Alan borrows $18,000 for eight years and agrees to make quarterly payments of $7

Question

Alan borrows $18,000 for eight years and agrees to make quarterly payments of $770. Each of these payments consists of interest for the just completed quarter and a deposit to a sinking fund that has a nominal interest rate of 6% convertible quarterly. For the first six years, each year the lender receiver 8% nominal interest convertible quarterly. For the remaining two years, the lender receives 12% nominal interest convertible quarterly. Find the amount by which the sinking fund is short of repaying the loan at the end of eight years.

The answer is $2,835.71

Please DO NOT use excel, formulas instead

Explanation / Answer

FV = PMT*((1+r)n - 1 ) / r

quarterly rate = annual rate / 4

number of periods = number of years * 4

Quarterly interest for first 6 years = 18000*8%/4 = 360

hence siniking fund deposits are = 770 - 360 = 410

Value of sinkinfund after 6 years = 410*((1+1.5%)24 - 1) / 1.5% = 11739.74

Value after 8 years = 11739.74*(1+1.5%)^8 = 13224.73

Quarterly interest for last 2 years = 18000*12%/4 = 540

hence siniking fund deposits are = 770 - 540 = 230

Value of sinkinfund at 8th year = 230*((1+1.5%)8 - 1) / 1.5% = 1939.55

Total Value after 8 years = 13224.73 + 1939.55 = 15164.29

Shortage = 18000 - 15164.29 = 2835.71

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