How do I calculate Sharpe ratio from P&L?

Say I have a market-making strategy that trades intraday. I start with a flat position and finish flat too. I end up with a daily P&L $p_{today}$. Over a year of trading I get $\vec{p} = (p_1,\dots,p_{252})$.

There is no way to calculate returns here. As such I calculate $$Sharpe = S(\vec{p}) = \sqrt{252} \cdot \frac{\mathbb{E}[\vec{p}]}{\sqrt{\mathbb{V}[\vec{p}]}} = \sqrt{252} \cdot \frac{mean(p)}{sd(p)}$$

My questions are :

1. Am I right to do it like this?
2. Do you usually bootstrap your Sharpe? (I do not but I am interested in your view of it.)
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Why is there no way to calculate returns? What about $(p_{i+1}-p_i)/p_i$? – user1157 Jan 27 '14 at 22:05
The returns here have nothing to do with pnl, please avoid downvoting without reading properly the post – statquant Jan 27 '14 at 22:59
I would like to understand the question. Why is there no way to calculate returns? Could you explain that in your question please? – user1157 Jan 28 '14 at 7:31

There is no way to calculate returns here.

Let me stop you right there. You didn't open a brokerage account with zero dollars. The money you put-up for margin is your starting position. After a year of trading, you have a stopping position represented by a different amount of money in your account. The change from your starting position to your stopping is your return.

Am I right to do it like this?

Your formula for annualized Sharpe ratio is correct, assuming you didn't introduce more margin into your brokerage account to do bigger trades. For a fair comparison using P&L, you must have the same amount of capital that you started with.

Do you usually bootstrap your Sharpe?

I've never heard of resampling applied to performance metrics like this. At least not by industry practitioners.

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Not sure what you really mean by "You did not open a brokerage..." as a practictioner I do not open anything. I start flat I use some capital provided by the firm, and I end up flat at the end of the day. It doesn't make sense (should you even find a way) to calculate return in that case I think. – statquant Nov 19 '13 at 14:31
@statquant How much capital did the firm have at the end of the day? More? Less? How does your employer determine your compensation? – chrisaycock Nov 19 '13 at 15:00
Sorry I don't get what you are saying, can you show me how you would calculate the return (I think you'll see the problem then) – statquant Nov 19 '13 at 15:04
@statquant Let's say your firm posts \$10M with the prime broker. And let's say the firm's P&L at the end of the year is \$1M. That's a 10% return. As I stated in my first paragraph above, returns are computed based on the capital under management. If your employer felt they could get more than 10% returns from an index fund, then surely they would shut the company down and put that \\$10M in an ETF. You personally might not consider returns in that light, but the backers of your firm certainly do. – chrisaycock Nov 19 '13 at 15:26
Return on collateral posted isn't really a metric for making business decisions in an HFT environment. P&L generation is limited primarily by strategy capacity rather than available capital (else HFT shops would have raised more capital from outsiders). Mostly one is making capital budgeting decisions on e.g. infrastructure spend (e.g. microwave towers, accessing new venues, new/upgraded hardware). This generates spend that effectively becomes a sunk cost - there's no easy reallocation of this "capital invested" to an ETF. – afekz Oct 17 '14 at 11:29

It is true that intraday/market-making strategies don't have a reasonable "return" metric. For this reason you can't characterize them with the Sharpe Ratio, which depends on a capital basis and how that basis is leveraged (not to mention the risk-free rate on the capital basis).

What you're asking is how to characterize the performance of a daily stream of dollar income that doesn't have a capital basis. Typically I would start with mean, standard deviation, and skewness. Or I might ask for %Winning days and AvgWin/AvgLoss, or Profit Factor. Bootstrapping your data does not benefit any of these measures.

Then I would go to other metrics where bootstrapping (i.e., resampling the returns to generate different return paths) could be beneficial. E.g., max drawdown, or max time to recover (i.e., return to high water mark).

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That is simply incorrect. Of course do any trading strategies allow for the computation of return metrics. How else do you think some hft houses generate and publish 7+ Sharpe ratio performance metrics? With all due respect but you do not sound like a market practitioner at all. Resampling is certainly not done by professionals in this context. And why do you talk about drawdowns or recovery periods when commenting on risk adjusted return metrics? – Matt Wolf Jul 15 '14 at 15:39
I am a professional quant, and we most certainly do bootstrap to generate drawdown estimates from limited data sets. I worked at one HFT shop, and we would generate "Sharpe ratios" more as a joke, or to satisfy unsophisticated stakeholders. Sharpe ratio does not make sense in such a context for the reasons described in my answer. – feetwet Jul 15 '14 at 16:04
Of course it makes sense why would it not, it is important to know whether a high frequency trading strategy has sufficient turnover and generates sufficiently positive returns in the context of risk, taken, whether we talk daily Sharpe or annualized Sharpe.And as said bootstrapping, resampling, drawdowns, nor avg win/loss has anything to do with the question at hand. Anything that generates income (even your salary) is somehow capitalized, hence the ability to calcultate returns. – Matt Wolf Jul 15 '14 at 16:10
@Matt: Sharpe is an extremely a poor metric and IMO downright misleading for answering whether a HF strat generates "sufficiently positive returns in the context of risk taken". Daily standard deviation of P&L isn't remotely reflective of risks taken. The highest risks with HF are usually operational, not financial("Knightmare"). P&L is more related to strategy capacity than it is to any capital invested metric. And which capital invested metric do you choose? Cash invested(huge infrastructure spend, with large sunk costs), realisable value, sum of collateral held at PB's/settlement agents? – afekz Oct 17 '14 at 11:03
Straw man: I didn't argue that as a trader(/PM) one thinks "all day long" about pricing op risk, but that 1) because of this (significant) tail risk (and the asymmetry, which I didn't write above), Sharpe can never show whether a strat generates sufficiently positive returns "in the context of risk taken" and 2) that even calculating Sharpe ratios requires some kind of decision-making of what constitutes "capital invested", given that collateral (recall "zero"(low) overnight positions) forms an unusually small portion of capital invested relative to "asset managers" ordinarily assessed with – afekz Oct 20 '14 at 6:27