Worked Return and Risk Analysis Across Scenarios

From Investment Outcomes to Return and Risk

An investment scenario may describe an ending price, cash income, and the probability of that outcome. To compare scenarios, first express each outcome as a return over the same period. Then use the scenario probabilities to summarize the distribution of returns. This sequence matters: calculating risk from prices or cash amounts without first putting outcomes on a comparable return basis can produce misleading comparisons.

The analysis in this tutorial treats scenarios as a discrete model for a single investment over one holding period. The scenarios are mutually exclusive, their probabilities sum to one, and each return includes the investment’s price change and cash income during the period. The probabilities are model inputs; the results are conditional on those inputs and are not guarantees about future outcomes.

Definition: The holding-period return is the total investment gain over a specified period, divided by the investment’s value at the start of that period. If the beginning value is $P_0$, the ending value is $P_1$, and cash income received during the period is $C$, then

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

This simple formula assumes the cash income is included once and is not reinvested to earn an additional return within the period. If there are several cash payments, $C$ is their total when the formula is being used for a single-period total return. For investments with multiple periods, returns must be measured consistently over the period being analyzed; a one-period return should not be mistaken for an annualized return.

Definition: For a discrete return distribution with possible returns $r_1,\ldots,r_n$ and corresponding probabilities $p_1,\ldots,p_n$, where $p_j\geq 0$ and $\sum_{j=1}^{n}p_j=1$, the expected return is $\mu_R=\sum_{j=1}^{n}p_jr_j$. The probability-weighted variance is $\sigma_R^2=\sum_{j=1}^{n}p_j(r_j-\mu_R)^2$, and the standard deviation is $\sigma_R=\sqrt{\sigma_R^2}$.

Returns are entered as decimals in these formulas: for example, a return of 8% is $0.08$. Variance is therefore measured in squared decimal-return units. Standard deviation has the same units as return and is commonly reported as a percentage. The coefficient of variation, when the expected return is positive, is

\operatorname{CV}=\frac{\sigma_R}{\mu_R}.

It expresses standard deviation relative to expected return. Unlike standard deviation, the coefficient of variation is not generally meaningful if expected return is zero or negative, and it can become unstable when expected return is positive but close to zero.

Why the Probability-Weighted Variance Formula Works

Variance is the expected squared distance from the mean. For each possible return $r_j$, the squared deviation from the mean is $(r_j-\mu_R)^2$. Weighting each such deviation by its scenario probability and adding gives the average squared deviation under the stated distribution. Thus a scenario with probability $p_j$ contributes $p_j(r_j-\mu_R)^2$, not simply its squared deviation.

The computational formula is a convenient equivalent. Expanding the squared deviation and using the fact that the probabilities sum to one gives:

\begin{aligned} \sigma_R^2 &=\sum_{j=1}^{n}p_j(r_j-\mu_R)^2\\ &=\sum_{j=1}^{n}p_j(r_j^2-2r_j\mu_R+\mu_R^2)\\ &=\sum_{j=1}^{n}p_jr_j^2 -2\mu_R\sum_{j=1}^{n}p_jr_j +\mu_R^2\sum_{j=1}^{n}p_j\\ &=\sum_{j=1}^{n}p_jr_j^2-2\mu_R^2+\mu_R^2\\ &=\sum_{j=1}^{n}p_jr_j^2-\mu_R^2. \end{aligned}

The third line uses $\sum p_jr_j=\mu_R$ and $\sum p_j=1$. This proves that variance can be calculated either from probability-weighted squared deviations or from the expected squared return minus the squared expected return. The scenario probabilities describe a specified distribution, so this is a probability-weighted variance, not a sample estimate from historical observations.

Worked Scenario Analysis

A reliable order of operations is to calculate each scenario return, check the probabilities, and only then calculate the summary measures. The examples use that sequence and retain unrounded values in later calculations.

Worked Example: Calculating Return and Risk for One Investment

A hypothetical investment costs $100. At the end of one year, it pays $2 of cash income in every scenario. Its ending value is $88 in a recession scenario, $106 in a steady scenario, and $118 in an expansion scenario. The scenario probabilities are 0.25, 0.50, and 0.25, respectively. Calculate the return in each scenario, expected return, variance, standard deviation, and coefficient of variation.

Method: calculate each holding-period return first. The beginning value is $P_0=100$, and the cash income is $C=2$. The returns are:

\begin{aligned} r_{\text{recession}}&=\frac{88-100+2}{100}=-0.10,\\ r_{\text{steady}}&=\frac{106-100+2}{100}=0.08,\\ r_{\text{expansion}}&=\frac{118-100+2}{100}=0.20. \end{aligned}

Thus the scenario returns are $-10\%$, $8\%$, and $20\%$. The probabilities are valid because $0.25+0.50+0.25=1.00$.

