Tutorials › VEE Accounting and Finance › Measuring Deviations From Expected Return

Home  /  Tutorials  /  VEE Accounting and Finance / Return Measurement and Risk

VEE Accounting and Finance · Return Measurement and Risk · Tutorial 513 of 996

Measuring Deviations From Expected Return

Calculate and interpret how far each possible return lies above or below an investment’s expected return.

Beginner 10 min read

What You'll Learn

  • Calculate an outcome’s deviation from expected return using a consistent return scale.
  • Interpret positive, negative, and zero deviations in context.
  • Distinguish a return deviation from both a return and its scenario probability.
  • Verify that probability-weighted deviations sum to zero.
  • Explain why signed deviations are useful inputs to measuring dispersion but do not by themselves summarize risk.

What a Deviation Measures

An expected return gives a probability-weighted center for a set of possible returns. It does not tell us how far each possible outcome lies from that center. A deviation from expected return answers that more specific question: for a particular outcome, it is the outcome’s return minus the expected return.

A deviation is not another scenario return and is not the probability of that scenario. It is a signed difference, expressed in the same units as the returns. A positive deviation means the outcome is above the expected return; a negative deviation means it is below. A deviation of zero means the outcome equals the expected return.

Definition: Suppose a one-period investment has a discrete set of possible returns \(r_1,\ldots,r_n\), with mutually exclusive and exhaustive scenarios having probabilities \(p_1,\ldots,p_n\). The probabilities satisfy \(p_j \geq 0\) and \(\sum_{j=1}^{n}p_j=1\). The expected return is \(E[R]=\sum_{j=1}^{n}p_jr_j\). The deviation associated with scenario \(j\) is \(r_j-E[R]\).

The expected return and deviations must use a consistent return scale. One can work in decimals, such as \(0.08\) for an 8% return, or in percentage points, such as \(8\%\). The subtraction gives the same economic difference when the scale is applied consistently. For instance, \(0.11-0.08=0.03\), which is a deviation of 3 percentage points. It is not a 3% probability.

Calculate Deviations Scenario by Scenario

The method is short, but its order matters. First find the expected return from the scenario probabilities and returns. Then subtract that expected return from each scenario return. Do not subtract a scenario probability from a return: those quantities have different meanings.

1
Specify the scenarios.
List each possible return and its probability. Confirm that the scenarios are mutually exclusive, exhaustive, and that their probabilities sum to 1.
2
Find the expected return.
Multiply each return by its scenario probability and add the weighted returns.
3
Subtract the expected return.
For each scenario, calculate return minus expected return, preserving the sign.
4
Interpret and check.
Read a positive deviation as above expectation and a negative deviation as below expectation. The probability-weighted deviations should sum to zero, apart from any rounding.

Worked Example: Deviations for a Three-Scenario Investment

A hypothetical investment has a one-year return of \(-12\%\) with probability \(0.25\), \(6\%\) with probability \(0.50\), or \(18\%\) with probability \(0.25\). The three scenarios are mutually exclusive and exhaustive. Find the expected return and each scenario’s deviation.

Method. Calculate the probability-weighted expected return first, then subtract it from each possible return. Work in percentage points throughout.

Expected return.

$$ E[R]=(0.25)(-12\%)+(0.50)(6\%)+(0.25)(18\%) =-3\%+3\%+4.5\%=4.5\%. $$

The probabilities check: \(0.25+0.50+0.25=1.00\).

Scenario deviations. For the \(-12\%\) outcome, the deviation is \(-12\%-4.5\%=-16.5\) percentage points. For the \(6\%\) outcome, it is \(6\%-4.5\%=1.5\) percentage points. For the \(18\%\) outcome, it is \(18\%-4.5\%=13.5\) percentage points.

The negative outcome lies 16.5 percentage points below the expected return; the other two outcomes lie above it. These deviations describe positions relative to the expected return, not the chance of those outcomes. As a check, the probability-weighted deviations are \((0.25)(-16.5\%)+(0.50)(1.5\%)+(0.25)(13.5\%)=-4.125\%+0.75\%+3.375\%=0\).

Why the Weighted Deviations Balance to Zero

The expected return is the probability-weighted mean of the scenario returns. This definition implies that the probability-weighted deviations from that mean cancel exactly. The result is useful both as a check and as a way to understand the role of signed deviations.

$$ \begin{aligned} \sum_{j=1}^{n}p_j\bigl(r_j-E[R]\bigr) &=\sum_{j=1}^{n}p_jr_j-E[R]\sum_{j=1}^{n}p_j\\ &=E[R]-E[R](1)\\ &=0. \end{aligned} $$

The first equality distributes the probability across the difference. The second uses both defining facts: \(\sum_{j=1}^{n}p_jr_j=E[R]\) and \(\sum_{j=1}^{n}p_j=1\). Thus outcomes above the expected return balance outcomes below it when deviations are weighted by their probabilities. This does not mean that every deviation is zero, or that the outcomes are equally likely. It means only that the signed weighted deviations have a zero total.

Worked Example: A Deviation Is Not a Scenario Probability

A hypothetical fund has a one-year return of \(2\%\) with probability \(0.20\), \(8\%\) with probability \(0.50\), and \(14\%\) with probability \(0.30\). Find the expected return and the deviation for each outcome. Then identify a common unit error.

