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Sharpe ratio: return per unit of risk

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Start with the idea

Sharpe measures average excess return relative to return variability. It is sensitive to horizon, financing conventions, serial correlation and rare losses. A smooth path from stale pricing can create a deceptively high ratio.

Symbols, units & horizon

R_p: portfolio return per observation period
R_f: matching risk-free return
σ_p: SD of excess returns, using a constant risk-free rate in the displayed identity
E: expected return, estimated with sample mean
A or 252: assumed observations per year
Sharpe: mean excess return divided by its SD
√A: IID square-root annualisation approximation

When and why to use this

Use Sharpe to compare risk-adjusted histories only after aligning currency, frequency, financing and costs. Pair it with drawdown, tail risk and confidence intervals.

Sharpe = \frac{𝔼 [R_{p}] - R_{f}}{σ_{p}}, annualised = {Sharpe}_{daily} \times \sqrt{252}

Algebra and arithmetic

Derive square-root annualisation

For IID periodic excess return with mean m and SD s, an A-period sum has mean Am and variance As². The ratio is $\frac{A m}{(s \sqrt{A})} = \sqrt{A} (\frac{m}{s})$ .
This concerns additive excess returns; annual compounded returns and serial dependence require additional care. Using n IID normal observations, an asymptotic annualised SE is $\sqrt{\frac{A}{n} + \frac{S_{a n n}^{2}}{(2 n)}}$ .

Work it by hand

Daily excess mean=.0005, SD=.008: daily ratio=.0625 and annual approximation=.0625√252=.992. For monthly Sann=.89 and n=36, approximate SE=√(12/36+.89²/72)=.587.

Sharpe is the number that lets you compare a 5%-vol bond strategy with a 40%-vol crypto strategy: Sharpe normalises return by volatility. Equal Sharpe does not imply equivalent liquidity, tail risk, financing or feasible leverage. There is no universal institutional deployment threshold.

Sharpe	what it feels like	years of data to be confident it is > 0
0.5	long, grinding drawdowns; hard to hold	≈ 16
1.0	roughly a losing year in six	≈ 4
2.0	losing months, rarely losing quarters	≈ 1
3.0+	HFT / market making; capacity-limited	months

The last column comes from the CLT: the standard error of an annualised Sharpe estimate over $T$ years is roughly $\sqrt{\frac{A}{n} + \frac{{Sharpe}^{2}}{(2 n)}}$ . Here n is the number of IID approximately normal observations and A is observations per year. With n = AT and modest daily Sharpe, SE is approximately 1/√T. Dependence and selection can materially increase uncertainty.

Mean daily return 0.05%, daily σ 0.8%, risk-free ≈ 0. Annualised Sharpe?

Daily Sharpe $= \frac{0.0005}{0.008} = 0.0625$ ; times $\sqrt{252} = 15.87$ gives 0.99.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from statistics import mean, stdev
from math import sqrt

def sharpe(excess_returns, periods_per_year=252):
    """Supply aligned portfolio minus risk-free returns, all as decimals."""
    sd = stdev(excess_returns)
    if sd == 0:
        raise ValueError("Sharpe undefined for zero sample SD")
    periodic = mean(excess_returns)/sd
    return periodic, periodic*sqrt(periods_per_year)

print(sharpe([.01, -.005, .002, .008]))

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