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Put stochastic discount factors to work in pricing assets, after showing how they are constructed.

FRM® Part II · Risk Management and Investment Management · Factor Theory · FRM2.IM.1.e

The idea

A stochastic discount factor, written m, is a random variable that prices any payoff x as follows: p = E(m × x). The risk-free gross return, written Rf, is the reciprocal of the average discount factor: Rf = 1 ÷ E(m). Because the price splits into E(m) × E(x) + cov(m, x), an asset's expected excess return, written Excess, is: Excess = −cov(m, R) ÷ E(m). Assets whose payoffs covary negatively with m, meaning they pay little when m is high, earn a positive premium. m is high in bad states, when marginal utility is high. It can be built from preferences, with β the time discount and u′ marginal utility: m = β × u′(c1) ÷ u′(c0). With power utility, where γ is risk aversion and c1 ÷ c0 is consumption growth, this becomes: m = β × (c1 ÷ c0)^(−γ). It can also be built from state prices divided by probabilities, or as a linear function of factors, which reproduces the CAPM and multifactor models. Absence of arbitrage implies that a positive m exists. The usual mistake is ignoring covariance and discounting at the risk-free rate.

Formulas

Price with a stochastic discount factor

p = E(m × x)

m
the stochastic discount factor in each state
x
the asset's payoff in each state

Weight each state by its probability. It equals E(m) × E(x) + cov(m, x), so discounting E(x) at the risk-free rate alone ignores the covariance that carries the risk adjustment.

Gross risk-free return from the discount factor

Rf = 1 ÷ E(m)

m
the stochastic discount factor

A gross return, one plus the rate. A sure unit payoff costs E(m), so the risk-free return is its reciprocal, not E(1 ÷ m).

Expected excess return from the discount factor

Excess = −cov(m, R) ÷ E(m)

cov(m, R)
the covariance of the discount factor with the asset's gross return
E(m)
the expected discount factor

Mind the minus sign: an asset that covaries positively with m pays well in bad states, is a hedge and earns a negative premium.

Discount factor under power utility

m = β × (c1 ÷ c0)^(−γ)

β
the time discount factor
c1
next period's consumption
c0
today's consumption
γ
the coefficient of relative risk aversion

Here β is time preference, not a CAPM beta. Faster consumption growth lowers marginal utility and so lowers m.

A practice question

From the question bank, with the reason each wrong option is wrong.

In SDF pricing, the gross risk-free return 1 + rf equals:

  1. E(m)Why not: Confuses the gross return with the discount factor, which is its inverse.
  2. cov(m, R)Why not: Covariance measures risk adjustment, not the level of the risk-free rate.
  3. E(1 / m)Why not: Takes the expectation of the inverse instead of the inverse of the expectation; these differ by Jensen's inequality.
  4. 1 / E(m) - correct

A riskless unit payoff has price E(m) x 1, so the risk-free gross return is 1 / E(m).

A harder one

There are two equally likely states next year. The SDF is 0.8 in the good state and 1.1 in the bad state. An asset pays USD 100 in the good state and USD 60 in the bad state. Its price is closest to:

  1. USD 146.0Why not: Sums m x payoff across states without multiplying by the probabilities.
  2. USD 73.0 - correct
  3. USD 76.0Why not: Discounts the expected payoff of 80 at the risk-free rate (E(m) = 0.95), ignoring the risk adjustment.
  4. USD 79.0Why not: Attaches the high SDF to the good-state payoff and the low SDF to the bad state.

p = 0.5 x 0.8 x 100 + 0.5 x 1.1 x 60 = 40 + 33 = 73. The bad state has the higher m, so its payoff is worth more per unit.

The hardest tier

Two states next year: expansion (probability 0.6, SDF 0.85) and recession (probability 0.4, SDF 1.20). An asset pays USD 120 in expansion and USD 90 in recession. Its expected return in excess of the risk-free rate is closest to:

  1. 2.4% - correct
  2. 3.4%Why not: Reports the asset's expected return and does not subtract the risk-free rate of 1.0%.
  3. 0.0%Why not: Treats the asset as risk-neutral: discounting its expected payoff at the risk-free rate gives no premium.
  4. 1.8%Why not: Weights the states equally when pricing instead of using probabilities 0.6 and 0.4.

Price = 0.6 x 0.85 x 120 + 0.4 x 1.2 x 90 = 104.4. E(x) = 108, so E(R) = 3.45%. E(m) = 0.99, so rf = 1.01%. Excess return = 2.4%.

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