An Intertemporal CAPM Approach to Evaluate Mutual
Fund Performance
JOW-RAN CHANG
Department of Quantitative Finance, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Rd., Hsinchu, Taiwan Tel.: 866-3-5742420, Fax: 866-3-5715403
E-mail: jrchang@mx.nthu.edu.tw.
MAO-WEI HUNG
College of Management, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei, Taiwan Tel.: 886-2-23630231 ext 2988, Fax: 886-2-23690833
E-mail: hung@mba.ntu.edu.tw
CHENG-FEW LEE∗
Department of Finance and Economics, Rutgers Business, Rutgers University, 94 Rockafeller Road, Piscataway, NJ 08854-8054, USA Tel.: 732-445-3530, Fax: 732-445-5927
E-mail: lee@rbs.rutgers.edu
Abstract. Merton (1973) and Campbell (1993) have demonstrated that if an investor anticipates information shifts, he will adjust his portfolio choice today in an attempt to hedge these shifts. Exploiting these insights, we construct a new performance measure to evaluate fund managers’ hedging ability. This new measure is different from two widely adopted performance evaluation measures: securities selectivity and market timing. Moreover, an econometric methodology is developed to simultaneously estimate the magnitudes of these three portfolio performance evaluation measures. The results show that mutual fund managers are on average with positive security selection and negative market timing ability. Furthermore, the mutual funds with investment style classified as “Asset Allocation” generally have positive hedging timing ability.
Key words: intertemporal CAPM, mutual fund, performance evaluation JEL Classification: G23
1. Introduction
Since mutual fund has been a major investment vehicle in U.S. as well as other coun-tries across the world, the evaluation of mutual fund’s performance has attracted enor-mous attention from both practitioners and academics. The most widely used measure is Jensen’s (1968) (1969)α measure, which uses the security market line to evaluate fund’s performance. However, there are some defects in Jensen’sα measure. One is that errors in inference may arise when the fund manager is a market timer. For instance, Jensen (1972) and Dybvig and Ross (1985) demonstrate that Jensen’s α measure may assign a negative performance when the fund manager possesses and utilizes superior timing information.
Fama (1972) indicates that there are two ways for fund managers to obtain abnormal returns. The first one is security analysis, which is the ability of fund managers to identify the potential winning securities. The second one is market timing, which is the ability of portfolio managers to time market cycles and take advantage of this ability in trading securities. Several market timing and selectivity models have been developed in two lines. Treynor and Mazuy (1966) develop the first quadratic market timing model to examine market timing ability of fund managers. The intuition behind the model is that a fund manager with market timing ability is on average to increase the stock portion of the managed portfolio when stock returns are high and reduce it when stock returns are low. A formal treatment of the quadratic market timing model is found in Jensen (1972), who develops theoretical structures for the evaluation of market timing performance of fund managers. Bhattacharya and Pfliederer (1983) extend the work of Jensen model by minimizing the variance of the forecasting error. Furthermore, Admati, Bhattacharya and Pfliederer (1986) use the portfolio approach and factor approach to assess the market timing ability of fund managers.
The second line of market timing research is initiated by Henriksson and Merton (1981). They use a free put option on the market portfolio with its exercise price equal to the risk-free rate. In their model, market timers forecast either that equities outperform bonds or vise versa. This implies that the probability of receiving an up or a down signal does not depend upon how far the market will be up or down. Henriksson (1984) employs the Henriksson and Merton’s model to evaluate mutual fund performance, and the empirical results did not support the hypothesis that fund managers are able to time the return on the market portfolio successfully. Jagannathan and Korajczyk (1986) offer explanations for apparent perverse timing involving possible option-like characteristics of mutual fund returns.
Numerous papers have employed different proxies for the benchmark portfolio in the CAPM framework to measure the performance of mutual funds. However, Roll (1977) argues that there is no appropriate benchmark portfolio to compute market beta and when inefficient market indices are used for the performance measure, any evaluation value can be assigned to a mutual fund. Therefore, Connor and Korajczyk (1986) develop a theory of portfolio performance measurement using the Arbitrage Pricing Theory (APT). Lehmann and Modest (1987) employ a variety of APT benchmark to investigate the sensitivity of the chosen benchmark to measure normal performance. To address the observed dynamic behavior of returns, Ferson and Schadt (1996) have proposed a conditional performance evaluation in which the conditional expected return on an asset is related to the conditional expected return on the market portfolio with a conditional time-varying beta and conditional alphas. However, their approach is a conditional CAPM which lacks a sound theoretical foundation in dynamic economics.
In this paper, we develop a new performance measure which explores an equilibrium version of dynamic asset pricing model proposed by Campbell (1993) to measure mutual fund performance. It is often believed that if a fund manager anticipates information shifts, he will adjust his portfolio to hedge these shifts in a dynamic economy. We extend Campbell’s framework to capture this hedging demand of mutual fund managers. In this model, hedging timing performance is developed to detect timing ability of hedging portfolio return as well as future return, market timing performance is used to detect the timing ability of market
return, and selectivity performance is constructed to detect the selectivity ability of mutual fund managers.
This paper contains several contributions. First, we extend the dynamic asset pricing model of Campbell to a multiple regression framework which argues that the realized excess returns on any portfolio can be represented as a linear function of its market risk premium and hedging risk premium. Second, we construct a new performance measure based on this multiple regression framework. This implies that our performance measure can explain the dynamic behavior of fund managers. Finally, we present an empirical examination of the selectivity, market timing, and hedging timing performance for a sample of U.S. mutual funds. The results suggest that mutual fund managers are on average better with selectivity ability than with market timing ability. We also find evidence for the hedging timing ability of the fund managers. This implies that these fund managers use the forecasts of the aggregate forward-looking factor in their hedging strategies.
