Evaluating Government Bond Fund Performance with Stochastic Discount Factors

Transcription

1 Evaluating Government Bond Fund Performance with Stochastic Discount Factors by Wayne Ferson, Tyler Henry and Darren Kisgen March 13, 2004 * Ferson is the Collins Chair in Finance at Boston College, 140 Commonwealth Ave, Chestnut Hill, MA , and a Research Associate of the NBER. He may be reached at (617) , fax , wayne.ferson@bc.edu, www2.bc.edu/~fersonwa. Henry and Kisgen are Ph.D students at the University of Washington School of Business Administration, Department of Finance and Business Economics Box Seattle, Washington , fax: Kisgen may be reached at kisgen@u.washington.edu. Henry may be reached at thenry@u.washington.edu. We are grateful to Warren Bailey, Jeffrey Busse, Yong Chen, Edie Hotchkiss, Clifton Green, Eric Jacquier, Raymond Kan, Anthony Lynch, Shingo Goto, Steven Mann, Meijun Qian, Richard Roll, Rudi Schadt and Russ Wermers for help or suggestions. This paper was presented at Boston College, Harvard, the University of Maryland, MIT, Princeton, the University of South Carolina and at the 2003 Conference on Financial Economics and Accounting, the 2003 Northern Finance Association meetings, and the 2004 Conference on Emerging Markets: Innovations in Portfolio Management, at the University of Viginia. An earlier pilot study benefited from workshops at Babson College, Boston University, Brandeis, the Universities of California at Los Angeles and Riverside, Cornell, the Federal Reserve Bank of Atlanta, the London Business School, London School of Economics, New York University, Northwestern, the Said Business School at Oxford University, the University of Texas at Dallas, Utah, the Warwick Business School and Wharton. Portions of this work were also presented at the 2003 Berkeley Program in Finance, the 2003 Boston College Finance Advisory Conference, the 2003 Chicago Quantitative Alliance conference, the 2003 European Finance Meetings and the 2003 McGill Conference on Global Asset Management. The authors appreciate the financial support from the Gutmann Center for Portfolio Management at the University of Vienna and the Institute for Quantitative Research in Finance.

2 ABSTRACT Evaluating Government Bond Fund Performance with Stochastic Discount Factors This paper shows how to evaluate the performance of managed portfolios using stochastic discount factors (SDFs) from continuous-time term structure models. Our approach addresses a bias in performance measurement, described by Goetzmann, Ingersoll and Ivkovic (2000) and Ferson and Khang (2002), that arises when fund managers may trade within the return measurement interval. The solution gives rise to empirical factors formed as time-averages of the underlying state variables in the model. We find that these empirical factors contribute explanatory power in factor model regressions and reduce the pricing errors of the SDF models for dynamic strategy returns. We illustrate our approach on a sample of U.S government bond funds during This example represents the first conditional performance evaluation for US fixed income mutual funds. During government bond funds as a group returned less on average than bond portfolios that don't pay expenses. Their returns vary more across term structure states than across characteristics-grouped portfolios formed on the basis of fund size, age, expenses and other common characteristics. High spot rates, high term structure slopes and low term structure convexity states predict higher conditional expected returns. However, after risk adjustment none of these performance differences is economically significant.

3 1. Introduction Recent years have witnessed an explosion of research on the performance of mutual funds, with most of the attention focussed on equity-style funds. The amount of academic work on fixed income funds is small in relation to the importance of these funds and assets in the economy. As of June 2002 there were 2,057 bond funds in the US, representing 25% of all mutual funds. Total assets under management by these funds totalled just over $1 trillion, or 15% of the $6.6 trillion in mutual fund assets. (These figures exclude balanced funds, which hold a mix of bonds and stocks.) Fixed income funds have also seen rapid growth over the recent decade, with the number of funds and assets under management increasing 97% and 245% respectively, between 1990 and One may expect that the variation in fixed income fund performance should be small, relative to that of equity funds. However, given the relatively low volatility of fixed income returns, the standard errors associated with measures of performance are also relatively small. Fixed income managers have a variety of tools to use in their quest to outperform their peer funds and benchmarks. They can tune the interest rate sensitivity (duration, convexity) of the portfolio to time changes in the level or the shape of the yield curve in response to economic developments. They can vary the allocation to asset classes (e.g. Treasury versus agency securities such as those issued by FNMA and FHLMC). They can attempt to exploit liquidity differences, such as those associated with on-the-run or off-the-run issues. They can strategically manage their securities lending operations. They can trade a host of interest rate derivative products in implementing these strategies. Active fixed income fund management would seem to present a rich field of research opportunities. The relatively small amount of research on fixed income funds may reflect 1 Sources: the Investment Company Institute, Trends in Mutual Fund Investing, June 2002, and 2002 Mutual Fund Handbook.

