Question

1) Carlos has borrowed $8,000 for 8 years at 6% compounded semi-annually. He will repay interest every 6 months plus principal at maturity. He will also deposit X every 6 months into a sinking fund paying 5% compounded semi-annually to pay off the principal at maturity.

a) Find X.

Carlos goes bankrupt at the end of year 6, just after making his interest payment and sinking fund deposit. The bank confiscates the money in the sinking fund, but gets no further payments.

b) How much money does the bank lose as a result of the loan default at the end of year 6?

c) Over the lifetime of the loan, how much money did the bank collect? Was it more or less than the amount of the original loan?

d) Assuming the bank re-invested all of Carlos payments at 6% (compounded semi-annually), how much money does the bank have at the end of 6 years? What is the equivalent yield (compounded semi-annually) they made on their initial loan?

Answer 1

Part (a)

FV of annuity X over n = 2 x 8 = 16 periods at an interest rate of i = 5% / 2 = 2.5% will be equal to the principal at maturity.

Hence, X / i x [(1 + i)ⁿ - 1] = 8,000

Hence, X / 2.5% x [(1 + 2.5%)¹⁶ - 1] = 8,000

Hence, 19.38X = 8,000

Hence, X = 8,000 / 19.38 = $ 412.80

Part (b)

FV of the sinking fund deposits over 6 years = X / i x [(1 + i)ⁿ¹ - 1] = 412.80 / 2.5% x [(1 + 2.5%)^{2 x 6} - 1] = 5,695

The money the bank loses as a result of the loan default at the end of year 6 = Principal - FV calculated above = 8,000 - 5,695 = $ 2,305

Part (c)

Over the lifetime of the loan, the money collected by the bank = Total interest paid over 6 years + FV of the sinking fund annuities = 8,000 x 6% x 6 + 5,695 = $ 8,575

It is more than the amount of the original loan.

Part (d)

FV of all the interest payments = A / r x [(1 + r)^t - 1]

A = interest payment per period = semi annual interest = 8,000 x 6% / 2 = 240

r = interest rate per period = 6% / 2 = 3%

t = time periods elapsed = 6 years = 2 x 6 = 12 periods

Hence, FV of all interest = 240 / 3% x [1.03¹² - 1] = $  3,406

So, the total money with the bank = FV of interests + Amount lying in the sinking fund = 3,406 + 5,695 = 9,101

Hence, if y is the equivalent yield (compounded semi-annually) they made on their initial loan, then

8,000 x (1 + y/2)^{2 x 6} = 9,101

Hence, (1 + y/2) = (9,101 / 8,000)^1/12 = 1.0108

Hence, y = 2 x (1.0108 - 1) = 2.16%