Question

An amount say `$800` is invested at a rate of interest of `5%` for `30` years. What is the amount at the end of the period if it is?

(a) compounded every year
(b) compounded monthly
(c) compounded every week
(d) compounded continuously.

Answer 1

(a) `$3457.55` (b) `$3574.20`
(c) `$3582.77` (d) `$3585.35`

Explanation:

When we invest an amount say `P` at a rate of interest of `r` for `t` years, the amount at the end of the period (at simple interest) becomes `P(1+(rt)/100)`.

However, when compounded, it depends on the fixed period after which it is compounded or rather how many times in a year interest is compounded. If it is compounded after every `1/n` year,

the amount becomes `P(1+r/(nxx100))^(nt)`

Here we have `P=$800`, `r=5%` and `t=30` years

(a) If compounded yearly i.e. once a year, as `n=1`, it becomes

`800(1+5/100)^30=800xx1.05^30=800xx4.321942=$3457.55`

(b) If compounded monthly i.e. `12` times a year, as `n=12`, it becomes

`800(1+5/1200)^(30xx12)=800xx1.004166^360=800xx4.467744=$3574.20`

(c) If compounded weekly i.e. `52` times a year, as `n=52`, it becomes

`800(1+5/5200)^(30xx52)=800xx1.00096154^1560=800xx4.47846=$3582.77`

(d) If compounded continuously, it approximates `Pe^((rt)/100)` i.e.

`800e^((5xx30)/100)=800xxe^1.5=800xx4.481689=$3585.35`