  # Re: How to compute Sharpe ratio for a high frequency multi-asset long/short strategy

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#2044

Thank you for the detailed explanation.

I think my question should be changed to “how to compute returns for an active portfolio?”.

In order to compute the returns (or differential returns as in Sharpe’s articles http://www.stanford.edu/~wfsharpe/art/sr/sr.htm), you need the original capital $$x$$, i.e.,
$$R = frac{x’}{x} – 1$$
where $$R$$ is the return, and $$x'$$ is the new portfolio value.

It’s easy to compute $$R$$ at regular intervals if the capital for the portfolio is the portfolio value at time 0.
However, when the strategy is not a buy-and-hold strategy, but an actively managed portfolio (i.e., capital is actively injected/withdrawn anytime, in other words, $$x$$ is changing over time), how should one compute the returns at regular (or irregular) intervals?

For example, here are some trades from a strategy:
At t1, BUY 1 unit of A at $10 At t2, SELL 2 unit of B at$20
At t3, SELL 2 unit of A at $14 At t4, BUY 1 unit of B at$18
At t5, BUY 1 unit of B at $16 At t6, BUY 1 unit of A at$11

We can still compute the P&L at t1, t2, …, but how should one compute the capital, hence the returns at these time points?

At t1, the capital is $10 x 1 =$10.
At t2, the capital becomes $10 +$20 x 2 = $50 (correct? what is the capital for short selling?) At t3, the capital becomes$50 + ($14 x 1 – ($14 – $10) x 1) =$60?

Should I just find the maximum of the running capital at all time points and take it as the capital?
The reason for this is that, I treat all these trades as asset reallocation for a fixed-capital portfolio.

So, suppose the price of A rose from $10 at t1 to$12 at t2, the P&L of the strategy at t2 is $2. Since the maximum running capital of these trades is$60, the return for the period [t1, t2] is $2/$60 = 3.33%

Another problem is that, Sharpe ratio is time dependent. The distribution of the time points does matter the ratio. Comparing 2 SR with different time point distributions seems to be meaningless.

For example,
Strategy X:
At t1, buy 1 unit at $1 At t2, buy 1 unit at$1
At t3, sell 2 units at $2 Strategy Y: At t1, buy 2 units at$1
At t3, sell 2 units at \$2

Suppose t1, t2, t3 are at regular intervals.
If we compute SR at regular time points t1, t2, t3, the ratios for both strategies are the same.
However, if we compute SR at each trade time, the ratios for these 2 strategies are different.
In this case, which one is correct?

What do you think?