Expected Returns Versus Realized Returns

Expected and Realized Returns Answer Different Questions

An investment decision is made before the future outcome is known. At that point, an analyst can describe possible returns and assign probabilities to them. After the period ends, one outcome has occurred and its return can be calculated. These are different stages of return measurement: the first is ex ante (before the outcome), and the second is ex post (after the outcome).

The distinction matters because an expected return is a probability-weighted summary of possible outcomes, not a promise about what one investment period will produce. A realized return is the result for the particular outcome that occurred. It can be above, below, or equal to the expected return.

Definition: Let $R$ be the return over a specified investment period, with possible values $r_1,\ldots,r_n$ and probabilities $p_1,\ldots,p_n$. The expected return is the ex ante probability-weighted mean, $E[R]=\sum_{j=1}^{n}p_jr_j$, where $p_j\geq 0$ and $\sum_{j=1}^{n}p_j=1$. The realized return is the return corresponding to the outcome that actually occurs. It is one observed value of $R$.

In these calculations, all possible returns must refer to the same investment and the same period, and the scenario probabilities must describe mutually exclusive outcomes that cover all possibilities. Returns are expressed as rates: for example, a return of $0.08$ is $8.00\%$. When a starting investment amount is supplied, the corresponding dollar gain or loss is that amount multiplied by the return.

Why an Expected Return Need Not Occur

The expected return is a weighted mean, so its value need not be one of the possible outcomes. To see why the weighted-mean formula holds, define an indicator $I_j$ to equal 1 if scenario $j$ occurs and 0 otherwise. Exactly one scenario occurs, so the realized return can be written as $R=\sum_{j=1}^{n}r_jI_j$. Since the probability that scenario $j$ occurs is $p_j$, $E[I_j]=p_j$. Taking expectations gives the result:

E[R] =E\!\left[\sum_{j=1}^{n}r_jI_j\right] =\sum_{j=1}^{n}r_jE[I_j] =\sum_{j=1}^{n}p_jr_j.

This calculation describes the mean across the probability model; it does not say that a single observation must equal that mean. For instance, if the possible returns are $-20.00\%$, $7.00\%$, and $17.00\%$, an expected return of $7.30\%$ would not itself be a possible realized return. Even if the expectation is one of the listed outcomes, the other outcomes remain possible.

The earlier material on expected return as a long-run weighted mean gives another useful interpretation: over many repetitions of the same probability model, the average return tends toward the expected return under the usual conditions for the law of large numbers. That long-run statement does not guarantee that any one period, or a short sequence of periods, will equal the expectation.

Worked Example: A Loss in One Period Despite a Positive Expectation

A hypothetical one-year investment has a $0.1000$ probability of earning $-20.00\%$, a $0.6000$ probability of earning $7.00\%$, and a $0.3000$ probability of earning $17.00\%$. The scenarios are mutually exclusive and exhaustive. Calculate the ex ante expected return. If the investment earns $-20.00\%$, state the realized return and explain what that observation establishes.

Method. Multiply each scenario return by its probability and add the weighted returns. First, verify the probabilities:

$$ 0.1000+0.6000+0.3000=1.0000. $$

The scenario contributions are $0.1000(-20.00\%)=-2.00$ percentage points, $0.6000(7.00\%)=4.20$ percentage points, and $0.3000(17.00\%)=5.10$ percentage points. Therefore:

E[R]=-2.00\%+4.20\%+5.10\%=7.30\%.

If the loss scenario occurs, the realized return is $-20.00\%$. That one observation does not disprove the $7.30\%$ expected return: the expectation is calculated from all possible outcomes and their probabilities, whereas the realized return records just the outcome that occurred. Check: the weighted contributions sum to $7.30\%$, and the probability weights sum to 1.0000.

Common mistake avoided: Do not replace the expected return with the result of the realized scenario. Here, reporting $-20.00\%$ as the expectation because it occurred confuses an ex post observation with the ex ante weighted mean.

