Correlation and covariance measure how investment returns move together. Correlation expresses that relationship on a scale from −1 to +1. Covariance preserves the size of the return movements, making it a direct input to portfolio risk calculations.
The distinction matters because portfolio volatility is not simply the weighted average of each holding’s volatility. It also depends on how those holdings behave together. Two portfolios can hold equally volatile assets in identical proportions and still produce different risk estimates. Within the mathematics of portfolio construction, covariance connects individual holdings to the risk of the combined portfolio.
What Is the Difference Between Correlation and Covariance?
Covariance measures joint deviations from average returns
Covariance measures whether two assets tend to deliver returns above or below their respective averages at the same time.
A positive covariance means their deviations tend to share the same sign: both above average or both below average. A negative covariance means an above average return for one tends to accompany a below average return for the other. This concerns deviations from averages, not simply whether both investments gained or lost money.
For paired historical returns, the NIST sample covariance formula becomes:
sAB = Σ[(rA,t − r̄A)(rB,t − r̄B)] ÷ (n − 1)
Here, rA,t and rB,t are the returns for the same observation period, r̄ denotes each asset’s arithmetic average return, and n is the number of paired observations. Subtract each average, multiply the paired deviations, add those products, then divide by n − 1.
Covariance has no fixed numerical range. Its magnitude depends on both the strength of the relationship and the volatility of the assets. Its units also depend on how returns are entered, which makes consistency important.
Correlation removes the effect of scale
Pearson correlation divides covariance by the product of the assets’ standard deviations. Using sample estimates:
ρAB = sAB ÷ (sAsB)
The result measures the strength and direction of a linear relationship. The NIST definition of Pearson correlation distinguishes perfect positive correlation at +1, perfect negative correlation at −1, and zero linear correlation at 0.
A correlation of +0.80 indicates a stronger positive linear relationship than +0.20. Neither figure tells you how volatile the assets are. Two relatively stable investments and two highly volatile investments can have the same correlation but very different covariances.
Zero correlation does not prove independence. Assets can have a nonlinear relationship that Pearson correlation misses. Nor does a correlation of −0.50 mean one investment rises whenever the other falls. It describes a statistical relationship across observations, not a promise for every trading session.
Use correlation to compare relationships. Use covariance to calculate how those relationships affect portfolio variance.
How Covariance Enters Portfolio Risk
Consider a fully invested portfolio containing two assets, A and B. Their weights, wA and wB, sum to 1. For fixed weights over the return period, portfolio variance is:
σp2 = wA2σA2 + wB2σB2 + 2wAwBCov(A,B)
Because Cov(A,B) = ρABσAσB, the same calculation can use correlation and individual volatilities. Both forms appear in William Sharpe’s Stanford material on two asset portfolios.
The first two terms capture each holding’s own variance, adjusted by its squared weight. The final term captures their interaction. The factor of 2 appears because expanding the squared portfolio return produces two matching cross terms.
Take the square root of portfolio variance to obtain portfolio standard deviation, commonly called portfolio volatility. Variance and volatility are related, but they are not interchangeable.
A worked example: a 60/40 portfolio
Suppose a hypothetical portfolio holds 60% in Asset A and 40% in Asset B. Assume annual return volatilities of 20% and 10%, respectively, with a correlation of +0.25. These are illustrative assumptions, not estimates for a named stock or bond market.
Enter the volatilities as decimals. Covariance is:
Cov(A,B) = 0.25 × 0.20 × 0.10 = 0.005
Substitute the weights, variances and covariance:
σp2 = (0.602 × 0.202) + (0.402 × 0.102) + (2 × 0.60 × 0.40 × 0.005)
σp2 = 0.0144 + 0.0016 + 0.0024 = 0.0184
σp = √0.0184 ≈ 13.56%
The weighted average of the individual volatilities would be 16%: 60% × 20% plus 40% × 10%. Portfolio volatility is lower because the returns are not perfectly positively correlated.
However, 13.56% is still higher than Asset B’s standalone volatility of 10%. Diversification reduces volatility relative to the perfectly correlated combination; it does not automatically make every blend less volatile than its safest holding.
Changing correlation while holding everything else constant
Keep the same 60/40 weights and individual volatilities, but change the assumed correlation. The resulting estimates are:
| Correlation | Portfolio volatility |
|---|---|
| +1.00 | 16.00% |
| +0.25 | 13.56% |
| 0.00 | 12.65% |
| −0.50 | 10.58% |
| −1.00 | 8.00% |
For these positive weights, lower correlation produces lower portfolio variance. Negative correlation is not required for diversification benefits: the +0.25 case already produces less volatility than the +1.00 case.
Even perfect negative correlation does not eliminate volatility at these weights. At −1, volatility equals |wAσA − wBσB|. Here, that is |0.12 − 0.04|, or 8%. Cancellation would require different weights as well as a perfectly negative relationship.
