Portfolio volatility: Definition & Examples

Portfolio volatility is the standard deviation of a portfolio's return over a stated horizon. It measures dispersion around the portfolio's expected return, in the same units as return. Portfolio managers use it to compare allocations, set risk limits and size exposures—but they must account for how the holdings move together.

A 60/40 portfolio does not generally have volatility equal to 60% of one asset's volatility plus 40% of the other's. Returns combine linearly; standard deviations generally do not.

Calculate variance first, then take the square root

Let $R_p$ denote portfolio return. With beginning-of-period weights $\mathbf{w}$ and the matching return covariance matrix $\Sigma$ ,

\sigma_p=\sqrt{\text{Var}(R_p)} =\sqrt{\mathbf{w}^{\top}\Sigma\mathbf{w}}.

This is the square root of portfolio variance. For a held portfolio, the weighted simple-return identity assumes consistent return definitions and currency treatment, with no interim trading or external cash flows. It does not require a normal distribution; finite second moments suffice. Log returns do not combine through the same weighted-return identity.

For two assets, write the covariance in terms of their correlation:

\sigma_p= \sqrt{w_A^2\sigma_A^2+w_B^2\sigma_B^2 +2w_Aw_B\rho_{AB}\sigma_A\sigma_B}.

The squared weights multiply individual variances. The final term captures co-movement, including both equal cross terms from squaring the weighted return.

A one-year 60/40 example

Consider two hypothetical sleeves with stipulated population standard deviations of their one-year simple total returns:

Input	Sleeve A	Sleeve B
Beginning-of-year capital weight	60%	40%
One-year return volatility	20%	10%

Assume correlation $0.25$ , one reporting currency, no interim trades, no external cash flows, and a consistent reinvestment convention for distributions. These are synthetic model inputs, not historical estimates or forecasts for named assets.

Using decimal returns, covariance is $0.25(0.20)(0.10)=0.005$ . Therefore,

\begin{aligned} \sigma_p^2 &=(0.6)^2(0.20)^2+(0.4)^2(0.10)^2 +2(0.6)(0.4)(0.25)(0.20)(0.10)\\ &=0.0144+0.0016+0.0024\\ &=0.0184,\\ \sigma_p&=\sqrt{0.0184}\approx0.1356466. \end{aligned}

Portfolio volatility is 13.56% over one year, rounded to two decimal places. The variance is 0.0184 in squared decimal-return units. Equivalently, variance is 184 squared percentage points, whose square root is about 13.56 percentage points. Reporting the variance itself as a 1.84% volatility would confuse the units.

The calculation

0.6(20\%)+0.4(10\%)=16\%

instead describes these positive-weight exposures at perfect positive correlation. With correlation 0.25, diversification lowers volatility relative to that 16% case, although the portfolio remains more volatile than sleeve B alone.

For the same 60/40 holdings and one-year sleeve volatilities of 20% and 10%, portfolio volatility is 8% at correlation minus one, 13.56% at correlation 0.25, and 16% at correlation one.

Original synthetic one-year population comparison. Only correlation changes; the highlighted case has variance 0.0184 and volatility 13.56%.

At correlation $-1$ , the weighted centered-return exposures oppose each other, giving $\lvert0.6(20\%)-0.4(10\%)\rvert=8\%$ . Perfect negative correlation does not eliminate risk at these weights: the two exposures have unequal sizes. At correlation $+1$ , they reinforce each other and give 16%. Negative correlation is unnecessary for a diversification benefit relative to that upper bound. These statements concern positive weights; a long–short portfolio changes the sign of the covariance contribution.

For a foundation in these moments and the two-asset formula, Expectation, Variance and Covariance includes worked calculations. The full lesson is free with a signed-in account; a public preview is available.

Read a volatility number with its measurement choices

A historical estimate and a model forecast answer different questions. With $T\ge2$ aligned portfolio-return observations, a common historical estimator is

\widehat\sigma_p= \sqrt{\frac{1}{T-1}\sum_{t=1}^{T}(R_{p,t}-\bar R_p)^2}.

Here $\bar R_p$ is the sample mean. Applying today's weights to past asset returns estimates a hypothetical fixed-weight portfolio; it need not reproduce the realised returns of a portfolio whose holdings changed. Missing dates, currency conversion and price-versus-total-return choices also affect the result.

The horizon belongs beside the number. The example already uses one-year returns, so it needs no annualisation. Square-root-of-time scaling requires assumptions about dependence and the return quantity being aggregated; it is not an automatic conversion from daily volatility to the exact volatility of compounded annual simple returns.

Finally, volatility counts favourable and unfavourable deviations alike. A 13.56% volatility is neither a maximum loss nor a loss probability. Tail shape and the expected return remain unspecified; value at risk and drawdown address different aspects of risk.

Sources

Elton, E. J., Gruber, M. J., Brown, S. J. and Goetzmann, W. N., Modern Portfolio Theory and Investment Analysis, 9th edition, Wiley, 2014, Chapter 4, “Characteristics of Portfolios in General,” pp. 50–54: weighted portfolio returns, covariance and two-asset variance; p. 52 distinguishes population moments from historical sample estimates.
Eric Zivot, University of Washington, Portfolio Theory with Matrix Algebra, Spring 2015, slides 6–10: portfolio weights, returns and the covariance-matrix variance formula. The slides use a normal-return model; the variance identity itself does not need normality.

Calculate variance first, then take the square root

A one-year 60/40 example

Read a volatility number with its measurement choices

Sources

Related terms

Correlation

Diversification

Drawdown

Portfolio variance

Standard deviation

Value at risk