Work From the Scenario Model to the Observation

A reliable solution keeps the forecast calculation separate from the record of what happened. Start with the full scenario distribution to calculate the expectation. Then use the observed outcome—not the expectation—to report the realized return. If dollar values are given, check that the return calculation and the dollar gain or loss agree.

Set the measurement period.
Confirm that every scenario return covers the same period and uses the same return definition.

Check the scenario probabilities.
Confirm that each probability is nonnegative and that the probabilities sum to 1.

Calculate the ex ante expectation.
Multiply each possible return by its probability and add the products.

Identify the ex post realization.
Select the return for the outcome that actually occurred; do not substitute the expectation.

Interpret the comparison.
State whether the observation is above, below, or equal to the expectation, without treating one result as a guarantee about other periods.

Worked Example: Expected Return and Dollar Outcome

A hypothetical fund begins a one-year period with an investment of $\$25{,}000.00$. Its modeled total return is $-5.00\%$ with probability $0.2000$, $8.00\%$ with probability $0.5000$, and $14.00\%$ with probability $0.3000$. Assume the stated returns include all distributions for the year. Find the expected return and expected dollar gain. If the $8.00\%$ scenario occurs, calculate the realized dollar gain and ending value.

Method. First weight the returns by their probabilities. The weights are valid because $0.2000+0.5000+0.3000=1.0000$. The contributions to expected return are $0.2000(-5.00\%)=-1.00$ percentage point, $0.5000(8.00\%)=4.00$ percentage points, and $0.3000(14.00\%)=4.20$ percentage points. Thus:

E[R]=-1.00\%+4.00\%+4.20\%=7.20\%.

The expected dollar gain is the initial investment multiplied by the expected return:

\$25{,}000.00(0.072)=\$1{,}800.00.

If the $8.00\%$ scenario occurs, its return—not $7.20\%$—is the realized return. The realized dollar gain is $\$25{,}000.00(0.08)=\$2{,}000.00$, and the ending value is $\$25{,}000.00+\$2{,}000.00=\$27{,}000.00$. Check: the three scenario dollar gains are $-\$1{,}250.00$, $\$2{,}000.00$, and $\$3{,}500.00$; their probability-weighted mean is $-\$250.00+\$1{,}000.00+\$1{,}050.00=\$1{,}800.00$, consistent with the expected return.

Common mistake avoided: The expected dollar gain is not the gain in the scenario that occurs. The $7.20\%$ expectation corresponds to $\$1{,}800.00$ as a probability-weighted amount; the $8.00\%$ realized scenario produces $\$2{,}000.00$.

Worked Example: Including a Cash Distribution in the Realized Return

A hypothetical share costs $\$80.00$ at the start of a one-year period and pays a $\$4.00$ cash distribution at the end of that period. Its possible ending prices are $\$72.00$, $\$88.00$, and $\$100.00$, with respective probabilities $0.2500$, $0.5000$, and $0.2500$. Calculate the expected total return. If the ending price is $\$88.00$, calculate the realized total return.

Method. For each outcome, include both the price change and the cash distribution. The one-period total return is the ending price minus the starting price plus the distribution, all divided by the starting price. For an ending price of $\$72.00$, the return is $(\$72.00-\$80.00+\$4.00)/\$80.00=-\$4.00/\$80.00=-5.00\%$. For $\$88.00$, it is $(\$88.00-\$80.00+\$4.00)/\$80.00=\$12.00/\$80.00=15.00\%$. For $\$100.00$, it is $(\$100.00-\$80.00+\$4.00)/\$80.00=\$24.00/\$80.00=30.00\%$.

The probabilities sum to $0.2500+0.5000+0.2500=1.0000$. Weight the three returns to obtain the ex ante expectation:

E[R]=0.2500(-5.00\%)+0.5000(15.00\%)+0.2500(30.00\%) =-1.25\%+7.50\%+7.50\% =13.75\%.

