Variance of Returns and Portfolio Variance
How investors use variance: variance and standard deviation of returns, annualising volatility, and the two-asset portfolio variance formula.
Calculate Portfolio Variance Directly From Return Series
You need to know how much an investment’s returns bounce around, and whether putting two assets together actually lowers the overall risk. The answer is portfolio variance. Calculate it for a single stock or a two-asset portfolio using the same core method: average the squared deviations from the mean return, then, for a portfolio, add the covariance term. Here is exactly how to do that, with the numbers and the formula side by side.
Portfolio variance quantifies the dispersion of returns around the expected return of a portfolio. Higher variance means the returns are more spread out, indicating higher risk. First get the variance of each individual asset’s returns. In finance you almost always use sample variance because historical returns are a sample of future behavior.
Variance Of Returns: The Sample Calculation From Price Data
To get the variance of returns for a single stock, start with a series of periodic returns, daily, monthly, or annual. Suppose you have 12 monthly returns for a stock: 2%, -1%, 3%, 0%, -2%, 4%, 1%, -3%, 2%, -1%, 3%, 0%. The mean return is 0.67%. The sample variance (s²) is the sum of the squared deviations from that mean, divided by n‑1 (11, for 12 months). The sum of squared deviations (SS) is 0.000664 in decimal terms (or 6.64 in percentage‑squared units). Divide by 11 gives a sample variance of 0.0000604, or 0.0604%². The standard deviation, the square root, is about 0.00777, or 0.777%. That is the stock volatility you would quote to a client.
If you used n instead of n‑1, the variance would be 0.0000553, or 0.0553%², a 9% underestimate. That is Bessel’s correction: using n‑1 makes the sample variance an unbiased estimator of the true population variance. The variance of returns is always in squared units, never in the same units as the returns themselves.
From Variance To Volatility: Standard Deviation And Annualising By √Time
Standard deviation, the square root of variance, is what you use for practical comparisons. If a stock has a monthly variance of 0.0604%², the monthly standard deviation is 0.246%. To convert that to an annualised volatility figure, multiply the monthly standard deviation by √12 (about 3.46).This is the method used in the CFA Institute curriculum reading on portfolio risk: volatility scales with the square root of time, not linearly. Do not multiply variance by 12; that would give an annualised variance of 0.7248%², which is not the same as 0.7248%².
The rule applies regardless of the period: daily volatility × √252 for annual, weekly × √52, monthly × √12. It holds only when returns are independent and identically distributed, a common assumption in finance that is approximately true for liquid assets.
Two-Asset Portfolio Variance Formula: w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov
The portfolio variance for two assets is the engine behind modern portfolio theory, first formalised by Markowitz H in "Portfolio Selection", Journal of Finance 1952 (volume 7, issue 1, pages 77-91). The formula appears in Bodie, Kane & Marcus, Investments (13th edition, 2021) as:
σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)
where w₁ and w₂ are the portfolio weights (they sum to 1), σ₁² and σ₂² are the variances of asset 1 and asset 2, and Cov(R₁,R₂) is the covariance between the two assets’ returns. Cov(R₁,R₂) can also be written as ρ₁₂σ₁σ₂, where ρ₁₂ is the correlation coefficient. The units are squared percent (or squared decimal).
The covariance term is the key: if the two assets move together (positive covariance), the portfolio variance increases; if they move in opposite directions (negative covariance), the portfolio variance decreases. This is why diversification works: combining assets with low or negative correlation can reduce overall portfolio variance without sacrificing expected return.
The Minimum Variance Portfolio
Given two assets, you can solve for the weight w₁ that minimises portfolio variance. The minimum‑variance portfolio weight is:
w₁* = (σ₂² - Cov(R₁,R₂)) / (σ₁² + σ₂² - 2Cov(R₁,R₂))
and w₂* = 1 - w₁*. This gives the combination that produces the lowest possible variance for the pair, regardless of expected return.
Worked Example: Two‑Asset Portfolio Variance With Illustrative Returns
These returns are made up for illustration. Stock X has monthly returns: 1%, 3%, -2%, 4%, 0%, 2%, -1%, 3%, 1%, -2%, 2%, 1%. Stock Y has monthly returns: -1%, 0%, 3%, 1%, 4%, 2%, 0%, -1%, 2%, 3%, 1%, 2%.
Step 1: Calculate each stock’s sample variance. For Stock X, the mean return is 1.0%. The sum of squared deviations (SS) is 0.000400 in decimal. Divide by n‑1 (11) gives sample variance σ₁² = 0.00003636, or 0.003636%². The standard deviation σ₁ = sqrt(0.00003636) = 0.00603, or 0.603%.
For Stock Y, mean return is 1.33%. SS = 0.000320. Sample variance σ₂² = 0.00002909, or 0.002909%². Standard deviation σ₂ = 0.00539, or 0.539%.
Step 2: Calculate the covariance between X and Y. Cov(R₁,R₂) = (1/(n‑1)) × Σ[(xᵢ - x̄)(yᵢ - ȳ)]. The sum of cross‑products is 0.000140. Divide by 11 gives covariance = 0.00001273, or 0.001273%² (positive, they tend to move together).
