# Capital Asset Pricing Modeling for Consumption

Suppose Asset $A$ has a variance of 7 and its returns are negatively correlated with consumption, while Asset $B$ has a variance of 3 and its returns are uncorrelated with consumption.

According to the Consumption CAPM, which asset should yield the higher return?

I believe it is Asset $A$ because CAPM asserts that you should choose an asset that is negatively correlated with consumption.

What do you guys think about my answer?

• What does choosing have to do with higher returns? Nov 16, 2015 at 21:21
• @denesp Returns depend on price, price depends on consumption choices. Nov 17, 2015 at 2:12

## 1 Answer

I guess the question is : which asset would yield the higher risk premium or expected return?

Risk premium depends indeed on the intrinsic characteristics of the asset, and on the pricing kernel, in particular the covariance between the pricing kernel (which is proportional to marginal utility) and the asset return. Therefor the risk premium will depend on the utility function you specify. But the general idea is that an asset that pays high when consumption is high will have a lower price and then a higher return to compensate for that risk. Under complete market, the general formula is : $$E(R^{e}_{t+1}) = -R_f\cdot cov(m, R^e_{t+1})$$

where $R^e_{t+1}$ is the excess return, $R_f$ the risk-free rate, and $m$ the stochastic discount factor. $m$ will depend on the utility you specify but will be proportional to the marginal utility. Because of the (generally assumed) concavity of utility function, $m$ will be low when consumption is high and high when consumption is low.

In your example, the expected excess return of asset B would be 0, otherwise stated the expected return of asset B would be the risk-free rate because its return is uncorrelated with consumption.

Asset A pays high when consumption is low, so when marginal utility is high, so the risk premium will be lower than the risk-free rate. The main idea is that since this asset can be use as insurance against variation in consumption, agents will want to hold it, so its price will be higher and then its return lower.

So you were almost right, asset A will have a higher price than B, but a lower expected return.