Probability-Weighted Variance of Returns

Why Probability-Weighted Variance Matters

An investment’s expected return summarizes its probability-weighted average outcome, but that average does not describe how widely possible returns are spread around it. Two hypothetical investments can have the same expected return while assigning different probabilities to unusually low or high outcomes. Variance measures the average squared distance of outcomes from the expected return, with more probable outcomes receiving more weight.

The earlier material on expected returns establishes how to find a probability-weighted mean. Here, the task is to use that mean as a center and calculate the weighted average of squared deviations. The calculation depends on the probabilities, not merely on how many scenarios are listed.

Definition: Suppose a one-period return $R$ can take the distinct values $r_1,r_2,\ldots,r_n$, with corresponding probabilities $p_1,p_2,\ldots,p_n$. The probabilities satisfy $p_j \geq 0$ and $\sum_{j=1}^{n}p_j=1$. The expected return is $\mu_R=E[R]=\sum_{j=1}^{n}p_jr_j$. The probability-weighted variance of return is $$ \operatorname{Var}(R)=\sum_{j=1}^{n}p_j(r_j-\mu_R)^2. $$ Each squared deviation is weighted by the probability of the return that produced it.

The outcomes must describe the same investment horizon and use the same return convention. For example, do not combine a one-month return in one scenario with an annual return in another. Scenario outcomes should also be mutually exclusive and collectively exhaustive, so exactly one listed outcome occurs and their probabilities sum to one.

How the Weighting Works

A deviation $r_j-\mu_R$ records how far a scenario’s return lies above or below the expected return. Squaring the deviation makes both positive and negative departures contribute positively. The probability $p_j$ then determines how much that scenario contributes to the overall variance. A return with a large squared deviation may still contribute modestly if it is unlikely; a smaller deviation may matter more if its probability is large.

The formula can be justified by thinking of many repetitions of the same investment opportunity under the stated scenario model. If scenario $j$ occurs about a fraction $p_j$ of the time, then its squared deviation contributes about that fraction of the average squared deviation. Adding the contributions from all scenarios gives the probability-weighted average.

Find the center.
Calculate $\mu_R=\sum p_jr_j$, using each scenario’s probability.

Measure each departure.
For every outcome, subtract the expected return: $r_j-\mu_R$.

Square the departure.
Calculate $(r_j-\mu_R)^2$, so below-average outcomes do not cancel above-average outcomes.

Apply its probability and add.
Multiply each squared deviation by its own $p_j$, then sum all contributions.

A useful check is that variance cannot be negative: every squared deviation is nonnegative, and every probability is nonnegative. Variance is zero only when every outcome with positive probability equals the expected return. Otherwise at least one positive-probability outcome has a nonzero squared deviation, making the weighted sum positive.

Worked Example: Weighting Unequal Scenario Probabilities

A hypothetical investment committee models a one-year holding-period return as follows: a return of $-10\%$ has probability $0.20$, a return of $8\%$ has probability $0.50$, and a return of $18\%$ has probability $0.30$. Find the expected return and the probability-weighted variance. Report the variance both in percentage-points squared and in decimal-return squared.

Method. First calculate the probability-weighted mean, then calculate and weight each squared deviation. This method avoids treating the three scenarios as equally likely merely because there are three of them.

Step 1: Check the probabilities and calculate the mean. The probabilities sum to $0.20+0.50+0.30=1.00$. Using percentage returns, $$ \mu_R=(0.20)(-10\%)+(0.50)(8\%)+(0.30)(18\%) =-2.0\%+4.0\%+5.4\%=7.4\%. $$

Step 2: Find the deviations and their squares. The deviations from $7.4\%$ are $-10\%-7.4\%=-17.4$ percentage points, $8\%-7.4\%=0.6$ percentage points, and $18\%-7.4\%=10.6$ percentage points. Their squared deviations are $302.76$, $0.36$, and $112.36$ percentage-points squared, respectively.

