Lecture 3

Question 1

What are the assumptions around the CCAPM

Answer 1

1. The market is in competitive equilibrium
2. Two-period model
3. All assets are tradeable
4. No market frictions
5. Investors are homogenous and maximize a utility function

Question 2

Define the basic properties of a utility function

Answer 2

• satiation: u() is increasing and concave
• Impatience: 0<ß<1 is the subjective discount factor
• intertemporal substitution
• Risk aversion

Question 3

Give the utility maximization problem for the CCAPM. Solve for the Euler and then for the central asset pricing formula.

Answer 3

max u(c_t) + ß E_t (u(c_t+1))
s.t. c_t = e_t + p_t ξ
c_t+1 = e_t+1 + x_t+1 ξ

where
ξ: quantity of a risky asset
e: endowment
x: random payoff

Euler: p_t u’(c_t) = ß E_t( u’(c_t+1) x_t+1)
Central asset pricing formula:
p_t = ß E_t( u’(c_t+1)/u’(c_t) x_t+1)

Question 4

Define the stochastic discount factor based on the basic consumption-based pricing equation. Give some intution.

Answer 4

Central asset pricing formula:
p_t = ß E_t( u’(c_t+1)/u’(c_t) x_t+1)
Then: p_t = E_t (m_t+1 x_t+1) where m_t+1 = ß u’(c_t+1)/u’(c_t)

The SDF is relatively low (high discount) when
* tomorrow is bright (c_t+1 is high, so u’(c_t+1) is low)
* today is bad (c_t is low, so u’(c_t) is high)

Question 5

What is the one bigh advantage of the SDF

Answer 5

You only need one SDF for all financial assets

Question 6

Define the interest rate without uncertainty. Give some intuition.

Answer 6

• Price of a risk-free asset that delivers 1 unit of consumption tomorrow:
  p_t = E_t( m_t+1) x 1
• The risk-free rate is given by: R_tf = 1/p_tf = 1/ E_t(m_t+1)
• With power utility and without uncertainty about future consumption:
  R_t = 1/ß (c_t+1/ c_t)^γ
• Real risk-free interest rates are:
• high when people are impatient (low ß)
• high when consumption growth is high (high c_t+1/c_t)
• more sensitive to consumption growth if agents are more risk averse (high γ)

Question 7

Define the logarithmizes form of the real interest rate without uncertainty.

Answer 7

r_tf = δ + γ μ_t - γ^2/2 σ^2_t

where
δ: impatience
-> real risk-free rates are low when consumption volatility is high

Question 8

Using the definition of covariance, rewrite the SDF.

Answer 8

Covariance: cov(m,x) = E(m,x) - E(x)E(m)
Then: p = E(m) E(x) + Cov (m,x)

Question 9

Payoffs positively correlation with consumption growth have ____ prices as compensation for risk.
Provide some intuition

Answer 9

LOWER

recall: R_f = 1/E(m)
and p = E(x) / R_f + cov (m,x)

Intuition: Payoffs positively correlated with consumption growth implies negatively correlated with the stochastic discount factor m

Question 10

An asset’s price is _ if its payoff covaries positively with consumption.
Provide intuition

Answer 10

Lowered
p = E(x)/R_f + cov (ß u’ (c_t+1), x)/u’(c_t)

Assets negatively correlated with consumption could have negative returns -> only systematic risk matters

Question 11

Define the beta representation based on the covariance formula and the SDF.