Independent solution

How to solve this Loan Amortization question

Setup

Setup

Translate the unusual post-first-payment balance into a relation between the unknown original principal and interest rate.

B1=L−100+iL=L+25⟹iL=125B_1=L-100+iL=L+25\Longrightarrow iL=125

Model

Model

The full loan value equals a 16-year payment level of 300 offset by 200 during the first eight years.

L=300a16‾∣−200a8‾∣L=300a_{\overline{16}|}-200a_{\overline8}|

Compute

Compute

Multiplying by i and using iL = 125 produces a quadratic in v to the eighth power; the stated bound selects the root 0.5.

300v16−200v8+25=0,v8=0.5300v^{16}-200v^8+25=0,\quad v^8=0.5

Answer

Answer

Immediately after payment eight, only eight payments of 300 remain, whose value is about 1,657, choice A.

B8=300a8‾∣=1657(A)\boxed{B_8=300a_{\overline8}|=1657\quad\text{(A)}}

Calculator reproduction

BA II Plus keystrokes

Check END/BGN, period, sign, TVM, and cash-flow setup

  1. 2nd CLR TVM; 2nd I/Y; 1 ENTER; ↓; 1 ENTER; 2nd CPT; 2nd PMT; if BGN is displayed, 2nd ENTER; 2nd CPT; 8 N; 9.050773 I/Y; 300 +/- PMT; 0 FV; CPT PVPV = 1657.32END mode; I/Y is the annual rate implied by the eight-year discount factor one half.