How to Solve Interest Theory — Time Value of Money — accumulation across piecewise effective rates Questions on Exam FM
Sample Practice Problem
ID: #SXWIR
Two accounts each receive one unit at time \(0\). Fund \(Y\) has accumulation function \(A_Y(t)=(t^2+2t+4)/4\) on \(0\le t\le1\), corresponding to force \(\delta_t=(2t+2)/(t^2+2t+4)\). Fund \(X\) earns simple interest at the annual effective rate equivalent to Fund \(Y\) over its first year. Calculate the time \(t\) at which the balance in Fund \(X\) minus the balance in Fund \(Y\) is greatest.
📖 Worked Solution & Strategy
Fund \(Y\) grows by \(A_Y(1)=(1+2+4)/4\), so its first-year effective rate is \(i=(2+1)/4\). Therefore \(A_X(t)=1+((2+1)/4)t\). The balance difference is \(D(t)=1+((2+1)/4)t-(t^2+2t+4)/4=(t-t^2)/4\); the terms containing \(2\) cancel. Thus \(D'(t)=(1-2t)/4=0\) at \(t=1/2\), and \(D''(t)=-1/2<0\). The maximum occurs at \(t=0.5000\).