The Sharpe ratio has a strict definition: SR = (E[R(p)] - r_f) / σ(R(p)). Again, see Wikipedia if need be: https://en.wikipedia.org/wiki/Sharpe_ratio
If Q insist on defining a return to risk ratio using: E[R(p)] / σ(R(p)), they can, but then, this is not a Sharpe ratio. And therefore, Q should use another name for it, whatever they want. They could revert back to what it was originally called in the 60's: a "reward-to-variability" ratio as @Vladimir also raised.
But, Q should understand that it is not comparable to what has been defined as a Sharpe ratio for the past 50 years, a very common term used by the whole portfolio management industry.
The historical long-term Sharpe is about 0.40. It goes like this: SR = (0.10 – 0.035)/0.16 = 0.40625. If we eliminate the risk-free rate from the equation we get: 0.625. That is a 50.8% overstatement. And therefore, becomes not that closely comparable to what is considered the industry standard: the Sharpe ratio.
The Sharpe ratio is designed to compare portfolio returns across portfolios. And if we distort its value, we can still compare things, but we would not be comparing portfolios in general on some Sharpe, but some subset following some other definition like: E[R(p)] / σ(R(p)).
For the case where a strategy is 99% in cash, its calculation should go as follows:
E[R(p)] = 0.99∙r_f + 0.01∙β∙(E[R(m)] - r_f). If beta tends to zero, we get: E[R(p)] → 0.99∙r_f
If beta tends to 1.00, we get: E[R(p)] = 0.99∙r_f + 0.01∙(E[R(m)] - r_f) which also tends to E[R(p)] = 0.99∙r_f + 0.01∙(E[R(m)] - r_f) → 0.99∙r_f + 0.000625 → 0.0353
If beta tends to 2.00, we get: E[R(p)] = 0.99∙r_f + 0.01∙2∙(E[R(m)] - r_f) which tends to
E[R(p)] = 0.99∙r_f + 0.01∙2∙(E[R(m)] - r_f) → 0.99∙r_f + 0.0013 → 0.03595
If totally hedged, we get: E[R(p)] → 0.99∙r_f + 0.01∙β∙(E[R(m)] - r_f) – 0.01∙β∙E[R(m)] = 0.99∙r_f +0.01∙β∙r_f. Again, that is not an interesting scenario either.
Am I trying to say that being 99% in cash is not that great a reward for a portfolio?
If you add an execution premium to the equation, meaning that your trading strategy can generate some real alpha, you would get: E[R(p)] = 0.99∙r_f + 0.01∙β∙(E[R(m) + α] - r_f), in fact saying that the alpha would not matter much if 99% in cash. However, just playing the game, you would get a small part of the historical long-term drift.
For a fully invested scenario with market neutral hedging as per the contest you would get:
E[R(p)] = r_f + β∙(E[R(m)] - r_f) – β∙E[R(m)] = r_f – β∙r_f. Again, not a great scenario. Total hedging would give you a beta of zero which would translate to: E[R(p)] = r_f. So, no wonder we see low CAGRs in the contest ranking.
There is only if you succeed in adding some real alpha to the mix that you might outperform, such as in: E[R(p)] = r_f + β∙(E[R(m) + α] - r_f) – β∙E[R(m)] = r_f + β∙ α – β∙r_f. However, as always, real alpha is hard to get. The CAPM sets the alpha at zero. So, you need a lot of skills to put some in.
Quantopian's alpha might be partly an illusion, since a part of it might reside between (E[R(p)] - r_f) / σ(R(p)) and E[R(p)] / σ(R(p)). Note again that E[R(p)] / σ(R(p)) could be overstated by as much as 50% compared to its nemesis.