The paper is organized as follows. Section 2 derives linear multiple regression function of the dynamic asset pricing model by extending Campbell’s equilibrium version of the dynamic asset pricing framework. In Section 3, we construct a new performance evaluation and derive its relationship to the general equilibrium structure of dynamic asset pricing model. Section 4 describes empirical methodology. Section 5 reports the data and the main empirical evidences. Conclusion is presented in the last section.
2. Dynamic asset pricing model
The pricing model adopted in the paper is based on a competitive equilibrium version of intertemporal asset pricing model derived in Campbell (1993).1The dynamic asset pricing model incorporates hedging risk as well as market. This section contains a brief review of the model and introduces the notation that we use throughout paper.
This model uses a loglinear approximation to the budget constraint to substitute out consumption from a standard intertemporal asset pricing model. Therefore, asset risk premia are determined by the covariances of asset returns with the market return and with news about the discounted value of all future market returns. Formally, the pricing restrictions on asset i imported by the conditional version of the model are
Etri,t+1− rf,t+1= −
Vii
2 + γ Vi m + (γ − 1)Vi h (1)
where Etri,t+1 − rf,t+1 is log return on asset i minus log return on riskless asset. Vii
denotes Vart(ri,t+1), γ is the agent’s coefficient of relative risk aversion, Vi m denotes
Covt(ri,t+1, rm,t+1), and Vi h = Covt(ri,t+1, (Et+1− Et),
∞
j=1ρ jr
m,t+1+ j), the parameter
ρ = 1 − exp(c − w) and c − w is the mean log consumption to wealth ratio.
Equation (1) states that the expected excess log return in an asset, adjusted for a Jensen’s inequality effect, is a weighted average of two covariances: the covariance with the return from the market portfolio and the covariance with news about future returns on invested wealth. The intuition for Eq. (1) is that assets are priced using their covariances with the return on invested wealth and with news about future returns on invested wealth. In addition,
we can explain rh,t+1as the hedging portfolio return which is an analogue of Merton’s (1973)
continuous time model.
Proposition. If there exists a single, infinitely lived agent with non-expected utility and his coefficient of relative risk aversion is constant across different assets, then intertemporal asset pricing model of (1) can be represented as
Etri,t+1− rf,t+1+ Vii 2 = βi m Etrm,t+1− rf,t+1+ Vmm 2 + βi h Etrh,t+1− rf,t+1+ Vhh 2 (2)
where market beta, βi m and hedging beta, βt h, are also the coefficients of the multiple
regression in Eq. (2). Proof: See Appendix.
Equation (2) implies that in additional to market factor in the traditional CAPM, there exists another factor, hedging factor, to explain expected asset return. Hedging factor is a portfolio which hedging the news about future returns on the market return.
3. A new performance evaluation measure
In this section we outline the foundations of the performance evaluation analysis and its relationship to the general equilibrium structure of asset pricing model derived in the last section. Let EtRi,t+1= Etri,t+1− rf,t+1+ Vii 2 , EtRm,t+1= Etrm,t+1− rf,t+1+ Vmm 2 , (3) EtRh,t+1= Etrh,t+1− rf,t+1+ Vhh 2 .
where Rj,t+1∀ j = i, m, h denotes the gross simple return on asset i, market portfolio, or
hedging portfolio and lowercase letters are used for logs of gross simple returns. Therefore,
Vj j/2 is a Jesen’s inequality effect.
Then, Eq. (1) can be expressed as
EtRi,t+1= βi mEt(Rm,t+1)+ βi hEt(Rh,t+1) (4)
and using the law of iterative expectations, we have
The formulation of dynamic market model is denoted as
Ri,t+1= βi mRm,t+1+ βi hRh,t+1+ εi,t+1 (6)
Equation (6) denotes that the realized excess returns on any portfolio can be represented as a linear function of its market risk and hedging risk. The random error,εi,t+1, has an
expected value of zero.
If the fund manager is an informed forecaster, he will tend to select securities which realizeεi,t+1> 0. Hence, his fund will earn more than the normal risk premium for its level
of risk. Allowing for the existence of a nonzero constant in Eq. (6), we have
Ri,t+1= αi+ βi mRm,t+1+ βi hRh,t+1+ ei,t+1 (7)
The new error term, ei,t+1, is assumed to have zero expectation and to be independent of
Rm,t+1and Rh,t+1. If the fund manager has the security selection ability, the intercept,αi,
will be positive. On the other hand, a passive strategy should be expected to yield a zero intercept.
Except for forecasts of price movements of selected individual stock, the fund manager will attempt to capitalize on any expectation he may have regarding the behavior of the market return in the next period and hedging portfolio return in the forward looking periods. That is, he should add his forecasts of the market factor,πm,t+1= Rm,t+1− E(Rm), and the
hedging factor,πh,t+1= Rh,t+1− E(Rh), to maximize his utility. The forecasts,πm∗,t+1and
π∗
h,t+1, based on the information set Ii,t, which is available to the manager i at time t, are
π∗
m,t+1= E(πm,t+1| Ii,t)= E(Rm,t+1| Ii,t)− E(Rm)
(8)
π∗
h,t+1= E(πh,t+1| Ii,t)= E(Rh,t+1| Ii,t)− E(Rh)
Bhattacharya and Pfleiderer (1983) assume that the manager observes a signal,πt+1+ υt+1
at time t whereυt+1is the difference between market factor and signal. It is easy to show
that the optimal forecasts in our model are
π∗
m,t+1= φm(πm,t+1+ ut+1)
(9)
π∗
h,t+1= φh(πh,t+1+ vt+1)
In Eq. (9),φmandφhcan be used to measure the quality of the manager’s market timing
and hedging timing information, respectively. The error term, ut+1andvt+1are assumed to
have zero expectation and to be independent ofπm,t+1andπh,t+1, respectively.
