Question

A financial analyst has recently argued that portfolio managers who rely on asset allocation techniques
spend too much time trying to estimate expected returns on different classes of securities and not enough
time on estimating the correlations between their returns. The correlations are important, he argues,
because a change in correlation, even with no change in expected returns, can lead to changes in the
optimal portfolio. In particular, he argues that as the correlation between stock and bond returns ranges
from 0.2 to 0.6, “the allocation to stocks remains fairly constant…but there are major asset shifts between
cash and bonds”.
See if you can illustrate this point with the following example: A portfolio manager is considering three
categories assets: stocks, bonds and cash. The expected returns, E(r), and standard deviations of returns
for these assets are as follows:
E(r) Std. Dev.
Stock 0.14 0.17
Bonds 0.10 0.09
Cash 0.08
The average degree of risk aversion of the portfolio’s clients is A= 4.
(a) What is the optimal complete portfolio composition if the correlation between stock and bond returns
is 0.2? (10 points)
(b) What is the optimal complete portfolio composition if the correlation between stock and bond returns
is 0.6? (10 points)
(c) Are your answers to (a) and (b) consistent with the analyst’s point? How would you explain what is
happening as we move from the conditions in part (a) to those of part (b)? (10 points)

Answer 1

(a) The utility factor U is given by

U = E(p) - 0.5*A*Sd(p)*Sd(p)

Where E(p) is the expected return on the portfolio and Sd(p) is the standard deviation of the portfolio

An investor tries to maximize this function.

For finding the optimal complete portfolio, we use an excel solver.

The expected return on the portfolio is the weighted average of the individual asset returns. The standard deviation is found using the formula (as calculated)

Here, cash is considered to be risk-free asset

We set the solver to maximize the utility function

Solving, we get

Weight in Stock	Weight in Bond	Weight at cash	Expected return	Standard deviation
47.26%	43.88%	8.87%	0.11713	9.635%

Utility function

0.098564797

b) If the correlation between stock and bond returns is 0.6

Setting the correlation = 0.6 and the same excel parameter as in part a), we get

Weight in Stock	Weight in Bond	Weight at cash	Expected return	Standard deviation
50.46%	4.54%	45.00%	0.11118	8.830%

Utility function

0.095592294

c)

As we move from part a) to part b), the correlation coefficient between stock and bond increases from 0.2 to 0.6

Yes, the answers are consistent with the analyst's point.

Here,as we observe that the expected return is about same ~11-11.7% in both the cases. The weight in stocks also remains fairly similar `47%-50%

However, similar to the analyst's say that there would be major major asset shifts between cash and bonds we observe that the weight in bonds decreases from ~44% to 4.5% and the weight of cash in the portfolio increases from ~9% to 45%.

As they move from part a to part b, the correlation between stock and bond increases. Due to this, for the same allocation weights, the overall risk, and hence the utility for the investor decreases. Lower the correlation, lower the risk, and hence higher the utility function. As correlation increases, more weight is given to cash as more overlapping between bond and stock leads to higher risk.