The core idea
Money has a time value for two reasons. First, money can earn interest, so cash in hand can grow. Second, inflation and risk make future money worth less than the same amount today. Time value of money (TVM) turns that into arithmetic: it lets you move any amount of money to a different point in time, so cash flows that happen at different times can be compared fairly.
Moving money forward in time is compounding. Moving it back is discounting. The interest rate does the work in both directions. Before you calculate anything, draw a timeline.
The timeline habit
Time: 0 1 2 3 Cash: -1,000 300 300 300
Mark time 0 as today, mark each cash flow with its sign (money out is negative, money in is positive) and write the rate per period. Most mistakes come from skipping this step.
The formulas you need
| What you want | Formula | Variables |
|---|---|---|
| Future value of a single sum | FV = PV x (1 + r)^n | PV present value, r rate per period, n number of periods |
| Present value of a single sum | PV = FV / (1 + r)^n | Same variables |
| Present value of an ordinary annuity | PV = PMT x [1 - (1 + r)^-n] / r | PMT equal payment at the end of each period |
| Future value of an ordinary annuity | FV = PMT x [(1 + r)^n - 1] / r | Same variables |
| Annuity due (payments at the start) | Multiply the ordinary annuity result by (1 + r) | Rent and insurance are common examples |
| Perpetuity | PV = PMT / r | Payments forever, with no end date |
| Growing perpetuity | PV = PMT1 / (r - g) | g is the constant growth rate and must be less than r |
Two rules prevent errors. The rate and the period must match: use a monthly rate with months and an annual rate with years. And an ordinary annuity pays at the end of each period, while an annuity due pays at the start.
Worked examples: single sums
Future value
You invest $5,000 for 8 years at 6 percent a year. FV = 5,000 x 1.06^8 = 5,000 x 1.5938 = $7,969.24.
Present value
You will receive $20,000 in 5 years. At an 8 percent discount rate it is worth today 20,000 / 1.08^5 = 20,000 / 1.4693 = $13,611.66.
Solving for time and rate
Time: how long for $10,000 to grow to $15,000 at 5 percent? n = ln(1.5) / ln(1.05) = 0.4055 / 0.0488 = 8.31 years.
Rate: what annual return turns $8,000 into $12,000 in 6 years? r = (12,000 / 8,000)^(1/6) - 1 = 1.5^0.1667 - 1 = about 7.0 percent.
A handy shortcut for checking: the rule of 72 says money doubles in about 72 divided by the rate in percent years. At 8 percent, that is 9 years, and 1.08^9 is indeed about 2.0.
Worked examples: annuities
Present value of an annuity
An investment pays $1,000 at the end of each year for 5 years. At 7 percent, PV = 1,000 x [1 - 1.07^-5] / 0.07 = 1,000 x (1 - 0.71299) / 0.07 = 1,000 x 4.1002 = $4,100.20.
If the payments came at the start of each year (an annuity due), multiply by 1.07 to get $4,387.21.
Future value of an annuity
You save $2,000 at the end of each year for 10 years at 5 percent. FV = 2,000 x [1.05^10 - 1] / 0.05 = 2,000 x (0.62889 / 0.05) = 2,000 x 12.5779 = $25,155.79.
A perpetuity
A preferred share pays $4 a year forever. At a required return of 8 percent, its value is 4 / 0.08 = $50. If the dividend is expected to grow at 3 percent a year, starting with $4 next year, value is 4 / (0.08 - 0.03) = $80.
Compounding frequency: APR and effective rates
Interest is often compounded more often than yearly. The quoted annual rate (APR) is then divided by the number of periods, and the effective annual rate (EAR) is higher than the APR.
EAR = (1 + APR / m)^m - 1, where m is the number of compounding periods a year.
| Quoted rate | Compounding | Periods (m) | Effective annual rate |
|---|---|---|---|
| 12 percent | Yearly | 1 | 12.00 percent |
| 12 percent | Half-yearly | 2 | 12.36 percent |
| 12 percent | Quarterly | 4 | 12.55 percent |
| 12 percent | Monthly | 12 | 12.68 percent |
| 12 percent | Daily | 365 | 12.75 percent |
For monthly compounding, check the arithmetic: 1.01^12 is 1.126825, so the EAR is 12.68 percent. When a problem gives a nominal rate and a compounding frequency, convert to a rate per period and use the number of periods, not years. For example, $1,000 for 3 years at 12 percent compounded monthly is 1,000 x 1.01^36 = $1,430.77.