Theorem. If fund manager’s optimal forecasts of market return and hedging return are π∗
m,t+1= φm(πm,t+1+ ut+1) andπh∗,t+1= φh(πh,t+1+ vt+1), then we can rewrite (7) as
Ri,t+1= αi+ βi mRm,t+1+ βi hRh,t+1+ ei,t+1
= η0+ η1Rm,t+1+ η2Rh,t+1+ η3Rm2,t+1
where
p limη0= αi
p limη1= κ1E(Rm)(1− φm)− κ2E(Rh)(1− φh)
p limη2= κ3E(Rh)(1− φh)− κ2E(Rm)(1− φm)
p limη3= κ1φm
p limη4= κ3φh
p limη5= −κ2(φh+ φm)
Proof: See Appendix.
Equation (10) is similar to a quadratic market timing regression in the Treynor and Mazuy (1996) except for hedging portfolio, hedging timing and interaction of market portfolio and hedging portfolio. Hedging portfolio in Eq. (10) reflects the prediction of news about future market return. The manager’s response of change the hedging portfolio is controled by hedging timing term. In the dynamic model, there exists interaction effect between market portfolio and hedging portfolio. For the completeness, we write down the special case when there is no hedging portfolio in the following corollary.
Corollary. If there is no hedging demand, then the security selection and market timing can be represented as follows:
Ri,t+1= αi+ βi mRm,t+1+ ei,t+1 = η0+ η1Rm,t+1+ η3Rm2,t+1+ wt+1 where p limη0= αi p limη1= κ1E(Rm)(1− φm) p limη3= κ1φm
The disturbance term in Eq. (10) has the following expression:
wt+1= (κ1φmRm,t+1− κ2φmRh,t+1)ut+1− (κ2φhRm,t+1− κ3φhRh,t+1)vt+1+ et+1
(11) The first term inwt+1contains the information needed to quantify the manager’s market
timing ability. However, the second term inwt+1contains the information needed to quantify
the manager’s hedging timing ability.
4. Econometric methodology
To implement empirical investigation, we first construct a state variable system to estimate the hedging portfolio. We adopt the Vector Auto-Regressive (VAR) approach of Campbell
(1991). We assume that the real market index return as the first element of a K -element state variable vector zt. The other element of ztare variables that are known to the market at the end of the period t and are related to forecasting future market return. In addition, we assume that the vector ztfollows first order VAR
zt+1= zt+ ξt+1 (12)
where is a K × K matrix which is known as the companion matrix of the VAR. We can use the first order VAR to generate simple multi-period forecasts of future returns as
Etzt+1+i = i+1zt (13)
In addition, we define a K -element constant vector e1. The first element of e1 is one and the other elements are all zero. Therefore, we can write rm,t as rm,t = e1zt and
rm,t+1− Etrm,t+1 = e1ξt+1. It follows that the discounted sum of forecast revisions in
market return of Eq. (1) can now be represented as
(Et+1− Et) ∞ j=1 ρj rm,t+1+ j = e1 ∞ j=1 ρjjξ t+1 (14) = e1ρ(I − ρ)−1ξ t+1 = λ hξt+1
whereλhis defined as e1ρ(I−ρ)−1which measure the importance of each state variable in forecasting future returns on world market.
To obtain efficient and consistent estimates of parameters, a Quasi-Maximum Likelihood (QML) procedure with heteroscedasticity is used. We assume that the variance of the error term in Eq. (10) is as follows.
Ht+1= (θ1φmRm,t+1− θ2φmRh,t+1)2σu2+ (θ2φhRm,t+1− θ3φhRh,t+1)2σv2+ σe2
(15) where Ht+1, σu2, σv2, andσe2are variance ofwt+1, ut+1, vt+1, and et+1.
We use Eqs. (10) and (15) as the benchmark model. Under the assumption of conditional normality, log-likelihood function can be written as
ln L(ψ) = −TN 2 ln 2π − 1 2 T t=1 ln|Ht(ψ)| − 1 2 T t=1 wt(ψ)Ht(ψ)−1wt(ψ) (16)
whereψ are the unknown parameters in the model.
Since the normality assumption is often violated in financial time series, this paper estimates the model and computes all tests using QML approach proposed by Bollerslev and Wooldridge (1992). QML with heteroscadasticity estimation provides efficient and consistent estimates of the population parameters of interest. The standard errors for the
estimated coefficients that are calculated under the normal assumption need not be corrected if the true data are non-normal.
5. Data and empirical results
To detect selection ability and timing ability of a mutual fund manager, monthly returns from January 1980 to September 1996 (201 months) for a sample of 65 U.S. mutual fund are used in this study. The random sample of mutual funds is provided by the MorningStar Company. The MorningStar Company segregates mutual funds into four basic investment styles on the basis of manager’s portfolio characteristics. Our sample consists of 8 Asset Allocation, 14 Aggressive Growth, 10 Equity Income, 16 Growth, and 17 Growth Income mutual funds. The monthly returns on the S&P 500 Index were used for market return. Monthly observations of the 30-day Treasury bill rate were used as a proxy for the risk-free rate.
Table 1 contains summary statistics for returns of the equity funds. All values are com-puted in excess of the return on the U.S. T-bill closest to 30 days to maturity. Panel A in Table 1 contains mean, standard deviation, maximum, and minimum. Each investment style average shows that the asset allocation style has the smallest expected return and it also has the smallest standard deviation. However, the aggressive growth style has the largest maxi-mum return but it also has the smallest minimaxi-mum return and the largest standard deviation. In other word, the more aggressive the funds are, the more volatility of the fund returns will be. Panel B is the abbreviation of investment style index.
