V1 Module 5 Errata: Statistical Characteristics of Asset Returns

1. The Mean–Median Ordering Is Interpreted in the Wrong Direction

Curriculum location: Measures of Central Tendency and Dispersion, German rate-spread case, p. 195.

"That the median is below the mean suggests that there is a small number of large negative changes in the spread between the interest rates."

The ordering itself is reported correctly: the mean is $-0.12$ bp and the median is $-0.20$ bp, so the median is below the mean. The error is the inference. A few large negative observations would pull the mean downward, often below the median. Under the usual Level I heuristic, a mean above the median points toward positive or right-tail influence, although the ordering alone does not prove skewness.

Correct reading: The mean–median comparison does not support the stated negative-outlier conclusion; under the usual heuristic it points in the opposite direction. Use the full distribution, not the ordering alone, to diagnose tail behavior.

Following the printed inference can reverse the standard mean-versus-median skewness heuristic.

2. Tail Percentages Mix Total and Per-Tail Conventions

Curriculum location: Measures of Central Tendency and Dispersion, trimmed-mean definition and Outliers and the Price of Oil case, pp. 196, 199–200.

"A 1% trimmed mean would exclude the lowest 0.5% and the highest 0.5% of values, and the mean is computed using the remaining 99% of the data."

"Trimming 1% removes two of the extreme observations, −79.82% and 33.51%, respectively, and the trimmed average changes to 0.83%. Trimming 10% removes 10 observations, and the trimmed average becomes 0.99%."

"The outliers are replaced by the respective 1% and 10% percentile values."

Printed p. 196 defines the percentage as the total split across both tails. For 100 observations, removing or replacing one observation in each tail affects two observations, so it is 2% total—not 1%. The conflicting label is repeated in the case prose and exhibit labels on pp. 199–200. The winsorization paragraph also names the 5th and 95th percentiles for its 10% series, then calls them “10% percentile values.”

Correct reading: Apply the printed-page 196 total-percentage convention consistently. In 100 observations, one observation per tail is 2% total; 10% total is five observations per tail. Update the prose, exhibit labels, and comparisons together. For winsorization, refer to the lower and upper cutoff percentiles, and use “winsorizing,” not “trimming,” for Exhibit 8.

The printed wording can lead candidates to remove or replace the wrong number of observations and to enter the wrong percentage in spreadsheet functions.

3. Sample Standard Deviation Uses Population-Variance Notation

Curriculum location: Measures of Central Tendency and Dispersion, variance discussion, p. 208.

"Note that, perhaps counterintuitively, the sample standard deviation, $s$, calculated as the square root of the variance, $\sqrt{{\sigma }^2}$, remains biased despite using Bessel’s correction!"

The display crosses the sample statistic $s$ with the population variance ${\sigma }^2$.

Correct reading: Write $s=\sqrt{s^2}$. The statement that the usual sample standard deviation remains biased may remain.

The symbol mismatch can make candidates confuse sample and population notation.

4. The Gold Case Uses Inconsistent Indices for Ten Returns

Curriculum location: Measures of Central Tendency and Dispersion, gold case, pp. 210-211.

"$\text{Average }\mu ={\sum }_{i=1}^{10}\frac{r_1}{n}=-2.85\%$"

"$r={\left({\prod }_{t=0}^{10}(1+r_t)\right)}^{1/10}-1$"

The arithmetic-mean formula repeats $r_1$ rather than using the running index $r_i$. The geometric product’s inclusive limits contain eleven indices even though the table and exponent use ten returns.

Correct reading: Use $r_i$ in both arithmetic-mean sums, and index the ten geometric-return factors consistently as $t=1,\dots ,10$ or $t=0,\dots ,9$.

Following the printed indices could repeat the first return or multiply eleven holding-period relatives instead of ten.

5. The Gold Case Reverses the Frequency Comparison

Curriculum location: Measures of Central Tendency and Dispersion, root-of-time discussion, p. 212.

"We show that the observed variance and standard deviation in lower-frequency data can be higher than expected from scaling longer-horizon variance or standard deviation; this is a typical occurrence."

The following case instead shows observed daily, higher-frequency volatility exceeding a daily estimate scaled down from annual, longer-horizon data.

Correct reading: Observed higher-frequency or shorter-horizon dispersion can exceed the value implied by scaling from lower-frequency or longer-horizon data.

The printed direction can invert a candidate’s interpretation of the case.

6. Positive Skew Is Said to Make High Returns More Probable

Curriculum location: Measures of Shape of a Distribution, skewness discussion, p. 217.

"A positively skewed return distribution of an investment suggests that there is a higher probability of achieving high returns compared to low returns."

Positive skew describes the relative contribution of the right tail to the third standardized moment. It does not determine how often an undefined “high return” occurs relative to an undefined “low return.” A distribution can be positively skewed because a few very large gains pull the right tail outward even when most observations are below the mean.

