Is volatility the same as risk?

No. Volatility measures how widely returns swing in both directions. Risk in the everyday sense, losing money or not having it when needed, also depends on the average return, the time horizon and the depth and length of declines from peaks. Volatility is one input to that picture, not the whole of it.

How do you annualize monthly volatility?

Multiply the standard deviation of monthly returns by √12. For daily returns, multiply by √252, the usual number of trading days. The rule assumes each period's return is independent of the previous ones.

Why divide by N − 1 rather than N?

The average used in the formula is estimated from the same data, which makes the deviations slightly smaller than they would be around the true average. Dividing by N − 1 corrects that bias in the squared deviations. Over decades of monthly data the difference is tiny.

What is the difference between volatility and downside deviation?

Volatility counts every month's distance from the average, up or down. Downside deviation counts only returns below a target, zero on MarketHeist, so a holding is not marked down for large gains. The Sortino ratio uses downside deviation in place of volatility.

Why does higher volatility lower compound growth?

Because losses and gains do not cancel: a fall of a given size needs a larger rise to get back to even. The compound growth rate sits below the average return by roughly half the square of the volatility, so with the same average return, the more volatile holding ends with less.

What is volatility? Annualized standard deviation, explained

By Jeffrey Lin · Published 2026-10-27 · Updated 2026-10-27 · Data through Sep 2026

On this page

Key takeaways

Volatility measures how widely a holding's returns swing around their average: the bigger the typical swing, the higher the volatility.
It is quoted as a yearly figure. Scaling up from monthly swings assumes each month is independent of the last; when returns trend or reverse, the true yearly swing is larger or smaller.
Volatility counts rises and falls alike. Two holdings with the same volatility can have very different worst months, worst years and declines from a peak.
It is not the probability of losing money over a holding period: that also depends on the average return and on how long the money stays invested.
MarketHeist measures it from monthly total returns, dividends included, and quotes it as a yearly figure.

Definition

Volatility is the standard deviation of an investment's returns: the typical distance of a period's return from its average, usually scaled to a yearly figure. It measures how widely returns swing in both directions, not how likely or how large a loss is.

Also known as: volatility, standard deviation of returns, annualized volatility, historical volatility, realized volatility, stdev, sigma.

How it works

Take a holding's return in each month. Find the average. For each month, measure how far its return sat from that average, square the distances so that rises and falls both count as positive, average the squares and take the square root. That is the monthly standard deviation: the size of a typical month's surprise.

A yearly figure is easier to compare with yearly returns, so the monthly number is scaled up. If months are independent, it is the squared spreads that add up across the twelve months, not the spreads themselves: the yearly spread is √12 times the monthly one, not twelve times. That is the source of the √12 in every annualized volatility on monthly data, and of √252 on daily data.

Volatility is a measure of uncertainty around the average, not of loss. A fund that rises steadily and sometimes jumps higher has volatility even though it never falls; a fund that drifts down slowly and smoothly can have low volatility while it loses money. What an investor experiences as risk, a deep fall from a peak, a long wait to recover, money needed at the wrong moment, depends on the average return, on the order of the returns and on the time horizon as much as on the spread.

Volatility is like the chop on a lake: it tells you how rough the crossing feels, but not whether the boat is heading for the far shore or drifting toward the rocks.

It still earns its place. Volatility is stable enough to estimate from a few years of data, it feeds directly into how holdings combine (with correlation, it sets a portfolio's overall swing), and it costs compound growth: the larger the swings, the further a holding's compound growth rate falls below its average return.

Formula

Sample standard deviation of monthly returns

s = \sqrt{\frac{1}{N - 1} \sum_{t = 1}^{N} {(r_{t} - \bar{r})}^{2}}

s: monthly standard deviation
rₜ: total return of month t (dividends reinvested)
r̄: average monthly return
N: number of months; dividing by N − 1 corrects for measuring the average from the same data

Annualized volatility

σ_{annual} = s \times \sqrt{12}

σ_annual: volatility per year
s: monthly standard deviation
√12: square root of the number of months in a year

Why √12, and when it fails: the spread of a year of correlated months

σ_{annual}^{2} = 12 s^{2} [1 + 2 \sum_{k = 1}^{11} (1 - \frac{k}{12}) ρ_{k}]

ρₖ: correlation between a month's return and the return k months later (autocorrelation at lag k)
k: lag in months
s, σ_annual: as above

For the sum of twelve monthly returns (log returns add exactly). With every ρₖ = 0 the bracket is 1 and the √12 rule holds; positive autocorrelation (trending) makes the yearly spread larger than √12 times the monthly one, negative (mean reversion) smaller.