Step 3: Assume a portfolio with 60% in Stock X (w₁ = 0.6) and 40% in Stock Y (w₂ = 0.4). Apply the portfolio variance formula:
σ²_p = (0.6)²(0.00003636) + (0.4)²(0.00002909) + 2(0.6)(0.4)(0.00001273)
= 0.00001309 + 0.00000466 + 0.00000611 = 0.00002386
Portfolio variance = 0.00002386 (in decimal), or 0.002386%². The portfolio standard deviation = sqrt(0.00002386) = 0.004885, or 0.4885%.
Compare to the weighted average of individual standard deviations: 0.6×0.603% + 0.4×0.539% = 0.577%. The actual portfolio standard deviation is 0.4885%, about 15% lower. That reduction is the diversification benefit.
| Variable | Stock X | Stock Y | Combined Portfolio |
|---|---|---|---|
| Mean Return | 1.0% | 1.33% | — |
| Sample Variance | 0.003636%² | 0.002909%² | 0.002386%² |
| Standard Deviation | 0.603% | 0.539% | 0.4885% |
| Covariance | — | — | 0.001273%² |
| Weight | 0.6 | 0.4 | 1.0 |
Why Diversification Lowers Variance: The Role Of Correlation
The entire reason diversification works is captured in the covariance term of the two asset portfolio variance formula. When two assets have a correlation coefficient ρ₁₂ less than +1, the portfolio variance is less than the weighted sum of individual variances. If ρ₁₂ = 0 (uncorrelated), the covariance term is zero, and the portfolio variance is simply w₁²σ₁² + w₂²σ₂², already lower than the weighted average of variances because of the squares. If ρ₁₂ is negative, the covariance term is negative, further reducing portfolio variance.
In the worked example above, the correlation ρ₁₂ = Cov(R₁,R₂) / (σ₁σ₂) = 0.00001273 / (0.00603 × 0.00539) = 0.00001273 / 0.0000325 = 0.392. That moderate positive correlation still allowed a 15% reduction in standard deviation because the weights are not equal and the variances differ. The maximum benefit comes from perfect negative correlation (ρ₁₂ = -1), where you can theoretically achieve zero portfolio variance with the right weights. No two real assets are perfectly negatively correlated, but the principle holds: adding assets with low correlation to the portfolio reduces overall variance.
Limits Of Variance As A Risk Measure: Downside And Fat Tails
Variance treats upside volatility and downside volatility identically. A stock that jumps 20% one month and falls 20% the next has the same variance as one that rises 5% and falls 5%, but investors hate the big loss more than they like the big gain. That is why professionals supplement variance with semi‑variance (which only measures downside deviations) or Value at Risk (VaR). Variance also assumes returns are normally distributed, but real financial returns have fat tails: extreme events occur more often than a normal distribution predicts. The 2008 crash and the 2020 COVID sell‑off are examples where variance alone understated tail risk.
Another limit: variance is not comparable across assets with different mean returns. That is where the coefficient of variation (CV = σ/μ) comes in, but only for ratio‑scale data with a positive mean. For portfolios with negative expected returns, the CV is meaningless.
Common Questions
What is the difference between sample variance and population variance for returns?
Sample variance uses n-1 in the denominator (Bessel's correction) to provide an unbiased estimate of the true population variance. Population variance uses n. In finance, you almost always compute sample variance because your historical return data is a sample of the future. Using population variance on a sample understates the true variance.
How do I calculate portfolio variance for more than two assets?
For n assets, portfolio variance is σ²_p = wᵀΣw, where w is an n×1 weight vector and Σ is the n×n covariance matrix. The sum of all weights must equal 1. This matrix form automates the pairwise covariances and is the standard method in the CFA Institute curriculum reading on portfolio risk.
Why is annualised volatility calculated with √12 and not just ×12?
Variance scales with time, but standard deviation, the square root of variance, scales with the square root of time. Monthly variance × 12 gives annual variance, then you take the square root to get annual standard deviation. Multiplying monthly standard deviation by √12 is a shortcut that gives the same result directly.
Can portfolio variance ever be zero?
Yes. Portfolio variance is zero if all assets have zero variance (e.g., cash) or if the assets are perfectly negatively correlated and the weights are chosen to cancel all risk. In practice, zero variance is only achievable with a risk‑free asset like a Treasury bill.
What is the minimum variance portfolio and how do I find it?
The minimum variance portfolio is the combination of two assets that produces the lowest possible portfolio variance, ignoring expected return.This is derived from setting the derivative of the portfolio variance formula to zero.
What mistake do most people make when calculating variance of returns?
The most common mistake is using n instead of n-1 in the denominator, which biases the variance low by a factor of (n-1)/n. The second most common is confusing variance with standard deviation and reporting variance in original percentage units instead of squared percentage units.
Does high variance always mean high risk?
Not necessarily. Variance treats upside and downside volatility equally, but investors dislike downside risk more. A stock that gains 20% and then loses 20% has high variance but may not be riskier than a stable stock if you are investing for the long term. That is why professionals also use semi‑variance, Value at Risk, and tail‑risk measures.