# 6 Bond fund performance measurement.

6.1 Fixed Income Models

Elton et al. (1993; 1995) were the seminal academic studies of the performance of bond style mutual funds. They used versions of the classical multibeta model alphas described above, where the factors are selected to address the risks most likely to be important for fixed income portfolios.

Ferson et al. (2006a) brought modern term structure models to the problem of measuring bond fund performance. These 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 = [micro]([X.sub.t])dt + [sigma]([X.sub.t])dw, (6.1)

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 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. Term structure models based on Equation (6.1) imply stochastic discount factors of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.2)

where [r.sub.s] is the instantaneous interest rate at time s. The notation [sub.t][m.sub.t+1] is chosen to emphasize that the SDF refers to a discrete time interval, say 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.sub.t+1] and [C.sub.t+1] for each state variable. A particular term structure model specifies the state variables, X, and the functions [micro](*),[sigma](*), and q(*).

One advantage of a continuous time model is that the SDF should correctly price dynamic strategies that trade as functions of the state variables, [X.sub.t]. In particular, the monthly returns of mutual funds that result from interim trading can be handled without concern. Thus, the interim trading biases discussed above should not be a problem in this class of models.

Ferson et al. (2006a) use a first-order Euler approximation to Equation (6.1) in order to implement the models:

X(t + [DELTA]) - X(t) [approximately equal to] [micro]([X.sub.t])[DELTA] + [sigma]([X.sub.t])[w(t + [DELTA]) - w(t)]. (6.3)

The period between t and t + 1 is divided into 1/[DELTA] increments of length [DELTA]. The period is one month, to match the mutual fund returns, divided into increments of one day. For a given model, we can use daily data on X(t + [DELTA]) and X(t) and the functions [micro]([X.sub.t]) and [sigma]([X.sub.t]) are specified. The approximate daily values of [w(t + [DELTA]) - w(t)] are inferred from Equation (6.3). The terms [A.sub.t+1], [B.sub.t+1] and [C.sub.t+1] in Equation (6.2) are approximated using daily data by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.4)

Farnsworth (1997) and Stanton (1997) evaluate the accuracy of these first order approximation schemes. Stanton concludes that with daily data, the 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. He also evaluates higher-order approximation schemes, and finds that with daily data they offer negligible improvements.

Ferson et al. (2006a) show that a popular class of three-factor affine term structure models, include the models of Cox et al. (1985) and Vasicek (1977) as special cases, imply the following reduced form expression for the approximated SDF:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6.5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The coefficients {a,b, c ... } are constant functions of the underlying fixed parameters of the models. The two-factor affine model is nested in the general three factor model of 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.sub.t+1] - [r.sub.t]], and an average of the daily short-rate levels over the month. The two-factor 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 time averaged terms in (6.5) are needed to control the interim trading bias. Thus the OE portfolio needs to include maximum correlation portfolios for time-averaged factors, in order to correct for interim trading biases. In practice we are limited by the data, and with daily data it is implicitly assumed that managers trade only at the end of each day. Funds actually engage in intradaily trading, of course, so the accuracy of this approach, using available data, remains an open empirical question.
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