V1 Module 4 Errata: The Time Value of Money in Finance

1. The Compounding Frequency Is Labeled as the Interest Rate

Curriculum location: Time Value of Money in Fixed Income and Equity, continuous-compounding introduction, printed p. 128.

"In the limit, as the compounding frequency, r, approaches infinity, we compound the initial cash flow on a continuous basis as follows:"

The annual rate $r$ remains fixed in the exponential accumulation formula; it is the number of compounding periods per year, conventionally $m$, that increases without bound.

Correct reading: In the limit, as the compounding frequency $m$ approaches infinity, $FV_t=PV_0e^{rt}$, where $r$ is the annual rate and $t$ is measured in years.

Candidates need to distinguish the limiting frequency from the rate that drives the accumulated value.

2. Monetary Interest Is Confused with a Return Rate

Curriculum location: Time Value of Money in Fixed Income and Equity, fixed-income cash-flow patterns, printed p. 129.

"This amount is equivalent to the yield-to-maturity and should correspond to the investor’s required rate of return for the valuation relationship to be accurate."

Curriculum location: Implied Return and Growth, implied return for fixed-income instruments, printed p. 149.

"the difference between the undiscounted value of that principal cash flow and the currently prevailing price of the asset represents its implied return."

Both passages treat a currency amount as though it were a return rate. The difference $FV_t-PV_0$ is total interest earned; a yield also depends on the investment horizon and is expressed as a rate.

Correct reading: The monetary difference is the total interest amount. For a single future cash flow, $r=(FV_t/PV_0)^{1/t}-1$ is the return per unit of the time basis used for $t$; it is an annualized return when $t$ is measured in years.

Candidates who equate an amount with a yield can omit the time dimension and select the wrong return calculation.

3. The Mortgage Spreadsheet Line Uses the Wrong Function, Principal, and Payment

Curriculum location: Time Value of Money in Fixed Income and Equity, Time-Value-of-Money Calculations Using Excel case, mortgage item function signature, printed p. 141.

"=PV(rate, nper, pmt, [fv], [type])"

Curriculum location: Time Value of Money in Fixed Income and Equity, Time-Value-of-Money Calculations Using Excel case, mortgage item PMT calculation, printed p. 141.

"−4,4417.63 = PMT(0.004375, 360, 2800000, 0, 0)."

The line labels the calculation with the $PV$ signature even though it uses $PMT$, enters USD2,800,000 rather than the USD800,000 amount financed, and prints an extra 4 in the payment.

Correct reading: Use $PMT(rate,nper,pv,[fv],[type])$. With $rate=0.0525/12$, $nper=360$, and $pv=800{,}000$, the payment is approximately $-$USD4,417.63 under Excel's cash-flow sign convention.

Entering the printed setup would produce a payment for the wrong loan amount.

4. Equation 13 and Exhibit 6 Double-Count Dividend Growth

Curriculum location: Time Value of Money in Fixed Income and Equity, Exhibit 6, Year-5 label, printed p. 143.

"Div₅ = Div₅ × (1 + g)⁵"

Curriculum location: Time Value of Money in Fixed Income and Equity, constant dividend growth, Equation 13, printed p. 143.

"PV₀ = ∑ᵢ₌₁∞ Divᵢ(1 + g)ⁱ/(1 + r)ⁱ"

Because $Div_i$ already denotes the dividend at time $i$, multiplying it by another $(1+g)^i$ applies the same growth twice. Exhibit 6 repeats that error in the Year-5 label.

Correct reading: In Exhibit 6, use $Div_5=Div_0(1+g)^5$. In Equation 13, write the numerator as $Div_0(1+g)^i$, so $PV_0=\sum_{i=1}^{\infty}Div_0(1+g)^i/(1+r)^i$. Equivalently, discount each already-grown $Div_i$ without multiplying by $(1+g)^i$ again.

The printed figure label and formula can both overstate a future dividend.

5. Equation 15 Uses the Wrong Low-Growth Exponent

Curriculum location: Time Value of Money in Fixed Income and Equity, two-stage dividend growth, Equation 15, printed p. 144.

"PV₀ = ∑ᵢ₌₁ⁿ Div_t(1 + g_s)ⁱ/(1 + r)ⁱ + ∑ⱼ₌ₙ₊₁∞ Div_{t+n}(1 + g_l)ʲ/(1 + r)ʲ"

Starting from the time-$n$ dividend, the first low-growth dividend at $j=n+1$ has only one low-growth period, not $n+1$.

Correct reading: From $Div_{t+n}$ to time $j$, use the exponent $j-n$: the low-growth term is $Div_{t+n}(1+g_l)^{j-n}$.