Downside deviation, for contrast

σ_{D} = \sqrt{\frac{1}{N} \sum_{t = 1}^{N} \min {(r_{t}, 0)}^{2}} \times \sqrt{12}

σ_D: annualized downside deviation: only losing months count, rising months count as zero
rₜ, N: as above

Worked example

The S&P 500 (SPY) and long-term US Treasuries (TLT, bonds maturing in 20 years or later), 484 monthly returns from Jun 1986 to Sep 2026.

The S&P 500's annualized volatility was 15.2%; TLT's was 11.6%. A fund of US government bonds, which carry very little risk of default, swung about three quarters as much as the stock market, because the price of a long bond moves a lot when interest rates change.

Volatility treats rises and falls alike, and the shape behind the two figures differed. The S&P 500's monthly returns had a skew of -0.69: its most extreme months were mostly falls. TLT's skew was 0.21, tilted slightly the other way. Counting only the losing months, the downside deviation was 10.1% for the S&P 500 and 7.2% for TLT.

Over the 40 years the more volatile fund grew faster: the S&P 500 compounded at 10.8% a year, TLT at 5.5%. Volatility measures the spread of returns around their average, not whether that average is positive, so on its own it says little about the chance of ending a long holding period with a loss.

Volatility does cost compound growth. The S&P 500's average return, the monthly mean times 12, was 11.5%, against the 10.8% it compounded at; for TLT the two were 6.0% and 5.5%.

Data through Sep 2026. Reproduce id 9a9e423f89af5449 (this guide, its lab setup and the data month).

Interactive lab

Annualized volatility beside the downside deviation, skew, compound growth and Sharpe ratio of both picks over the same months. Switch to a bond fund, gold or a balanced portfolio to see how much of each one's volatility came from falls.

Annualized Sharpe and Sortino ratios of monthly returns. Jun 1986 to Sep 2026.

Probability that the true Sharpe ratio exceeds 0.00 as the record grows. The dashed line marks 95%.

Measure	SPY	TLT
Sharpe ratio	0.76	0.52
Sortino ratio	1.15	0.84
Probabilistic Sharpe ratio (against 0.00)	100.0%	100.0%
Skewness of monthly returns	-0.69	0.21
Excess kurtosis of monthly returns	1.97	1.88
Downside deviation (annualized)	10.1%	7.2%
Volatility (annualized)	15.2%	11.6%
Average return (mean monthly return x 12)	11.5%	6.0%
CAGR	10.8%	5.5%
Volatility drag (annualized)	1.2%	0.7%
Omega ratio (threshold 0%)	1.76	1.49
Months in the window	484	484

Risk-free rate 0, 12 periods a year. The probabilistic Sharpe ratio is the probability that the true Sharpe ratio exceeds the benchmark, given the record's length, skewness and kurtosis. Volatility drag: 12 times the arithmetic mean monthly return minus the geometric mean monthly return. Omega ratio: the sum of the monthly gains over the sum of the monthly losses, at a threshold of 0% a month. Jun 1986 to Sep 2026.

How MarketHeist computes it

Every choice an implementation has to make, as the code makes it today. The page is checked against the code each time it is built.

Choice	How it is computed	In the code
Data	Monthly total returns (dividends reinvested). Funds younger than the period shown are extended backward with older funds, spliced on returns, not prices.
Window	The worked example and the lab use the common window of the two portfolios compared, Jun 1986 to Sep 2026 (484 months), ending at the data month of this page. Another pick in the lab is read over its own common window.
Portfolio streams	A fund is its proxy-extended monthly return stream. A built-in portfolio is read as the MarketHeist MCP reads it: its composed stream with its own rebalancing rule and overlay. A custom mix is rebalanced yearly.
Ratios	Risk-free rate 0, 12 periods a year. Sharpe: mean of the monthly returns over their sample standard deviation, times the square root of 12. Sortino: the same mean over the downside deviation, the root of the average squared negative return over all N months. The probabilistic Sharpe ratio uses the per-period Sharpe ratio, the skewness and the kurtosis of the monthly returns; the benchmark Sharpe ratio is divided by the square root of 12 first.
Volatility	The sample standard deviation (divides by N − 1) of monthly total returns over the window, times √12. Returns 0 when it cannot be computed (fewer than two months).	`annualizedStdev`packages/portfolio/engine/metrics_extra.ts
Standard deviation	Ignores missing values, needs at least two, and divides the sum of squared deviations from the mean by N − 1.	`nanstd`packages/portfolio/engine/utils.ts
Portfolios	A portfolio's volatility is measured on its own simulated monthly returns, after rebalancing, so it includes the effect of correlation between the holdings.	`buildMetrics`packages/portfolio/engine/portfolio.ts
Risk tables	Volatilities inside the risk-contribution and weighting tables come from the covariance matrix, which divides by N rather than N − 1, and from target weights rather than drifting ones, so they can differ slightly from the headline volatility of the same portfolio.	`covarianceMatrix`packages/portfolio/engine/risk.ts
Downside deviation on this page	√(Σ min(rₜ, 0)² / N) × √12: losing months only, divided by all N months.	`ratiosSide`content-engine/lab/compute.ts
Backtester	Uses the returns of its bars and the factor of its bar frequency (√252 daily, √52 weekly, √12 monthly), so a daily backtest's volatility is not computed from the same returns as a monthly portfolio's.	`periodsPerYear`packages/shared/types.ts
Engine functions	The functions the portfolio app and the MarketHeist MCP tools run, called here on the same monthly returns.	`sharpe`packages/shared/engine/metrics.ts`sortino`packages/shared/engine/metrics.ts`probabilisticSharpeRatio`packages/shared/engine/metrics.ts`skewness`packages/shared/engine/metrics.ts`excessKurtosis`packages/shared/engine/metrics.ts

