Covariance: Measuring How Investment Returns Vary Together

What is Covariance?

Covariance measures whether two variables tend to move above and below their averages together.

For investment returns, a common sample covariance is based on the paired deviations of two return series from their respective means:

1Covariance(A,B)
2≈ average of [(Return A - Mean A) × (Return B - Mean B)]

A positive covariance means the two return series tended to move in the same direction relative to their averages. A negative covariance means they tended to move in opposite directions. A covariance near zero means little linear co-movement was measured in the sample.

The sign is easy to interpret. The magnitude takes more care.

A small example

Suppose Asset A and Asset B both tend to post above-average returns in the same months and below-average returns in the same months.

Their paired deviations often have the same sign:

1positive × positive = positive
2negative × negative = positive

Those observations push covariance upward.

If one asset tends to be above average when the other is below average, the products are often negative and covariance falls.

That logic makes covariance useful in portfolio construction because it captures an important fact that standalone Volatility cannot: how holdings behave together.

Covariance is not standardized

Covariance is not standardized.

Its units depend on the units used for the underlying variables. If returns are expressed as decimals, the covariance is in squared decimal-return units. If someone instead uses percentage-point returns, the numerical covariance changes by a large scale factor even though the economic relationship is the same.

That makes raw covariance magnitudes awkward to compare across different pairs.

Correlation solves this by dividing covariance by the two assets' standard deviations:

1Correlation(A,B)
2=
3Covariance(A,B)
4-----------------------------
5Standard Deviation(A) × Standard Deviation(B)

Correlation is therefore a standardized expression of linear co-movement, bounded from -1 to +1. Covariance is not bounded in the same way.

The sign does not prove a causal relationship

Positive covariance can arise because two businesses depend on the same economic cycle, because two bonds respond to the same interest-rate changes, because two commodities share supply drivers, or because a short sample happened to produce similar moves.

Negative covariance can reflect genuine hedging behavior or temporary market conditions.

Neither sign proves that one asset causes the other to move.

Covariance describes a statistical relationship in the measured data. Economic interpretation requires additional evidence.

Covariance is a direct input to portfolio risk

For two assets, Portfolio Variance can be written as:

1σp²
2= wA² σA²
3+ wB² σB²
4+ 2 wA wB Cov(A,B)

The first two terms represent weighted standalone variances. The third is the covariance term.

Suppose two portfolios hold the same assets with the same weights and standalone volatilities. If one pair has lower covariance, that portfolio can have lower total variance.

This is the mathematical core of Diversification: the portfolio outcome depends on cross-asset relationships, not merely on how risky each holding looks by itself.

A covariance matrix scales the idea to many assets

With more than two securities, portfolio calculations use a covariance matrix.

A simplified three-asset matrix looks like this:

1             A          B          C
2A        Var(A)     Cov(A,B)   Cov(A,C)
3B        Cov(B,A)   Var(B)     Cov(B,C)
4C        Cov(C,A)   Cov(C,B)   Var(C)

The diagonal contains each asset's variance. The off-diagonal entries contain pairwise covariances.

Portfolio variance can then be written compactly as:

1Portfolio Variance = w' Σ w

where w is the vector of portfolio weights and Σ is the covariance matrix.

This compact expression sits behind many mean-variance optimization systems.

More assets create many more relationships

A portfolio of 100 securities does not require only 100 volatility estimates. It also contains thousands of pairwise relationships.

The number of unique pairwise covariances is:

1n(n - 1) / 2

For 100 assets, that is 4,950 pairwise covariances.

Estimating all of those relationships from limited historical data introduces noise. The problem becomes especially difficult when there are many assets but relatively few independent return observations.

This is one reason professional risk models often use factor structures, shrinkage, or other techniques instead of taking every raw sample covariance at face value.

Historical covariance can move sharply

A covariance estimate depends on:

• the sample period;
• return frequency;
• volatility during the sample;
• changing Correlation;
• market structure and liquidity;
• the economic regime; and
• changes in the underlying holdings or businesses.

Even if correlation stays constant, covariance can rise when the assets become more volatile because covariance combines correlation with both standard deviations:

1Cov(A,B) = Corr(A,B) × σA × σB

This matters in crises. Volatility can jump at the same time correlations rise, causing covariance estimates to increase substantially.

Covariance estimates can destabilize optimization

An Efficient Frontier and a Minimum Variance Portfolio depend on estimated covariance relationships.

If small estimation changes produce large weight changes, the optimizer may be fitting noise rather than discovering a durable portfolio.

Practical controls can include:

• limiting position sizes;
• grouping related assets into factors;
• shrinking extreme estimates;
• testing multiple lookback windows;
• imposing turnover limits;
• stress-testing higher correlations and volatility; and
• comparing optimized portfolios with simpler allocations.

The goal is not to pretend uncertainty disappears. It is to keep noisy inputs from producing implausibly precise outputs.

Covariance is different from tracking error and beta

Covariance appears in several investment statistics, but the concepts are not interchangeable.

Beta standardizes an asset's covariance with a benchmark by the benchmark variance:

1Beta = Covariance(Asset, Benchmark) / Variance(Benchmark)

Tracking Error measures the standard deviation of active returns relative to a benchmark.

Covariance itself only describes paired co-movement. It does not tell you whether a manager added value, how much absolute risk a portfolio carries, or whether a benchmark is appropriate.

Covariance is not a tail-risk measure

A covariance matrix is central to many portfolio models, but it can summarize only the relationships represented by the chosen return distribution and estimation method.

It does not directly tell an investor:

• the worst historical drawdown;
• the probability of an extreme nonlinear loss;
• liquidity under stress;
• the loss threshold at a chosen confidence level; or
• the average loss beyond that threshold.

Those questions belong to other measures, including Maximum Drawdown, Value at Risk, and Expected Shortfall.

How investors should use covariance

A disciplined review asks:

1. What return units were used?
2. What frequency and sample period produced the estimate?
3. Is this sample covariance or a forecast from a risk model?
4. How volatile were the assets during the estimation period?
5. Did correlations change materially across subperiods?
6. Are the assets driven by common factors that a raw matrix may obscure?
7. How sensitive are portfolio weights to small covariance changes?
8. What happens under a stressed covariance matrix?

Grizzly Bulls' Models can be evaluated with those questions without turning this article into live covariance authority. The Macroeconomic Conditions Index adds separate regime context, but it does not replace the return data and estimation choices required for a covariance matrix.

Sources and further reading

• CFA Institute, 2026, Portfolio Risk and Return: Part I: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/portfolio-risk-return-part-1
• CFA Institute, 2026, Portfolio Mathematics: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/portfolio-mathematics
• CFA Institute, 2026, Using Multifactor Models: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/using-multifactor-models

Explore more topics in the Financial Research Encyclopedia.