Independent solution

How to solve this Loan Balances and Amortization question

Setup

Setup

For a four-payment unit-payment loan, principal portions are v to powers four through one, and interest portions are one minus the corresponding principal portions.

Pt=v5−tP_t=v^{5-t}
It=1−v5−tI_t=1-v^{5-t}

Model

Model

Insert the specified sums into their relation and simplify in terms of v.

(1−v2)+(1−v)=(1+i)2(v4+v3)(1-v^2)+(1-v)=(1+i)^2(v^4+v^3)
v2+v−1=0v^2+v-1=0

Compute

Compute

The positive root is v = 0.61803399, giving i = 0.61803399.

i=v−1−1=0.6180339887i=v^{-1}-1=0.6180339887

Answer

Answer

The annual effective rate is approximately 0.618, selecting choice D.

i≈0.618(D)\boxed{i\approx0.618\quad\text{(D)}}