Standard Deviation (Finance)

Standard deviation, in finance, is a statistical measure of how widely an investment's returns spread around their average. A fund whose yearly returns sit close to its average has a low standard deviation; one whose returns swing from large gains to large losses has a high one. It is the most common way of putting a number on volatility, and therefore on investment risk.

The measure comes from general statistics, where it describes the spread of any data set. In investing it appears on fund fact sheets, in the Sharpe ratio, in portfolio theory and in banks' risk models. It is related to, but different from, beta: standard deviation captures all the ups and downs of an investment, while beta captures only the part linked to the overall market.

Think about a daily commute. If the journey always takes between 28 and 32 minutes, you can plan your morning with confidence. If it sometimes takes 15 minutes and sometimes an hour, the average may still be 30 minutes, but the average tells you much less about tomorrow.

Standard deviation is a way of saying how far, typically, each day lands from the usual. For investments, the "journey" is the yearly or monthly return. A low figure means returns tend to land near the average, so the investor knows roughly what to expect. A high figure means results are scattered, so a bad year can be much worse than the average suggests. It measures how unpredictable the ride is, not whether the destination is good.

For a sample of past returns, standard deviation is the square root of the average squared distance from the mean:

• $\sigma$ is the standard deviation of returns.
• $R_t$ is the return in period $t$.
• $\bar{R}$ is the average (mean) return over all periods.
• $N$ is the number of periods; dividing by $N - 1$ rather than $N$ is the usual correction when working from a sample.

Take four yearly returns: 10%, −5%, 15% and 0%. The mean is 20 ÷ 4 = 5%. The distances from the mean are 5, −10, 10 and −5 points. Their squares are 25, 100, 100 and 25, which add to 250. Dividing by 3 gives about 83.3, which is the variance. The square root of 83.3 is about 9.1, so the standard deviation is about 9.1%.

Returns measured monthly are usually converted to an annual figure by multiplying by the square root of the number of periods in a year:

A monthly standard deviation of 4% becomes about 4% × 3.46 ≈ 13.9% a year. Daily figures are usually scaled with the square root of about 252 trading days.

If returns followed a normal, bell-shaped distribution, about two-thirds of periods would fall within one standard deviation of the average and about 95% within two. A fund averaging 8% a year with a standard deviation of 15% would then return between −7% and 23% in roughly two years out of three, and between −22% and 38% in about 19 years out of 20.

Typical patterns, which change over time, are:

• Cash, money market funds and Treasury bills have very low standard deviations.
• High-quality bonds sit higher, with long-dated bonds more volatile because of interest rate risk.
• Broad share indices such as the S&P 500 sit higher still.
• Single shares, small-cap stocks, emerging markets and cryptocurrencies are usually the most volatile.

Standard deviation underpins much of modern finance. Harry Markowitz's 1952 portfolio theory measured risk as the variance (the square of standard deviation) of portfolio returns and showed that combining assets that do not move together can cut a portfolio's standard deviation without cutting its expected return. That is the mathematical basis of diversification and asset allocation.

Fund data services report a fund's standard deviation, commonly over trailing three-year periods of monthly returns, so investors can compare it with peers and with their own risk tolerance. The standard deviation of the difference between a fund's and its benchmark's returns is called tracking error. Option traders quote option prices in terms of implied volatility, which is an annualised standard deviation, and banks use it in value-at-risk models to estimate potential losses on trading books.

• It counts gains as risk. A fund that occasionally jumps sharply upwards shows a higher standard deviation, even though investors do not mind such surprises. Downside measures such as the Sortino ratio address this.
• Real returns have fat tails. Market crashes such as October 1987 and the 2008 financial crisis produced moves far larger than a normal distribution would predict, so standard deviation understates the chance of extreme losses.
• It is backward-looking. Volatility tends to cluster, with calm periods followed by turbulent ones, so a figure from a quiet period can mislead.
• It ignores the order of returns. Two funds with the same standard deviation can have very different peak-to-trough losses, which matter greatly to someone who must sell during a bear market.
• Smoothed prices understate it. Assets that are valued infrequently, such as private equity, show deceptively low figures.

• Standard deviation (Investor.gov, U.S. Securities and Exchange Commission)
• Measures of Scale (NIST/SEMATECH e-Handbook of Statistical Methods)