4 2 differences in the empirical models available for fixed income and equity returns. 2 Standard models for expected equity returns lend themselves naturally to measures of risk-adjusted "abnormal" returns. For example, an "alpha" is measured as the difference between the average return of a fund and the expected return predicted by a model on the basis of the fund's beta risk. Fixed income models, in contrast, are typically directed at the problem of solving for the prices of derivative claims. If a portfolio is formed with unobserved weights, such as a mutual fund, the value of the portfolio of claims is difficult to model. This paper measures the performance of government bond mutual funds in a stochastic discount factor (SDF) framework. The approach has several advantages. Popular term structure models identify stochastic discount factors, and we show how these are easily temporally aggregated for pricing monthly returns. The time aggregation generates empirical "factors" from the theory whose use for pricing returns would not otherwise be obvious. Such factors are appealing in comparison with recently-popular asset pricing factors for equities that arise from empirical regularities (e.g., Fama and French, 1996). 3 We find that the empirical factors implied by time aggregation of the continuous-time models contribute to an improved performance in explaining conditional returns. The approach in this paper addresses an "interim trading bias" in returns-based 2 Studies that focus on US fixed income funds include Blake, Elton and Gruber (1993), Elton, Gruber and Blake (1995) and Kang (1995). Cornell and Green (1991) and Gudikunst and McCarthy (1992, 1997) examine low-grade bond funds, Stock (1982) and Kihn (1996b) examine municipal bond portfolios and Kihn (1996a) examines convertible bond funds. Duke, et al. (1993), Schadt (1996), Gallo et al. (1997), Fjelstad (1999), Detzler (1999) and Silva and Cortez (2002) study international fixed income fund performance. Dahlquist, Engstrom and Soderlind (2000) include bond and money market funds in their sample of Swedish funds, and Massa (2003) includes them in his study of fund families. Fung and Hsieh (2002) compare the styles of fixed income hedge funds and mutual funds. Additional related studies include D'Antonio et al. (1997), Dietz (1981), Fong, Pearson and Vasicek (1983), Grantier (1988), Kahn (1991) and Shyy and Lieu (1994). 3 Critiques by Lo and MacKinlay (1990), MacKinlay (1995) and Ferson, Sarkissian and Simin (1999) illustrate the pitfalls of asset pricing factors motivated by empirical regularities.

5 3 performance measures that arises when monthly returns are measured, while managers trade on public information during the month (Goetzmann, Ingersoll and Ivkovic (2000), Ferson and Khang, 2002). Ferson and Khang propose a weight-based measure of performance to avoid interim trading bias. With our approach, portfolio weights are not needed to address the problem. This is convenient since the holdings of mutual funds are typically available on a less-frequent basis than are the funds' returns. The stochastic discount factor approach is appealing for other reasons as well. For a given SDF a measure of abnormal return or "SDF alpha" is easily constructed (Chen and Knez, 1996). The approach lends itself naturally to "Conditional Performance Evaluation," where funds' alphas are conditioned on ex-ante economic states. Term structure models in particular prescribe the state variables to condition on. This removes some of the ambiguity in instrument selection that is typical of the conditional asset pricing literature. This paper is the first to use the stochastic discount factor approach and continuous-time models of the term structure to evaluate fund performance. The main goal of the paper is to motivate and illustrate this approach. This is also the first paper to present a conditional evaluation of US fixed income funds. We first evaluate the SDF models using benchmarks that we construct to represent both passive and dynamic bond portfolio strategies. We find that the returns and volatility of these benchmarks vary over the states of the term structure, which motivates a conditional performance analysis. The SDF models provide good control of these conditional expected returns. For example, the conditional expected returns of the Lehman Government bond index vary between 61 and 132 basis points per month over the period, depending on the term structure state. After risk adjustment with SDF models the conditional alphas are less than 14 basis points, and close to zero with the best models. A two-factor affine model outperforms a single factor model, which includes the models of Vasicek (1977) and Cox, Ingersoll and Ross (1985a). A two-factor Brennan

6 4 and Schwartz (1979) model performs similarly to the two-factor affine model, while a three factor affine model incorporating convexity has slightly better pricing performance. We illustrate the use of the SDF models with a sample of U.S. government bond funds. During the funds returned less on average than benchmark portfolios that don't pay expenses. Conditioning on the level, slope and convexity of the term structure, we find more variation in funds' average returns across the states than across groups formed according to characteristics like expense ratios, size, age and past returns. When we adjust for risk using the stochastic discount factor models, we find that abnormal conditional performance is usually negative but economically small. The rest of the paper is organized as follows. Section 2 presents the models for the stochastic discount factors, following term structure theory, and describes how we operationalize them to handle monthly mutual fund data. Section 3 discusses the interim trading bias. Section 4 describes our estimation methods. Section 5 describes the data. Section 6 presents a comparison of our approach with linear factor models, and results on the estimation of the stochastic discount factor models for benchmark strategies. Section 7 evaluates government bond fund performance. Section 8 offers concluding remarks. 2. The Stochastic Discount Factor Models Most asset pricing models posit the existence of a stochastic discount factor, m(φ) t+1, which is a scalar random variable that depends on data observed up to time t+1 and parameters φ, such that the following equation holds: E t (m(φ) t+1 R t+1 )= 1, (1) where R t+1 is an N-vector of gross (i.e., one plus) "primitive" asset returns, 1 is an N-vector of ones and E t (.) denotes the conditional expectation, given the information in the model at

