Consider an investor with separable utility over consumption streams:
U(c_t,c_(t+1) )=u(c_t )+ρE[u(c_(t+1) )]
Derive the stochastic discount factor. What is its defining property?
We derived a formula for the risk premium of an asset in terms of the stochastic discount factor. Write down the formula, and then explain what determines the sign of an asset’s risk premium. What is the intuitive explanation?
Suppose the period utility is given by a power function:
u(c)=c^(1-γ)/(1-γ)
where γ is a number greater than 1.
Note that
u^' (c)=c^(-γ)
Show the stochastic discount factor is m_(t+1)=〖ρ(c_t/c_(t+1) )〗^γ
Suppose there is no uncertainty about future consumption (so that E[m]=m). Derive a formula for the risk-free rate and use your formula to show that the risk-free rate is high when either i) investors are more impatient or ii) consumption growth is high. Does this make sense?
Read the article The Equity Premium: Why Is It a Puzzle? by R. Mehra, posted on the course website. Then answer/discuss the following:
State the puzzle and clearly explain why it is a puzzle.
Describe the “risk-free rate puzzle.”
Summarize the various attempts to resolve the puzzle and the author’s response to each.
What do you think is the most likely explanation or resolution?
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