- Compound interest is most frequently seen in terms of investments.
- Certificates of deposit are the most common consumer level
- A simple example
I invest $1000 for three years at 2.5% interest.
Using the simple interest formula, I would receive
i = prt
p = 1000, r = 2.5%, t = 3
i = 1000 x .025 x 3
= $75
But what if I were clever, I went to the bank and withdrew the money each year
Year 1
i = prt
p = 1000, r = 2.5%, t = 1
i = 1000 x .025 x 1
= $25
Year 2
i = prt
p = 1025 (why) r = 2.5%, t = 1
i = 1025 x 0.025 x 1
= $25.63 (why)
Year 3
i = prt
p = 1,050.63 r = 2.5%, t = 1
i = 1,050.63 x 0.025 x 1
= 26.27
So the total is 1,076.90
- There is a formula to compute compound interest.
- A = p(1+r/n) ^ (nt)
- p r, and t are as in the simple interest formula
- n is the number of compoundings per year.
- Compounded once a year (annually) n = 1
- Compounded twice a year (semiannually) n = 2
- Compounded quarterly n = 4
- Compounded monthly n = 12
- Compounded daily n = 360
- In our case, n = 1, t = 3, r = 2.5%, p = 1000
A = p(1+r/n)^(nt)
A = 1000 (1+.025/1)^(1*3)
= 1,076.89
- Problem 7, page 627
$2500 for 4 years at 1.2% compounded monthly
p = 2500
r = 1.2%
t = 4 years
n = 12
A = 2500(1+0.012/12)^(12*4)
= $2,622.86
How much interest was paid?
A = i + p
i = A-p
= 2,622.86 - 2,500
= 122.86
- We might also want to know how much to invest now to have a given amount in the future.
- Just solve the compound interest equation for p
- p = A/(1+r/n)^(nt)
- Problem 15 page 627
The desired accumulated amount is $50,000 after 10 years
invested in an account with 5% interest compounded annually.
n = 1
t = 10
A = 50,000
r = 5%
p = ?
p = A/(1+r/n)^(nt)
= 50000/(1+0.05/1)^(1*10)
= $30,695.66