Portfolio Variance (Two Assets)

In one sentence: Portfolio variance says how much a two-asset mix’s risk comes from each part and from how the parts move together, and it shows that mixing assets can lower risk.

$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$

Read it out loud: “Sigma p squared equals w 1 squared times sigma 1 squared, plus w 2 squared times sigma 2 squared, plus two times w 1 times w 2 times rho 12 times sigma 1 times sigma 2. Sigma p squared is the variance of the portfolio, the squared size of its typical ups and downs. The first two terms are each asset's own risk, and the last term is the link between the two assets.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Weights: A weight is the share of the money in one asset, written as a fraction between 0 and 1. A weight of 0.5 means half of the money. The weights of a portfolio add to 1, which is the same as 100%, because all of the money is invested somewhere in the mix.
Volatility and sigma: Volatility is the typical size of the ups and downs of an investment’s returns over a year. The Greek letter sigma, $\sigma$ , is the usual symbol for it. A volatility of 0.20 means 20% a year, and a bigger number means a bumpier ride.
Variance and squares: Variance is volatility squared, so $\sigma^2$ is the variance. Squaring a number multiplies it by itself, so $0.2^2 = 0.04$ . Variance is useful in formulas because squared pieces can be added together, and its square root brings the answer back to the scale of the returns.
Square root: The square root of a number is the value that, multiplied by itself, gives the number back. The square root of 0.0225 is 0.15, because $0.15 \times 0.15 = 0.0225$ . A calculator has a square-root function for finding it.
Correlation and rho: Correlation, written $\rho$ (the Greek letter rho), is a number between $-1$ and $1$ that measures how closely two things move together. A correlation of 1 means they always move together, 0 means there is no link, and a negative value means they tend to move in opposite directions.

What each symbol means

Symbol	Meaning
$\sigma_p^2$	The variance of the portfolio, a measure of its risk. It is the volatility squared, so taking its square root gives the volatility. With returns written as decimals, it is a plain number with no everyday unit.
$w_1$	The weight of asset 1: its share of the money, as a fraction. A weight of 0.5 means half of the money is in asset 1.
$w_2$	The weight of asset 2, its share of the money as a fraction. The two weights add to 1, because all of the money is in the two assets.
$\sigma_1$	The volatility of asset 1: the typical size of its ups and downs per year, written as a decimal. Sigma is the Greek letter s. A volatility of 0.20 means 20% a year.
$\sigma_2$	The volatility of asset 2, the typical size of its ups and downs per year, written as a decimal in the same way.
$\rho_{12}$	The correlation between asset 1 and asset 2: a number from -1 to 1 that says how closely the two move together. Rho is the Greek letter r. A value of 1 means they move exactly together, 0 means no link, and -1 means they move in opposite directions.

The formula piece by piece

$w_1^2 \sigma_1^2$: Asset 1’s own risk, scaled by its weight squared. The weight squared shrinks the part of the risk that comes from asset 1, and $\sigma_1^2$ is the variance of asset 1.
$w_2^2 \sigma_2^2$: Asset 2’s own risk, scaled by its weight squared in the same way, with $\sigma_2^2$ as the variance of asset 2.
$2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$: The link term: how the two assets move together, scaled by both weights and both volatilities. It is positive when the correlation is positive and negative when the correlation is negative.
$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$: The variance of the portfolio equals asset 1’s own risk, plus asset 2’s own risk, plus the link between the two assets.
$\sigma_p = \sqrt{\sigma_p^2}$: The volatility of the portfolio is the square root of its variance, which puts the answer back in the same units as the returns.

What the formula is doing, and why

The formula adds three things: asset 1’s own risk, asset 2’s own risk, and a link term that depends on how the two assets move together. What goes in: the weights, the volatility of each asset and the correlation between them. What comes out: the portfolio’s variance. Variance is in squared units, so the square root of the result is the portfolio’s volatility, which is in the same units as the returns.

Why square the weights and volatilities? Variance is built from squared quantities. If an asset makes up a share $w$ of the money, its moves are scaled by $w$ , and the variance of a scaled quantity is scaled by $w^2$ . That is why each asset’s own variance is multiplied by its weight squared. The volatility is squared as well, because $\sigma_1^2$ is the variance of asset 1.

Why the link term? When two assets move together, their ups and downs add up and the risk grows. When they move in opposite directions, the moves partly cancel and the risk shrinks. The correlation $\rho_{12}$ says which case applies. The factor 2 appears because squaring a sum of two moves produces the cross term twice.

Why does diversification work? If the correlation is 1, the link term is as large as it can be, and the volatility of the mix is the weighted average of the two volatilities. Any correlation below 1 makes the link term smaller, so the risk of the mix is lower than that average. A mix with the same weights can therefore carry less risk when its parts do not move in step.

This formula is a model. Correlations are estimated from past data and can change over time, so a mix that looked well diversified may behave differently in a difficult market.

An everyday picture

Think of two people carrying a long table together. If they sway in step, the table tilts a lot and their separate wobbles add up. If one leans left while the other leans right, the wobbles partly cancel and the table stays level. Correlation is how in step the two people are, and diversification is pairing two partners who do not sway in the same way.

