Homework 6

Due 2/27

1. (40 points) You are considering investing in 3 risky assets and a riskless asset with expected returns, E covariance matrix and covariance matrix inverse as given below:

   E=

   IBM .12

   Polaroid .15

   Gold .08

   Riskless .06

   =

   .0625 .05 .0037

   .05 .16 .006

   .0037 .006 .0225

   =

   21.3 -6.6 -1.7

   -6.6 8.34 -1.14

   -1.7 -1.14 44.9

   1. What are the proportions in the optimal portfolio of risky assets (for IBM, Polaroid, and Gold)? Show some of your calculations.

      
      
      where
      
      
      
      
      
      and
      
      Therefore
      
      and
      
      Finally
      
      Then
      
      
      
      and
      
      
      
      

   2. If you decided that you wish to have a portfolio standard deviation of 30%, what is the maximum expected return that you can achieve? (Hint: compute the capital market line first.)

      The CML is now
      
      So you must be
      
      if you want

   3. What portfolio would you hold to achieve your goal in (b)?

      We know first
      
      
      
      
      
      and
      
      So
      
      This means we will buy this portfolio with 174.97% and sell the risk-free asset with 0.7497%.

   4. If someone else looked at these same assets and agreed with you on the covariance, but thought that the mean returns could be (.20 .15 .06 .06), respectively, what optimal mix of risky assets should she hold? Compare this portfolio to that in (a). Is it different in sensible ways?
      
      where
      
      
      
      
      
      and
      
      Therefore
      
      and
      
      Finally
      
      Then
      
      
      
      and
      
      means

      she will buy 127.43% on IMB, sell on Polaroid and sell 18.175% on Gold.