Question

How long will it take monthly payments of $600 to repay a $65,000 loan if the interest rate on the loan is 9.5% compounded semiannually? How much will the time to repay the loan be reduced if the payments are $50 more per month?

Answer 1

Monthly Payments = $ 600 | Loan amount = $ 65,000 | Rate = 9.5% compounded semiannually.

First we need to find the monthly effective rate to be used in the calculate.

As the rate is compounded semiannually, hence, 9.5% / 2 becomes the 6-month effective rate.

Using this, we can calculate the monthly rate.

(1+monthly rate)⁶ - 1 = 6 month EAR

(1+monthly rate) = (1 + 4.75%)^1/6

Monthly rate = (1 + 4.75%)^1/6 - 1

Monthly rate = 0.007764 or 0.7764%

Now using the Loan amount, monthly rate, payment and time as T, we can find use the Annuity formula to solve for T.

PV of Annuity = (PMT / R)*(1 - (1+R)^-T)

PV = 65,000 | PMT = 600 | R = 0.7764% | Time = T

=> 65000 = (600 / 0.7764%)*(1 - (1+0.7764%)^-T)

=> 65000 / (600 / 0.7764%) = (1 - (1+0.7764%)^-T)

=> 0.8411 = 1 - 1 / (1+0.7764%)^T

=> 1 / (1+0.7764%)^T = 1 - 0.8411

=> 1 / 0.1589 = (1+0.7764%)^T

=> 6.2933 = (1+0.7764%)^T

Putting Log10 on both sides of the equation.

=> Log10 (6.2933) = T* Log10 (1.007764)

=> T = Log10 (6.2933) / Log10 (1.007764)

T = 237.84 or 238 months

It will take 237.84 or 238 months to repay the loan of 65000 at 600 monthly payment.

The 238 months are 19 years and 10 months.

With $ 50 more per month means payment becomes $ 650. Except payment everything remains the same.

Monthly rate = 0.7764% | Loan Amount = 65000 | Let Time be T

PV of Annuity = (PMT / R)*(1 - (1+R)^-T)

=> 65000 = (650 / 0.7764%)*(1 - (1+0.7764%)^-T)

=> 65000 / (650 / 0.7764%) = (1 - (1+0.7764%)^-T)

=> 0.7764 = 1 - 1 / (1+0.7764%)^T

=> 1 / (1+0.7764%)^T = 0.2236

=> (1+0.7764%)^T = 1 / 0.2236

Putting Log10 on both sides

=> T * Log10 (1.007764) = Log10 (4.4723)

=> T = Log10 (4.4723) / Log10 (1.007764)

=> T = 0.650531 / 0.003359

T = 193.68 or 194 months

If payments per month increases by $ 50 per month, then time will reduce by = 237.84 - 193.68 = 44.16 or 44 months

44 months in terms of years is 3 years and 8 months.