What it means
Variance is a statistical measure of spread. For a single investment it tells you how widely returns have scattered around the average, and for a portfolio it does the same for the combined result.
The bigger the number, the bumpier the ride an investor should expect. The key insight is that a portfolio's risk is not simply the weighted average of its parts.
Two investments that do not move in lockstep partly cancel each other's swings, so the combined variance can be lower than either holding on its own. This is the mathematical reason behind the advice not to put all your eggs in one basket.
The ingredient that captures how investments move together is covariance (a measure of whether two returns tend to rise and fall at the same time). Covariance is often converted to correlation, a number between -1 and 1.
The lower the correlation between holdings, the more the portfolio benefits from diversification. Because variance is in squared units, which are hard to picture, most people take the square root to get the standard deviation.
A portfolio with a standard deviation of 12% is easier to discuss than one with a variance of 0.0144. Fund managers use the figure to compare portfolios, set risk limits and build the efficient frontier (the set of portfolios giving the best return for each level of risk).
There are limits to the measure. It treats upside and downside swings as equally bad, it relies on past data to estimate future relationships, and correlations tend to rise in a crisis just when investors need them to stay low.
Treat the result as a guide to the usual range of outcomes rather than a promise.
In practice
Real-world examples.
Example
A pension fund trustee compares a portfolio of only domestic shares with one that adds government bonds. The bond holding has a low correlation with shares, so portfolio variance falls even though the bonds earn less.
Example
A founder who sold her company for $3,000,000 puts all of it into one sector fund. Her adviser shows that splitting it across three unrelated sectors would cut variance sharply with little change in expected return.
Example
An insurance company checks whether adding a property fund to its investments will push portfolio variance above the limit set by its board. The model shows the limit is still met because property returns have only a modest correlation with its bond holdings.
Formula
Calculation
For two assets: Portfolio variance = (w1^2 x s1^2) + (w2^2 x s2^2) + (2 x w1 x w2 x Cov12), where w is the weight, s is the standard deviation, and Cov12 = correlation x s1 x s2.
A portfolio holds 60% in Fund A (standard deviation 20%) and 40% in Fund B (standard deviation 10%), with a correlation of 0.3.
Step 1: Fund A term = 0.6^2 x 0.2^2 = 0.36 x 0.04 = 0.0144.
Step 2: Fund B term = 0.4^2 x 0.1^2 = 0.16 x 0.01 = 0.0016.
Step 3: Covariance = 0.3 x 0.2 x 0.1 = 0.006, so the cross term = 2 x 0.6 x 0.4 x 0.006 = 0.00288.
Step 4: Portfolio variance = 0.0144 + 0.0016 + 0.00288 = 0.01888.
Step 5: Standard deviation = the square root of 0.01888, which is about 13.7%. A simple weighted average of the two risks would give 16%, so diversification has cut the risk by roughly 2.3 percentage points.Case study
Seen in the real world.
Larkfield Partners is a fictional family office that held 70% in a single technology share fund and 30% in cash. Its adviser calculated the variance of the portfolio and found that almost all the risk came from the one fund.
The family agreed to move 30% of the portfolio into infrastructure and healthcare funds with lower correlation to technology. In this illustrative case, the calculated standard deviation dropped from about 17% to about 12% while expected return slipped only slightly, and the family accepted the trade because the portfolio was now much easier to live with in a downturn.
Watch out
Common mistakes.
- Averaging the individual risks by weight and calling the result portfolio risk, which ignores the effect of correlation.
- Assuming a low variance means a low chance of loss, when it only describes the usual spread of returns.
- Using correlations measured in calm markets and assuming they hold in a crisis, when they often rise sharply.
Questions
People also ask.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance, which puts the figure back into the same percentage units as returns.
Can portfolio variance be lower than every holding's own variance?
Yes, when the holdings have low enough correlation the combined portfolio can be less volatile than its steadiest member.
Do I need the formula for more than two assets?
The same idea applies, but you add a cross term for every pair of assets, which is why professionals use spreadsheets or software.
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