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VEE Accounting and Finance · Return Measurement and Risk · Tutorial 516 of 996

The Computational Formula for Return Variance

Learn to calculate return variance efficiently from scenario probabilities while keeping return units and probability weights consistent.

Beginner 10 min read

What You'll Learn

  • Derive the identity that expresses variance using the first and second moments of return
  • Calculate expected squared returns from discrete scenarios and their probabilities
  • Subtract the squared expected return to obtain variance
  • Convert variance consistently between decimal-return and percentage-point units
  • Check a computational result against the definition based on squared deviations
  • Recognize errors caused by omitting probability weights or mixing units

Why Use a Computational Formula?

Return variance measures the probability-weighted spread of returns around their expected return. Its definition is direct: find the expected return, calculate each scenario’s squared deviation from that mean, and weight those squared deviations by their probabilities. For a discrete distribution, a mathematically equivalent formula can be quicker: calculate the expected squared return, then subtract the square of the expected return.

The computational formula is especially useful when scenario returns and probabilities are already listed. It avoids repeatedly subtracting the mean from each possible return. However, it is not a different measure of risk: it produces exactly the same variance as the definition, provided the same return units and probabilities are used throughout.

Definition: If return \(R\) has possible values \(r_1,\ldots,r_n\) with probabilities \(p_1,\ldots,p_n\), where \(p_j\geq 0\) and \(\sum_{j=1}^{n}p_j=1\), then its expected return is \(\mu_R=E(R)=\sum_{j=1}^{n}p_jr_j\). Its variance is \(\operatorname{Var}(R)=\sum_{j=1}^{n}p_j(r_j-\mu_R)^2\). The computational formula is \(\operatorname{Var}(R)=E(R^2)-[E(R)]^2=\sum_{j=1}^{n}p_jr_j^2-\left(\sum_{j=1}^{n}p_jr_j\right)^2\).

Here \(R^2\) means the square of the return, not the return on a second investment. \(E(R^2)\) is the probability-weighted average of the squared returns. It is generally not equal to \([E(R)]^2\); their difference is the variance.

Why the Identity Holds

Let \(\mu_R=E(R)\). Start with the definition of variance as the expected squared deviation from the mean. Expanding the square and using the linearity of expectation gives the computational identity:

$$ \begin{aligned} \operatorname{Var}(R) &=E\left[(R-\mu_R)^2\right]\\ &=E\left[R^2-2\mu_RR+\mu_R^2\right]\\ &=E(R^2)-2\mu_RE(R)+\mu_R^2\\ &=E(R^2)-2\mu_R^2+\mu_R^2\\ &=E(R^2)-[E(R)]^2. \end{aligned} $$

The fourth line uses \(E(R)=\mu_R\). For a finite discrete distribution, the expectation of any function \(g(R)\) is \(\sum_jp_jg(r_j)\). In particular, \(E(R^2)=\sum_jp_jr_j^2\), which gives the scenario-based version of the formula. The derivation requires a defined expected return and a finite second moment; a finite list of returns with probabilities summing to one satisfies these conditions.

Applying the Formula to Discrete Scenarios

A reliable calculation keeps the steps separate: check the probabilities, find \(E(R)\), find \(E(R^2)\), then subtract \([E(R)]^2\). Returns may be entered as decimals or percentage points, but every step must use the same scale. The resulting variance is in the square of the chosen return unit.

Worked Example: Three Possible Investment Outcomes

A hypothetical fund has a one-period return of \(-10\%\), \(5\%\), or \(20\%\), with respective probabilities \(0.20\), \(0.50\), and \(0.30\). Calculate the expected return and variance using the computational formula, with returns expressed as decimals.

Method. Use the scenario probabilities to calculate the first moment \(E(R)\) and the second moment \(E(R^2)\), then subtract the square of the first moment. First check that the probabilities form a distribution:

$$ 0.20+0.50+0.30=1.00. $$

Write the returns as \(-0.10\), \(0.05\), and \(0.20\). The expected return is:

$$ \begin{aligned} E(R) &=0.20(-0.10)+0.50(0.05)+0.30(0.20)\\ &=-0.020+0.025+0.060\\ &=0.065. \end{aligned} $$

Thus the expected return is \(6.5\%\). Next square each return before applying its probability:

$$ \begin{aligned} E(R^2) &=0.20(-0.10)^2+0.50(0.05)^2+0.30(0.20)^2\\ &=0.20(0.0100)+0.50(0.0025)+0.30(0.0400)\\ &=0.00200+0.00125+0.01200\\ &=0.01525. \end{aligned} $$

The squared expected return is \((0.065)^2=0.004225\). Therefore:

$$ \operatorname{Var}(R)=0.01525-0.004225=0.011025. $$

Check. Using the definition, the deviations from \(0.065\) are \(-0.165\), \(-0.015\), and \(0.135\). The weighted squared deviations are \(0.20(0.027225)=0.005445\), \(0.50(0.000225)=0.0001125\), and \(0.30(0.018225)=0.0054675\). Their sum is \(0.011025\), as required. The common mistake to avoid is squaring each deviation or return but forgetting to multiply it by its scenario probability.

Worked Example: Keeping Percentage Units Consistent

A hypothetical bond strategy has returns of \(-5\%\), \(10\%\), and \(25\%\), with probabilities \(0.25\), \(0.50\), and \(0.25\). Calculate the variance in percentage-points squared and convert it to decimal-return squared.

