Stochastic discount factor

A Stochastic discount factor (SDF) is a concept in financial economics and mathematical finance.

If there are n assets with initial prices $p_1, ..., p_n$ at the beginning of a period and payoffs $\tilde{x}_1, ..., \tilde{x}_n$ at the end of the period (all x's are random variables), then SDF is any random variable $\tilde{m}$ satisfying

$E(\tilde{m}\tilde{x}_i) = p_i, \quad \forall i.$

This definition is of fundamental importance in asset pricing. The name "stochastic discount factor" reflects the fact that the price of an asset can be computed by "discounting" the future cash flow $\tilde{x}_i$ by the stochastic factor $\tilde{m}$ and then taking the expectation.^[1]

Properties

If each $p_i$ is positive, by using $R_i = \tilde{x}_i / p_i$ to denote the return, we can rewrite the definition as

$E(\tilde{m}\tilde{R}_i) = 1, \quad \forall i,$

and this implies

$E[\tilde{m} (\tilde{R}_i - \tilde{R}_j)] = 0, \quad \forall i,j.$

Also, if there is a portfolio made up of the assets, then the SDF satisfies

$E(\tilde{m}\tilde{x}) = p, E(\tilde{m}\tilde{R}) = 1.$

Notice the definition of covariance, it can also be written as

$1 = cov (\tilde{m}, \tilde{R}) + E(\tilde{m}) E(\tilde{R}).$

Suppose there is a risk-free asset. Then $\tilde{R} = R_f$ implies $E(\tilde{m}) = 1/R_f$. Substituting this into the last expression and rearranging gives the following formula for the risk premium of any asset or portfolio with return $\tilde{R}$:

$E(\tilde{R}) - R_f = -R_f cov (\tilde{m}, \tilde{R}).$

This shows that risk premiums are determined by covariances with any SDF.^[1]

The existence of an SDF is equivalent to the law of one price.^[1]

The existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities.

Other names

The stochastic discount factor is sometimes referred to as the pricing kernel. This name comes from the fact that if the expectation

$E(\tilde{m}\,\tilde{x}_i)$

is written as an integral, then $\tilde{m}$ can be interpreted as the kernel function in an integral transform.^[2]

Other names for the SDF sometimes encountered are the marginal rate of substitution (the ratio of utility of states, when utility is separable and additive, though discounted by the risk-neutral rate), a change of measure, or a state-price density.^[2]

References

1. Kerry E. Back (2010). Asset Pricing and Portfolio Choice Theory. Oxford University Press.
2. Cochrane, John H. (2001). Asset Pricing. Princeton University Press. p. 9.