Independent solution

How to solve this Loan Amortization question

Setup

Setup

Use monthly rate 0.6% and value every payment at loan issue.

LA=ma48‾∣+1.5mv48a48‾∣L_A=ma_{\overline{48}|}+1.5m v^{48}a_{\overline{48}|}

Model

Model

Loan A uses month-end annuities, while Loan B uses beginning-of-month annuities.

LB=1.2ma¨48‾∣+0.9mv48a¨48‾∣L_B=1.2m\ddot a_{\overline{48}|}+0.9m v^{48}\ddot a_{\overline{48}|}

Compute

Compute

Common factors m and the 48-payment annuity cancel from the ratio, leaving the staged multipliers and timing adjustment.

LALB=1+1.5v48(1.2+0.9v48)(1.006)=1.12668\frac{L_A}{L_B}=\frac{1+1.5v^{48}}{(1.2+0.9v^{48})(1.006)}=1.12668

Answer

Answer

The loan-amount ratio is approximately 1.127, corresponding to choice A.

LA/LB≈1.127(A)\boxed{L_A/L_B\approx1.127\quad\text{(A)}}