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

Future Value and Present Value of a Single Cash Flow

Updated 7 October 2026 · Fact-checked

Future value compounds a lump sum forward: FV = PV(1 + r)^N. Present value discounts it back: PV = FV ÷ (1 + r)^N. Match the rate to the period: with m periods a year, use r ÷ m and N × m. To find rate or periods, rearrange or use the TVM keys.

Understand Future Value and Present Value of a Single Cash Flow

Money today is worth more than the same amount later, because today's money can earn interest. This is the time value of money. Everything in this topic follows from that one idea.

Future value (FV) is what a lump sum grows to if it earns interest and the interest is reinvested. This is compounding: you earn interest on the original amount and on the interest already earned. That is why growth is faster than simple interest.

Present value (PV) is the reverse. It is today's worth of a single amount you will receive later. You discount the future amount by dividing by the same growth factor. PV and FV are the same relationship read in two directions.

The rate must match the period. If interest is added monthly, the rate per period is the annual rate ÷ 12 and the number of periods is years × 12. The stated annual rate is not the same as the effective annual return once you compound more than once a year.

There are four variables: PV, FV, the rate per period (r) and the number of periods (N). Give the exam any three and you can find the fourth. Most questions are just a choice of which one is missing.

Key formulas to remember

Future value of a lump sum: FV = PV × (1 + r)^N; r is the rate per period and N is the number of periods. (1 + r)^N is the future value factor.
Present value of a lump sum: PV = FV ÷ (1 + r)^N = FV × (1 + r)^−N; Discounting is the inverse of compounding. A higher r or longer N gives a lower PV.
Non-annual compounding: FV = PV × (1 + rs ÷ m)^(m × N); rs is the stated annual rate, m is compounding periods per year, N is years. Use rs ÷ m per period and m × N periods.
Solving for the rate: r = (FV ÷ PV)^(1 ÷ N) − 1; N is the number of compounding periods. The result is the rate per period. Multiply by m to restate it as a stated annual rate if periods are shorter than a year. This is not the effective annual rate.
Solving for the number of periods: N = ln(FV ÷ PV) ÷ ln(1 + r); N is in periods, not necessarily years. ln is the natural logarithm.
Continuous compounding: FV = PV × e^(rs × N); Used when compounding is continuous. e is about 2.71828. PV = FV × e^(−rs × N).

How to solve Future Value and Present Value of a Single Cash Flow questions

Use this method for any lump-sum question. The main risk is mismatching the rate and the period.

1Identify the three known variables among PV, FV, r and N, and mark which one is missing.
2Check the compounding frequency. Find m, the number of compounding periods per year.
3Convert to per-period terms: rate per period = stated annual rate ÷ m, and N = years × m.
4Write the formula for the missing variable. Compounding forward needs FV; moving back needs PV.
5Calculate with the TVM keys or a formula. Keep full precision until the final step.
6Run a sense check. FV must be above PV for a positive rate, and PV must be below FV.
7If the answer needs converting, restate it per year, for example multiply the periodic rate by m, or divide N by m to get years.

Quickest way: TVM keys on the calculator

When to use it: Use for any single-cash-flow question with a whole or fractional number of periods, especially when you must solve for r or N.

BA II Plus: clear old data with 2ND CLR TVM. Note that 2ND CLR TVM does not reset P/Y, so check it separately: press 2ND P/Y, set P/Y to 1, press ENTER, then 2ND QUIT. Then work with per-period inputs.
Enter known values with the keys N, I/Y, PV, PMT, FV. Enter I/Y as a percent per period (for example 1.5, not 0.015). Set PMT = 0 for a lump sum.
Use opposite signs for PV and FV. Money paid out is negative and money received is positive. Same signs usually give an error.
Press CPT then the key of the missing variable, for example CPT FV or CPT N.
HP 12C: clear with f CLEAR FIN. Enter n, i, PV, then press FV (or the missing key). If n is fractional, the HP 12C uses simple interest for the fractional period unless the C flag is displayed, so the result can differ from the compounding formula. Set the C flag with STO EEX, or use the formula with the y^x key. Also, when the HP 12C solves for n, it rounds the result up to the next whole number, so for a fractional answer use the formula with ln instead.
With the BA II Plus, I/Y is per period when P/Y = 1. For monthly compounding, enter annual rate ÷ 12 and N in months.

