Portfolio Management
Homework 2
LAST NAME: _______________________________
SURNAME: ___________________________
Answer the following questions by copying and pasting results in the appropriate
space. Use the same spaces to write comments when required. Note: using
bootstrap simulations the probability that your results are exactly identical to any
of your colleagues is zero.
1. Suppose that you live in a world where there are only two risky assets: equity and
real estate. Use a simple IID scheme (with replacement) to bootstrap 20,000 alternative
scenario of (simple) returns between time t and t+1. Compute means, variances,
standard deviations, and the pairwise linear correlation from the 20,000 simulations,
copy and paste them below. Moreover, perform a comparison between the bootstrapped
means, variances, etc. and their sample analogs.
2. Use the boostrapped returns to compute the optimal, constrained (in the interval [0, 1])
weights under negative exponential (with CARA coefficient equal to 0.5), power (with CRRA
coefficient equal to 4), and quadratic (with  coefficient equal to 1/3). Copy and paste the
weights are and compute the corresponding, ex-ante expected portfolio return, expected
wealth, expected utility, the standard deviation of portfolio return, and the standard deviation
of wealth.
3. For the case of negative exponential utility and including all the assets available in the data
set that was given during the lectures, fix the coefficient of absolute risk aversion () to 0.5 and
use the bootstrap with 20,000 simulations to assess the effects of imposing the following
configurations of constraints: (i) no constraints, (ii) non-negative weights for all assets but the
1-month T-bill, (iii) all weights in [0, 1]. Copy and paste your results here. Is imposing nonnegative constraints and constraints of maximum 100%-weight equivalent to just setting to
zero or 100% all the weights that would be otherwise negative or in excess of 100%, or is it
more complex than that?
4. Compute optimal, constrained (in the interval [0, 1]) portfolios under power utility
function for  that changes between 0 and 20 in steps of 0, 0.1, 0.5, 1, 1.5, 2, 3, 5, 10, 15, and
20. Plot the optimal weights in stocks and real estate as a function of the coefficient of
constant relative risk aversion . Why are the weights changing the way they do?