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Curriculum errata

CFA Level I 2027 Errata: The Return and Risk of a Financial Portfolio

Quantitative Methods · Module 8

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

15 explained errors20-item complete index15 min read

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

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  1. 1. Cumulative Return Is Added Instead of Compounded
  2. 2. Example 3 Switches between Covariance and Correlation
  3. 3. Exhibit 3 Reverses the Direction of Expected Return
  4. 4. Equation 19 Uses the Wrong Scenario Subscript
  5. 5. Equation 20 Omits the Equal-Weight Condition
  6. 6. The Large-N Limit of 1 − 1/N Is Reversed
  7. 7. The Minimum-Variance Formula Ignores Its No-Short Constraint
  8. 8. Example 11 Drops One Covariance Subtraction
  9. 9. The Efficient-Frontier Direction Depends on Which Asset Has the Higher Return
  10. 10. A Complete Portfolio Is Misnamed the Optimal Risky Portfolio
  11. 11. Equation 27’s Risk Effect Is Stated without Conditioning on A
  12. 12. Exhibit 22 Mixes Fixed-Risk and Fixed-Return Comparisons
  13. 13. Absolute and Relative Risk Aversion Are Not Defined by Risk Amounts
  14. 14. Exhibit 24 Conflates Two Different Tangencies
  15. 15. Beta Is Not Generally the Ratio of Two Volatilities
  16. Complete Module 8 Body-Text Errata Index
  17. References

1. Cumulative Return Is Added Instead of Compounded

Curriculum location: Calculating Portfolio Statistics, Returns of a Portfolio, Equation 7, printed p. 388.

“The cumulative or total arithmetic portfolio return over T periods with changing weights is”

Curriculum location: Calculating Portfolio Statistics, Example 2, printed p. 390.

“The cumulative or total arithmetic return, using Equation 7, is”

Adding periodic returns does not measure the change in wealth over multiple periods. The curriculum’s own Equation 9 uses the compounding structure that a cumulative return requires.

Correct reading: Equation 7 is a sum of periodic arithmetic returns, not a cumulative return. The cumulative return is ∏t=1T(1+rP,t)−1\prod_{t=1}^{T}(1+r_{P,t})-1.

A candidate who adds rather than compounds can report the wrong multi-period portfolio performance.

2. Example 3 Switches between Covariance and Correlation

Curriculum location: Calculating Portfolio Statistics, Example 3, printed pp. 392–393.

“Furthermore, assume that the covariance of the returns between these two indices is 0.0050.”

“σ_P = √0.0228 = 0.1510 = 15.10%.”

Curriculum location: Calculating Portfolio Statistics, Exhibit 2, Portfolio 9, printed p. 393.

PortfolioS&P 500 Index (%)MSCI Emerging Markets Index (%)Expected Return (%)Standard Deviation (%)
9802011.5814.59

The 15.10% calculation inserts 0.0050 as covariance. The exhibit and chart instead use 0.0050 as correlation, which produces approximately 14.59%.

Correct reading: Treat 0.0050 as correlation, first compute covariance as ρσAσB\rho\sigma_A\sigma_B, and then calculate portfolio standard deviation. The 80/20 portfolio result is approximately 14.59%.

This distinction determines whether a candidate inserts a dimensionless correlation or a covariance into the portfolio-variance formula.

3. Exhibit 3 Reverses the Direction of Expected Return

Curriculum location: Calculating Portfolio Statistics, Exhibit 3 interpretation, printed p. 393.

“… the portfolio risk and return decreases first (from P11 to P9) before increasing.”

The exhibit shows standard deviation falling from 16.21% to 14.59% from P11 to P9, but expected return rises from 9.93% to 11.58%.

Correct reading: From P11 to P9, portfolio risk decreases while expected return increases. Beyond P9, both risk and expected return increase.

A candidate could otherwise trace the return axis in the wrong direction.

4. Equation 19 Uses the Wrong Scenario Subscript

Curriculum location: Calculating Portfolio Statistics, Equation 19, printed p. 400.

∑M′=1MpM′∑i=1Nwiri,M\sum_{M'=1}^{M}p_{M'}\sum_{i=1}^{N}w_i r_{i,M}

The return inside the M′M' expectation retains the outer index MM. That prevents the inner term from ranging across scenarios.

Correct reading: The final return term must be ri,M′r_{i,M'}: ∑M′pM′∑iwiri,M′\sum_{M'}p_{M'}\sum_i w_i r_{i,M'}.

Using the printed subscript can make a candidate calculate deviations from the wrong expected return.

