Measuring the Timing Ability of Fixed Income Mutual Funds

Transcription

1 Measuring the Timing Ability of Fixed Income Mutual Funds by Yong Chen Wayne Ferson Helen Peters first draft: November 25, 2003 this revision: March 9, 2006 * Ferson and Peters are Professors of Finance at Boston College, 140 Commonwealth Ave, Chestnut Hill, MA , and Ferson is a Research Associate of the NBER. He may be reached at (617) , fax , wayne.ferson@bc.edu, www2.bc.edu/~fersonwa. Peters may be reached at (617) , helen.peters.1@bc.edu. Chen is a Ph.D student at Boston College, and may be reached at yong.chen@bc.edu. We are grateful to participants in workshops at Baylor University, Brigham Young, Boston College, DePaul, the University of Florida, Georgetown, the University of Iowa, the University of Massachusetts at Amherst, Michigan State University, the University of Michigan and Virginia Tech. and the Office of Economic Analysis, Securities and Exchange Commission for feedback, as well as to participants at the following conferences: The 2005 Global Finance Conference at Trinity College, Dublin, the 2005 Southern Finance and the 2005 Western Finance Association Meetings. We also appreciate helpful comments from Nadia Voztioblennaia.

2 ABSTRACT Measuring the Timing Ability of Fixed Income Mutual Funds This paper evaluates the ability of US fixed income mutual funds to time common factors related to bond markets. A methodological challenge is to control for potential non-timing-related sources of nonlinearity in the relation between fund returns and common factors. Nonlinearities may arise from dynamic trading strategies or derivatives, funds' responses to public information, they may appear in the underlying assets held by the fund, or they may be related to systematic patterns in stale pricing. Controlling for these effects we find that funds' returns are more typically nonlinear in nine common factors than are fixed-weight benchmark returns. This is an incomplete draft

3 1. Introduction The relative share of academic work on bond funds is dwarfed by the share and importance of these funds and assets in the economy. Elton, Gruber and Blake (1993, 1995) and Ferson, Henry and Kisgen (2006) study US bond mutual fund performance, concentrating on the funds' risk-adjusted returns, or alphas. These studies find that the typical average performance after costs is negative and on the same order of magnitude as the funds' expenses. However, the total performance may be decomposed into components, such as timing and selectivity ability. If investors place value on timing ability, for example a fund that can mitigate losses in down markets, they may accept a total performance cost for this service. To better understand performance we need to examine its components. We argue that a logical place to start, in the case of fixed income funds, is to study timing ability. 1 That is the goal of this paper. This is one of the first papers to comprehensively study the ability of US bond funds to time their markets. 2 Timing ability on the part of a fund manager is the ability to use information about the future realizations of common factors affecting security returns. Selectivity refers to the use of security-specific information. If common market factors explain a relatively large part of the variance of a typical bond return, it follows that a relatively large fraction of the potential performance of bond funds is likely attributed to timing. Thus, a logical step in looking at performance is to concentrate on timing ability. However, measuring 1 We do not explicitly study market timing in the sense recently taken to mean trading by investors in a fund to exploit stale prices reflected in the fund's net asset values, or late trading -- potentially fraudulent trading after the close of the market. But we will see that these issues can affect measures of a fund manager's ability. 2 Brown and Marshall (2001) develop an active style model and an attribution model for fixed income funds, isolating managers' bets on interest rates and spreads. Comer, Boney and Kelly (2005) study timing ability in a sample of 84 high quality corporate bond funds, , using variations on Sharpe's (1992) style model. Comer (2006) measures bond fund sector timing using portfolio weights. Aragon (2005) studies the timing ability of balanced funds for bond and stock indexes.

4 2 the timing ability of fixed income funds is a subtle problem. Traditional models of market timing ability rely on convexity in the relation between the fund's returns and the common factors. 3 In bond funds, perhaps even more clearly than in equity funds, convexity or concavity can arise for various reasons unrelated to timing ability. Our empirical analysis controls for other sources of nonlinearity. These include convexity or concavity that may be inherent in the benchmark assets held by a fund, the use of dynamic trading strategies or derivatives, portfolios that respond to publicly available information and stale prices in the measured returns. Most versions of the models with the controls for non-timing-related nonlinearities find that funds' returns are more typically nonlinearly related to nine common factors than are fixed-weight benchmark returns. We find evidence of positive timing ability for some factors, including credit spreads and the relative value of the dollar. There is also evidence of "negative" timing ability for some factors, including interest rates, mortgage spreads and bond market liquidity. Negative timing is similar to what previous studies have found for equity funds relative to the stock market. However, unlike the case of equity funds, conditional models (e.g. Ferson and Schadt, 1996) do not explain the nonlinearity. The rest of the paper is organized as follows. Section 2 describes the models and methods. Section 3 describes the data. We study timing relative to a set of seven bond market factors and two equity market factors. Section 4 presents our empirical results and Section 5 offers some concluding remarks. 3 The alternative approach is to examine managers' portfolio weights and trading decisions to see if they can predict returns and factors (e.g. Grinblatt and Titman, 1989). Comer (2006) is a first step in this direction for bond funds.

