Why Scenario Covariance Matters
An asset’s expected return and its variance describe its return on its own. To assess how two assets move together, we need a measure that uses their returns in the same scenarios. Covariance does this by multiplying the two assets’ deviations from their respective expected returns in each scenario, then averaging those products using the scenario probabilities.
The pairing matters. A table must show which return for Asset A occurs alongside which return for Asset B in each state of the world. Two separate lists of possible returns, without their joint scenario pairings, are not enough to calculate covariance. This tutorial focuses on a discrete probability model in which each scenario has a specified probability and a return for each of two assets.
The returns in this definition must use a consistent unit. If returns are entered as percentages, such as \(5\) for \(5\%\), covariance is in squared percentage points. If returns are entered as decimals, such as \(0.05\) for \(5\%\), covariance is in decimal-return squared. The numerical values differ by a factor of \(10{,}000\), since converting a percentage-point return to a decimal return divides it by \(100\), and covariance multiplies two returns.
Calculating Covariance from a Scenario Table
For each asset, first calculate its probability-weighted expected return. Then, in each scenario, subtract each asset’s own expected return from its scenario return. Multiply those two deviations together and weight the product by the scenario probability. The sum of the weighted products is the covariance.
A positive product occurs when both assets’ deviations have the same sign in a scenario: both are above expectation or both are below. A negative product occurs when one asset is above expectation while the other is below. Covariance adds these signed, probability-weighted contributions. Its sign gives a direction of joint variation in the stated model; the magnitude depends on the return units and scales.
Confirm that the probabilities are nonnegative and sum to one, and that both assets’ returns are paired in each scenario for the same period.
For each asset, multiply each scenario return by its probability and add the results.
Subtract the appropriate expected return from each scenario return, then multiply the two deviations within that scenario.
Multiply each deviation product by its scenario probability and add all contributions. Report the unit used for the returns.
Worked Example: Three Economic Scenarios
A hypothetical investment analysis assigns probabilities \(0.20\), \(0.50\), and \(0.30\) to three economic scenarios. The two assets’ one-year returns are shown as percentages. Calculate their covariance. Use the same percentage-point units throughout, then also express the result in decimal-return squared.
| Scenario | Probability | Asset A return | Asset B return |
|---|---|---|---|
| Weak conditions | 0.20 | -10% | 20% |
| Stable conditions | 0.50 | 5% | 0% |
| Strong conditions | 0.30 | 15% | -5% |
Method. Calculate each expected return, then sum the probability-weighted products of paired deviations. This method retains the scenario matching instead of treating the assets’ return lists separately.
Step 1: Check probabilities. Their sum is \(0.20+0.50+0.30=1.00\), so they form a probability distribution.
Step 2: Calculate expected returns. For Asset A, the contributions are \(0.20(-10)=-2\), \(0.50(5)=2.5\), and \(0.30(15)=4.5\) percentage points. Thus \(E[R_A]=-2+2.5+4.5=5\%\). For Asset B, the contributions are \(0.20(20)=4\), \(0.50(0)=0\), and \(0.30(-5)=-1.5\), giving \(E[R_B]=2.5\%\).
Step 3: Find deviations and weighted products. Returns and deviations below are expressed in percentage points.
| Scenario | Asset A deviation | Asset B deviation | Deviation product | Probability-weighted product |
|---|---|---|---|---|
| Weak | \(-10-5=-15\) | \(20-2.5=17.5\) | \((-15)(17.5)=-262.5\) | \(0.20(-262.5)=-52.5\) |
| Stable | \(5-5=0\) | \(0-2.5=-2.5\) | \(0(-2.5)=0\) | \(0.50(0)=0\) |
| Strong | \(15-5=10\) | \(-5-2.5=-7.5\) | \((10)(-7.5)=-75\) | \(0.30(-75)=-22.5\) |
Step 4: Sum and convert units. The covariance is \(-52.5+0-22.5=-75\) squared percentage points. Since each percentage-point return is \(1/100\) of a decimal return, divide by \(100^2=10{,}000\) to express covariance in decimal-return squared: \(-75/10{,}000=-0.0075\).
