Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

The first-payment and last-payment value dates are separated by n minus one half-year periods.

7968.89(1+j)n−1=19549.257968.89(1+j)^{n-1}=19549.25

Model

Model

Use their ratio to replace the terminal discount factor inside the annuity-due present-value expression.

7968.89=1000(1+1−7968.89/19549.25j)7968.89=1000\left(1+\frac{1-7968.89/19549.25}{j}\right)

Compute

Compute

That substitution isolates the half-year rate at 8.5%.

j=0.085j=0.085

Answer

Answer

Compounding twice gives annual effective rate 17.72%, which rounds to choice E.

i=(1.085)2−1=17.72%(E)\boxed{i=(1.085)^2-1=17.72\%\quad\text{(E)}}