What a Discrete Return Distribution Represents
An investment return is uncertain when its value at the end of a period is not known at the start. A discrete return distribution represents that uncertainty by listing the possible returns and assigning a probability to each one. It is a structured description of outcomes—not a statement that any one outcome is certain, and not a calculation of the average return.
The distribution is only meaningful if the outcomes are defined consistently. Returns should refer to the same investment, measurement period, and return convention. For a one-period simple holding-period return, include any cash income received during the period as well as the change in the investment’s value. If the initial value is \(V_0\), the ending value is \(V_1\), and cash income received is \(C\), the return is:
This is a percentage change in total value relative to the initial value. For example, a return of \(-0.10\) is a loss of 10% over the stated period. The formula assumes that income is measured over that same period and that the initial value is positive. It is not, by itself, an annualized return when the period is not one year. The earlier material on holding-period returns develops these measurement conventions; here, the focus is on representing the possible return outcomes and their probabilities.
A probability belongs to an outcome, not to the return value as a percentage. The probabilities express how likely the modeled outcomes are according to the stated assumptions or evidence. They must not be confused with the magnitudes of the returns: a large positive return does not automatically have a large probability.
Constructing a Distribution from Investment Scenarios
A scenario description often gives an economic condition, a probability for that condition, and financial values under that condition. To build the return distribution, first calculate the return in each scenario using one consistent return definition. Then check the probabilities and identify whether different scenarios give the same return. If they do, combine the probability attached to those scenarios when presenting a distribution of return values.
State the investment, the start and end of the period, and whether cash income is included.
Check that the scenarios cannot occur together and cover all outcomes in the model.
Use the same return formula and units in every scenario.
Confirm that probabilities are nonnegative and sum to one; combine probabilities for scenarios with identical returns.
Worked Example: Including Income in Each Scenario
A hypothetical fund share costs $80 at the beginning of a one-year period. An analyst specifies three mutually exclusive, exhaustive scenarios. In a weak market, the share ends at $70 and pays a $2 distribution; the scenario probability is 0.20. In a steady market, it ends at $84 and pays $2; the probability is 0.50. In a strong market, it ends at $92 and pays $2; the probability is 0.30. Construct the discrete return distribution.
Method. Calculate the one-year simple holding-period return in each scenario, including the cash distribution, and pair it with the stated scenario probability.
Weak market. The ending share value plus income is \(70+2=\$72\). The dollar change in total value is \(72-80=-\$8\), so the return is \(-8/80=-0.10\), or \(-10\%\).
Steady market. The ending share value plus income is \(84+2=\$86\). The dollar change is \(86-80=\$6\), so the return is \(6/80=0.075\), or \(7.5\%\).
Strong market. The ending share value plus income is \(92+2=\$94\). The dollar change is \(94-80=\$14\), so the return is \(14/80=0.175\), or \(17.5\%\).
The probabilities are all nonnegative, and their sum is \(0.20+0.50+0.30=1.00\). The resulting distribution is:
| One-year return | Probability |
|---|---|
| \(-10\%\) | 0.20 |
| \(7.5\%\) | 0.50 |
| \(17.5\%\) | 0.30 |
Check. Each return uses the same initial value and includes the distribution. The three scenario probabilities total 1.00, so the model accounts for all possible outcomes. A common mistake would be to omit the $2 cash distribution; that would produce different returns and would not measure the stated total holding-period return.
Why the Probability Conditions Matter
The requirements for a valid distribution follow from the meaning of probability. If the listed scenarios are mutually exclusive and collectively exhaustive, then exactly one of them occurs. The probability of the union of disjoint outcomes is the sum of their probabilities, and the probability of the complete set of possible outcomes is one. Therefore, the scenario probabilities must add to one. A negative probability is not a valid probability of an outcome.
This reasoning also explains how to handle repeated returns. Suppose two distinct scenarios \(A\) and \(B\) cannot occur together, but both produce the same return \(r\). The event \(R=r\) occurs if either scenario occurs. Its probability is therefore \(P(A)+P(B)\). The return distribution lists the distinct return values, so those scenario probabilities belong together in one row.
Worked Example: Validating Probabilities and Combining Equal Returns
A hypothetical investment has an initial value of $200 and is held for one year. A scenario model lists four mutually exclusive outcomes: contraction, probability 0.15, ending value $176 and income $4; stable conditions, probability 0.55, ending value $204 and income $6; a separate income event, probability 0.05, ending value $210 and no income; and expansion, probability 0.25, ending value $224 and income $8. Construct and check the return distribution.
Method. Calculate each total holding-period return, check the four scenario probabilities, and combine rows if distinct scenarios produce an identical return.
Contraction. Total ending value including income is \(176+4=\$180\). The return is \((180-200)/200=-20/200=-0.10\), or \(-10\%\).
Stable conditions. Total ending value is \(204+6=\$210\). The return is \((210-200)/200=10/200=0.05\), or \(5\%\).
Separate income event. Total ending value is \(210+0=\$210\). The return is also \((210-200)/200=0.05\), or \(5\%\).
Expansion. Total ending value is \(224+8=\$232\). The return is \((232-200)/200=32/200=0.16\), or \(16\%\).
