Calculating Expected Return From Scenarios

Introduction

When an investment can produce several possible returns, a simple average of those returns generally does not account for how likely each outcome is. A return that occurs with probability 0.80 should contribute more to the expectation than one that occurs with probability 0.10. The calculation that accounts for these unequal probabilities is the expected return.

This tutorial considers a finite set of scenarios for one investment over one specified period. It focuses on the calculation and interpretation of the probability-weighted return, including situations where a scenario return must first be calculated from an investment’s price change and cash distribution.

Definition: Suppose an investment has $n$ mutually exclusive and exhaustive scenarios. Let $r_j$ be the investment return in scenario $j$, and let $p_j$ be the probability of that scenario, for $j=1,\ldots,n$. The expected return is the probability-weighted sum $$ E[R]=\sum_{j=1}^{n}p_jr_j, $$ where $p_j\geq 0$ for every $j$, and $\sum_{j=1}^{n}p_j=1$. Returns must be expressed on a consistent basis, such as all as percentages or all as decimals.

Set Up the Scenario Calculation

A scenario is one possible outcome used in the return analysis. The scenario set must cover all outcomes being considered, and no two scenarios should overlap. These conditions matter because each probability must describe one distinct case, and the probabilities must account for the full set of possible cases.

Before multiplying anything, check the scenario table. Each probability should be between zero and one, inclusive, and the probabilities should sum to one. If a probability is missing but the scenarios are exhaustive, it can be found by subtracting the known probabilities from one. If the stated probabilities do not add to one, do not silently treat them as complete: identify whether a probability is missing or whether the inputs need clarification.

The return in each scenario must also refer to the same investment and time period. For example, a one-year return should not be averaged directly with a two-year return. If scenario returns are given in percentage form, keep them in percentage form throughout the weighted sum; if they are written as decimals, keep them as decimals. The final answer can then be reported in the requested form.

Check the scenarios and probabilities.
Confirm that the scenarios are distinct and cover the outcomes under consideration, then verify that the probabilities sum to one.

Establish each scenario return.
Use the stated return, or calculate it from the investment cash flows and initial value. Keep the time period and return convention consistent.

Weight and add.
Multiply each return by its scenario probability, then sum those products.

Why the Weighted-Sum Formula Holds

The formula follows from expressing the return as a variable that takes one value in each scenario. Let $I_j$ equal 1 if scenario $j$ occurs and 0 otherwise. Exactly one scenario occurs, so the return can be written as $R=\sum_{j=1}^{n}r_jI_j$. The indicator $I_j$ has expected value $p_j$: it equals 1 with probability $p_j$ and 0 otherwise.

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

The second equality uses the linearity of expectation: a constant can be taken outside an expectation, and the expectation of a finite sum is the sum of the expectations. This justification does not require the scenario returns to be equal or the probabilities to be equal. It does require that the scenario probabilities and returns describe the same complete scenario set.

Worked Example: Unequal Probabilities and Three Possible Returns

A hypothetical investment has a one-year return of 18% in a strong market, 6% in a steady market, and −10% in a weak market. The respective scenario probabilities are 0.25, 0.50, and 0.25. Calculate its expected one-year return. Use percentage returns in the calculation and round the final result to two decimal places.

Method. Apply the probability-weighted sum to the stated returns. First, check the probabilities: $0.25+0.50+0.25=1.00$. The three scenarios therefore have probabilities that sum to one.

Multiply each return by its probability:

\begin{aligned} E[R] &=(0.25)(18\%)+(0.50)(6\%)+(0.25)(-10\%)\\ &=4.50\%+3.00\%-2.50\%\\ &=5.00\%. \end{aligned}

The expected return is 5.00%. As a check, the three contributions sum to 5.00 percentage points. The negative-return scenario reduces the total by 2.50 percentage points; it is not omitted just because its return is below zero.

Common mistake avoided. Taking the unweighted average gives $(18\%+6\%-10\%)/3=4.67\%$ to two decimal places. That calculation incorrectly treats all three scenarios as equally likely. The stated probabilities are unequal, so each return must be multiplied by its own probability.

Calculate Returns Before Weighting When Needed

Sometimes a scenario gives an ending value and a cash distribution rather than a return. For a one-period holding-period return, use the same initial investment value in every scenario and include both the ending value and the cash received during the period. If the initial value is $P_0$, the ending value is $P_1$, and the cash distribution is $D$, the return is

R=\frac{P_1-P_0+D}{P_0}.

This formula calculates the investment’s gain relative to the amount initially invested. Once each scenario return is found, use its scenario probability to calculate the expected return. Do not weight the ending values in place of returns unless the calculation specifically calls for an expected ending value.

Worked Example: Price Changes and a Cash Distribution

A hypothetical share costs $100 at the start of a one-year period. It pays a $4 cash distribution during that period. At year-end, its price is $112 in a high-price scenario, $104 in a middle scenario, and $88 in a low-price scenario. The probabilities are 0.20, 0.50, and 0.30, respectively. Calculate the expected one-year holding-period return, rounding the final result to two decimal places.

Method. Calculate the holding-period return in each scenario by adding the price change and the $4 distribution, then dividing by the initial price of $100. The probabilities check: $0.20+0.50+0.30=1.00$.

In the high-price scenario, the gain is $\$112-\$100+\$4=\$16$, so the return is $\$16/\$100=16\%$. In the middle scenario, the gain is $\$104-\$100+\$4=\$8$, so the return is $\$8/\$100=8\%$. In the low-price scenario, the gain is $\$88-\$100+\$4=-\$8$, so the return is $-\$8/\$100=-8\%$.

