CFA Level I Exam · The Time Value of Money in Finance

Implied Rates, Growth Rates and Cash Flow Additivity

Updated 7 October 2026 · Fact-checked

Implied rates and growth rates use the same TVM equation: FV = PV(1 + r)^N. Fill in the three known values and solve for the rate, r = (FV ÷ PV)^(1/N) − 1. To compare cash flows at different dates, move each to one common date at the same rate, then add them. That is cash flow additivity.

Understand Applications: Implied Rates, Growth Rates and Equivalence

Every time value of money problem is one equation with four moving parts: present value, future value, the rate per period, and the number of periods. You normally solve for FV or PV. But you can also solve for the rate or for the number of periods. When you solve for the rate, the answer is called an implied rate. It is the return that makes today's amount grow into the later amount.

A compound growth rate is the same calculation. Replace PV with the starting value (sales, earnings, dividends, a fund balance) and FV with the ending value. The rate you find is the constant growth per period that would take you from start to end. It is a geometric idea, not a simple average of yearly changes.

Money at different dates cannot be compared or added directly. ₹ or $ received in year 1 and the same amount received in year 5 are different things. Equivalence means two cash flows, or two sets of cash flows, have the same value once you move them to the same date using the same rate. You move forward by multiplying by (1 + r) for each period, and backward by dividing.

The cash flow additivity principle says that amounts of money indexed to the same date can be added. So the value of a stream of cash flows at any date is the sum of each cash flow's value at that date. This is why you can break a messy stream into simple pieces, value each piece, and add. It also lets you check that two different payment plans are equivalent: compare their totals at one date.

One warning. The rate and the period must match. If cash flows are quarterly, N counts quarters and r is the quarterly rate. Convert to an annual figure only at the end, and only in the way the question asks.

Key formulas to remember

Future value of a single sum: FV = PV × (1 + r)^N; The base equation. Every other formula here is a rearrangement of it.
Implied rate of return or growth rate: r = (FV ÷ PV)^(1/N) − 1; N is the number of compounding periods between the two values, not the number of data points. Use the starting value as PV and the ending value as FV.
Implied number of periods: N = ln(FV ÷ PV) ÷ ln(1 + r); Use this to find how long it takes an amount to reach a target. On the calculator, enter I/Y, PV and FV and compute N.
Moving a cash flow to another date: Value at date t = CF at date s × (1 + r)^(t − s); If t is later than s, you compound forward. If t is earlier, the exponent is negative, which means you discount.
Cash flow additivity: Value at date t of a stream = Σ [CFₛ × (1 + r)^(t − s)]; Only cash flows valued at the same date, at the same rate, can be added.
Equivalence of two payment plans: Plan A is equivalent to Plan B if value of A at date t = value of B at date t; Pick any single date. The comparison gives the same answer at every date if the rate is constant.
Rule of 72 (approximation): Years to double ≈ 72 ÷ rate in percent; A rough check for doubling-time questions only. It works best for rates around 6% to 10%. Do not use it where the question asks for an exact answer.

How to solve Applications: Implied Rates, Growth Rates and Equivalence questions

Use this method for any implied rate, growth rate, equivalence or additivity question.

1Draw a quick timeline. Mark each cash flow and the date it occurs, with time 0 as today.
2Identify what is unknown: the rate, the number of periods, a value at one date, or a single equivalent amount.
3Match the period and the rate. If flows are semiannual or quarterly, set N in those periods and use the rate per period.
4For an implied rate or growth rate, set PV as the earlier value and FV as the later value. Count N as the number of intervals between them. Five yearly figures from 2020 to 2024 give N = 4.
5Solve with r = (FV ÷ PV)^(1/N) − 1, or use the calculator. Convert to a percent and keep at least four decimals until the end.
6For equivalence or additivity, choose one comparison date, usually time 0 or the last cash flow. Move every cash flow to that date at the same rate.
7Add the moved values. Compare totals, or solve for the missing amount.
8Check the answer for reasonableness. The compound growth rate lies between the lowest and highest single-period growth rates over the interval. When all cash flows are positive and r is positive, a present value must be less than the sum of the undiscounted flows.

Quickest way: TI BA II Plus shortcut for implied rates and growth rates

When to use it: Use it when the question gives a start value, an end value and a number of periods, and options are rounded to one or two decimals.

Clear the TVM registers: press 2ND, then CLR TVM.
Enter N, for example 5.
Enter the earlier value in PV as a negative number, for example 120 then +/- then PV. The sign matters: one of PV and FV must be negative.
Enter the later value in FV as a positive number, for example 198 then FV.
Press CPT then I/Y. The result is the rate per period in percent, here about 10.53.
On the HP 12C: 5 n, 120 CHS PV, 198 FV, then i gives the rate per period.
To find a missing N instead, enter I/Y, PV and FV and press CPT then N.
Use the result to eliminate options. Then do a quick sanity check by compounding one option forward and comparing with the end value.

