My main point is that [tex]V_t[/tex] changes if and only if we upscale or downscale this particular trading strategy. If [tex]V_t[/tex] is constant, we can simply use PnL to substitute percentage returns ([tex]R_t[/tex] using Ken’s notation). If not, we can still calculate the Sharpe ratio by using PnL divided by scaling factors.
The magnitude of [tex]V_t[/tex] is unimportant – only the relative scaling matters. In my example, we upscale the strategy by a factor of 2 at [tex]t = 4[/tex], so the scaling vector becomes (1, 1, 1, 2, 2). Regardless of whether we calculate returns at regularly spaced time or not, Sharpe ratio should be scale independent.
1. $3.67 is simply the sample standard deviation of ($2, $1, $5, -$5, $2). Sorry, I didn’t define SR explicitly in my previous post.
2. As for a strategy that involves a basket of assets (e.g. pairs trading), we don’t need to calculate the Sharpe ratios of individual assets. We may simply treat them as a whole and calculate the SR for this strategy. E.g. Assume [tex]Z = X – alpha Y[/tex], [tex]alpha = -1[/tex], and the returns from [tex]X[/tex] and [tex]Y[/tex] [tex](t = 1, ldots, 9)[/tex] are listed in the table below:
X: $1 NA $2 NA -$1 NA NA $2 -$1
Y: $1 $1 NA -$1 NA $3 NA -$1 $0
To calculate the SR of the strategy, simple use the sample mean divided by the sample standard deviation of [tex]Z[/tex]:
Z: $2 $1 $2 -$1 -$1 $3 $0 $1 -$1
[tex]SR_Z = (2/3) / (1.5) = 4/9.[/tex] We can then compare different strategies (traded at similar frequencies) by using annualized (or some other time scales, e.g. daily) SR.
P.S. If the strategy is NOT scalable (i.e. the expected monetary return is not proportional to the scaling factor due to low liquidity, large transaction cost, etc), we should treat the scaled strategy as different (competing) strategy rather than using the scaling vector approach described above.