The effect of survival bias in fund performance measure is presented by Brown et al. (1992) and Hendricks et al. (1997). Since the funds in our samples contain only survival funds, it may exist survival bias. However, our empirical results show that compared to traditional performance measures, selectivity and market timing become poorer; but the additional hedging timing is important in dynamic performance measure. These results cannot seemly be explained by survivor bias.
Descriptive statistics for the state variables are reported in the Table 2. We select a set of instruments that have been widely used in the asset pricing literature. The instruments include the month to month change in the U.S. term premium which is equal to rate on the U.S. Treasury note in excess of the three-month T-bill rate; the dividend yield (DIV) which is the monthly S&P dividend yield, and the U.S. one month T-bill rate (TB). The term premium (TERM) follows De Santis and Gerard (1997) and others. Dividend yield (DIV) is a component of the return of stocks and hence it is a good forecasting variable for capturing predictions of stock returns. Campbell (1996) finds that the dividend yield has some predictive power for future stock returns. The short-term bill rate (TB) which has been used by Fama and Schwert (1977), Ferson (1989), and Ferson and Harvey (1991) is capable of predicting monthly returns of bonds and stocks.
We start our investigation by constructing the dynamic behavior of the state variables. Table 3 reports the coefficients in a one lag VAR. The matrix of coefficients in the VAR companion matrix is denoted by in Eq. (12). The first row of Table 1 shows that the monthly forecasting equation for the excess log return of market portfolio, rm,t−rf,t. There
is a minimal serial correlation in monthly market log return, but the coefficient on lagged
Table 1. Summary statistics for excess returns of the mutual funds
Investment Standard
Fund Name Style Mean Deviation Maximum Minimum
Panel A
General securities Aa 0.477 5.084 15.389 −17.151
Franklin asset allocation Aa 0.407 3.743 10.424 −19.506
Seligman income A Aa 0.394 2.414 8.474 −7.324
Usaa income Aa 0.316 2.024 9.381 −5.362
Valley forge Aa 0.293 1.803 9.980 −5.573
Income fund of America Aa 0.566 2.552 9.166 −8.836
FBL growth common stock Aa 0.273 3.599 10.466 −24.088
Mathers Aa 0.220 3.910 14.405 −14.750
Asset Allocation Average Aa 0.391 2.550 8.962 −9.464
American heritage Ag −0.905 6.446 28.976 −33.101 Alliance quasar A Ag 0.644 6.547 15.747 −39.250 Keystone small co grth (S-4) Ag 0.433 7.053 19.250 −38.516 Keystone omega A Ag 0.473 6.112 18.873 −33.240 Invesco dynamics Ag 0.510 6.009 17.378 −37.496 Security ultra A Ag 0.222 6.940 16.297 −43.468 Putnam voyager A Ag 0.808 5.781 17.179 −29.425
Stein roe capital opport Ag 0.578 6.783 17.263 −32.135
Value line spec situations Ag 0.145 6.240 13.532 −37.496 Value line leveraged GR inv Ag 0.601 4.970 14.617 −29.025
WPG tudor Ag 0.726 6.010 14.749 −33.658
Winthrop aggressive growth A Ag 0.476 5.596 17.012 −34.921
Delaware trend A Ag 0.787 6.536 14.571 −42.397
Founders special Ag 0.564 5.900 12.905 −31.861
Aggressive Growth Average Ag 0.459 5.814 13.142 −35.335
Smith barney equity income A Ei 0.601 3.270 7.813 −18.782 Van Kampen AM cap eqty-inc A Ei 0.510 3.530 12.292 −22.579
Value line income Ei 0.423 3.357 9.311 −18.242
United income A Ei 0.714 4.037 11.852 −13.743
Oppenheimer equity-income A Ei 0.555 3.422 10.071 −16.524
Fidelity equity-income Ei 0.706 3.612 10.608 −19.627
Delaware decatur income A Ei 0.547 3.615 10.269 −20.235
Invesco industrial income Ei 0.601 3.705 9.349 −20.235
Old dominion investors Ei 0.360 3.699 11.498 −21.092
Evergreen total return Y Ei 0.508 3.220 8.074 −13.857
Equity Income Average Ei 0.527 3.238 9.094 −18.718
Guardian park avenue A G 0.740 4.391 11.321 −27.965
Founders growth G 0.718 4.986 13.055 −25.108
Fortis growth A G 0.724 5.983 14.520 −30.771
Franklin growth I G 0.570 4.050 12.907 −11.706
Fortis capital A G 0.682 4.791 12.818 −21.585
Growth fund of America G 0.625 4.722 12.226 −23.962
Hancock growth A G 0.484 5.381 15.708 −25.236
Franklin equity I G 0.469 5.156 12.818 −32.135
Table 1. (Continued).
