Lesson 7: Modern Portfolio Theory
Economics of Finance
School of Economics, UNSW
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Expected utility
The mean-variance expected utility takes the form: v s2
• e is the expected return
• s is the standard deviation of the expected return,
• t = 2/c is the investor’s risk tolerance and c is risk aversion.
• t or c can be time-varying and wealth-dependent, but for simplicity we assume they are constant
Eu = e − t = e − t ,
Optimal Portfolio Choice
• Investor will maximize the utility (blue indifference curves) • Given the e − s opportunities (red) available on the market
– efficient frontier of a portfolio
Why is efficient frontier concave?
Efficient Frontier: many securities
Efficient frontier is as the most “optimal” portfolio of portfolios of all risky securities
Combine portfolios of risky assets with a risk-free asset
The point of tangency T is called “Market portfolio”, the best portfolio of risky assets on the market you can use to combine with the risk-free asset.
What about combinations with F and E portfolios?
Sharpe ratio
proposed the following Sharpe ratio S = e − ef
where e − ef is excess return and s is standard deviation (risk)
of a risky asset/portfolio.
• reflect the tradeoff between excess return and risk • simple way to compare different stocks/portfolios • usually S > 1 is acceptable, S > 2 is very good
Sharpe ratio of the Market portfolio
SM = eM −ef is the slope to the tangent line and therefore the sM
best Sharpe ratio available on the market
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