Common mistakes in Applications: Implied Rates, Growth Rates and Equivalence

Using the number of data points as N when finding a growth rate.
A table of six yearly sales figures looks like six periods.
Fix: Count the gaps between the first and last figure. Six yearly figures have five intervals, so N = 5.
Adding cash flows that occur at different dates.
The amounts look like plain numbers, so it feels natural to sum them.
Fix: Move every cash flow to one common date first. Only then apply additivity.
Getting a zero or error from the calculator because PV and FV have the same sign.
The TVM keys treat money paid out and money received as opposite signs.
Fix: Enter one of PV and FV as negative. If the calculator shows an error, check the signs first.
Mixing periodic and annual rates.
The problem gives an annual rate but flows are quarterly, or N is in years while the rate is monthly.
Fix: Put N and r in the same time unit. Divide the annual rate by the number of periods per year only when the question states that compounding convention.
Finding growth by averaging the yearly percentage changes.
The arithmetic mean is the more familiar average.
Fix: Use (end ÷ start)^(1/N) − 1. The arithmetic average of yearly changes overstates compound growth when returns vary.
Rounding the exponent or the ratio too early.
Numerical options sit close together, so small rounding errors push you to the wrong option.
Fix: Keep full calculator precision through the calculation. Round only the final answer.

Worked examples

Example 1

A company's revenue was $120 million in 2020 and $198 million in 2025. What is the compound annual growth rate over the period? A) 8.5% B) 10.5% C) 12.5%

Show the solution

Timeline: the start value is at 2020 and the end value is at 2025. There are 5 yearly intervals, so N = 5.
Set PV = 120 and FV = 198 (in $ millions).
Apply r = (FV ÷ PV)^(1/N) − 1 = (198 ÷ 120)^(1/5) − 1 = 1.65^0.2 − 1.
Compute: ln(1.65) = 0.50078. Divide by 5 to get 0.10016. The exponential of 0.10016 is 1.1053.
So r = 1.1053 − 1 = 0.1053, or about 10.5%.
Check: 1.105^5 ≈ 1.6474, close to 1.65. Calculator: 5 N, 120 +/- PV, 198 FV, CPT I/Y gives 10.53.

Answer: B) 10.5%

Example 2

An investor will receive €5,000 at the end of year 2 and €8,000 at the end of year 4. The discount rate is 6% per year. What single payment today is equivalent to these two cash flows? A) €9,860 B) €10,787 C) €11,300

Show the solution

Timeline: €5,000 at t = 2 and €8,000 at t = 4. Compare at t = 0, so discount each flow back to today.
PV of the first flow = 5,000 ÷ 1.06² = 5,000 ÷ 1.1236 = €4,449.98.
PV of the second flow = 8,000 ÷ 1.06⁴ = 8,000 ÷ 1.262477 = €6,336.75.
By cash flow additivity, add the two present values: 4,449.98 + 6,336.75 = €10,786.73, about €10,787.
Check at t = 4: 5,000 × 1.1236 = 5,618, plus 8,000 = 13,618. Discounting 13,618 by 1.262477 gives €10,787, so the two dates agree.

Answer: B) €10,787

Exam tips

Read the question for what is being solved. A growth rate question gives two values and a time gap. A trap option often comes from using the number of data points as N.
Check signs before you press CPT. One of PV and FV must be negative, otherwise the rate computation fails.
Expect equivalence questions to have a single trap: comparing cash flows at different dates. Move everything to one date, and choose the date that needs the least arithmetic.
To check a growth rate answer, compound the start value forward at one option's rate for N periods and see whether you reach the end value. This can often rule out options that are far apart. Use the Rule of 72 only for doubling-time questions. With no penalty for wrong answers, never leave a question blank.
Keep the period and the rate in the same unit. If the question mentions semiannual or quarterly flows, convert N first.

Practice questions from The Time Value of Money in Finance

Applications: Implied Rates, Growth Rates and Equivalence: frequently asked questions

What is the cash flow additivity principle in CFA Level I?

It says that cash flows indexed to the same date can be added together. So you can value each cash flow separately at a chosen date, using the same rate, and sum the results to get the value of the whole stream. It is the reason you can value an uneven stream piece by piece.

How do I calculate compound growth rate using TVM?

Treat the starting value as PV and the ending value as FV. Enter N as the number of intervals between them, then compute the rate with CPT I/Y, or use (FV ÷ PV)^(1/N) − 1. Remember that one of PV and FV must be entered as negative on the calculator.

What is an implied rate of return?

It is the rate that makes a present amount grow to a stated future amount over a stated number of periods. You find it by solving the TVM equation for r. It is the same calculation as a compound growth rate, just applied to an investment.

Why is N one less than the number of years listed in a table?

Growth happens between observations. Six yearly figures, such as 2019 to 2024, contain five yearly intervals. Compounding applies once per interval, so N = 5.