Step 3: Weight and sum. $$ \operatorname{Var}(R) =(0.20)(302.76)+(0.50)(0.36)+(0.30)(112.36) =60.552+0.180+33.708=94.44 $$ percentage-points squared.

As a check on the units, the returns expressed as decimals are $-0.10$, $0.08$, and $0.18$, and the expected return is $0.074$. Therefore the variance in decimal-return squared is $94.44/10{,}000=0.009444$. The weighted contributions are all nonnegative, and the probabilities sum to one, as required.

Units and Interpretation

Variance has squared return units. If returns are entered as percentages, such as $8$ for $8\%$, the variance is in percentage-points squared. If returns are entered as decimals, such as $0.08$, the variance is in decimal-return squared. These two numerical results differ by a factor of $10{,}000$, because each percentage-point value is one hundred times the corresponding decimal value, and the deviations are squared.

It is valid to use either scale, but all returns, the mean, and the deviations must use one scale throughout a calculation. The variance is a measure of dispersion around the expected return; it is not itself a return. In particular, do not report a variance such as $94.44$ percentage-points squared as though it were a return of $94.44\%$.

Worked Example: Keeping the Return Scale Consistent

A hypothetical fund’s one-year return is $2\%$ with probability $0.25$, $6\%$ with probability $0.45$, and $14\%$ with probability $0.30$. Calculate its variance in decimal-return squared. Show how a calculation in percentage points gives the same result after conversion.

Method. Calculate the expected return and weighted squared deviations using percentage points, then convert the final variance. This avoids mixing decimal returns with percentage-point deviations.

Step 1: Find the expected return. The probabilities sum to $0.25+0.45+0.30=1.00$, and $$ \mu_R=(0.25)(2\%)+(0.45)(6\%)+(0.30)(14\%) =0.50\%+2.70\%+4.20\%=7.40\%. $$

Step 2: Calculate squared deviations. Relative to $7.40\%$, the deviations are $-5.40$, $-1.40$, and $6.60$ percentage points. Their squares are $29.16$, $1.96$, and $43.56$ percentage-points squared.

Step 3: Weight each squared deviation. $$ \operatorname{Var}(R) =(0.25)(29.16)+(0.45)(1.96)+(0.30)(43.56) =7.290+0.882+13.068=21.240 $$ percentage-points squared.

Step 4: Convert the variance. Since the deviations in decimal form are one hundredth of their percentage-point values, their squares are one ten-thousandth as large: $$ 21.240 \div 10{,}000=0.002124 $$ decimal-return squared. The calculation is consistent: $0.002124$ is the same variance expressed on the decimal-return scale, not a second measure with a different meaning.

When Outcomes Repeat and Why Probabilities Matter

A scenario table may list distinct economic conditions that happen to produce the same return. In the variance calculation, such scenarios can be left as separate rows, each with its own probability, or combined into one return outcome by adding their probabilities. Either approach gives the same result because the squared deviation is identical for those rows.

For a finite distribution, the probabilities are the weights. Dividing by the number of scenarios is not part of the definition. That distinction matters whenever the scenario probabilities are unequal. Even when they are equal, the variance is still calculated as a probability-weighted average; the equal probabilities happen to give each scenario the same weight.

Worked Example: Combining Scenarios With the Same Return

A hypothetical renewable-energy investment has a one-year return of $-12\%$ with probability $0.15$, $4\%$ with probability $0.35$ in one operating condition, $4\%$ with probability $0.20$ in a different operating condition, and $16\%$ with probability $0.30$. Find the expected return and variance in percentage-points squared. Verify that the two $4\%$ scenarios can be combined.

Method. Calculate the weighted mean and squared deviations, retaining the two $4\%$ rows initially. Then combine their probabilities as a check. This avoids incorrectly treating repeated return values as separate equally likely outcomes.

