Homework 2

The Risk-Return Trade-off: Theory

1Question 1

Lets specialize the concept of returns we have seen generically for securities to stocks. The gross return of stock i between time t and t + 1 in state s (given available information at t) is

Ri,t:t+1(s) Ri,t+1(s) Pi,t+1(s) + Di,t+1(s) Pi,t

where

P_i,t: price at time t of stock i

D_i,t+1(s): dividend paid at time t + 1 by stock i in state s

P_i,t+1(s): price of stock i at time t + 1 in state s (after the dividend is paid; ex-dividend price)

Answer the following two conceptual questions

Why P_i,t is not a function of the states of the world?

Given the definition of linear return r_i,t:t+1(s) r_i,t+1(s) R_i,t+1(s) 1, if we define capital gain/loss as CGi,t+1(s) = Pi,t+1(s)Pi,t Pi,t what is the implied definition of dividend yield

DYi,t+1(s)?

Suppose ABC stock price is currently $100. Assume that the (ex-dividend) stock price will either be $105 or $90 next month. (The state will be either good or bad, g or b.) Either way, the firm will pay a $5 dividend.

Compute the stock capital gain/loss, CG_ABC,t+1(s), dividend yield, DY_ABC,t+1(s) and linear return r_ABC,t+1(s) in the good and the bad state

Without computing the log return r_ABC,t^ln ₊₁(s), do you expect it to be higher or lower than rABC,t+1(s) ? Why?

Given a risk-free rate of 0.0025 compute the excess return r_ABC,t^e ₊₁(s) in the good and the bad state

2Question 2

Consider the following setup:

3 states of the world: p₁ = Prob(s₁) = 0.3,p₂ = Prob(s₁) = 0.5 and p₃ = Prob(s₁) = 0.2

Asset As price today is P_A,0 = 100, in 6 months it pays for sure a dividend D_A,0.5 = 2 and its ex-dividend price will be state-dependent either P_A,0.5(s₁) = 110,P_A,0.5(s₂) = 103 and PA,0.5(s3) = 90

Asset Bs price today is P_B,0 = 50, in 6 months it pays for sure a dividend D_B,0.5 = 3 and its ex-dividend price will be state-dependent either P_B,0.5(s₁) = 52,P_B,0.5(s₂) = 57 and PB,0.5(s3) = 35

the risk-free interest rate during this 6 months is 0.01

Compute the following

The expected returns, risk premia, volatilities and correlations of the three assets

The expected return and volatility of a portfolio invested 60% (i.e. with a weight of 0.6) in asset A, 30% in asset B and 10% in the risk-free asset

Hint: for question 2 first compute the (31) vector of expected return ₀ and the (33) covariance matrix V₀

3Question 3

Given generic asset 1, with expected return ₁ and risk ₁, and generic asset 2, with expected return ₂ and risk ₂, let _1,2 be the covariance between the two assets.

Derive the formulas for a generic proper portfolio ^p expected return and risk as a function of asset 1 weight ^p1

Derive the formulas for a generic long-short portfolio ^ls expected return and risk as a function of asset 1 weight ^ls1

What is the condition under which the two portfolios have the same expected value?

What is the condition under which the two portfolios have the same risk?

Are the two above conditions mutually exclusive? Why?

Given ₁ = 0.16,₂ = 0.12,₁ = 0.2 and ₂ = 0.2

if the proper portfolio has an expected return of 0.15, what is the proportion invested in asset 1?

if the long-short portfolio has an expected return of 0.15, what is the proportion invested in asset 1?