1. 1 PORTFOLIO How should you invest your money? What ’ s the best investment portfolio? How do you maximize your return without losing money?

2. 2 PORTFOLIO How should you invest your money? You should diversify your investment — spread it out among many assets in order to lower the risk. A portfolio is an answer for this queation.

3. 3 PORTFOLIO A portfolio is a set of stocks or other financial assets. Suppose that between 1990-99 you held a portfolio invested 50% in GM and 50% in MSFT.

4. 4 Portfolio mean and variance for a two-asset portfolio

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7. 7 We built a table of the portfolio statistics using the formulas. In the table we vary the proportion of GM stock in the portfolio from 0% to 100%

8. 8 Portfolio mean and variance for a two-asset portfolio

9. 9 Portfolio statistics for multiple assets Suppose that we have N stocks, and that for each stock i, we have computed the mean E(r i ) and the variance σ i ^2 = Var(r i ) of the stock ’ s returns. Furthermore, suppose that for each pair of stocks i and j, we have calculated the covariance of the returns Cov(r i,r j ).

10. 10 Portfolio statistics for multiple assets

11. 11 PORTFOLIO What ’ s the best investment portfolio? There is no single best investment portfolio. It all depends on your willingness to trade off return for additional risk. What may surprise you, however, is that we can say a lot about how not to invest. W develop the notion of the efficient frontier — this is the set of all portfolios that you would consider as investment portfolios.

12. 12 PORTFOLIO How do you maximize your return without losing money? To some extent the efficient frontier answers this question: It shows us which portfolios are so bad that you can improve both the return and the risk.

13. 13 PORTFOLIO If the correlation between asset returns is +1 then diversification will not reduce portfolio risk. If the correlation between asset returns is -1 then we can create a risk-free asset. When asset returns are partially correlated, diversification can lower risk, though it cannot completely eliminate it.

14. 14 PORTFOLIO RETURNS THE EFFICIENT FRONTIER

15. 15 PORTFOLIO RETURNS THE EFFICIENT FRONTIER

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17. 17 PORTFOLIO RETURNS THE EFFICIENT FRONTIER

18. 18 PORTFOLIO RETURNS THE EFFICIENT FRONTIER Looking at the graph it is easy to see that some portfolios are better than others. A portfolio on the top part of the graph which has the same standard deviation and higher expected return.

19. 19 PORTFOLIO RETURNS THE EFFICIENT FRONTIER There is a clear risk-return tradeoff between these two portfolios — it is impossible to say that one is unequivocally better than the other. The portfolio with the higher return also has the higher standard deviation of returns. All of the portfolios on the top part of the graph have this property.

20. 20 PORTFOLIO RETURNS THE EFFICIENT FRONTIER This top part of the graph is called the efficient frontier. The efficient frontier is the set of those portfolios which offer the highest expected return for a given standard deviation.

21. 21 PORTFOLIO RETURNS THE EFFICIENT FRONTIER In order to calculate the efficient frontier, we have to find its starting point, the portfolio with the minimum standard deviation of returns, which is called the minimum-variance portfolio.

22. 22 PORTFOLIO RETURNS THE EFFICIENT FRONTIER

23. 23 PORTFOLIO RETURNS THE EFFICIENT FRONTIER Minimum variance portfolios using calculus: There ’ s actually a formula for the minimum variance portfolio: Homework : Derive this formula and use this formula in Excel to give the same answer as that given by Solver.