Step 1: Check the probability total and find the mean. The probabilities sum to $0.15+0.35+0.20+0.30=1.00$. Thus $$ \mu_R=(0.15)(-12\%)+(0.35)(4\%)+(0.20)(4\%)+(0.30)(16\%) =-1.80\%+1.40\%+0.80\%+4.80\%=5.20\%. $$

Step 2: Calculate the squared deviations. The deviations from $5.20\%$ are $-17.20$, $-1.20$, $-1.20$, and $10.80$ percentage points. Their squares are $295.84$, $1.44$, $1.44$, and $116.64$ percentage-points squared.

Step 3: Weight each contribution. $$ \operatorname{Var}(R) =(0.15)(295.84)+(0.35)(1.44)+(0.20)(1.44)+(0.30)(116.64) $$ $$ =44.376+0.504+0.288+34.992=80.160 $$ percentage-points squared.

Step 4: Verify by combining identical returns. The probability of a $4\%$ return is $0.35+0.20=0.55$. Its contribution is therefore $(0.55)(1.44)=0.792$, equal to $0.504+0.288$. Keeping the $-12\%$ and $16\%$ contributions unchanged gives $44.376+0.792+34.992=80.160$, matching the original calculation.

Common Mistakes and a Reliable Check

A tempting but incorrect shortcut is to average squared deviations equally across the listed scenarios. In the first worked example, that calculation would be $$ \frac{302.76+0.36+112.36}{3}=138.4933\ldots $$ percentage-points squared. It is wrong because the scenario probabilities are $0.20$, $0.50$, and $0.30$, not three equal weights. The correct calculation is $94.44$ percentage-points squared, found by multiplying each squared deviation by its stated probability.

Do not omit the probability weights. The variance uses $p_j(r_j-\mu_R)^2$ for each outcome, not just the squared deviations.
Do not use the wrong center. Calculate the probability-weighted expected return first. An unweighted average of the returns will generally give the wrong deviations.
Do not let positive and negative deviations cancel. Square each deviation before weighting and summing.
Do not mix scales. If returns are entered as percentages, keep the expected return and deviations in percentage points too; convert the resulting variance only after the calculation.
Check the probability total. If probabilities do not sum to one, the scenario model is incomplete or the weights need correction before using the variance formula.

A practical final check is to review the three ingredients separately: the probabilities sum to one, the expected return is the weighted mean of the listed outcomes, and each variance contribution is a probability times a squared deviation from that mean. The result must be nonnegative and expressed in squared return units.

Key takeaway: For a discrete return distribution, first find the probability-weighted expected return. Then square each outcome’s deviation from that mean, multiply by the outcome’s probability, and add the weighted contributions. Use a consistent return scale from start to finish.

Check Your Understanding

For each hypothetical one-year return distribution, calculate the expected return and variance. Unless stated otherwise, give variance in percentage-points squared.

A small infrastructure note returns $3\%$ with probability $0.40$ and $9\%$ with probability $0.60$. Find the expected return and variance. (Answer: expected return $6.6\%$; variance $8.64$ percentage-points squared.)
A hypothetical investment has returns of $-5\%$, $5\%$, and $15\%$ with probabilities $0.20$, $0.50$, and $0.30$, respectively. Find its expected return and variance. (Answer: expected return $6.0\%$; variance $49.00$ percentage-points squared.)
A project’s return is $0\%$ with probability $0.25$, $8\%$ with probability $0.50$, and $12\%$ with probability $0.25$. Find its expected return and variance in decimal-return squared. (Answer: expected return $7.0\%$; variance $0.001900$ decimal-return squared.)
A hypothetical commodity-linked asset returns $-8\%$ with probability $0.30$, $2\%$ with probability $0.20$, and $10\%$ with probability $0.50$. Find its expected return and variance. (Answer: expected return $3.0\%$; variance $61.00$ percentage-points squared.)
A speculative investment returns $-20\%$, $0\%$, $10\%$, or $25\%$ with respective probabilities $0.10$, $0.30$, $0.40$, and $0.20$. Find its expected return and variance. (Answer: expected return $7.0\%$; variance $156.00$ percentage-points squared.)

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