Time value of money explained: present value, future value and annuities
Money you have today is worth more than the same amount promised later. You can invest it and earn interest, and a promise carries the risk it will not be kept. Time value of money is the arithmetic that puts a price on that gap. Almost everything else in finance, from bond pricing to valuing a company, is this idea applied to a stream of cash flows.
The one equation behind every problem
Every time-value problem has the same five ingredients: the number of periods (N), the interest rate per period (I/Y), a present value (PV), a payment each period (PMT) and a future value (FV). Four of them are known and one is not. Given a periodic rate r, they are linked by a single equation:
PV + PMT × [1 − (1 + r)−N] / r + FV × (1 + r)−N = 0
Cash you pay out is negative and cash you receive is positive, so the equation balances what goes in against what comes out. The time value of money calculator solves for whichever one you leave empty.
Future value: compounding forward
Put 1,000 in an account paying 5% a year and leave it for ten years. Interest is earned on interest, so the balance grows by a factor of 1.05 each year:
FV = PV × (1 + r)N = 1,000 × 1.0510 = 1,628.89
Simple interest would give only 1,500. The extra 128.89 is interest earned on interest, and it grows quickly as the rate or the time rises.
Present value: discounting back
Going the other way is discounting. What is 10,000 due in five years worth today if you could otherwise earn 8%?
PV = FV / (1 + r)N = 10,000 / 1.085 = 6,805.83
The 8% here is the discount rate: the return you give up by waiting, plus compensation for any risk. A higher discount rate or a longer wait makes the present value smaller.
Annuities: equal payments
An annuity is a series of equal payments at equal intervals. The present value of five annual payments of 1,000 at 6% is:
PV = 1,000 × [1 − 1.06−5] / 0.06 = 4,212.36
That assumes an ordinary annuity, where each payment comes at the end of its period. If each payment comes at the start, it is an annuity due, and every payment is one period closer, so the value is higher by a factor of (1 + r): 4,212.36 × 1.06 = 4,465.11.
Typical annuities are loan repayments, rent and bond coupons. The future value of an annuity runs the other way: saving 500 a year for ten years at 7% builds to 500 × [(1.0710 − 1) / 0.07] = 6,908.22.
Perpetuities: payments that never end
A perpetuity pays the same amount forever. Its value is the payment divided by the rate:
PV = PMT / r
A perpetuity paying 50 a year at a 4% discount rate is worth 50 / 0.04 = 1,250. If the payment grows at a constant rate g, the first payment is divided by the gap between the rate and the growth: 50 / (0.08 − 0.03) = 1,000. This growing perpetuity is the core of the dividend discount model, and it only works when r is above g.
Uneven cash flows
When the payments differ, discount each one separately and add them. Receiving 100, 200 and 300 at the end of years one, two and three, discounted at 5%:
100 / 1.05 + 200 / 1.052 + 300 / 1.053 = 535.80
There is no shortcut formula, but a calculator's cash-flow function does the same sum.
Compounding more often than yearly
A stated annual rate compounded monthly earns interest twelve times a year. Divide the rate by twelve and count the months. The effective annual rate is higher than the stated rate because interest is earned on interest within the year; the effective annual rate guide covers the conversion.
Mistakes that cost marks
- Mismatched periods. If payments are monthly, N is the number of months and the rate is the monthly rate. Mixing years with a monthly payment is the single most common slip.
- Wrong signs. A positive PV with a positive FV has no solution on the calculator. Money in and money out must have opposite signs.
- Ordinary versus due. Rent is usually paid at the start of the period; loan repayments at the end. Read the wording.
- A stale calculator. Clear the registers and reset the payment mode between questions.
- Rate above growth. A growing perpetuity formula gives nonsense when the growth rate is at or above the discount rate.
Where this is tested
Time value of money is the foundation of the quantitative methods and fixed income material at every level, and it underlies equity valuation and corporate finance. The practice questions below it in the syllabus make you apply it until the setup is automatic.