5 3 2. Models and Methods A traditional view of performance separates timing ability from security selection ability, or selectivity. For equity funds, timing is closely related to asset allocation, where funds rebalance the portfolio among asset classes and cash. Selectivity essentially means picking good stocks within the asset classes. Like equity funds, fixed income funds engage in activities that may be viewed as selectivity or timing. Fixed income funds may attempt to predict institution-specific supply and demand or changes in credit risks associated with particular bond issues. Funds can also attempt to exploit liquidity differences across bonds, for example on-the-run versus off-the-run issues. These trading activities are akin to security selection. In addition, managers may tune the interest rate sensitivity (e.g., duration) of the portfolio to time changes in the level of interest rates or the shape of the yield curve in anticipation of the influence of economic developments. They may vary the allocation to asset classes (e.g. corporates versus governments versus mortgages) that are likely to have different exposures to economic factors. These activities may naturally be considered as attempts at "market timing," since they relate to anticipating market-wide factors. Classical models of market-timing ability consider an equity manager who chooses a portfolio consisting of a market index and Treasury bills. The manager observes a signal about the future relative performance of the market over bills, and adjusts the market exposure of the portfolio. If the response is assumed to be a linear function of the signal as modelled by Admati, Ross and Plfeiderer (1986), the portfolio return is a convex quadratic function of the market return as in the model of Treynor and Mazuy (1966). If the manager shifts the portfolio weights discretely, as in the model of Merton and Henriksson (1981), the resulting convexity may be modelled with call options. Jiang (2003) develops a timing measure based on the probability, but not the magnitude of a convex relation. In

6 4 each case the convexity in the relation between the fund return and the market return indicates the timing ability. Our models modify this classical set up to accommodate nonlinearities unrelated to managers' timing ability. 2.1 Classical Market Timing Models The classical market-timing regression of Treynor and Mazuy (1966) is: r pt+1 = a p + b p f t+1 + Λ p f t u t+1, (1) where r pt+1 is the fund's portfolio return, measured in excess of a short-term Treasury bill. With equity market timing, as considered by Treynor and Mazuy, f t+1 is the excess return of the stock market index. Treynor and Mazuy (1966) argue that Λ p >0 indicates markettiming ability. The logic is that when the market is up, the successful market-timing fund will be up by a disproportionate amount. When the market is down, the fund will be down by a lesser amount. Therefore, the fund's return bears a convex relation to the market factor. In a fixed income context it seems natural to replace the market excess return with systematic factors for bond returns. However, if a factor is not an excess return, the appropriate sign for the coefficient might not be obvious. For example, bond returns move in the opposite direction as interest rates, so a signal that interest rates are about to rise means bond returns are likely to be low. We show below that market timing ability still implies a positive coefficient on the squared factor. Stylized market-timing models confine the fund to a single risky-asset portfolio and cash. This makes sense theoretically, from the perspective of the Capital Asset Pricing Model (CAPM, Sharpe, 1964). Under that model's assumptions there is two-fund

7 5 separation and all investors hold the market portfolio and cash. But two-fund separation is generally limited to single-factor term structure models, and there is no central role for a "market portfolio" of bonds in most fixed income models. In practice, however, fixed income funds often manage to a "benchmark" portfolio that defines the peer group or investment style. We use style-specific benchmarks to replace the market portfolio. Assume that the fund manager combines a benchmark portfolio with return R B and a short-term Treasury security or "cash" with known return R f, using the portfolio weight x(s), where s is the private timing signal. The managed portfolio return is R p = x(s)r B + [1- x(s)]r f. The signal is observed and the weight is set at time t, the returns are realized at time t+1, and we suppress the time subscripts for simplicity. In the simplest example the factor and the benchmark's excess return are related by a linear regression (we allow for nonlinearities below): r Bt+1 = μ B + b B f t+1 + u Bt+1, (2) where r B = R B - R F is the excess return, μ B =E(r B ), the factor is normalized to have a mean of zero and u Bt+1 is independent of f t+1. Assume that the signal s = f + v, where v is an independent, mean zero noise term with variance, σ v2. The manager is assumed to maximize the expected value of an increasing, concave expected utility function, E{U(r p ) s}. Finally, assume that the random variables (r,f,s) are jointly normal and let σ 2 f = Var(f). With these assumptions the optimal portfolio weight of the market timer is: 4 4 The first order condition for the maximization implies: E{U'(.)r B s} = 0 = E{U'(.) s} E{r B s} + Cov(U'(.),r B s}, where U'(.) is the derivative of the utility function. Using Stein's (1973) lemma, write the conditional covariance as: Cov(U'(.),r B s} = E{U''(.) s} x(s) Var(r B s). Solving for x(s) gives the result, with λ = -E{U'(.) s}/e{u''(.) s} > 0.

