1 VI. Sample Techniques for Computing Optimal Portfolios

2 Techniques developed to simplify inputs 1. Sharpe Single Index Model 2. Mean Models Led to better forecasts We will now show that these models 1.Simplify the computation of the efficient frontier 2.Allow an unambiguous ranking of securities 3.Produce an intuitively appealing explanation of why a security enters the efficient portfolio

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4 Equation A N equations N unknowns Done but still need to solve simultaneous equations. Z’s on both sides of equation. Can solve once and for all and have a closed form solution.

5 Multiply by  i Sum over all stocks Substitute into equation A

6 where

7 1) C o is a number dependent on the population of stocks under consideration 3) Unique ranking 4) Ranking is intuitively appealing

8 Problem R i R f = 5

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10 SHORT SELLING NOT ALLOWED Kuhn-Tucker Conditions Equations hold but

11 Same mathematically except apply assuming first security in, then second, until for security x, Z x turns negative. Then first x-1 securities are included in the optimal portfolio and C is defined over the first x-1 securities.

12 Constant Correlation Case Multiply by, sum, and substitute for

13 But So

14 SUMMARY Models that were developed to simplify the inputs to portfolio analysis so that analysts could deal with it have unexpectedly 1) led to better forecasts and the selection of better portfolios 2) Simplified computation 3) led to an intuitive understanding of why securities enter into optimal portfolios

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/ / / /10 _________ Sum 32.5/10 1 2/100 2/10 2/ / /10 7/ / /10 47/ / / /100 ______ Sum 54.5/100

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18 Back to example 4 th not in C k = 5.175

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