I think I understand Ken’s concerns. Here’s my two cents worth:
1. (Calculation of Returns)
When dealing with a zero-investment strategy (zero outlay and +/-/0 returns), we can always associate an initial notional value [tex]V[/tex] (this could be the maximum allowable short position/capital for this particular strategy or some other sensible values) at the start of the trading period. The percentage return at time [tex]t[/tex] is then calculated as [tex]R_t = x_t / V[/tex], where [tex] x_t[/tex] denotes the absolute return (measured in the same unit as [tex]V[/tex]) at time [tex]t[/tex]. I won’t suggest to use the “maximum running capital”, since unlike in a buy-and-hold strategy, we should only update the value of [tex]V[/tex] when upscaling or downscaling of the zero-investment strategy is required. This will be discussed next and we will see that Sharpe ratio should be SCALE INDEPENDENT.
2. (Scale Independence)
To incorporate time varying [tex]V_t[/tex], let [tex]R_t = x_t / V_t[/tex].
a) Sharpe ratio is independent of the initial notional value [tex]V[/tex]. E.g. Assume that the maximum allowable capital for a stategy is $200 and we do not upscale or downscale this strategy, the returns in the first 5 time periods are [tex]x_t = ($2, $1, $5, -$5, $2)[/tex]. If we choose [tex]V_t = 200[/tex], then the Sharpe ratio is 0.5% / 1.84% = 0.272. If we have totally ingored the notional value (i.e. [tex]V_t = 1[/tex]) and used the PnL directly in our calculation (as in Haksun’s post), the Sharpe ratio remains unchanged at 0.272 (= $1 / $3.67). In view of this, we can simply use the PnL (measure in the same unit as V, e.g. dollars) instead of percentage return for a zero-investment strategy, since [tex]SR(R_t) = SR(x_t)[/tex] if we do not upscale or downscale this strategy.
b) Next assume the strategy is scalable (i.e. liquid and very low transaction cost, so that the return is proportional to the capital injection). In this case, as long as we are consistent with our choice of [tex]V_t[/tex], Sharpe ratio is still (scale-)independent of the notional values [tex]V_t[/tex]. Continue with the previous example and consider the following senario for instance: after observing very stable and good performance in the first 3 time periods, the management decides to upscale the strategy by doubling the maximum allowable capital at [tex]t = 4[/tex], so that the monetary returns become [tex]x_t = ($2, $1, $5, -$10, $4[/tex]). We should calculate the Sharpe ratio of this particular strategy by using [tex]V_t = (1, 1, 1, 2, 2)[/tex] instead, as if no upscaling was involved. The returns remain unchanged at [tex]R_t = (2, 1, 5, -5, 2) / V[/tex], which yields a Sharpe ratio of 0.272, same as before.
This also applies when we deal with irregular time intervals.
3. (Time Independence)
The beauty of Sharpe ratio is that it’s TIME INDEPENDENT, so that strategies (traded in similar frequencies) over different time horizons can be compared; see Sharpe’s (1966, 1975) original papers for details. I totally agree with Ken that it is meaningless to compare strategies traded at different frenquencies (e.g. compare daily strategies with high-frequency ones). As for Ken’s last example (Stragies X and Y at time [tex]t_1, t_2[/tex] and [tex]t_3[/tex]), there is no single “correct” SR to use, and we should only compare them on the same time scale. E.g. calculate the returns at [tex]t_1, t_2[/tex] and [tex]t_3[/tex] and obtain the annualized Sharpe ratio using these returns and a suitable scaling vector [tex]V_t[/tex].
Please let me know if this answers your questions or if you have other questions/concerns. Thanks.