Topic 3 Asset Pricing

Question 1

Assume investors get utility from consumption, and we model investors by their utility from current and future consumption

Answer 1

consumption next period 𝑐_(𝑡+1) is random, the period utility 𝑢(∙) is increasing and concave the parameter 𝜌 measures investor impatience

Question 2

The amount that the investor buys maximizes his expected utility, resulting in FOC for optimal consumption

Let 𝑥_(𝑡+1)=𝑝_(𝑡+1)+𝑑_(𝑡+1) be the payoff of an asset.

Assume the investor can freely buy or sell as much of the asset as he wishes, at a price 𝑝_𝑡.

Answer 2

𝑝_𝑡 = 𝜌𝐸_𝑡 [(𝑢′ (𝑐_(𝑡+1) ))/(𝑢′ (𝑐_𝑡 ) ) 𝑥_(𝑡+1)]

Question 3

Let 𝑚_(𝑡+1) ≔ 𝜌 (𝑢′ (𝑐_(𝑡+1) )) / (𝑢′ (𝑐_𝑡 ) )

Stochastic discount factor (MRS)

Answer 3

so that 𝑝_𝑡=𝐸_𝑡 [𝑚_(𝑡+1) 𝑥_(𝑡+1)]

Note that 𝑚_(𝑡+1) does not depend on the particular payoff 𝑥_(𝑡+1)

the same stochastic disc. factor 𝑚_{<span>(𝑡+1)</span>} prices all assets

Question 4

Marginal rate of substitution

𝑚_(𝑡+1)

also the stochastic discount factor

Answer 4

Is the rate at which the investor is willing to substitute consumption at time 𝑡 for consumption at time 𝑡+1

Question 5

It is often more convenient to divide both sides of

𝑝=𝐸[𝑚𝑥] by 𝑝:

Answer 5

1=𝐸 [𝑚 x]

Because the risk-free rate 𝑅𝑓 is known ahead of time, 𝑅𝑓= 1 / (𝐸[𝑚])

Question 6

definition of covariance, we can write the fundamental relation as

𝐸[𝑅]−𝑅𝑓=− 𝑅𝑓 𝑐𝑜𝑣(𝑚,𝑅)

Answer 6

This tells us that the risk premium of an asset is determined by the covariance of the asset’s return with the discount factor.

Assets that covary positively with the discount factor

(i.e., and negatively with consumption) are so desirable that they are priced to offer negative risk premiums

Question 7

For example, the CAPM takes f to be the return on the market portfolio 𝑅^𝑀. In this case, it can be shown the risk premium is

Answer 7

−” “ 𝑅^𝑓 𝑐𝑜𝑣(𝑚,𝑅)=(𝑐𝑜𝑣(𝑅,𝑅^𝑀))/(𝑣𝑎𝑟(𝑅^𝑀)) (𝐸[𝑅^𝑀]−𝑅^𝑓 ) and therefore 𝐸[𝑅]−𝑟_𝑓=𝛽(𝐸[𝑅_𝑀]−𝑟_𝑓 ) where β=(𝑐𝑜𝑣(𝑅,𝑅_𝑀))/(𝑣𝑎𝑟(𝑅_𝑀))

Question 8

Growth Stocks

Answer 8

High Price/Book

High Price/Earnings

Market/Book

Question 9

Value Stocks

Answer 9

Low Price/Book

Low Price/Earnings

Market prices are believed to be below the intrinsic value

outperform growth in the long run

Question 10

Arbitrage Pricing Theory

APT portfolio construction

Question 11

How many factors to use in the APT

Answer 11

Principal Component Analysis

• Compute the covariance matrix of all available securities
• compute it’s (the cov. matrix) eigenvalues and corresponding eigenvectors
• The relative sizes of eigenvalues determine how many factors to use

Question 12

APT Example

Question 13

Equity Premium Puzzle

Question 14

First order condition for asset pricing

Answer 14

p_tu’ (c_t ) =ρE_t [u′ (c_(t+1) ) x_(t+1)]

Question 15

Accounting for Total Risk

Sharpe ratio