Question

How long will it take payment of $500 per quarter to amortize a loan of $8,000 at 16% compounded quarterly? Approximate your answer in terms of years and months. How much less time will it take if loan payments are made at the beginning of each quarter rather that at the end?

Answer 1

Formula for PV of annuity can be used to compute required number of periods as:

PV = P x [1 – (1+r)-n]/r

PV = Present value of loan = $ 8,000

P = Periodic payment = $ 500

r = Rate per period = 16 % p.a. or 0.16/4 = 0.04 p. q.

n = Number of periods

$ 8,000 = $ 500 x / [1 – (1+0.04)-n]/0.04

   = $ 500 x [1 – (1.04)-n]/0.04

$ 8,000 / $ 500 = [1 – (1.04)-n]/0.04

[1 – (1.04)-n]/0.04 = 16

1 – (1.04)-n = 16 x 0.04 = 0.64

(1.04)-n = 1 – 0.64

(1.04)-n = 0.36

Taking log of both sides and solving for n, we get:

-n log 1.04 = log 0.36

-n x 0.0170333392988= -0.443697499233

n = 0.443697499233/0.0170333392988

   = 26.04876774 periods or 6 years 6.15 months

It will take 6 years 6.15 months to pay off the loan.

Formula for PV of annuity due can be used to compute required number of periods if payments are made at the beginning of periods.

PV = P + P [1 – (1+r)-n-1/r]

$ 8,000 = $ 500 + $ 500 [ 1 – (1+0.04)1-n/0.04]

$ 8,000 - $ 500 = $ 500 [ 1 – (1.04)1-n/0.04]

$ 7,500 = $ 500 [ 1 – (1.04)1-n/0.04]

15 = 1 – (1.04)1-n/0.04

1 – (1.04)1-n = 15 x 0.04 = 0.6

(1.04)1-n = 1 – 0.6

(1.04)1-n = 0.4

Taking log of both sides and solving for n, we get:

1-n x log 1.04 = log 0.4

1-n x 0.0170333392988 = -0.397940008672

n -1 = 0.397940008672/0.0170333392988

n – 1 = 23.36241894

n = 23.36241894 + 1 = 24.36241894 periods or 6 years 1.09 months

It will take 6 years 1.09 months to pay off the loan if payments are made at the beginning of periods.

Time difference = 26.04876774 - 24.36241894 = 1.686348799 periods

= 5.059046396 months or 5.06 months

It will take 5.06 months less time if payments are made at the beginning of periods.