Independent solution

How to solve this Perpetuity Payment Frequency question

Setup

Setup

Let v_m be the one-month discount factor. The two-month discount factor is its square.

vm=(1+i)−1/12v_m=(1+i)^{-1/12}
v2m=vm2v_{2m}=v_m^2

Model

Model

A perpetuity-due paying X each payment period has price X divided by one minus the period discount factor.

198,000=X1−vm198{,}000=\frac{X}{1-v_m}
100,000=X1−vm2100{,}000=\frac{X}{1-v_m^2}

Compute

Compute

Divide the equations to cancel X and factor one minus v squared.

1.98=1−vm21−vm=1+vm1.98=\frac{1-v_m^2}{1-v_m}=1+v_m
vm=0.98v_m=0.98
i=0.98−12−1=0.274345i=0.98^{-12}-1=0.274345

Answer

Answer

The annual effective rate is 27.4%, choice E.

i≈27.4%(E)\boxed{i\approx27.4\%\quad\text{(E)}}