Why Use Two Measures of Return Dispersion?
Two investments can have different expected returns and different amounts of variation around those returns. Standard deviation measures the absolute spread of returns, while the coefficient of variation expresses that spread relative to the expected return. These answer different questions, so their rankings can differ without either calculation being wrong.
For example, an investment with a higher expected return may also have a larger standard deviation than another investment. That tells us its returns vary more in absolute percentage-point terms. It does not, by itself, tell us whether the variability is large or small compared with the return expected. The coefficient of variation makes that relative comparison.
Both measures describe variability, not certainty about what return will occur. A standard deviation does not say that a particular range of returns must occur, and the coefficient of variation does not establish that one investment is preferable. Their usefulness depends on the comparison being made.
Absolute and Relative Comparisons
Use standard deviation when the question concerns the absolute size of return fluctuations. If one investment has a standard deviation of 6 percentage points and another has a standard deviation of 3 percentage points, the first has twice the absolute dispersion under the model being used.
Use coefficient of variation when the question concerns variability relative to expected return. A coefficient of variation of 0.40 means that the standard deviation is 0.40 times the expected return. It is a ratio, not a return rate or a probability.
The calculation must use consistent units. If the expected return and standard deviation are both expressed as percentages, dividing one by the other gives the same coefficient of variation as expressing both as decimals. For example, \(4\%/8\%=0.5\), just as \(0.04/0.08=0.5\). Do not divide a percentage by a decimal.
| Measure | Question answered | Units |
|---|---|---|
| Standard deviation, \(\sigma_R\) | How large is the absolute spread of returns? | Same units as the return |
| Coefficient of variation, \(CV_R\) | How large is the spread relative to expected return? | None; a ratio |
How the Comparison Works
Suppose two investments have positive expected returns. Investment A has expected return \(\mu_A\) and standard deviation \(\sigma_A\); investment B has expected return \(\mu_B\) and standard deviation \(\sigma_B\). The absolute comparison is based on \(\sigma_A\) versus \(\sigma_B\). The relative comparison is based on \(\sigma_A/\mu_A\) versus \(\sigma_B/\mu_B\).
These comparisons can point in different directions because each coefficient of variation uses its own expected return as the denominator. A larger standard deviation can be offset by a sufficiently larger expected return when measuring relative dispersion.
The relationship follows directly from division by the expected return. For positive \(\mu_A\) and \(\mu_B\), the relative comparison is equivalent to comparing cross-products:
Thus, an investment can have the larger absolute standard deviation but the smaller coefficient of variation. The denominator matters; do not infer the relative ranking from standard deviations alone.
Worked Comparisons
Worked Example: Equal Relative Dispersion, Different Absolute Risk
Two hypothetical funds each have two possible annual returns, with probability \(0.5\) for each outcome. Fund A returns \(4\%\) or \(12\%\); Fund B returns \(6\%\) or \(18\%\). Calculate each expected return, standard deviation, and coefficient of variation. Compare the funds under both measures.
Method. For a discrete return distribution, calculate the probability-weighted mean, then the square root of the probability-weighted squared deviations from that mean. Use percentage-point units throughout.
For Fund A, the expected return is
Its variance is
Therefore, \(\sigma_A=\sqrt{16}=4\) percentage points and \(CV_A=4\%/8\%=0.50\).
For Fund B, the expected return is
Its variance and standard deviation are
Thus, \(CV_B=6\%/12\%=0.50\). Comparison: Fund B has greater absolute dispersion, 6 rather than 4 percentage points. The relative dispersion is equal: both coefficients of variation are 0.50. The common mistake to avoid is treating the larger standard deviation as proof that Fund B also has greater relative dispersion.
Worked Example: Larger Standard Deviation but Smaller Coefficient of Variation
Two hypothetical investments have the following two-outcome annual return distributions, with probability \(0.5\) for each outcome. Investment C returns \(7\%\) or \(13\%\); Investment D returns \(3.5\%\) or \(8.5\%\). Find and compare their expected returns, standard deviations, and coefficients of variation.
For C, the mean is \(\mu_C=0.5(7\%)+0.5(13\%)=10\%\). Its two deviations from the mean are \(-3\) and \(3\) percentage points, so
The coefficient of variation is \(CV_C=3\%/10\%=0.30\).
For D, the mean is \(\mu_D=0.5(3.5\%)+0.5(8.5\%)=6\%\). The deviations are \(-2.5\) and \(2.5\) percentage points, giving
Therefore, \(CV_D=2.5\%/6\%=0.4167\), to four decimal places. Comparison: C has greater absolute dispersion, because \(3>2.5\), but smaller relative dispersion, because \(0.30<0.4167\). If the comparison is about percentage-point fluctuation, D varies less. If it is about dispersion per unit of expected return, C has the smaller ratio. Neither conclusion alone determines which investment is appropriate.
The trap is to select the investment with the lower standard deviation and report that it has the lower risk under every interpretation. That conclusion ignores the expected returns and the comparison objective.