7 5 time t. We say that the SDF "prices" the primitive assets if equation (1) is satisfied. Rearranging equation (1) reveals that the expected return is determined by the SDF model as: E t (R t+1 ) = [E t (m t+1 )] -1 + Cov t { -m t+1 /E t (m t+1 ); R t+1 }, (2) where Cov t (.,.) is the conditional covariance given the information at time t and [E t (m t+1 )] -1 is the risk-free or "zero-beta" return. Thus, predicted excess returns differ across funds in proportion to the conditional covariances of their returns with the SDF. We allow that a mutual fund, with return R p,t+1, may not be priced exactly by the SDF. Its SDF alpha is defined as α pt E t (m t+1 R p,t+1-1), following Chen and Knez (1996) and Farnsworth, et. al (2002). The SDF alpha is proportional to the traditional alpha in a beta pricing representation, when the SDF is linear in the factors. For example, in the case of the Capital Asset Pricing Model (Sharpe, 1964), the SDF is linear in the market return and α p is proportional to Jensen's (1968) alpha Stochastic Discount Factors from Term Structure Models Popular term structure models specify a continuous-time stochastic process for the underlying state variable(s). For example, let X be the state variable following a diffusion process: dx = µ(x t ) dt + σ(x t ) dw, (3) where dw is the local change in a standard Wiener process. The state variable may be the level of an interest rate, the slope of the term structure, etc. The model also specifies the 4 See Ferson (1995, 2003) and Cochrane (2001) for more discussion and interpretation of SDF alphas.

8 6 form of a market price of risk, q(x), associated with the state variable. The market price of risk is the expected return in excess of the instantaneous interest rate, per unit of state variable risk. 5 The literature has directed a lot of firepower at accurately modelling continuoustime processes like Equation (3). For example, when X is a short term interest rate and σ(x) = X γ, studies following Chan, Karolyi, Longstaff and Sanders (1992) debate whether γ is 0.5, 1.0, 1.5 or some other number. Other studies ask whether the drift of the process is linear or nonlinear in X (see, e.g, Ait-Sahalia (1996), Stanton (1997), Chapman and Pearson (2000), Balduzzi and Eom (2002), Duarte (2003), Jones, 2003). As we show below, our approach does not require all of the moment conditions implied by a process like Equation (3), and should therefore be robust to some misspecifications. Term structure models based on Equation (3) can be shown (using Girsanov's Theorem, see Cox, Ingersoll and Ross (1985b) or Farnsworth, 1997) to imply stochastic discount factors of the following form: tm t+1 = exp( - A t+1 - B t+1 - C t+1 ), where A t+1 = t t+1 r s ds, (4) B t+1 = t t+1 q(x s ) dw s C t+1 = (1/2) t t+1 q(x s ) 2 ds, where r s is the instantaneous interest rate at time s. The notation t m t+1 is chosen to 5 Multiple-state-variable models are described below. The models we study are based on timehomogeneous diffusions; that is, the functions µ(.) and σ(.) in Equation (3) depend on time only through the level of the state variable at a point in time. In contrast, interest rate models such as Hull and White (1990) allow time variation in the functions, choosing them to fit closely the term structure of bond prices at each time t. Such models are attractive for the practice of pricing interest-dependent derivative securities, among other reasons, because by fitting the term structure at each date the models can avoid derivative prices that allow arbitrage at the current bond prices.

9 7 emphasize that the SDF refers to a discrete time interval, in our case one month, that begins at time t and ends at time t+1. When there are multiple state variables there is a term like B t+1 and C t+1 for each state variable. Note that, unlike beta pricing models where the SDF is linear in the factors, the SDF in (4) is nonlinear. Dietz, Fogler and Rivers (1981) argue that tests of bond portfolio performance should allow for nonlinearity. 2.2 Discretizations We use a simple first order Euler approximation to Equation (3) in order to use the term structure models with monthly mutual fund data: X(t+ ) - X(t) µ(x t ) + σ(x t )[w(t+ ) - w(t)]. (5) The period between t and t+1 is divided into 1/ increments of length. The period is one month, to match the mutual fund returns, and we divide it into increments of one day. For a given model we have daily data on X(t+ ) and X(t) and the functions µ(x t ) and σ(x t ) are specified. We infer the approximate daily values of [w(t+ ) - w(t)] from equation (5). The terms A t+1, B t+1 and C t+1 in Equation (4) are approximated using daily data by: A t+1 Σ i=1,...1/ r(t+(i-1) ) B t+1 Σ i=1,...1/ q[x(t+(i-1) )] [w(t+i ) - w(t+(i-1) )] (6) C t+1 (1/2) Σ i=1,...1/ q[x(t+(i-1) )] 2. Farnsworth (1997) and Stanton (1997) evaluate the accuracy of similar first order approximation schemes. Stanton concludes that with daily data, these approximations are almost indistinguishable from the true functions over a wide range of values, and the approximation errors should be small when the series being studied is observed monthly.

10 He also evaluates higher order approximation schemes, and finds that with daily data they offer negligible improvements over the first order approximations The Term Structure Models A term structure model specifies the state variables, X, and the functions µ(.), σ(.) and q(.). We illustrate our approach on a collection of classical term structure models. These models were developed prior to the period covered by our sample of mutual funds, which starts in This protects us against concerns about "data mining" variables that work for a given sample period. The first model is a single-state variable model in the affine class, where the short term interest rate r t is the state variable. dr = K(θ - r) dt + σ(r) dw, (7) σ(r) = (γ + δ r) ½, q(r) = λ(γ + δ r) ½. Equation (7) includes the single-factor models of Vasicek (1977), where δ=0, and of Cox, Ingersoll and Ross (1985a), where γ=0. We include two versions of two-factor term structure models. The first is the Two-factor Affine Model: dr = K 1 (θ 1 - r t ) dt + [(q 1 /λ 1 ) dw 1 + ρ(q 2 /λ 2 ) dw 2 ], (8) dl = K 2 (θ 2 - l t ) dt + [ρ(q 1 /λ 1 ) dw 1 + (q 2 /λ 2 ) dw 2 ], q 1 = λ 1 (α 1 + β 1 r t + γ 1 l t ) ½, 6 Sun (1997) evaluates the approximations in (6), used to estimate the parameters of continuous time models of the short rate, and finds that the approximation errors can be improved upon with a test function method. However, Gourioux, Montfort and Polimenis (2002) show that if we start with the assumption that an affine model holds for the discrete period of length, then the summations in Equation (6) apply to the longer holding period returns without any approximation error.

