2 p r0 p 2 j p j r0 separation

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Unformatted text preview: P 0 ) ( 2" ai! i , j = # µ j $ r0 i =1 2! 2 = " ( µ P # r0 ) P ( 2! j , P = " µ j # r0 ) 2! 2 = " ( µ P # r0 ) P ( 2! j , P = " µ j # r0 ) Separation or Two Fund Theorem Formal Optimization Problem n ( r0 + " a j µ j ! r0 = µ P 2! = " ( µ P # r0 ) 2 P ) Any efficient portfolio can be obtained as a combination of two investments: P and the riskless investment. Mean return P r Mean return standard deviation Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as sj = µ j ! r0 "j Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as Find all assets j µ j ! r0 with the same sj = "j Sharpe ratio as A. µ Asset A r0 ! Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as Find all assets j µ j ! r0 with the same sj = "j Sharpe ratio as A. µ Asset A r0 ! Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as sj = µ j ! r0 "j Sharpe Ratio Sharpe Ratio Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as sj = µ j ! r0 "j µ j = r0 + s j! j Let j be an asset (or a portfolio of assets). Then the Sharpe ratio sj of asset j is defined as Find all assets j µ j ! r0 with the same sj = "j Sharpe ratio as A. µ Asset A µ j = r0 + s j! j r0 ! Sharpe Ratio sj = µ j ! r0 "j µ Sharpe Ratio µ j = r0 + s j! j sj = Along which line do assets have a higher Sharpe ratio? r0 µ j ! r0 "j µ j = r0 + s j! j µ er igh h io rat e arp Sh Along which line do assets have a higher Sharpe ratio? r0 ! ! Sharpe Ratio sj = µ j ! r0 "j Sharpe Ratio µ j = r0 + s j! j µ er igh h sj = io rat e arp Sh What is better: A higher or a lower Sharpe ratio? r0 µ j = r0 + s j! j µ er igh h io rat e arp Sh Higher Sharpe Ratio: Offers Higher return with same stdv. r0 ! Sharpe Ratio ! Estimating Sharpe Ratios Efficient Portfolios have the highest Sharpe ratio among all feasible Portfolios! Mean return µ j ! r0 "j P r standard deviation We now estimate Sharpe Ratios for MSFT and AAPL, using weekly data from Feburary 25, 2011 to February 25, 2013. Estimating Sharpe Ratios Estimating Sharpe Ratios We now estimate Sharpe Ratios for MSFT and AAPL, using weekly data from Feburary 25, 2011 to February 25, 2013. We now estimate Sharpe Ratios for MSFT and AAPL, using weekly data from Feburary 25, 2011 to February 25, 2013. Assume r0=2%, i.e., the weekly rate is 1.021/52"1. Assume r0=0.2%, i.e., the weekly rate is 1.0021/52"1. Sharpe Ratios: MSFT 0.081 AAPL 0.046 Looks like Apple gets higher risk-adjusted return. Sharpe Ratio µ j ! r0 sj = "j Estimating Sharpe Ratios µ j = r0 + s j! j AAPL has higher Sharpe ratio µ Problem: There is uncertainty about the estimates. Need confidence bands... In other words, it could be pure “luck of the draw” that resulted in a higher Sharpe Ratio for Apple. PL AA r0 Sharpe Ratios: MSFT 0.046 AAPL 0.081 Looks like Apple is better managed than Microsoft - Apple gets higher risk-adjusted return. T MSF !...
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This note was uploaded on 09/12/2013 for the course ECON 490 taught by Professor Gottheil during the Spring '12 term at University of Illinois, Urbana Champaign.

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