Making sense of negative Sharpe ratios

I'm teaching an in-house Fundamentals of Performance class this week in Canada, and, as usual, we touch upon the Sharpe ratio, and how negative Sharpe ratios can produce results which appear inconsistent with our expectations.

To help try to communicate what's going on, I constructed the following graphic:

What you're seeing are two different cases: one where we're dealing with a positive Sharpe ratio, and the other where we have a negative. In both cases, the portfolio's risk exceeds that of the benchmark, and in both cases the portfolio's return equals that of the benchmark. On the positive side, given the higher risk, we would expect a higher return for the portfolio; but because it failed to do that, we end up with a lower Sharpe ratio. On the negative side, we would expect to see a lower return, given the higher risk; but failing to see this occur, we are rewarded with a higher Sharpe ratio.

This may not be clear enough to comprehend, and I will take the subject up later this month, in our monthly newsletter. So, consider this a "warm up"!