11 q 2 = λ 2 (α 2 + β 2 r t + γ 2 l t ) ½, E(dw 1 dw 2 )=0, 9 where {K 1, θ 1, K 2, θ 2, λ 1, λ 2, ρ, α 1, β 1, α 2, β 2, γ 1, γ 2 } are constant parameters. In this model, l t is the level of a long-term interest rate at time-t, and ρ controls the correlation. The drifts, the squared diffusion terms and the squared prices of risk are affine functions of the two state variables r t and l t. 7 Our second two-factor model is the Brennan and Schwartz (1979) two-factor model, which falls outside of the affine class: dr = r t [α ln(l t / κ r t )] dt + r t σ 1 dw 1, dl = [l 2 t - r t l t + l t σ q 2 l t σ 2 ] dt + l t σ 2 dw 2, (9) q 1, q 2 are constant, E(dw 1 dw 2 ) = ρ dt, and the fixed parameters are {α, κ, σ 1, σ 2, q 1, q 2, ρ}. We also consider a three-factor affine model where a measure of convexity is the third factor. The motivation for the three-factor model is provided by studies such as Litterman and Sheinkman (1988). This model extends equation (8) in the obvious way: There is an additional convexity factor with stochastic differential dc, an additional Wiener process dw 3 and an additional market price of risk for convexity, q 3. The three squared market prices of risk are assumed to be affine in r t, l t and c t, where c t is the 7 Dybvig, Ingersoll and Ross (1996) show that infinite-maturity forward rates and zero coupon yields can never fall, in order to avoid arbitrage in frictionless term structure models. The long rates in the models examined here can move up or down stochastically, so they should be interpreted as rates for a finitematurity index. Empirically we use a ten year fixed-maturity yield as the long rate.

12 10 convexity at time t. 2.4 Empirical SDF Models Affine: The empirical SDF models are written in reduced form, as follows: tm(φ) t+1 = exp(a + b A r t+1 + c[r t+1 -r t ] + d A l t+1 + e[l t+1 -l t ] + f[c t+1 - c t ] + ga c t+1), (10a) Brennan and Schwartz: tm(φ) t+1 = exp(a + b A r t+1 + c A l t+1 + d D r t+1 + e D l t+1 + g D rl t+1), (10b) where: A r t+1 = Σ i=1,...1/ r(t+(i-1) ), A l t+1 = Σ i=1,...1/ l (t+(i-1) ), A c t+1 = Σ i=1,...1/ c(t+(i-1) ), D r t+1 = Σ i=1,...1/ {r(t+i )/r(t+(i-1) ) -1}, D l t+1 = Σ i=1,...1/ {l (t+i )/ l (t+(i-1) ) -1}, D rl t+1 =Σ i=1,...1/ ln[r(t+(i-1) )/ l (t+(i-1) )]. The coefficients {a,b,c,...} are constant functions of the underlying fixed parameters of the models, and they differ across the models. The two-factor affine model is nested in the general, three factor model of (10a) by setting f=g=0. The single-factor affine model is nested in the two-factor affine model by further setting d=e=0. Note that the single-factor model actually depends on two short rate "factors." Because of the effects of time aggregation, there is both a discrete change in the short rate, [r t+1 - r t ], and an average of the daily short-rate levels over the month. The two-factor

13 11 affine model depends on the monthly changes in the long and short rates and on monthly averages of the long rate and short rate levels. The three-factor model adds a discrete change in convexity and an average convexity factor. The Brennan and Schwartz model uses no discrete rate changes at all, only the daily averages D r t+1, D l t+1 and D rl t+1. The time-aggregated version of the Brennan and Schwartz two-factor model actually uses five empirical "factors." (We still refer to the models according to the number of instantaneous factors.) 8 3. Addressing Interim Trading Bias Stochastic discount factors like Equation (4) address the interim trading bias discussed by Goetzmann, Ingersoll and Ivkovic (2000) and Ferson and Khang (2002). This problem arises when discrete period (say, monthly) returns data are available while managers trade during the month. If public information about returns is revealed during the month and a manager trades on this information, Ferson and Khang show that standard performance measures are likely to record "superior" information or ability. Consistent with the literature on conditional performance evaluation, the models here assume that a manager who merely trades mechanically, in reaction to the publicly available state variables, should not record superior performance. A related problem is the nonlinearity in managed portfolio returns induced by dynamic trading or option-like positions. For example, holding a call option-like position may be confused with market timing ability (e.g. Jagannathan and Korajczyk, 1986). The interim trading problem is resolved in this paper in the following way. 8 In continuous time the Brennan and Schwartz model can admit arbitrage when l t is an infinite maturity long rate (Dybvig, Ingersoll and Ross, 1996). However, our reduced-form application of the model does not admit arbitrage based on monthly trading strategies. (No arbitrage is equivalent to the existence of a strictly positive SDF, and the SDF in Equation (10b) is strictly positive.)

