Stochastic discount factor


The stochastic discount factor is a concept in financial economics and mathematical finance. The name "stochastic discount factor" reflects the fact that the price of an asset can be computed by "discounting" the future cash flow by the stochastic factor and then taking the expectation. This definition is of fundamental importance in asset pricing.
If there are n assets with initial prices at the beginning of a period and payoffs at the end of the period, then SDF is any random variable satisfying
The stochastic discount factor is sometimes referred to as the pricing kernel. This name comes from the fact that if the expectation is written as an integral, then can be interpreted as the kernel function in an integral transform. Other names for the SDF sometimes encountered are the "marginal rate of substitution", a "change of measure", "state-price deflator" or a "state-price density".

Properties

The existence of an SDF is equivalent to the law of one price; similarly, the existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities. This being the case, then if is positive, by using to denote the return, we can rewrite the definition as
and this implies
Also, if there is a portfolio made up of the assets, then the SDF satisfies
By a simple standard identity on covariances, we have
Suppose there is a risk-free asset. Then implies. Substituting this into the last expression and rearranging gives the following formula for the risk premium of any asset or portfolio with return :
This shows that risk premiums are determined by covariances with any SDF.