Why Convert Between Covariance and Correlation?
Covariance and correlation both describe how two returns vary together, but they answer that question on different scales. Covariance retains the units of the two returns, so its numerical value changes when returns are recorded as decimals rather than percentages. Correlation removes those units by scaling covariance by the returns’ standard deviations. That makes correlation easier to compare across pairs of assets with different levels of volatility.
The conversion is useful when a problem supplies one measure but asks for the other. It is also a consistency check: a valid correlation must lie between negative one and positive one. The calculation is meaningful only when the covariance and standard deviations refer to the same pair of returns, observation period, and method of calculation.
Here \(\operatorname{Cov}(R_A,R_B)\) is the expected product of the returns’ deviations from their respective means. A positive covariance indicates a tendency for deviations to have the same sign; a negative covariance indicates a tendency for them to have opposite signs. A covariance near zero indicates little linear co-movement, but does not by itself establish that the returns are independent.
The Conversion Formulas and Their Conditions
To calculate correlation, divide covariance by the product of the two standard deviations. To calculate covariance, multiply correlation by that product. Both standard deviations must be positive: if either return has zero standard deviation, it is constant, and the correlation is undefined because the denominator is zero.
For random returns with finite variances, correlation satisfies \(-1\leq \rho_{A,B}\leq 1\). A correlation of \(1\) means that one return is an exact positive linear function of the other; \(-1\) means an exact negative linear function. Values between these endpoints describe less-than-perfect linear association. The sign indicates the direction of linear co-movement, while the magnitude indicates its strength.
The units provide a practical guide to the arithmetic. If both returns are expressed as decimals, their standard deviations are decimals and covariance is in decimal-return-squared units. If both returns are expressed in percentage points, covariance is in percentage-point-squared units. The numerical value of covariance changes between these representations, but correlation does not, provided all quantities in the calculation use a consistent representation.
Why the Formula Works
Standardizing a return means subtracting its mean and dividing by its standard deviation. The standardized return has mean zero and standard deviation one. Correlation is the covariance of these two standardized returns, which explains why dividing covariance by the product of the original standard deviations removes the units.
To see the conversion directly, use the definition of covariance and the fact that the standard deviations are positive constants:
The middle equality follows because a constant factor can be taken outside an expectation. The final equality is the definition of correlation. Thus correlation is precisely the covariance after each return has been measured in standard-deviation units. The bound on correlation follows from the Cauchy–Schwarz inequality: the absolute covariance of two finite-variance variables cannot exceed the product of their standard deviations. Dividing that inequality by the positive product gives \(|\rho_{A,B}|\leq 1\).
Worked Conversions
Worked Example: Converting a Negative Correlation to Covariance
A hypothetical growth fund has an annual return standard deviation of 12%, and a hypothetical inflation-sensitive fund has an annual return standard deviation of 8%. Their annual returns have correlation \(-0.25\). Find the covariance in decimal-return-squared units and in percentage-point-squared units. Explain what its sign means.
Method. Use \(\operatorname{Cov}(R_A,R_B)=\rho_{A,B}\sigma_A\sigma_B\), with all inputs first expressed in the same units. The rates are annual, so both standard deviations and the correlation must describe the same annual return period.
Convert the standard deviations to decimals: \(\sigma_A=0.12\) and \(\sigma_B=0.08\). Their product is \(0.12(0.08)=0.0096\). Therefore, \[ \operatorname{Cov}(R_A,R_B)=(-0.25)(0.0096)=-0.0024. \]
To express the result in percentage-point-squared units, note that one decimal return equals 100 percentage points. The covariance is therefore multiplied by \(100^2=10{,}000\): \(-0.0024(10{,}000)=-24\) percentage-point-squared units.
Check and interpretation. Dividing \(-0.0024\) by \(0.12(0.08)=0.0096\) returns \(-0.25\). The negative covariance indicates a tendency for the two returns’ deviations from their means to have opposite signs. It does not mean that one fund’s return is always the negative of the other’s. A common mistake is to report \(-0.0024\) as percentage-point-squared units; that number is in decimal-return-squared units. The correct percentage-point-squared value is \(-24\).
Worked Example: Recovering Correlation From Covariance
For two hypothetical monthly investment returns, the covariance is \(0.0018\) in decimal-return-squared units. Their monthly standard deviations are 6% and 10%. Find the correlation and state whether the value is within the permitted range.