14 12 Consider a model with state vector X and assume that an SDF, t m t+, prices the returns on the primitive assets for the period t to t+. Equation (1) implies this SDF must also price all portfolios of the primitive assets formed at time t and held until t+, with weights that may depend on the information generated by the state variables at time t. Similarly, at time t+, there is an SDF t+ m t+2 that prices all portfolios formed at time t+ and held until t+2, with weights that may be any function of X t+. From the Euler equation (1) and the law of iterated expectations, it follows that the SDF t m t+2 = ( t m t+ )( t+ m t+2 ) prices all returns measured over the t to t+2 period, including all dynamic strategies that trade the primitive assets at t and t+ as functions of the public information at these two dates. 9 As approaches zero in the continuous-time limit, the SDF should correctly price all dynamic strategies that are nonanticipating functions of the state variables. Of course, in practice we are limited by the data, and with daily data we implicitly assume that managers trade only at the end of each day. In the term structure models t m t+2 is the product of exponentials which implies summing the exponents, this generating the time-averaged terms. 10 The time-averaged terms eliminate arbitrage opportunities that arise when interim trading is allowed. To see this, consider the special case when there is no risk. Consider two trading strategies over the t to t+1 period. The first strategy holds the "long-term," one-period bond with price 9 Let w(x t ) be a portfolio weight vector that sums to 1.0 and consider the two-period strategy with gross return tr pt+2 =[w(x t )' t R t+ ][w(x t+ )' t+ R t+2 ]. At time t+ the Euler equation implies 1=E t+ { t+ m t+2 [w(x t+ )' t+ R t+2 ]}. At time t we have: 1 = E t { t m t+ [w(x t )' t R t+ ] 1 } = E t { t m t+ [w(x t )' t R t+ ]E t+ { t+ m t+2 [w(x t+ )' t+ R t+2 ]}} = E t {( t m t+ )( t+ m t+2 )[ t R pt+2 ]}. 10 In theory many models can resolve interim trading bias. Consider a consumption-based asset pricing model in which t m t+ = β (C t+ /C t ) -α and C t is the aggregate consumption at time t. This model should price dynamic trading strategies using the SDF t m t+2 = ( t m t+ )( t+ m t+2 ) = β 2 (C t+2 /C t ) -α because the intermediate consumption at t+ cancels out. A similar result would occur whenever t m t+ is a ratio of observables at the two dates.

15 13 E t ( t m t+1 ) = t m t+1 and gross return equal to 1/ t m t+1. The second strategy rolls over shortterm bonds, earning a gross return of exp[ t t+1 r s ds]. To avoid arbitrage the returns of the two strategies must be equal, implying t m t+1 = exp[- t t+1 r s ds]. The no-arbitrage SDF depends on the time-average of the short term interest rate. In the special case of no risk, q(x)=0 and we see that Equation (4) gives the same solution. An SDF with only timeaveraged terms should exactly price a default-free bond of any maturity, assuming no arbitrage and risk neutrality. Time-averaged factors also arise in discrete-time models. Consider the example of a discrete time affine model with state variable, X t, where t m t+ = exp(a+bx t +cx t+ ) and t+ m t+2 = exp(a+bx t+ +cx t+2 ) for some constants {a,b,c}. Therefore, ( t m t+ )( t+ m t+2 ) = exp(2a + 2(b+c)[X t +X t+ ]/2 + c[x t+2 -X t ]). The SDF for the period t to t+2 depends both on the discrete change in the state variable and also on its time-averaged value. The timeaveraged value captures the levels of the state variable over the period where interim trading may occur. 11 Thus, pricing dynamic strategies and avoiding interim trading bias provides intuition for the time-aggregated factors in our empirical SDFs. It is interesting that the models imply a central role for time-averaged data. A common instinct in empirical finance is to focus on end-of-period data. The CRSP data files and many other financial data bases are organized this way. The preceding illustrates that such an approach relies on the assumption that the primitive trading interval in the model is of the same length as the period over which the data are measured. In contrast, many macroeconomic data series are reported as time averages. These are often considered to be biased or "noisy" series, to be adjusted or otherwise viewed with 11 In this example we would not need the time-averaged term to model buy-and-hold bond strategies. The natural log of the price of a discount bond is an affine function of the state vector at each date. The continuously-compounded return therefore depends only on the levels of the state variables at the beginning and end of the period, not on any time averaged terms. (See Gourieroux, Montfort and Polimenis (2002) for a general discussion.)

