Portfolio theory Lecture 7

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

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

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

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

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

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

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

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.