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 X-A · Chapter 14: Introduction to Modern Portfolio Theoryintroduced here
- Series V-A · Chapter 10: Risk, Return and Performance of Fundsintroduced here
- Series XXI-B · Chapter 15: Introduction to Modern Portfolio Theoryintroduced here
- 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 XIX-C · Chapter 9: Fee Structure and Fund Performanceintroduced here
- Series XVII · Chapter 5: Evaluating Fund Performance & Fund Selectionintroduced here
- Series XIX-E · Chapter 7: Fee Structure and Fund Performanceintroduced here
- Series XXI-A · Chapter 10: Performance Measurement and Evaluation of Portfolio Managersintroduced here
- Series XXI-B · Chapter 20: Performance Measurement and Evaluation of Portfolio Managers
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.
- 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.
- Maximum DrawdownThe largest peak-to-trough fall in a Category III AIF's AUM over a reporting period, quoted as a percentage of the peak — the worst loss an investor could have suffered in that window.
- SkewnessA measure of how far a fund's return distribution departs from symmetry around its mean. Zero is symmetrical; positive has a long right tail; negative has a long left tail and is riskier.
- KurtosisA measure of how peaked a return distribution is and how fat its tails are. The workbook takes a normal distribution as kurtosis 3; above 3 (leptokurtic) signals outliers and higher risk.
- 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.
- VarianceA measure of how widely a scheme's periodic returns fluctuate around its own average return — the square of standard deviation, and a risk measure relevant to both debt and equity schemes.
- M² measureA portfolio's return re-scaled to the market's level of risk, so it can be compared with the market return directly, in percentage points.
- Sortino RatioExcess return over the risk-free rate divided by semi-standard deviation — a Sharpe-style ratio that counts only downside volatility as risk.
- Parametric VaRValue at Risk estimated from just two parameters — expected return and standard deviation — assuming returns are normally distributed.
- Risk Aversion IndexThe 'A' in the Markowitz utility formula U = E(r) − 0.5 × A × σ² — a number capturing how heavily an individual investor's utility is penalised for taking on variance.
- Semi varianceA downside-risk measure that looks only at the dispersion of returns below the mean (or a target return), ignoring the upside swings that standard deviation counts equally alongside the downside ones.
- Significance levelThe probability, equal to 1 minus the confidence level, that a variable's true value falls outside the chosen confidence range — the basis for picking the z-score used in Value at Risk.
- Z-scoreThe number of standard deviations a value lies from the mean of a standard normal distribution — the cut-off figure used to compute Value at Risk at a chosen confidence level.
- Rolling returnsThe average annualised return over many consecutive holding periods inside an evaluation period — say every one-year return in three years — used to remove the distortion of an unusually high or low start or end point.
- 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.
- CovarianceA measure of the degree to which the returns of two assets move together relative to their own averages — the building block of portfolio risk, standardised into the correlation coefficient.
- Ex-ante returnThe forward-looking expected rate of return: the sum of the possible returns multiplied by their probabilities. The opposite of ex-post return, which is computed from historical data.
- Ex-post returnA return calculated from historical data — what an investment actually earned — as distinct from the forward-looking, probability-weighted ex-ante return.
- Variance-covariance matrixThe grid of every asset's variance and every pair's covariance that is needed to compute the risk of a portfolio holding more than two securities.
- 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.
- Historical volatilityVolatility measured backwards — the standard deviation of the underlying's past percentage price changes, as against implied volatility, which is worked out forwards from the option price.
- Fund factsheetThe monthly scheme summary an AMC publishes voluntarily — portfolio, performance, risk measures and investment details in one document. It is an AMFI best practice, not a statutory requirement.
- Net selectivityFama's absolute performance measure: the fund's return above the risk-free rate, less the market premium the portfolio's own total risk would have earned on its own.
- Probability distribution of expected returnsThe list of returns an investment might deliver, each with the chance of it happening — the form in which Modern Portfolio Theory assumes an investor sees every investment choice.
- 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.
- Downside riskThe probability of a loss on account of falling asset prices in changing market conditions — the half of volatility investors actually mind, measured by maximum drawdown and value at risk.
- LeptokurticA return distribution with kurtosis above 3 (excess kurtosis above zero): values bunch near the mean or sit far from it — a high peak with fat tails, which the workbook reads as riskier.
- Expected Portfolio ReturnThe weighted average of the expected returns of the holdings in a portfolio, each weight being that asset's share of portfolio value — the return half of Modern Portfolio Theory.
- Variance of a SecurityThe probability-weighted average of the squared gaps between a security's possible returns and its expected return — the forward-looking measure of a single asset's risk in Modern Portfolio Theory.
- OutliersExtreme returns in a fund's track record, whether highly positive or highly negative — the thing skewness and kurtosis are computed to detect.
- PlatykurticA return distribution with kurtosis below 3, so excess kurtosis is negative: values cluster at consistent frequencies around the mean with no significant deviations.
- 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.
- Defensive stocksShares of companies whose product demand and revenues barely respond to the business cycle — such as medicines and health services — used to reduce a portfolio's unsystematic risk.
- Risk adjusted returnThe return an investment earns for each unit of risk it takes on, rather than the return looked at on its own — in this paper, measured through the Sharpe ratio.
- Value at RiskA statistical estimate of the most a position is likely to lose over a set holding period at a chosen confidence level — the idea behind the initial margin a clearing corporation charges on a derivatives trade.
- CorrelationA measure of the strength and direction of the relationship between two variables, running from -1 to +1, and the single factor that determines how much risk diversification actually removes.
- Bollinger bandsA technical indicator that plots bands a set number of standard deviations either side of a moving average, treating prices at the upper band as overbought and at the lower band as oversold.
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