Investment Standard
Fund Name Style Mean Deviation Maximum Minimum
Nationwide growth G 0.598 4.370 11.444 −27.570
Neuberger & Berman focus G 0.434 4.366 12.187 −25.108
MSB G 0.517 4.665 13.452 −31.178
Neuberger & Berman partners G 0.661 3.612 9.311 −19.385 Neuberger & Berman Manhattan G 0.606 5.095 11.574 −30.500
Nicholas G 0.710 4.067 10.125 −19.385
Oppenheimer A G 0.225 5.234 11.321 −31.451
New England growth A G 0.727 5.802 19.120 −37.207
Growth Average G 0.608 4.505 11.121 −26.081
Pioneer II A Gi 0.517 4.386 10.912 −29.693
Pilgrim America Magnacap A Gi 0.611 3.949 10.843 −22.704
Pioneer Gi 0.410 4.339 12.293 −28.361
Philadelphia Gi 0.244 4.004 11.074 −23.457
Penn square mutual A Gi 0.504 3.907 11.852 −20.724
Oppenheimer total return A Gi 0.507 4.451 13.861 −22.829
Vanguard/Windsor Gi 0.726 4.078 10.746 −18.542
Van Kampen Am Cap GR & Inc A Gi 0.570 4.781 15.349 −32.135 Van Kampen Am Cap Comstock A Gi 0.599 4.539 13.167 −34.921 Winthrop growth & income A Gi 0.430 3.987 10.717 −24.088 Washington mutual investors Gi 0.723 3.882 11.409 −20.113
Safeco equity Gi 0.587 4.797 14.263 −31.042
Seligman common stock A Gi 0.553 4.224 11.785 −23.331
Salomon bros investors O Gi 0.583 4.194 11.785 −24.980
Security growth & income A Gi 0.233 3.825 10.161 −19.674
Selected American Gi 0.650 3.969 13.142 −19.385
Putnam fund for GRTH & Inc A Gi 0.637 3.540 8.456 −22.081
Growth Income Average Gi 0.544 3.940 10.380 −24.469
Classifications Investment Style
Panel B Aa Asset allocation Ag Aggressive growth Ei Equity income G Growth Gi Growth income
Note: Monthly mutual funds are from January 1980 to September 1996 for a sample of 65 U.S. mutual funds.
The data are from Morningstar Company.
Term premium, TERM, and the U.S. one month T-bill rate, TB have negative signs. The remaining rows of Table 3 reports the monthly dynamics of the forecasting variables. To a first approximation, TERM, DIV, and TB behave like persistent AR(1) process with coefficients of 0.82, 0.87, and 0.94, respectively.
After constructing the VAR system, we can use Table 3 to calculate long-run forecasts of future market return. Revisions in these forecasts are linear combinations of shocks to
Table 2. Summary statistics of instrument variables
Mean Std. Dev. Maximum Minimum
TERM 0.1653 0.1055 0.3658 −0.1591
DIV 0.3142 0.0848 0.5266 0.1833
TB 0.5732 0.2578 1.4050 0.2250
Summary statistics for state variables (in percentages per month) for period 1980:1-1996–99. The state variable include the U.S. term premium (TERM) which is equal to yield on 10-year U.S. T-notes in excess of the yield of the 3-month U.S. T-bill, the dividend yield (DIV) is the monthly S&P dividend yield, and the 30-day U.S. T-bill return (TB).
Table 3. Estimates of VAR
Dep. Regressors Variable rm,t−1− rf,t−1 TERM DIV TB
rm,t− rf,t 0.2138 −13.1686 28.8475 −12.6126 (0.0657) (3.4079) (5.5284) (2.2553) TERM −0.0020 0.8227 0.1090 −0.0484 (0.0010) (0.0532) (0.0864) (0.0352) DIV −0.0008 0.0410 0.8753 0.0477 (0.0002) (0.0124) (0.0201) (0.0082) TB 0.0061 −0.0026 0.0667 0.9479 (0.0016) (0.0838) (0.1360) (0.0554)
We adopt the Vector Auto-Regressive (VAR) approach of Campbell (1991). We assume that the real market index return as the first element of a K-element state variable vector Zt. The other element of Ztare
variables that are known to the market at the end of the period t and are related to forecasting future market return. The state variables include the U.S. term premium (TERM) which is equal to yield on 10-year U.S. T-notes in excess of the yield of the 3-month U.S. T-bill, the dividend yield (DIV) is the monthly S&P dividend yield, and the 30-day U.S. T-bill return (TB). In addition, we assume that the vector Ztfollows a
first order VAR zt−1= Φzt+ ut+1. Standard errors are in parentheses.
the state variables. The combinations that are defined by the vectorλh in Eq. (14). Thus,
we can derive a hedging portfolio return, rh,t+1= (Et+1− Et)
∞
j=1ρjrm,t+1+ j = λhξt+1,
to proceed our empirical implementation.
Table 4 reports the estimates of security selectivity, market timing and hedging timing for our dynamic model. In order to compare our results to those in the previous studies, a traditional model without hedging timing is also estimated. The results are shown in Table 5. The marketing timing coefficients in the traditional model indicate 7 funds have significantly negativeφmat 10% level and 10 funds have significantly negativeφmat 10%
level. But, we find a different picture in our dynamic model. Of the 65 estimates of market timing coefficient, there are only 1 positive and 6 negative significantly at 10%. Therefore, in presence of hedging timing, the market timing becomes less significant. This may be due to the fact that the hedging timing captures part of importance of the traditional market timing.
