留学生考试辅导 Notes on the stochastic discount factor, SDF

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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