Quizara
Open navigation
Log In

Curriculum errata

CFA Level I 2027 Errata: Statistical Distributions for Financial Asset Prices and Returns

Quantitative Methods · Module 6

A candidate-focused review of the module's most consequential body-text errors, with corrected readings and a complete errata index for the topics covered.

15 explained errors20-item complete index15 min read

This is independent candidate-focused analysis. It is not an official curriculum errata notice.

Back to all articles
On this page
  1. 1. Sample Size Does Not Turn a Discrete Variable into a Continuous One
  2. 2. CDFs, PDFs, and Interval Probabilities Are Not Interchangeable
  3. 3. The Discrete-Uniform Variance Formula Needs a Unit-Spacing Condition
  4. 4. The Stock-Price and Return Distributions Are Reversed
  5. 5. Brownian Increments Must Be Scaled by the Square Root of the Time Step
  6. 6. A Final Price of 11.09 Does Not Imply an 11.09% Return
  7. 7. Log Returns Are Unbounded
  8. 8. The Log-Normal Price Case Mixes Drift, Mean Log Return, and Median Price
  9. 9. The Uniform-CDF Spreadsheet Formula Is Invalid
  10. 10. Conditional Expectation Is Misidentified as a Conditional PDF
  11. 11. Conditional Expectations Repeatedly Omit the Conditioning Bar
  12. 12. The Total-Expectation Formula Uses a Probability Where It Needs an Expected Value
  13. 13. The Depression Mean Has the Wrong Sign in the Variance Calculation
  14. 14. The Unconditional Variance Omits Within-Scenario Risk
  15. 15. Equal Priors Do Not Reverse Conditional Probabilities
  16. Complete Module 6 Body-Text Errata Index
  17. References

1. Sample Size Does Not Turn a Discrete Variable into a Continuous One

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

"When the number of observations is large, the distributions will converge to continuous distributions."

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. CDFs, PDFs, and Interval Probabilities Are Not Interchangeable

Curriculum location: Random Variables and Unconditional Expectations, Exhibit 3, printed p. 255.

"The CDF ... illustrates the ... cumulative proportion of returns, summing to a total area of 1."

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

pX(x)==∫abfX(x) dxp_X(x)==\int_a^b f_X(x)\,dx

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

"Equation 23 is the joint density function."

These statements exchange density, accumulated probability, and interval probability. A PDF is a density; its integral over an interval is a probability. A CDF is the accumulated probability up to a point. In the joint example, f12f_{12} is the density and its double integral is F12F_{12}, the CDF.

Correct reading: FX(x)=P(X≤x)F_X(x)=P(X\le x), fX(x)=dFX(x)/dxf_X(x)=dF_X(x)/dx when differentiable, and P(a≤X≤b)=∫abfX(x) dxP(a\le X\le b)=\int_a^b f_X(x)\,dx. The mound in Exhibit 21 is a joint PDF, while Equation 23 defines the joint CDF.

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

3. The Discrete-Uniform Variance Formula Needs a Unit-Spacing Condition

Curriculum location: Key Statistical Distributions in Finance, Discrete Uniform Distribution and Exhibit 18, printed pp. 266 and 279.

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

σ2=(b−a+1)2−112\sigma^2=\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 nn equally spaced points from aa to bb, Var⁡(X)=(b−a)2(n+1)12(n−1)\operatorname{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+1n=b-a+1.

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

4. The Stock-Price and Return Distributions Are Reversed

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

5. Brownian Increments Must Be Scaled by the Square Root of the Time Step

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

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

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

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

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

6. A Final Price of 11.09 Does Not Imply an 11.09% Return

Curriculum location: Key Statistical Distributions in Finance, Monte Carlo Simulation of Stock Prices, Exhibit 14 discussion, printed 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%(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.

7. Log Returns Are Unbounded

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

"Logarithmic returns ... are bounded"

For strictly positive prices, ln⁡(P1/P0)\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.

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

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

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

ln⁡(S1)=ln⁡(S0)+(μ−σ2/2)t\ln(S_1)=\ln(S_0)+(\mu-\sigma^2/2)t, followed by S1=e4.6652=106.1868S_1=e^{4.6652}=106.1868

The case then subtracts σ2/2\sigma^2/2, which is appropriate when μ\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 μ\mu is mean log return, E[ln⁡ST]=ln⁡S0+μTE[\ln S_T]=\ln S_0+\mu T. If μ\mu is GBM arithmetic drift, E[ln⁡ST]=ln⁡S0+(μ−σ2/2)TE[\ln S_T]=\ln S_0+(\mu-\sigma^2/2)T, Median⁡(ST)=S0e(μ−σ2/2)T\operatorname{Median}(S_T)=S_0e^{(\mu-\sigma^2/2)T}, and E[ST]=S0eμTE[S_T]=S_0e^{\mu T}.

