# Some basic Consumption CAPM questions

Say we are in a world described by the consumption CAPM. All investors in this world have quadratic utility. Also, assume that consumption is as follows: $$c_{t+1} = (1+m_t)c_t + s_t c_t e_t$$ where the shocks $$e_t$$ are iid with $$E_t(e_{t+1})=0$$ and $$V_t(e_t) = 1$$. Now $$m_t$$ and $$s_t$$ depend on the business cycle so that in down times $$m_t$$ and $$s_t$$ are high, but in boom periods $$m_t$$ and $$s_t$$ are low. Assume that the risk-free rate is independent of time.

I have a few questions as follows and was wondering if my answers make sense or if someone can correct me.

Say there is a company (e.g., luxury goods) that is very sensitive to the business cycle. Will this company have high or low returns?

My answer: Investors don't like uncertainty about consumption. You consume more luxury good in good times and less in bad times. Since it varies positively with consumption, you will require a low price and hence luxury goods will have high returns. Is my explanation right here?

Now consider a security that provides insurance against bad consumption shocks so that in down periods it pays off more. What can we say about the price of this security? Will it have a high/low price when consumption is high? Will it have high/low price during a down period?

My answer: This security would smooth consumption since it varies negatively with consumption. It pays off good in bad times and does not pay off in good times. As a result during a recession/down period, this security would have a high price and during a boom (when consumption is high), it will have a low price. Is my explanation right here?

Using the consumption CAPM, rank the expected returns on the following types of companies: i) computer software; ii) luxury goods; iii) basic consumer products; iv) oil and gas; v) high-growth company.

We need to compare the consumption beta of each company. The higher the consumption beta, the higher the expected return, all else equal. But I'm not quite sure how to roughly compare the betas of the above companies. Could anyone help?

Now for the above ranking, consider other risk factors that could explain expected returns, e.g., Fama-French 3 factors etc. How would the rankings change?

Again, I know I need to think about the covariance between the returns of the companies and how they co-vary with the risk factors, but I am unsure of the direction of the covariance. Could someone help here?