Why Joint Return Variation Matters
Two assets can each have substantial return variability and still behave differently from one another. When one asset’s return is above its expected return, the other might also tend to be above its own expected return, or it might tend to be below its expected return. Covariance summarizes this joint pattern by averaging the products of the assets’ deviations from their respective expected returns.
This tutorial focuses on the meaning, sign, magnitude, and units of covariance. It does not require the two returns to have the same expected value or the same variability. The important comparison is whether each return is above or below its own expectation during the same observation period.
The definition applies to returns measured on a consistent basis: for example, both assets’ returns over the same year, with both expressed as decimals or both as percentage points. The returns can be modeled as random variables, and the expectations are taken under the same probability model. Finite means and variances ensure that the expected product of deviations is well-defined.
Reading the Sign
The product inside the expectation determines the basic interpretation. If both deviations are positive, their product is positive. If both are negative, their product is also positive. If one deviation is positive and the other negative, their product is negative. Covariance averages these signed products, so its sign describes the balance of joint movements—not whether either asset has a positive expected return.
| Deviation for asset A | Deviation for asset B | Product of deviations |
|---|---|---|
| Above its expected return | Above its expected return | Positive |
| Below its expected return | Below its expected return | Positive |
| Above its expected return | Below its expected return | Negative |
| Below its expected return | Above its expected return | Negative |
Positive covariance means positive cross-products dominate on average: the assets’ deviations tend to point in the same direction. This does not mean every observation has matching signs. Negative covariance means negative cross-products dominate on average: the deviations tend to point in opposite directions. Again, it does not require opposite signs in every observation.
A covariance of zero means the expected product of the deviations is zero. Positive and negative cross-products may offset, or some deviations may be zero. Zero covariance does not, in general, establish that the returns are independent; it only states that their average cross-product of centered deviations is zero.
Worked Example: Interpreting a Positive Covariance
A hypothetical investment analysis reports annual decimal returns for two assets over the same population of possible market outcomes. Their expected returns are 8% for asset A and 5% for asset B, and their covariance is \(0.0012\). Interpret the sign and units, and explain what the value does and does not establish.
Method. Read the covariance as the expected product of deviations from the two separate means. The returns are stated as decimals, so \(8\%=0.08\) and \(5\%=0.05\).
When asset A returns more than 0.08 and asset B returns more than 0.05, both deviations are positive, giving a positive product. When both return less than their respective means, both deviations are negative and the product is again positive. The reported positive covariance indicates that these same-direction products dominate the opposite-direction products on average.
The value \(0.0012\) is in decimal-return-squared units. It is not a return of 0.12%, nor does it say that asset A earns 0.12% more when asset B does. Covariance is an average of products of deviations, not a percentage change in one return caused by the other.
The value also does not imply that the assets’ deviations have matching signs in every outcome. It summarizes an average across the probability model. Common mistake avoided: interpreting a positive covariance as a guarantee that both returns will be above their means in each period. The definition supports a tendency in the average cross-product, not an outcome-by-outcome guarantee.
An Equivalent Formula and Why It Holds
The definition can be expanded into a form that is useful when the expected product of the two returns is available. This identity follows by multiplying out the centered returns and then applying the linearity of expectation.
In the third line, \(\mu_A\) and \(\mu_B\) are constants, so they can be taken outside the expectation. In the final line, \(E[R_A]=\mu_A\) and \(E[R_B]=\mu_B\); thus the middle terms combine with the constant term to leave \(-\mu_A\mu_B\). The result is valid when the expectations exist, as they do under the finite-second-moment condition stated above.
This is a computational identity, not a different definition. It also shows that covariance depends on the joint expected product \(E[R_AR_B]\) as well as on each return’s mean. Knowing the two means alone is not enough to determine covariance.
Worked Example: Finding Covariance From Return Moments
For two hypothetical assets, both measured as decimal annual returns under the same probability model, suppose \(E[R_A]=0.10\), \(E[R_B]=0.08\), and \(E[R_AR_B]=0.0098\). Find and interpret the covariance.
Method. Use the equivalent formula \(\operatorname{Cov}(R_A,R_B)=E[R_AR_B]-E[R_A]E[R_B]\). The required moments are supplied, so a scenario-by-scenario calculation is not needed.
First multiply the means: $$ E[R_A]E[R_B]=(0.10)(0.08)=0.0080. $$
Then subtract this product from the expected product: $$ \operatorname{Cov}(R_A,R_B)=0.0098-0.0080=0.0018. $$
The covariance is positive, so the expected product of the centered returns is positive. In this model, same-direction deviations dominate in the average cross-product. Since both returns were expressed as decimals, the covariance is in decimal-return-squared units. Check: adding back the product of the means recovers the supplied moment: \(0.0018+0.0080=0.0098\).
Common mistake avoided: subtracting one mean from the expected product, or treating \(E[R_AR_B]\) as the covariance itself. The required subtraction is the product of the two means, \(0.10\times0.08\).
Why Covariance Has Squared-Return Units
A return deviation has the same unit as the return. The definition multiplies one deviation by another, so the product has the return unit squared. For decimal returns, the unit is decimal-return-squared; for returns expressed in percentage points, it is percentage-points-squared.
