Linear Factor Models: CAPM

Question 1

CAPM background

Answer 1

The Capital Asset Pricing Model (CAPM) is the single-factor model with risk factor RW

RW is the return on total wealth in the economy

this includes all stocks, bonds, derivatives, etc.
… but in principle also illiquid and private assets (e.g. real estate, private equity) … and other forms of wealth such as human capital and public infrastructure

In practice, people often proxy RW by a large index of investable securities (often stocks) R W is then also called the market return (and sometimes written R M )

Question 2

CAPM core equation

Answer 2

Question 3

Why is the CAPM important

Answer 3

The CAPM is a linear factor model that tells us exactly what the factor should be

If we believe assumptions leading to CAPM: asset pricing is all about covariance with RW

RW is difficult to define and measure empirically, but
there are proxies that are relatively easy to measure (e.g. stock indices)

the CAPM has been an empirical success for a long time
and it is still the most widely applied asset pricing model today

The CAPM is also of historical significance: it was the first modern general equilibrium pricing model

Question 4

Illustration: Empirical Fit of the CAPM

Answer 4

Question 5

CAPM and stochastic discount factors

Answer 5

from Lecture 4a: a linear pricing model is equivalent to a linear SDF

The CAPM is equivalent to the discount factor model
m = a + bRW

So the CAPM really says two things (interpreted through the lens of the consumption-based model)

1. RW is a sufficient proxy for marginal utility (for the purpose of asset pricing)
2. the relationship between marginal utility growth and RW is linear

We can try to derive the CAPM in general equilibrium by showing these two properties

Question 6

Consumption-based Model with Quadratic Utility

Answer 6

Question 7

Combining SDF and Budget Constraints Yields Factor Model

Answer 7

Question 8

Market equilibrium

Answer 8

Let’s close the model, suppose:
there are I investors, all are identical and have quadratic utility

there is a fixed supply of assets with aggregate payoff xW t+1

In equilibrium, the sum of payoffs to all investors must equal the supply

Question 9

Market Equilibrium Leads to the CAPM

Answer 9

Question 10

Alternative Derivation: Exponential Utility and Normal Distributions

Answer 10

Question 11

Portfolio Choice under Mean-Variance Preferences

Answer 11

Investors are said to have mean-variance preferences if they always choose optimal portfolios on the mean-variance frontier

any investor who
only cares about mean and variance of consumption

dislikes consumption variance
has mean-variance preferences

oes CAPM always result from mean-variance preferences? want to show next that the answer is yes

Question 12

What assumptions/environment implies 1. RW must be the tangency portfolio return
2. RW is on the mean-variance frontier?

Answer 12

all investors have mean-variance preferences
→ they all choose a portfolio on the mean-variance frontier

there is a risk-free asset
→ all investors choose the same tangency portfolio of risky assets

the risk-free asset is in zero net supply (claims of borrowers and lenders net to zero)
→ market supply of all assets = supply of risky assets

investors do not have any outside income
→ total wealth = market supply of all assets

Question 13

Mean-Variance Preferences ⇒ RW Is on the Frontier

Answer 13

We do not need to assume the existence of a risk-free asset in zero net supply

Suppose that only the following two assumptions are satisfied:

all investors have mean-variance preferences

investors do not have any outside income

Then the essential conclusions from previous slide remain valid:

• all investors choose portfolios on the frontier (not necessarily the same)
• the combined portfolio of all investors is again on the frontier
  (because X mv is a linear space)

because there is no outside income, this combined portfolio equals the total wealth portfolio

Conclusion from previous argument: RW is on the mean-variance frontier

Question 14

Theorem: Any Frontier Return Can Be Used as a Factor

Answer 14

RW is on the mean-variance frontier

How does this help to conclude the CAPM?

Answer: because any mean-variance efficient return can be used as a factor

Let Rmv be a return on the mean-variance frontier that is different from the global minimum variance return. Then asset returns satisfy a linear one-factor model with factor Rmv.

Question 15

Proof of the Theorem: Any Frontier Return Can Be Used as a Factor

(need risk-free asset)

Answer 15

Question 16

Mean-Variance Preferences Imply the CAPM

Answer 16

On the previous slides, we have established the following implication chain

investors have mean-variance preferences & no outside income

⇒ RW is on the mean-variance frontier

⇒ returns satisfy a one-factor model with factor RW

Hence, the CAPM results from any model that features mean-variance preferences

(and no non-tradeable outside income)

In any such model, a side implication is that RW is on the mean-variance frontier

Question 17

Equivalence: CAPM Holds ⇔ RW Is on the Frontier

Answer 17

Question 18

Summary

Answer 18

CAPM: linear factor model with factor RW

Simple derivations as special case of consumption-based model

all investors have quadratic utility

all investors have exponential utility and normally distributed asset returns

Connections between CAPM and mean-variance analysis

Question 19

Useful details on the budget constraint

Answer 19