Portfolio Selection Lab 1 PDF

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1. Optimal Mean Variance Portfolios and the SIM-Model. In this assignment I started out by calculating the monthly returns for the index and the five stocks for every month between 1987 and 2016 using the formula: (PriceCurrent Month-PricePrevious Month))/PricePrevious Month. Then I was given an annual risk free rate of 4% which I converted to a monthly risk free rate using two separate methods, a linear approximation and a geometric average. When calculating the linear approximation I simply divided the annual 4% by 12 (# of months) which gave me a monthly rate of about 0.33%. And when calculating the geometric average I did the following: (1+4%)(1/12)-1, which gave me a monthly rate of about 0.327%. The geometrical average is the real monthly rate while the linear approximation is just an approximation. If I’d use the linear approximation to calculate the annual interest it would overestimate it due to compounding effects. I then calculated the geometrical and arithmetical monthly mean returns for the stocks and the index. To calculate the geometric mean I first calculated the total return over the 357 months by taking the value of the last month minus the value the first month divided by the value of the first month. I then took the 157th root out of the total return in order to get the monthly average. To calculate the arithmetical mean I simply used the sum of the 157 returns divided by 157. The excess returns were calculated by subtracting the linear approximation of the monthly risk free rate from the arithmetic mean returns. The excess returns were then divided by the standard deviations of each return in order to get the Sharpe ratios. The variances, standard deviations, kurtosis’ and skew’s were calculated using pre existing formulas in excel. It is sensible that the arithmetic means I calculated overestimate the monthly returns since it does not factor in the compounding effect which are the real logarithmic returns. Enjoy my amateur sketch:

Then on question 4 I computed a covariance matrix between aapl & msft which I used when calculating the minimum variance portfolio. To solve the weights I used the following formula: (Varaapl  -(correlaapl,  msft*stdvaapl  *stdvmsft  ))/(varmsft  +varaapl  -(2*stdvaapl  *stdvmsft  )), this gave me the weights 71,3% and 28,7% for MSFT and AAPL respectively. I then wanted to find the weights for a maximized Sharpe ratio by first using the following equation: ((ExcessMSFT*varAAPL)-(ExcessAAPL*CorrelAAPL, MSFT*stdv  AAPL*stdv  MSFT))/((Excess  AAPL*var  MSFT)+(Excess  MSFT*var  AAPL)-((return  AAPL+return  MSFT-2*rf)*C  orrelAAPL, MSFT*stdvMSFT*stdvAAPL)), which gave the weights 68,4% and 31,6% for MSFT and AAPL respectively. I got about the same results when using the Excel solver. This exercise did increase the portfolio's Sharpe ratio from 0,194796 to 0,194962. For the two individuals with different risk aversion I came up with for individual A: weightPortfolio=212,63% and weightrf=-112.63% and an expected return of 4,132% and a variance of 0,038 and a utility of 0,022338. And for individual B: weightportfolio=106,31%, weightrf=-6,31%, variance= 0,0095, E(r)=2,234% and a utility of 0,01238. The reason the percentages for both individuals is larger than 100% in the portfolio is because in order to maximize their utility they need leveraged position in the portfolio which also means a short position in the risk free asset. For question five I computed a covariance matrix for all five stocks and essentially underwent the same process as in question 4, but more complex thanks to the increased number of assets. First I calculated the optimal weights for maximizing the sharpe ratio using the matrix equation which gave me the following weights: AAPL

22%

EXXON

24,2%

GE

-19,3%

J&J

28,2%

MSFT

44,8%

(the negative position means it is a short position) These weights gave me a portfolio expected return of 1,747%, and a sharpe ratio of 0,20979. Then I solved for optimal weighting using the Excel Solver without restrictions on short selling, which gave me the following weights: AAPL

26,9%

EXXON

28,4%

GE

-8%

J&J

28%

MSFT

24,8%

The solver solution differs from the matrix formula solution mainly by having a smaller short position in GE and a smaller long position in MSFT. When I imposed a restriction on short selling and solved with the solver I got the following weights: AAPL

24,9%

EXXON

26,2%

GE

0%

J&J

25,8%

MSFT

23,1%

When short selling is not allowed the expected return was 1,452% and the sharpe ratio 0,1866. On question 6 I computed a matrix with each individual stocks covariance with the market, the results were pretty similar to the other covariance matrix. I then calculated the optimal shares numerically and got an expected return of 2,12% and a sharpe ratio of 0,01648. This gave me a higher return but a lower sharpe ratio than in question 5....