# Python代写 | B9339 Homework Assignment #2

B9339 Homework Assignment #2

1. (Sharpe ratio enhancement) In this problem, you are to investigate conditions under which
additional investment strategies are beneficial to an existing portfolio. Suppose that you
already have a portfolio and that you are considering adding n new strategies to it. Your goal
is to improve your existing portfolio, and you expect the new strategies to be “beneficial.” A
natural question that arises in this situation is what is the “relationship” between your
existing portfolio and the new strategies that is necessary so that the addition of the new n
strategies is beneficial for your existing portfolio.
To approach this question, you make the following simplifying assumptions:
(a) Your notion (criterion) of “beneficial to your portfolio” amounts to increasing the Sharpe
ratio (defined as the ratio of return over volatility) of your existing portfolio, SRold;
(b) You can adequately describe the correlation structure of the new n strategies by a single
parameter, their average pairwise correlation ρnew;
(c) You can adequately describe the correlation structure of your existing portfolio and the
new n strategies by a single parameter, the average correlation ρold,new of all the pairwise
correlations between your existing portfolio and each of the new strategies;
(d) All the new strategies you are thinking of adding have equal Sharpe ratios, SRnew;
(e) Your portfolio allocation ensures that in your portfolios (existing and new), each
component (strategy) contributes an equal amount of 1 unit of marginal volatility (i.e., in
your new portfolio, your existing portfolio would be contributing an equal amount of
volatility as each of the new strategies).
With this framework:
1. Explain how you can achieve the portfolio allocation of assumption (e) above.
2. Derive an upper bound for the parameter ρold,new in terms of SRold, n, ρnew, SRnew , so
that your new portfolio is indeed an improvement over your existing portfolio.
3. When you are adding only one strategy (n=1), what does your bound in (i) reduce to?
4. Plot your bound for n=1, as a function of SRnew for the typical values of strategy Sharpe
ratios we have seen so far: SRold = 0.25, 0.50, 0.75, 1.00, 1.25. Explain why your plots
make/do not make intuitive sense.
Explain how much more complicated your analysis (bound) becomes if, keeping all other
assumptions intact:
5. All the new strategies do not have the same Sharpe ratio as in assumption (d) above.
6. You cannot adequately describe the correlation structure of your existing portfolio
and the new n strategies by a single parameter as in assumption (c) above.
7. You cannot adequately describe the correlation structure of the new n strategies by a
single parameter as in assumption (b) above.
8. Give an example of another notion (criterion) for “beneficial to your portfolio” that is
different from the Sharpe ratio assumed in assumption (a) above. Explain the
rationale behind your alternative criterion, and how it differs from the Sharpe ratio.
Outline how you could approach the question of the problem for your alternative
criterion (only a brief description is needed; you do not need to carry out the analysis).