Questions:
1. Find the minimum variance and tangency portfolios of the industries. (hint: you will need to compute the means (arithmetic average), standard deviations, variances, and covariance matrix of the industries. The risk-free rate is given in the spreadsheet.) Comment on the different weights applied to each industry under the MVP and Tangent portfolios.
a) Compute the means and standard deviations of the MVP and Tangent portfolios. Plot the efficient frontier of these 10 industries and plot the 10 industries as well on a mean-standard deviation diagram. Why does the efficient frontier exhibit the shape that it does (i.e., why is it a parabola)?
b) Comment on the reliability of the mean return estimates for each industry. Then, artificially change the mean return estimates of each industry by a one standard error increase. How much does the Tangent portfolio change? Does the efficient frontier change a lot or a little?
c) Comment on the reliability of the covariance matrix estimate. First, assume that all covariances are zero and recompute the efficient frontier using the diagonal matrix of variances as the covariance matrix. Then, assume very simply that the covariance matrix is just the identity matrix (i.e., a matrix of ones along the diagonal and zeros everywhere else). Does the mean-variance frontier change a lot or a little, relative to b)? How important are the covariance terms relative to the variance terms?
d) EXTRA CREDIT. Run some simulations. Using the mean and covariance matrix you calculated in sample from the historical returns, use these parameters to simulate data under a multivariate normal distribution.
• Draw a random sample of 10 (N) returns from this distribution 947 times (T = the number of months). This gives you one simulation.
• Calculate the tangency and minimum variance portfolio weights from these simulated data. Then, apply these weights to the actual (NOT SIMULATED) returns on the industries (e.g., the weights come from the simulated returns, but they are applied to true/actual returns on the industries).
• Then repeat 1,000 times and save the mean and standard deviation of each MVP and Tangency portfolio you calculated under each simulation of data used to get the weights and applied to actual returns.
• On two separate plots of mean-standard deviation space, plot the simulated MVP and Tangency portfolios relative to the ones calculated using weights estimated from the real data. (One plot for MVP and one for Tangency portfolios, each plot will contain 1001 data points).
• These plots indicate the estimation error (under a normal distribution) of the Tangency and MVP weights. Which portfolio (MVP or Tangency) is estimated with less error? Why?
e) EXTRA CREDIT. Now run some simulations under the empirical distribution of returns rather than the normal distribution. This is called a bootstrap simulation.
• Draw a random sample of 10 (N) returns from the empirical distribution 947 times (T = the number of months) with replacement. The way to do this is to randomly sample a particular month by selecting a number (integer) at random from 1 to 947. Pick the 10 industry returns corresponding to the month chosen randomly between 1 and 947. These 10 returns become your first data point of 10 industry returns (effectively this becomes month t=1 in the simulation). Then, pick another number from 1 to 947, even if it is the same number (this is what resampling with replacement means), and repeat. This becomes month t=2 in the simulation. Repeat 947 times and this gives you one simulation.
(Hint: in Matlab you can create 947 random numbers between 1 and 947 and then simply pull off the rows in the industry return matrix (which is 947 X 10) that correspond to the 947 random numbers chosen. A Matlab file is posted to help you out.)
• Calculate the tangency and minimum variance portfolio weights from these simulated data. Then, apply these weights to the actual (NOT SIMULATED) returns on the industries (e.g., the weights come from the simulated returns, but they are applied to true/actual returns on the industries).
• Then repeat 1,000 times and save the mean and standard deviation of each MVP and Tangency portfolio you calculated under each simulation of data used to get the weights and applied to actual returns.
• On two separate plots of mean-standard deviation space, plot the simulated MVP and Tangency portfolios relative to the ones calculated using weights estimated from the real data. (One plot for MVP and one for Tangency portfolios, each plot will contain 1001 data points).
• These plots indicate the estimation error (under the empirical distribution) of the Tangency and MVP weights. How does the estimation error compare under the empirical simulations versus the normal distribution simulations of question d)?
Mean Variance Mathematics
2. Refer to the background pdf for the problem set on Mean Variance Mathematics. For this question, assume that no riskless asset exists.
a) An interesting portfolio to identify for any given Minimum Variance (i.e., efficient) portfolio is its orthogonal portfolio. This orthogonal portfolio is defined as that portfolio which has a zero covariance with a given minimum variance portfolio. Prove that for any minimum variance portfolio with mean return 1, its orthogonal portfolio has a mean return
μ2 = (C - Bμ1)/(B -Aμ1)
(except for the global minimum variance portfolio) where C, B, and A are the mean-variance constant matrices.
Covariance of the two portfolios being equal to zero here means that
w1TSw2 = 0
where S is the sample covariance matrix.
(Hint - Use equation 1.6 and 1.7 from supplement.)
b) One interesting characteristic of the minimum variance portfolio is that the covariance of its returns with any other portfolio p is equal to 1/1TS-11 its variance, which is equal to . Prove this. Thus, prove that
Cov(μg - μp) = Var(μg) = 1/1TS-11
c) Related to this, another interesting characteristic of the minimum variance portfolio is that any portfolio's returns regressed on it must have a beta equal to one. Show that the beta of any portfolio's return regressed on the return of the minimum variance portfolio is one.
3. Solve by hand the following. There are three securities A, B, C with mean returns of 17%, 13%, and 9%, respectively. Furthermore, their standard deviations are 20%, 40%, and 15%, respectively. The correlation between A and B is 0.50, between B and C is 0.30, and between A and C is zero. The risk-free rate is 5%.
a) Find the MVP and Tangent portfolios of these three assets, and calculate each of the portfolio return means and standard deviations.
b) Write the equation for the efficient frontier of these three assets.
c) Find the portfolio of A, B, C that gives the lowest possible variance for a return of 13%, and find the portfolio that gives the highest possible return for a standard deviation of 15%. Calculate the Sharpe ratios of these two portfolios.
d) Repeat c) allowing an investor to also invest in the riskless asset. Find the portfolio that gives the lowest possible variance for a return of 13%, and find the portfolio that gives the highest possible return for a standard deviation of 15%. Again, calculate the Sharpe ratios of these two portfolios. Compare your answers to those in c). Illustrate graphically (in a mean-standard deviation diagram) what is going on.
Attachment:- ProblemSet3.xls