8 6 x(s) = λ E(r B s)/σ B2, (3) where λ > 0 is the Rubinstein (1976) measure of risk tolerance, which is assumed to be a fixed parameter, and σ 2 B = Var(r B s), which is a fixed parameter under normality. The implied regression for the managed portfolio's excess return follows from the optimal timing weight x(s) and the regression (2). We have r p = x(s) r B, then substituting from (2) and (3) and using E(r B s) = μ B + b B [σ f2 /(σ 2 f + σ v2 )] (f+v), we obtain equation (1), where a p = λμ B2 /σ B2, b p = (λμ B b B /σ B2 ) [1 + σ f2 /(σ f2 +σ v2 )] and Λ p = (λ/σ B2 )b 2 B [σ f2 /(σ f2 +σ v2 )]. The error term u pt+1 in the regression is a linear function of u B, v, vu B, fu B and vf. The assumptions of the model imply that the regression error is well specified, with E(u p f) = 0 = E(u p ) = E(u p f 2 ). The simple model illustrates that timing ability implies convexity between the fund's return and a systematic factor that is linearly related to the benchmark return. Note that if λ>0 the coefficient Λ p 0, independent of the sign of b B. Also, if the manager does not receive an informative signal, then Λ p =0, as E(r B s) and x(s) are constants in that case. However, the simple model does not consider nonlinearity in the fund's return that may arise for reasons unrelated to timing ability. In order to obtain reliable measures of timing ability, it is necessary to extend the model to control for these other potential sources of nonlinearity. 2.2 Nonlinearity Unrelated to Timing Convexity or concavity is likely to arise when the underlying assets held by a fund bear a nonlinear relation to the common factors. While this may occur in equity funds, such nonlinearities are likely in bond funds. Even simple bond returns are nonlinearly

9 7 related to interest rate changes, and securities such as mortgage products are described in terms of their "negative convexity" with respect to interest rate factors. A second potential cause of nonlinearities is "interim trading," as described by Goetzmann, Ingersoll and Ivkovic (2000), Ferson and Khang (2002) and Ferson, Henry and Kisgen (2006). This refers to a situation where fund managers trade more frequently than the fund's returns are measured. With monthly fund return data, interim trading definitely occurs. Derivatives may often be replicated by dynamic trading, so the use of derivatives is a closely related issue. For example, a fund that holds call options bears a convex relation to the underlying asset, and that can be confused with market timing ability (Jagannathan and Korajczyk, 1986). 5 A third reason for nonlinearity, unrelated to timing ability, arises if there is public information about future asset returns, and if managers' portfolios respond to this public information. As shown by Ferson and Schadt (1996), even if the conditional relation is linear, the response to public information can induce nonlinearities in the unconditional relation between the fund and a benchmark return. In this case the portfolio weights at the beginning of the period are correlated with returns over the subsequent period, due to their common dependence on the public information, and the products of the weights and the returns are nonlinear. Nonlinearity from public information trading can occur even if there is no interim trading problem. The fourth potential reason for nonlinearity unrelated to timing ability is stale pricing of a fund's underlying assets. Thin or nonsynchronous trading in a portfolio has 5 In another example, Brown et al. (2004) explore arguments that incentives and behavioral biases can induce managers to engage in option-like trading within performance measurement periods, even if they have no superior information. The fund's investors are "short" any option-like compensation provided to fund managers, and the fund's measured return, which is net of manager compensation, could bear a nonlinear relation to the benchmark.

10 8 long been known to bias beta estimates for the portfolio (e.g., Scholes and Williams, 1977). We show that if the degree of nonsynchronous trading is related to a common factor, this "systematic" stale pricing can create additional spurious concavity or convexity. The following subsections describe our approaches to controlling for these issues. 2.3 Addressing Nonlinearity in Benchmark Assets We generalize the classical market timing model to handle nonlinearity in the relation between the benchmark asset returns and the common factors. To do this we replace the linear regression (2) with a nonlinear regression: r Bt+1 = a B + b B (f t+1 ) + u Bt+1, (4) where b B (f) is a nonlinear function, specified below, and we assume that u B and b B (f) are normal and independent. The manager's market-timing signal is now assumed to be s = b B (f) + v, where v is normal independent noise with variance, σ v2. This captures the idea that the manager focusses on the return implications of information about the common factor. With these modifications the optimal weight function in (3) obtains with: E(r B s) = μ B [σ v2 /(σ *2 f + σ v2 )] + [σ *2 f /(σ *2 f + σ v2 )][a B + b B (f) + v], where σ *2 f =Var(b B (f)). Substituting as before we derive the nonlinear regression for the portfolio return: r pt+1 = a p + b p [b B (f t+1 )] + Λ p [b B (f t+1 )] 2 + u t+1, (5) with: a p = λ a B (μ B σ 2 v + a B σ *2 f )/[σ B2 (σ *2 f +σ v2 )], b p = λ(μ B σ 2 v + 2a B σ *2 f )/[σ B2 (σ *2 f +σ v2 )], and Λ p = (λ/σ B2 )[σ *2 f /(σ *2 f +σ v2 )].

11 9 The model restricts the coefficients in the nonlinear regression of the fund's return on the factors. The intuition is that nonlinearity of the benchmark return partly determines the nonlinearity of the fund's return. If there is no market-timing signal, then x(s) is a constant, Λ p =0 in (5) and the nonlinearity of the fund's return mirrors that of the benchmark. A successful market timer has a convex relation, relative to the benchmark, and thus Λ p >0. For example, if the benchmark is convex in the factor, a market-timing fund is more convex than the benchmark. We combine equations (4) and (5), and estimate the model by the Generalized Method of Moments (Hansen, 1982). One of the forms for b B (f) that we examine is a quadratic function. This has an interesting interpretation in terms of systematic co-skewness. Asset-pricing models featuring systematic coskewness with a market factor are studied, for example, by Kraus and Litzenberger (1976). Equation (1) is, in fact, equivalent to the quadratic "characteristic line" used by Kraus and Litzenberger, when the factor is the market portfolio excess return. The coefficient on the squared market factor measures the systematic coskewness risk of the portfolio, and this may have nothing to do with market timing. Thus, a fund's return can bear a convex relation to the factor because it holds assets with positive coskewness risk. Equation (5) allows the benchmark to have coskewness risk when b B (f) is quadratic. When Λ p =0 and the manager has no timing ability, the fund inherits its coskewness from the benchmark. If the fund has timing ability, Λ p >0 measures the additional convexity generated by the timing ability. 2.4 Addressing Interim Trading Interim trading means that fund managers trade more frequently than the fund's returns are measured. This can lead to spurious inferences about market timing ability, as

