I'd like to thank Mark Johnson for his prior posts that got me thinking about this stuff. I think I ended up with a pretty important improvement to my own measures as a result.
In a private message Mark Johnson wrote:The different ways we picked to compute the S.R., sent me on a web hunt. I found Professor Sharpe's website
http://www.stanford.edu/~wfsharpe/art/sr/sr.htm
These paragraphs were helpful to me. They appear right below his equation (10).
In practice, the situation is likely to be more complex. Multiperiod returns are usually computed taking compounding into account, which makes the relationship more complicated. Moreover, underlying differential returns may be serially correlated. Even if the underlying process does not involve serial correlation, a specific ex post sample may.
It is common practice to "annualize" data that apply to periods other than one year, using equations (7) and ( 8 ). Doing so before computing a Sharpe Ratio can provide at least reasonably meaningful comparisons among strategies, even if predictions are initially stated in terms of different measurement periods.
To maximize information content, it is usually desirable to measure risks and returns using fairly short (e.g. monthly) periods. For purposes of standardization it is then desirable to annualize the results.
I have a long-time and well-earned reputation for disagreeing with experts. Let this occasion not pass me up.
I believe that the Sharpe Ratio for monthly data is damn close to useless for measuring what I care about. I really don't care whether or not the monthy returns are smooth.
I'd much rather have returns of 10%, 25%, 10%, 25%, 10%, 25%, than 10%, 12%, 10%, 12%, 10%, 12%, guess which one has the better Sharpe Ratio? Annualizing the monthly numbers, as Sharpe suggests, doesn't help this situation since the basis is still the standard deviation of the monthly returns.
Now, all's not entirely bad with the Sharpe Ratio. The issue is that one should really be using timeframes for the Sharp Ratio that represent the time horizon the investor cares about.
After thinking about it for a bit, I realized that a one year Sharpe Ratio works pretty well for me. I would much prefer a system that returns 50%, 50%, 50%, 50%,... over one that returns 200%, -25%, 200%, -25%, ... even though they compound to the same return.
I'm even okay with using monthly datapoints, one can simply use the trailing 12 month geometric average return each month. This takes out the rather arbitrary western calendar year effect. A 12 month trailing Sharpe Ratio is much better for me.
One could even combine everything into a formula like:
Return X 2.5
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(Avg worst 5 DD + 2 Std. Dev. ) X Std. Dev. of trailing 12 month Return
which comes pretty close to incorporating all I care about in one number.
If anyone is curious about the 2.5 in the numerator that comes from:
- 1 for the MAR Ratio like Return / Drawdown
- 1 for the Sharpe Ratio like Differential Return / Drawdown
- 0.5 because contrary to what the various ratios would have you believe higher return is better, so it gets a little extra weight.
So we really have:
1/2 return X MAR Ratio X Sharpe-like Ratio using trailing 12 months
I'm calling it Sharpe-like since I don't use differential returns. I'm not sure that the risk-free rate reduction used by Sharpe is a big factor for the kinds of returns I demand from my trading systems, so I prefer the simplification.
MAR-like accounts for using the average of the five worst drawdowns plus two standard deviations to get a better estimate of the potential worst case than what a single drawdown would suggest. This is a modification to Mark's improvement to the MAR ratio.
He uses the average of the five worst drawdowns. I thought that adding a couple of standard deviations would improve it as a measure of worst-case scenarios.