Solutions to Problem Set for Lecture 11:
1
A credit union advertises that its monthly interest rate on loans is 1%. Calculate the annual
effective interest rate and using interpolation find the P/F factor for n=8. (do not use the formula
for P/F conversion for this problem)
SOLUTION: To find the effective annual interest rate: i = (1+0.01)12 -1 = 0.1268 =
12.68%. Now we can find from the tables the factors for P/F, n=8 for i=12% and i=13%,
and interpolate: For 12% it is 0.4039; for 12.68% it is unknown; for 13% it is 0.3762. So
(P/F,12.68%,8) = 0.4039 - (12.68 – 12)(0.4039 - 0.3762)/(13-12) = 0.4039 (0.68)(0.0188) = 0.4039 – 0.0188 = 0.3851.
2
You plan to place money in a certificate of deposite (CoD) that pays 18% per year compounded
daily. What effective rate will you receive (a) yearly and (b) semiannually?
SOLUTION: (a) i = (1 + 0.18/365)365 – 1 = 0.19716 = 19.716% per year
(b) i = (1 + 0.09/182)182 – 1 = 0.09415 = 9.415% per 6 months
3
A bank is offering a loan of $20,000 with a nominal interest rate of 12%, payable in 48 months?
a Calculate the monthly payments
b This bank also charges a loan fee of 4% of the amount of the loan, payable at the time the
loan is given to the borrower. What is the effective interest rate the bank is charging?
ANSWER: a.) the monthly payments: n = 48; i = 12%/12 = 1% per period (month).
A=P(A/P, i, n) = 20,000(A/P,1%,48) = 20,000(0.0263) = $526
b.) actual money received = 20,000 – 0.04(20,000) = 19,200. But A = 526.
Then: 526 = 19,200(A/P, i, 48)
(A/P, i, 48) = 526/19,200 = 0.02739
for i = 1.25%, (A/P, 1.25%, 48) = 0.0278
for i = 1 %, (A/P, 1, 48) = 0.0263
by interpolation: i = 1% + 0.25(0.0263 - 0.02739)/ (0.0263 - 0.0278) =
= 1.1817%. The effective interest rate = (1 + 0.011817)12 – 1 = 0.1514
ANSWER: a.) $526 b.) 15.14%
4
Assume that you borrowed $50,000 at an interest rate of 1% per month, to be repaid in uniform
monthly payments for 30 years. In the 163rd payment, how much of it would be interest (NB:
not interest rate, but interest), and how much of it would be principal?
SOLUTION: To determine the interest paid on loan at time t, we need to multiply the
effective interest rate times the outstanding principal after the payment at time t-1. So we
need to compute the outstanding principal at time t = 162. This can be done either by
substracting the future worth of 162 payments from the future wrth at time 162 of the
amount borrowed, or, alternatively, by computing the present worth at time 162, of the
198 payments remaining. The uniform payments are 50,000(A/P,1%,360) = $514.31,
thus P(t=162) = 50,000(F/P,1%,162) – 514.31(F/A,1%,162) = 514.31(P/A.1%,198) =
$44,259.78. The interest is: 0.01(44,259.78) = $442.59, and the principal is: 514.31 –
442.59 = $71.72