How to Solve Interest Theory — Time Value of Money — equations for two funds earning different rates Questions on Exam FM
Sample Practice Problem
ID: #GL75J
A coastal engineering cooperative borrows \(9,000\). The debt is repaid by ten payments one year apart, with the first payment due one-half year after the loan is issued. The first payment is one-half of each later payment. Interest is \(5\%\) annually effective during the first \(4.5\) years and \(3\%\) annually effective thereafter. Calculate the amount of the first payment.
📖 Worked Solution & Strategy
Let the first payment be \(X\), so each later payment is \(2X\). For a payment at time \(t\le 4.5\), its time-zero discount factor is \(1.05^{-t}\); after time \(4.5\), it is \(1.05^{-4.5}1.03^{-(t-4.5)}\). Denote this piecewise factor by \(d(t)\). The equation of value is \(9,000=X(1.05)^{-0.5}+2X\sum_{k=1}^9 d(0.5+k)\), which gives \(X=590.1347\) and \(2X=1180.2694\). Immediately after the fifth payment, five level payments remain at the new rate, so the prospective balance is \(2X a_{\overline{5}|0.03}=5405.2883\). Therefore the requested value is \(590.1347\), which matches the correct option.