Worked example: Two funds with a correlation of 0.2

We split 1000 dollars equally between two funds, so each weight is 0.5. Fund 1 is a stock fund with a volatility of 0.20, which is 20% a year. Fund 2 is a bond fund with a volatility of 0.10, which is 10% a year. The two funds have a correlation of 0.2, which means they move together only a little. All numbers are made up for the example. We want the volatility of the portfolio, and we compare it with the case where the two funds move in perfect step, a correlation of 1.

Write down the inputs. Each fund gets half of the money, so $w_1 = 0.5$ and $w_2 = 0.5$ . The volatilities are $\sigma_1 = 0.20$ and $\sigma_2 = 0.10$ , and the correlation is $\rho_{12} = 0.2$ . The weights add to 1, as they must, because all 1000 dollars is in the two funds.
Square each volatility to get each variance. Variance is volatility squared, so $\sigma_1^2 = 0.20^2 = 0.04$ and $\sigma_2^2 = 0.1^2 = 0.01$ . These are the variances of the two funds on their own, before they are mixed together.
Work out the own-risk terms. Each fund’s own variance is multiplied by its weight squared, and $0.5^2 = 0.25$ . So $w_1^2 \sigma_1^2 = 0.25 \times 0.04 = 0.01$ and $w_2^2 \sigma_2^2 = 0.25 \times 0.01 = 0.0025$ . Their sum, 0.0125, is the variance the mix would have if the correlation were zero, because the link term would then vanish.
Work out the link term. The link term is $2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2$ . Multiply the numbers step by step: $2 \times 0.5 \times 0.5 = 0.5$ , then $0.5 \times 0.2 = 0.1$ for the correlation, then $0.1 \times 0.2 = 0.02$ for the first volatility, and finally $0.02 \times 0.1 = 0.002$ for the second volatility. The correlation is positive, so the link term adds a little to the risk.
Add the terms to get the variance. Adding the own-risk terms and the link term gives $\sigma_p^2 = 0.01 + 0.0025 + 0.002 = 0.0145$ . This is the portfolio variance. It is a squared quantity, so it is not yet in the same units as the returns.
Take the square root to get the volatility. The square root brings the number back to the scale of the returns: $\sigma_p = \sqrt{0.0145} \approx 0.1204$ . That is about 12.0% a year, the typical size of the portfolio’s ups and downs. The square root is taken once, at the end.
Compare with perfect correlation. If the correlation were 1, the link term would be $2 \times 0.5 \times 0.5 \times 1 \times 0.2 \times 0.1 = 0.01$ , and the variance would be $0.0125 + 0.01 = 0.0225$ . Its square root is 0.15, or 15% a year. The weights did not change, so only the link between the two funds differs, and the lower correlation gives the lower risk of 12.0%.

Answer: The portfolio’s volatility is about 12.0% a year, compared with 15% a year if the two funds moved in perfect step. Because the correlation is below 1, the mix is less risky than the weighted average of the two volatilities, which is also 15%.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Adding the weighted volatilities and calling the sum the risk. That is the risk only in the special case of correlation 1. With a lower correlation the true risk is smaller.
Forgetting to square the weights or the volatilities at the start, or forgetting the square root at the end. The first step gives variance and the last step gives volatility.
Leaving out the factor 2 in the link term. That makes the risk too small, and it hides the effect of the correlation.
Using a correlation outside the range from -1 to 1. Correlation can never be larger than 1 or smaller than -1.

Where you will meet it

Measuring the risk of a mix of two funds, such as a stock index fund and a bond fund, in an investment plan.
Choosing a stock and bond split, where the risk of each possible mix is compared for a given expected return.
Mean-variance portfolio optimization, which chooses weights that give the highest expected return for a given variance. Harry Markowitz introduced this approach in 1952.
Risk models for portfolios with many assets, which use the same structure with a term for every pair of assets.

Questions people ask

What is rho?

Rho is the Greek letter r, and in finance it stands for correlation. It is a number between -1 and 1. A value of 1 means two assets move exactly together, 0 means their moves are unrelated, and -1 means they move in exact opposite directions.

Why take the square root at the end?

Variance is in squared units. Its square root is the volatility, which is in the same units as the returns and easy to read. For example, a variance of 0.0225 has the square root 0.15, which is 15% a year.

Does adding more assets always lower risk?

Usually, when the assets are not perfectly correlated. The two-asset formula shows the effect for two assets, and more assets follow the same idea with more pairs. A set of assets that all move together gives little benefit.

Is the risk of a mix the average of the risks of its parts?

Only when the correlation is 1. For any lower correlation, the risk of the mix is smaller than the weighted average of the parts’ volatilities.

Volatility (Standard Deviation of Returns): Volatility says how far an investment’s returns usually land above and below their average, so a bigger number means a bumpier ride.
Expected Return of a Portfolio: The expected return of a portfolio is the average of its parts’ expected returns, where each part counts in proportion to how much of the money it holds.
Beta: Beta says how much a stock tends to move when the whole market moves: 2 means it moves about twice as far as the market, and 0.5 means about half as far.