# Is market price of risk always negative?

I might have a gap in understanding, so clarifying:

Basic pricing equation

$$E(R) = - cov(m, R)$$ where $$R$$ = excess return and $$m$$ = stochastic discount factor

(I think this is continuous case, in discrete case it $$E(R) = - Rf*cov(m, R)$$ where $$Rf$$ = risk free rate)

Modifying the above

$$E(R) = - \frac{cov(m, R)}{var(m)} * var(m)$$

$$= > E(R) = - Beta_{(R, m)} * var(m)$$

And we label $$-var(m) = \lambda (m)$$ = Market price of risk for m.

Since $$var(m)$$ is a positive number, is market price of risk always negative?

• Not in electricity futures, where contracts are often backwardated, due to increasing electricity supply from increasingly cheap sources (renewables). Recall, unlike other commodities, electricity must be consumed immediately and monthly contracts are for monthly consumption rather than point-in-time delivery. Dec 10, 2020 at 2:46

## Summary

Yes, the market price derived from the moments of the SDF is negative. Assets pay high returns (are risky) if they negatively correlate with the SDF (and thus positively correlate with business cycles). Remember that the SDF is high in recessions.

For different specifications of the SDF (e.g. C-CAPM), we get a more intuitive positive market price of consumption risk. The standard CAPM gives a positive market price of risk derived from the market portfolio.

## What does the stochastic discount factor measure?

In most consumption-based models, the SDF relates to marginal utility: if marginal utility is high, so is the SDF. When is marginal utility high? In bad states of nature (recessions), when the level of consumption is low. Marginal utility is kind of how hungry investors are and if consumption is low, people value consumption a lot. Of course, marginal utility decreases with consumption levels (utility functions are concave!). If the economy is booming, the SDF is low.

Essentially, if $$\mathbb{C}\text{ov}\left(R,M\right)>0$$, the returns on your asset covary negatively with business cycles and the general state of the economy. That's the opposite of a risky asset and rightfully results in negative expected returns.

## Economic Intuition

When does an asset have high returns? When it's very risky. An asset is risky if it pays a lot when marginal utility is high and if it pays nothing (or has a negative payoff) if marginal utility is low.

Prime examples are stocks: when the economy is booming and you're doing well in your job, you get high dividends. But when the economy tanks, you lose your job and you need money, then your stock portfolio also loses in value and you're worse off. Thus, stock returns and the SDF correlate negatively.

Insurances, on the other hand, pay you a lot when your house just burned down and your marginal utility high and you really need some extra consumption. Their returns correlate positively with the SDF.

As you see, stocks have high risk premia and insurances have typically negative expected returns.

## Risk Premia and Betas

In discrete time we have \begin{align*} 1=\mathbb{E}_t[M_{t+1}R_{i,t+1}] &=\mathbb{E}_t[M_{t+1}]\mathbb{E}_t[R_{i,t+1}]+ \mathbb{C}\text{ov}_t\left(R_{i,t+1},M_{t+1}\right) \\ \implies\hspace{1cm}\mathbb{E}_t[R_{i,t+1}]-R_{f,t} &= -R_f\mathbb{C}\text{ov}_t\left(R_{i,t+1},M_{t+1}\right) \\ \implies\hspace{1cm}\mathbb{E}_t[R_{i,t+1}]-R_{f,t} &= \frac{\mathbb{C}\text{ov}_t\left(R_{i,t+1},M_{t+1}\right)}{\mathbb{V}\text{ar}_t\left[M_{t+1}\right]}\cdot\left(-\frac{\mathbb{V}\text{ar}_t\left[M_{t+1}\right]}{{\mathbb{E}_t\left[M_{t+1}\right]}}\right) =\beta_{i,t}^M\cdot\lambda_{t}^M, \end{align*} where indeed the market price of risk $$\lambda_t^M<0$$ is negative.

If you prefer continuous time, \begin{align*} \mathbb{E}_t[\text{d}R_{i,t}]-r_{f,t}\text{d}t = -\mathbb{C}\text{ov}_t\left(\text{d}R_{i,t},\frac{\text{d}\Lambda_t}{\Lambda_t}\right)=\frac{\mathbb{C}\text{ov}_t\left(\text{d}R_{i,t},\frac{\text{d}\Lambda_t}{\Lambda_t}\right)}{\mathbb{V}\text{ar}_t\left[\frac{\text{d}\Lambda_t}{\Lambda_t}\right]} \cdot \left(-\mathbb{V}\text{ar}_t\left[\frac{\text{d}\Lambda_t}{\Lambda_t}\right]\right)=\beta_{i,t}^\Lambda \lambda_t^\Lambda, \end{align*} where again $$\lambda_t^\Lambda<0$$.

These equations have several important implications. Most importantly, only covariance with the SDF is priced (contributes to the risk premium). This is often called systematic risk. Idiosyncratic variance bears no premium and is not priced. Thus, a volatile stock does not necessarily need to provide high returns.

## Consumption CAPM

Under power utility, an agent's FOC gives rise to $$M_{t+1}=\beta\left(\frac{u'(c_{t+1})}{u'(c_t)}\right)=\beta \left(\Delta c_{t+1}\right)^{-\gamma}$$ where $$\beta<1$$ is the subjective discount factor and $$\gamma>0$$ is the risk aversion coefficient. Thus, $$M_{t+1}$$ is high is marginal consumption is high and consumption levels are low.

In continuous time, if we assume $$c_t$$ follows a geometric Brownian motion, \begin{align*} \mathbb{E}_t[\text{d}R_{i,t}]-r_{f,t}\text{d}t = -\mathbb{C}\text{ov}_t\left(\text{d}R_{i,t},\frac{\text{d}\Lambda_t}{\Lambda_t}\right)=\gamma\mathbb{C}\text{ov}_t\left(\text{d}R_{i,t},\frac{\text{d}c_t}{c_t}\right). \end{align*}

This motivates the following approximation for discrete time \begin{align*} \mathbb{E}_t[R_{i,t+1}]-R_{f,t} &=\gamma\mathbb{C}\text{ov}_t\left(R_{i,t+1},\Delta c_{t+1}\right), \end{align*}

So again, if your asset has high returns when consumption growth is high, your asset is risky and thus has high returns. Your asset's returns are procyclical. On the other hand, if the asset's returns relate negatively to consumption growth, your asset acts as an insurance and has low returns.

Consider the $$\beta$$ representation of the C-CAPM, \begin{align*} \mathbb{E}_t[R_{i,t+1}]-R_{f,t} &= \frac{\mathbb{C}\text{ov}_t\left(R_{i,t+1},\Delta c_{t+1}\right)}{\mathbb{V}\text{ar}_t\left[\Delta c_{t+1}\right]} \cdot\left(\gamma \mathbb{V}\text{ar}_t\left[\Delta c_{t+1}\right]\right)=\beta_{i,t}^c\cdot\lambda_t^c, \end{align*} where the market price of consumption risk is positive, $$\lambda_t^c>0$$. Also, $$\lambda_t^c$$ is higher if uncertainty is high ($$\mathbb{V}\text{ar}_t\left[\Delta c_{t+1}\right]$$ is high) and if people dislike risk a lot ($$\gamma$$ is high)

• Kev, you are an epic contributor, I enjoy reading your posts. +1 Dec 10, 2020 at 8:21
• @JanStuller Thank you so much, very flattering indeed! :) I think it's a great community and we all learn a great deal from each other :) Dec 10, 2020 at 11:24