Regarding to security selectivity, it can be seen that the number of funds which is signif-icantly positive at 5% level is reduced from 40 to 30 when we include the hedging timing
Table 4. Estimates of selectivity, market timing and hedging timing Ri,t+1= αi+ βi mRm,t+1+ βi hRh,t+1+ ei,t+1
= η0+ η1Rm,t+1+ η2Rh,t+1+ η3Rm,t+12 + η4Rh,t+12 + η5Rm,t+1Rh,t+1+ wt+1
where
p limη0= α
p limη1= κ1E(Rm)(1− φm)− κ2E(Rh)(1− φh)
p limη2= κ3E(Rh)(1− φh)− κ2E(Rm)(1− φm)
p limη3= κ1φm
p limη4= κ3φh
p limη5= −κ2(φh+ φm)
Investment Market Hedging
Fund Name Style Selectivityα Timingφm Timingφh
General securities Aa −0.0016 0.0200** 2.9833
Franklin asset allocation Aa 0.0010 −0.0056 3.0258*
Seligman income A Aa 0.0008 −0.0000 0.0348*
USAA income Aa 0.0020* −0.0005 1.0737
Valley forge Aa 0.0012* 0.0000 0.0697**
Income fund of America Aa 0.0039** −0.0051 0.0071
FBL growth common stock Aa 0.0017 −0.0008 0.9670**
Mathers Aa −0.0030* −0.0005 5.2469 American heritage Ag −0.0130** −0.0001 −0.1811 Alliance quasar A Ag 0.0039 −0.0000 1.7905 Keystone small co GRTH (S-4) Ag 0.0022 −0.0000 0.3931 Keystone omega A Ag 0.0036 −0.0000 1.3998 Invesco dynamics Ag 0.0033 −0.0001 11.5533** Security ultra A Ag 0.0021 −0.0000 0.1640 Putnam voyager A Ag 0.0052* −0.0001 0.2948
Stein roe capital opport Ag 0.0049 −0.0092* 0.0054
Value line spec situations Ag −0.0019 −0.0001 −0.0000
Value line leveraged Gr Inv Ag 0.0053** −0.0000 0.0374
WPG tudor Ag 0.0050* −0.0000 2.6563**
Winthrop aggressive growth A Ag 0.0030 0.0000 0.1682
Delaware trend A Ag 0.0069** −0.0001** 11.4132
Founders special Ag 0.0031 −0.0000 8.9994
Smith Barney equity income A Ei 0.0057** −0.0096 0.0412
Van Kampen Am Cap Eqty-Inc A Ei 0.0049** −0.0106 0.1417**
Value line income Ei 0.0033* −0.0130** −0.3991
United income A Ei 0.0058** −0.0034 0.0429*
Oppenheimer equity-income A Ei 0.0049** −0.0091 0.0037
Fidelity equity-income Ei 0.0059** −0.0075 0.0320
Delaware decatur income A Ei 0.0048** −0.0067 0.0856**
Invesco industrial income Ei 0.0056** −0.0118* −0.2267
Old dominion investors’ Ei 0.0028 −0.0047 0.3276
Evergreen total return Y Ei 0.0035** −0.0114** −1.8007
Guardian park avenue A G 0.0063** 0.0000 0.1559
Founders growth G 0.0053* −0.0000 0.0270
Fortis growth A G 0.0055* −0.0000 3.0069
Franklin growth I G 0.0039* 0.0001 6.7099
Fortis capital A G 0.0046** 0.0001 0.0018
Growth fund of America G 0.0046** −0.0000 0.0334
Table 4. (Continued).
Investment Market Hedging
Fund Name Style Selectivityα Timingφm Timingφh
Hancock growth A G 0.0027 −0.0000 1.7042
Franklin equity I G 0.0042* −0.0000 5.0368
Nationwide growth G 0.0053** 0.0000 0.0661
Neuberger & Berman focus G 0.0025 −0.0000 0.7693
MSB G 0.0037* −0.0001 0.0825
Neuberger & Berman partners G 0.0044** −0.0001 1.2095 Neuberger & Berman Manhattan G 0.0045* −0.0000 0.2221
Nicholas G 0.0048** −0.0001 0.0010
Oppenheimer A G 0.0013 −0.0000 6.4330
New England growth A G 0.0048* −0.0002 0.8900**
Pioneer II A Gi 0.0046** −0.0054 0.0114
Pilgrim America Magnacap A Gi 0.0063** −0.0082* 0.0135
Pioneer Gi 0.0030 −0.0005 0.0858
Philadelphia Gi 0.0016 −0.0047 0.0046
Penn square mutual A Gi 0.0043** 0.0009 0.0291
Oppenheimer total return A Gi 0.0045 −0.0015 0.1079
Vanguard/Windsor Gi 0.0065** −0.0033 0.0135
Van Kampen Am Cap GR & Inc A Gi 0.0021 −0.0036 −30.5100 Van Kampen Am Cap Comstock A Gi 0.0041** −0.0000 −34.1823 Winthrop growth & income A Gi 0.0040** −0.0034 0.0414
Washington mutual investors Gi 0.0067** −0.0035 0.0091
Safeco equity Gi 0.0059** −0.0044 0.0305
Seligman common stock A Gi 0.0042** −0.0032 0.0017
Salomon Bros investors O Gi 0.0047** −0.0016 0.0160
Security growth & income A Gi 0.0005 −0.0013 1.9871
Selected American Gi 0.0057** −0.0050 0.0070
Putnam fund for Grth & Inc A Gi 0.0057** −0.0064 0.0006
∗Significant at the 0.10 level. ∗∗Significant at the 0.05 level.
to the traditional model. In addition there are 10 funds which have a significantly positive hedging timing,φm. This implies that some fund managers use the expected information
to implement their hedging strategy and hence the effect of the market timing is partly absorbed into the effect of the hedging timing.
For each different style of investment, the Asset Allocation funds generally have a sig-nificant positive hedging timing coefficient (50% at the 0.1 sigsig-nificant level). The Equity Income funds have a significant positive hedging timing coefficient (30% at the 0.10 sig-nificant level). The Aggressive Growth funds have a sigsig-nificant positive hedging timing coefficient (14% at the 0.05 significant level). The Growth funds have a significant positive hedging timing coefficient (6% at the 0.05 significant level). However, the Growth Income funds have no significant positive hedging timing coefficient. This empirical evidence is consistent with the facts that the managers of Asset Allocation funds focus more on forecast-ing the future investment opportunity as well as adjustforecast-ing their hedgforecast-ing strategy than other styles of funds. The less aggressive funds are, the more hedging strategy fund managers will do.
