non annual compounding.

Question

9. The following questions illustrate nonannual compounding.
a) One hundred dollars is placed in an account that pays 12 percent. How much will be in the account after one year if interest is compounded annually, semiannually, or monthly? b) One hundred dollars is to be received after one year. What is the present value of this amount if you can earn 12 percent compounded annually, semiannually, or monthly?

Answer 1

We use the formula:

 

A = P(1+r/n)^nt

 

A = total amount

P = principle investment

r = interest rate

n = # of time intervals in a year

t = # of years

(the letters used to represent these may differ for your class, but use of the formula is the same,)

 

Part a:

P = 100

r = 12% = .12

t = 1

A is unknown

 

For annual interest, n=1 (there is 1 year in a year) so we can solve:

A = 100(1+.12/1)^1*1= $112

 

Semiannually n=2 (2 semi years in a year.)

A = 100(1+.12/2)^2*1= $112.36

Monthly n=12 (12 months in a year)

A = 100(1+.12/12)^12*1 = 112.682503 ≈ $112.68

 

b.  The same formula, but slightly different given values.

A = 100

t = 1

r = 12% = .12

P is unknown.

 

Annually

100 = P(1+.12/1)^1*1

100 = P(1.12)

P = 89.28571429 ≈ $89.29

 

Semiannually

100 = P(1+.12/2)^2*1

100 = P(1.06)²

100 = P(1.1236)

P = 88.999644 ≈ $89.00

 

Monthly

100 = P(1+.12/12)^12*1

100 = P(1.01)¹²

100 = P(1.12682503)

P = 88.74492253 ≈ $88.74