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Get an instant quoteLoan payments and amortization
A loan with equal payments is the present value of an annuity. To find the payment, rearrange the formula: PMT = PV x r / [1 - (1 + r)^-n].
A car loan (hypothetical)
You borrow $20,000 at 6 percent APR, with monthly payments over 4 years. The monthly rate is 0.5 percent and there are 48 payments. PMT = 20,000 x 0.005 / [1 - 1.005^-48] = 100 / 0.2129 = $469.70.
Total paid is 469.70 x 48 = $22,545.60, so total interest is $2,545.60.
An amortization table shows how each payment splits between interest and principal. Interest in a given month is the opening balance times the monthly rate. The rest of the payment repays principal.
| Month | Opening balance | Payment | Interest | Principal | Closing balance |
|---|---|---|---|---|---|
| 1 | 20,000.00 | 469.70 | 100.00 | 369.70 | 19,630.30 |
| 2 | 19,630.30 | 469.70 | 98.15 | 371.55 | 19,258.75 |
| 3 | 19,258.75 | 469.70 | 96.29 | 373.41 | 18,885.34 |
Early payments are mostly interest, and the principal share rises over time. This is a favorite exam point.
Practice problems with solutions
Problem 1: future value
You invest $12,000 for 5 years at 7 percent compounded annually. What is it worth at the end?
Solution. FV = 12,000 x 1.07^5 = 12,000 x 1.40255 = $16,830.62.
Problem 2: present value of quarterly payments
A contract pays $800 at the end of each quarter for 3 years. The discount rate is 8 percent a year, compounded quarterly. What is it worth today?
Solution. The quarterly rate is 8 / 4 = 2 percent and there are 12 payments. PV = 800 x [1 - 1.02^-12] / 0.02 = 800 x 10.5753 = $8,460.27.
Problem 3: a mortgage
You borrow $150,000 at 6 percent a year for 30 years with monthly payments. Find the payment and the total interest.
Solution. The monthly rate is 0.5 percent and there are 360 payments. PMT = 150,000 x 0.005 / [1 - 1.005^-360] = 750 / 0.83396 = $899.33. Total paid is 899.33 x 360 = $323,758.80, so total interest is about $173,759, more than the amount borrowed. This is why a shorter term or an extra payment matters so much.
Check your answers in Excel
Spreadsheet functions give you a fast check. They use a sign convention: money you pay out is negative and money you receive is positive, so you often need a minus sign.
| Task | Excel formula | Result for the examples |
|---|---|---|
| Future value | =FV(0.06,8,0,-5000) | 7,969.24 |
| Present value | =PV(0.08,5,0,-20000) | 13,611.66 |
| Annuity present value | =PV(0.07,5,-1000) | 4,100.20 |
| Annuity future value | =FV(0.05,10,-2000) | 25,155.79 |
| Loan payment | =PMT(0.06/12,48,20000) | -469.70 (shown negative because you pay it) |
| Number of periods | =NPER(0.05,0,-10000,15000) | 8.31 |
| Rate | =RATE(6,0,-8000,12000) | 6.99 percent |
| Effective annual rate | =EFFECT(0.12,12) | 12.68 percent |
If you set the optional type argument to 1, Excel treats payments as annuity due. See our guide to Excel formulas for business students for more on setting up models with clear inputs.
Common mistakes
- Mismatched rate and period A monthly payment needs a monthly rate and a number of months. Divide the annual rate by 12 and multiply years by 12.
- Mixing ordinary and annuity due Check whether payments are at the start or end of each period.
- Sign errors Keep outflows negative and inflows positive, especially in Excel.
- Forgetting to draw a timeline A diagram exposes timing errors before you calculate.
- Rounding the rate early Keep full precision until the final step, then round the answer.
- Using the wrong formula for growth A growing perpetuity needs r greater than g and uses next year's payment.
- Ignoring inflation when asked about real returns Adjust the nominal rate if the problem asks for a real rate.
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