Topic 2 Asset Pricing Models

Question 1

Efficient portfolio

Answer 1

Portfolios that lie on the investment opportunity set. This set shows the relationship between expectation and minimum achievable σ of return Mean-variance optimized

Question 2

What does investment opportunity set look like?

Answer 2

If one of the assets is riskless, it is a straight line. (the CAL) If all assets are risky, then the shape of the investment opp. set depends on the covariances (or correlations) between the assets.It will be a hyperbola; the fact that it is curved reflects the benefits of diversification.

Question 3

Formulas for efficient portfolios

Answer 3

If there are N assets, then there are N vectors W whose components sum to 1 : 𝑤′1 ⃗ = 1 If Σ denotes the covariance matrix of the N asset returns, the variance of the portfolio is Var(R)=𝑤′Σ𝑤 If 𝜇 ⃗ denotes the expected returns of the N assets, the expected return of the portfolio is E[R]=𝑤′𝜇 ⃗

Question 4

Formulas for efficient portfolios of N risky assets only

Answer 4

So, if all assets are risky, you solve the following problem: min 𝑤′Σ𝑤 s.t. i) 𝑤′1 ⃗ = 1 and ii) 𝜇=𝑤′𝜇 ⃗ This can be solved using the method of Lagrange multipliers

Question 5

Formulas for efficient portfolios N risky assets and a riskless asset.

Answer 5

If one of the assets is riskless, you can find the efficient portfolios two ways: Solve the previous minimization problem, and then find the optimal risky portfolio It will be the tangency portfolio, the portfolio with the largest Sharpe ratio the efficient frontier (including riskless asset) is a straight line (CAL) that joins the tangency portfolio to the riskless asset

Question 6

Formulas for Efficient Portfolios N risky assets and a riskless asset

Answer 6

Or: Solve the optimization problem directly: keep the riskless asset separate let w denote the weights on the risky assets only note that the riskless asset does not contribute to the variance of the portfolio the expected return on the portfolio = w 𝜇 + (1−𝑤′1 ⃗ ) 𝑟𝑓 min⁡ 𝑤′ Σ𝑤 s.t. 𝑤′ 𝜇 +(1−𝑤′1 ⃗ ) 𝑟𝑓=𝜇o ({𝑤})

Question 7

If the market portfolio is efficient, then excess expected returns

Answer 7

𝐸[𝑅]−𝑟𝑓=(𝑐𝑜𝑣(𝑅,𝑅𝑀))/(𝑣𝑎𝑟(𝑅𝑀)) x(𝐸[𝑅𝑀]−𝑟𝑓 )