.2091 0.0758 0.1120 0.1255Telcm 0.0145 0.1737 0.0889 0.0364 0.1147Utils 0.0262 0.0756 0.0828 0.0409 0.1484Shops 0.0263 0.1115 0.1055 0.0697 0.0969Hlth 0.0073 0.0809 0.1157 0.0265 0.1036Money 0.0346 0.1253 0.1061 0.0501 0.0660Other 0.0335 0.1461 0.0906 0.0868 0.0660

S&P 500 0.0232 0.1235 0.1041 0.0676

Table 3Implied risk sensitivities for the selected mutual funds

Name Imp. risk sens. Fit. error Perf. index Sharpe R.

Chase Vista 2.2164 0.0102 0.0697 0.0102Fidelity 1.2739 0.0037 0.0173 0.0394First Amer. 1.2628 0.0013 0.0190 0.0412Franklin 2.3223 0.0171 0.0867 0.0024Morg. Stan. 1.5840 0.0054 0.0261 0.0195Perrit Micro 0.7842 0.0090 0.0084 0.0808Wells Fargo 1.7163 0.0109 0.0451 0.0097Vanguard S&P 500 2.1596 0.0093 0.0519 0.0116



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With the estimated model parameters, we now nd the implied risk sensitivity. Themutual funds selected for our study consist of different investment mandates. The compu-tational results are presented in Table 3 .

All selected funds appear to out-perform the S&P 500 price index. However, from thenumerical results in Table 3 , we nd all selected mutual funds under performed their

0 200 400 600 800 1000 1200

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Time Period (September 2001 December 2005)

Comparisons among the Actual Return, the Benchmar, and the Fitted Return

Fitted Portfolio Value

FRANKLIN EQ.INC.FD.S&P 500 Price Index

0 200 400 600 800 1000 12000.4

0.2

0

0.2

0.4

0.6

0.8

1


W e i g h

t s i n P e r c e n t a g e

Fund Allocation for Franklin

Growth Optimum PortfolioBenchmark PortfolioCash

W e i g h


Fig. 1. The upper gure presents the performance of the mutual fund, Franklin Eq. Inc. Fd. While the fundpicked up all the market momentum and had a similar performance to the benchmark, it is a little bit under-performing the model suggested portfolio. The lower gure shows the dynamics of the portfolio weights if thefund follows the suggested strategies.



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model projected portfolios, as the performance indices (fourth column in Table 3 ) are allnegative. This is consistent with the ndings in mutual fund research. The performanceindex seems to be a proper performance measure for ranking mutual funds, as it is very

0 200 400 600 800 1000 1200

0.8

1

1.2

1.4

1.6

1.8

2

2.2


P o r

t f o l i o G r o s s

R e t u r n

Comparisons Among the Actual Return, the Benchmark, and the Fitted Return

Fitted Portfolio Value

PERRIT MICRO CAP.OPPS. FD.

S&P 500 Price Index

0 200 400 600 800 1000 12000.6

0.4

0.2

0

0.2

0.4

0.6

0.8


W e i g h


Fund Allocation for Perrit Micro

Growth Optimum Portfolio

Benchmark Portfolio

Cash

Fig. 2. The upper gure presents the performance of the mutual fund, Perrit Micro. This fund had the bestperformance for the period from September 2001 to December 2005. It outperformed the benchmark throughoutthe period. But it still under-performed the model suggested portfolio. The tted portfolio is more stable andtherefore the fund has a lower lambda (riskier) than any fund selected. The lower gure depicts the portfolio turnover throughout the period.



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comparable to the Sharpe Ratio. If the performance index P k is used, the descending orderof ranks is: Perrit Micro Fidelity First Amer. Morg. Stan. Wells Fargo VanguardS&P 500 Chase Vista Franklin. If Sharpe Ratio is used, then the corresponding rank-ing is: Perrit Micro First Amer. Fidelity Morg. Stan. Vanguard S&P 500 ChaseVista Wells Fargo Franklin. Since this performance is based on one trajectory and isnot related to the risk sensitivity, the ranking may not be entirely fair. If the risk sensitivityis high (low lambda), the exceptional performance may be wiped out by a downward mar-ket trend for a static portfolio. But if mangers are able to dynamically control for risk, thesituation of an upward or downward market trend does not really matter. The best per-forming fund, Perrit Micro, has a low lambda, and therefore are very sensitive to risk.The implied risk sensitivity in ascending order is: Perrit Micro Fidelity First Amer. Morg. Stan. Wells Fargo Vanguard S&P 500 Chase Vista Franklin, which is thesame as the performance ranking using P k with the only exception that positions of FirstAmer. and Fidelity are exchanged.