Worked Example: Comparing Both Measures from Scenario Probabilities
A hypothetical investment X has annual returns of \(-5\%\), \(10\%\), and \(25\%\), with probabilities \(0.20\), \(0.50\), and \(0.30\). A hypothetical investment Y has returns of \(2\%\), \(8\%\), and \(14\%\), with probabilities \(0.25\), \(0.50\), and \(0.25\). Calculate and compare both risk measures. Treat the stated probabilities as the complete distributions for this one-period comparison.
For X, the probabilities sum to \(0.20+0.50+0.30=1\). Its expected return is
The deviations from the mean are \(-16.5\), \(-1.5\), and \(13.5\) percentage points. Therefore,
Hence \(\sigma_X=\sqrt{110.25}=10.5\) percentage points, and \(CV_X=10.5\%/11.5\%=0.9130\), to four decimal places.
For Y, the probabilities also sum to 1. Its expected return is
The deviations are \(-6\), \(0\), and \(6\) percentage points. Its variance and standard deviation are
Thus \(CV_Y=4.2426\%/8\%=0.5303\), to four decimal places. Comparison: X has both a higher expected return and greater absolute dispersion. It also has greater relative dispersion, since \(10.5>4.2426\) and \(0.9130>0.5303\). The square roots are rounded only at the final step to four decimal places. A calculation check is that each variance is nonnegative and each standard deviation is its nonnegative square root.
Why the Two Measures Matter
Standard deviation is stated in return units, so it is directly interpretable as an absolute spread around the expected return. The coefficient of variation removes those units by dividing the standard deviation by the expected return. This makes it useful for a relative comparison across investments whose positive expected returns differ.
The two measures are not substitutes for expected return. For example, a low coefficient of variation does not mean an investment has a high expected return; it only describes the ratio of its standard deviation to that return. Similarly, a low standard deviation does not indicate whether the expected return is sufficient for an investor's objective.
A useful derivation explains why the coefficient of variation does not change when all returns are multiplied by the same positive scale factor. If \(R^*=cR\), where \(c>0\), then \(E[R^*]=c\mu_R\) and \(\sigma_{R^*}=c\sigma_R\). Consequently,
Both absolute measures, however, are multiplied by \(c\). This is why the coefficient of variation describes relative dispersion rather than the original percentage-point spread.
Common Comparison Errors
- Ranking by standard deviation when relative dispersion is requested. Calculate both coefficients of variation using each investment's own expected return.
- Ranking by coefficient of variation when absolute spread is requested. A ratio does not replace the standard deviation's percentage-point interpretation.
- Mixing units. Use percentages for both numerator and denominator, or decimals for both. The ratio must be unchanged.
- Calling a smaller measure “better” without context. The measures summarize different aspects of return variability; the preferred investment depends on the purpose and constraints of the decision.
- Using the ratio without checking its domain. This comparison is defined here for positive expected returns; do not apply the formula mechanically when that condition is absent.
Check Your Understanding
For each problem, compare absolute dispersion using standard deviation and relative dispersion using coefficient of variation where the expected returns are positive.
- Fund M has annual returns of \(2\%\) and \(10\%\), each with probability \(0.5\). Fund N has returns of \(5\%\) and \(11\%\), each with probability \(0.5\). Find each mean, standard deviation, and coefficient of variation, then compare the rankings. (Answer: M: mean \(6\%\), standard deviation \(4\) percentage points, \(CV=0.6667\); N: mean \(8\%\), standard deviation \(3\) percentage points, \(CV=0.3750\). M has greater absolute and relative dispersion.)
- Investment P has expected return \(9\%\) and standard deviation \(3\) percentage points. Investment Q has expected return \(15\%\) and standard deviation \(6\) percentage points. Find both coefficients of variation and state which has lower absolute and relative dispersion. (Answer: \(CV_P=0.3333\), \(CV_Q=0.4000\); P has lower absolute and relative dispersion.)
- A hypothetical fund has returns of \(0\%\), \(8\%\), and \(16\%\) with probabilities \(0.25\), \(0.50\), and \(0.25\). Calculate its expected return, standard deviation, and coefficient of variation. (Answer: mean \(8\%\), variance \(32\) percentage-points squared, standard deviation \(5.6569\) percentage points, \(CV=0.7071\).)
- Investment R returns \(-2\%\), \(10\%\), or \(22\%\) with probabilities \(0.25\), \(0.50\), and \(0.25\). Investment S returns \(4\%\), \(8\%\), or \(12\%\) with probabilities \(0.25\), \(0.50\), and \(0.25\). Find each expected return and compare standard deviations and coefficients of variation. (Answer: R has mean \(10\%\), standard deviation \(8.4853\) percentage points, and \(CV=0.8485\); S has mean \(8\%\), standard deviation \(2.8284\) percentage points, and \(CV=0.3536\). R has greater absolute and relative dispersion.)
- Two hypothetical investments have positive expected returns. Investment A has standard deviation \(5\) percentage points and expected return \(10\%\); investment B has standard deviation \(6\) percentage points and expected return \(15\%\). Which has greater absolute dispersion, and which has greater relative dispersion? (Answer: B has greater absolute dispersion; A has greater relative dispersion because \(CV_A=0.5000\) and \(CV_B=0.4000\).)