Method. Convert each standard deviation to decimal form, then divide the covariance by their product. The covariance and standard deviations must describe the same monthly returns; annual standard deviations would not be suitable inputs here.
The standard deviations are \(\sigma_A=0.06\) and \(\sigma_B=0.10\). Their product is \(0.06(0.10)=0.006\). Thus, \[ \rho_{A,B}=\frac{0.0018}{0.006}=0.30. \]
Unit check. The same calculation can be done using percentage points. The standard deviations are 6 and 10 percentage points. Convert the covariance by multiplying by \(10{,}000\): \(0.0018(10{,}000)=18\) percentage-point-squared units. Then \(18/(6(10))=18/60=0.30\).
Check and interpretation. The result, \(0.30\), is between \(-1\) and \(1\), as required. It indicates positive but not perfect linear co-movement. A common mistake is to divide \(0.0018\) by \(6(10)=60\), mixing decimal covariance with percentage-point standard deviations. That inconsistent calculation gives \(0.00003\), not the correlation. Either use decimal units throughout or convert all the quantities consistently.
Worked Example: Handling Covariance Reported in Percentage-Point Units
A hypothetical pair of annual returns has standard deviations of 6% and 5%. A report gives their covariance as \(-18\) percentage-point-squared units. Calculate correlation using both a percentage-point calculation and a decimal calculation, then check that the results agree.
Method. Since the covariance is reported in percentage-point-squared units, first calculate directly in percentage points. Then convert the covariance and standard deviations to decimal units as an independent check.
In percentage points, the standard deviations are 6 and 5. Therefore, \[ \rho_{A,B}=\frac{-18}{6(5)}=\frac{-18}{30}=-0.60. \]
For the decimal check, convert the standard deviations to \(0.06\) and \(0.05\). Convert the covariance by dividing by \(10{,}000\): \(-18/10{,}000=-0.0018\). Then \[ \rho_{A,B}=\frac{-0.0018}{0.06(0.05)} =\frac{-0.0018}{0.003} =-0.60. \]
Check and interpretation. Both consistent unit systems give the same correlation, and \(-0.60\) lies within the required range. The negative value signals an inverse linear tendency, not a guaranteed opposite movement in every year. A common mistake is to use \(-18\) with decimal standard deviations: \(-18/(0.06(0.05))\) mixes units and has no valid correlation interpretation.
Using Correlation Carefully
Because correlation is unit-free, it helps compare linear co-movement across return pairs whose scales differ. But it does not describe how volatile either return is on its own. Two pairs can have the same correlation and very different covariances if their standard deviations differ. Conversely, a covariance’s size alone is difficult to interpret without knowing the return units and standard deviations that produced it.
Correlation also captures linear association rather than every possible dependence. A correlation of zero means zero covariance, but does not generally prove that two returns are independent or unrelated in every way. In addition, correlation is not itself a statement about causation. When interpreting a result, identify the time period, the return units, the sign, and whether the relationship is perfect or imperfect.
For historical data, use a covariance and standard deviations calculated from the same observations and with the same convention. For example, a sample covariance paired with standard deviations calculated from a different sample or time window may not produce a meaningful sample correlation. With a consistent sample, the conversion works in the same way as for population quantities.
Check Your Understanding
For each problem, keep the return period and units consistent. Answers are given at the end of each question.
- A hypothetical pair of quarterly returns has standard deviations of 4% and 7%, with correlation \(0.50\). Find covariance in decimal-return-squared units. (Answer: \(0.0014\))
- Two hypothetical annual returns have standard deviations of 5% and 8%, and covariance \(0.0020\) in decimal-return-squared units. Find their correlation and check its range. (Answer: \(0.50\), within \([-1,1]\))
- A report gives covariance \(-12\) percentage-point-squared units. The corresponding standard deviations are 3 and 8 percentage points. Find correlation and state its direction. (Answer: \(-0.50\), negative linear co-movement)
- Two monthly returns have correlation \(-0.40\) and standard deviations of 2% and 5%. Find covariance in decimal-return-squared units and in percentage-point-squared units. (Answer: \(-0.0004\) decimal-return-squared units; \(-4\) percentage-point-squared units)
- A pair of returns has covariance \(0.0042\) in decimal-return-squared units and standard deviations \(0.06\) and \(0.08\). A proposed correlation is \(1.10\). Check the conversion and explain what the result implies. (Answer: \(0.0042/(0.06(0.08))=0.875\); \(1.10\) is inconsistent with the supplied values and outside the permitted range)