Check. The weak and strong scenarios each have deviations with opposite signs, so their products are negative; the stable scenario contributes zero because Asset A is exactly at its expected return. The negative result is consistent with those contributions. The covariance is \(-75\) squared percentage points, or \(-0.0075\) decimal-return squared.
Why the Formula Uses Deviations from Expected Returns
Covariance is the expected value of the product of the two centered returns. Centering means subtracting each variable’s own expected value. For discrete scenarios, an expected value is a probability-weighted sum, so applying that rule to the centered product gives the scenario formula directly.
A useful check follows by expanding the product. Write \(\mu_A=E[R_A]\) and \(\mu_B=E[R_B]\). Then
The final equality uses \(E[R_A]=\mu_A\) and \(E[R_B]=\mu_B\). Thus an equivalent computational check is \(\operatorname{Cov}(R_A,R_B)=E[R_AR_B]-E[R_A]E[R_B]\). In a discrete model, \(E[R_AR_B]\) is found by multiplying the two returns paired in each scenario, weighting that product by the scenario probability, and summing. This identity is a check on the centered-deviation calculation; it does not remove the need for correct joint scenario pairings.
Worked Example: Four Unequally Likely Scenarios
A hypothetical analyst evaluates two assets over the same period under four mutually exclusive scenarios. Calculate covariance in squared percentage points and verify it with the computational identity.
| Scenario | Probability | Asset A return | Asset B return |
|---|---|---|---|
| 1 | 0.10 | 2% | 6% |
| 2 | 0.40 | 8% | 3% |
| 3 | 0.30 | -4% | 12% |
| 4 | 0.20 | 10% | -2% |
Method. Use the probability-weighted deviations formula, with each probability applied once to the product for its paired scenario.
Step 1: Check probabilities and means. The probabilities sum to \(0.10+0.40+0.30+0.20=1.00\). Asset A’s expected return is \(0.10(2)+0.40(8)+0.30(-4)+0.20(10)=0.2+3.2-1.2+2=4.2\%\). Asset B’s expected return is \(0.10(6)+0.40(3)+0.30(12)+0.20(-2)=0.6+1.2+3.6-0.4=5.0\%\).
Step 2: Calculate each contribution. All deviations and products are in percentage-point units.
| Scenario | Asset A deviation | Asset B deviation | Product | Weighted contribution |
|---|---|---|---|---|
| 1 | \(2-4.2=-2.2\) | \(6-5=1\) | \(-2.2\) | \(0.10(-2.2)=-0.22\) |
| 2 | \(8-4.2=3.8\) | \(3-5=-2\) | \(-7.6\) | \(0.40(-7.6)=-3.04\) |
| 3 | \(-4-4.2=-8.2\) | \(12-5=7\) | \(-57.4\) | \(0.30(-57.4)=-17.22\) |
| 4 | \(10-4.2=5.8\) | \(-2-5=-7\) | \(-40.6\) | \(0.20(-40.6)=-8.12\) |
Step 3: Sum contributions. The covariance is \(-0.22-3.04-17.22-8.12=-28.60\) squared percentage points.
Step 4: Verify with the computational identity. The scenario return products are \(2(6)=12\), \(8(3)=24\), \((-4)(12)=-48\), and \(10(-2)=-20\). Therefore \(E[R_AR_B]=0.10(12)+0.40(24)+0.30(-48)+0.20(-20)=1.2+9.6-14.4-4=-7.6\) squared percentage points. Also, \(E[R_A]E[R_B]=(4.2)(5.0)=21\). The difference is \(-7.6-21=-28.6\), which matches the centered calculation.
Common mistake avoided. Giving each scenario equal weight would ignore the stated probabilities. For example, scenario 3 has probability \(0.30\), not \(0.25\); its weighted contribution is \(0.30(-57.4)=-17.22\), not \(0.25(-57.4)\). The correct covariance is \(-28.60\) squared percentage points.