12 10 shown by Goetzmann, Ingersoll and Ivkovic (2000) and Jiang, Yao and Yu (2005), and can also lead to incorrect inferences about total performance, as shown by Ferson and Khang (2002). Ferson, Henry and Kisgen (2006) propose a solution, starting with a continuoustime asset pricing model. The continuous-time model prices all portfolio strategies that may trade within the period, provided that the portfolio weights are nonanticipating functions of the state variables in the model. If the continuous-time model can price derivatives by replication with dynamic strategies, the use of derivatives is also covered by this approach. Ferson, Henry and Kisgen derive the stochastic discount factor (SDF) for a set of popular term structure models, that applies to discrete-period returns and allows for interim trading during the period from time t to time t+1 6 : tm t+1 = exp(a - A r t+1 + b'a x t+1 + c'[x t+1 - x t ]), (6) where x is the vector of state variables in the model. The terms A x t+1 = Σ i=1,...1/δ x(t+(i-1)δ)δ approximate the integrated levels of the state variables. The measurement period between t and t+1 is one month, matching observed fund returns, and the period is divided into pieces of length Δ=one trading day. A r t+1 is the similarly time-averaged level of the short-term interest rate. The empirical "factors," f t+1, in the SDF thus include the discrete monthly changes in the state variables, their time averages and the time-averaged short term interest rate: f t+1 = {[x t+1 -x t ], A x t+1, A r t+1}. With the approximation e f 1+f, we can consider a model in which the SDF is linear in the expanded set of empirical factors. Using a linear function for the SDF is 6 A stochastic discount factor is a random variable, t m t+1, that "prices" assets through the equation E t { t m t+1 R t+1 }=1.

13 11 equivalent to using a beta pricing model for expected returns, written in terms of the relevant factors (Dybvig and Ingersoll (1982), Ferson, 1995). This motivates including the time-averaged variables in regressions like (5), as additional factors to control for interim trading effects Addressing Public Information A nonlinear relation between a fund's return and common factors may arise if managers respond to public information and there is time variation in factor risk premiums. Such time variation may be even more likely for bond returns than for stock returns. Conditional timing models have been designed to control for these effects by allowing portfolio weights and funds' betas to vary over time with public information. Using equity funds, Ferson and Schadt (1996) and Becker, et al. (1999) find that such conditional timing models are better specified than models that do not control for public information. In particular, Ferson and Schadt (1996) propose a conditional version of the market timing model of Treynor and Mazuy (1966): 8 r pt+1 = a p + b p r Bt+1 + C p '(Z t r Bt+1 ) + Λ p r Bt u t+1. (7) 7 Equation (6) also motivates the use of an exponential function for b B (f) in Equation (4). Under an exponential function the squared term in a market timing model like (5) is perfectly correlated with the linear term, so we don't estimate this case separately. An exponential function for b B (f) is closely approximated by a linear function when the factor changes are small numbers. 8 Ferson and Schadt (1996) also derive a conditional version of the market timing model of Merton and Henriksson (1981), which views successful market timing as analogous to producing cheap call options. This model is considerably more complex than the conditional Treynor-Mazuy model, and they find that it produces similar results.

14 12 In Equation (7), the new term C p '(Z t r Bt+1 ) controls for nonlinearity due to the public information, Z t. The expected benchmark return and the manager's optimal portfolio weights both depend on Z t. This induces common time-variation in the benchmark return and the fund's "beta" on the benchmark. The term C'(Z t r Bt+1 ) captures this common variation, controlling for the "spurious" timing effect due to public information. In a regression like (7) we measure timing with respect to the benchmark return, r B. However, we are also interested in timing with respect to interest rates and other economic factors that may bear a nonlinear relation to the benchmark, as described above. We study timing ability for these factors by replacing r B in Equation (7) with the nonlinear function b B (f) and estimating the equations (4) and (7) together in a system. 2.6 Addressing Stale Prices Thin or nonsynchronous trading in a portfolio biases estimates of the portfolio beta (e.g., Scholes and Williams, 1977). A similar effect occurs when the measured value of a fund reflects stale prices (e.g. Getmansky, Lo and Makarov, 2004). If the extent of stale pricing is related to a common factor, we call it "systematic" stale pricing. Systematic stale pricing can create spurious concavity or convexity and thus a spurious impression of market timing ability. To address the stale pricing issue we use a simple model in the spirit of Getmansky, Lo and Makarov. Let p t be the natural log of the fund's "true" net asset value per share with the last period's dividends reinvested, so that r t = p t - p t-1 is the true return. The true return would be the observed return if no prices were stale. We assume r t is independent over time with mean μ. The measured price, p * t is given by δ t p t-1 + (1-δ t ) p t, where the coefficient δ t ε [0,1] measures the extent of stale pricing during the month t. The