Table 5. Estimates of selectivity and market timing in the traditional model Ri,t+1= αi+ βi mRm,t+1+ ei,t+1 = η0+ η1Rm,t+1+ η3R2m,t+1+ wt+1 where p limη0= αi p limη1= κ1E(Rm)(1− φm) p limη3= κ1φm
Fund Name Investment Style Selectivityα Market Timingφm
General securities Aa 0.0034 −0.0010
Franklin asset allocation Aa 0.0028* 0.0000
Seligman income A Aa 0.0026** 0.0000
USAA income Aa 0.0024* 0.0119
Valley forge Aa 0.0025** −0.0057
Income fund of America Aa 0.0043** 0.0000
FBL growth common stock Aa 0.0012 0.0001*
Mathers Aa 0.0005 −0.0003 American heritage Ag −0.0114** 0.0004 Alliance quasar A Ag 0.0047* 0.0001** Keystone small Co Grth (S-4) Ag 0.0040 −0.0063 Keystone omega A Ag 0.0037 0.0001 Invesco dynamics Ag 0.0037 0.0001 Security ultra A Ag 0.0025 0.0001** Putnam voyager A Ag 0.0070** −0.0045
Stein Roe capital opport Ag 0.0059* −0.0079
Value line spec situations Ag 0.0012 −0.0071
Value line leveraged GR Inv Ag 0.0047** 0.0000
WPG tudor Ag 0.0074** −0.0082
Winthrop aggressive growth A Ag 0.0022 0.0001
Delaware trend A Ag 0.0096** −0.0125*
Founders special Ag 0.0035 0.0000**
Smith Barney equity income A Ei 0.0058** −0.0097*
Van Kampen Am Cap Eqty-Inc A Ei 0.0042** 0.0001
Value line income Ei 0.0046** −0.0129*
United income A Ei 0.0063** −0.0054
Oppenheimer equity-income A Ei 0.0051** −0.0080
Fidelity equity-income Ei 0.0066** −0.0080
Delaware Decatur income A Ei 0.0054** −0.0099*
Invesco industrial income Ei 0.0060** −0.0103
Old Dominion investors’ Ei 0.0034* −0.0093
Evergreen total return Y Ei 0.0049** −0.0098*
Guardian park avenue A G 0.0053** 0.0000*
Founders growth G 0.0069** −0.0081
Fortis growth A G 0.0052* 0.0001
Franklin growth I G 0.0038** 0.0000
Fortis capital A G 0.0048** 0.0000
Growth fund of America G 0.0045** 0.0000**
Hancock growth A G 0.0043 −0.0065
Franklin equity I G 0.0043* −0.0078
Nationwide growth G 0.0051** −0.0061
Neuberger & Berman focus G 0.0056** −0.0155**
Table 5. (Continued).
Fund Name Investment Style Selectivityα Market Timingφm
MSB G 0.0031* 0.0001
Neuberger & Berman partners G 0.0061** −0.0078
Neuberger & Berman Manhattan G 0.0043* 0.0000*
Nicholas G 0.0064** −0.0075
Oppenheimer A G 0.0028 −0.0110*
New England growth A G 0.0058** 0.0001
Pioneer II A Gi 0.0038** 0.0000
Pilgrim America Magnacap A Gi 0.0068** −0.0132**
Pioneer Gi 0.0038* −0.0087
Philadelphia Gi 0.0026 −0.0107
Penn square mutual A Gi 0.0047** −0.0085
Oppenheimer total return A Gi 0.0033 0.0000
Vanguard/Windsor Gi 0.0068** −0.0075
Van Kampen Am Cap Gr & Inc A Gi 0.0044** 0.0001
Van Kampen Am Cap Comstock A Gi 0.0051** 0.0001
Winthrop growth & income A Gi 0.0043** −0.0103*
Washington mutual investors Gi 0.0058** 0.0001**
Safeco equity Gi 0.0044** 0.0000
Seligman common stock A Gi 0.0052** −0.0086
Salomon Bros investors O Gi 0.0052** −0.0072
Security growth & income A Gi 0.0005 0.0000
Selected American Gi 0.0062** −0.0086
Putnam fund for Grth & Inc A Gi 0.0068** −0.0128**
∗Significant at the 0.10 level. ∗∗Significant at the 0.05 level. 6. Conclusion
In this paper, we identify a hedging factor in the equilibrium asset pricing model and use this benchmark to construct a new performance measure. Based on this measure, we are able to evaluate fund managers hedging timing ability in addition to more traditional security selectivity and market timing. While security selectivity performance involves forecasts of price movements of selected individual stock, market timing measures the forecasts of next period realizations of the market portfolio. However, hedging timing refers to forecasts of future realizations of the hedging portfolio.
Our approach can be contrasted with Ferson and Schadt (1996), who empirically em-ployed a conditional CAPM to capture dynamic behavior of fund managers. However, ours is a theoretical dynamic asset model. Exploiting the insights of the dynamic behaviors of returns in our framework, we are able to construct a hedging timing, which is different from the traditional security selection and market timing.
The empirical evidence in the paper indicates that the selectivity measure is positive on average and the market timing measure is negative on average. However, the hedg-ing timhedg-ing measure is positive on average. Moreover, mutual funds with Asset Allo-cation and Equity Income styles show significantly positive hedging timing. This re-sult is consistent with the facts that the funds classified as Asset Allocation style are
more concerned with hedging demand than those classified as other kinds of investment style.
