Maths and SIMPLE interest?

Question

I don't understand this problem it's just wierd to me. Can somebody help out.


1) We placed $100 into a bank account with an annual compound interest rate of 8%. How much money
do we have in the account after 20 years if the bank compounds
(a) annually
(b) semi-annually
(c) monthly
(d) daily
(e) continuously?

Answer 1

Let
A = amount after 20 years
i = annual interest rate = .08

1a) compound annually
A = 100(1 + i)^20 = 100(1.08)^20 = $466.10
All answers will be to the nearest cent.

1b) compound semi-annually
Instead of compounding 8% once a year, compound 8%/2 = 4% twice a year.

A = 100(1 + i/2)^(2*20) = 100(1.04)^40 = $480.10
This is a little more than annual compounding as we would expect.

1c) compound monthly
Instead of compounding 8% once a year, compound 8%/12 twelve times a year. In banking, for the purpose of calculating interest, it is standard practice to assume all months are of equal length.

A = 100(1 + i/12)^(12*20) = 100(1+ .08/12)^240 = $492.68
This is a little more than semi-annual compounding as we would expect.

1d) compound daily
Instead of compounding 8% once a year, compound 8%/360 daily. In banking, for the purpose of calculating interest, it is standard practice to assume a year consists of 12 months of 30 days each for a total of 360 days in a year.

A = 100(1 + i/360)^(360*20)
= 100(1+ .08/360)^7200 = $495.22
This is a little more than monthly compounding as we would expect.

1e) compound continuously
Instead of compounding 8% once a year, compound as follows:

limit (1 + i/n)^(nt) = e^(it)
n→∞

A = 100e^(.08*20) = 100e^(1.6) = $495.30.
This is a just a little bit more than daily compounding as we would expect.

Clearly, the more often you compound the higher the effective rate of interest.

Answer 2

Use a simple recursive formula:
U0=100
Un=Un-1(1+.08) is the equation for a where U0 is the first term, Un is just a term number. If you want to make it semi-annually, then divide (1+.08) by 2. For monthly, by 12, for daily, by 365, and continuously, I don't really know. This could make absolutely no sense.

Answer 3

(a)
Start with 100
After year 1, have 100 * 1.08 = 108
After year 2, have ((100 * (1.08))*1.08) = 100 * (1.08)^2 = 116.64
After year 20, have 100 * (1.08)^20 = 466.0957

(b) start with 100

After 6 months, have 100 * 1.04 = 104
After 12 months, have 100 * (1.04)^2 = 108.16
After 20 years, have 100 * (1.04)^40 = 480.1021

(c) start with 100

After 1 month, have 100 * 1.00667 = 100.6667
After 12 months have 100 * (1.00667)^12 = 108.3
After 20 years, have 100 * (1.00667)^240 = 492.6803

(d) Start with 100

After 1 day, have 100 * 1.000219 = 100.0219
After 1 year, have 100 * 1.000219^365 = 108.3278
After 20 years, have 100 * 1.00219^7300 = 495.2164

not sure how to do it continuously...Hope it helps!

Answer 4

Go to the URL below and download the 40-year investment calculator.

Put in your figures and look at what it tells you.

Do your own homework. You can give a fish to a hungry person, but then they'll be hungry again tomorrow. How about I teach you to fish so you can eat forever?