Volatility (annualized standard deviation) compared with its neighbours

Volatility and the risk measures around it
Measure	What it answers	Counts rises as risk	Uses the order of returns
Volatility (standard deviation)	How widely do returns swing around their average?	Yes	No
Downside deviation	How large are the losing periods, typically?	No	No
Value at risk (historical)	How bad is a bad month, at a chosen percentile?	No	No
Maximum drawdown	How far did the value fall from a peak, at worst?	No	Yes
Ulcer index	How deep and how long were the declines from peaks?	No	Yes
Beta	How much does it move with the market?	Yes	No

Limitations

It is backward-looking. Volatility comes in regimes: calm years and turbulent years cluster, so a figure measured over a calm stretch understates what the next turbulent one brings.
It is symmetric and blind to the order of returns. A steady decline and an equal-sized jumble of ups and downs can have the same volatility; a low-volatility holding can still suffer a deep, slow fall, as long Treasuries did when interest rates rose quickly.
The √12 scaling assumes independent months. Trending returns, mean reversion, or smoothed prices of illiquid assets break it, in opposite directions.
Monthly data hide moves that reversed within a month, so monthly volatility can be lower than volatility measured from daily returns over the same years.
Long histories are extended backward with older funds or indexes where a fund is younger than the window, so early months approximate the fund rather than reproduce it.

Frequently asked questions

Is volatility the same as risk?: No. Volatility measures how widely returns swing in both directions. Risk in the everyday sense, losing money or not having it when needed, also depends on the average return, the time horizon and the depth and length of declines from peaks. Volatility is one input to that picture, not the whole of it.
How do you annualize monthly volatility?: Multiply the standard deviation of monthly returns by √12. For daily returns, multiply by √252, the usual number of trading days. The rule assumes each period's return is independent of the previous ones.
Why divide by N − 1 rather than N?: The average used in the formula is estimated from the same data, which makes the deviations slightly smaller than they would be around the true average. Dividing by N − 1 corrects that bias in the squared deviations. Over decades of monthly data the difference is tiny.
What is the difference between volatility and downside deviation?: Volatility counts every month's distance from the average, up or down. Downside deviation counts only returns below a target, zero on MarketHeist, so a holding is not marked down for large gains. The Sortino ratio uses downside deviation in place of volatility.
Why does higher volatility lower compound growth?: Because losses and gains do not cancel: a fall of a given size needs a larger rise to get back to even. The compound growth rate sits below the average return by roughly half the square of the volatility, so with the same average return, the more volatile holding ends with less.

References

Markowitz, H. (1952) Portfolio Selection. Journal of Finance 7(1). Standard deviation of return as the measure of portfolio risk.
Lo, A. W. and MacKinlay, A. C. (1988) Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test. Review of Financial Studies 1(1). The variance ratio test of the square-root-of-time rule.
Lo, A. W. (2002) The Statistics of Sharpe Ratios. Financial Analysts Journal 58(4). Annualizing under serial correlation.
Sortino, F. A. and van der Meer, R. (1991) Downside Risk. Journal of Portfolio Management 17(4).
Mandelbrot, B. (1963) The Variation of Certain Speculative Prices. Journal of Business 36(4). Fat tails and volatility clustering in price changes.

Try it on your own holdings

See TLT's full history: volatility, worst years and declines →

Or ask an assistant connected to the MarketHeist MCP server (https://marketheist.io/mcp), which computes the same figures:

Using MarketHeist.io, compare the annualized volatility, downside deviation, skew and compound annual growth of SPY and TLT over the same months, and explain why the more volatile one grew faster

Ask an AI assistant

ChatGPT Claude Google AI Mode Perplexity Grok

The prompt opens in a new tab, already filled in. An assistant without the MarketHeist connector answers from its own knowledge, not from the figures on this page.

Historical simulation, not advice. Computed by the MarketHeist portfolio engine from monthly total returns, data through Sep 2026. Markdown version of this page: volatility.md.