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CFA L1 2027

V1 Module 9 Errata: Simulation of Financial Asset Prices and Returns

Volume 1 · Quantitative Methods

Published by Quizara
9 independently reviewed issues8 issues explained8 min read

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

Back to the 2027 errata indexAll blog articles
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  1. 1. The Daily Standard Deviations Are Not Several Orders of Magnitude Larger
  2. 2. The Parametric VaR Uses a Two-Sided Critical Value for a One-Sided Tail
  3. 3. The Worked Percentile Chain Contains Both a Rank-Convention Error and an Arithmetic Error
  4. 4. The Option-Payoff Objective Refers to the Wrong Equation
  5. 5. A Lower Sample Standard Deviation Does Not Confirm Simulation Smoothness
  6. 6. Ordinary Bootstrap Resampling Does Not Automatically Reduce Outlier Influence
  7. 7. The Risk-Neutral Drift Adjustment Has the Wrong Sign in the Prose
  8. 8. One Asian Call Subtype Is Presented as the Definition of All Asian Options
  9. Complete Module 9 Body-Text Errata Index
  10. References

1. The Daily Standard Deviations Are Not Several Orders of Magnitude Larger

Curriculum location: Historical Simulation, historical VaR case, Exhibit 9 discussion, printed p. 463.

"Second, the standard deviation of daily returns dominates the mean daily return, typically by several orders of magnitude."

In Exhibit 9, the standard-deviation-to-absolute-mean ratios are approximately 1.3948/0.0607=22.981.3948/0.0607=22.98 and 0.9741/0.0611=15.940.9741/0.0611=15.94. Those ratios are a little over one base-10 order of magnitude, not several.

Correct reading: In Exhibit 9, the daily-return standard deviations are roughly 1616–2323 times the absolute mean returns. They still dominate the means enough to support the passage's practical zero-mean approximation.

Candidates should not translate a roughly twentyfold difference into a multi-order gap.

2. The Parametric VaR Uses a Two-Sided Critical Value for a One-Sided Tail

Curriculum location: Historical Simulation, historical VaR case, parametric comparison, printed p. 463.

VaR⁡95%=z95%×σp=1.96×1.0656=HKD 2.0885 million.\operatorname{VaR}_{95\%}=z_{95\%}\times\sigma_p=1.96\times1.0656=\mathrm{HKD}\,2.0885\text{ million}.

The case defines 95% VaR from the lower 5% tail. The one-sided standard-normal magnitude for that tail is approximately 1.6451.645; 1.961.96 is the 97.5th percentile used for a two-sided 95% interval.

Correct reading: Use VaR⁡95%=1.645×1.0656≈HKD 1.7529 million\operatorname{VaR}_{95\%}=1.645\times1.0656\approx\mathrm{HKD}\,1.7529\text{ million}.

Candidates can otherwise apply a two-sided confidence-interval multiplier to a one-tailed risk measure.

3. The Worked Percentile Chain Contains Both a Rank-Convention Error and an Arithmetic Error

Curriculum location: Historical Simulation, sustainable-energy startup case, Step 2, printed p. 465.

"2.5th percentile (between ranks 2 and 3) using the linear interpolation formula:"

Curriculum location: Historical Simulation, sustainable-energy startup case, Step 2 substitution, printed p. 466.

x=−25.60+2.5−23−2×(−25.56−(−25.60))=−25.58%.x=-25.60+\dfrac{2.5-2}{3-2}\times(-25.56-(-25.60))=-25.58\%.

Curriculum location: Historical Simulation, sustainable-energy startup case, Step 3 downside, printed p. 466.

−25.58%×123,897.22/0.01≈−3,170,000.-25.58\%\times123{,}897.22/0.01\approx-3{,}170{,}000.

The same case maps ranks 48 and 49 to the 96th and 98th percentiles. Under that convention, ranks 1 and 2 map to 2% and 4%, so the 2.5th percentile lies between ranks 1 and 2. The same worked chain also has a separate upper-tail arithmetic error.