Reading a Portfolio Covariance Matrix
A covariance matrix organizes the same information for several assets. Each row and column represents a holding. The diagonal entries contain individual variances; entries away from the diagonal contain covariances between pairs.
In the worked example, Asset A’s variance is 0.0400, Asset B’s variance is 0.0100, and both matching off diagonal entries are 0.0050. The matrix is symmetric because Cov(A,B) equals Cov(B,A).
For a portfolio of many assets, the compact formula is:
σp2 = wTΣw
Here, w is the vector of portfolio weights, wT is its transpose, and Σ is the covariance matrix. This is shorthand for multiplying every variance and covariance by the relevant weights and adding the results. MIT’s risk analytics and portfolio theory notes present this weighted variance and covariance structure.
A correlation matrix follows a similar layout, but its diagonal entries are 1 for assets with nonzero volatility. It is easier to scan for strong relationships, though it cannot determine portfolio volatility without individual volatility estimates.
Capital weights alone therefore give an incomplete picture of risk. A 20% allocation to a volatile asset can affect the portfolio more than a larger allocation to a steadier holding. Covariance also matters, which is why risk contribution and risk parity allocation require more than comparing investment amounts.
Estimating Correlation and Covariance From Return Data
The formulas are straightforward. Preparing comparable inputs requires more care. A spreadsheet will happily calculate a precise answer from mismatched data, it will not warn you that the question was wrong.
Use returns, not raw price levels
For portfolio risk analysis, calculate relationships between returns rather than share prices or index levels. Two price series can rise over time without having a stable relationship between their period by period returns.
Use a consistent return definition. If the analysis concerns total investment performance, include distributions such as dividends and interest where applicable. Check the data provider’s treatment of splits and distributions rather than assuming every adjusted price series follows the same method.
For straightforward portfolio aggregation, simple returns provide a clear starting point: each asset’s period return is multiplied by its beginning weight.
Match dates, frequency and currency
Pair returns covering the same periods. A monthly return for one asset should not be matched with a daily return for another. Different holidays, closing times and missing observations also deserve attention.
Measure returns in the currency relevant to the investor. A US investor holding an unhedged foreign asset receives a dollar return affected by both the asset and the exchange rate. Correlations calculated entirely in local currencies answer a different question.
Keep covariance and volatility on the same time basis. Monthly covariance cannot be inserted alongside annual volatility estimates without a consistent conversion method and explicit assumptions.
Check the sample and numerical scale
A practical calculation sequence is:
- Build aligned return observations for every holding over a common period.
- Choose a consistent sample convention, such as covariance using n − 1.
- Calculate covariance, standard deviation and correlation from those observations.
- Apply the proposed portfolio weights and test alternative assumptions.
With a common return sample and unchanged weights, the standard deviation of the weighted portfolio return series should match the covariance based result, apart from rounding. This provides a useful calculation check.
Also distinguish decimal returns from percentage point entries. Entering 20 rather than 0.20 multiplies that series by 100. If both series are scaled this way, covariance increases by 10,000, while correlation stays unchanged. A large covariance is not meaningful until its units are clear.
What These Measures Cannot Tell You
A historical correlation describes the chosen sample. It is not a contractual relationship between assets. Changing the observation window or return frequency can change the estimate, so compare results across reasonable alternatives rather than treating one coefficient as permanent.
Use scenario analysis to ask what happens if relationships become less favorable. For positive portfolio weights, increasing correlation while holding volatility constant raises the covariance contribution to variance. The separate guide to rising correlations during market stress examines that problem without assuming every asset pair behaves identically.
Correlation and covariance also do not specify maximum losses. A portfolio volatility estimate of 13.56% does not mean losses stop at 13.56%. Variance summarizes dispersion around an average; it does not describe the entire return distribution or every risk attached to an investment.
Do not confuse the number of holdings with the number of independent exposures, either. Several funds may own many of the same securities. The SEC’s guidance on asset allocation and diversification highlights the need to check overlapping holdings and warns that narrowly focused funds do not necessarily provide diversification.
Using the Numbers to Assess an Allocation
Start with the proposed weights, not with a search for the lowest correlation available. Combine those weights with comparable volatility and covariance estimates, then calculate the resulting portfolio risk.
Next, change the assumptions. Does the allocation still look acceptable if correlation rises, one holding becomes more volatile, or the historical window changes? An allocation that looks attractive only under one favorable estimate deserves caution.
Correlation is also not an expected return measure. A negatively correlated investment can reduce estimated volatility while offering an unattractive expected return, high costs or an unsuitable exposure. Choosing an allocation requires evaluating those tradeoffs together. That is the next step in balancing expected return and risk on the efficient frontier.
The useful question is not simply whether two investments move together. It is how their relationship, volatility and position sizes change the behavior of the portfolio you actually plan to hold.