Method. Work in percentage points first, so the returns are \(-5\), \(10\), and \(25\). The probabilities sum to \(1\). The expected return is:

$$ E(R)=0.25(-5)+0.50(10)+0.25(25)=-1.25+5+6.25=10. $$

Thus the mean return is \(10\%\). For the second moment, square the percentage-point values and weight them:

$$ \begin{aligned} E(R^2) &=0.25(25)+0.50(100)+0.25(625)\\ &=6.25+50+156.25\\ &=212.5\text{ percentage-points}^2. \end{aligned} $$

The squared mean is \(10^2=100\) percentage-points squared. Hence the variance is \(212.5-100=112.5\) percentage-points squared. To convert to decimal-return squared, divide by \(100^2=10{,}000\):

$$ 112.5\div10{,}000=0.01125. $$

Check. Relative to the mean of \(10\%\), the deviations are \(-15\), \(0\), and \(15\) percentage points. Their weighted squared deviations are \(0.25(225)+0.50(0)+0.25(225)=112.5\) percentage-points squared. The common mistake is to combine percentage-point returns with a decimal mean, or to divide by \(100\) rather than \(10{,}000\) when converting a variance.

Worked Example: Finding the Missing Scenario Probability First

A hypothetical equity investment has possible returns of \(0\%\), \(4\%\), and \(8\%\). The probabilities of the first two outcomes are \(0.10\) and \(0.60\); the probability of the third outcome is not stated. Find the missing probability and calculate the variance using decimal returns.

Method. A discrete return distribution’s probabilities must sum to one. The missing probability is \(1-0.10-0.60=0.30\), so the three probabilities are \(0.10\), \(0.60\), and \(0.30\). Use returns \(0\), \(0.04\), and \(0.08\).

First calculate the expected return:

$$ \begin{aligned} E(R) &=0.10(0)+0.60(0.04)+0.30(0.08)\\ &=0+0.024+0.024\\ &=0.048. \end{aligned} $$

The expected return is \(4.8\%\). The second moment is:

$$ \begin{aligned} E(R^2) &=0.10(0^2)+0.60(0.04^2)+0.30(0.08^2)\\ &=0+0.60(0.0016)+0.30(0.0064)\\ &=0+0.00096+0.00192\\ &=0.00288. \end{aligned} $$

The squared mean is \((0.048)^2=0.002304\), so:

$$ \operatorname{Var}(R)=0.00288-0.002304=0.000576. $$

Check. The deviations from \(0.048\) are \(-0.048\), \(-0.008\), and \(0.032\). The weighted squared deviations are \(0.10(0.002304)=0.0002304\), \(0.60(0.000064)=0.0000384\), and \(0.30(0.001024)=0.0003072\). Their sum is \(0.000576\). The common mistake is to use the stated probabilities without first accounting for the missing outcome, which would leave the distribution incomplete.

Checks and Exam Traps

The computational formula shortens arithmetic, but its terms have distinct roles. \(E(R^2)\) is found by squaring each possible return and then weighting. \([E(R)]^2\) is found by weighting the returns first and then squaring their mean. Reversing those operations changes the answer.

Key takeaway: For a discrete return distribution, calculate \(\sum_jp_jr_j^2\) and subtract \(\left(\sum_jp_jr_j\right)^2\). Keep the return scale fixed, confirm that probabilities sum to one, and remember that variance has squared return units.

Variance cannot be negative because its definition is an expected value of squared deviations, and squares are nonnegative. This provides a useful arithmetic check: if the computational formula produces a negative value, investigate the probability weights, the squaring, and the subtraction. A small negative result caused by rounding should prompt a more precise calculation, not be interpreted as a negative measure of risk.

The probability-weighted formula applies when the listed scenarios represent a return distribution with known probabilities. Its weights are probabilities, not merely counts of past observations. When the task instead concerns estimating variance from historical observations, the method and interpretation differ; do not substitute a probability-distribution calculation without checking what the data represent.

1
Validate the scenario distribution.
Check that each probability is nonnegative and that the probabilities sum to one; determine any missing probability before calculating.
2
Calculate the first moment.
Multiply each return by its probability and add the products to obtain \(E(R)\).
3
Calculate the second moment.
Square each return, multiply each square by its probability, and add the products to obtain \(E(R^2)\).
4
Subtract and check.
Calculate \(E(R^2)-[E(R)]^2\). Confirm that the result is nonnegative and, if needed, verify it with the weighted squared deviations.

Check Your Understanding

For each discrete return distribution, calculate variance using the computational formula. Keep the units specified in the question and check that the probabilities form a valid distribution.

  1. A hypothetical asset returns \(-4\%\) with probability \(0.25\) and \(6\%\) with probability \(0.75\). Find the variance in decimal-return squared. (Answer: \(0.001875\))
  2. A hypothetical investment returns \(0\%\), \(5\%\), or \(15\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\). Find the variance in percentage-points squared. (Answer: \(31\) percentage-points squared)
  3. A hypothetical fund returns \(-10\%\), \(0\%\), or \(20\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\). Find the expected return and variance in percentage-points squared. (Answer: expected return \(4\%\); variance \(124\) percentage-points squared)
  4. A hypothetical strategy returns \(2\%\), \(8\%\), or \(12\%\), with probabilities \(0.30\), \(0.40\), and \(0.30\). Find the variance in decimal-return squared. (Answer: \(0.001524\))
  5. A hypothetical asset returns \(-8\%\), \(4\%\), or \(16\%\), with probabilities \(0.20\), an unknown probability, and \(0.30\), respectively. Determine the missing probability and the variance in percentage-points squared. (Answer: missing probability \(0.50\); variance \(70.56\) percentage-points squared)

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