V6 Modules 11–13 Errata: Duration, Convexity, and Interest-Rate Risk

1. Module 11 — FRN Duration Mixes Coupon Periods with Years

Curriculum location: Money Duration and Price Value of a Basis Point, FRN discussion and Knowledge Check, pp. 283–285; Practice Problem 7 solution, p. 296.

The FRN formula is introduced as a:

"fraction of a period"

Equation 10 is correct in coupon-period units: $(T-t)/T$ is the fraction of the current reset period remaining. The error appears when the semiannual period fractions are placed beside annualized fixed-bond durations without conversion. The comparison table reports an FRN value of:

$0.8306$

while the accompanying explanation caps the FRN's duration at:

"six months"

The units cannot all be years: 0.8306 years is more than six months. Read as coupon periods, 0.8306 is valid but is not comparable with the annualized fixed-bond column. The later Practice Problem 7 solution repeats the mixed basis by comparing the fixed bond's 3.6747 years with the FRN value:

$0.5$

For a reset frequency of $m$ times per year, the annualized duration is

$$D_{\mathrm{FRN},\mathrm{years}}=\frac{T-t}{T}\times \frac{1}{m}.$$

With semiannual resets, $(180-57)/180=0.683333$ coupon periods becomes $0.341667$ years. Likewise, 0.8306 periods becomes 0.4153 years, and 0.5 periods becomes 0.25 years. The Practice Problem 7 key remains unchanged because 3.6747 years is still greater than 0.25 years.

Correct reading: Divide each semiannual coupon-period fraction by two before comparing it with an annualized fixed-bond duration.

This avoids a factor-of-two error in FRN interest-rate risk and keeps all compared durations in the same unit.

2. Module 11 — Exhibit 4 Extends Duration Rankings Without a Position-Size Normalization

Curriculum location: Properties of Duration, opening paragraph and the sentence after Exhibit 4, p. 288.

The opening paragraph says:

"the same characteristics hold for modified duration, money duration, and price value of basis point"

The sentence after Exhibit 4 repeats:

"the same characteristics hold for modified duration, money duration, and price value of basis point"

The module's own Practice Problem solutions show why that blanket extension fails. For Bonds One, Two, and Three, the p. 296 solution reports modified durations of $3.539$, $3.652$, and $3.585$, but PVBPs of $0.04069$, $0.03150$, and $0.03585$; the rankings differ. The same page's money-duration solution also shows the contrasting fixed-market-value normalization: with $50$ invested in each bond, money duration equals $50$ times modified duration. The normalization must therefore be stated.

The extension is valid for modified duration, but not as a blanket rule for money duration or PVBP. Both dollar-risk measures scale with the bond's full price. When par is held fixed, the price effect can reverse a directional relationship even though modified duration moves exactly as Exhibit 4 says.

For a direct fixed-par example, consider five-year annual-pay bonds with par $100$ and a $5\%$ yield. Raising the coupon from 4% to 6% lowers modified duration from approximately 4.400307 to 4.264525, as expected. But price rises from approximately 95.670523 to 104.329477, so money duration rises from approximately 420.979673 to 444.915661. Exact symmetric 1 bp repricing likewise raises PVBP from approximately 0.042098 to 0.044492; the first-order $0.0001\times \mathrm{MoneyDur}$ approximation gives the same direction. The reversal comes from price scaling under fixed par, not from the choice between those PVBP calculations.

Correct reading: Do not apply Macaulay or modified-duration directional rules mechanically to money duration or PVBP. State whether par amount or market value is held fixed. For a fixed-par comparison, evaluate annualized modified duration together with full price; exact symmetric repricing and the first-order BPV approximation give the same coupon-direction result here.

This prevents a candidate from reversing a dollar-risk ranking when two bonds have different coupons or prices.

3. Module 12 — Knowledge Check Adds Duration and Yield Instead of Multiplying Them

Curriculum location: Bond Risk and Return Using Duration and Convexity, Knowledge Check question 3, p. 310.

$$\%\Delta P{V}^{\mathrm{Full}}\approx (-4.9535+0.01)+[\frac{1}{2}\times 29.918\times (0.01{)}^2]$$

The general equation immediately above this substitution multiplies modified duration by the yield change. The stated final result, $-4.8039\%$, is also obtained only by multiplication. Adding $-4.9535$ and $0.01$ is neither the duration term nor a calculation that can reproduce the worked answer.

Correct reading: Use $-4.9535\times 0.01$ for the duration contribution. Then

$$-4.9535\times 0.01+\frac{1}{2}\times 29.918\times (0.01{)}^2\approx -0.048039,$$

or $-4.8039\%$.

