V1 Module 6 Errata: Statistical Distributions for Financial Asset Prices and Returns

1. Large Samples Are Said to Make Distributions Continuous

Curriculum location: Random Variables and Unconditional Expectations, Random Variables and Their Properties, p. 253.

"When the number of observations is large, the distributions will converge to continuous distributions, as it does in the next example."

The number of observations and the variable's support are different ideas. A binomial count remains discrete even with a very large sample, while a measured return can be modeled continuously even in a small sample.

Correct reading: A random variable is discrete when its possible values are countable and continuous when its possible values fill a continuum. Increasing sample size does not change that classification.

A candidate who follows the printed rule could misclassify count data solely because the dataset is large.

2. The Discrete-Uniform Variance Formula Omits the Unit-Spacing Condition

Curriculum location: Key Statistical Distributions in Finance, Discrete Uniform Distribution, p. 266.

"there are $n-2$ evenly spaced outcomes between $a$ and $b$ with spacing $(b-a)/(n-1)$."

"${\sigma }^2=\frac{(b-a+1{)}^2-1}{12}$."

Curriculum location: Key Statistical Distributions in Finance, Exhibit 18, p. 279.

"$\frac{(b-a+1{)}^2-1}{12}$"

The prose permits any equal spacing, but the displayed variance is the special formula for consecutive integers.

Correct reading: For $n$ equally spaced points from $a$ to $b$, $\mathrm{Var}(X)=\frac{(b-a{)}^2(n+1)}{12(n-1)}$. The printed formula applies only when the spacing is one, so $n=b-a+1$.

A candidate could use the wrong variance when the support is equally spaced but not consecutive.

3. The Stock-Price and Return Distributions Are Reversed

Curriculum location: Key Statistical Distributions in Finance, Monte Carlo Simulation of Stock Prices, p. 272.

"For stock prices, this target distribution is often the normal distribution, due to the assumption of log-normal returns."

The standard log-normal price model uses the opposite relationship: the logarithm of the price relative is normal, so the price level is log-normal.

Correct reading: Under geometric Brownian motion, stock price levels are log-normal and continuously compounded returns are normal.

A candidate could select the normal distribution for a price level when the log-normal distribution is required.

4. The Brownian-Increment Step Omits $\sqrt{\Delta t}$

Curriculum location: Key Statistical Distributions in Finance, Monte Carlo Simulation of Stock Prices, p. 273.

"Use the result from Step 2 as $W_t$ in the GBM formula."

A standard-normal draw has unit variance, while a Brownian increment over $\Delta t$ has variance $\Delta t$.

Correct reading: Draw $Z\sim N(0,1)$ and set $\Delta W=\sqrt{\Delta t}Z$. For $\Delta t=1/250$, the printed price 9.818 follows only after scaling the draw $-1.48$ by $\sqrt{1/250}$.

A candidate who inserts $Z$ directly would overstate one-day volatility by a factor of $\sqrt{250}$.

5. Exhibit 14 Treats Price 11.09 as an 11.09% Return

Curriculum location: Key Statistical Distributions in Finance, Monte Carlo Simulation of Stock Prices, Exhibit 14 discussion, p. 273.

"The average simulated final price is 11.09, implying an 11.09% return"

The exhibit starts the simulated paths at 10. A price level and a return are different objects: the return is the change from the initial price divided by that initial price.

Correct reading: With an initial price of 10 and a final price of 11.09, the return is $(11.09-10)/10=0.109=10.9\%$.

A candidate could otherwise mistake the final price level for a percentage and overstate the simulated gain.

6. Log Returns Are Called Bounded

Curriculum location: Key Statistical Distributions in Finance, Log-Normal Distribution, p. 275.

"Logarithmic returns, on the other hand, use the natural logarithm of the price ratio, $\ln P_1/P_0$, and are bounded, making them additive and more appropriate for longer periods."

For strictly positive prices, $\ln (P_1/P_0)$ can be any real number. It is the price level that is bounded below by zero, not the log return.

Correct reading: Log returns are unbounded. Their multi-period advantage is additivity: log returns sum across periods.

A candidate could otherwise confuse the support of a price level with the support of its log return.

7. The Log-Normal Price Case Mixes Drift, Mean Log Return, and Median Price

Curriculum location: Key Statistical Distributions in Finance, Modeling Asset Prices case, p. 276.

"$\mu$, the mean of the logarithmic annual return $=8\%$"

"$\ln (S_1)=\ln (S_0)+\left(\mu -\frac{{\sigma }^2}{2}\right)\times t=\ln (100)+\left(0.08-\frac{0.04}{2}\right)\times 1$"

"$=4.6052+0.06=\mathrm{4.6652.}$"

"The future price is $S_1=e^{4.6652}=106.1868$."

The case subtracts ${\sigma }^2/2$, which is appropriate when $\mu$ denotes GBM arithmetic drift, not when it already denotes the mean log return. It also exponentiates the mean log price and calls the result the future price, although that exponential is the log-normal median.

Correct reading: If $\mu$ is mean log return, $E[\ln S_T]=\ln S_0+\mu T$. If $\mu$ is GBM arithmetic drift, $E[\ln S_T]=\ln S_0+(\mu -{\sigma }^2/2)T$, $\mathrm{Median}(S_T)=S_0{e}^{(\mu -{\sigma }^2/2)T}$, and $E[S_T]=S_0{e}^{\mu T}$.