The printed exponent causes candidates to overgrow every dividend in the low-growth stage.

6. Equation 16 Equates Values Measured at Different Dates

Curriculum location: Time Value of Money in Fixed Income and Equity, two-stage dividend growth, Equation 16, printed p. 144.

"∑ⱼ₌ₙ₊₁∞ Div_{t+n}(1 + g_l)ʲ/(1 + r)ʲ = Div_{t+n}(1 + g_l)/(r − g_l) = Div_{t+n+1}/(r − g_l)"

The sum on the left is discounted to time 0, while the Gordon value on the right is located at time $n$. They cannot be equated without moving one side to the other's valuation date.

Correct reading: If the left side is a time-0 present value, then $\sum_{j=n+1}^{\infty}Div_{t+n}(1+g_l)^{j-n}/(1+r)^j=Div_{t+n+1}/[(r-g_l)(1+r)^n]$. Alternatively, express the left-hand sum at time $n$, using discount exponent $j-n$.

The printed equality can cause candidates to add a terminal value at the wrong date.

7. The Shipline Terminal Value Is Assigned to Year 4 Instead of Year 3

Curriculum location: Time Value of Money in Fixed Income and Equity, Shipline two-stage dividend-growth case, Step 2, printed p. 146.

"E[S₄]/(1 + r)³, with E[S₄] = Div₄/(r − g_l)."

$Div_4$ is the first dividend after the three-year high-growth period. Therefore, $Div_4/(r-g_l)$ is the terminal value at the end of Year 3, consistent with discounting it by three periods.

Correct reading: Label the terminal value $E[S_3]=Div_4/(r-g_l)$, then discount $E[S_3]$ by $(1+r)^3$.

Candidates otherwise see a one-year mismatch between the terminal-value label and its discounting date.

8. Four Worked Solutions Cite the Wrong Equations

Curriculum location: Time Value of Money in Fixed Income and Equity, Question Set 1, Mylandia Question 4 solution, printed p. 148.

"Third, we calculate the sum of the present values of these expected dividends using Equation 16:"

Curriculum location: Implied Return and Growth, Question Set 2, Swiss zero-coupon bond solution, printed p. 160.

"Using Equation 18,"

Curriculum location: Cash Flow Additivity, Forward Interest Rate Changes case, Step 1, printed p. 166.

"Using Equation 18, we solve for each market discount rate, r."

Curriculum location: Cash Flow Additivity, Forward Interest Rate Changes case, Step 2, printed p. 167.

"Using these market implied discount rates, we can now solve for the respective forward rates, F₁,₂. Doing so requires rearranging Equation 25:"

The displayed calculations do not match the cited equation numbers. The Mylandia calculation uses the complete two-stage model, the Swiss and forward-rate Step 1 calculations use the implied-return formula, and forward-rate Step 2 rearranges the equality following the forward-rate definition.

Correct reading: Cite Equation 18 for Mylandia, Equation 19 for the Swiss zero-coupon solution, Equation 19 for forward-rate Step 1, and Equation 26 for forward-rate Step 2.

Wrong cross-references send candidates to unrelated formulas even when the worked arithmetic is otherwise usable.

9. A Forward-Rate Inequality Is Given the Wrong Condition

Curriculum location: Cash Flow Additivity, Forward Interest Rate Changes case, Exhibit 16 discussion, printed p. 167.

"Note that in a rising rate environment, F₁,₂ > r₂, as shown in Exhibit 16 comparing r₁, r₂, and F₁,₂ on 31 May (lower rates) to 15 June (higher rates)."

Whether $F_{1,2}$ exceeds $r_2$ depends on the cross-sectional slope of the spot curve at a given date. A general increase in rates from one date to another does not establish that inequality.

Correct reading: For an upward-sloping spot curve with $r_2>r_1$, $F_{1,2}>r_2$. Exhibit 16 has that shape on both dates; a rise in the whole curve over time is not the operative condition.

Candidates need to separate a change in rate levels over time from the shape of the term structure at one date.

10. The FX Forward Amount and Quotation Label Are Both Wrong

Curriculum location: Cash Flow Additivity, FX Forward Rates in a Changing Interest Rate Environment case, 15 June foreign strategy, printed p. 169.

"returns USD1,294.27 = 1,260.20e⁰·⁰²⁶⁶⁷ in one year."

Curriculum location: Cash Flow Additivity, FX Forward Rates in a Changing Interest Rate Environment case, 15 June forward quote label, printed p. 169.

"The no-arbitrage USD/GBP forward rate as of 15 June is"

The exponential future value is misrounded, and the case's USD-per-GBP quotation is labeled in the reverse order.