15 13 measured return on a fund, r t*, is then given by: r t * = δ t-1 r t-1 + (1-δ t )r t, (8) Assume that the market or factor return r mt, with mean μ m and variance σ m2, is iid over time and measured without error. 9 To model stale pricing that may be systematic, consider a regression of the extent of stale pricing at time t on the market factor: δ t = δ 0 + δ 1 (r mt - μ m ) + ε t, (9) where we assume that ε t is independent of the other variables in the model. We are interested in moments of the true return, like Cov(r t,r mt2 ), which measures the timing ability. Straightforward calculations relate the moments of the observable variables to the moments of the unobserved variables, as follows. E(r * ) = μ Cov(r t*,r mt ) = Cov(r,r m )(1- δ 0 + 2δ 1 μ m ) - δ 1 {μ σ 2 m + Cov(r,r m2 )} Cov(r t*,r mt2 )= Cov(r,r m2 )(1- δ 0 + δ 1 μ m ) - δ 1 { μe(r m3 ) + Cov(r,r m3 ) - [Cov(r,r m )+μμ m ](μ m2 +σ m2 )} Cov(r t*,r mt ) + Cov(r t*,r mt-1 ) = Cov(r,r m ) Cov(r t*,r mt2 ) + Cov(r t*,r mt-12 ) = Cov(r,r m2 ) Cov(r t*,r mt3 ) + Cov(r t*,r mt-13 ) = Cov(r,r m3 ) (10a) (10b) (10c) (10d) (10e) (10f) 9 We have considered models in which the benchmark return r m also has stale pricing, and find that the number of parameters proliferates to the point of intractability. Thus, our estimates of this model should in practice capture fund staleness relative to any staleness in the benchmark return.

16 14 Equation (10a) shows that stale prices will not affect the measured average returns in this model, even if stale pricing is systematic. 10 Equation (10b) captures the bias in the measured covariance with the market. The market beta, of course, is this covariance divided by the market variance. If δ 1 =0 stale pricing is not systematic, and the effect on the measured beta reduces to an attenuation bias as in the model of Scholes and Williams (1977). If δ 1 is not zero stale pricing is systematic, and the measured beta is affected by both stale pricing and the true timing ability. Equation (10c) shows that the measured timing coefficient is also biased. If the stale pricing is not systematic, however, the bias reduces to a scale factor. If δ 1 =0 we can therefore test the null hypothesis of no timing ability using the measured covariance, as the true timing coefficient is zero if and only if the measured covariance with the squared market factor is zero (assuming that δ 0 1). When stale pricing is systematic the bias of the timing coefficient is complex as it depends on the true beta, the value of δ 1, higher moments and other parameters. We need to control for this complicated bias in order to measure the timing ability. Fortunately, equation (10e) reveals a simple way to control for a biased market timing coefficient due to stale prices. The sum of the covariances of the measured return with the squared market factor and the lagged squared factor equals the true covariance or timing coefficient. This is similar to the bias correction for betas in the models of Scholes and Williams (1977) and Dimson (1979). If we append moment conditions to the system (10) to identify μ m, σ 2 m and E(r m3 ) we have a system of nine equations that allow us to identify the nine parameters, {μ m, σ m2, 10 Qian (2005) studies the effects of stale prices on the measured average performance of equity style funds, and finds that if staleness is correlated with new money flows there can be effects on average returns. See Zietowitz (2003) and Edelin (1999) for estimates of the size of potential losses to buy-and-hold investors, attributed to other investors trading on stale net asset values.

17 15 Cov(r,r m ), Cov(r,r m2 ), E(r m3 ), Cov(r,r m3 ), δ 0, δ 1, and μ}. We estimate the system by the generalized method of moments (GMM, Hansen, 1982). We measure the timing ability, adjusted for the effects of stale pricing, via the parameter Cov(r,r m2 ). 2.7 Combining the Effects We have models for four different sources of convexity or concavity unrelated to timing ability, and we may wish to control for all of these effects. The models for interim trading and public information suggest control variables that may be included in the market timing regressions, while the nonlinear-benchmark models imply a system of equations. The general form of the models is a system including Equation (4) and Equation (11): r pt = a + β'x t + Λ p [b B (f t ) 2 ] + u pt, (11) where X t is a vector of control variables observed at t or before, and which includes b B (f t ), the benchmark's nonlinear projection on the factor. In the presence of stale pricing we observe r * pt and not the true return, r pt. However, according to Equation (10e) we can still estimate the timing coefficient easily if we do not require estimates of the deeper structural parameters of the model. Simply replacing b B (f t ) 2 in (11) with the sum, b B (f t ) 2 + b B (f t-1 ) 2, delivers a coefficient that should be proportional to Λ p. 3. The Data We first describe our sample of bond funds. We then describe the interest rate and other economic data that we use to construct the common factors relative to which we study