Expected return. Weight each return by its probability:

\mu_R=(0.25)(-0.10)+(0.50)(0.08)+(0.25)(0.20) =-0.025+0.040+0.050=0.065.

The expected return is $6.50\%$.

Variance and standard deviation. The deviations from $0.065$ are $-0.165$, $0.015$, and $0.135$. Their squared deviations are $0.027225$, $0.000225$, and $0.018225$. Therefore,

\begin{aligned} \sigma_R^2 &=(0.25)(0.027225)+(0.50)(0.000225)+(0.25)(0.018225)\\ &=0.00680625+0.00011250+0.00455625\\ &=0.011475. \end{aligned}

Taking the square root gives $\sigma_R=\sqrt{0.011475}\approx0.1071214$, or $10.71\%$. As a check, the computational formula gives $(0.25)(0.10^2)+(0.50)(0.08^2)+(0.25)(0.20^2)-(0.065)^2=0.015700-0.004225=0.011475$, the same variance.

Coefficient of variation. Since the expected return is positive, calculate the ratio using the unrounded standard deviation:

\operatorname{CV}=\frac{\sqrt{0.011475}}{0.065} \approx\frac{0.1071214}{0.065} \approx1.6480.

Check and common mistake avoided. The returns are total returns, so the $2 cash payment is included once in every scenario. Omitting it would produce returns of $-12\%$, $6\%$, and $18\%$, which do not match the specified cash flows. The independent computational-formula check confirms the reported variance.

Worked Example: Comparing Two Investments with the Same Expected Return

Two hypothetical investments have different scenario distributions. Investment A costs $200 and pays income of $6, $10, or $12 in its recession, stable, or strong-growth scenario. Its ending values are $184, $210, and $238, with probabilities 0.40, 0.40, and 0.20. Investment B costs $100 and pays $2 in every scenario. Its ending values are $100, $105, and $110, with probabilities 0.20, 0.60, and 0.20. Compare their expected returns, standard deviations, and coefficients of variation.

Method: convert each price-and-income outcome to a return. For A, the returns are $(184-200+6)/200=-0.05$, $(210-200+10)/200=0.10$, and $(238-200+12)/200=0.25$. For B, they are $(100-100+2)/100=0.02$, $(105-100+2)/100=0.07$, and $(110-100+2)/100=0.12$. Each investment’s probabilities sum to one.

Expected returns. For A,

\mu_A=(0.40)(-0.05)+(0.40)(0.10)+(0.20)(0.25) =-0.02+0.04+0.05=0.07.

For B, $\mu_B=(0.20)(0.02)+(0.60)(0.07)+(0.20)(0.12)=0.004+0.042+0.024=0.07$. Both expected returns are $7.00\%$.

Risk for A. Its deviations from $0.07$ are $-0.12$, $0.03$, and $0.18$. Thus

\begin{aligned} \sigma_A^2 &=(0.40)(0.12^2)+(0.40)(0.03^2)+(0.20)(0.18^2)\\ &=0.00576+0.00036+0.00648=0.01260,\\ \sigma_A&=\sqrt{0.01260}\approx0.1122497,\\ \operatorname{CV}_A&=\frac{\sqrt{0.01260}}{0.07} \approx\frac{0.1122497}{0.07}\approx1.6036. \end{aligned}

Risk for B. Its deviations from $0.07$ are $-0.05$, $0$, and $0.05$. Hence

\begin{aligned} \sigma_B^2 &=(0.20)(0.05^2)+(0.60)(0^2)+(0.20)(0.05^2)\\ &=0.00050+0+0.00050=0.00100,\\ \sigma_B&=\sqrt{0.00100}\approx0.0316228,\\ \operatorname{CV}_B&=\frac{\sqrt{0.00100}}{0.07} \approx\frac{0.0316228}{0.07}\approx0.4518. \end{aligned}

Investment B has lower standard deviation and lower coefficient of variation under these scenario assumptions, while the expected returns are equal. The standard deviations are about $11.22\%$ for A and $3.16\%$ for B. The CV calculations use the unrounded square roots; dividing rounded displayed percentages can create small discrepancies.

Check and common mistake avoided. Do not compare the raw ending values or cash payments: the investments have different beginning values. Expressing each scenario as a return makes the comparison relative to capital invested. This comparison is conditional on the stated probabilities and does not establish that B will have less risk under other assumptions.

Worked Example: Interpreting a Negative Expected Return

A hypothetical asset has three one-year scenario returns: $-12\%$, $-4\%$, and $8\%$, with probabilities 0.25, 0.50, and 0.25. Calculate the expected return, variance, standard deviation, and numerical ratio of standard deviation to expected return. Explain whether that ratio is a useful coefficient of variation.

