To pay his tution, sam borrowed money at 3% per annum, compounded semi-annually. For the debt, he ows $ 5000 to be paid 2 years.......
.... from now. He earned more at his summer job than he expected, so he wants to pay off the loan at its present value. How much would he pay?
Sam borrowed money at 3% per annum compounded semi-annually. To clear his loan he has to pay $5000 after a period of two years. The present value of the amount to be repaid after two years can be determined using the interest rate for the amount borrowed and the time left for it to be repaid. As the interest is compounded semi-annually, the interest rate per term is `(3%)/2 = 1.5%` and the number of terms in 2 years is 2*2 = 4.
The present value of 5000 is `5000/(1+0.015)^4 ~~ 4710.92`
Sam has to pay back $4710.92 to clear the debt right now.
You need to use the compound interest formula such that:
`A = P(1 + (r/m))^n`
A represents the final value
P represents the initial value
r represents the annual interest rate
m represents the number of compounding periods per year
n represents the number of periods
The problem provides the information that the annual interest rate is of `3%` , the number of periods in years is n = 4, the final value is `A = 5000` and the number of compounding periods per year is `m = 2` , hence, you may evaluate the final value such that:
`5000 = P(1 + 3/(100*2))^4 => P = 5000/(1 + 0.03/2)^4`
`P ~~ 5000/(1.0613) => P ~~ $4711`
Hence, evaluating the present value of the loan that Sam needs to give back yields `P ~~ $4711.`