# Behavioral SDF: modelling sentiment risk premium

With reference to Behavioral Asset Pricing models, I know that the discount factor (or required rate of return) is equal to:

Discount rate = Risk-free rate + Fundamental risk premium + Sentiment risk premium

The first 2 components are the same as in Traditional Finance.

Sentiment Risk Premium, which should capture "not-so-rational" beliefs not captured by other factors, can be proxied by the dispersion of analysts' forecasts. However I couldn't find a paper getting more specific on the calculation of the Sentiment Risk Premium.

Maybe it is not exactly what you are looking for, but you can take a look at this paper by Kozak, Nagel and Santosh. Roughly speaking, we know that the first order conditions of arbitrageurs must be satisfied, i.e. the following Euler equation should be satisfied for any gross return $R_{t+1}^i$ $$1 = \widetilde{E}_t[M_{t+1}R^i_{t+1}] = \sum_{\omega\in\Omega} \widetilde{\pi}(\omega)M_{t+1}(\omega)R^i_{t+1}(\omega)$$ where $M_{t+1}$ is the stochastic discount factor and $\widetilde{\pi}(\omega)$ are subjective probabilities that don't need to coincide with the "true" probabilities $\pi(\omega)$. We can always rewrite the previous equation in terms of "true" probabilities: $$1 = \sum_{\omega\in\Omega} \pi(\omega)\frac{\widetilde{\pi}(\omega)}{\pi(\omega)}M_{t+1}(\omega)R^i_{t+1}(\omega) = E_t\left[\frac{\widetilde{\pi}}{\pi}M_{t+1}R^i_{t+1}\right]$$ This is equivalent to a model with a stochastic discount factor $\widetilde{M}_{t+1} = \frac{\widetilde{\pi}}{\pi}M_{t+1}$. This in turn implies that $$E_t[R^i_{t+1} - R_f] \propto -Cov_t\left(\frac{\widetilde{\pi}}{\pi}M_{t+1}, R^i_{t+1}\right)$$$$= -E_t\left[\frac{\widetilde{\pi}}{\pi}\right]\underbrace{Cov_t\left(M_{t+1}, R^i_{t+1}\right)}_{Fundamental Risk Premium} -E_t\left[M_{t+1}\right]\underbrace{Cov_t\left(\frac{\widetilde{\pi}}{\pi}, R^i_{t+1}\right)}_{Sentiment Risk Premium} - E_t\left[(M_{t+1}-R_f^{-1})\left(\frac{\widetilde{\pi}}{\pi}-E\left[\frac{\widetilde{\pi}}{\pi}\right]\right)(R^i_{t+1}-E[R^i_{t+1}])\right]$$

The paper also show that the Hansen-Jagannathan bound tells us that the maximum squared Sharpe ratio is approximately equal to: $$\max_i \left(\frac{E_t[R_{t+1}^i]-R_f}{\sigma_{it}}\right)^2 \approx Var_t\left(\widetilde{M}_{t+1}\right)= Var_t\left(\frac{\widetilde{\pi}}{\pi}M_{t+1}\right)$$ To avoid "near-arbitrage opportunities" (i.e. too high Sharpe ratios) we need to require $Var_t\left(\frac{\widetilde{\pi}}{\pi}M_{t+1}\right)$ to be relatively low (say between $0.5^2$ and $1.5^2$ in annual terms). The main result of Kozak et al. is to show that low Variance implies a strong factor structure in returns. This, in turn, does not tell us anything regarding the sources of premia, i.e. whether it comes from fundamentals or from wrong beliefs.

Kozak et al. use analyst forecast biases as a proxy for $\frac{\widetilde{\pi}}{\pi}$ and show that they align with the principal components in equity returns, implying that they represent a significant part in the determination of risk premia.

• @fnic, amazing! Just a quick question: shouldn't there be the risk-free rate multiplied to the covariance in the third equation? $$E_t[R^i_{t+1}−R_f] = − R_f × Cov_t(...)$$ May 9, 2018 at 22:19
• Yes, you're right. I've changed $=$ with $\propto$
– fni
May 9, 2018 at 23:45
• super-helpful 😊 I confess I need to do some homework to get my head around that covariance decomposition 😅 May 10, 2018 at 8:06
• I believe $Cov(XY,Z) = E[X]COV(Y,Z) + E[Y]Cov(X,Z)+ E[(X-\mu_X)(Y-\mu_Y)(Z-\mu_Z)]$
– fni
May 10, 2018 at 20:07