Standard deviation
Also written Sigma
A measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
In plain language
Two funds both averaged 12% a year over three years. The first returned 11%, 12% and 13%. The second returned −8%, 12% and 32%.
The average is identical and the experience of owning them is not remotely comparable. Standard deviation is the number that tells them apart: it measures the typical distance between a year's return and the average return.
How it works
Take each year's deviation from the mean, square it so that overshoots and undershoots do not cancel, average the squares to get the variance, then take the square root to return to percentage units.
If returns are roughly normally distributed, the result reads directly as a range: about two years in three fall within one standard deviation of the mean, and about nineteen in twenty within two. A fund averaging 12% with a standard deviation of 20% should therefore be expected to land between −8% and +32% in two years out of three — and worse than −28% in one year in twenty.
The formula
σ = √[ Σ (Rᵢ − R̄)² ÷ (n − 1) ]
Variance = σ²
Divide by n − 1 for a sample and by n for a full population. NISM questions almost always intend the sample form.
A worked example
Two equity funds, five years of annual returns, both averaging exactly 12%:
| Year | Fund A | Fund B |
|---|---|---|
| 1 | 14% | 38% |
| 2 | 9% | −14% |
| 3 | 12% | 26% |
| 4 | 11% | 4% |
| 5 | 14% | 6% |
| Mean | 12% | 12% |
Fund A: deviations 2, −3, 0, −1, 2
squares 4, 9, 0, 1, 4 = 18
variance 18 ÷ 4 = 4.5 σ = 2.1%
Fund B: deviations 26, −26, 14, −8, −6
squares 676, 676, 196, 64, 36 = 1,648
variance 1,648 ÷ 4 = 412 σ = 20.3%
Same average return, ten times the volatility. On Rs 10 lakh, Fund A's worst year still returned Rs 90,000; Fund B's worst year lost Rs 1.4 lakh. An investor who needed the money in year two experienced two completely different products.
Why NISM asks about it
Chapter 12 (Fundamentals of Risk and Return) introduces standard deviation as the measure of total risk, and it returns as the denominator of the Sharpe ratio. Expect a short series of returns to compute from, and questions that turn on the difference between variance and standard deviation.
Common exam traps
- Variance is in squared units and standard deviation is its square root. Answer options routinely offer both; only one is comparable to a return.
- Standard deviation is total risk, beta is systematic risk. A question asking for the risk that diversification cannot remove is not asking for this.
- It treats upside and downside identically. A fund whose surprises are all pleasant still scores as risky.
- A portfolio's standard deviation is not the weighted average of its holdings'. Correlation below 1 makes the portfolio less volatile than its parts — which is the mathematics of diversification.
- Comparing standard deviations computed over different periods, or on monthly against annual data, is meaningless.
Where this is taught
- Series XIX-B · Chapter 6: Fees Structure, Fund Performance and Benchmarkingintroduced here
- Series XV · Chapter 12: Fundamentals of Risk and Returnintroduced here
- Series V-D · Chapter 10: Risk, Return and Performance of Fundsintroduced here
- Series V-A · Chapter 10: Risk, Return and Performance of Fundsintroduced here
- Series X-A · Chapter 14: Introduction to Modern Portfolio Theoryintroduced here
- Series XVII · Chapter 5: Evaluating Fund Performance & Fund Selectionintroduced here
- Series XIX-C · Chapter 9: Fee Structure and Fund Performanceintroduced here
Related terms
- BetaHow sharply a share moves relative to the market index — beta 1 moves with the index, above 1 amplifies it, below 1 dampens it. The standard measure of systematic risk.
- CAPMA model that prices the return an investor should demand from a share: the risk-free rate plus beta times the market risk premium.
- Sharpe ratioReturn earned above the risk-free rate divided by standard deviation — how much reward an investment produced for each unit of total risk its holder had to live with.
- Systematic riskThe part of an investment's risk that comes from economy-wide forces moving every asset at once — it cannot be diversified away, and it is the only risk the market pays you to carry.
- RiskThe possibility that actual returns turn out different from what was expected — measured as the dispersion of returns around their own average, and not the same thing as uncertainty.
- Modern Portfolio TheoryMarkowitz's framework for building portfolios on expected return and risk together, in which the co-movement between holdings — not their individual riskiness — decides the risk of the whole.
- Unsystematic riskThe part of an investment's risk that belongs to one company or one issuer — a strike, a fraud, a downgrade — and which diversification can remove, unlike market-wide systematic risk.
- Treynor ratioRisk premium per unit of market risk — the return a scheme earned above the risk-free rate, divided by its beta rather than by its standard deviation.
- AlphaThe return a fund earned above what its beta and the benchmark say it should have earned — the slice of performance left over once the market has been given credit for its share.
- CAGRThe single smoothed annual rate at which a starting value would have to grow, compounding each year, to reach the ending value over a given period.
- BenchmarkThe independently published index a scheme's performance is measured against, chosen to match its investment objective, asset allocation and strategy, and disclosed in the Scheme Information Document.
- DiversificationSpreading an exposure across holdings that do not move together, so that total risk falls by more than total return does — minimising risk per unit of return.