18 16 timing ability. Finally, we describe the fund style-related benchmark returns. 3.1 Bond Funds The fund data are from the Center for Research in Security Prices (CRSP) mutual fund data base, and include returns for the period from January of 1962 through December of We select open-end funds whose stated objectives indicate that they are bond funds. 11 We exclude money market funds and municipal securities funds. There are a total of 23,281 fund-year records in our initial sample. In order to address back-fill bias we remove the first year of returns for new funds, and any returns prior to the year of fund organization, a total of 1345 records. Data may be reported prior to the year of fund organization, for example, if a fund is incubated before it is made publicly available (see Elton, Gruber and Blake (2001) and Evans, 2004). Extremely small funds are more likely to be subject to back-fill bias. We delete cases where the reported total net assets of the fund is less than $5 million. This removes 3205 records. We delete all cases where the reported equity holdings at the end of the previous year exceeds 10%. This removes another 644 records. After these screens we are left with 18,087 fund-years. The number of funds with some monthly return data in a given year is five at the beginning of 1962, rises to 17 at the end of 1973, to 617 by 1993 and to 1756 at the end of 11 Prior to 1990 we consider funds whose POLICY code is B&P, Bonds, Flex, GS or I-S or whose OBJ codes are I, I-S, I-G-S, I-S-G, S, S-G-I or S-I. We screen out funds during this period that have holdings in bonds plus cash less than 70% at the end of the previous year. In 1990 and 1991 only the three digit OBJ codes are available. We take funds whose OBJ is CBD, CHY, GOV, MTG or IFL. If the OBJ code is other than GOV, we delete those funds with holdings in bonds plus cash totalling less than 70%. After 1991 we select funds whose OBJ is CBD, CHY, GOV, MTG, or IFL or whose ICDI_OBJ is BQ, BY, GM or GS, or whose SI_OBJ is BGG, BGN, BGS, CGN, CHQ, CHY, CIM, CMQ, CPR, CSI, CSM, GBS, GGN, GIM, GMA, GMB, GSM or IMX. From this group we delete 116 fund years for which the POLICY code is IG or CS.

19 17 the sample in December of We group the funds into equally-weighted portfolios according to fund style. We define seven styles on the basis of the various objective codes on the CRSP database. The styles are Global, Short-term, Government, Mortgage, Corporate, High Yield and Other. 12 Summary statistics for the style-grouped funds are reported in Panel A of Table 1. The mean returns are all between 0.31 and 0.56% per month. The standard deviations of return range between 0.47% per month, for Short-term Bond funds, to 1.66% per month for High-yield funds. The first-order autocorrelations of the returns range from 8%, for Global Bond funds, to 30% for High Yield funds. The minimum return across all of the style groups in any month is -6.7%, suffered in October of 1979 by the corporate bond funds. The maximum return is 10.3%, also earned by the Corporate bond funds in November of Table 1 also reports the second order autocorrelations of the fund returns. The stylized thin trading model assumes that thin trading is resolved (all the assets trade) within two months. This implies that the measured returns have an MA(1) time-series structure, and the second order autocorrelations should be zero. The largest second order autocorrelation in the panel is 7.1%, with an approximate standard error of 1/ T = 1/ 144 = 8.3%. For the portfolio of all funds, where the number of observations is the greatest, the second order autocorrelation is -2.2%, with an approximate standard error of 1/ 492 = 4.5%. Thus, none of the second order autocorrelations is significantly different from zero, 12 Global funds are coded SI_OBJ=BGG or BGN. Short-term funds are coded SI_OBJ=CSM, CPR, BGS, GMA, GMS or GBS. Government funds are coded OBJ=GS POLICY=GOV, ICDI_OBJ=GS or GB, or SI_OBJ=GIM or GGN. Mortgage funds are coded ICDI_OBJ=GM, OBJ=MTG or SI_OBJ=GMB. Corporate funds are coded as OBJ=CBD, ICDI_OBJ=BQ, POLICY=B&P or SI_OBJ=CHQ, CIM, CGN or CMQ. High Yield funds are coded as ICDI_OBJ=BY, SI_OBJ=CHY or OBJ=CHY or OBJ=I-G and Policy=Bonds. Other funds are defined as funds that we classify as bond funds (see the previous footnote), but which meet none of the above criteria.

20 18 consistent with the assumptions of the model. We also group the funds into equally-weighted portfolios according to various fund characteristics, measured at the end of the previous year. Since the characteristics are likely to be associated with fund style, we form groups within each of the style classifications. The characteristics include fund age, total net assets, percentage cash holdings, percentage of holdings in options, reported income yield, turnover, load charges, expense ratios, the average maturity of the funds' holdings, and the lagged return for the previous year. The Appendix provides the details. 3.2 Common Factor Data We use daily and weekly data to construct monthly empirical factors, including the discrete changes measured from the ends of the months and the time-averaged values used as controls for interim trading bias. Most of the data are from the Federal Reserve (FRED) and the Center for Research in Security Prices (CRSP) databases. The daily interest rates are from the H.15 release. The factors reflect the term structure of interest rates, credit and liquidity spreads, exchange rates, a mortgage spread and two equity market factors. Three factors represent the term structure of Treasury yields: A short-term interest rate, a measure of the term slope and a measure of the curvature of the yield curve. Studies such as Litterman, Sheinkman and Weiss (1991) find that three factors explain most of the variance of Treasury bond returns across the maturity spectrum, and that level, slope and curvature provide a convenient way to represent three factors. The short-term interest rate is the three-month Treasury rate, converted from the bank discount quote to a continuously compounded yield. Chapman, Pearson and Long (2001) argue that the three month Treasury rate is a good empirical proxy for the

