For returns less than the risk-free rate, more volatility will actually improve the Sharpe ratio (push it closer to 0 when it is negative.) Is this a valid limitation? Example:
Risk free rate = 5%.
Invest in a money market fund yielding 1%, volatility (s.d.) 0.01% - Sharpe ratio = (1-5)/0.01 = -400
Invest in a stock returning -10%, volatility (s.d.) 15% - Sharpe ratio = (-10-5)/15 = -1.
Which investment is better as judged by Sharpe ratio?
first question … when something is giving you below the “risk free” rate - why would anybody invest in that piece? You are “risk free” doing practically nothing. Then you want to judge yourself with a Sharpe Ratio measure for investing in that “highly risky” investment - when you were practically better off doing nothing?
The question is not whether you would invest in a manager that will perform below the risk-free rate. The answer is obviously no, but that’s if you knew s/he was going to do that. You don’t know the manager is going to do so. When you look at returns of a manager and try to calculate a Sharpe Ratio, you are using historical data. Any manager that has a negative month or other period will “underperform” the risk-free rate.
If you increase risk, the Sharpe Ratio comes down if it is a positive number. But as the OP said, if Sharpe Ratio is negative, then increasing risk actually increases Sharpe Ratio. This is a limitation of the use of the Sharpe ratio. I don’t recall if it is one of the limitations in the curriculum though, which is what matters for our purposes!
I would suspect that the main response as a limitation is it expects a normally distrubuted chance of return outcomes and is therefore not applicable for PE who typically used derivatives - skewing the curve.
For the exam, there is zero chance they will give us a Rp < Rf because as cpk123 said no-one with that information up front would invest with such a manager