1. MANY RISKY SECURITIES èWith many risky securities, principles of portfolio optimization are still the same as in Portfolio Problem #2 èSolution is also same: optimal proportion for asset x = sum of hedging plus speculative demand èProblem: how to calculate portfolio variance?

2. SIMPLE CASE: N IDENTI CAL SECURITIES Securities have identical std. dev.  Any pair of securities has correlation  Equal portfolio weights: 1/N for each security

3. ASIDE Diversification works in two ways: èCombine securities whose returns have low correlations with one another (see Portfolio Problem #2) èCombine many securities with less than perfectly correlated returns

4. MATRIX VERSION OF PORTFOLIO VARIANCE

5. GENERAL CASE: PORTFOLIO PROBLEM #3 a and a’ are the weight vector and its transpose r is the vector of expected returns  is the variance- covariance matrix

6. ADDING A RISKLESS ASSET

7. PORTFOLIO PROBLEM #4 èThe optimal combination of risky assets, c, is the one that maximizes the slope of the capital allocation line èThe optimal combination of risky assets does not depend on investor utility èInvestor utility determines how we combine the riskless asset with the optimal risky portfolio

8. SOLUTION TO PORTFOLIO PROBLEM #4

9. CAPITAL MARKET EQUILIBRIUM èIn equilibrium, security supplies and demands are equated èSuppose investors have the same estimates of expected returns, variances, covariances èLet an investor combine a single risky asset with the market index, using Portfolio Problem #4 èImplication?

10. COMBINING SECURITY X WITH INDEX PORTFOLIO (Result: CAPM)

11. ARBITRAGE PRICING THEORY (APT) Assume: Security returns have the form shown b coefficients represent security i’s sensitivity to economic factors F e represents a random error term

12. AN ARBITRAGE OPPORTUNITY

13. ARBITRAGE PORTFOLIO

14. EQUILIBRIUM RETURNS  -r f represents a risk premium for the systematic risk caused by a particular factor Expected returns must look like this to eliminate arbitrage

15. EQUILIBRIUM RETURNS: EXAMPLE

16. TRADEOFFS AMONG FACTORS èAPT equilibrium equation tells us about tradeoffs among factors èI.e., how much more of one type of risk would you have to accept to get rid of some other type of risk without sacrificing return? èNote that an index portfolio locks you in to fixed proportions of any type of risk

17. FEASIBLE RISK PROFILES