16 14 suspicion (e.g. Working (1960), Breeden, Gibbons and Litzenberger, 1989). When dynamic trading strategies are involved it should be useful to refine this view. 4. Empirical Methods We estimate the conditional performance of a fund and the parameters of the SDF model simultaneously using the following system of moment conditions and the Generalized Method of Moments (GMM, see Hansen, 1982). E{ [ t m(φ) t+1 R t+1-1] D t } = 0 (11a) E{ [ t m(φ) t+1 R pt α p 'D t ] D t } = 0. (11b) Equation (11a) says that the SDF prices the primitive asset returns, R t+1, while Equation (11b) identifies the fund's SDF alpha, α pt = α p 'D t. D t is the Conditioning Dummy Variable, a vector of (0,1) variables for discrete states of the term structure. For example, we define a conditioning dummy variable indicating whether the term structure slope is "normal," steeper or flatter than normal, as described below. The SDF alpha for a fund then has three distinct values. In this way, we measure the expected abnormal performance of a fund conditional on the slope of the term structure being steep, flat, or normal. Instead of a single alpha representing the average conditional performance of a fund, as in Ferson and Schadt (1996) for example, we allow the conditional performance to vary over time with the state of the term structure. Christopherson et al. (1998) allow time-varying alphas, assuming the conditional alphas are linear functions of lagged instruments. When the instrument is a conditioning dummy variable the performance measure is "nonparametric," avoiding a functional form assumption. Using conditioning dummy variables and a small number of states we obtain simplicity and interpretability. The cost is a coarse representation of the conditioning information. Of course, one can

17 15 define more dummies to refine the information, relative to the examples we use here to illustrate the approach. Farnsworth et. al. (2002) show that estimating a system like (11a, 11b) for one fund at a time produces the same point estimates and standard errors for alpha as a system that includes an arbitrary number of funds. This is convenient, as the number of available funds exceeds the number of monthly time series, and joint estimation with all of the funds is therefore not feasible. 12 Farnsworth, et. al. (2002) find small biases in SDF alphas for equity funds. To the extent that there are biases and they are similar for the fund and a benchmark return R B,t+1, the biases may be controlled by measuring the abnormal performance of a fund relative to that of the benchmark return. This is accomplished by replacing equation (11b) with the expression E{ [ t m(φ) t+1 (R pt+1 -R Bt+1 ) - α p 'D t ] D t } = 0. Measuring performance relative to a benchmark may also increase the precision of the alpha, because the variance of the excess return is smaller than the raw return. Of course, if the model correctly prices the benchmark return, the point estimate of the fund's alpha is not changed by the introduction of the benchmark. We present both versions of the conditional alphas. The number of parameters that can be identified in the reduced form models is always smaller than the number of underlying parameters in the theoretical term structure models, even with the empirical factors that arise from time aggregation. For example, the continuous time one-factor model of Equation (7) has five parameters (four, in the special cases of the Vasicek and Cox-Ingersoll-Ross models), while only three parameters can be identified using the version of (10a) with d=e=f=g=0. It would be possible to incorporate additional moment conditions, derived from the 12 Farnsworth et. al. (2002) provide the invariance result for the special case where there is only a constant in D t, so the alpha is a constant. An Appendix to this paper, available by request, refines and extends the result for a general time-varying alpha.

18 16 theoretical interest rate processes, and thereby identify additional model parameters. 13 However, if we use the extra moment conditions and the interest rate process is misspecified, the misspecification will spill over into the estimated performance measures. To measure performance it is sufficient to work with the smaller number of parameters identified by (10). This means that our approach is robust to some misspecifications of the underlying theoretical interest rate models. 5. The Data First we describe our sample of US government bond mutual funds. We then describe the conditioning dummy variables for the states of the term structure and the behavior of bond returns across the term structure states. A data appendix describes the daily interest rate data that we use to construct the empirical factors implied by the models, the monthly bond return data sources, the dynamic interim-trading strategies that we use for model diagnostics and the mutual fund performance benchmarks. 5.1 Government Bond Mutual Fund Data The government bond fund data are from the Center for Research in Security Prices (CRSP) mutual fund data base. We select funds whose objectives indicate that they are primarily US government bond funds. 14 The number of funds with some monthly return data in a given year is less than 40 prior to Our fund sample therefore starts in 13 For example, in the Cox-Ingersoll-Ross model the first and second moments of the discrete changes, r t+1 - r t, conditional on the current value of the state variable r t, may be expressed as a function of r t and the parameters of the square root interest rate process. We could append these moment conditions to the system to identify all of the model's parameters. See Farnsworth (1997) for an illustration. Sun (1997) shows how a test function approach can be used to estimate all of the parameters of CIR-type models by equating additional empirical moments of interest rates to those implied by the model's parameters. 14 The Government bond funds include the ICDI_OBJ code GS, OBJ codes GOV or LTG, POLICY code of GS or SI_OBJ codes of GGN, GIM, GSM, GMB or GS.

19 17 January of 1986 where the number of funds, based on the objective codes for year-end 1985, is 67. The number of funds rises to 878 in June of Starting in the year 2000 many funds report quarterly data on fund characteristics. For these cases we use only the data for the last quarter of the year (which includes all of the monthly returns). There are a total of 6552 fund-year records. We subject the sample of funds to a number of screens. To address back-fill bias we delete years prior to and including the year of fund organization. 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, 2003). This screen removes 539 records. We also delete all cases where the total net assets of the fund is reported to be less than five million dollars. This removes another 809 records. 15 We delete all cases where the reported equity holdings at the end of the previous year exceeds 10% (555 records), and all cases where the reported holdings of bonds plus cash is less than 85% (1273 records). After these screens we are left with 3376 fund-years. The screens no doubt reduce the cross-sectional variation of performance in our sample, but provide an example that is more consistent with our focus on default-free term structure factors. The number of funds with some monthly return data varies from only one in 1986, three during , 31 in 1989, 168 in 1993 and a high of 306 at the end of We examine an equally weighted portfolio of all the funds from 1986 through We also group the funds into portfolios according to fund characteristics measured at the end of the previous year, starting in 1988 (returns for 1989). The characteristics include fund age, total net assets, percentage cash holdings, income yield, annual turnover, total load charges, annual expense ratios, a measure of total cost (load divided by five plus the 15 Without these last two screens, the youngest third of the funds have higher average returns, by 9 bp per month, lower volatility, and significant positive alphas after risk adjustment.

