# 3 Conditional performance evaluation.

Traditional measures of risk-adjusted performance for mutual funds compare the average return of a fund with an OE benchmark designed to control for the fund's average risk. For example, Jensen's (1968) alpha is the difference between the return of a fund and a portfolio constructed from a market index and cash with fixed weights. The portfolio has the same average market exposure, or "beta" risk as the fund. The returns and beta risks are typically measured as averages over the evaluation period, and these averages are taken "unconditionally," or without regard to variations in the state of financial markets or the broader economy. One weakness of this unconditional approach relates to the likelihood of changes in the state of the economy. For example, if the evaluation period covers a bear market, but the period going forward is a bull market, the performance evaluation may not have much validity.In the Conditional Performance Evaluation (CPE) approach, fund managers' risk exposures and the related market premiums are allowed to vary over time with the state of the economy. The state of the economy is measured using predetermined, public information variables. Provided that the estimation period covers both bull and bear markets, we can estimate expected risk and performance in each type of market. This way, knowing that we are now in a bull state of the market for example, we can estimate the fund's expected performance given a bull state.

Problems associated with variation over time in mutual fund risks and market returns have long been recognized (e.g., Jensen, 1972; Grant, 1977), but CPE draws an important distinction between variation that can be tracked with public information and variation due to private information on the state of the economy. CPE takes the view that a managed portfolio strategy that can be replicated using readily available public information should not be judged as having superior performance. For example, in a conditional approach, a mechanical market timing rule using lagged interest rate data is not a strategy that requires investment ability. Only managers who correctly use more information than is generally publicly available, are considered to have potentially superior investment ability. CPE is therefore consistent with a version of semi-strong-form market efficiency as described by Fama (1970). While market efficiency can motivate the null hypothesis that conditional alphas are zero, one need not ascribe to market efficiency to use CPE. By choosing the lagged variables, it is possible to set the hurdle for superior ability at any desired level of information.

In addition to the lagged state variables, CPE like any performance evaluation requires a choice of benchmark portfolios. The first measures used a broad equity index, motivated by the CAPM. Ferson and Schadt (1996) used a market index and also a multifactor benchmark for CPE. Current practice is more likely to use a benchmark representing the fund manager's investment style.

3.1 Motivation and Example

The appeal of CPE can be illustrated with the following highly stylized numerical example. Assume that there are two equally-likely states of the market as reflected in investors' expectations; say, a "Bull" state and a "Bear" state. In a Bull market, assume that the expected return of the S&P500 is 20%, and in a Bear (1) market, it is 10%. The risk-free return to cash is 5%. Assume that all investors share these views--the current state of expected market returns is common knowledge. In this case, assuming an efficient market, an investment strategy using as its only information the current state, will not yield abnormal returns.

Now, imagine a mutual fund which holds the S&P500 in a Bull market and holds cash in a Bear market. Consider the performance of this fund based on CPE and Sharpe's (1964) CAPM. Conditional on a Bull market, the beta of the fund is 1.0, the fund's expected return is 20%, equal to the S&P500, and the fund's conditional alpha is zero. (2) Conditional on a Bear market, the fund's beta is 0.0, the expected return of the fund is the risk-free return, 5%, and the conditional alpha is, again, zero. A conditional approach to performance evaluation correctly reports an alpha of zero in each state. This is essentially the null hypothesis of a CPE analysis.

By contrast, an unconditional approach to performance evaluation incorrectly reports a nonzero alpha for our hypothetical mutual fund. Without conditioning on the state, the returns of this fund would seem to be highly sensitive to the market return, and the unconditional beta of the fund (3) is 1.5. The unconditional expected return of the fund is 0.5(0.20) + 0.5(0.05) = 0.125. The unconditional expected return of the S&P500 is 0.5(0.20) + 0.5(0.10) = 0.15, and the unconditional alpha of the fund is therefore: (0.125 - 0.05) - 1.5(0.15 - 0.05) = -7.5%. The unconditional approach leads to the mistaken conclusion that the manager has negative abnormal performance. But the manager's performance does not reflect poor investment choices or wasted resources, it merely reflects common variation over time in the fund's conditional risk exposure and the market premium. The traditional model over adjusts for market risk and assigns the manager a negative alpha. However, investors who have access to information about the economic state would not use the inflated risk exposure and would therefore not ascribe negative performance to the manager.