Now weight the three returns:

\begin{aligned} E[R] &=(0.20)(16\%)+(0.50)(8\%)+(0.30)(-8\%)\\ &=3.20\%+4.00\%-2.40\%\\ &=4.80\%. \end{aligned}

The expected one-year holding-period return is 4.80%. The calculation includes the distribution in all three scenarios, and the weighted contributions sum to 4.80 percentage points.

Common mistake avoided. Using only the price change would give scenario returns of 12%, 4%, and −12%, omitting the $4 distribution. The expected return from those incomplete returns would be $2.40\%+2.00\%-3.60\%=0.80\%$, not 4.80%. The distribution is part of the one-period investment return.

Using Conditional Probabilities

Scenario information is sometimes presented in stages. For example, the probability of an economic state may be given first, followed by probabilities for a project outcome conditional on that state. In that case, multiply the probability of the state by the conditional probability of the outcome to obtain the joint probability of that combined scenario:

P(\text{state and outcome}) =P(\text{state})P(\text{outcome}\mid\text{state}).

Use each joint probability with the return for its combined scenario. This prevents a conditional probability from being mistaken for the probability of the outcome across all states.

Worked Example: Returns With Conditional Outcome Probabilities

A hypothetical investment’s return depends on both the economic state and whether a service contract is renewed. The economic states are growth, flat, and contraction, with probabilities 0.30, 0.50, and 0.20. Given growth, the probabilities of renewal and no renewal are 0.70 and 0.30; the corresponding returns are 18% and 4%. Given flat conditions, those conditional probabilities are 0.40 and 0.60, and the returns are 10% and 2%. Given contraction, the conditional probabilities are 0.25 and 0.75, and the returns are 6% and −14%. Calculate the expected return, rounding the final result to two decimal places.

Method. Find each combined scenario’s joint probability by multiplying the economic-state probability by the conditional probability. The six joint probabilities are:

\begin{aligned} P(\text{growth, renewal})&=(0.30)(0.70)=0.21,\\ P(\text{growth, no renewal})&=(0.30)(0.30)=0.09,\\ P(\text{flat, renewal})&=(0.50)(0.40)=0.20,\\ P(\text{flat, no renewal})&=(0.50)(0.60)=0.30,\\ P(\text{contraction, renewal})&=(0.20)(0.25)=0.05,\\ P(\text{contraction, no renewal})&=(0.20)(0.75)=0.15. \end{aligned}

Check that they sum to one: $0.21+0.09+0.20+0.30+0.05+0.15=1.00$. Now weight each return by its joint probability:

\begin{aligned} E[R] &=(0.21)(18\%)+(0.09)(4\%)+(0.20)(10\%)\\ &\quad +(0.30)(2\%)+(0.05)(6\%)+(0.15)(-14\%)\\ &=3.78\%+0.36\%+2.00\%+0.60\%+0.30\%-2.10\%\\ &=4.94\%. \end{aligned}

The expected return is 4.94%. The check on the joint probabilities confirms that all six combined scenarios have been included.

Common mistake avoided. Weighting each return by its conditional probability alone would ignore how likely its economic state is. For example, the probability of renewal given contraction is 0.25, but the joint probability of contraction and renewal is $(0.20)(0.25)=0.05$. The joint probability—not 0.25—is the weight for the 6% return.

Interpret the Result Carefully

An expected return is a weighted summary of the scenario returns under the probabilities used in the calculation. It need not equal the return in any one scenario. In the first worked example, 5.00% is not one of the three listed outcomes, but it is still the probability-weighted result.

The answer depends on both parts of the scenario information: the returns and the probabilities. If either changes, the weighted sum may change. A return with probability zero contributes nothing; a scenario with a larger probability has a larger weight, but its contribution can still be negative if its return is negative. Do not remove a loss scenario from the calculation simply because it lowers the result.

A final check is to confirm that the result uses the same return definition and period throughout, that every scenario return has been calculated correctly, and that the probabilities sum to one. These checks help catch common errors such as averaging scenarios equally when their probabilities differ, leaving out a cash distribution, or using conditional probabilities as though they were joint probabilities.

Key takeaway: For a complete set of discrete scenarios, calculate expected return by multiplying each scenario return by that scenario’s probability and adding the products. Unequal probabilities must be respected, and any return components needed to define the scenario return must be included before weighting.

Check Your Understanding

For each problem, check the probability total and keep the returns on a consistent basis.

An investment has returns of 12%, 3%, and −9% with probabilities 0.20, 0.50, and 0.30. Calculate its expected return. (Answer: 1.20%)
A hypothetical fund has returns of 20%, 5%, and −10% in three scenarios. Their probabilities are 0.15, 0.55, and 0.30. Calculate its expected return. (Answer: 2.75%)
A hypothetical asset costs $250 and pays a $5 distribution during the year. Its year-end prices are $275, $255, and $220 in three scenarios with probabilities 0.25, 0.45, and 0.30. First calculate each holding-period return, then find the expected return. (Answer: 1.80%)
A project faces high, middle, and low market conditions with probabilities 0.25, 0.50, and 0.25. Conditional on those respective conditions, the probabilities of contract completion are 0.80, 0.60, and 0.20. Returns if the contract is completed are 14%, 7%, and −3%; returns if it is not completed are −6%, −9%, and −18%. Find the expected return using joint scenario probabilities. (Answer: −0.95%)
A hypothetical investment costs $80. Its year-end price and cash distribution are, respectively, $105 and $3 in the first scenario, $86 and $3 in the second, and $70 and $3 in the third. The first two scenario probabilities are 0.20 and 0.30; the third is the remaining probability. Calculate the return in each scenario and the expected return. (Answer: 6.00%)

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