107 Stat Spring 2012 Homework #3 Solutions
1. The Portfolio which invests 35.28% in stock A and the rest in stock B gives a higher mean return (42.62%) for the same standard deviation. This is shown below: stock A stock B Portfolio mean 14.25% 62.72% 45.62% variance 6.38% 14.43% 4.32% sigma 25.25% 37.99% 20.78% covariance -5.52% weights 0.352773 0.647227 1
2. A single run gave a correlation of -0.006206696. A more comprehensive summary is given below for 10000 runs of this scenario:
Min. 1st Qu. Median Mean 3rd Qu. Max. -6.08E-02 -1.15E-02 -2.82E-05 1.95E-04 1.21E-02 6.66E-02 For any single run, the correlation is close to zero. However, it is obvious that the investing schemes are dependent since we know that the daily returns for both managers are exactly the same in magnitude.
3. a) and b)
QUESTION 1
Mean Return (%) 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00 26.00
Unrestricted Standard Deviation (%) 22.81 21.48 20.29 19.26 18.41 17.77 17.36 17.20 17.30 17.65 18.23 19.04 20.03 21.18 22.48 23.88 25.39
QUESTION 2 Restricted Standard Deviation (no short sales) (%) No Solution 22.33 20.64 19.36 18.42 17.77 17.36 17.20 17.41 18.36 20.07 22.52 No Solution No Solution No Solution No Solution No Solution
30.00 25.00 20.00 15.00 10.00 5.00 0.00 0.00 5.00 10.00
Efficient Frontier
Question 3a Question 3b
15.00
20.00
25.00
30.00
In 3b, some of the returns do not have an optimal solution when we are restricted to no short sales. This is because those mean returns lie outside the range of mean stock returns. (i.e. they are either smaller
than the smallest mean stock return or larger than the largest) and thus no convex combination (i.e. the weights are between 0 and 1) of the stocks will ever achieve those mean returns. c) Global Minimum Variance portfolio:
US 0.3757 Germany 0.1971 UK 0.0725 Japan 0.2075 Australia 0.1144 Canada 0.0347 France -0.0019
weights
Portfolio Variance Portfolio std Portfolio expected return
295.76 17.20% 17.12%
d) Maximum Return Portfolio (with no short sales): If short sales are not permitted, the maximum return that can be achieved will be the return of the stock with the maximum return. This will be the mean return of Germany. The volatility is the standard deviation. The details are given below.
Portfolio Variance Portfolio std Portfolio expected return 625.00 25.00 21.70
Maximum Return Portfolio (with short sales): If short sales are permitted, the maximum return that can be achieved is undefined due to the weights being infinite (In Solver, there will be no convergence, and any weights obtained are only due to finite machine precision). The volatility here is likewise undefined. The result from Solver is however given here.
Portfolio Variance Portfolio std Portfolio expected return 4197602989166900000.00 2048805258.97 674846681.37
e) short With sales permitted, the following were obtained for the Tangent portfolio:
Portfolio Variance Portfolio std Portfolio expected return Riskless rate: Sharpe ratio: 442.14 21.03 22.87 5.5 0.826177
With the following weights,
US Germany UK Japan Australia Canada France
weights
0.6462
0.5400
0.2629
0.1979
0.1495
-0.6360
-0.1605
f) With no short sales, the Tangent portfolio is given by:
Portfolio Variance Portfolio std Portfolio expected return Riskless rate: Sharpe ratio: 339.90 18.44 19.06 5.5 0.735257
With these weights:
US 0.2245 Germany 0.4417 UK 0.1769 Japan 0.1546 Australia 0.0023 Canada 0.0000 France 0.0000
weights
It can be seen that the weights change a lot from 3e. g) i) We can't create such a portfolio since the desired standard deviation (12%) is smaller than the standard deviation of our minimum variance portfolio (even with short sales allowed). ii) This problem simply needs us to find the weights w for the client's portfolio, such that: Portfolio= (1-w)*Riskfree + w*Tangent portfolio ...and Std(Portfolio)=12% Where the Tangent portfolio is the same one obtained in 3f. We can easily use solver to obtain this by maximizing the Portfolio mean return subject to the constraints that: 1. w is positive; and 2. std(portfolio)=12 This produces the following results:
weights 0.650888009 0.349111991
T-portfolio Riskless
i.e. Invest 65.09% of your capital in the Tangent portfolio and use the rest to buy the bank Certificate of Deposit. iii)
Portfolio std Portfolio mean return 12.00 14.32
h) i) Yes this is easily obtained in solver. Note that there won't be a unique solution, since the standard deviations will be different for different weights combinations that lead to a mean return of 30%. For this question, they are all correct. However the optimal (minimum variance) weight combination is given below:
US 0.9811 Germany 0.9643 UK 0.4989 Japan 0.1864 Australia 0.1932 Canada -1.4669 France -0.3569
weights
.. with the following mean and volatility:
Portfolio Variance Portfolio std Portfolio expected return 1029.74 32.09 30.00
ii) Somewhat similar to 3g, this problem simply needs us to find the weights w for the client's portfolio, such that: Portfolio= (1-w)*Riskfree + w*Tangent portfolio ...and mean(Portfolio)=30% Where the Tangent portfolio is the same one obtained in 3e (NOT 3f this time). We can easily use solver to obtain this by minimizing the Portfolio standard deviation subject to the constraint that: 1. mean(portfolio)=30 Note here that w is unconstrained. This produces the following results:
weights 1.410246391 -0.410246391
T-portfolio Riskless
i.e. borrow 41.02% of your capital (or short the bank Certificate of deposit) and then add that to your original capital and invest everything in the tangent portfolio. iii)
Portfolio std 29.65
Portfolio mean return
30.00