Standard deviation in finance

Most risk numbers quoted for a stock or a fund are the standard deviation of its returns, rescaled or divided by something. Volatility, the Sharpe ratio, beta and a simple value at risk all start from the same two quantities that sdtoolkit computes: the mean and the sample standard deviation.

The data

Ten daily returns, in percent, for a stock and for the market index it trades against (illustrative numbers):

day stock market
1 1.2 0.8
2 −0.8 −0.5
3 0.5 0.3
4 2.1 1.2
5 −1.5 −1.1
6 0.9 0.6
7 −0.3 −0.2
8 1.4 0.9
9 −0.6 −0.4
10 0.7 0.4 
>>> import sdtoolkit as sd
>>> stock = [1.2, -0.8, 0.5, 2.1, -1.5, 0.9, -0.3, 1.4, -0.6, 0.7]
>>> market = [0.8, -0.5, 0.3, 1.2, -1.1, 0.6, -0.2, 0.9, -0.4, 0.4]
>>> sd.mean(stock)
0.36
>>> round(sd.sample_variance(stock), 6)
1.267111
>>> round(sd.sample_stdev(stock), 4)
1.1257

Returns are a sample of what the stock could do, not the whole population, so the n − 1 versions are the right ones (see sample-vs-population.md). If you want to see where the n − 1 comes from, the variance formula page derives both versions side by side.

Volatility

Volatility is the standard deviation of returns, quoted on a yearly scale. Daily returns are close enough to independent that their variances add, so the yearly variance is about 252 times the daily one (252 trading days), and the yearly standard deviation is √252 times the daily one:

annual volatility = daily SD × √252
                  = 1.1257 % × 15.875
                  ≈ 17.9 %

>>> import math
>>> round(sd.sample_stdev(stock) * math.sqrt(252), 2)
17.87

Use √52 for weekly returns and √12 for monthly ones. Ten days is far too short for a real estimate; a year of daily data is the usual minimum. A volatility calculator does the annualising from a column of prices, including the step from prices to returns.

Sharpe ratio

The Sharpe ratio asks how much return you got per unit of standard deviation, after subtracting what a risk-free asset would have paid:

Sharpe = mean(return − risk-free) / SD(return − risk-free)

With a risk-free rate of 0.02 % a day:

>>> excess = [r - 0.02 for r in stock]
>>> daily = sd.mean(excess) / sd.sample_stdev(excess)
>>> round(daily, 4)
0.302
>>> round(daily * math.sqrt(252), 2)
4.79

The mean scales by 252 and the SD by √252, so the ratio scales by √252. An annual Sharpe of 4.8 would be extraordinary; it comes from ten good days, not from a good strategy, which is why the sample length matters as much as the number. To compare funds over the same period, a Sharpe ratio calculator takes the returns and the risk-free rate directly.

Beta

Beta measures how much the stock moves when the market moves. It is the covariance of the two return series divided by the variance of the market:

β = Cov(stock, market) / Var(market)

sdtoolkit has no covariance function, but the formula is one line on top of the means:

>>> ms, mm = sd.mean(stock), sd.mean(market)
>>> cov = sum((a - ms) * (b - mm) for a, b in zip(stock, market)) / (len(stock) - 1)
>>> round(cov, 6)
0.815556
>>> round(sd.sample_variance(market), 6)
0.528889
>>> round(cov / sd.sample_variance(market), 3)
1.542

Both terms use n − 1, so the choice cancels; what matters is using the same divisor in both. A beta of 1.54 means the stock has moved about 1.5 % for each 1 % move in the index. A covariance calculator shows the cross-products step by step, and a stock beta calculator goes straight from the two price series to β.

A one-day value at risk

If daily returns were normal, the worst day in twenty would sit about 1.645 standard deviations below the mean:

VaR(95 %) = mean − 1.645 × SD
          = 0.36 − 1.645 × 1.1257
          ≈ −1.49 %

The 1.645 comes from the normal distribution; a z-table lists the value for any other confidence level (2.326 for 99 %). Real returns have fatter tails than the normal curve, so this figure understates how bad the bad days get; treat it as a floor on the risk, not a ceiling.

Summary

measure built from value here
daily SD sample SD of returns 1.13 %
annual volatility daily SD × √252 17.9 %
Sharpe ratio (annual) mean excess ÷ SD, × √252 4.79
beta covariance ÷ market variance 1.54
95 % one-day VaR mean − 1.645 × SD −1.49 %

For more of these tools in one place, the finance calculators index groups them by topic.