20 18 annual expense ratio, in percent) and the annual return over the previous year. Each year we sort the funds with nonmissing characteristic data from high to low and break them into thirds. We form equally weighted portfolio returns from the funds in the high group and the low group for each month of the next year. The returns are based on the end-ofmonth net asset values of the funds, plus any distributions. Investors can trade open-end mutual funds at their net asset values per share at the close of each trading day, regardless of when the underlying assets of the funds may trade. Figure 1 plots the end-of-year time-series of the cutoff values for the fund characteristics that define the upper and lower thirds of the distributions. Government bond funds have experienced some trends different from equity funds. The age distribution was the youngest at the beginning of the period, then grew older. In contrast, the age distribution of equity funds over the same period has grown younger, reflecting the large number of new equity funds starting in the mid-1980s (e.g. Ferson and Qian, 2004). The total net assets (TNA) per fund shows a downward trend. This also is different from the pattern for equity funds, where total net assets per fund (among funds with TNA in excess of $5 Million) have trended mildly upwards, according to Ferson and Qian. The turnover distribution is also an interesting contrast to equity funds, in that there is no trend over the period. Equity fund turnover has risen over this period. Figure 1 shows that the government bond funds have similar patterns with respect to their load fees as are found for equity funds, but expenses have behaved differently. Total load fees have declined, with the upper cutoff at 4.5% in 1988, ending at 3% in 2000, and the lower-third cutoff holding steady at zero. Expense ratios have remained relatively stable, with the upper-third cutoff near one percent per year and the lower-third cutoff near 0.8%. The decline in load fees has resulted in a slight decline in the total cost measure over the period. Equity funds, in contrast have experienced increases in both expense ratios and turnover over the period.

21 19 Income yields for the government bond funds diminished over the sample period as interest rates fell. The cutoff values for the low and high thirds were 77 and 92 basis points per year at the end of 1988, narrowing to 50 and 58 basis points in Few funds in this sample report cash holdings larger than zero before 1990, and the distribution remains flat after that, with upper and lower-third cutoffs fluctuating around 1% and 6%, respectively. Finally, the lagged annual returns show more random variation over time than the other characteristics, with relatively little dispersion across funds. The time series highlights the good years for government bond funds in 1989, 1991, 1995 and 2000 and the bad years in 1994 and Summary statistics for the fund returns are reported in Table 1. Not surprisingly, the returns look very different from equity mutual fund returns. The means are all between 0.5 and 0.7% per month and the first-order sample autocorrelations are all about 0.2. The standard deviations are between 0.98 to 1.3% per month, about 1/10 the values of equity style mutual funds. The minimum return for a portfolio including all the funds, in any month since January of 1986, is -3.4%, suffered in April of The minimum return across all the characteristics-grouped portfolios, in any month since January of 1989, is - 3.2%, earned by the high lagged-return funds in March of The highest return, 4.8%, was earned by the low-lagged-return funds in May of October of 1987 was a relatively high return month, where a portfolio of all the government bond funds earned 2.9%. On average the government bond funds earned average returns of 0.60% per month, less than the Lehman Government bond index (described below), which was percent per month over the period. The largest average return differences observed between the high and low characteristic-groups are for high versus low income funds, high versus low expense ratio funds and high versus low lagged return funds. There is no indication of momentum, as the low lagged return funds return slightly more than the high lagged-return funds. The

22 20 larger mean return differences across characteristics are on the order of 7 to 17 basis points per month. The standard errors of the mean are approximately 8 basis points, so none of the differences are strongly significant. The lack of dispersion across the fund groups is consistent with the impression from the lagged return cutoffs in Figure Measuring Term Structure States Most previous studies of conditional performance use a standard set of lagged instruments consisting of dividend yields, Treasury yields and yield spreads, following the empirical studies of Fama and French (1988, 1989), Campbell (1987) and others. One of the appeals of the models used here is that term structure theory suggests the relevant state variables. In the Cox-Ingersoll-Ross and Vasicek models, the level of the short term interest rate is the relevant conditioning information. In the two-factor models we use the short rate and a term spread, measured as the difference between a ten-year and the threemonth yield. In the three-factor model we add convexity as the third state variable. One innovation of our study is the use of conditioning dummy variables, D t, to describe conditional performance. Consider the D t for the monthly short rate series, r t. We first convert the short rate into a deviation from its average level over the last 60 months: x t = r t - (1/60)Σ j=1,...60 r t-j. We then use the last 60 months to estimate a rolling standard deviation, σ(r t ). The dummy variable D t,hi for a "higher than normal" level of the spot rate is defined as the indicator function: I{[x t /σ(r t )] > 1}. Similarly, the dummy variable D t,lo for a "lower than normal" level of the spot rate is I{[x t /σ(r t )] < -1}. The lagged instrument in Equation (11) is defined as: D t = (1,D t,lo,d t,hi ). Dummy variables for the other state variables are similarly defined. Summary statistics for the lagged instruments used to construct the conditioning dummy variables are presented in Panel D of Table 1. The average end-of-month short rate over the period is 0.46% per month, or about 5.6% annualized. The slope of

