Computational Issues in the Stochastic Discount Factor Framework for Equity Risk Premium

Abstract

The Stochastic Discount Factor (SDF) methodology is a general and convenient framework for asset pricing. SDF encapsulates all the modeling uncertainties and its advantage is that we do not require the knowledge of investors’ preferences. Suitable specification of SDF is, therefore, critical. It has been based on single or multiple factors and also on observable factors as well as latent factors. The variables required to proxy for the factors may be both macroeconomic as well as behavioral. In this article we show how we can incorporate such variables for empirical implementation of equity risk premium with daily frequency. Practical issues crop up to define the dependence between the asset return and the SDF. Here we show how copula can be used in this context and solve some of the analytical complexities for software implementation.

Appendix A: Bivariate Log-Normal Copula

The intrinsic relations between bivariate distributions and their marginal distributions can be clearly characterised by copulas. The bivariate log-normal distributions play important roles in areas other than being described here. In this appendix, log-normal copula is derived. These formulas are in terms of the Gaussian Q-function, being supported by MATLAB. Thus the copula evaluation process can be expedited both analytically and numerically.

The bivariate log-normal probability density function for a pair of random variables $X$ and $Y$, is given by (A.3), where $(x,y)$ are transformed from bivariate normal distribution variables $(x',y')$ as:

$$\begin{aligned} x=A \exp (mx'), \end{aligned}$$

(A.1)

$$\begin{aligned} y=B \exp (ny'). \end{aligned}$$

(A.2)

With $\rho$ as correlation between $(x',y')$ and $(x>0, y>0, A>0, B>0)$:

$$\begin{aligned} f_{x,y}(x,y)&=\frac{1}{2mn\pi \sigma _X\sigma _Yxy\sqrt{1-\rho ^2}}\times \exp \bigg \{\frac{-1}{2(1-\rho ^2)}\bigg [\Big (\frac{\ln (x/A)-m\mu _X}{m\sigma _X}\Big )^2 \\&\quad -2\rho \Big (\frac{\ln (x/A)-m\mu _X}{m\sigma _X}\Big )\Big (\frac{\ln (y/B)-n\mu _Y}{n\sigma _Y}\Big )+\Big (\frac{\ln (y/B)-n\mu _Y}{n\sigma _Y}\Big )^2\bigg ]\bigg \} \end{aligned}$$

(A.3)

Referring to (A.1) and (A.2), $x$ and $y$ are log-normal variables. With respect to the SDF formulation, we may consider $x$ as the SDF and $y$ as the gross return as discussed in Sect. 3. In this formulation, $x'$ may be represented as a function of observable determinants and similarly for $y'$. The additional parameters $(A,m,B,n)$ in (A.1) and (A.2) are used to make these transformations as general as possible.

The marginal PDFs associated with (A.3) take the following forms:

$$\begin{aligned} f_X(x)=\frac{1}{mx\sigma _X\sqrt{2\pi }}\exp \bigg [-\frac{1}{2}\bigg (\frac{\ln (x/A)-m\mu _X}{m\sigma _X}\bigg )^2\bigg ], \end{aligned}$$

(A.4)

$$\begin{aligned} f_Y(y)=\frac{1}{my\sigma _Y\sqrt{2\pi }}\exp \bigg [-\frac{1}{2}\bigg (\frac{\ln (y/B)-n\mu _Y}{n\sigma _Y}\bigg )^2\bigg ]. \end{aligned}$$

(A.5)

The corresponding marginal distributions are:

$$\begin{aligned} F_X(x)=1-Q\bigg (\frac{\ln (x/A)-m\mu _X}{m\sigma _X}\bigg ), \end{aligned}$$

(A.6)

$$\begin{aligned} F_Y(y)=1-Q\bigg (\frac{\ln (y/B)-n\mu _Y}{n\sigma _Y}\bigg ). \end{aligned}$$

(A.7)

Here, $Q(\cdot )$ is referred to as the Gaussian Q-function with the following definition:

$$\begin{aligned} Q(z)=\frac{1}{2\pi }\int \limits ^\infty _z \exp \Big (-\frac{t^2}{2}\Big )dt. \end{aligned}$$

(A.8)

With change of variable, the Gaussian Q-function may be written as (over finite interval):

$$\begin{aligned} Q(z)=\frac{1}{\pi }\int \limits ^{\pi /2}_z \exp \Big (-\frac{z^2}{2\sin ^2\theta }\Big )d\theta . \end{aligned}$$

(A.9)

here $Q(\cdot )$ is computable using MATLAB built in function. For specific values of $m (=2)$ and $n (=2)$ the elements of the covariance matrix for the pair of random variables $X$ and $Y$ are given below:

$$\begin{aligned} \mathrm{var}(X)=A^2\exp (4\mu _X)\exp (4\sigma ^2_X)[\exp (4\sigma ^2_X)-1], \end{aligned}$$

(A.10)

$$\begin{aligned} \mathrm{var}(Y)=B^2\exp (4\mu _Y)\exp (4\sigma ^2_Y)[\exp (4\sigma ^2_Y)-1], \end{aligned}$$

(A.11)

$$\begin{aligned} \mathrm{cov}(X,Y)=AB\exp (2\mu _X+2\mu _Y)\exp (2\sigma ^2_X+2\sigma ^2_Y)[\exp (4\rho \sigma _X\sigma _Y)-1]. \end{aligned}$$

(A.12)

From (A.6) and (A.7) we can write:

$$\begin{aligned} x=F^{-1}_X(u)=A\exp [m\mu _X+m\sigma _XQ^{-1}(1-u)]:= a_1, \end{aligned}$$

(A.13)

$$\begin{aligned} y=F^{-1}_Y(w)=B\exp [n\mu _Y+n\sigma _YQ^{-1}(1-w)]:= a_2, \end{aligned}$$

(A.14)

where $Q^{-1}(\cdot )$ is the inverse Gaussian Q-function and this is available as a built-in function in MATLAB. Now, we are in a position to write the bivariate log-normal copula distribution function as follows:

$$\begin{aligned} C(u,w)=F_{XY}[F^{-1}_X(u),F^{-1}_Y(w)]=F_{XY}(a_1,a_2)=\int \limits ^{a_1}_0\int \limits ^{a_2}_0f_{X,Y}(x,y)dydx. \end{aligned}$$

(A.15)

The simplification of the double integral in (A.15) is at the core issue in empirical implementation. The quantity $f_{X,Y}(x,y)$ is given by Eq. (A.3) and Liu (2010) shows how to convert this to a single integral which can be readily implemented in MATLAB. In this appendix we just quote the final expression for the copula distribution function.

$$\begin{aligned} C_{u,w}=\frac{1}{\sqrt{2\pi }}\int \limits ^{Q^{-1}(1-u)}_{-x}\exp \Big (-\frac{r^2}{2}\Big )Q\Big (\frac{\rho r-Q^{-1}(1-w)}{\sqrt{1-\rho ^2}}\Big )dr \end{aligned}$$

(A.16)

$$\begin{aligned} (0<u<1,0<w<1). \end{aligned}$$

Liu (2010) suggests further avenue to reduce computational burden.