Historical simulation approach to the estimation of stochastic discount factor models

Abstract

We propose an approach to the estimation of the parameters of stochastic discount factor (SDF) models which is based on the idea that the next period joint distribution of the variables in a SDF and asset returns can be well approximated by their joint historical distribution. The estimates of the SDF parameters may therefore be found as the values of the parameters at which the mean of the historical distribution of the product of the SDF with an asset return equals one. Each time period, the estimates are updated using the most recent periods of data and hence can change over time. This method can be viewed as an alternative to the approaches that specify a particular functional form relating the SDF parameters to proxies for the state of the world.

Keywords:

• Asset pricing
• Applied econometrics
• Applied finance
• Capital asset pricing

Acknowledgements

I thank Narayana Kocherlakota and the three anonymous referees for their comments and constructive suggestions. I am also grateful to the Social Sciences and Humanities Research Council of Canada (SSHRC) for financial support.

Notes

†In the Value-at-Risk literature, the historical simulation approach consists of assuming that the historical distribution of an asset return can be viewed as an adequate proxy for the unconditional distribution of the next period returns.

‡If the probability of at least one state is zero, the investor will short this state security and no unconstrained equilibrium will result.

†When using the consumption CAPM with power utility, Bakshi and Chen (1996) and Gordon and St-Amour (2004), for example, assume the coefficient of RRA to be a decreasing function of the individual's wealth, while Sundaresan (1989), Constantinides (1990) and Campbell and Cochrane (1999) introduce time-varying attitudes towards risk through habit formation. Becker and Mulligan (1997) assume that the time preference discount factor is an increasing and concave function of the resources expended on activities devoted to expanding the imagination.

‡The forward joint distribution of the SDF variables and asset returns can be derived directly from their joint historical distribution or may be assumed to be analytic. When the pricing kernel comes from the absence of arbitrage condition for the optimal consumption and savings choice problem, for example, some researchers assume joint conditional lognormality and homoskedasticity of asset returns and consumption growth when estimating the parameters of the SDF in the consumption CAPM (for example, Hansen and Singleton (1983)). Because asset returns are not, in fact, conditionally lognormally distributed and homoskedastic, this assumption can lead to unreliable estimates of the SDF parameters.

†See also Tversky and Kahneman (1974) and Carroll (1978), for example.

‡See Boudoukh et al. (1998) for more details on weighted historical simulation.

§As w  → 1, π _t−s  → 1/S and thus weighted historical simulation becomes equivalent to historical simulation.

†See Epstein and Zin (1989, 1991). When  ϕ = 1, we obtain the model for the representative investor with power utility, i.e. E _t−1[δ(C _t /C _t−1)^α−1 R _i,t ]  = 1. In the case of logarithmic risk preferences (α  = 0,  ρ ≠ 0), we obtain .

†Because we use the consumption growth rate lagged one period as an instrument and the first observation in our data set corresponds to January 1959, the first 240-month estimation window covers the sample period February 1959 through January 1979.

†Stock and Wright (2000) also encountered this problem when estimating the consumption CAPM with Epstein–Zin preferences by GMM using U.S. monthly data from January 1959 to December 1990.

†The returns on the five NYSE, AMEX, and NASDAQ industry portfolios are from Kenneth R. French's web page.

‡Campbell et al. (1997) argue that the small variation in conditional expected returns and consumption growth makes it adequate to calculate the moments of innovations in the series as the moments of the raw series.

†The first value of the consumption growth rate lagged one period that we take as an instrument will correspond to that in January 1959.

‡There are two opposite effects of the increase in the risk-free rate on consumption growth. On the one hand, as the risk-free rate goes up, future consumption becomes cheaper relative to current consumption. This leads an agent to reduce his current consumption and increase savings in order to consume more in the next time period. This is the substitution effect. On the other hand, a raise of the risk-free rate increases overall wealth, which leads to an increase in current consumption and a decrease in savings. This is the income effect, leading to the smoothing of consumption over time. When the elasticity of intertemporal substitution is greater than unity, the substitution effect is dominant.

†When doing this exercise, we do not consider the case when the length of the estimation window is 240 because, in this case, the estimates of the coefficient of RRA are very imprecise and therefore are not very informative.

‡Since the SDF is unique for any asset (or portfolio of assets), we can use the estimates of the agent's preference parameters obtained from the joint estimation of the Euler equations for the 10 industry portfolios and the risk-free return to forecast the return on any traded asset (or portfolio of assets), including the return on the value-weighted market portfolio.

§To compare the forecasting abilities of different models, Lettau and Ludvigson (2001), for example, use the mean-squared error from the set of one-step-ahead forecasts. Akgiray (1989) and Balaban et al. (2006) evaluate the out-of-sample volatility forecasts of stock returns using the mean absolute error. The advantage of the mean absolute percentage error is that it produces a measure of relative overall fit (clearly, the same forecast error is less important when the absolute value of the variable of interest is greater). We follow Balaban et al. (2006) and Hsieh and Ritchken (2005) and employ the mean absolute percentage error to evaluate the model performance.

†, m  = 5,10. is asymptotically distributed as .