Appendix
Proof of Proposition: When the asset return under consideration is a market return rm,t+1,
we can obtain the following expected market return pricing formula
Etrm,t+1− rf,t+1+
Vmm
2 = γ Vmm+ (γ − 1)Vmh (A1)
Similarly, the hedging return can be represented as
Etrh,t+1− rf,t+1+
Vhh
2 = γ Vhm+ (γ − 1)Vhh (A2)
Combining Eqs. (2), (A1) and (A2), we can obtain
γ = Eri− rf +V2ii + Vi h Vi m+ Vi h = Erm− rf +Vmm2 + Vmh Vmm+ Vmh = Erh− rf +V2hh + Vhh Vhm+ Vhh (A3)
Using the third equality of Eq. (A3), expected market return can be represented as
Erm− rf + Vmm 2 = Vmm+ Vmh Vhm+ Vhh Erh− rf + Vhh 2 + Vhh − Vmh (A4)
Notice that we can define the coefficient of relative risk aversion as
γ = 1 Vmm− VmhVhh−1Vhm Erm− rf + Vmm 2 − VhmVmm−1 Vhh− VhmVmm−1Vmh Erh− rf + Vhh 2 (A5) γ − 1 = − VmhVhh−1 Vmm− VmhVhh−1Vhm Erm− rf + Vmm 2 + 1 Vhh− VhmVmm−1Vmh Erh− rf + Vhh 2 (A6)
After substituting Eqs. (A5) and (A6) into Eq. (2), we obtain Etri,t+1− rf,t+1+ Vii 2 = Vmi − VmhVhh−1Vhi Vmm− VmhVhh−1Vhm Etrm,t+1− rf,t+1+ Vmm 2 + Vhi− VhmVmm−1Vmi Vhh− VhmVmm−1Vmh Etrh,t+1− rf,t+1+ Vhh 2 = βi m Etrm,t+1− rf,t+1+ Vmm 2 + βi h Etrh,t+1− rf,t+1+ Vhh 2 (A7) where market beta, βi m, and hedging beta, βi h, are also the coefficients of the multiple
regression in Eq. (A7). That is to say, expected rates of return on risky assets are related to rates of returns on market portfolio and hedging portfolio in a linear way.
Proof of Theorem: We denote the conditional variances of the market factor and the hedging factor as σ2 i(πm,t+1)= Var(πm,t+1| Ii,t) (A8) σ2 i(πh,t+1)= Var(πh,t+1| Ii,t)
Letδm,t be the fraction invested in the market portfolio,δh,t be the fraction invested in
the hedging portfolio, and 1− δm,t− δh,t the fraction invested in the risk-free asset at time
t. The expected excess return and variance of return on the portfolio are E(Ri,t+1)= δm,t[E(Rm)+ πm∗,t+1]+ δh,t[E(Rh)+ πm∗,t+1]
(A9) V (Ri,t+1)= δm2,tσ 2 i(πm,t+1)+ δ2h,tσ 2 i(πh,t+1)+ 2δm,tδh,tσi(πm,t+1, πh,t+1)
It can be seen that the fund manager’s maximization problem is as follows. max δm,tδh,tU [E(Ri,t+1), V (Ri,t+1)]= maxδm,tδh,tU δmt[E(Rm)+ πmt∗+1] + δht[E(Rh)+ πht∗+1], δ2mtσm2 + δ2 mtσm2+ 2δmtδhtσmh (A10) whereσ2
m,σh2, andσmhdenote Var(πm,t+1| Ii,t), Var(πh,t+1| Ii,t), and Cov(πm,t+1, πh,t+1|
Ii,t), respectively.
The solution to this problem can be obtained as
δmt = 1 2σ2 mσh2− σmh2 dV(Ri,t+1) dE(Ri,t+1) ×[E(Rm)+ πmt∗+1]σh2− [E(Rh)+ πht∗+1]σmh (A11)
δht = 1 2σ2 mσh2− σmh2 dV(Ri,t+1) dE(Ri,t+1) [E(Rh)+ πht∗+1]σm2− [E(Rm)+ πmt∗ +1]σmh where d V (Ri,t+1) d E(Ri,t+1) = − ∂U/∂ E(Ri,t+1) ∂U/∂V (Ri,t+1) > 0.
Since βi m,t = δmt andβi h,t = δht, the manager’s optimal choices of market risk and
hedging risk for the portfolio are given by
βi m,t = µi,t σ2 hE(Rm)− σmhE(Rh) + µi,t σ2 hπmt∗ +1− σmhπht∗+1 (A12) βi h,t = µi,t σ2 mE(Rh)− σmhE(Rm) + µi,t σ2 mπht∗+1− σmhπmt∗+1 whereµi,t = (1/2)∗[1/(σm2σh2− σmh2 )]∗[d V (Ri,t+1)/dE(Ri,t+1)].
Given the objective of the manager and assumption that the conditional distribution of
πm,t+1andπh,t+1is a bivariate normal distribution, and following the lines of Jensen (1972),
it can be shown that
βi m,t = βi mT+ [κ1πmt∗ +1− κ2πht∗+1]
(A13)
βi h,t = βi hT+ [κ3πht∗+1− κ2πmt∗ +1]
whereκ1= µi,tσh2,κ2= µi,tσmh,κ3= µi,tσm2, andβi mT = µi,t[σh2E(Rm)− σmhE(Rh)] as
well asβi hT = µi,t[σm2E(Rh)− σmhE(Rm)] can be considered as fund manager’s “target”
risk level, given the unconditional expected returns on the market portfolio and hedging portfolio.
After substituting Eqs. (8) and (A13) into Eq. (6), we obtain Eq. (10).
Note
1. Campbell (1993) uses a log-linear approximation to the budget constraint to derive an intertemporal asset pricing formula that makes no reference to consumption. The formula is a discrete-time version of Merton’s (1973) continuous-time model but is much easier to implement empirically. Hence, we adopt this discrete-time model to derive a dynamic model of performance evaluation.
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