Time value of money: practice problems with step-by-step solutions

Also called: compound interest · present value · future value

Free Time value of money practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: future value, compounding and discounting, annuities and loans.

Free practice problemEasy
years012n = ?€2 500€7 500

You deposit €2 500 at 5% interest per year, compounded annually (interest is added at the end of each year). After how many whole years n does the balance first reach at least €7 500?

PV = €2 500FV = €7 500i = 5%
years

A whole number of years.

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What you practise

  1. Easyfuture value
  2. Mediumcompounding and discounting
  3. Hardannuities and loans

How to solve Time value of money problems

  1. Draw a timeline with every cash flow and its timing.
  2. Match the rate to the period: a monthly rate for monthly payments.
  3. Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
  4. Check the sign and the order of magnitude of the answer.

Common mistakes

  • Using the annual rate with monthly periods.
  • Using simple instead of compound interest.
  • Counting the number of periods wrong.

Key formulas

Future value
FV = PV·(1 + i)n
Present value
PV = FV/(1 + i)n
Several periods per year
i = r/m, N = m·n
Annuity, present value
PV = PMT·[1 − (1 + i)−n]/i
Annuity, future value
FV = PMT·[(1 + i)n − 1]/i

Learn Time value of money from scratch

1Why money grows over time

If you put €1000 in a savings account that pays 5% interest per year, you have €1050 after one year. The extra €50 is the interest.

In the second year you earn 5% on €1050, not on €1000. So the interest is €52.50 and the balance becomes €1102.50.

Earning interest on earlier interest is called compound interest. With simple interest you would only ever earn 5% of the first €1000.

0123€1000€1050€1102.50FV

Because money can grow like this, €1000 today is worth more than €1000 in three years. This is the time value of money.

Every savings plan, loan and mortgage is calculated with this idea.

Self-check You deposit €100 at 10% per year, compounded annually. How much is in the account after 2 years?
  1. €110
  2. €120
  3. €121
  4. €200

Answer: €121. Year 1: 100 · 1.10 = 110. Year 2: 110 · 1.10 = 121. €120 is simple interest: it forgets the interest earned on the first €10 of interest.

2Future value and present value

Each period the balance is multiplied by (1 + i), where i is the interest rate per period as a decimal. After n periods:

FV = PV · (1 + i)n

PV (present value) is the amount today. FV (future value) is the amount after n periods.

Going backwards is called discounting. It tells you how much you need today to reach a target later:

PV = FV / (1 + i)n

Example: to have €1102.50 in 2 years at 5%, you need 1102.50 / 1.05² = €1000 today.

Banks often quote a yearly nominal rate r but add interest m times a year: m = 4 for quarterly, m = 12 for monthly. Then you work per period:

i = r / m N = m · n
Self-check A rate of 6% per year is compounded monthly for 3 years. What are i and N?
  1. i = 0.005 and N = 36
  2. i = 0.005 and N = 3
  3. i = 0.06 and N = 36
  4. i = 0.5 and N = 36

Answer: i = 0.005 and N = 36. i = 0.06 / 12 = 0.005 per month and N = 12 · 3 = 36 months. Using the yearly 0.06 with 36 months is the most common mistake: it counts 6% every month.

3Annuities: equal payments

An annuity is a series of equal payments A at the end of every period, for example €250 saved each month or a monthly loan repayment.

0123…NAAAAPV

You could compound or discount every payment on its own and add them. The annuity formulas do this sum in one step.

What you have at the end if you save A every period:

FV = A · [(1 + i)N − 1] / i

What all the payments are worth today:

PV = A · [1 − (1 + i)−N] / i

A loan is paid back with an annuity: the amount borrowed P equals the present value of the payments. Solve that for the payment:

A = P · i / [1 − (1 + i)−N]

Example: €10 000 borrowed at 6% per year, compounded monthly, over 3 years. i = 0.005 and N = 36, so A = 10 000 · 0.005 / (1 − 0.8356) = €304.22 per month.

4The method step by step

  1. Draw a timeline. Mark what is known (an amount today, a target later, equal payments) and what you must find.
  2. Find the rate per period i = r/m and the number of periods N = m · n. Write the rate as a decimal.
  3. Choose the formula: one amount (FV or PV) or a series of equal payments (annuity).
  4. Calculate. Work out the power (1 + i)N first and keep at least 4 decimals.
  5. Check the size of the answer: a future value is larger than the deposit, and a present value is smaller than the target.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

You invest €2000 at a nominal rate of 4% per year, compounded quarterly. Find the value FV of the investment after 3 years.

1

Draw the timeline

012…12€2000FV = ?

One amount goes in today. You want its value after 3 years, which is 12 quarters.

2

Rate and number of periods

Interest is added 4 times a year, so work in quarters.

i = 0.04 / 4 = 0.01
N = 4 · 3 = 12
3

Growth factor

Each quarter the balance is multiplied by 1.01.

(1 + i)N = 1.0112 = 1.126825
4

Future value

FV = 2000 · 1.126825 = 2253.65
5

Check the answer

With yearly compounding you would get 2000 · 1.04³ = €2249.73. Quarterly compounding gives a little more, because interest starts earning interest sooner. €2253.65 fits.