A candidate could apply the volatility adjustment to the wrong parameter or call a median an expectation.

8. Exhibit 19 and Equation 23 Interchange Densities, CDFs, and Probabilities

Curriculum location: Key Statistical Distributions in Finance, Exhibit 19, p. 280.

"$p_X(x)=={\int }_a^b{f}_X(x)dx$"

Curriculum location: Conditional Expectations, Variances, and Covariances, Exhibit 21 and Equation 23, p. 283.

"Equation 23 is the joint density function."

The first expression equates a pointwise density with an interval probability. In the joint example, $f_{12}$ is the density and its double integral is $F_{12}$, the CDF.

Correct reading: For differentiable $F_X$, $f_X(x)=d{F}_X(x)/dx$, while $P(a\le X\le b)={\int }_a^b{f}_X(x)dx$. The mound in Exhibit 21 is a joint PDF, and Equation 23 defines the joint CDF.

A candidate could otherwise integrate, differentiate, or label the wrong probability function.

9. Exhibit 20 Prints an Invalid Uniform-CDF Spreadsheet Formula

Curriculum location: Key Statistical Distributions in Finance, Exhibit 20 and following note, pp. 280–281.

"$\text{BETA.DIST}(x-a)/(b-a)$"

The expression omits required distribution arguments and then divides the incomplete function result by the interval width.

Correct reading: For $a\le x\le b$, use $(x-a)/(b-a)$, with 0 below $a$ and 1 above $b$. If BETA.DIST is used, distinguish the valid Google Sheets syntax on p. 281 from Excel's required cumulative argument.

A candidate could enter a formula that errors or returns the wrong probability.

10. The Total-Expectation Formula Uses a Probability Where It Needs an Expected Value

Curriculum location: Conditional Expectations, Variances, and Covariances, Equation 30, p. 287.

"$E[X]={\sum }_{i=1}^{n}P(X\mid S_i)\times P(S_i)$."

The rule recombines scenario-conditioned means. $P(X\mid S_i)$ is an event-probability expression, not the conditional mean of a random variable.

Correct reading: $E[X]={\sum }_{i=1}^{n}E[X\mid S_i]P(S_i)$.

A candidate could substitute probabilities for the returns or other numerical outcomes being averaged.

11. The Depression Mean Has the Wrong Sign in the Variance Calculation

Curriculum location: Conditional Expectations, Variances, and Covariances, portfolio scenario case, p. 288.

"$(0.1575-0.0483{)}^2$"

The case previously calculated the depression mean as $-15.75\%$, so the deviation must begin with $-0.1575$.

Correct reading: Use $(-0.1575-0.0483{)}^2$. With the other two scenario means, the between-scenario variance is about $0.0336$, not $0.0234$.

A candidate could reproduce the wrong sign and understate the between-scenario variance.

12. The Unconditional Variance Omits Within-Scenario Risk

Curriculum location: Conditional Expectations, Variances, and Covariances, portfolio scenario case, p. 288.

"The unconditional variance is:"

$${\sigma }_1^2=\left(\frac{1}{3}\times (0.1575-0.0483{)}^2\right)+\left(\frac{1}{3}\times (0.015-0.0483{)}^2\right)+\left(\frac{1}{3}\times (0.2875-0.0483{)}^2\right)=\frac{0.1092^2+(-0.0333{)}^2+0.2392^2}{3}=\frac{0.0119+0.0011+0.0572}{3}=\frac{0.0702}{3}=\mathrm{0.0234.}$$

Dispersion of the conditional means is only one component of total variance. The case has already computed nonzero variances inside each scenario, and those must also be included.

Correct reading: $\mathrm{Var}(R)=E[\mathrm{Var}(R\mid S)]+\mathrm{Var}(E[R\mid S])$. Using the displayed inputs gives a total variance of about $0.0484$.

A candidate could understate portfolio risk by omitting all within-scenario dispersion.

13. The Diffuse-Prior Paragraph Equates Reverse Conditional Probabilities

Curriculum location: Bayesian Updating, diffuse-prior paragraph, p. 299.

"When the prior probabilities are equal, the probability of information given an event equals the probability of the event given the information."

Equal priors remove relative prior weighting, but the posterior still divides each likelihood by the sum of likelihoods across the exhaustive scenarios.

Correct reading: With equal priors, $P(S_i\mid B)=P(B\mid S_i)/{\sum }_{j}P(B\mid S_j)$. In general, $P(B\mid S_i)\ne P(S_i\mid B)$.

A candidate could reverse the conditioning direction in Bayes' theorem.

Complete Module 6 Errata Index

Scope: Module 6, pp. 249–307 (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: Probability distributions and related probability functions
• Penn State STAT 414: Discrete random variables and CDFs
• Penn State STAT 414: Continuous random variables, variance, and uniform distributions
• Penn State STAT 414: Mathematical expectation and variance
• NIST/SEMATECH: Binomial distribution
• NIST/SEMATECH: Lognormal distribution
• MIT OpenCourseWare: Brownian motion defining properties
• Microsoft Support: BETA.DIST function
• Google Docs Editors Help: BETA.DIST function
• Berkeley Statistics: Conditional variance and the law of total variance
• Penn State STAT 414: Bayes' theorem

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 6 Statistical Distributions for Financial Asset Prices and Returns. It is not an official CFA Institute errata notice, and inclusion here must not be read as CFA Institute confirmation, endorsement, or approval.