5. Equation 20 Omits the Equal-Weight Condition

Curriculum location: Calculating Portfolio Statistics, The Benefits of Diversification, printed p. 405.

“If all assets have the same variance, σ², and the average correlation between assets is ρ̄, then the portfolio variance simplifies to:”

The displayed coefficients σ2/N\sigma^2/N and (1−1/N)ρˉσ2(1-1/N)\bar{\rho}\sigma^2 also depend on every asset having weight 1/N1/N.

Correct reading: Equation 20 applies to an equally weighted NN-asset portfolio whose assets have the stated common variance and average pairwise correlation.

Without that condition, a candidate could use the shortcut for an unequal-weight portfolio.

6. The Large-N Limit of 1 − 1/N Is Reversed

Curriculum location: Calculating Portfolio Statistics, The Benefits of Diversification, printed p. 406.

“When N becomes large, the first term, σ²/N, the individual asset variance contribution, becomes negligible. This also applies to (1 − 1/N).”

Only σ2/N\sigma^2/N becomes negligible. The factor 1−1/N1-1/N approaches 1.

Correct reading: As N→∞N\to\infty, σ2/N→0\sigma^2/N\to0 and 1−1/N→11-1/N\to1, so portfolio variance approaches ρˉσ2\bar{\rho}\sigma^2.

Treating 1−1/N1-1/N as negligible would incorrectly imply that the common-covariance component disappears from a large portfolio.

7. The Minimum-Variance Formula Ignores Its No-Short Constraint

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, two-asset setup, printed p. 416.

“the portfolio weights must be non-negative: w_A, w_B ≥ 0.”

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, Equations 23–24 conclusion, printed p. 417.

“These weights minimize the portfolio’s variance”

Equations 23–24 give the unconstrained interior stationary point. If a calculated weight is negative or above 1, it is infeasible under the stated constraint.

Correct reading: Use the formula directly only when both weights lie in [0,1][0,1]. Otherwise, the constrained minimum occurs at a feasible boundary, so compare the endpoint portfolios.

A candidate could otherwise present a prohibited short position as the no-short minimum.

8. Example 11 Drops One Covariance Subtraction

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, Example 11, printed p. 418.

wA=0.1096−0.01070.0263+0.1096−0.0107=0.09890.1144≈0.864w_A=\dfrac{0.1096-0.0107}{0.0263+0.1096-0.0107}=\dfrac{0.0989}{0.1144}\approx0.864

The denominator formula requires subtracting twice the covariance. With only one subtraction, the denominator is 0.1252, not 0.1144.

Correct reading: Write the denominator as 0.0263+0.1096−2(0.0107)≈0.11450.0263+0.1096-2(0.0107)\approx0.1145. The resulting weight remains approximately 0.864.

The omitted factor can lead a candidate to a different minimum-variance weight.

9. The Efficient-Frontier Direction Depends on Which Asset Has the Higher Return

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, printed p. 421.

“If w_A is to be positive, r_P must be at least as large as r_B.”

“Covering all possible weights for w_A from w_min,A to 1 will generate the entire efficient frontier.”

From wA=(rP−rB)/(rA−rB)w_A=(r_P-r_B)/(r_A-r_B), both the sign condition and the direction of the upper frontier depend on the sign of rA−rBr_A-r_B. In the module’s own asset assignment, rA<rBr_A<r_B.

Correct reading: If rA>rBr_A>r_B, wA≥0w_A\ge0 requires rP≥rBr_P\ge r_B, and the efficient branch moves from wmin⁡,Aw_{\min,A} toward 1. If rA<rBr_A<r_B, wA≥0w_A\ge0 requires rP≤rBr_P\le r_B, and the efficient branch moves from wmin⁡,Aw_{\min,A} toward 0. The module’s own asset assignment is the second case.

The printed rule can send a candidate onto the inefficient branch.

10. A Complete Portfolio Is Misnamed the Optimal Risky Portfolio

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, transition to Example 13, printed p. 422.

“We need the risk-free asset to create the optimal risky portfolio, as the following example shows.”

Curriculum location: The Minimum–Variance Portfolio and the Efficient Frontier, Example 13, printed p. 422.

“A portfolio manager creates an optimal risky portfolio by combining the risky portfolio, proxied by the MSCI World Index, and a risk-free asset”

The optimal risky portfolio is the risky-asset tangency portfolio. Combining a risky portfolio with the risk-free asset creates a complete portfolio on a capital allocation line.

Correct reading: Identify the tangency portfolio first. Investor risk aversion then determines the allocation between that risky portfolio and the risk-free asset, producing the investor’s complete portfolio.

