Financial Economics — Expected Utility, Risk Premiums and the Stochastic Discount Factor

Financial Economics — Risky Cash Flows With the Same Expected Value Have Different Prices

Dividing the state prices $q_s$ of chapter 1 by probabilities gives the stochastic discount factor (SDF). This chapter looks at why that value is larger in recessions, and at which assumptions make the CAPM used in corporate finance a special case.

1. One won is dearer in a state of high marginal utility

Suppose there is a representative consumer with time-separable utility. The price of giving up 1 won of consumption today to get 1 more won in state $s$ is the ratio of marginal utilities.

Stochastic discount factor

m_s = \beta u'(c_s) / u'(c_0)

In states such as recessions, where consumption is low and marginal utility high, m is larger, and cash flows in that state receive a higher present value.

The price of an asset is the expected value of its cash flow times $m$ .

Asset price

p = E[m x]

Assets whose x is large in recessions (insurance, government bonds) have a large E[m x], so their prices are high and expected returns low.

Rewritten in terms of expected returns, $E[R] - r_f = -(1+r_f)\,\mathrm{Cov}(m, R)$ . Assets that move with the market have a negative covariance with $m$ , so their risk premium is positive. The risk that enters prices is not “high variance” but “losing money in bad states”.

2. In numbers, insurance is dearer than equity

Let the risk-free rate be 2% and the probabilities of boom and recession 50% each. In the boom $m=0.90$ and in the recession $m=1.06$ . The mean is $E[m]=0.98$ , so $1/E[m]≈1.0204$ , consistent with the risk-free rate.

A share pays 140 won in the boom and 70 won in the recession. Its price is $0.5×(0.90×140 + 1.06×70) = 100.1$ won. The expected cash flow is 105 won, but the price is lower. An insurance policy pays 0 won in the boom and 100 won in the recession. Its price is $0.5×1.06×100 = 53$ won, more than the expected cash flow of 50 won. Even with the same expected value of 50 won, a payoff concentrated in recessions is bought at a premium.

The channels through which risk enters prices
Measure	What it captures	What it misses
Variance, standard deviation	The size of fluctuations	In which states the fluctuations occur
Market beta	How much the asset moves with the market	Income risk outside the market
Covariance with the SDF	Losses in states of high marginal utility	Cases where preference or complete-market assumptions fail

3. The CAPM arises when the SDF is linear in the market return

With mean-variance preferences or normally distributed returns, choices depend only on the mean and variance. The SDF is then linear in the return on the market portfolio, and risk premiums are proportional to beta.

CAPM

E[R_i] = r_f + \beta_i (E[R_m]-r_f)

Beta is the covariance of the asset with the market divided by the market variance. The claim is that idiosyncratic risk, once diversified, is not priced.

How firms use the CAPM in practice is covered in corporate finance chapter 4. The important limits here are three: the market portfolio cannot be observed, consumption risk can differ from market returns, and non-traded risks such as labour income and housing remain outside beta.

Check your understanding

Is the expected return on an asset that moves against the market higher than the risk-free rate? No. It pays off in states of low consumption, so it has a positive covariance with the SDF; its risk premium is negative and its expected return lower than the risk-free rate. The insurance in section 2 is bought for 53 won and pays 50 won on average, an expected return of about −5.7%.

References

John Cochrane, Asset Pricing, ch. 1
Rajnish Mehra and Edward Prescott, “The Equity Premium: A Puzzle,” Journal of Monetary Economics (1985)
MIT OpenCourseWare, 15.401 Finance Theory