21 19 theoretical short-term interest rate. The slope of the term structure is the ten-year yield less the one-year yield. The curvature measure is the concavity: y 3 - (y 7 + 2y 1 )/3, where y j is the j-year fixed-maturity yield. Since some of our funds hold corporate bonds subject to default risk and mortgage backed securities subject to prepayment risks, we construct associated factors. Our credit spread series is the yield of Baa corporate bonds minus Aaa bonds, from the FRED. This series is measured as the weekly averages of daily yields. We use the averages of the weeks in the month for our time-averaged version of the spread. For the discrete changes in the spread we use the first differences of the last weekly values for the adjacent months. The first difference series may not be as clean as with daily data, but we are limited by the data available to us. Our mortgage spread is the difference between the average contract rate on new conventional mortgages, also available weekly from the FRED, and the yield on a three-year, fixed maturity Treasury bond. Here we use daily data on the Treasury bond and weekly data on the mortgage yield to construct the time averages and discrete changes. Liquidity is likely to be a relevant factor for bond funds. Liquidity varies both across issues and over time. For example, on-the-run Treasury issues are more liquid than off-the-run issues. Also, credit markets have periodic episodes where liquidity gets tight or loose. For market timing we are interested in market-wide fluctuations in liquidity. Our measure follows Gatev and Strahan (2005), who advocate a spread of commercial paper over Treasury yields as a measure of short term liquidity in the corporate credit markets. We use the yield difference between three-month nonfinancial corporate commercial paper rates and the three month Treasury yield. The commercial paper rates are measured weekly, as the averages over business days, and we convert them from the bank discount basis to continuously compounded yields.

22 20 Some of the funds in our sample are global bond funds, so we include a factor for currency risks. Our measure is the value of the US dollar, relative to a trade-weighted average of major trading partners, from the FRED. This index is measured weekly, as the averages of daily figures, and we treat it the same way we treat the credit spread data. Corporate bond funds, and high-yield funds in particular, may be exposed to equity-related factors. We therefore include two equity market factors in our analysis. We measure equity volatility with the VIX-OEX index implied volatility. This series is available starting in January of We also include an equity market valuation factor, measured as the log dividend ratio for the CRSP value-weighted index. The dividends are the sum of the dividends over the past twelve months, and the value is the cum-dividend value of the index. The level of this ratio is a state variable for valuation levels in the equity market, and its monthly first difference is used as a factor. The first difference of this series approximates the (negative of the) continuously-compounded capital gains component of the index return. Table 2 presents summary statistics for the monthly common factor series starting in February of 1962 or later, depending on data availability, and ending in December Missing values are excluded and the units are percent per year (except for the dollar index). Panel A presents the levels of the variables, panel B the monthly first differences and Panel C presents the time-averaged values. The average term structure slope was positive, at just under 80 basis points during the sample period. The average credit spread was about one percent, but varied between 32 basis points to more than 2.8%. The average mortgage spread over Treasuries was just over 2% for the period starting in 1971, and varied between about -1% to more than 7%. The liquidity spread averaged 4.7%, with a range between 1.3% and 6.8% over the period.

23 21 In their levels the factors are highly persistent time series, as indicated by the first order autocorrelation coefficients. Six of the nine are in excess of 90%. The time averaged factors are also highly persistent, with autocorrelations greater than 84%. Moving to first differences, the series look more like innovations Style Index Returns We construct style-specific benchmarks for the funds from a set of asset-class returns, using a methodology similar to Sharpe (1992). The problem is to combine the asset class returns, R i, using a set of portfolio weights, {w i }, so as to minimize the "tracking error" between the return of the fund (or fund style group), R p, and the style-matched benchmark portfolio, Σ i w i R i. The portfolio weights are required to sum to 1.0 and must be non-negative, which rules out short positions: Min {wi} Var[R p - Σ i w i R i ], (13) subject to: Σ i w i = 1, w i 0 for all i, where Var[.] denotes the variance. We solve the problem numerically, deriving a set of weights for each fund style group. The asset class returns include US Treasury bonds of three maturity ranges from CRSP (less than 12 months, less than 48 months and greater than 120 months), the Lehman Composite Global bond index, the Lehman US Aggregate 13 Given the relatively high persistence and the fact that some of the factors have been studied before exposes us to the risk of spurious regression compounded with data mining, as studied by Ferson, Sarkissian and Simin (2003). However, Ferson, Sarkissian and Simin (2005) find that biases from these effects are largely confined to the coefficients on the persistent regressors, while the coefficients on variables with low persistence are well behaved. Thus, the slopes on our persistent control variables and thus the intercepts may be biased, but the timing coefficients should be well behaved.

24 22 Mortgage Backed Securities index, the Merrill Lynch High Yield US Master index and the Lehman Aaa Corporate bond index. 14 Panel B of Table 1 presents the style-index weights for each of the style portfolios. The weights reported in the table do not sum exactly to 1.0 due to rounding errors, but we carry many more digits of precision in the return calculations. The weights present sensible patterns in most cases, suggesting that both the style classification of the funds and Sharpe's procedure are reasonably valid. The global funds load most heavily on corporate bonds and global bonds, then on low quality bonds. Short term funds have most of their weight in bonds with less than 48 months to maturity. Mortgage funds have their greatest weights on mortgage backed securities and shorter term bonds. Corporate funds have more than 50% of their weight in high or low grade corporate bonds. High yield funds have 70% of their weight in low grade corporate bonds. Government funds load highly on mortgage-backed securities (33%). This is consistent with the portfolio weights described by Comer (2006), who finds that government style bond funds allocate about 1/3 of their portfolios to mortgage related securities. 4. Empirical Results We first examine the empirical relations between the various factors, the fund groups and passive investment strategies proxied by the Sharpe style benchmarks. We estimate the model of stale pricing and evaluate the effects of the other controls for nonlinearities unrelated to timing. Finally, we apply the adjusted timing measures to individual funds. 14 We splice the Blume, Keim and Patel (1991) low grade bond index prior to 1991, with the Lehman High Yield index after that date. We splice the Ibbotson Associates 20 year government bond return series for , with the CRSP greater than 120 month government bond return after 1971.