A candidate copying the printed substitution would use the wrong operator and fail to reproduce the result.

4. Module 12 — The PRICE Call Uses $3.45\%$ for a $50$-bp Yield Decrease

Curriculum location: Bond Risk and Return Using Duration and Convexity, Question Set question 4, p. 311.

$$P{V}^{\mathrm{Full}}=\mathrm{PRICE}(\mathrm{DATE}(2025,6,30),\mathrm{DATE}(2032,6,30),0.0295,0.0345,100,2,0)=103.198$$

The bond starts at a 2.95% yield, and the question asks for a 50 bp decrease. That means a new yield of 2.45%, not 3.45%. The curriculum's own result confirms the direction: the call with 2.45% gives approximately 103.198, while 3.45% gives approximately 96.914.

Correct reading: Use $0.0245$ as the PRICE yield argument:

$$\mathrm{PRICE}(\mathrm{DATE}(2025,6,30),\mathrm{DATE}(2032,6,30),0.0295,0.0245,100,2,0)\approx \mathrm{103.198.}$$

This prevents a candidate from reversing the yield shock inside the pricing function.

5. Module 12 — Practice Problem 1 Labels a $5$-bp Repricing Shock as $50$ bps

Curriculum location: Bond Convexity and Convexity Adjustment, Practice Problem 1, p. 316.

Status: Officially corrected by CFA Institute.

"For a 50 bps increase and decrease in yield-to-maturity, $P{V}_{+}$ and $P{V}_{-}$ are 99.82283 and 100.177546, respectively."

Curriculum location: Bond Convexity and Convexity Adjustment, Practice Problem 1 solution, p. 318.

$$\mathrm{ApproxCon}=\frac{100.177546+99.82283-(2\times 100)}{(0.005{)}^2\times 100}=15.04$$

The stated prices are 5 bp repricings, but the original stem labels the shock as 50 bps and the original solution uses $0.005^2$. CFA Institute's official correction changes the shock to 5 bps and the denominator to $0.0005^2$, retaining the stated prices. The corrected calculation is $[100.177546+99.82283-(2\times 100)]/(0.0005^2\times 100)=15.04$, closest to keyed option B.

Correct reading: The yield shock is 5 bps, not 50 bps. Retain $P{V}_{+}=99.82283$ and $P{V}_{-}=100.177546$, and use $0.0005$ as the yield change; approximate convexity is $15.04$, closest to $15.045$.

This keeps the repriced values and denominator on the same shock scale and restores the keyed calculation.

6. Module 13 — The BRWA Yield-Based Comparison Credits $-4.47\%$ to $ApproxModDur$ Alone

Curriculum location: Bond Risk and Return Using Curve-Based Duration and Convexity, BRWA yield-based comparison paragraph, p. 331.

"the previously derived estimates of $-4.47\%$ and $4.71\%$ for this BRWA bond when using $ApproxModDur$ alone and $ApproxModDur$ and $ApproxCon$, respectively"

The module's p. 327 inputs are $ApproxModDur=4.587$ and $ApproxCon=24.239$. For a $+100$-bp shift, duration alone gives $-4.587\%$; adding the convexity adjustment of approximately $+0.1212\%$ gives $-4.466\%$, which rounds to $-4.47\%$. Both reported percentages can be reproduced only by adding the approximate-convexity adjustment to the signed duration term. $ApproxModDur$ alone gives about $-4.59\%$, not $-4.47\%$, so the paragraph’s method attribution conflicts with its own calculation.

Correct reading: Both $-4.47\%$ and $4.71\%$ use $ApproxModDur$ and $ApproxCon$ at $\pm 100$ bps.

A candidate could omit convexity from the upward-shift estimate and learn an asymmetric method that the calculation does not use.

7. Module 13 — Question 5’s Solution Mislabels the Downward Shift as Upward

Curriculum location: Bond Risk and Return Using Curve-Based Duration and Convexity, Question Set question 5 solution, p. 333.

"For the upward shift, the duration contribution is positive and convexity contribution is negative, offsetting each other."

For an upward shift, the signed duration term is negative and the negative-convexity term is also negative, so they reinforce rather than offset. The terms offset only for a downward shift.

Correct reading: Read the sentence as ‘For the downward shift’; the positive duration term is then offset by the negative convexity term.

A candidate could reverse the sign logic used to interpret duration and negative convexity.

8. Module 13 — The Callable-Bond KRD Calculation Is Not Supported by the Inputs Provided

Curriculum location: Key Rate Duration as a Measure of Yield Curve Risk, callable-bond two-year key-rate example, p. 335.

