What is the Sharpe ratio, and how is Sharpe different?

William Sharpe introduced it in 1966 as the reward-to-variability ratio. It takes the average return of a portfolio above the risk-free rate and divides it by the standard deviation of those returns,

where $\bar{R}$ is the average return per period, $R_f$ the risk-free return over the same period, and $\sigma$ the standard deviation of returns. Two portfolios that made the same 20% are not equal if one swung twice as hard to get there, and the ratio says which one paid less risk for the same result. Higher is better.

Take daily returns over the period, find their mean and their standard deviation, and divide one by the other. Then annualise, because a daily ratio is a small number that is hard to compare. Stock markets use the square root of 252 trading days. Crypto trades on every day of the year, so the convention is the square root of 365, about 19.1.

Say a trader's 90 days average a return of 0.2% a day with a standard deviation of 4%, and the risk-free rate is set to zero for simplicity. The daily ratio is 0.002 ÷ 0.04 = 0.05, and the annual ratio is 0.05 × 19.1, about 0.96. Fund managers would call that respectable.

The first problem is the yardstick. The ratio measures return above cash. In a year when crypto rises, a trader who carried three times Bitcoin's risk for three times its return shows a strong ratio, although the trading added nothing that a bigger bet on Bitcoin would not have. For someone trading crypto, the alternative to trading is holding crypto, and a measure against cash never asks whether the trading beat it.

Ninety days can also prove very little. Andrew Lo showed in 2002 that the ratio has a standard error, and that it is large over short periods. For the trader above, the error over 90 days works out to about 2.0 in annual terms. A measured 0.96 is consistent with a true ratio well below zero, and also with a very good trader. The number on its own does not tell them apart, and a ranking by raw Sharpe ratio puts the luckiest short records at the top.

Shape is the last problem. The ratio treats a large winning day as risk, the same as a large losing day, and its standard error assumes returns are spread evenly around their mean. Crypto returns are not. Small-token trading in particular produces a few very large days among many ordinary ones, and those few days can carry the whole record.

Sharpe compares each day of a record with Bitcoin's return on the same day. The mean of those daily differences, divided by their standard deviation, is an information ratio against Bitcoin (Goodwin, 1998), the Sharpe ratio's relative that uses a benchmark in place of cash. The differences are taken in logs, so that a record which swung hard and lost money cannot average its way to a good score.

The ratio is then discounted by half its own standard error, estimated without assuming that returns are spread evenly, and annualised. The result is placed on a ladder as Edge. A record at 2,000 Edge or above, the Bitcoin line, finished ahead of holding Bitcoin by enough to clear half its uncertainty. Beside every rank, Sharpe shows Confidence, the probability that the record's true excess over Bitcoin is above zero, which is the probabilistic Sharpe ratio of Bailey and López de Prado (2012).

For comparing two portfolios or funds against cash over a long record, the Sharpe ratio is a fair measure. For the question a crypto trader usually means, whether their trading did better than holding Bitcoin, the benchmark has to be Bitcoin, and the measure has to say how much evidence stands behind the answer. The methodology has every equation Sharpe uses, and any Solana or Ethereum wallet can be pasted into Sharpe to see its own.