3.2 Conditional Alphas

Conditional alphas are developed as a natural generalization of the traditional, or unconditional alphas. In the CPE approach the risk adjustment for a bull market state may be different from that for a bear market state, if the fund's strategy implies different risk exposures in the different states. Let [r.sub.m,t+1] be the excess return on a market or benchmark index. For example, this could be the S&P 500, a "style" index such as "small cap growth," or a vector of excess returns if a multi-factor model is used.

The model proposed by Ferson and Schadt is:

[r.sub.p,t+1] = [[alpha].sub.p] + [[beta]/sub.o][r.sub.m,t+1] + [beta]'/sub.o][r.sub.m,t+1] + [cross product] [Z.sub.t]] + [u.sub.pt+1,] (3.1)

where [r.sub.p,t+]1 is the return of the fund in excess of a short term "cash" instrument, and [Z.sub.t] is the vector of lagged conditioning variables, in demeaned form. The symbol [cross product] denotes the kronecker product, or element-by-element multiplication when [r.sub.m,t+1] is a single market index. A special case of Equation (3.1) is the classical CAPM regression, where the terms involving [Z.sub.t] are omitted. In this case, [[alpha].sub.p] is Jensen's (1968) alpha.

To see how the model in Equation (3.1) arises, consider a conditional "market model" regression allowing for a time-varying fund beta, [beta]([Z.sub.t]), that may depend on the public information, [Z.sub.t]:

[r.sub.p,t+1] = [[alpha].sub.p] + [beta]([Z.sub.t])[r.sub.m,t+1] + [u.sub.pt+1,] (3.2)

with E([u.sub.pt+1,] | [Z.sub.t]) = E{[u.sub.pt+1,] [r.sub.p,t+1] | [Z.sub.t]) = 0. Now assume that the time-varying beta can be modeled as a linear function: [beta]([Z.sub.t]) = [[beta].sub.o] + [beta]'[Z.sub.t]. The coefficient [[beta].sub.o] is the average conditional beta of the fund (as Z is normalized to mean zero), and the term [beta]'[Z.sub.t] captures the time-varying conditional beta. The assumption that the conditional beta is a linear function of the lagged instrument can be motivated by a Taylor series approximation, or by a model such as Admati et al. (1986), in which an optimizing agent would endogenously generate a linear conditional beta by trading assets with constant betas, using a linear portfolio weight function.

Substituting the expression for the conditional beta into Equation (3.2), the result is Equation (3.1). Note that since E([Z.sub.t]) = 0, it follows that:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.] (3.3)

where the second equality follows from representing [r.sub.m,t+1] = E([r.sub.m,t+1]| [Z.sub.t]) + [u.sub.m,t+1], with Cov{[u.sunb.m,t+1,][beta]([Z.sub.t])} = 0. Equation (3.3) says that the interaction terms [beta][[r.sub.m,t+1] [cross product] [Z.sub.t] in the regression (3.1) control for common movements in the fund's conditional beta and the conditional expected benchmark return. This was the cause of the "bias" in the unconditional alpha in the example we used to motivate CPE. The conditional alpha, [[alpha].sub.p] in (3.1) is thus measured net of the effects of these risk dynamics.