Curriculum location: Historical Simulation, sustainable-energy startup case, Step 2 upper-tail interpolation, printed p. 466.

x=5.49+97.5−9698−96×(7.30−5.49)=6.89%.x=5.49+\dfrac{97.5-96}{98-96}\times(7.30-5.49)=6.89\%.

The displayed substitution equals 6.8475%6.8475\%, which rounds to 6.85%6.85\%, not 6.89%6.89\%.

Correct reading: For the lower tail, interpolate between −28.56%-28.56\% and −25.60%-25.60\%: x=−28.56+2.5−24−2×[−25.60−(−28.56)]=−27.82%x=-28.56+\dfrac{2.5-2}{4-2}\times[-25.60-(-28.56)]=-27.82\%. The downside impact is −0.2782×123,897.22/0.01=−GBP 3,446,820.66-0.2782\times123{,}897.22/0.01=-\mathrm{GBP}\,3{,}446{,}820.66, approximately −GBP 3.447 million-\mathrm{GBP}\,3.447\text{ million}, and the resulting NPV is approximately −GBP 2.447 million-\mathrm{GBP}\,2.447\text{ million}. For the upper tail, use 6.85%6.85\%, giving approximately GBP 849,000\mathrm{GBP}\,849{,}000 of upside and an NPV of approximately GBP 1.849 million\mathrm{GBP}\,1.849\text{ million}.

Candidates can otherwise combine incompatible percentile coordinates, propagate the wrong downside valuation, and reproduce an independent upper-tail arithmetic error.

4. The Option-Payoff Objective Refers to the Wrong Equation

Curriculum location: Bootstrapping, Brighton Rock option case, Step 1 setup, printed p. 472.

"Objective: Calculate the option value at maturity, given by Equation 4:"

Equation (4) is the earlier percentile-interpolation formula. The call-option payoff immediately preceding this setup is labeled Equation (5).

Correct reading: Calculate the option value at maturity using Equation (5).

Candidates following the printed cross-reference are sent to an unrelated formula.

5. A Lower Sample Standard Deviation Does Not Confirm Simulation Smoothness

Curriculum location: Bootstrapping, Brighton Rock 500-versus-10,000 run comparison, printed p. 476.

"Comparing the results of the 10,000-run simulations and the 500-run simulations, we observe smoother distribution curves in the 10,000-run graphs, as confirmed by the lower standard deviation."

Increasing the run count reduces sampling noise in the empirical histogram and stabilizes estimated summaries. It does not reduce the volatility of the underlying outcome distribution. The small differences between the reported sample standard deviations are simulation-to-simulation variation.

Correct reading: The 10,000-run histograms are smoother and their tail estimates are more stable because the larger run count reduces sampling noise; the lower reported sample standard deviation does not confirm that improvement.

Candidates should distinguish precision of an estimate from dispersion of the modeled outcome.

6. Ordinary Bootstrap Resampling Does Not Automatically Reduce Outlier Influence

Curriculum location: Bootstrapping, strengths, robust variability estimates, printed p. 480.

"Bootstrapping also reduces the impact of outliers in the data, relative to historical simulation."

Curriculum location: Bootstrapping, Question Set 2, item 2 solution, printed p. 481.

"The correct response is B. Bootstrapping, sampling with replacement, lessens the impact of outliers in the data."

In an ordinary size-nn bootstrap, every original observation has probability 1/n1/n on each of nn draws, so its expected multiplicity is one. An outlier may be omitted from one resample and duplicated in another; resampling with replacement is not itself an outlier-robust method.

Correct reading: Ordinary bootstrap resampling inherits the observed sample's outliers. Outlier resistance requires a robust statistic or a specifically robust resampling procedure; it does not follow merely from sampling with replacement.

Candidates should not choose bootstrapping as an automatic cure for contaminated data.

7. The Risk-Neutral Drift Adjustment Has the Wrong Sign in the Prose

Curriculum location: Monte Carlo Simulation, European call case, risk-neutral drift, printed p. 484.

"the risk-neutral drift—equal to the risk-free rate plus an adjustment for the volatility."

Equation (6), the worked arithmetic, and the later geometric-Brownian-motion formula all subtract one-half of the variance. The word “plus” reverses the sign of that adjustment.

