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 800, a seventh installment of (800 + K), with each subsequent installment increasing by K over the previous payment. The period of the loan is not changed. Determine the revised amount of the last installment.

the last installment = $1,239.42

Explanation:

renegotiated agreement:

year          payment

1                 $1,000

2                $1,000

3                $1,000

4                $1,000

5                $1,000

6                $800

7                $800 + K

8                $800 + 2K

9                $800 + 3K

10               $800 + 4K

11                $800 + 5K

12               $800 + 6K

13               $800 + 7K

14               $800 + 8K

15               $800 + 9K

we must first determine the original loan and to do that we need the PV of the original payment schedule:

PV = $1,000 x 10.380 (PV annuity factor, 5%, 15 periods) = $10,380

now we find the present value of the first 5 installments:

PV = $1,000 x 4.3295 (PV annuity factor, 5%, 5 periods) = $4,329.50

$10,380 - $4,329.50 = $6,050.50

now to find K:

$6,050.50 = $800/1.05⁶ + ($800 + K)/1.05⁷ + ($800 + 2K)/1.05⁸ + ($800 + 3K)/1.05⁹ + ($800 + 4K)/1.05¹⁰ + ($800 + 5K)/1.05¹¹ + ($800 + 6K)/1.05¹² + ($800 + 7K)/1.05¹³ + ($800 + 8K)/1.05¹⁴ + ($800 + 9K)/1.05¹⁵ = 596.97 + 568.55 + 0.71K + 541.47 + 1.35K + 515.69 + 1.93K + 491.13 + 2.46K + 467.74 + 2.92K + 445.47 + 3.34K + 424.26 + 3.71K + 404.05 + 4.04K + 384.81 + 4.33K = $4,840.14 + 24.79K

$6,050.50 = $4,840.14 + 24.79K

$6,050.50 - $4,840.14 = 24.79K

$1,210.36 = 24.79K

K = $48.82

the last installment = $800 + 9K = $800 + (9 x $48.82) = $1,239.42

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