Confusing the two steps can make a candidate treat investor risk aversion as determining the risky fund itself.

11. Equation 27’s Risk Effect Is Stated without Conditioning on A

Curriculum location: Optimal Portfolio Selection, Utility functions, printed p. 430.

“… utility … decreases with risk and the investor’s coefficient of risk aversion, suggesting that investors experience diminishing marginal utility as risk increases.”

For U=E[r]−12Aσ2U=E[r]-\tfrac12A\sigma^2, the marginal effect of variance is −A/2-A/2. It is negative for A>0A>0, zero for A=0A=0, and positive for A<0A<0; it is also constant per unit of variance.

Correct reading: Utility falls with variance only for a risk-averse investor with A>0A>0. It is independent of variance for A=0A=0 and rises with variance for A<0A<0. This formula does not imply diminishing marginal utility in variance.

A candidate could otherwise reverse the model’s prediction for risk-neutral or risk-seeking investors.

12. Exhibit 22 Mixes Fixed-Risk and Fixed-Return Comparisons

Curriculum location: Optimal Portfolio Selection, Exhibit 22, printed p. 431.

“Higher return and higher risk at any given level of return”

“Higher return and lower risk at any given level of return”

The risk-seeking and risk-averse cells say that return is both higher and held fixed. They have merged two different dominance comparisons into one contradictory phrase.

Correct reading: At a given level of risk, an investor prefers higher expected return. At a given expected return, a risk-seeking investor prefers higher risk, while a risk-averse investor prefers lower risk.

A candidate could otherwise hold the wrong variable constant when classifying investor preferences.

13. Absolute and Relative Risk Aversion Are Not Defined by Risk Amounts

Curriculum location: Optimal Portfolio Selection, risk-aversion definitions, printed p. 431.

“Absolute risk aversion measures how much an investor’s utility changes with each incremental increase in risk, holding wealth constant.”

“Relative risk aversion measures the proportion of wealth an investor is prepared to risk”

These are not the formal Arrow–Pratt definitions. Both measures use the curvature of utility with respect to wealth.

Correct reading: Absolute risk aversion is A(W)=−u′′(W)/u′(W)A(W)=-u''(W)/u'(W). Relative risk aversion is R(W)=−W u′′(W)/u′(W)R(W)=-W\,u''(W)/u'(W).

The printed descriptions can cause a candidate to confuse formal preference measures with risk capacity or portfolio allocation.

14. Exhibit 24 Conflates Two Different Tangencies

Curriculum location: Optimal Portfolio Selection, Exhibit 24 description, printed p. 435.

“Exhibit 24 shows the highest attainable utility curve that is tangent to the efficient frontier.”

The investor’s highest attainable indifference curve is tangent to the capital allocation line (CAL). A different tangency determines the optimal CAL: the CAL is tangent to the risky-asset efficient frontier.

Correct reading: First identify the risky tangency portfolio where the optimal CAL touches the risky-asset efficient frontier. Then identify the investor’s complete portfolio where the highest attainable indifference curve touches that CAL.

Conflating the two tangencies can make a candidate select the wrong portfolio point.

15. Beta Is Not Generally the Ratio of Two Volatilities

Curriculum location: Optimal Portfolio Selection, CML-to-beta transition, printed p. 437.

“Along the CML, the ratio of the standard deviation of an asset or a portfolio, σ_p, to … σ_m, indicates how sensitive the portfolio’s returns are”

Curriculum location: Optimal Portfolio Selection, CAPM and its estimation, printed p. 439.

“β_i is the beta value of asset i and is equal to σ_p/σ_m.”

For a general asset, beta also depends on correlation with the market. The volatility ratio alone equals beta only in the unit-correlation case. The capital market line (CML) is the special line formed by the risk-free asset and the market portfolio.

Correct reading: Use βi=ρi,mσi/σm\beta_i=\rho_{i,m}\sigma_i/\sigma_m for a general asset. For a positive combination of the market portfolio and the risk-free asset on the CML, correlation with the market is 1, so the volatility ratio shortcut applies.

Using volatility alone can make a candidate rank total risk as though it were systematic risk.

Complete Module 8 Body-Text Errata Index

Errata scope

  • Curriculum: CFA Level I 2027 Curriculum
  • Volume: Volume 1, Quantitative Methods
  • Module: Module 8, The Return and Risk of a Financial Portfolio
  • Topics covered: Calculating Portfolio Statistics; The Minimum–Variance Portfolio and the Efficient Frontier; Optimal Portfolio Selection
  • Source reviewed: CFA Level I 2027 Curriculum, Volume 1, Module 8 PDF

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 question-set issues are excluded.