25 Factor Models We begin the empirical analysis with regressions of the Sharpe style benchmark returns on changes and squared changes in the common factors, looking for convexity or concavity in these relations. Any nonlinearity would appear in a naive application of the timing regression (1) for funds, if funds simply held the benchmarks. These regressions are shown in Panel A of Table 3. Panel B of Table 3 shows regressions for the mutual funds' returns, grouped by the same style categories. Comparing the funds with the style benchmarks provides a feel for the affects of active management. The table shows the coefficients on the squared terms for the regressions, run one factor at a time. The coefficients for the style benchmarks in Panel A of Table 3 present some interesting patterns. Out of 63 cases (seven styles x 9 factors) there are only five heteroskedasticity-consistent t-ratios on the squared factors with absolute values larger than two. However, 21 of the absolute t-ratios are between 1.6 and 2.0, which provides some evidence of nonlinearity. Most of the coefficients on the squared factors are positive -- only eight of the 63 are less than zero. This means that we would expect to measure weak convexity or timing ability, based on regression (1), if funds simply held the benchmarks. The coefficients for the mutual funds in Panel B of Table 3 tell a different story. We find 13 absolute t-ratios larger than 2.0. This is more than expected by chance; thus, the mutual fund returns are significantly nonlinearity related to the factors. 15 Furthermore, the coefficients on the squared factors are negative in about half the cases; whereas, most 15 Using a simple binomial model assuming independence, the t-ratio associated with finding 13 "rejections," when the probability of observing a rejection is 5%, is (13/ )/(.05(.95)/63) 0.5 = 5.69.

26 24 of the coefficients were positive for the benchmarks. Thus, overall the fund returns appear more concave in relation to the factors than the benchmark returns. This suggests poor market timing ability on the part of the mutual funds. Alternatively, the concavity could reflect derivatives or dynamic trading, public information or stale prices. The right-hand column of Table 3 presents the R-squares of the factor model regressions, using all of the factors simultaneously. The fractions of the fund return variances explained by the factors are interesting because large fractions suggest that the ability to time common factors could represent a large portion of funds' potential performance. In the first row of R-squares the changes in the factors are the independent variables. In the second row the squared factor changes are also included in the regression. The R-squares for the funds in Panel B are between 33% and 62%, using all the factors, and exceed 45% for five of the seven styles. The squared factors contribute more to the explanatory power of the regressions for the funds than they do for the benchmark returns (Panel A). The adjusted R-squares rise by almost 20% for global bond funds, 8.6% for short term funds and 3.5% for government funds when the squared factor changes are included in the regressions. This makes sense if trading by fund managers leads to nonlinearity in the funds' returns, relative to the benchmarks. Panel A of Table 3 reports the adjusted R-squares for the benchmark returns, which range from 28% to 72%. Including the squared terms typically reduces the adjusted R- squares a little. The R-squares are somewhat higher if we include the lagged values of the state variables in the regressions to capture expected return variation over time. 4.2 Stale pricing This section summarizes the results of estimating the stale pricing model, system

27 25 (10), for the excess returns of the funds grouped by style. The Sharpe benchmark excess return for the fund style group serves as r m. We also estimate the model for the excess returns of the characteristics-based fund groups for each style. Table 4 shows the estimated coefficients in Equation (9), the model for systematic stale pricing, along with the estimate of the timing coefficient adjusted for stale pricing effects. We scale the timing coefficients similar to a regression coefficient, reporting the estimates of Cov(r,r m2 )/E{r m3 }. The right-hand column of the table reports a Wald test of the hypothesis that there is no stale pricing effect: δ 0 =δ 1 =0. The δ 0 coefficients, which measure the average level of stale pricing, are positive and between and One interpretation of the model is that, on average, the reported prices for 13-27% of the securities held by the funds are stale at the end of the month. The government style funds are the least stale by this measure and the corporate bond funds are the most stale. However, the standard errors of these estimates are large, and none are significantly different from zero. The δ 1 coefficients measure the sensitivity of stale pricing to the return of the benchmark index. The δ 1 coefficients are not statistically significant. The hypothesis that δ 1 =0 says that stale pricing is not systematic, and implies that we can test for timing ability using the measured covariance with the squared benchmark return. The Wald test does reject the hypothesis that there are no stale pricing effects (δ 0 =δ 1 =0) for five of the seven styles. Examining 147 cases, including funds grouped by characteristics within each style, we find p-values of the Wald test less than 5% in 79 of the cases. Thus, the results leave open the possibility that stale pricing might effect the estimates of timing ability. The timing coefficients should measure timing ability purged of the stale pricing effects. The standard errors of these coefficients are also large, and none are significantly different from zero. However, all of the values in the table are positive, which suggests