Method. Use the probabilities only to calculate the weighted mean. Use returns, not probabilities, in the subsequent subtraction.

The expected return is

$$ E[R]=(0.20)(2\%)+(0.50)(8\%)+(0.30)(14\%) =0.4\%+4.0\%+4.2\%=8.6\%. $$

The probabilities sum to \(0.20+0.50+0.30=1.00\). The deviations are \(2\%-8.6\%=-6.6\) percentage points, \(8\%-8.6\%=-0.6\) percentage points, and \(14\%-8.6\%=5.4\) percentage points.

Common mistake and correction. For the 14% outcome, subtracting its probability from its return would give \(14\%-0.30\), which mixes a return with a probability and has no valid interpretation. The correct calculation is \(14\%-8.6\%=5.4\) percentage points: subtract the expected return, not the probability. The weighted-deviation check is \((0.20)(-6.6\%)+(0.50)(-0.6\%)+(0.30)(5.4\%)=-1.32\%-0.30\%+1.62\%=0\).

Deviations Are Inputs to Measuring Dispersion

Looking at signed deviations shows which outcomes are above or below expectation, and how far away they are. It also reveals why deviations can be an input to a measure of dispersion: their sizes contain information about how spread out the scenario returns are. But simply adding signed deviations is not a useful summary of that spread. By the result above, their probability-weighted sum is always zero, even when some outcomes are far from the expected return.

For example, consider two hypothetical investments. Each has outcomes with probabilities \(0.25\), \(0.50\), and \(0.25\). Investment A has returns \(4\%\), \(8\%\), and \(12\%\); Investment B has returns \(-8\%\), \(8\%\), and \(24\%\). Both have expected return \(8\%\):

$$ E[R_A]=(0.25)(4\%)+(0.50)(8\%)+(0.25)(12\%)=8\%, \qquad E[R_B]=(0.25)(-8\%)+(0.50)(8\%)+(0.25)(24\%)=8\%. $$

Investment A’s deviations are \(-4\), \(0\), and \(4\) percentage points. Investment B’s are \(-16\), \(0\), and \(16\) percentage points. Each set has probability-weighted deviations of zero, but the scenario returns for B lie farther from the shared expected return in the noncentral outcomes. This comparison illustrates why the signed total alone cannot describe dispersion. A separate method is needed to summarize the distances without allowing positive and negative signs to cancel; the deviations are the starting quantities for that analysis.

A deviation is also different from a realized return. Before the outcome is known, one can calculate a deviation for every scenario in the assumed distribution. Once one scenario occurs, its return can be compared with the expected return to identify the realized deviation. The expectation remains the reference point; the realized deviation describes how the particular outcome differed from it. Neither quantity changes the probability that the scenario was assigned in the model.

Common Interpretation Checks

  • Keep the subtraction order fixed. Use scenario return minus expected return. Reversing the order reverses the sign and changes the meaning.
  • Keep the units consistent. A return written as \(0.10\) and an expected return written as \(8\%\) must first be converted to a common scale.
  • Do not call a deviation a return outcome. A \(+3\)-percentage-point deviation means an outcome is 3 percentage points above expectation; it does not mean the investment returned 3%.
  • Do not interpret the sign as likelihood. Positive means above the expected return, not probable; negative means below, not improbable.
  • Use the weighted-zero identity as a check, not as a substitute for calculation. The identity does not reveal the individual deviations or their sizes.
Key takeaway: For each scenario, deviation from expected return equals scenario return minus the probability-weighted expected return. Its sign tells whether the outcome is above or below expectation, while its magnitude measures the distance in return units. Probability-weighted signed deviations sum to zero, so dispersion requires more than adding them.

Check Your Understanding

For each discrete one-period return model, calculate what is requested and retain the signs of the deviations.

  1. A hypothetical asset returns \(-5\%\), \(7\%\), or \(13\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\), respectively. Find its expected return and all three deviations. (Answer: Expected return \(6.4\%\); deviations \(-11.4\), \(+0.6\), and \(+6.6\) percentage points.)
  2. A project has returns \(4\%\), \(10\%\), and \(16\%\), with probabilities \(0.25\), \(0.50\), and \(0.25\). Calculate the expected return and deviations, then state which outcome has zero deviation. (Answer: Expected return \(10\%\); deviations \(-6\), \(0\), and \(+6\) percentage points; the 10% outcome has zero deviation.)
  3. A hypothetical portfolio has returns \(3\%\), \(9\%\), and \(21\%\), with probabilities \(0.40\), \(0.40\), and \(0.20\). Find the expected return and the deviation for the 21% outcome. (Answer: Expected return \(9\%\); the 21% outcome’s deviation is \(+12\) percentage points.)
  4. An investment has returns \(-10\%\), \(5\%\), and \(20\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\). Calculate each deviation and verify that the probability-weighted deviations sum to zero. (Answer: Expected return \(6.5\%\); deviations \(-16.5\), \(-1.5\), and \(+13.5\) percentage points; weighted sum \(-3.30\%-0.75\%+4.05\%=0\).)
  5. A hypothetical fund has returns \(2\%\), \(8\%\), and \(17\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\). If the 17% outcome occurs, calculate its deviation from expected return and interpret the sign. (Answer: Expected return \(9.5\%\); deviation \(+7.5\) percentage points, so the realized return is above expectation.)

Previous

Next

Back to Tutorials
Clinical Biostats