The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow
\tilde{x}i
\tilde{m}
If there are n assets with initial prices
p1,\ldots,pn
\tilde{x}1,\ldots,\tilde{x}n
\tilde{m}
E(\tilde{m}\tilde{x}i)=pi,fori=1,\ldots,n.
The stochastic discount factor is sometimes referred to as the pricing kernel as, if the expectation
E(\tilde{m}\tilde{x}i)
\tilde{m}
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 (see Fundamental theorem of asset pricing). This being the case, then if
pi
\tilde{R}i=\tilde{x}i/pi
E(\tilde{m}\tilde{R}i)=1, \foralli,
E\left[\tilde{m}(\tilde{R}i-\tilde{R}j)\right]=0, \foralli,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.
By a simple standard identity on covariances, we have
1=\operatorname{cov}(\tilde{m},\tilde{R})+E(\tilde{m})E(\tilde{R}).
Suppose there is a risk-free asset. Then
\tilde{R}=Rf
E(\tilde{m})=1/Rf
\tilde{R}
E(\tilde{R})-Rf=-Rf\operatorname{cov}(\tilde{m},\tilde{R}).
This shows that risk premiums are determined by covariances with any SDF.