The OE portfolio in this CPE setting is the "naive" dynamic strategy, formed using the public information [Z.sub.t], that has the same time-varying conditional beta as the portfolio to be evaluated. This strategy has a weight at time t on the market index equal to [[beta].sub.o] + [beta]'[Z.sub.t], and {1 - [[beta].sub.o] -[beta]'[Z.sub.t]} is the weight in safe asset or cash. Using the same logic as before, Equation (3.1) implies that ap in the Ferson and Schadt model is the difference between the unconditional expected return of the fund and that of the OE strategy.

3.3 Conditional Market Timing 123

Christopherson et al. (1998a,b) propose a refinement of (3.1) to allow for a time-varying conditional alpha:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.] (3.4)

In this model, [[alpha].sub.p0] + [[alpha]'.sub.p] [Z.sub.t] measures the time-varying, conditional alpha, and the OE portfolio is the same as in the previous case, but the time-varying alpha is now the difference between the conditional expected return of the fund, given [Z.sub.t], and the conditional expected return of the OE portfolio strategy. This refinement of the model may have more power to detect abnormal performance if performance varies with the state of the economy. For example, if a manager generates a large alpha when the yield curve is steep, but a small or negative alpha when it is shallow, the average abnormal performance may be close enough to zero that it cannot be detected using regression (2.7). In such a case regression (3.4) could track the time-variation in alpha and record this as a nonzero coefficient ap on the instrument for the term structure slope. (4)

3.3 Conditional Market Timing

The classical market-timing model of Treynor and Mazuy (1966) was presented in Equation (2.7). Treynor and Mazuy (1966) argued that a market-timing manager will generate a return that bears a convex relation to the market. However, a convex relation may arise for a number of other reasons. One of these is common time-variation in the fund's beta risk and the expected market risk premium, related to public information on the state of the economy. Ferson and Schadt propose a CPE version of the Treynor-Mazuy model to handle this situation:

[r.sub.pt+1] = [[alpha].sub.p] + [b.sub.p][r.sub.mt+1] + C'p ([Z.sub.t][r.sub.m,t+1]) + [[DELTA].sub.p][r.sup.2.sub.mt+1 + [w.sub.t+1]. (3.5)

In Equation (3.5), the term [C'.sub.p] ([Z.sub.t][r.sub.m,t+1]) controls for common time-variation in the market risk premium and the fund's beta, just like it did in the regression (3.1). A manager who uses [Z.sub.t] linearly to time the market has no conditional timing ability, and thus [[DELTA].sub.p] = 0. The coefficient [[DELTA].sub.p] measures the market timing ability based on information beyond that contained in [Z.sub.t].

Interpreting the intercept in (3.5) raises issues similar to those in the classical Treynor-Mazuy regression. As described above, [[alpha].sub.p] is not the difference between the fund's return and that of an OE portfolio. The model may be generalized, given a conditional maximum correlation portfolio to [r.sup.2.sub.m], along the lines previously described. Such an analysis has not yet appeared in the literature.

Merton and Henriksson (1981) and Henriksson (1984) describe the alternative model of market timing in which the quadratic term is replaced by an option payoff, Max(0,[r.sub.m,t+1]), as described earlier. Ferson and Schadt (1996) develop a conditional version of this model as well.

In theoretical market-timing models the timing coefficient is shown to depend on both the precision of the manager's market-timing signal and the manager's risk tolerance. For a given signal precision, a more risk tolerant manager will implement a more aggressive timing strategy, thus generating more convexity. Similarly, for a given risk tolerance a manager with a more precise timing signal will be more aggressive. Precision probably varies over time, as fund managers are likely to receive information of varying uncertainty about economic conditions at different times. Effective risk aversion may also vary over time, according to arguments describing mutual fund "tournaments" for new money flows (e.g, Brown et al., 1996), which may induce managers to take more risks when their performance is lagging and to be more conservative when they want to "lock in" favorable recent performance. Therefore, it seems likely that the timing coefficient which measures the convexity of a fund's conditional relation to the market is likely to vary over time. Ferson and Qian (2004) allow for such effects by letting the timing coefficient vary over time as a function of the state of the economy. Replacing the fixed timing coefficient above with [[DELTA].sub.p] = [[DELTA].sub.0p] + [[DELTA]'.sub.1p][Z.sub.t] we arrive at a conditional timing model with time-varying performance:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.] (3.6)