The scenario probabilities sum to \(0.15+0.55+0.05+0.25=1.00\). The two distinct scenarios that return 5% have a combined probability of \(0.55+0.05=0.60\). Thus, the distribution of distinct return values is:
| One-year return | Probability |
|---|---|
| \(-10\%\) | 0.15 |
| \(5\%\) | 0.60 |
| \(16\%\) | 0.25 |
Check. The distribution probabilities sum to \(0.15+0.60+0.25=1.00\), matching the sum before combining rows. A common mistake is to leave two separate 5% rows in a table intended to show distinct return outcomes, or to discard one scenario because its return duplicates another. The scenarios are different, but their probabilities both contribute to the probability of a 5% return.
Scenario Probabilities and Observed Frequencies
A probability distribution can be built from forward-looking scenarios or from observed returns. These are different descriptions. In a scenario model, probabilities are assigned to possible future outcomes using stated assumptions, analysis, or judgment. They are not necessarily based on how many times each outcome occurred in the past.
For historical observations, the relative frequency of a return is its count divided by the total number of observations. The resulting table describes the data collected. It can be useful to organize historical results, but a past relative frequency is not automatically the probability of that outcome in a future period. The observations may be limited, and future conditions may differ from the historical period.
Worked Example: Building a Relative-Frequency Distribution
A hypothetical investment has the following eight observed annual returns, recorded to the nearest percentage point: \(-6\%,-6\%,-1\%,2\%,2\%,2\%,5\%,8\%\). Build a relative-frequency distribution and state what it represents.
Method. Count each distinct return, divide each count by the total of eight observations, and check that the frequencies sum to one.
The return \(-6\%\) occurs twice, so its relative frequency is \(2/8=0.25\). The return \(-1\%\) occurs once, giving \(1/8=0.125\). The return \(2\%\) occurs three times, giving \(3/8=0.375\). The return \(5\%\) occurs once, giving \(1/8=0.125\). The return \(8\%\) occurs once, also giving \(1/8=0.125\).
| Observed annual return | Count | Relative frequency |
|---|---|---|
| \(-6\%\) | 2 | 0.25 |
| \(-1\%\) | 1 | 0.125 |
| \(2\%\) | 3 | 0.375 |
| \(5\%\) | 1 | 0.125 |
| \(8\%\) | 1 | 0.125 |
Check. The counts total \(2+1+3+1+1=8\), and the relative frequencies total \(0.25+0.125+0.375+0.125+0.125=1.00\). This table summarizes these eight observed years. It does not establish that the next year has a 37.5% probability of a 2% return. Treating observed frequencies as guaranteed future probabilities is a common interpretation error.
Common Checks Before Using a Return Distribution
Before relying on a distribution, verify both its numerical structure and its interpretation. In particular, check that:
- Every return uses the same time period and return definition.
- Cash income is included or excluded consistently, as specified.
- Every probability is between zero and one, inclusive.
- The listed outcomes are mutually exclusive and collectively exhaustive under the stated model.
- The probabilities sum to one; if they do not, revisit the scenario list or the supplied probabilities rather than silently changing them.
- Repeated return values are combined when the table is meant to show distinct return outcomes.
- A historical relative frequency is identified as a summary of observed data, not automatically as a forecast probability.
These checks matter because later calculations based on a distribution inherit its assumptions. A table with omitted outcomes, inconsistent return definitions, or invalid probabilities cannot accurately represent the uncertainty it is intended to describe. The distribution is the input to further analysis; constructing and validating it is a separate task from calculating a summary of its returns.
Check Your Understanding
For each problem, construct or check the requested discrete distribution. Treat scenario probabilities as model inputs unless the problem explicitly describes observations.
- A hypothetical share costs $125 at the start of a one-year period. In a downside scenario, probability 0.30, it ends at $115 and pays $7 income. In an upside scenario, probability 0.70, it ends at $132 and pays $5 income. Find each return and check the distribution. (Answer: downside \( -2.4\% \) with probability 0.30; upside \(9.6\%\) with probability 0.70; probabilities sum to 1.00.)
- A model assigns probabilities 0.18, 0.47, and 0.30 to three scenarios. Are they a valid complete distribution as stated? What additional probability would be required if the listed scenarios are exhaustive but the supplied probabilities are incomplete? (Answer: No; they sum to 0.95, so a missing 0.05 probability mass must be accounted for.)
- An investment has an initial value of $500. Three mutually exclusive scenarios have probabilities 0.20, 0.50, and 0.30. Their ending values and cash incomes are, respectively, $470 and $10; $515 and $5; and $540 and $10. Construct the return distribution and check it. (Answer: \(-4\%\) with probability 0.20, \(4\%\) with probability 0.50, and \(10\%\) with probability 0.30; probabilities sum to 1.00.)
- A record contains six observed annual returns: \(-3\%,0\%,0\%,4\%,4\%,4\%\). Give the relative-frequency distribution and explain what its probabilities describe. (Answer: \(-3\%\): \(1/6\); \(0\%\): \(2/6\); \(4\%\): \(3/6\); these are the proportions among the six observations, not automatically future scenario probabilities.)
- Two mutually exclusive scenarios each produce a return of \(6\%\), with probabilities 0.22 and 0.13. A third scenario produces \(-2\%\) with probability 0.65. Write the distribution by distinct return and verify it. (Answer: \(-2\%\) with probability 0.65 and \(6\%\) with probability \(0.35\); the probabilities sum to 1.00.)