Portfolio standard deviation calculator

The two-asset formula

σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ σp = √σp²

w₁ and w₂ are the fractions of the portfolio in each asset, σ₁ and σ₂ are the standard deviations of their returns, and ρ is the correlation between those returns. The first two terms are each asset's variance scaled by its weight squared. The third is twice the weighted covariance, since ρσ₁σ₂ is the covariance of the two return series.

On the defaults above — 60% in an asset with an 18% standard deviation, 40% in one with 7%, correlation 0.25 — the three terms come to:

0.6² × 0.18² = 0.011664 0.4² × 0.07² = 0.000784 2 × 0.6 × 0.4 × 0.25 × 0.18 × 0.07 = 0.001512 σp² = 0.01396 σp = √0.01396 = 0.1182, or 11.8%

The weight on each asset is squared, which is why a small position contributes far less to portfolio variance than its weight suggests. The 40% position here contributes 0.000784 to the variance against 0.011664 from the 60% position — about a fifteenth as much, not two thirds, because its own standard deviation is smaller and the squaring compounds the gap.

Why it is not the weighted average

The tempting shortcut is to average the two standard deviations by weight: 0.6 × 0.18 + 0.4 × 0.07 = 0.136, or 13.6%. The true figure is 11.8%. The difference of 1.8 percentage points is the diversification benefit, and it exists because the two assets do not move in lockstep.

Set ρ = 1 in the calculator and the cross term becomes 2 × 0.6 × 0.4 × 0.18 × 0.07 = 0.006048, the variance rises to 0.018496 and the standard deviation becomes exactly 0.136. With perfect correlation, the formula collapses to the weighted average — the expression under the square root is a perfect square, (w₁σ₁ + w₂σ₂)². That is the ceiling on portfolio risk for any pair of weights.

Set ρ = −1 and the cross term flips sign: the variance drops to 0.0064 and the standard deviation to 0.08, or 8%. The perfect square is now (w₁σ₁ − w₂σ₂)², so the risk can be pushed all the way to zero by choosing weights that make the two products equal. On these inputs that means w₁ = σ₂ / (σ₁ + σ₂) = 0.07 / 0.25 = 0.28 in the first asset and 0.72 in the second. Two perfectly negatively correlated assets, held in the right proportions, form a portfolio whose return does not vary at all.

Real assets sit between those extremes. The size of the benefit depends on three things: how far the correlation is below 1, how balanced the weights are (the cross term contains w₁w₂, which is largest at 50/50), and how similar the two standard deviations are. Two assets of equal risk and zero correlation, held equally, have a portfolio standard deviation of σ/√2 — about 71% of either asset on its own.

The general formula for n assets

With more than two holdings, every pair of assets contributes a covariance term. The portfolio variance is the sum, over every ordered pair of assets including each asset with itself, of the product of the two weights, the two standard deviations and the correlation between them:

σp² = Σᵢ Σⱼ wᵢ wⱼ σᵢ σⱼ ρᵢⱼ (ρᵢᵢ = 1)

For two assets that double sum has four terms — the two diagonal terms w₁²σ₁² and w₂²σ₂², and the cross term counted twice, once as (1, 2) and once as (2, 1) — which is exactly the formula at the top of the page. For n assets it has n² terms and needs a full n × n correlation matrix: three correlations for three assets, six for four, 45 for ten. The number of correlations grows much faster than the number of assets, and each one has to be estimated from data.

This calculator handles two assets. It is possible to chain it: take the portfolio standard deviation of assets A and B as the "standard deviation" of a single combined holding, then enter that alongside asset C. But that only works if the correlation you enter is the correlation between C and the combined A-and-B return, not C's correlation with A or with B individually. Without that number the chained answer is wrong, so for three or more assets the honest route is the full matrix.

Where the inputs come from

An asset's standard deviation is the standard deviation of its periodic returns — a list of daily, weekly or monthly percentage changes. Paste those returns into the standard deviation calculator and use the sample formula, dividing by n − 1, since a run of historical returns is a sample of the process that produced them rather than the whole of it. Twelve monthly returns give a monthly standard deviation; 252 daily returns give a daily one.

The correlation comes from the same two return series lined up period by period. The correlation coefficient calculator takes the paired returns and gives ρ directly. Both series must cover the same dates at the same frequency: a monthly correlation from 2019 to 2024 cannot be combined with a daily standard deviation from last year.

Standard deviations are normally quoted annualised. If you have a daily figure, multiply it by √252, the number of trading days in a year; a weekly figure by √52; a monthly figure by √12. A monthly standard deviation of 5% becomes 5% × 3.464 ≈ 17.3% annualised; a daily 1.2% becomes 1.2% × 15.87 ≈ 19.0%. The square root appears because variances add across independent periods while standard deviations do not. Correlation does not need annualising — it is a pure number with no time unit.

Standard deviation as volatility

In trading, "volatility" is the standard deviation of returns, and the two words are used interchangeably. When a stock is described as having 25% volatility, that is its annualised return standard deviation. Option prices are quoted in the same units: an implied volatility of 30% is the market's estimate of the standard deviation of the underlying's return over the life of the option.

A one-standard-deviation move is therefore a concrete quantity. If a stock's annual volatility is 20%, a one-SD annual move is 20 percentage points either side of its expected return, and under a normal distribution about 68% of years should land inside that band — the empirical rule. Divide the annual figure by √252 for a one-day SD: 20% ÷ 15.87 ≈ 1.26% per day. A 4% single-day drop is then a z-score of about −3.2, which the normal distribution says should happen about once in thirteen hundred trading days.

It happens more often than that. Which is the point of the next section.

What the number does not tell you

• Correlations rise in a crisis. The diversification benefit depends on ρ staying low, and historically it has not stayed low when it mattered most. Assets that showed a correlation of 0.3 over calm years have moved together at 0.8 or higher during sharp sell-offs, at which point a portfolio behaves much closer to the weighted-average case than the calculation suggested.
• Returns are fat-tailed. The empirical rule assumes a normal distribution. Financial returns produce three- and four-SD moves far more often than the normal curve predicts, so a standard deviation understates the chance of a large loss even when it describes the typical year accurately.
• Past SD is not future SD. Volatility clusters: calm periods follow calm periods and turbulent ones follow turbulent ones, and the level shifts over time. A standard deviation measured over one window is an estimate, with sampling error, of a quantity that was not constant even within that window.

None of this makes the calculation useless. It is the standard first measure of portfolio risk and the input to most of the others. It is a description of the past, not a forecast.

The Sharpe ratio

The most common use of a portfolio's standard deviation is as the denominator of the Sharpe ratio: the portfolio's return in excess of the risk-free rate, divided by its standard deviation. It expresses return per unit of risk, so two portfolios with different volatilities can be compared on the same scale. A portfolio returning 9% with an 11.8% standard deviation, against a 3% risk-free rate, carries 6 points of excess return for 11.8 points of risk. The denominator is the annualised figure this page computes.