In this model the coefficient [[DELTA].sub.1p] on the interaction term ([Z.sub.t][r.sup.2.sub.m,t+1]) captures the variability in the managers timing ability, if any, over the states of the economy. By examining the significance of the coefficients in [[DELTA].sub.1p], it is easy to test the null hypothesis that the timing ability is fixed against the alternative hypothesis that timing ability varies with the economic state. Ferson and Qian (2004) find evidence that market timing ability varies with the economic state.

Becker et al. (1999) further develop conditional market-timing models. In addition to incorporating public information, their model features explicit performance benchmarks for measuring the relative performance of fund managers. In practice, performance benchmarks represent an important component of some fund managers' incentives, especially for hedge funds that incorporate explicit incentive fees. Even for mutual funds, Schultz (1996) reports that Vanguard included incentive-based provisions in 24 of 38 compensation contracts with external fund managers at that time. Elton et al. (2003) find that about 10% of the managers in a sample of US mutual funds are compensated according to incentive contracts. These contracts determine a manager's compensation by comparing fund performance to that of a benchmark portfolio. The incentive contracts induce a preference for portfolio return in excess of the benchmark. The model of Becker et al. refines the conditional market timing models of Ferson and Schadt (1996) in two ways. First, it allows for explicit, exogenous performance benchmarks. Second, it allows for the separate estimation of parameters for risk aversion and the quality of the market timing signal, conditional on the public information.

Starks (1987), Grinblatt and Titman (1989b) and Admati and Pfleiderer (1997) present models of incentive-based management contracts, focusing on agency problems between managers and investors. Chiu and Roley (1992) and Brown et al. (1996), among others, examine the behavior of fund managers when relative performance is important. Heinkel and Stoughton (1995) present unconditional market-timing models with benchmark investors.

3.4 Conditional Weight-Based Measures

The weight-based performance measures discussed in Chapter 2 are unconditional, meaning they do not attempt to control for dynamic changes in expected returns and volatility. Like the classical returns-based performance measures, unconditional weight-based measures have problems handling return dynamics. It is known that unconditional weight-based measures can show performance when the manager targets stocks whose expected return and risk have risen temporarily (e.g., stocks subject to takeover or bankruptcy); when a manager exploits serial correlation in stock returns or return seasonalities; and when a manager gradually changes the risk of the portfolio over time, as in style drift. (5) These problems may be addressed using a conditional approach.

Ferson and Khang (2002) develop the Conditional Weight-based Measure of performance (CWM) and show that it has a number of advantages. Like other CPE approaches, the measure controls for changes in expected returns and volatility, as captured by a set of lagged economic variables or instruments. However, the CWM uses the information in both the lagged economic variables and the fund's portfolio weights.

The Conditional Weight Measure is the average of the conditional covariances between future returns and portfolio weight changes, summed across the securities held. It generalizes Equation (2.11) as follows:

[MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII.] (3.7)

The symbol [w.sub.j](Z,S) denotes the portfolio weight at the beginning of the period. The weights may depend on the public information, denoted by Z. The weights of a manager with superior information, denoted by S, may also depend on the superior information. Superior information, by definition, is any information that can be used to predict patterns in future returns that cannot be discerned from public information alone.

In Equation (3.7) the term [r.sub.j] - E([r.sub.j] | Z) denotes the unexpected, or abnormal future returns of the securities, indexed by j. Here, we define the abnormal return as the component of return not expected by an investor who only sees the public information Z at the beginning of the period. For example, if returns are measured over the first quarter, E(r|Z) is the expected return for the first quarter based on public information about the economy as of the last trading day of the previous December. The sum of the covariances between the weights, measured at the end of December, and the subsequent abnormal returns for the securities in the first quarter, is positive for a manager with superior information, S. If the manager has no superior information, S, then the covariance is zero.

