Notes on the stochastic discount factor, SDF
August 26, 2021
In the first lecture, we introduced the stochastic discount factor, SDF. We said that it is a “risk adjustment”. We also said that the stochastic discount factor amounts to a change of probabilities. I gave you the logic. The derivations below, however, contain a proof. The proof only uses simple properties of prob- abilities. Hence, it only uses basic statistics notions.
1 The SDF and probability changes
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We know that, in a world in which the risk-less rate is zero (recall our discussion in Lecture 1):
pt = Et[mt+1pt+1] = EtQ[pt+1]. Equivalently, I will show that
Y = E[mX] = EQ[X]
for three random variables Y,X and m (which is the SDF, in my notation). Note: without loss of generality, I will assume that Y,X and m are dis- crete random variables with J outcomes. The same logic applies to continuous
random variables.
Y = this is just the definition of expectation =
now, let us just multiply and divide by the true probs =
re-arranging, we have … =
x j p Qj j=1
Notice that the stochastic discount factor satisfies the following property
which is our stated result.
under the true (not the risk-adjusted) probabilities: E(m) = 1. In fact,
E(m) = mjpj
j=1 putting in the definition of m … = J pQj pj
j=1 … since the risk-adjusted probabilities also sum up to 1 = 1.
In a world in which the risk-less rate is not zero, we generally write the following (like in Lecture 2)
pt = Et[mt+1pt+1] = 1 EtQ[pt+1], 1+Rf
which is very intuitive: prices are discounted expectations of future cash flows taking risk into account. The two expressions are equivalent: I can discount expectations of future cash flows using an SDF (which takes risk and time into account). Alternatively, but equivalently, I can take risk into account by tweaking the probabilities before I discount at the risk-free rate.
J pQj xjpj j=1 pj
= mjxjpj j=1
Now, following the same steps as before we have
Y = this is just the definition of expectation =
now, let us just multiply and divide by the true probs =
re-arranging, we have … =
where the SDF m is a random variable with outcomes m = 1 pQj for j =
1, …, J. Note: the SDF takes risk and time into account.
This, however, implies that E(m) = 1 (rather than E(m) = 1), 1+Rf
which is the same expression in Lecture 2, slide 12.
1 EQ[X] 1+Rf
1+Rf J1pQjxjpj
j=11+Rf pj
mjxjpj j=1
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