Method: calculate the probability-weighted mean and deviations. The probabilities sum to one. The expected return is

\mu_R=(0.25)(-0.12)+(0.50)(-0.04)+(0.25)(0.08) =-0.03-0.02+0.02=-0.03.

The expected return is $-3.00\%$. The deviations from $-0.03$ are $-0.09$, $-0.01$, and $0.11$. Therefore,

\begin{aligned} \sigma_R^2 &=(0.25)(0.09^2)+(0.50)(0.01^2)+(0.25)(0.11^2)\\ &=0.002025+0.000050+0.003025=0.005100,\\ \sigma_R&=\sqrt{0.005100}\approx0.0714143=7.14\%. \end{aligned}

The numerical ratio is $\sqrt{0.005100}/(-0.03)\approx0.0714143/(-0.03)\approx-2.3805$. It is not an economically useful coefficient of variation: a negative ratio does not communicate “risk per unit of expected return” in the usual way. The standard deviation remains interpretable as dispersion around the expected return.

Check and common mistake avoided. The variance is positive because it averages squared deviations, even though the mean is negative. Reporting the negative ratio as if a lower CV were automatically preferable would reverse the intended interpretation. The mean’s sign must be checked before interpreting a coefficient of variation.

Interpreting the Measures Carefully

Expected return is the probability-weighted average return under the specified scenarios. It is not necessarily the return that will occur, nor does a single realization need to be close to it. Standard deviation measures the distribution’s probability-weighted spread around that mean. A larger standard deviation indicates more dispersion under the model, but it does not say whether that dispersion comes from unusually poor outcomes, unusually favorable outcomes, or both.

The coefficient of variation rescales standard deviation by expected return. When expected returns are positive and meaningfully separated from zero, it can help compare relative dispersion across investments with different expected returns. It does not replace standard deviation: two measures answer different questions. Standard deviation reports absolute variability in return units; CV reports variability relative to the expected return. When the mean is zero, the ratio is undefined. When it is negative, the ratio is generally not a useful relative-risk ranking. When it is close to zero, even a moderate standard deviation can produce a very large and unstable ratio.

Scenario results also depend on the completeness and calibration of the scenario model. If an adverse outcome is omitted or assigned too small a probability, expected return and risk measures can understate the implications of that outcome. The standard deviation treats positive and negative deviations symmetrically because both are squared. It is therefore not, by itself, a measure of downside loss. Nor does it describe the likelihood or size of every tail outcome in enough detail to replace examination of the scenario table.

Define the period and cash flows.
Use a consistent beginning value, ending value, and income amount for every scenario.

Calculate each scenario return.
Include price change and cash income once, and verify that the return is over the intended holding period.

Check the probabilities.
Confirm that scenario probabilities are nonnegative and sum to one before calculating summary measures.

Calculate and interpret the summaries.
Find the weighted mean and variance, take the square root for standard deviation, and use CV only when its mean-based interpretation is appropriate.

A frequent exam trap is to apply the variance formula to percentage figures such as $-10$, $8$, and $20$ while treating them as decimal returns. If returns are entered in percentage points, the resulting variance is scaled by $100^2$ relative to decimal-return variance. Choose one convention, use it throughout, and report standard deviation in the corresponding units. Another trap is rounding the standard deviation before calculating CV; retain the square root at full precision for the division and round only the final reported ratio.

Key takeaway: Scenario-based return and risk analysis requires a consistent holding-period return for each outcome, probabilities that form a valid distribution, and calculations that keep full precision until the final rounding. Interpret standard deviation as model-based dispersion and treat CV cautiously whenever expected return is not clearly positive.

Check Your Understanding

For each problem, calculate with returns as decimals and round reported percentages to two decimal places where needed.

A hypothetical investment begins at $160, ends at $170, and pays $6 of cash income over the holding period. Find its holding-period return. (Answer: $10.00\%$)
An asset has returns of $-4\%$, $6\%$, and $11\%$ with probabilities 0.30, 0.50, and 0.20. Find its expected return. (Answer: $4.00\%$)
A return distribution has outcomes $0\%$, $5\%$, and $10\%$, with probabilities 0.25, 0.50, and 0.25. Find the variance and standard deviation. (Answer: Variance $=0.00125$; standard deviation $\approx3.54\%$)
A hypothetical asset costs $250 and pays $5 in every scenario. Its ending values are $230, $260, and $290, with probabilities 0.20, 0.50, and 0.30. Find its scenario returns and expected return. (Answer: Returns $-6\%$, $6\%$, and $18\%$; expected return $=7.20\%$)
A return distribution has outcomes $-8\%$, $0\%$, and $8\%$, with probabilities 0.25, 0.50, and 0.25. Find its expected return and standard deviation, and state whether its coefficient of variation is defined. (Answer: Expected return $=0\%$; standard deviation $\approx5.66\%$; CV is undefined)

What You'll Learn