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

9. The Uniform-CDF Spreadsheet Formula Is Invalid

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

BETA.DIST(x-a)/(b-a)\texttt{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≤x≤ba\le x\le b, use (x−a)/(b−a)(x-a)/(b-a), with 0 below aa and 1 above bb. If BETA.DIST\texttt{BETA.DIST} is used, supply the complete current syntax for the relevant spreadsheet application.

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

10. Conditional Expectation Is Misidentified as a Conditional PDF

Curriculum location: Conditional Expectations, Variances, and Covariances, conditional-expectation section, printed p. 285.

"The calculation of the conditional PDF of XX given Y=yY=y"

The equations immediately below compute conditional expected values, not a density function.

Correct reading: The lead-in should say "the calculation of the conditional expectation of XX given Y=yY=y."

A candidate could otherwise confuse a conditional mean with a conditional density.

11. Conditional Expectations Repeatedly Omit the Conditioning Bar

Curriculum location: Conditional Expectations, Variances, and Covariances, conditional-expectation examples, printed pp. 285-286.

E[Spending Y=1]E[\text{Spending }Y=1]

Without the conditioning bar, the notation reads as juxtaposition or multiplication rather than an expectation under a condition.

Correct reading: Write E[X∣Y=y]E[X\mid Y=y] and E[Spending∣Y=1]E[\text{Spending}\mid Y=1].

A candidate could omit the operator that distinguishes a conditional mean from an ordinary expectation or product.

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

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

E[X]=∑iP(X∣Si)P(Si)E[X]=\sum_i P(X\mid S_i)P(S_i)

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

Correct reading: E[X]=∑iE[X∣Si]P(Si)E[X]=\sum_i E[X\mid S_i]P(S_i).

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

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

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

(0.1575−0.0483)2(0.1575-0.0483)^2

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

Correct reading: Use (−0.1575−0.0483)2(-0.1575-0.0483)^2. With the other two scenario means, the between-scenario variance is about 0.03360.0336, not 0.02340.0234.

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

14. The Unconditional Variance Omits Within-Scenario Risk

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

σ2=(0.1575−0.0483)2+(0.015−0.0483)2+(0.2875−0.0483)23\sigma^2=\frac{(0.1575-0.0483)^2+(0.015-0.0483)^2+(0.2875-0.0483)^2}{3}

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: Var⁡(R)=E[Var⁡(R∣S)]+Var⁡(E[R∣S])\operatorname{Var}(R)=E[\operatorname{Var}(R\mid S)]+\operatorname{Var}(E[R\mid S]). Using the displayed inputs gives a total variance of about 0.04840.0484.

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

15. Equal Priors Do Not Reverse Conditional Probabilities

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

"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(Si∣B)=P(B∣Si)/∑jP(B∣Sj)P(S_i\mid B)=P(B\mid S_i)/\sum_j P(B\mid S_j). In general, P(B∣Si)≠P(Si∣B)P(B\mid S_i)\ne P(S_i\mid B).

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

Complete Module 6 Body-Text Errata Index

Errata scope

  • Curriculum: CFA Level I 2027 Curriculum
  • Volume: Volume 1, Quantitative Methods
  • Module: Module 6, Statistical Distributions for Financial Asset Prices and Returns
  • Topics covered: Random Variables and Unconditional Expectations; Key Statistical Distributions in Finance; Conditional Expectations, Variances, and Covariances; Bayesian Updating
  • Source reviewed: CFA Level I 2027 Curriculum, Volume 1, Module 6, printed pp. 249-307

The table below lists all confirmed source-authored body-text errors identified across the in-scope topics. Repeated instances of the same defect are consolidated into one row. Import-only defects, Knowledge Check and question-set issues, and product metadata defects are excluded.

Page references use the printed curriculum page numbers.

