Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

First value the four quarterly payments within a representative year at that year's beginning.

j=(1+0.12/4)4−1=0.1255088j=(1+0.12/4)^4-1=0.1255088

Model

Model

The amounts 100 through 400 form an increasing annuity-immediate over four quarters.

B=100(Ia)4‾∣ 0.03B=100(Ia)_{\overline{4}|\,0.03}

Compute

Compute

Those yearly blocks repeat at times zero through nine, so their block values form an annual annuity-due.

P=Ba¨10‾∣ jP=B\ddot a_{\overline{10}|\,j}

Answer

Answer

Using the equivalent annual rate 12.55% gives the equation in choice E.

P=100(Ia)4‾∣ 3%a¨10‾∣ 12.55%(E)\boxed{P=100(Ia)_{\overline{4}|\,3\%}\ddot a_{\overline{10}|\,12.55\%}\quad\text{(E)}}