108156.pdf

Solutions 269

The capital repaid as part of thenth instalment is the difference between the

outstanding balance of the loan after the (n−1)st and after thenth instal-

ment. By (S.1) this difference is equal to

P(1 +r)

n−(1 +r)n− 1

(1 +r)^5 − 1

=Pr(1 +r)

n− 1

(1 +r)^5 − 1

. (S.3)

PuttingP to be $1,000 andrto be 15% in (S.1), (S.2) and (S.3), we can

compute the figures collected in the table:

t(years) interest paid ($) capital repaid ($) outstanding balance ($)

0 — — 1 , 000. 00

1 150. 00 148. 32 851. 68

2 127. 75 170. 56 681. 12

3 102. 17 196. 15 484. 97

4 72. 75 225. 57 259. 40

5 38. 91 259. 40 0. 00

2.14The amount you can afford to borrow is

PA(18%,10)× 10 ,000 =^1 −(1 + 0.18)

− 10

0. 18

× 10 , 000 ∼= 44 , 941

dollars.

2.15The present value of the balance after 40 years is

PA(5%,40)× 1 ,200 =^1 −(1 + 0.05)

− 40

0. 05 ×^1 ,^200

∼= 20 , 591

dollars. Dividing by the discount factor (1 + 0.05)−^40 , we find that the actual

balance after 40 years will be

20 , 591

(1 + 0.05)−^40

∼= 144 , 960

dollars.

2.16The annual payments will amount to

C=PA^100 (6%,^000 ,10)∼= 13 , 586. 80

dollars each. The outstanding balance to be paid to clear the mortgage after

8 years (once the 8th annual payment is made) will be

PA(6%,2)×C∼= 24 , 909. 93

dollars.

2.17Suppose that paymentsC, C(1+g),C(1+g)^2 ,...are made after 1 year, 2 years,

3 years, and so on. If the interest rate is constant and equal tor, then the

present values of these payments areC(1 +r)−^1 ,C(1 +g)(1+r)−^2 ,C(1 +

g)^2 (1 +r)−^3 ,.... The present value of the infinite stream of payments is,

therefore,

C

1+r+

C(1 +g)

(1 +r)^2

+C(1 +g)

2

(1 +r)^3

+...=r−Cg.