A borrower and a lender agree to the following arrangements: the borrower will p

Question

A borrower and a lender agree to the following arrangements: the borrower will pay annual interest to the lender for 9 years at 7%, at the end of each year. The borrowwer will pay 110% of the original loan amount to the lender at the end of 9 years by making 5 annual deposits in a sinking fund (S.F.) earning 5%. After making the 5 deposits, the S.F. grows with interest only. The total annual payment made by the borrower in the first five years is $13,000 (that is, each year, the borrower pays $13,000 which is distributed between the interest payment to the lender and the deposit to the S.F.). What is the amount of the loan

Explanation / Answer

Say the amount of Loan is x

So, interest on loan /year =0.07x

Total payment $13000

Invested in sinking fund each year =13000-0.07x

Fuiture vale of annuity =FV= A [ (1+k)n-1/k]

FV = Future anuuity value after 5 years

A = periodical payment =$13000-0.047x annually

K=interest rate=5% annually

N=periods=5 years

So FV = (13000-0.07x)[(1.05)5-1]/0.05 =(13000-0.07x)*5.52

After 5 years the value of sinking fund will be (13000-0.07x)*5.52

Compounding factor for 4 years @5% =1.216

So after 9 years fund value will be (13000-0.07x)*5.52*1.216

The value of sinking fund is 110% of loan.

So, 1.10x= (13000-0.07x)*5.52*1.216

Or 1.10x=(13000-0.07x)*6.712

Or 1.10x=87256 -0.47x

Or, 1.57x=87256

Or x= 55,577

So the loan amount is $ 55,577

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