23 21 the term structure averages 0.13% per month, or about 1.6% annualized, and is about 30 percent less volatile than the short rate. The convexity measure is the least volatile state variable, with about one eighth the volatility of the short rate. On average the term structure is slightly concave, with a mean convexity of -0.02% per month. The average monthly change in both short and long rates is slightly negative over this period, and the average change in convexity is slightly positive. The fact that we observe the fixed income funds over a falling-rate period implies a caveat about our performance analysis. The performance results for funds may not be valid in a rising interest rate environment. We would argue that using factors motivated by theoretical models is likely to be especially important in such a setting, as theoretical factors should have better external validity. Table A1 summarizes data for the longer sample period, August, 1974 through December 2000, which we use for model diagnostics that do not require mutual fund returns. With these data our diagnostics on the models include both rising and falling interest rate environments. Figure 2 presents plots of the three state variables and their associated conditioning dummies over time. The figure shows that the short rate is either normal or low over most of the sample period after 1985, with brief exceptions in 1989 and For the other state variables the high and low states, as we define them, are not concentrated in discrete subperiods, but occur in episodes throughout the sample. A significant feature of the term structure state variables is the high persistence of the short rate and slope, as indicated by the first order sample autocorrelations of 98% and 96%. The averaged short rate, long rate and convexity series, as well as the averaged relative slope D rl, are also highly persistent time series. High persistence raises concerns about small-sample biases (e.g. Stambaugh, 1999) and spurious regression problems (e.g. Ferson, Sarkissian and Simin, 2003). See Bekaert, Hodrick and Marshall (1997) for an analysis of finite sample issues in term structure regressions. One advantage of our

24 22 conditioning dummy variable approach is that the autocorrelations of the dummy variables are smaller than those of the underlying instruments. The maximum first order autocorrelation of a dummy variable, shown in Table 1, is 91% and the rest are below 80%. 16 Panel E of Table 1 shows the correlations among the conditioning dummy variables. The average absolute correlation is less than The three highest absolute correlations are between the high convexity and low slope dummies (0.710), low convexity and high slope (0.563), and the low short rate and low slope dummies (-0.501). 5.3 Bond Returns Across Term Structure States Table 2 presents conditional mean returns and standard errors of the mean for selected bond returns, conditional on the discrete states of the term structure. These results motivate our use of the dummy variables in a conditional analysis. The table shows that over the sample period of Panel B used for model diagnostics, high short-term interest rates predict relatively high and volatile short term bond returns, and lower and more volatile long-term bond returns. This is consistent with studies such as Fama and Schwert (1977) and Ferson (1989). Over the mutual fund sample period starting in 1986, high spot rates are associated with higher subsequent returns on the long-term as well as the short-term bonds. This illustrates again the special nature of the period over which we have mutual fund return data. A steeply sloped term structure is a weak negative signal about next month's short-term bill returns, but predicts high long-term bond returns in 16 Using the conditioning dummy variables lowers the autocorrelation of the lagged instruments in the GMM system, but it does not address the persistence of the variables in the empirical SDF. The main issue is one of finite sample performance, including the accuracy of the asymptotic properties of the GMM estimators in the presence of this persistence. Ferson and Foerster (1994) study finite sample properties of GMM estimators in models with persistent implied SDFs. They find that Hansen's J-statistic rejects too frequently and the coefficient estimators can be unreliable in a two-stage GMM approach, while an iterated GMM approach is reasonably accurate. We use iterated GMM in this paper.

25 23 both sample periods, consistent with Keim and Stambaugh (1986). A high term structure slope is also associated with higher returns on the dynamic strategies buyhiy and buyhir (described in the Appendix), as well as the Lehman index. This predictability reflects departures from the constant-premium version of the expectations hypothesis of the term structure (e.g. Campbell and Shiller, 1991), consistent with consumption-based model predictions as described by Breeden (1986) and Harvey (1988). Table 2 also shows that high convexity is generally associated with lower bond returns. This is consistent with the convexity/return relationship described in Grantier (1988), using a much earlier sample. Some simple calculations illustrate the statistical significance of the differences across the states in Table 2. We compute t-ratios for the null hypothesis that the conditional means are equal in the high and low states. We allow for heterogeneous variances but assume the returns in the high and low states are uncorrelated. This is a conservative but reasonable approximation, because the autocorrelations of the returns are not large and the high and low states are typically separated in time by a normal state, as illustrated in Figure 2. We find that the t-ratios for the differences across states are larger than two, in ten of the 42 comparisons. This is far above the two or three examples we would expect to find by chance. For example, assuming that each of the 42 comparisons is an independent Bernoulli trial with 5% probability, the t-statistic 17 for a sample with ten "rejections" is Alternatively, we use the conservative Bonferroni inequality to evaluate the maximum of the 42 t-ratios, which is The right-tail Bonferroni p-value using the Chi-squared distribution 6.5x10-8. Thus, the bond return patterns across the discrete states are significant and it seems unlikely that our conditioning dummy variables are weak instruments. 17 The calculation is [10/ ] / [.05(.95)/42] 1/2