Last month I took one funding-carry paper run and computed Sharpe three ways. Daily excess return averaged 6.4 basis points, with an 81-basis-point daily standard deviation. Times √252, that's 1.25. Times √365, that's 1.51. Sample the same equity curve every five minutes and multiply by √(365×288), and the note says 3.08. The first two numbers are a calendar argument. The third is what our research agents emit when they annualize at the frequency of the data file instead of the frequency of the decision. It isn't a Sharpe ratio in any sense I would defend.
Sharpe (ann.): 3.08. Five-minute returns, crypto_carry_v3.
I keep the 3.08. That is the one that needs explaining.
Why does a five-minute Sharpe double the daily one?
The square-root-of-time rule is a statement about independent draws. Add N i.i.d. returns and the mean scales with N, the standard deviation with √N, the ratio with √N. Five-minute crypto returns are not independent draws. The mid bounces off the bid. A stale mark ticks back. Funding does not accrue in 288 independent slices; it posts on a clock. A book that trades at 16:00 UTC and then sits overnight did not make 287 other decisions while you slept.
When short-horizon returns mean-revert, and microstructure noise mean-reverts by construction, a mean-reversion signal sampled every five minutes also gets to trade that bounce. Daily marks net it out. The extra 1.57 of annualized Sharpe is often a two-basis-point oscillation that never survived an eight-basis-point round trip. You did not find more edge. You sliced noise into more pieces and then multiplied by 324.
Andrew Lo wrote the √N failure down in 2002, for serial correlation in monthly hedge-fund returns. Different market. Same sin. We still treat √N as a formatting choice.
252 or 365 is the smaller fight
US equity NAVs skip weekends and holidays, which gives you about 252 points a year. Crypto perps print Saturday. If you mark a BTC book every calendar day, Saturday is a real P&L with real funding, and you have about 365 returns. Using 252 on that series understates the annualized number by about 20 percent, because √365/√252 is 1.20. I use 365 for a 24/7 book marked daily and 252 for a US equity book marked on session closes. I write the constant next to the Sharpe.
That discrepancy is real. It is also not how 1.51 became 3.08. Comparing a crypto 1.51 built on 365 to an equity 1.51 built on 252 without converting is how pitch decks get made. Convert both to a calendar-year excess return over the standard deviation of calendar-year returns, from daily NAVs, or don't compare them.
Annualize at the decision clock
If the strategy can trade every five minutes and actually does, five-minute returns are the honest series. You still should not blindly multiply by √105,120, because those returns are serially correlated. Build a daily NAV from the five-minute book, then annualize the daily Sharpe with √365. The daily mark folds the bounce into the same number you would read off a paper account.
If the strategy trades once a day, do not give it a five-minute Sharpe at all. And do not resample a daily strategy into minute bars so the equity curve looks smoother. A colleague once sent me a velvet-looking plot that was a daily book, forward-filled into minutes. The trades had not changed. The minute-file Sharpe was enormous. I still think about that curve when an agent says it checked multiple timeframes.
What belongs on the research note
A Sharpe is a property of a return series, not a property of a strategy. The note has to say which series.
| Convention | Multiplier | Honest when |
|---|---|---|
| Daily × √252 | 15.87 | US equities, session NAVs, weekends are not returns |
| Daily × √365 | 19.10 | Crypto (or any 24/7 book) marked every calendar day |
| Hourly × √(365×24) | 93.6 | Each hour is an independent decision, and you still believe √N |
| 5-minute × √(365×288) | 324.3 | Almost never, for a strategy that does not trade 288 times a day |
I want three things next to every Sharpe we print: the return frequency, the annualization constant, and whether weekend marks are in the sample. If an agent omits them, I throw the number out and recompute from the daily NAV.
Paper trading settles the argument in a boring way. Mark the paper account at 00:00 UTC. Annualize that series with the number of marks in the year. If the backtest only agrees after you change √N, you were never looking at the strategy.
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