"the rate on the two-year Treasury note, with modified duration of $1.99$—rise by $25$ bps"

On the same p. 335, Exhibit 5 lists the two-year government bond's modified duration as $1.990$ and its key rate duration as $0.711$, with the printed derivation $0.711=1.990\times (99.006/277.0)$. The curriculum thus demonstrates that modified duration and a two-year KRD are different quantities. Equation 6 estimates the callable bond's price response and therefore requires that bond's two-year key rate duration. The Treasury benchmark supplies the $+25$-bp rate change; its own modified duration is not the target bond's sensitivity. The displayed $-0.4975\%$ follows arithmetically from $-1.99\times 0.0025$, but the passage never supplies evidence that the callable bond's KRD is $1.99$.

Correct reading: Use the callable bond’s two-year key rate duration with the $+25$-bp change in the two-year Treasury benchmark rate. If that KRD is $1.99$, the estimate is $-0.4975\%$; otherwise, the passage does not provide enough information to verify the number.

A candidate could substitute a benchmark security's modified duration for the target bond's key-rate exposure.

9. Module 13 — Exhibit 6 Calls Effective Duration an Estimate of Modified Duration

Curriculum location: Empirical Duration, Exhibit 6 Effective Duration interpretation, p. 338.

"Curve-based method to estimate modified duration for complex bonds with uncertain cash flows"

The module also states that bonds with embedded options do not have well-defined yields to maturity, so Macaulay and modified duration are not appropriate interest rate risk measures for them. Exhibit 6 therefore cannot coherently describe effective duration as estimating modified duration for the same complex bonds with uncertain cash flows.

Correct reading: Read the cell as a curve-based method to estimate interest-rate risk for complex bonds with uncertain cash flows.

A candidate could collapse effective and modified duration into one measure and choose the wrong statistic for embedded-option bonds.

10. Module 13 — Question 1’s Solution Calls Empirical Models Non-Statistical

Curriculum location: Empirical Duration, Question Set question 1 solution, p. 340.

"estimated using historical data in non-statistical models"

Exhibit 6 and the main text on p. 338 define empirical duration as using statistical models. The Learning Module Self-Assessment solution on p. 324 gives the same item the correct explanation: statistical models are primarily used for empirical duration and convexity. ‘Non-statistical’ is appropriate only inside the deliberately false option; the p. 340 solution incorrectly carries that word into its affirmative correction.

Correct reading: Replace ‘non-statistical’ with ‘statistical’ in the solution; keep the deliberately incorrect option unchanged.

A candidate could memorize the opposite of the defining empirical method.

11. Module 13 — Practice Problem 3’s Stem Omits the $+100$-bp Curve Shift

Curriculum location: Bond Risk and Return Using Curve-Based Duration and Convexity, Practice Problem 3 stem, p. 342.

"Compare the interest rate risk of Bond A and Bond B."

Solution, p. 344:

"Bond B is riskier since the $100$ bps upward shift in the yield curve results in a greater percentage price decrease, owing to its negative effective convexity:"

The solution silently introduces a $+100$-bp shift. At $+10$ bps, Bond A has the slightly larger price decline; at $+100$ bps, Bond B does. The values in the stem therefore do not determine the keyed ranking without the omitted shock.

Correct reading: Ask the comparison for a $100$-bp upward shift in the benchmark yield curve.

Without the shock, a candidate cannot reproduce the keyed comparison and can reasonably challenge whether B is always riskier.

Complete Duration, Convexity, and Interest-Rate Risk Errata Index

Scope: Modules 11–13, pp. 267–344 (printed page numbers).

Reviewed modules. Module 11, pp. 267–296; Module 12, pp. 297–319; and Module 13, pp. 321–344.

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 volume's own data. It does not list spelling mistakes, equation-numbering slips, or wording that is loose but defensible.

The table lists every high-value, objectively confirmed curriculum error admitted for these modules. Repeated instances of the same defect are consolidated into one row. Extraction or import-fidelity problems, disputed readings, and low-value editorial issues are outside the public errata scope.

References

• CFA Institute — Yield-Based Bond Duration Measures and Properties
• Microsoft Support — PRICE function
• Investor.gov — Basis Point
• CFA Institute — 2027 CFA Program Level I Errata Notice

This article is an independent candidate-focused analysis of substantive errors identified in our review of the CFA Level I 2027 Curriculum, Volume 6 Fixed Income, Modules 11–13. It is not an official CFA Institute errata notice, and inclusion here must not be read as CFA Institute confirmation, endorsement, or approval.