In practice, just as with the unconditional weight-based measures, it may be useful to introduce a benchmark with weights, [w.sub.jb], that are included in the public information set Z at the beginning of the quarter. The modified measure is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

Ferson and Khang (2002) define the benchmark weights at the beginning of quarter t as the portfolios' actual weights lagged k periods, updating these with a buy-and-hold strategy. Thus, each manager's position, k quarters ago, defines his "personal" benchmark. The underlying model thus presumes that a manager with no investment ability follows a buy-and-hold strategy over the k quarters. A manager with investment ability changes the portfolio in order to beat a buy-and-hold strategy. The weight-based measures are in the units of an excess return of the fund over the benchmark. Because [w.sub.jb] is assumed to be known given Z, it will not affect the CWM in theory. However, the benchmark weights will affect the statistical properties of the measure.

The benchmark weights are assumed to be public information at time t. However, the date when past weights become public information will depend on the circumstances. Mutual funds' portfolio weights become publicly available at least every six months, by law, although with a reporting lag. Many funds now report their holdings monthly, or even more frequently. In application to pension funds, one may view the public information as that available to pension plan sponsors. If a plan sponsor wishes to know the current holdings of a portfolio, a manager is likely to respond within days, if not hours. However, plan sponsors systematically examine holdings data on a less frequent basis. A lag of one quarter may be the most reasonable assumption; however, one could argue for a longer period. For example, a careful review of the holdings may take place on an annual basis with a more cursory review at quarterly reporting periods. The time when the weights can legitimately be called public information is therefore not clear. Thus, Ferson and Khang (2002) use various lags, k, to evaluate the sensitivity of the measures to this issue.

(1) This differs from the conventional definition of a bear market, which some consider to be a 20% decline off of a previous high.

(2) The conditional alpha given a bull state, according to the CAPM, is the fund's conditional expected excess return over cash minus its conditional beta multiplied by the conditional expected market excess return over cash, which is equal to (0.20 - 0.05) - 1(0.20 - 0.05) = 0.

(3) The calculation is as follows. The unconditional beta is Cov(F,M)/Var(M), where F is the fund return and M is the market return. The numerator is:

Cov(F,M)=E{(F - E(F))(M - E(M))|Bull} x Prob(Bull)

+E{(F - E(F))(M - E(M))|Bear} x Prob(Bear)

= {(0.20 - 0.125)(0.20 - 0.15)} x 0.5 + {(0.05 - 0.125)(0.10 - 0.15)} x 0.5

= 0.00375.

The denominator is:

Var(M)=E{(M - E(M))2|Bull} x Prob(Bull)

+E{(M - E(M))2|Bear} x Prob(Bear)

= {(0.20 - 0.15)2} x 0.5 + {(0.10 - 0.15)2} x 0.5

= 0.0025.

The beta is therefore 0.00375/0.0025 = 1.5. Note that the unconditional beta is not the same as the average conditional beta, because the latter is 0.5 in this example.

(4) The regression (3.4) also has statistical advantages in the presence of lagged instruments that may be highly persistent regressors with high autocorrelation, as is often the case in practice. Ferson et al. (2007) show that by including the [[alpha]'.sub.p][Z.sub.t] term, the regression delivers smaller spurious regression biases in the beta coefficients. They also warn, however, that the t-statistics for the time-varying alphas are likely to be biased in this case.

(5) See Grinblatt and Titman (1993) for a discussion.

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Title Annotation: | Portfolio Performance Evaluation |
---|---|

Author: | Aragon, George O.; Ferson, Wayne E. |

Publication: | Foundations and Trends in Finance |

Geographic Code: | 1USA |

Date: | Apr 1, 2006 |

Words: | 3757 |

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