1. Portfolio theory
Lecture 7

2. Risk, returns, preferences and opportunities
Suppose we have 3 assets
Assets
Expected return
Volatility A
10% 8% B
14% 7% C 4%

3. Comparison of the assets
Prefer B over A
Prefer C over A
E(r) B C A δ
Portfolio theory is about maximizing return and minimizing the risk

4. Portfolio of 2 assets A&B
E(r)B> E(r)A Volatility of A > Volatility of B Volatility of Portfolio = 𝐴 2 ×𝜎 𝐴 2 + 𝐵 2 ×𝜎 𝐵 2 +2×𝐴×𝐵× 𝜎 𝐴 × 𝜎 𝐵 × 𝜌 𝐴𝐵 If 𝜌 𝐴𝐵 =+1, Volatility of Portfolio =A× 𝜎 𝐴 +𝐵× 𝜎 𝐵

5. If 𝜌 𝐴𝐵 =+1 Volatility of Portfolio =A× 𝜎 𝐴 +𝐵× 𝜎 𝐵
Feasible set is a red line, changing the proportion of A and B will move us along the line
E(r) B A δ

6. If 𝜌 𝐴𝐵 =-1 Volatility of Portfolio =± A× 𝜎 𝐴 −𝐵× 𝜎 𝐵 ≥0
Let 𝜎 𝑃 =0, A/B = 𝜎 𝐵 /𝜎 𝐴
E(r) B A δ

7. If 𝜌 𝐴𝐵 =0
Volatility of Portfolio = 𝐴 2 ×𝜎 𝐴 2 + 𝐵 2 ×𝜎 𝐵 2 <A× 𝜎 𝐴 +𝐵× 𝜎 𝐵
E(r) B A δ

8. What happens if we have more than 2 assets in a portfolio?
Let N ≥ 3
E(r)
Efficient frontier
Minimum variance portfolio
Mean variance frontier δ

9. Efficient frontier
is a modern portfolio theory tool that shows investors the best possible return they can expect from their portfolio, given the level of volatility they are willing to accept.