solve without excel Roy borrows $25,000 at an effective annual interest rate of

Question

solve without excel Roy borrows $25,000 at an effective annual interest rate of 12%. He has the following options for repayment: (c) Annual amortization method, with payments made at year-end for 10 years. Paying annual interest at year-end and building up a sinking fund (earning effective annual interest rate of 7%), by making level annual payments at year-end, to pay off the loan at the end of 10 years. Determine the absolute value of the difference between the total annual payment under option 1 and the total annual payment under option 2. (7 marks) A loan is being repaid by 15 annual installments of 1,000 each. Interest is at an effective annual rate of 5%. Immediately after the fifth installment is paid, the loan is renegotiated. The revised amortization schedule calls for a sixth installment of 700, a seventh installment of (700+X), with each subsequent installment increasing by X over previous payment. The period of the loan is not changed. Determined the revised amount of the last installment. (d) (8 marks) Total : 25 marks]

Explanation / Answer

c. Equated Installment formula = Loan Amount * [ r * (1+r)t] / [ (1+r)t - 1 ] ; where r is the interst rate and t is the loan tenure.

Option 1: Hence we have Equated installment = 25000 * [12% * (1+12%)10 ] / [(1+12%)10 - 1] = 4424.60. Thus in this case, the person will pay an absolute amount over 10 years = 44246.04

Option 2: Annual interest cost = 25000*12% = 3000 which for 10 years will be 30000.

Sinking fund formula = Annual Amount * [(1+r)t - 1]/r ; since we know the total value at the end of 10 years should be 25000, we can solve for the annual amount at 7%.

25000 = Annual Amount * [(1+7%)10 - ] / 7% ; solving we get Annual Amount = 1809.44

Thus the total absolute amount paid under option 2 over 10 years = 30000 + 18094.4 = 48094.4 which is more than option 1 by 3848.33

d. Using the equated installment formula we calculate the loan amount.

1000 = Loan Amount * [5% ( 1+5%)15 ] / [(1+5%)15 - 1] ; solving we get Loan Amount = 10379.66

Formula for residual loan balance after time period k : Loan Amount * [(1+r)t - (1+r)k]/[(1+r)t - 1]

So the loan balance after 5 years will be = 10379.66 * [(1+5%)15 - (1+5%)5] / [(1+5%)15 - ] = 7721.73

Now the proposed cash flows stream of 700 (Year 6) , 700 + X (Year 7) and so on till Year 15 which will be 700 + 9X

The present value of this cash flows stream should be 7721.73. The PV formula for the same is

Base Amount * [(1+r)t - 1]/[r * (1+r)t ] + Gradient * 1/r * [[(1+r)t - 1]/[r * (1+r)t ] - t/(1+r)t]

here Base Amount is 700 and gradient is X. Plugging in the values we get:

7721.73 = 700 * [(1+5%)10 - 1]/[5% * (1+5%)10] + X * 1/5% * [[(1+5%)10 - 1]/[5% * (1+5%)10] - 10/(1+5%)10] = 700 * 7.72 + X * 31.65

7721.73 = 5405.21 + 31.65X or X = 73.19

Hence Year 15 installment = 700 + 9*73.19 = 1358.68

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