Compound interest
Compound interest earns interest on interest. After n periods the amount is A = P(1 + i)ⁿ, where i is the rate per period.
Compound interest formulas
Year by year: Rs. 200,000 at 10% compounded yearly
| Year | Opening (Rs.) | Interest at 10% (Rs.) | Closing (Rs.) |
|---|---|---|---|
| 1 | 200,000 | 20,000 | 220,000 |
| 2 | 220,000 | 22,000 | 242,000 |
| 3 | 242,000 | 24,200 | 266,200 |
Compound vs simple interest
Rs. 200,000 is invested for 3 years at 10% a year. Compare compound interest (yearly) with simple interest.
- Compound: A = 200,000 × 1.1³ = 200,000 × 1.331 = 266,200.
- CI = 266,200 − 200,000 = 66,200.
- Simple: I = 200,000 × 0.10 × 3 = 60,000.
- Difference = 66,200 − 60,000 = 6,200: the interest earned on earlier interest.
Compound interest Rs. 66,200 vs simple interest Rs. 60,000.
More frequent compounding
Rs. 100,000 is invested for 2 years at 12% a year. Find the amount if interest is compounded (a) yearly, (b) quarterly, (c) monthly.
- (a) i = 0.12, n = 2: A = 100,000 × 1.12² = 100,000 × 1.2544 = 125,440.
- (b) i = 0.12 ÷ 4 = 0.03, n = 4 × 2 = 8: A = 100,000 × 1.03⁸ = 100,000 × 1.26677008 = 126,677.01.
- (c) i = 0.12 ÷ 12 = 0.01, n = 12 × 2 = 24: A = 100,000 × 1.01²⁴ = 100,000 × 1.269734649 = 126,973.46.
(a) Rs. 125,440 (b) Rs. 126,677.01 (c) Rs. 126,973.46. More frequent compounding gives more.
Effective annual rate
Which is better for a saver: 12% compounded monthly or 12% compounded quarterly?
- Monthly: (1 + 0.12/12)¹² − 1 = 1.01¹² − 1 = 1.126825 − 1 = 0.126825, i.e. 12.68%.
- Quarterly: (1 + 0.12/4)⁴ − 1 = 1.03⁴ − 1 = 1.12550881 − 1 = 0.12550881, i.e. 12.55%.
Monthly compounding is better: 12.68% effective against 12.55%.
How long will it take?
How many years does Rs. 50,000 take to grow to Rs. 80,000 at 8% a year compounded yearly?
- 50,000 × 1.08ⁿ = 80,000, so 1.08ⁿ = 1.6.
- n = log 1.6 ÷ log 1.08 = 0.20412 ÷ 0.033424 = 6.107.
- Interest is added only at year ends. After 6 years: 50,000 × 1.08⁶ = Rs. 79,343.72 (not enough).
- After 7 years: 50,000 × 1.08⁷ = Rs. 85,691.21.
6.107 years by formula, so the deposit first passes Rs. 80,000 at the end of year 7.
Sri Lankan fixed deposits and savings accounts compound interest. A deposit that compounds monthly grows faster than the same nominal rate paid yearly, so compare offers by the effective annual rate, using the rates the bank actually quotes.
Short answer: “Find the effective annual rate of 12% compounded quarterly.” 1.03⁴ − 1 = 0.1255, i.e. 12.55%.
Structured question: “Find the amount of Rs. 200,000 at 10% a year compounded half-yearly for 3 years.” i = 0.05, n = 6: 200,000 × 1.05⁶ = Rs. 268,019.13.
Essay: “Compare Rs. 100,000 for 2 years at 12%: simple interest, and compounding yearly, quarterly and monthly.” Rs. 124,000; Rs. 125,440; Rs. 126,677.01; Rs. 126,973.46.
Before you use A = P(1 + i)ⁿ, convert both: i = r ÷ m and n = m × years. Most lost marks come from using the yearly rate with monthly periods.