If the ending price is $\$88.00$, the realized total return is $15.00\%$, not $13.75\%$. Check: the scenario dollar profits per share are $-\$4.00$, $\$12.00$, and $\$24.00$. Their weighted mean is $0.2500(-\$4.00)+0.5000(\$12.00)+0.2500(\$24.00)=\$11.00$; dividing by the initial $\$80.00$ gives $13.75\%$.

Common mistake avoided: Ignoring the cash distribution would understate each scenario return. The distribution is part of the total return in every scenario, even though the ending price changes across scenarios.

Interpretation and Common Traps

The expected return is a feature of a specified probability model. If the probabilities or possible outcomes change, the expected return can change. The realized return, by contrast, is computed from the observed investment result over the stated period. A comparison between the two is informative, but the difference by itself does not show that the expectation was calculated incorrectly.

An expected return is also not necessarily the most likely return. In the share example, $15.00\%$ is the most likely scenario return because its probability is $0.5000$, while the expected return is $13.75\%$. The expectation averages all three scenarios; it does not select the scenario with the largest probability.

Confusing the mean with a guarantee. An expectation does not promise that the next realized return will equal it.
Using only the observed outcome to calculate the expectation. The ex ante calculation uses every modeled outcome and its probability.
Reporting an expected return as the realized return. Once an outcome is known, report that outcome's return as realized.
Leaving out part of the return definition. If the stated measure is total return, include relevant cash distributions as well as price change.
Mixing periods or probability weights. Scenario returns must refer to a common period, and valid exhaustive scenario probabilities must sum to 1.

Key takeaway: Expected return is calculated before the outcome as a probability-weighted mean. Realized return is calculated after the outcome from what actually happened. A difference between them is entirely compatible with the probability model.

Check Your Understanding

For each problem, distinguish the probability-weighted expectation from the return observed in the stated outcome.

A hypothetical one-year asset has returns of $-8.00\%$, $6.00\%$, and $18.00\%$, with probabilities $0.2500$, $0.5000$, and $0.2500$. Calculate its expected return. If the $18.00\%$ scenario occurs, report the realized return. (Answer: Expected return $5.50\%$; realized return $18.00\%$.)
A hypothetical $\$12{,}000.00$ investment has possible one-year returns of $-2.00\%$, $5.00\%$, and $20.00\%$, with probabilities $0.4000$, $0.4000$, and $0.2000$. Calculate the expected return and expected dollar gain. If the $5.00\%$ outcome occurs, give the realized dollar gain. (Answer: Expected return $5.20\%$; expected dollar gain $\$624.00$; realized dollar gain $\$600.00$.)
A hypothetical share costs $\$50.00$, pays a $\$2.00$ distribution at year-end, and has possible ending prices $\$45.00$, $\$54.00$, and $\$62.00$, with probabilities $0.2000$, $0.5000$, and $0.3000$. Calculate the scenario total returns and their expected return. If the ending price is $\$54.00$, state the realized return. (Answer: Scenario returns $-6.00\%$, $12.00\%$, and $28.00\%$; expected return $13.20\%$; realized return $12.00\%$.)
A hypothetical project has a $0.1000$ probability of returning $-30.00\%$, a $0.7000$ probability of returning $4.00\%$, and a $0.2000$ probability of returning $16.00\%$. Calculate the expected return. If the project realizes $-30.00\%$, does that single result disprove the expected return? (Answer: Expected return $3.00\%$; no, the one realized outcome does not by itself disprove the probability-weighted expectation.)
A hypothetical investment has possible returns of $-10.00\%$, $5.00\%$, and $20.00\%$, with probabilities $0.2000$, $0.5000$, and $0.3000$. Calculate the expected return and compare it with the realized return if the middle scenario occurs. Is the expected return itself one of the scenario outcomes? (Answer: Expected return $6.50\%$; realized return $5.00\%$; no, $6.50\%$ is not one of the listed outcomes.)

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