Independent solution

How to solve this Monthly Loan Equations question

Setup

Setup

Let j be the monthly effective rate corresponding to the unknown nominal rate. Price option 1 at the loan date.

225,000=1,960a180‾∣j225{,}000=1{,}960a_{\overline{180}|j}

Model

Model

After solving for j, option 2 is an annuity-due because every payment is made at the beginning of a month.

225,000=Xa¨360‾∣j225{,}000=X\ddot a_{\overline{360}|j}
a¨360‾∣j=(1+j)a360‾∣j\ddot a_{\overline{360}|j}=(1+j)a_{\overline{360}|j}

Compute

Compute

The first equation gives j = 0.005416723. The 360-payment annuity-due factor then gives X.

X=225,000a¨360‾∣0.005416723X=\frac{225{,}000}{\ddot a_{\overline{360}|0.005416723}}
X=1,414.501X=1{,}414.501

Answer

Answer

The monthly payment rounds to 1415, so the official answer is A.

X≈1,415(A)\boxed{X\approx1{,}415\quad\text{(A)}}

Calculator reproduction

BA II Plus keystrokes

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

  1. 2ND · CLR TVM; 180 · N; 225000 · PV; 1960 · +/− · PMT; 0 · FV; CPT · I/Y0.541672The displayed rate is monthly because the cash-flow period is one month.
  2. 2ND · PMT; 2ND · SET until BGN; 2ND · QUIT; 360 · N; 225000 · PV; 0 · FV; CPT · PMT−1414.501Retain the monthly I/Y value and switch to beginning mode.