Question

4.

Find the present value of the annuity.

Amount Per Payment: $4,725

Payment at End of Each: 6 months

Number of Years: 15

Interest Rate: 10%

Compounded: Semiannually


$72,634.83

$35,938.73

$32,242.03

$68,951.03

Answer 1

To solve this we are going to use the present value of annuity formula:
`PV=P[ (1-(1+ (r)/(n))^(-kt) )/( (r)/(n) ) ]`
where

`PV` is the present value

`P` is the periodic payment

`r` is the interest rate in decimal form

`n` is the number of times the interest is compounded per year

`k` is the number of payments per year

`t` is the number of years

We know from our problem that
`P=4725` and
`t=15`. To convert the interest rate to decimal form, we are going to divide it by 100%:

`r= (10)/(100)`

`r=0.1`
Since the interest is compounded semiannually, it is compounded 2 times per year; therefore,
`n=2`. Similarly, since the payment is made at the end of each 6 months, it is made 2 times per year; therefore,
`k=2`.
Lest replace the values in our formula:


`PV=P[ (1-(1+ (r)/(n))^(-kt) )/( (r)/(n) ) ]`

`PV=4725[ (1-(1+ (0.1)/(2))^(-(2)(15)) )/( (0.1)/(2) ) ]`

`PV=72634.83`

We can conclude that the correct answer is $72,634.83