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Annuities: Equal Payments, Future Value and Present Value

An annuity is a series of equal payments made at equal time gaps, like a monthly saving or a loan instalment. Each payment earns compound interest, so the payments form a geometric series. Future value of an ordinary annuity (payments at the end of each period): FV = P[(1+i)ⁿ − 1]/i. Present value (what the payments are worth today): PV = P[1 − (1+i)⁻ⁿ]/i. Loans use PV: the loan amount equals the present value of all instalments.

🎬 Step-by-step story

  1. Every year, at the end of the year, you put 1,000 into a savings account. Equal amounts, equal gaps. This is an annuity.
  2. The account pays 5% a year, compounded. The first payment earns interest for 4 years, the last for none. So the first stack grows the most.
  3. Add all the grown stacks: 1,000 + 1,050 + 1,102.50 + … They form a geometric series with ratio 1.05. The total is the future value: about 5,526.
  4. Now ask: what are those 5 payments worth today? Discount each one back to time 0. The later the payment, the more it shrinks. The total is the present value: about 4,329.
  5. A loan works the same way. A bank lends 10,000 today at 5%. The equal yearly instalment that repays it is the payment whose present value is 10,000: about 2,310.
  6. Your turn. Change the payment, the rate and the number of periods. Watch the future value and present value change.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why must the payments be equal?

Equal payments make the grown amounts a clean geometric series, so one formula adds them. Unequal payments must be grown one by one.

Why does the first payment grow most?

It sits in the account longest, earning interest on interest for n − 1 periods. Watch the leftmost stack in step 2.

Where does the formula come from?

From the geometric series sum a(rⁿ − 1)/(r − 1) with a = P and r = 1 + i. The 3D adds the stacks into the green total.

Why is the present value smaller than the total paid?

Money later is worth less today, because today's money could earn interest. Each later payment shrinks more when discounted.

Why do I pay back more than I borrowed?

The bank gets interest on the unpaid balance each period. The loan equals the PV of the instalments, but their simple total is bigger.

What happens when the rate is 0?

No growth or discount: FV = PV = n × P. Set the rate slider to 0 to see all stacks equal.

What is an annuity?

An annuity is a set of equal payments made at equal time gaps (every month, every year).

Words: P = payment each period, i = interest rate per period (as a decimal), n = number of payments. For monthly payments with yearly rate r: i = r/12, n = 12 × years.

Future value: a geometric series

With payments at the end of each period, the first payment earns interest for n − 1 periods, the last for 0. So

FV = P + P(1+i) + P(1+i)² + … + P(1+i)ⁿ⁻¹

This is a geometric series with first term P, ratio (1+i) and n terms. Using Sn = a(rⁿ − 1)/(r − 1):

FV = P · [(1+i)ⁿ − 1] / i

For an annuity due every payment gets one extra period of interest: FVdue = FV × (1+i).

A sinking fund asks the reverse: what payment P gives a target FV? P = FV · i / [(1+i)ⁿ − 1].

Present value of an annuity

Money today is worth more than the same money later, because today's money can earn interest. To bring a payment k periods away back to today, divide by (1+i)k. This is discounting.

PV = P/(1+i) + P/(1+i)² + … + P/(1+i)ⁿ — again a geometric series (ratio 1/(1+i)):

PV = P · [1 − (1+i)⁻ⁿ] / i

Check: PV × (1+i)ⁿ = FV. For a perpetuity n → ∞ and (1+i)⁻ⁿ → 0, so PV = P / i.

Loans and instalments (EMI)

When you borrow L today and repay with n equal instalments, the bank sets the instalment so that the present value of all instalments equals L:

P = L · i / [1 − (1+i)⁻ⁿ]

Each instalment pays the interest due on what is left, and the rest reduces the loan. Early on most of it is interest; near the end most of it is principal. Total paid = n × P, and total interest = nP − L.

To find how many periods are needed, solve (1+i)ⁿ for n with logarithms: n = log(…) / log(1+i), then round up.

Key formulas and definitions

Worked examples

1. 1,000 at the end of each year for 5 years at 5% p.a. compounded yearly. Find the future value.

FV = 1000 × (1.05⁵ − 1)/0.05. Step 1: 1.05⁵ = 1.27628. Step 2: 1.27628 − 1 = 0.27628. Step 3: ÷ 0.05 = 5.5256. Step 4: × 1000 = 5,525.63.

2. Same payments. Find the present value.

PV = 1000 × (1 − 1.05⁻⁵)/0.05. 1.05⁻⁵ = 0.78353. 1 − 0.78353 = 0.21647. ÷ 0.05 = 4.3295. PV = 4,329.48. Check: 4,329.48 × 1.27628 = 5,525.63 ✓.

3. 500 saved at the end of every month for 1 year at 12% p.a. compounded monthly. Find FV.

i = 0.12/12 = 0.01, n = 12. FV = 500 × (1.01¹² − 1)/0.01 = 500 × 12.6825 = 6,341.25.

4. A loan of 10,000 at 5% p.a. is repaid in 5 equal yearly instalments. Find the instalment.

P = 10000 × 0.05/(1 − 1.05⁻⁵) = 500/0.21647 = 2,309.75. Total paid = 11,548.74, interest = 1,548.74.

5. A loan of 200,000 at 9% p.a. for 3 years, monthly EMI. Find the EMI.

i = 0.09/12 = 0.0075, n = 36. 1.0075⁻³⁶ = 0.76415. P = 200000 × 0.0075/(1 − 0.76415) = 1500/0.23585 = 6,359.95.

6. 1,000 at the end of each year at 5%. After how many years will the fund first pass 10,000?

1000(1.05ⁿ − 1)/0.05 ≥ 10000 → 1.05ⁿ ≥ 1.5 → n ≥ log 1.5 / log 1.05 = 8.31. So 9 years.

7. A scholarship pays 5,000 every year for ever. Money earns 4% p.a. How much must be invested today?

Perpetuity: PV = P/i = 5000/0.04 = 125,000.

Common mistakes

Practice quiz

1. The payments of an annuity form which kind of sequence (after interest)?
2. FV of an ordinary annuity is:
3. Payments at the start of each period make an:
4. A loan amount equals the ___ of its instalments.
5. PV of a perpetuity of 600 per year at 6%:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is an annuity in simple words?

A plan of equal payments at equal time gaps, such as a monthly saving or a loan instalment.

What is the difference between an ordinary annuity and an annuity due?

In an ordinary annuity payments are at the end of each period; in an annuity due they are at the start, so each payment earns one extra period of interest.

How is EMI calculated?

EMI = L × i / [1 − (1+i)⁻ⁿ], where L is the loan, i the monthly rate and n the number of months. It makes the present value of all EMIs equal the loan.

Where this is taught

CBSE (India)Class 11Basics of Financial Mathematics
CBSE (India)Class 12Financial Mathematics
South Korea고등학교 2학년Numbers and the economy
South Korea고등학교 3학년Sequences and finance

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