For example, a deviation of \(0.02\) in decimal-return units is a two-percentage-point deviation. Multiplying it by another decimal-return deviation produces a value in decimal-return-squared units. If the same deviations are instead written as 2 and 1.5 percentage points, their product is 3 percentage-points-squared. Since one percentage point is \(0.01\) in decimal form, the numerical values differ by a factor of \(100^2=10{,}000\). The underlying return observations have not changed; only their measurement scale has.
Adding a constant to a return changes its mean by the same constant but leaves its deviations from the mean unchanged. Multiplying a return by a constant multiplies its deviations by that constant. Applying these facts to the definition gives:
where \(a,b,c,d\) are constants. The additions \(c\) and \(d\) disappear when deviations from the new means are formed; the scale factors \(a\) and \(b\) remain as factors in the product. This explains both why a change of origin does not alter covariance and why changing the return scale does.
Worked Example: Tracking Scale and Units
A hypothetical analysis reports \(\operatorname{Cov}(R_A,R_B)=0.0004\), where both returns are decimals. A revised measure defines \(R_A^*=1.5R_A+0.02\) and \(R_B^*=0.5R_B-0.01\). Find the revised covariance, then express it in squared percentage-point units.
Method. Apply the affine-transformation rule. The additive constants shift the returns but do not affect their deviations from their means.
The scale factors are 1.5 and 0.5, so $$ \operatorname{Cov}(R_A^*,R_B^*) =(1.5)(0.5)(0.0004) =0.75(0.0004) =0.0003. $$
This result is still in squared units of the revised decimal returns. To convert a covariance from decimal-return-squared units to percentage-point-squared units, multiply by \(10{,}000\), because each return value is multiplied by 100 and the product of two returns is therefore multiplied by \(100^2\): $$ 0.0003(10{,}000)=3.00. $$
The revised covariance is \(0.0003\) in decimal-return-squared units, or \(3.00\) percentage-points-squared. Check: converting back gives \(3.00/10{,}000=0.0003\). Common mistake avoided: treating the added constants as covariance multipliers. Only the two scale factors multiply the original covariance.
Interpreting Magnitude Carefully
The sign has a direct interpretation in terms of the direction of the average cross-product. The magnitude requires more care. Because covariance is measured in squared-return units and is affected by the scale of each return, its numerical value alone is not a unit-free measure of how closely two assets move together. A covariance can change numerically if returns are recorded in a different unit, even when the underlying observations are unchanged.
Magnitude also reflects the sizes of the deviations, not only whether their signs match. Larger deviations can produce larger products. Therefore, a larger covariance does not by itself establish a stronger same-direction relationship across two different pairs of assets, especially if their return scales or variability differ. Keep the comparison on a consistent return scale and interpret the value in the context of the returns being measured.
In portfolio analysis, the joint behavior of returns matters because an asset’s return deviation can occur alongside deviations in other assets. Covariance describes one part of that joint behavior. Its interpretation should remain distinct from each asset’s expected return and individual variance: the means describe expected levels, the variances describe each return’s own dispersion, and covariance describes paired deviations.
Common Exam Traps
- Using deviations from the wrong reference points. Center each return on its own expected return, not on the other asset’s mean or on a shared average.
- Reading the sign as a certainty. Positive covariance does not mean every pair of deviations has the same sign; negative covariance does not mean every pair has opposite signs.
- Calling covariance a return or percentage. It is measured in the product of return units, and in the same-unit case those units are squared.
- Inferring independence from zero covariance. Zero covariance only says that the expected product of centered returns is zero; it does not generally rule out other forms of dependence.
- Comparing raw magnitudes without checking units. A change from decimal returns to percentage-point returns changes the numerical covariance by a factor of 10,000.
Check Your Understanding
For each problem, use the stated return units and distinguish the sign interpretation from what covariance cannot establish.
- Two hypothetical assets have decimal expected returns of 7% and 4%, and \(E[R_AR_B]=0.0040\). Find their covariance and state its sign interpretation. (Answer: \(0.0012\) decimal-return-squared; positive, so same-direction deviation products dominate on average.)
- A reported covariance is \(-0.0009\) for two decimal annual returns. What does the negative sign indicate, and does it require opposite-signed deviations in every outcome? (Answer: Opposite-direction deviation products dominate on average; no, it does not require opposite signs in every outcome.)
- Returns are recorded in percentage points, and the covariance is \(12\). Express the covariance in decimal-return-squared units. (Answer: \(0.0012\), since \(12/10{,}000=0.0012\).)
- Suppose \(\operatorname{Cov}(R_A,R_B)=0.0005\), \(R_A^*=2R_A+0.03\), and \(R_B^*=0.5R_B-0.01\). Find the covariance of the revised returns. (Answer: \(0.0005\), since \(2(0.5)(0.0005)=0.0005\); the additive shifts do not affect covariance.)
- Two returns have zero covariance. What precise conclusion follows, and what stronger conclusion does not generally follow? (Answer: Their centered deviations have expected product zero; independence does not generally follow.)