# 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?)

• In the CCAPM, $M_{t,t+1}=\beta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}$. Using data on aggregate consumption and using parameter values for $\beta$ and $\gamma$, you get realisations of the SDF. Clearly, this is not a very sophisticated model and it doesn't fit the data well, but it is a very important model and it's arguably the simplest SDF. Commented Dec 20, 2022 at 19:50
• @Kevin, hm, let me see if I understand this. $M_{t,t+1}$ conditional on the information set $I_t$ is a random variable because $\beta$, $\gamma$ and $C_{t}$ are scalars when conditioned on $I_t$, and $C_{t+1}$ is a random variable. Since we have a single realization of $C_{t+1}$ in the data, we get a single realization of $M_{t+1}$. That is not much of an estimate of a random variable; we would want a distribution, would we not? But perhaps we could impose some structure on the time evolution of $M$ to borrow information from across time periods, as we usually do in time series models. Commented Dec 20, 2022 at 20:24
• You're right with everything you're saying. If we make assumptions about the distribution of consumption growth, we get the sought distribution of $M_{t,t+1}$. For example, you could assume that $C_t$ follows a geometric Brownian motion (iid consumption growth). Again, these are the simplest possible assumptions, not the most realistic ones. Commented Dec 20, 2022 at 21:35
• @Kevin, OK, so if we employ such an assumption, can we obtain an analytical expression of $M_{t,t+1}$? What would it be? Commented Dec 21, 2022 at 7:41
• What do you mean with "analytical expression"? The SDF is a random variable (better: a stochastic processs). I would call $M_{t,t+1}=\beta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}$ an analytical expression. Depending on your assumption about the distribution of consumption growth, you can then derive the probability distribution of the SDF. Commented Dec 21, 2022 at 8:41

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}

• Thanks! This is great. I would not derive utility from expectations, though. Utility of actual consumption would sound more reasonable. Now, is your example compatible with the CAPM? If not, would the simplest case that is in line with the CAPM be much more complicated? Commented Dec 21, 2022 at 9:28
• $U$ is more like the objective function of the agent. We should better call it expected lifetime utility from consumption'' (assuming the agent never dies). Commented Dec 21, 2022 at 9:32
• The example is totally compatible with the CAPM. In fact, this setup is a common way to derive the CAPM :) Commented Dec 21, 2022 at 9:33
• Frankly, none. The model is known poorly describe the data. The CAPM/CCAPM just don't work empirically. We want $\gamma$ to be small (1-5), $\sigma$ is low in the data (1-2%), .. but these parameters just don't fit the data. We need better models (habits, long run risks, disasters, production, etc.) These models are great to understand concepts, run simulations, and get an idea what an SDF looks like. But the CAPM/CCAPM aren't great at explaining asset prices Commented Dec 21, 2022 at 9:41
• @RichardHardy Perhaps these answers (1 and 2) also help. They are about plotting the SDF for a concrete example (Black-Scholes option pricing). Commented Dec 24, 2022 at 19:23