Answer

FV ≈ €2253.65 after 3 years.

6Common mistakes

  • Using the yearly rate with monthly or quarterly periods. Divide the rate by m and multiply the years by m.
  • Using simple interest, PV · (1 + i · n), instead of compound interest, PV · (1 + i)n.
  • Counting the number of periods wrong, for example 3 instead of 36 for three years of monthly payments.
  • Writing the rate as a percentage: 1.412 instead of 1.0412.
  • For a savings plan, only adding up the deposits. Each deposit also earns interest.
Self-check You invest €1000 at 6% per year, compounded monthly, for 3 years. Which calculation is correct?
  1. 1000 · (1 + 0.06 · 3) = €1180
  2. 1000 · 1.00536 = €1196.68
  3. 1000 · 1.0636 = €8147.25
  4. 1000 · 1.063 = €1191.02

Answer: 1000 · 1.00536 = €1196.68. Monthly compounding: i = 0.06/12 = 0.005 and N = 36. Using 1.06 for 36 periods charges 6% every month. 1.06³ ignores the monthly compounding, and 1180 is simple interest.

7How exam questions look

  • Easy: future value. One deposit with yearly compounding: FV = PV · (1 + i)n.
  • Medium: compounding and discounting. Quarterly or monthly compounding, asking for a future value or the deposit needed today.
  • Hard: annuities and loans. The final value of a monthly savings plan, or the monthly payment on a loan.

Always say what the number means: “after 3 years you have €2253.65”.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
years012n = ?€1 000€2 500

You deposit €1 000 at 8% interest per year, compounded annually (interest is added at the end of each year). After how many whole years n does the balance first reach at least €2 500?

PV = €1 000FV = €2 500i = 8%

Find n (years)

Show hints
  1. Write FV = PV·(1 + i)ⁿ and solve for the exponent n.
  2. Take the natural log of both sides: n = ln(FV/PV)/ln(1 + i).
  3. FV/PV = 2.5, ln(2.5) = 0.9163.
Show the answer and the step-by-step solution

Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.

Set up the equation

1 000·(1 + 0.08)n = 2 500 → (1.08)n = 2.5

Solve with logarithms

n = ln(2.5)/ln(1.08) = 0.9163/0.07696 = 11.91

Round up to whole years

Interest is only added at the end of each year, so after 11 years the balance is still below the target.

1 000·1.0811 = €2 331.64 < €2 500
1 000·1.0812 = €2 518.17 ≥ €2 500
n = 12

Answer

n = 12 years
Example 2Medium
periods0123411 + EAR = ?22% / 4

A credit card charges a nominal interest rate of 22% per year, compounded quarterly. Find the effective annual rate EAR in percent.

r = 22%m = 4

Find EAR (%)

Show hints
  1. Find the rate per period r/m first.
  2. EAR = (1 + r/m)ᵐ − 1.
  3. r/m = 0.055.
Show the answer and the step-by-step solution

Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.

Rate per period

r/m = 0.22/4 = 0.055

Compound over one year

EAR = (1 + 0.055)4 − 1 = 1.239 − 1 = 23.88%

Answer

EAR = 23.88%

The EAR is always above the nominal rate when interest compounds more than once a year.

Example 3Hard
months01…96…240€250 000AB = ?A3% / 12

A mortgage of €250 000 is repaid with equal monthly payments over 20 years at 3% per year, compounded monthly. How much is still owed just after the payment at the end of year 8? Find the outstanding balance B.

P = €250 0003% / 12N = 240k = 96

Find B (€)

Show hints
  1. First find the monthly payment A with the annuity formula.
  2. The balance equals the present value of the payments that are still to come.
  3. A = €1 386.49, remaining payments: 144.
Show the answer and the step-by-step solution

Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.

Monthly payment

i = 0.0025, N = 240
A = 250 000·0.0025/(1 − (1 + 0.0025)−240) = €1 386.49

Present value of the remaining payments

Whatever is still owed must be exactly paid off by the remaining payments.

N − k = 240 − 96 = 144
B = 1386·(1 − (1 + 0.0025)−144)/0.0025 = €167 494.12

Answer

B = €167 494.12

Questions and answers

What is Time value of money?

If you put €1000 in a savings account that pays 5% interest per year, you have €1050 after one year. The extra €50 is the interest.

In the second year you earn 5% on €1050, not on €1000. So the interest is €52.50 and the balance becomes €1102.50.

How do I solve Time value of money problems step by step?
  1. Draw a timeline with every cash flow and its timing.
  2. Match the rate to the period: a monthly rate for monthly payments.
  3. Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
  4. Check the sign and the order of magnitude of the answer.
What are common mistakes in Time value of money?
  • Using the annual rate with monthly periods.
  • Using simple instead of compound interest.
  • Counting the number of periods wrong.
What do the Time value of money exercises cover?
  • Easy: future value
  • Medium: compounding and discounting
  • Hard: annuities and loans

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

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Printable worksheet

Time value of money worksheet with answers
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