Correct reading: Positive skew indicates greater right-tail contribution or a longer/heavier right tail; it does not by itself mean that high returns occur more often than low returns.

Candidates should distinguish tail magnitude from event frequency.

7. Exhibit 25 Omits the Normalization in Skewness and Kurtosis

Curriculum location: Measures of Shape of a Distribution, Exhibit 25, p. 222.

"$\text{Skew}={\sum }_{i=1}^n{\left(\frac{X_i-\mu }{\sigma }\right)}^3$"

"$\text{Kurt}={\sum }_{i=1}^n{\left(\frac{X_i-\mu }{\sigma }\right)}^4$"

Both moment formulas are raw sums rather than averages, and their population bounds use $n$ rather than $N$.

Correct reading: Use $N^{-1}{\sum }_{i=1}^N((X_i-\mu )/\sigma {)}^3$ for population skewness and $N^{-1}{\sum }_{i=1}^N((X_i-\mu )/\sigma {)}^4$ for population raw kurtosis, with the corresponding $1/n$ sample approximations.

Omitting the averaging factor multiplies the reported moment by the population or sample size.

8. Question Set 2 Gives Three Incompatible Frequency Rules for Negative Skew

Curriculum location: Measures of Shape of a Distribution, Question Set 2, p. 223.

"Hence, daily returns of the front-month Brent crude oil futures returns are not symmetric and are characterized by infrequent, large daily negative returns and frequent, small daily positive returns."

"The distribution is skewed to the left, suggesting that there may be more negative daily returns than small, positive returns."

"The negative skew, −0.9393, specifically indicates that there are either frequent, small negative returns; infrequent, large negative returns; or both, rather than infrequent, large positive returns."

The keyed tail-direction explanation is sufficient, but the additional frequency claims disagree with one another and infer counts from a statistic that does not contain them.

Correct reading: Keep option B because the negative skewness value indicates a longer left tail. Remove the option A inference that negative skew means more negative returns. In the option C discussion, do not present frequent small negative returns with a few extreme positive returns as a negative-skew signature; that is a common positive-skew pattern.

The keyed answer survives, but the explanation teaches mutually incompatible rules.

9. Question Set 3 Switches from Growth to Levels

Curriculum location: Covariance and Correlation between Variables, Question Set 3, p. 233.

The solution says:

"there is a positive association between current earnings yield and expected dividend yield."

The stem and option B concern growth or changes, but the explanation switches to current and expected levels.

Correct reading: Edit only the explanation so it remains on growth in earnings yield and growth in dividend yield, as named in the stem. Option B is already correct and should remain unchanged.

Correlations of levels and growth are not interchangeable, even when the keyed direction is unchanged.

10. Equation 15 Mixes Population Notation with a Sample Convention

Curriculum location: Other Dispersion Measures, Equation 15 and following discussion, pp. 235-236.

"${\sigma }_{\text{Semi}}^2={\sum }_{i:x_i<\bar{X}}^n\frac{(X_i-\bar{X}{)}^2}{n}$"

"Because these are samples drawn from the population of all returns, both divide through by $n-1$, even though semi-deviation uses fewer observations."

The equation and prose disagree on the denominator. The equation also uses $\bar{X}$, a sample center, with the population symbol ${\sigma }^2$.

Correct reading: Under the sample convention stated in the text, write $s_{\text{Semi}}^2={\sum }_{i:X_i<\bar{X}}(X_i-\bar{X}{)}^2/(n-1)$.

A candidate otherwise has to choose between two printed denominators and two incompatible notation systems.

11. Question Set 4 Calls Semi-Deviation a Variance

Curriculum location: Other Dispersion Measures, Question Set 4, p. 238.

"The correct response is B. Semi-deviation is the variance for observations below the mean."

Semi-variance is the squared measure. Semi-deviation is its positive square root.

Correct reading: Semi-deviation is the square root of semi-variance and measures downside dispersion below the mean or a target.

The distinction matters because variance is in squared units while deviation is in the original return units.

Complete Module 5 Body-Text Errata Index

Scope: Module 5, pp. 187–240 (printed page numbers).

What this review covers. Errors in the printed curriculum that would change a candidate's answer or understanding: wrong numbers, wrong formulas, reversed logic, and statements that contradict the module's own data. It does not list spelling mistakes, equation-numbering slips, or wording that is loose but defensible.

References

• NIST/SEMATECH: Measures of location
• NIST/SEMATECH: Skewness and kurtosis
• NIST/SEMATECH: Standard deviation
• Microsoft Support: TRIMMEAN function
• Penn State STAT 500: Correlation

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 5 Statistical Characteristics of Asset Returns. It is not an official CFA Institute errata notice, and inclusion here must not be read as CFA Institute confirmation, endorsement, or approval.