Equivalences

Question 1

Motivation

Answer 1

We have discussed various mathematical descriptions for asset pricing models, e.g. stochastic discount factors
mean-variance frontier linear factor models

relationships between these descriptions

the three descriptions are equivalent

we can represent all pricing information by any of the following objects:

• a stochastic discount factor
• (almost) any return on the mean-variance frontier a linear factor pricing model
• knowing any one of these objects allows us to construct the others

Question 2

Relation between Three Views of Asset Pricing

Answer 2

Question 3

Assumptions made throughout

Answer 3

The payoff space X satisfies free portfolio formation

Asset prices are described by a price function p that leaves no arbitrage opportunities

recall from Lecture 2: this implies a (positive) SDF representation exists p(xˆ), p(x∗) ̸= 0, so that the returns Rˆ, R∗ exist

Question 4

Equivalence: Linear SDFs and Beta Representations

Answer 4

Suppose, conversely, that (∗) holds for all returns Ri and that γ ̸= 0.

Then there are a∈R,b∈RJ,such that m=a+bT f is a SDF that represents the price function p.

Question 5

Remark: relationship between a, b and γ, λ:

Answer 5

Question 6

Equivalence Theorem: Idea of Proof

Answer 6

Question 7

Conclusion 1: from SDF to Linear Factor Model

Answer 7

Suppose we have an asset pricing model for the SDF m = f (data)
Then we can obviously write m = 0+1·m

Question 8

Conclusion 2: from Linear Factor Model to SDF

Answer 8

Question 9

From SDF to a Return on the Mean-Variance Frontier

Answer 9

Question 10

From Factor Model to a Return on the Mean-Variance Frontier

Answer 10

Suppose we have linear factor model with factor vector f

We can first construct a SDF m = a + bf using the equivalence theorem

need to solve a J + 1 dimensional linear equation to find a and b

Can then construct R∗ as before

in general, need to project m onto the payoff space

can skip projection if m = x∗

this is the case if all factors are payoffs and there is a risk-free asset

Question 11

From Frontier Return to SDF

Answer 11

Remarks :
If there is a risk-free asset, all frontier returns Rmv but the risk-free rate work
(because then Rˆ = Rf )
The coefficients a and b can be determined by imposing that m prices any two assets

Question 12

From Frontier Return to SDF: Idea of Proof

Answer 12

Use the two funds theorem to express R∗ as a portfolio of Rˆ and Rmv

After multiplying by p(x∗), this yields a representation

x∗ =axˆ+bRmv
with some constants a, b
Because xˆ = proj(1 | X), the SDF m = a + bRmv also works

Question 13

Remark: Converse Result

Answer 13

Question 14

From Frontier Return to Linear Factor Model

Answer 14

Remarks :
If there is a risk-free asset, all frontier returns Rmv but the risk-free rate work
(because then Rgmv = Rf )

One can determine γ and λ in the factor model by imposing that it prices any two assets

A type of converse also holds: if R can be a factor, it must be on the frontier

Question 15

From Frontier Return to Linear Factor Model: Idea of Proof 1

Answer 15

We have discussed the proof in Lecture 4b for the special case that Rf exists

The general case works similarly, but now a different return needs to take the role of Rf
This return is the zero-beta return associated with Rmv:

Question 16

From Frontier Return to Linear Factor Model: Idea of Proof 2

Answer 16

Question 17

Summary

Answer 17

Three views of asset pricing are equivalent:

SDFs, beta representation, mean-variance frontiers

Any of the following objects fully describe a price function p a stochastic discount factor m

a set of factors f in a linear factor model

a single return Rmv on the mean-variance frontier provided Rmv ∈/ {Rˆ, Rgmv }

The equivalent results and their proofs are not just abstract theory they tell us how to move from one view to another one