Compound Interest

Compound Interest If the interest gained after a period of time is not withdrawn and added to the principal, the principal will then increase at the end of each deposit period. The interest obtained in this way is called the compound interest. … P1 A2 = P2+ I2 A1 = P1+ I1 after the 1st period of time after the 2nd period of time P3 P2 P = principal A = amount I = interest A1 becomes the principal of the 2nd period of time.

If we deposit principal $P for n periods with the interest compounded at r % per period, then the amount $A can be calculated by the following formula: A = P  (1 + r %)n Let’s consider the following example.

(a) If the interest rate is compounded yearly, then Raymond deposits $10 000 in a bank at 12% p.a. for 3 years. (a) If the interest rate is compounded yearly, then amount = $10 000  (1 + 12%)3 Substitute P = $10 000, r = 12 and n = 3 into the formula A = P  (1 + r %)n. = $10 000  1.123 = $14 049 (cor. to the nearest $1) compound interest = $(14 049 – 10 000) = $4049 (cor. to the nearest $1)

interest rate for half a year = 12%  If the interest rate is compounded half-yearly, then interest rate for half a year = 12%  2 1 number of half-year periods per year = 2 = 6% number of half-year periods in 3 years = 3  2 = 6 ∴ amount = $10 000  (1 + 6%)6 Substitute P = $10 000, r = 6 and n = 6 into the formula A = P  (1 + r %)n. = $10 000  1.066 = $14 185 (cor. to the nearest $1) ∴ compound interest = $(14 185 – 10 000) = $4185 (cor. to the nearest $1)

interest rate for a quarter = 12%  Compounded quarterly means ‘ compounded every 3 months’. (c) If the interest rate is compounded quarterly, then interest rate for a quarter = 12%  4 1 number of quarters per year = 4 = 3% number of quarters in 3 years = 3  4 = 12 ∴ amount = $10 000  (1 + 3%)12 Substitute P = $10 000, r = 3 and n = 12 into the formula A = P  (1 + r %)n. = $14 258 (cor. to the nearest $1) ∴ compound interest = $(14 258 – 10 000) = $4258 (cor. to the nearest $1)

interest rate for a month = 12%  (d) If the interest rate is compounded monthly, then interest rate for a month = 12%  12 1 number of months per year = 12 = 1% number of months in 3 years = 3  12 = 36 ∴ amount = $10 000  (1 + 1%)36 Substitute P = $10 000, r = 1 and n = 36 into the formula A = P  (1 + r %)n. = $14 308 (cor. to the nearest $1) ∴ compound interest = $(14 308 – 10 000) = $4308 (cor. to the nearest $1)

Follow-up question Annie deposits $5000 in a bank at 4% p.a. Find the amount and compound interest she will obtain after 2 years if the interest compounded (a) yearly (b) half-yearly (Give your answers correct to the nearest $1 if necessary.) Solution (a) Amount = $5000  (1 + 4%)2 = $5000  1.042 = $5408 Interest = $(5408 – 5000) = $408

Follow-up question (cont’d) Annie deposits $5000 in a bank at 4% p.a. Find the amount and compound interest she will obtain after 2 years if the interest compounded (a) yearly (b) half-yearly (Give your answers correct to the nearest $1 if necessary.) Solution (b) Interest rate for half a year = 4%  2 1 = 2% number of half-year periods in 2 years = 2  2 = 4

Follow-up question (cont’d) Annie deposits $5000 in a bank at 4% p.a. Find the amount and compound interest she will obtain after 2 years if the interest compounded (a) yearly (b) half-yearly (Give your answers correct to the nearest $1 if necessary.) Solution (b) ∴ Amount = $5000  (1 + 2%)4 = $5000  1.024 = $5412 (cor. to the nearest $1) Interest = $(5412 – 5000) = $412 (cor. to the nearest $1)