It is observed that the tted portfolio is very close to reality, since the total tted errors(third column in Table 3 ) are all small. To visually examine the deviation of the tted trajectory of the portfolio value from the actual outcome, Figs. 1 and 2 in Appendix depictthe best and worst performances within the selected funds. The dynamics of the tted port-folio weights are also provided in those gures. The wild portfolio turnover in the begin-ning of the period is largely due to the event on September 11, 2001.

6. Conclusion

This paper studied a dynamic model of portfolio management process related to theproblem of outperforming a benchmark. Assuming managers adopt a benchmarking pro-cedure for investment, the optimal portfolio policy is state dependent, being a function of time to the investment horizon, the return on the benchmark, and the return on the invest-ment portfolio itself. We obtained an explicit formula for calculating the optimal portfolioweights in a continuous time setting.

Based on risk sensitivities, we also studied a model for portfolio performance evalua-tion. Risk sensitivities are estimated using a dynamic matching approach which minimizesthe total error between the actual returns and the returns implied by the model. Empirical

analysis shows that there is a strong relation between the level of performance and a man-gers risk sensitivity.We briey discussed how this model can be used for evaluating portfolio perfor-

mance. We selected a sample of the U.S. funds to illustrate the implementation of ourmodel. The tted portfolio value is very close to reality with only a small margin of difference.

Appendix

Proof of Theorem 1. Since the utility function U is strictly concave, the optimal valuefunction J is also strictly concave in R . The proof of this statement is a direct observationof the conclusion in Theorem 2 . The rst order condition implies that the optimal solution,x * , to the embedded maximization problem in (11),



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sup x2 C T

r x> h RJ R q MJ M 12 x> rr > cc> xR2 J RR x> rb RMJ RM

12

b> b M 2 J MM & ':must satisfy the following equation

h RJ R rb RMJ RM rr > cc> x R2 J RR 0:

Since r c is of full row rank by assumption, rr > + cc> is an invertible matrix. There-fore, by the strict concavity of the function J (this implies that J RR 5 0) and noting thatg = ( rr > + cc> ) 1 h and p = ( rr > + cc> ) 1 rb , the optimal solution is

x J R

RJ RRg

MJ RM

RJ RRp

which completes the proof of Theorem 1 . h

Proof of Theorem 2. The relevant partial derivatives of the value function J are:

J t c0 b01 R b02 M 12 a

011 R

2 a012 RM 12 a022 M

2

J R b1 a11 R a12 M

J M b2 a12 R a22 M

J RR a11

J RM a12

J MM a22;

8>>>>>>>>>>>>>>>>>>>>>>>>>:

where 0 stands for the rst order derivative with respect to time t. Substituting the abovepartial derivatives into (13) yields

c0b21

2a11h> g b01 r h

> gb1 R b02 qb2 h> g pb1a12a11 M a011 r

12

h> g a11 R2 a012 r q h> g pa 12 RM a022 q

12

b> b a22 12 h rb > g pa212a11 M 2 0: 21Setting all coefficients of terms in R and M equal to zero in the above equation derives

the following systems of ordinary differential equation in time t



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c0 b21

2a11h> g

b01 r h> gb1

b02 qb2 h> g p

b1a12a 11

a011 r 12

h> g a11a012 r q h

> g pa12

a022 q 12

b> b a 22 12 h rb > g pa212a11 :

22

The boundary conditions are

cT 0; b1T 1; b2T 1; a11T a22T k; a 12T k:

The 2nd, 4th and 5th equation in (22) jointly imply

b1t eR T t r h> gd sa11t keR T t r 12h> gd sa12t keR

T

t r qh>

g pd s

8>>>>>>>>>:Substituting b1 , a 11 , and a 12 into the 1st, 3rd, and 6th equations in (22) and solving thedifferential equations, we obtain

ct 12k R T t h> geR T

ur 32h

> gd sdu

b2t eR T t qdt 1 R T t h> g peR T

ur 32h

> gh> p d sdu a22t keR

T

t q 1

2b> b

d s

1 12 R T t h rb > g peR

T

ur q3

2h> g2h> p1

2b> b

d s

du :

8>>>>>>>>>>>>>:

Proof of Theorem 3. From the results of Theorem 2 in Eqs. (15) and (16), the partial deriv-atives of the performance function J are obtained as

J R b1t a11t R a12 M

J M b2t a12t R a22 M

J RR a11t J RM a 12t

J MM a22t :

8>>>>>>>>>>>>>>>:



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Hence,

J R RJ RR

b1t

Ra12t 1

a12t M a11t R

MJ RM RJ RR

a 12 M a11 R

:

8>>>:Let s q 12 h> g h> p , then, x

J R RJ RR

g MJ RM RJ RR

p

b1t

a11t R 1

a12t M a11t R g a 12t M a11t R p

1

k R e

R T

t

1

2h> gd s

M R eR

T

t sd s

1 g M R eR

T

t sd s

p :

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