Deriving a Lognormal Stochastic Discount Factor from Consumption

Summary

The document gives a concrete consumption-based asset-pricing example of a stochastic discount factor (SDF). A household with constant relative risk aversion has a pricing kernel equal to discounted marginal utility: the discount factor over time multiplied by consumption raised to the negative risk-aversion parameter. The example connects expected stock returns to the risk-free rate and the covariance between consumption growth and returns.

Assuming consumption follows a process with constant drift and volatility and Brownian shocks, consumption is lognormally distributed. Its negative power, and therefore the SDF at a fixed time, is also lognormal; the document gives the corresponding normal distribution for the SDF's logarithm. This illustrates how an economic model implies a distribution rather than treating the SDF as an arbitrary named distribution. The result depends on the stated process and utility assumptions; it is a theoretical example, not an empirical estimate or evidence that observed SDFs are generally lognormal.

Key ideas

• A household's marginal utility provides a consumption-based pricing kernel.
• The SDF combines time discounting with consumption raised to negative risk aversion.
• Expected returns depend on the covariance between consumption growth and asset returns.
• Under the assumed diffusion process, consumption and the resulting SDF are lognormally distributed.
• The distributional conclusion depends on the model assumptions.

Tags

# A concrete example of a stochastic discount factor


# A concrete example of a stochastic discount factor












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 by Kevin (score 3, accepted)

https://quant.stackexchange.com/a/74150

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}

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}

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.