Interpreting Results and Avoiding Traps
The covariance is measured in the product of the two return units. For returns measured in the same units, it is a squared return unit. This makes the sign easier to interpret than the size: positive covariance means the weighted products of deviations sum to a positive value, while negative covariance means they sum to a negative value. A covariance of zero means the weighted signed products cancel in this model. It does not by itself establish that the assets are unrelated in every sense.
Covariance depends on the scale used to record returns. If all returns are converted from percentages to decimals, each deviation is divided by \(100\), and the covariance is divided by \(10{,}000\). Do not compare numerical covariance values across different return units without converting them first.
Another frequent error is to calculate the expected return of each asset correctly but then combine returns from different rows when forming deviation products. Covariance requires the two deviations from the same scenario. A negative product must remain negative; taking absolute values would erase the direction of joint variation. Similarly, scenario probabilities weight the products, not just one asset’s deviation.
Worked Example: A Percentage-versus-Decimal Check
A two-scenario hypothetical model assigns probability \(0.25\) to a recovery scenario and \(0.75\) to a contraction scenario. The returns are \(8\%\) and \(0\%\) for Asset A, and \(-4\%\) and \(4\%\) for Asset B, respectively. Calculate covariance and identify the unit conversion trap.
Step 1: Expected returns in percentage points. For Asset A, \(E[R_A]=0.25(8)+0.75(0)=2\%\). For Asset B, \(E[R_B]=0.25(-4)+0.75(4)=-1+3=2\%\).
Step 2: Center returns by scenario. In recovery, the deviations are \(8-2=6\) and \(-4-2=-6\), so their product is \(-36\). Its weighted contribution is \(0.25(-36)=-9\). In contraction, the deviations are \(0-2=-2\) and \(4-2=2\), so their product is \(-4\). Its weighted contribution is \(0.75(-4)=-3\).
Step 3: Sum and convert. The covariance is \(-9-3=-12\) squared percentage points. In decimal-return squared, it is \(-12/10{,}000=-0.0012\).
Check. The two deviations have opposite signs in both scenarios, so both weighted products are negative. A common unit trap is to report \(-12\) as decimal-return squared; that number is correct only when returns were measured in percentage points. In decimal units the covariance is \(-0.0012\).
Check Your Understanding
For each problem, calculate covariance from the paired scenario returns. Unless stated otherwise, report the result in squared percentage points.
- A model has two equally likely scenarios. Asset A returns \(0\%\) and \(10\%\); Asset B returns \(5\%\) and \(-5\%\), respectively. Find the covariance. (Answer: \(-25\) squared percentage points, or \(-0.0025\) decimal-return squared.)
- Three scenarios have probabilities \(0.20\), \(0.30\), and \(0.50\). Asset A returns \(-5\%\), \(5\%\), and \(9\%\); Asset B returns \(10\%\), \(0\%\), and \(2\%\). Find the covariance. (Answer: \(-16\) squared percentage points.)
- A hypothetical portfolio study assigns probabilities \(0.25\), \(0.25\), and \(0.50\) to three states. Asset A returns \(4\%\), \(-4\%\), and \(8\%\); Asset B returns \(-2\%\), \(10\%\), and \(2\%\), respectively. Calculate both expected returns and covariance. (Answer: \(E[R_A]=4\%\), \(E[R_B]=3\%\), and covariance \(=-16\) squared percentage points.)
- Four scenarios have probabilities \(0.10\), \(0.20\), \(0.30\), and \(0.40\). Asset A returns \(12\%\), \(4\%\), \(0\%\), and \(-3\%\); Asset B returns \(-4\%\), \(8\%\), \(3\%\), and \(1\%\), respectively. Find the covariance and state it in decimal-return squared as well. (Answer: \(-1.60\) squared percentage points, or \(-0.00016\) decimal-return squared.)