A concrete example of a stochastic discount factor

Question

Asset pricing uses the concept of a stochastic discount factor (SDF). I have read various things about it but have not seen a concrete example. Could you give a concrete example of an SDF, e.g. one that has been estimated in an academic paper or being used by practitioners? (E.g. could it be F-distributed? Lognormally distributed? Something else?)

Answer 1

Simplest example:

• Consinder a household with utility function \begin{align} U=\mathbb{E} \int_0^\infty e^{-\beta t}\frac{C_t^{1-\gamma}}{1-\gamma}\text{d}t \end{align}

• The pricing kernel (SDF) is \begin{align} \Lambda_t=e^{-\beta t}C_t^{-\gamma} \end{align}

• Expected stock returns are \begin{align} \mathbb{E}_t[\text{d}R_t]=r_f\text{d}t + \gamma\mathbb{E}_t\left[\frac{\text{d}C_t}{C_t}\text{d}R_t\right] \end{align}

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Assume iid consumption growth: $$\frac{\text{d}C_t}{C_t}=\mu\text{d}t+\sigma\text{d}W_t$$ Then, $C_t$ is log-normally distributed and so is $C_t^{-\gamma}$ and so is $e^{-\beta t}C_t^{-\gamma}$. Put differently, \begin{align} \ln(\Lambda_t)&=-\beta t-\gamma\ln(C_t) \\ &=-\beta t-\gamma\left(\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right) \end{align} Thus, the probability distribution of this SDF is \begin{align} \ln(\Lambda_t)\sim N\left(-\beta t-\gamma\left(\mu-\frac{1}{2}\sigma^2\right)t,\sigma^2t\right) \end{align}