A city has two sources to borrow $200,000.The first source is to borrow at j₁ = 9% and amortize the debt for 10 years with annual payments of $31,164.For the second source,the money can be borrowed at j₁ = 8% if the interest is paid annually and the principal is repaid in a lump sum at the end of 10 years.Assume that a sinking fund can earn money at j₁ = x%.At which of the following values of × will the second source be cheaper?
(a) 7% (b) 6% (c) 5%

Question

Quizplus · Accepted Answer

The second source will be cheaper if the present worth of the annual interest payments plus the present worth of the lump sum repayment at the end of 10 years is less than the present worth of the annual payments for the first source.The present worth of the annual interest payments for the second source is:$$PW = \frac{0.08 	imes 200,000}{x} 	imes \frac{1 - (1 + x)^{-10}}{x}$$The present worth of the lump sum repayment at the end of 10 years is:$$PW = \frac{200,000}{(1 + x)^{10}}$$The present worth of the annual payments for the first source is:$$PW = 31,164 	imes \frac{1 - (1 + 0.09)^{-10}}{0.09}$$Setting the present worth of the annual interest payments plus the present worth of the lump sum repayment for the second source equal to the present worth of the annual payments for the first source, we get:$$\frac{0.08 	imes 200,000}{x} 	imes \frac{1 - (1 + x)^{-10}}{x} + \frac{200,000}{(1 + x)^{10}} = 31,164 	imes \frac{1 - (1 + 0.09)^{-10}}{0.09}$$Solving for x, we get:$$x = 0.07$$Therefore, the second source will be cheaper if the sinking fund can earn money at 7%.

A City Has Two Sources to Borrow $200,000