Correct reading: $1{,}260.20e^{0.02667}=1{,}294.261728$, which rounds to USD1,294.26. Dividing by GBP1,015.74 gives approximately USD1.2742 per GBP, so the quotation label is GBP/USD.

The printed amount and label can propagate inconsistent inputs and an inverted currency convention.

11. Covered Interest Parity Is Presented as an Exchange-Rate Expectation

Curriculum location: Cash Flow Additivity, FX Forward Rates in a Changing Interest Rate Environment case, interpretation, printed p. 169.

"Stated differently, the expectation for US dollar depreciation on a forward basis versus the British pound would require a higher US dollar interest rate to attract investors to US dollars versus British pounds."

The case derives the forward rate through a fully hedged no-arbitrage replication. That covered-interest-parity relation prices the forward; it does not, without an additional assumption, forecast the future spot rate.

Correct reading: Under covered interest parity, a higher USD interest rate relative to the GBP interest rate makes the US dollar trade at a larger forward discount against GBP. The forward discount is not by itself an expectation of future spot depreciation.

Candidates should not use covered and uncovered interest parity as interchangeable claims.

12. The Real-Option Exercise Cost Is Counted Twice

Curriculum location: Cash Flow Additivity, Cash Flow Additivity case, real-option calculation, Outcome 2 label, printed p. 175.

"Outcome 2: The present value of the option:"

Curriculum location: Cash Flow Additivity, Cash Flow Additivity case, real-option calculation, discounted exercise-cost formula, printed p. 175.

"MYR0.5 million/1.06¹⁰ = MYR0.5 million/1.7908 = MYR0.28 million."

Curriculum location: Cash Flow Additivity, Cash Flow Additivity case, real-option calculation, conclusion lead-in, printed p. 175.

"Thus, the value of the real option is"

Curriculum location: Cash Flow Additivity, Cash Flow Additivity case, real-option calculation, final subtraction, printed p. 175.

"MYR3.15 million − MYR0.28 million = MYR2.87 million."

Curriculum location: Cash Flow Additivity, Cash Flow Additivity case, real-option calculation, follow-through sentence, printed p. 176.

"Should the company exercise the investment option in Year 10, the incremental value of the project, expressed as its net present value, would increase by MYR2.87 million."

MYR0.28 million is the discounted Year-10 investment cost, not the option value. The stated NPV with investment already includes that cost, so subtracting it again double-counts it; the next page then repeats the wrong MYR2.87 million result.

Correct reading: With the stated cash flows, the NPV without the investment is approximately $-$MYR2.12 million and the NPV with the investment, including the Year-10 MYR0.5 million cost, is approximately MYR1.03 million. The incremental real-option value is therefore $1.03-(-2.12)=$ MYR3.15 million; no further subtraction is required. The follow-through sentence on printed p. 176 should also state MYR3.15 million.

The printed calculation and its follow-through sentence understate the real-option value by MYR0.28 million.

13. A Protective-Put Explanation Names a Call and Reverses the Hedge Direction

Curriculum location: Cash Flow Additivity, Question Set 3, Question 4, Response C, printed p. 178.

"because the increase in stock value is offset by the declining value of the call option position."

The question specifies a purchased put. In the down state, the stock loses value while the put gains value; the explanation names the wrong option and describes the offset in the wrong direction.

Correct reading: In the down state, the increasing value of the purchased put offsets the decline in the stock position.

Candidates should recognize the payoff direction of a protective put even though the keyed response remains unchanged.

Complete Module 4 Body-Text Errata Index

Errata scope

• Curriculum: CFA Level I 2027 Curriculum
• Volume: Volume 1, Quantitative Methods
• Module: Module 4, The Time Value of Money in Finance
• Topics covered: Time Value of Money in Fixed Income and Equity; Implied Return and Growth; Cash Flow Additivity
• Source reviewed: The official 66-page Module 4 curriculum PDF, with teaching content on printed pp. 127–178

The table below lists all confirmed source-authored errors identified in the three official teaching topics. Embedded topic Question Sets through printed p. 178 are included. End-of-module Practice Questions beginning on printed p. 179 are excluded.

Page references use the printed curriculum page numbers.

References

• CFA Institute: Equity Valuation—Concepts and Basic Tools
• CFA Institute: Discounted Dividend Valuation
• CFA Institute: The Term Structure of Interest Rates—Spot, Par, and Forward Curves
• CFA Institute: Pricing and Valuation of Forward Commitments
• CFA Institute: Currency Exchange Rates—Understanding Equilibrium Value
• CFA Institute: Capital Investments and Capital Allocation
• Microsoft Support: PMT Function

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 4 The Time Value of Money in Finance. It is not an official curriculum errata notice.