Independent solution

How to solve this Loan Refinancing Rate question

Setup

Setup

Compute the original monthly payment on 400,000 over 180 months at 0.75% per month. Immediately after month 36, value the remaining 144 payments to obtain the balance.

R=400000/a180‾∣ 0.0075=4057.07R=400000/a_{\overline{180}|\,0.0075}=4057.07

Model

Model

The refinanced payment available for principal and interest is 4,057.07 − 409.88. Set the balance equal to the present value of 144 such payments at unknown monthly rate j/12.

B36=Ra144‾∣ 0.0075=356499.17B_{36}=Ra_{\overline{144}|\,0.0075}=356499.17

Compute

Compute

The root is j/12 = 0.00575, so the nominal annual refinance rate is 6.90%.

356499.17=(4057.07−409.88)a144‾∣ j/12⟹j=0.069356499.17=(4057.07-409.88)a_{\overline{144}|\,j/12}\Longrightarrow j=0.069

Answer

Answer

The calculation gives 6.90% for loan refinancing rate, matching published choice D.

j(12)=6.90%(D)\boxed{j^{(12)}=6.90\%\quad\text{(D)}}

Calculator reproduction

BA II Plus keystrokes

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

  1. 2nd CLR TVM; 180 N; 0.75 I/Y; 400000 +/- PV; 0 FV; CPT PMT4057.07Original monthly payment.
  2. 144 N; 0.75 I/Y; 4057.07 PMT; 0 FV; CPT PV−356499.17Balance immediately after payment 36.
  3. 144 N; 356499.17 +/- PV; 3647.19 PMT; 0 FV; CPT I/Y; × 12 =6.90Nominal annual refinance rate.