Common mistakes in Future Value and Present Value of a Single Cash Flow

Using the annual rate with a periodic number of periods, such as 6% with 12 quarters.
The compounding frequency in the stem is skipped, or only one of r and N is converted.
Fix: Always convert both. Rate per period = annual rate ÷ m and periods = years × m. Write both on your scratch paper first.
Getting an error or a wrong sign on the calculator because PV and FV have the same sign.
The TVM solver treats cash outflow and inflow as opposite signs. Entering both as positive breaks the equation.
Fix: Enter PV as negative if you invest it, and FV as positive. If the answer sign looks odd, it is usually a sign convention, not a wrong result.
Entering the rate as a decimal, for example 0.05 instead of 5.
Formulas use decimals, but the I/Y key expects a percent.
Fix: On the calculator key in 5. In a hand formula use 0.05. Check the result is reasonable.
Using simple interest, PV × (1 + r × N), instead of compounding.
It feels easier, and one distractor option is often built from it.
Fix: Unless the question says simple interest, use (1 + r)^N. Simple interest gives a lower FV, so it is often a trap option.
Rounding the factor early, such as 1.1956 to 1.20, before multiplying.
Trying to save time on a large PV.
Fix: Keep the calculator's full value and round only at the end. Small rounding in a factor can move the answer into another option.
Leaving the answer in periods when the question asks for years, or in a periodic rate when it asks for an annual rate.
The solved variable is per period, and the stem asks for another unit.
Fix: Reread what is asked. Divide N by m for years, and multiply the periodic rate by m for a stated annual rate.

Worked examples

Example 1

An investor deposits €20,000 in an account paying a stated annual rate of 6%, compounded quarterly. What is the value after 3 years? A. €23,600 B. €23,820 C. €23,912

Show the solution

Known: PV = 20,000, stated annual rate = 6%, m = 4, years = 3. Missing: FV.
Rate per period = 6% ÷ 4 = 1.5%. Number of periods = 4 × 3 = 12.
Formula: FV = 20,000 × (1.015)^12.
(1.015)^12 = 1.195618, so FV = 20,000 × 1.195618 = 23,912.36.
Calculator: N = 12, I/Y = 1.5, PV = −20,000, PMT = 0, CPT FV = 23,912.36.
Eliminate the others. €23,600 is simple interest (20,000 × 1.18). €23,820 uses annual compounding (20,000 × 1.06^3), which ignores the quarterly compounding.

Answer: C. €23,912 (about €23,912.36).

Example 2

At an annual rate of 5%, compounded annually, how many years will it take for $50,000 to grow to $80,000? A. 9.6 years B. 12.0 years C. 16.0 years

Show the solution

Known: PV = 50,000, FV = 80,000, r = 5% per year. Missing: N.
FV ÷ PV = 80,000 ÷ 50,000 = 1.6.
N = ln(1.6) ÷ ln(1.05) = 0.470004 ÷ 0.048790 = 9.633 years.
Calculator: I/Y = 5, PV = −50,000, FV = 80,000, PMT = 0, CPT N = 9.63.
Eliminate the others. 12.0 years comes from simple interest: 60% growth ÷ 5% per year. With compounding, interest earns interest, so it takes less than 12 years. 16.0 years comes from 80 ÷ 5, which has no basis in the formula.

Answer: A. 9.6 years (about 9.63).

Exam tips

Scan the stem for words such as monthly, quarterly or semiannual. They signal that the rate and the periods both need converting.
Estimate first. With a positive rate, FV must exceed PV, and the answer should be larger than a simple-interest estimate. This eliminates one or two options in seconds.
Distractors are usually built from common errors: simple interest, annual compounding instead of the stated frequency, or an unconverted rate. Work out which error produced each wrong option.
For 'solve for N' or 'solve for r', go straight to the TVM keys. On the HP 12C, remember that when it solves for n it rounds up to the next whole number, so use the ln formula if you need the fractional answer. If you enter a fractional n on the HP 12C, it uses simple interest for the fractional period unless the C flag is set (STO EEX), so use the y^x formula or set the C flag.
You have about 90 seconds per question. Clear the TVM registers every time so old values do not carry into the next question.

Practice questions from The Time Value of Money in Finance

Future Value and Present Value of a Single Cash Flow in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Future Value and Present Value of a Single Cash Flow: frequently asked questions

What is the difference between future value and present value?

Future value is what a sum today grows to after earning interest. Present value is what a future sum is worth today after discounting. They use the same factor, (1 + r)^N, once multiplying and once dividing.

How do I solve for the number of periods in a TVM question?

Use N = ln(FV ÷ PV) ÷ ln(1 + r). On the BA II Plus, enter I/Y, PV and FV with opposite signs, set PMT to 0 and press CPT N. Remember N is in compounding periods, so divide by m to get years.

How do I find the interest rate when PV, FV and N are known?

Use r = (FV ÷ PV)^(1 ÷ N) − 1, or enter N, PV and FV and press CPT I/Y. The result is the rate per period. If the periods are shorter than a year, multiply by m for a stated annual rate.

Why does my BA II Plus show an error when I compute N or I/Y?

The most common cause is that PV and FV have the same sign. Make one negative and the other positive. Also check that PMT is 0 and that old values have been cleared with 2ND CLR TVM.