Consumption Risk, Payoff Covariance, and Asset Prices

# Covariance between payoff and consumption in consumption model from Asset Pricing


# Covariance between payoff and consumption in consumption model from Asset Pricing












I have a question from reading the the first chapter of Asset Pricing by John H. Cochrane.

The basic setup for this chapter is that an investor wants to maximize his utility from consumption at time $t$ and $t+1$. Let $e_t$ and $e_{t+1}$ be his "original" consumption if he doesn't buy any investment asset. I interpret this in my head as his fixed stream of income at time $t$ and $t+1$, e.g. his salary, which he can spend all on consumption. Also let $u$ be the utility function and $\beta$ be his subjective discount factor, which expresses his impatience for utility.

So at time $t$, he has a maximization problem:

$$ \max_\xi u(c_t) + E_t[\beta u(c_{t+1})] $$ where $$ c_t = e_t - p_i\xi \\ c_{t+1} = e_{t+1} + x_{t+1}\xi $$ and $\xi$ is the number of units of the asset he buys, $p_t$ is the price, and $x_{t+1}$ is the payoff at time $t+1$. He needs to choose an optimal number $\xi$. I think $E_t$ is the expectation based on the information available at time $t$.

Assuming the derivative of the objective function with respect to $\xi$ at the optimal $\xi$ is $0$, we get $$ p_t = E_t\Big[\beta \frac{u'(c_{t+1})}{u'(c_t)}x_{t+1} \Big]. $$

Set (stochastic discount factor) $m = \beta \frac{u'(c_{t+1})}{u'(c_t)}$ and (risk-free rate) $R^f = 1/E[m]$ and (covariance) $cov(m, x) = E[mx] - E[m]E[x]$. (The subscripts were dropped to make nicer notations.)

With a bit of algebraic manipulation: $$ p = \frac{E[x]}{R^f} + cov(m, x). $$

Rewrite the last equation by substituting in the definition of $m$: $$ p = \frac{E[x]}{R^f} + \frac{ cov[\beta u'(c_{t+1}), x_{t+1}] }{ u'(c_t) }. $$ (I think we can pull $u'(c_t)$ out of the $cov$ or $E$ operator because we know $c_t$ at time $t$.) What confuses me is the explanation that comes after this equation.

> Marginal utility $u'(c)$ declines as $c$ rises. Thus, an asset's price is lowered if its payoff covaries positively with consumption.

How can the payoff $x_{t+1}$ not covary positively with the consumption $c_{t+1}$ when $c_{t+1} = e_{t+1} + x_{t+1}\xi$?

## Answer by mehman (score 0)

https://quant.stackexchange.com/a/82210

It is possible under the assumption of consumption smoothing. It is not directly visible under the one period model, but investors engage in consumption smoothing across time, i.e. they dislike the volatility of the consumption. Hence, if $c_{t+1}$ is predetermined by the investor for every state of the nature $s$ - it does necessarily have to be the same across all states $s$ -, then it might covary negatively or positively with $x_{t+1}$.