TopicCurriculum locationConfirmed curriculum errorCorrected reading
Random Variables and Unconditional Expectationsprinted p. 253, Random Variables and Their PropertiesA large sample makes a distribution continuous.A variable is discrete when its support is countable and continuous when it ranges over a continuum; sample size does not change that classification.
Random Variables and Unconditional Expectationsprinted p. 255, Exhibit 3 discussion and chartA CDF is an area-normalized density.The PDF integrates to 1; the CDF is the running probability F(x)=P(X≤x) and approaches 1.
Key Statistical Distributions in Financeprinted pp. 266, 279, Discrete Uniform Distribution and Exhibit 18The printed variance applies to any equally spaced n-point support.For n equally spaced points from a to b, Var(X)=(b−a)²(n+1)/[12(n−1)]. The printed formula is the unit-spacing special case.
Key Statistical Distributions in Financeprinted p. 272, Monte Carlo Simulation of Stock Prices caseStock prices are normally distributed because returns are log-normal.In the standard GBM model, stock price levels are log-normal and continuously compounded returns are normal.
Key Statistical Distributions in Financeprinted p. 273, Monte Carlo Simulation of Stock Prices case, Steps 2-3A standard-normal draw can be inserted directly as a one-day Brownian increment.Use ΔW=√Δt·Z with Z~N(0,1). For Δt=1/250 and Z=−1.48, the stated price 9.818 follows only after this scaling.
Key Statistical Distributions in Financeprinted p. 273, Exhibit 14 discussionA final price of 11.09 from an initial price of 10 implies an 11.09% return.The return is (11.09−10)/10=10.9%.
Key Statistical Distributions in Financeprinted p. 275, Log-Normal DistributionLog returns have finite lower and upper bounds.Log returns are unbounded; their multi-period usefulness comes from additivity, not boundedness.
Key Statistical Distributions in Financeprinted p. 276, Modeling Asset Prices caseThe same μ can be called mean log return, reduced by σ²/2, and then exponentiated as an unidentified future-price statistic.If μ is mean log return, E[ln ST]=ln S0+μT. If μ is GBM drift, E[ln ST]=ln S0+(μ−σ²/2)T, but exp of that value is the median; E[ST]=S0e^{μT}.
Key Statistical Distributions in Financeprinted p. 280, Exhibit 19 binomial PMFA binomial PMF may use k in the coefficient and x in the exponents.Write C(n,x)p^x(1−p)^(n−x), or use k consistently throughout.
Key Statistical Distributions in Financeprinted p. 280, Exhibit 19 PDF headerAn interval-probability integral defines the pointwise PDF.For differentiable F, fX(x)=dFX(x)/dx; and ∫_a^b fX(x)dx=P(a≤X≤b).
Key Statistical Distributions in Financeprinted pp. 280, 281, Exhibit 20 and following noteThe printed expression is a valid uniform-CDF spreadsheet formula.Use (x−a)/(b−a) for a≤x≤b, with 0 below a and 1 above b. If BETA.DIST is used, supply the platform's complete current syntax.
Key Statistical Distributions in Financeprinted p. 273, Monte Carlo Simulation of Stock Prices caseThe exhibit visualizes 10 simulated paths.Read 100 paths, not 10.
Conditional Expectations, Variances, and Covariancesprinted p. 283, Exhibit 21 and Equation 23The double integral defining F12 is itself a joint density.Exhibit 21's mound is a joint PDF. Equation 23 defines the joint CDF F12 by integrating that density.
Conditional Expectations, Variances, and Covariancesprinted p. 285, Conditional Expectations lead-inEquations 27-28 calculate a conditional PDF.The lead-in should say conditional expectation, not conditional PDF.
Conditional Expectations, Variances, and Covariancesprinted pp. 285, 286, Conditional expectation formulas and casesJuxtaposing a random variable and condition denotes conditional expectation.Write E[X|Y=y] and E[Spending|Y=1].
Conditional Expectations, Variances, and Covariancesprinted p. 287, Equation 30The total-expectation rule weights conditional probabilities of X.Write E[X]=Σ E[X|Si]P(Si).
Conditional Expectations, Variances, and Covariancesprinted p. 288, Portfolio scenario variance calculationThe depression conditional mean is +15.75%.Use (−0.1575−0.0483)². This changes the between-scenario variance from the printed 0.0234 to about 0.0336.
Conditional Expectations, Variances, and Covariancesprinted p. 288, Portfolio scenario variance calculationThe variance of conditional means alone is the portfolio's unconditional variance.Use Var(R)=E[Var(R|S)]+Var(E[R|S]). With the displayed inputs, the total is about 0.0484.
Bayesian Updatingprinted p. 295, Stock Market Returns and Technology Companies caseThe probabilities come from Exhibit 26.Read Exhibit 28, not Exhibit 26.
Bayesian Updatingprinted p. 299, Diffuse-prior paragraphEqual priors make reverse conditional probabilities equal.For equal priors over exhaustive scenarios, P(Si|B)=P(B|Si)/ΣjP(B|Sj); generally P(B|Si)≠P(Si|B).

References

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 curriculum errata notice.