Page references use the printed curriculum page numbers.

TopicCurriculum locationConfirmed curriculum errorCorrected reading
Calculating Portfolio Statisticspp. 388, 390, Equation 7 and Example 2A sum of periodic returns is labeled cumulative or total return.Compound period returns: ∏t(1+rt)−1\prod_t(1+r_t)-1.
Calculating Portfolio Statisticspp. 392–393, Example 3 and Exhibit 2The same 0.0050 input is treated as covariance in the example and correlation in the exhibit.Treat it as correlation, convert to covariance, and obtain about 14.59%.
Calculating Portfolio Statisticsp. 393, Exhibit 3 interpretationBoth risk and return are said to fall from P11 to P9.Risk falls while expected return rises.
Calculating Portfolio Statisticsp. 400, Equation 19The inner scenario sum uses ri,Mr_{i,M}.Use ri,M′r_{i,M'} inside the M′M' sum.
Calculating Portfolio Statisticsp. 405, Equation 20 introductionThe equal-weight condition is omitted.Add wi=1/Nw_i=1/N for every asset.
Calculating Portfolio Statisticsp. 406, large-N explanation1−1/N1-1/N is said to become negligible.1−1/N→11-1/N\to1; only σ2/N→0\sigma^2/N\to0.
Calculating Portfolio Statisticsp. 395, variance–covariance matrix introductionPairwise calculation growth is called exponential.The N(N−1)/2N(N-1)/2 covariance count grows quadratically.
The Minimum–Variance Portfolio and the Efficient Frontierpp. 416–417, Equations 23–24Unconstrained weights are presented as always solving the no-short problem.Require both weights in [0,1][0,1]; otherwise use a boundary.
The Minimum–Variance Portfolio and the Efficient Frontierp. 418, Example 11One covariance subtraction is omitted from the denominator.Use −2(0.0107)-2(0.0107); the weight remains about 0.864.
The Minimum–Variance Portfolio and the Efficient Frontierp. 421, frontier derivationThe positive-weight condition and efficient-frontier direction ignore the return ordering.If rA>rBr_A>r_B, require rP≥rBr_P\ge r_B and move wAw_A toward 1; if rA<rBr_A<r_B, require rP≤rBr_P\le r_B and move wAw_A toward 0.
The Minimum–Variance Portfolio and the Efficient Frontierp. 422, Example 13 transitionA risky/risk-free mix is called the optimal risky portfolio.It is a complete portfolio on the CAL.
The Minimum–Variance Portfolio and the Efficient Frontierp. 421, variable definitionsAsset B’s return is labeled rPr_P.Label Asset B’s return rBr_B; reserve rPr_P for the portfolio.
Optimal Portfolio Selectionp. 430, Equation 27 interpretationUtility is said to fall with risk for every investor type and to be diminishing in variance.The variance effect is −A/2-A/2 and depends on the sign of AA.
Optimal Portfolio Selectionp. 431, Exhibit 22The preference cells vary return while also claiming return is fixed.At fixed risk prefer higher return; at fixed return, risk-seeking prefers higher risk and risk-averse prefers lower risk.
Optimal Portfolio Selectionp. 431, risk-aversion definitionsAbsolute and relative risk aversion are described as risk amounts.Use −u′′/u′-u''/u' and −Wu′′/u′-Wu''/u'.
Optimal Portfolio Selectionp. 434, CAL assumptionsStandard deviation is compared directly with the risk-free rate.State σQ>0\sigma_Q>0 and E[rQ]>rfE[r_Q]>r_f separately.
Optimal Portfolio Selectionpp. 434–435, CAL setupThe risky portfolio changes symbols between PP and QQ.Use QQ for the risky portfolio and pp for the complete portfolio throughout.
Optimal Portfolio Selectionp. 435, Exhibit 24 descriptionThe indifference curve is said to be tangent to the efficient frontier.The indifference curve touches the CAL; the CAL touches the risky frontier.
Optimal Portfolio Selectionpp. 437, 439, beta discussionBeta is equated to a volatility ratio without correlation.Use βi=ρi,mσi/σm\beta_i=\rho_{i,m}\sigma_i/\sigma_m generally.
Optimal Portfolio Selectionp. 442, Example 17The CAPM formula contains an extra RR before rfr_f.Delete the stray RR.

References

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 8 The Return and Risk of a Financial Portfolio. It is not an official curriculum errata notice.