Correct reading: The risk-neutral log-return drift is (rf−σ22)s\left(r_f-\dfrac{\sigma^2}{2}\right)s: the one-half-variance adjustment is subtracted from the risk-free rate.

Candidates can otherwise reverse the drift adjustment in simulated returns.

8. One Asian Call Subtype Is Presented as the Definition of All Asian Options

Curriculum location: Monte Carlo Simulation, Asian Option Valuation Using Monte Carlo Simulation, printed p. 485.

"Asian-style options, also commonly referred to as Asian options, provide a payoff at maturity equal to the greater of zero or the difference between the underlying stock’s price at maturity and the average stock price over the life of the option."

The stated payoff is an average-strike, or floating-strike, Asian call. Average-price, or fixed-strike, Asian calls use the path average as the underlying value and compare it with a fixed strike.

Correct reading: The case studies an average-strike Asian call with payoff max⁡(ST−Sˉ,0)\max(S_T-\bar S,0). A fixed-strike Asian call instead has payoff max⁡(Sˉ−K,0)\max(\bar S-K,0).

Candidates need the strike convention before selecting or calculating an Asian-option payoff.

Complete Module 9 Body-Text Errata Index

Errata scope

  • Curriculum: CFA Level I 2027 Curriculum
  • Volume: Volume 1, Quantitative Methods
  • Module: Module 9, Simulation of Financial Asset Prices and Returns
  • Topics covered: Historical Simulation; Bootstrapping; Monte Carlo Simulation
  • Source reviewed: The official 56-page Module 9 curriculum PDF, with teaching content on printed pp. 453–502 and embedded topic Question Sets included

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 and end-of-module Practice Problems are excluded.

Page references use the printed curriculum page numbers.

TopicCurriculum locationConfirmed curriculum errorCorrected reading
Historical Simulationp. 463, Exhibit 9 discussionStandard deviations roughly 1616–2323 times the mean magnitudes are described as several orders larger.Describe the ratios as roughly 1616–2323 times, or a little over one order of magnitude.
Historical Simulationp. 463, parametric VaRA one-sided 95% VaR uses z=1.96z=1.96.Use z≈1.645z\approx1.645, giving approximately HKD 1.7529 million\mathrm{HKD}\,1.7529\text{ million}.
Historical Simulationpp. 465–466, lower-tail NPV caseThe 2.5th percentile uses ranks 2 and 3 and propagates −25.58%-25.58\%.Use ranks 1 and 2, −27.82%-27.82\%, approximately −GBP 3.447 million-\mathrm{GBP}\,3.447\text{ million} impact, and approximately −GBP 2.447 million-\mathrm{GBP}\,2.447\text{ million} NPV.
Historical Simulationp. 466, upper-tail interpolationThe displayed formula is reported as 6.89%6.89\%.It equals 6.8475%6.8475\%, displayed as 6.85%6.85\%; using that displayed value gives approximately GBP 849,000\mathrm{GBP}\,849{,}000 and GBP 1.849 million\mathrm{GBP}\,1.849\text{ million}.
Bootstrappingp. 472, Brighton Rock setupThe call-payoff formula is cited as Equation 4.Cite Equation 5.
Bootstrappingp. 476, simulation-count comparisonA lower sample standard deviation is said to confirm smoother curves.Attribute smoother curves and more stable estimates to reduced sampling noise from more runs.
Bootstrappingpp. 480–481, strengths and embedded Question Set 2Ordinary bootstrap is said to reduce outlier influence.Ordinary bootstrap inherits sample outliers; resampling with replacement is not automatic robustification.
Monte Carlo Simulationp. 484, risk-neutral driftThe prose says the volatility adjustment is added.Use (rf−σ22)s\left(r_f-\dfrac{\sigma^2}{2}\right)s.
Monte Carlo Simulationp. 485, Asian-option definitionThe average-strike call payoff is generalized to all Asian options.Distinguish max⁡(ST−Sˉ,0)\max(S_T-\bar S,0) from the fixed-strike payoff max⁡(Sˉ−K,0)\max(\bar S-K,0).

References

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 9 Simulation of Financial Asset Prices and Returns. It is not an official curriculum errata notice.

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