# A firm recently purchased a new facility costing $933,000. The firm financed this purchase with... ## Question: A firm recently purchased a new facility costing$933 thousand. The firm financed this purchase with an amortized loan at an interest rate of 9.6 per cent APR, with monthly payments of \$18.4 thousand. How long will it take to pay off this loan?

(Answer in months, accurate to two decimal places.)

## Monthly Loan Payment:

Monthly loan payment for a loan is calculated such that the present value of these payments, discounted at the loan's interest rate, is equal to the amount borrowed. The higher the interest rate, all else the same, the higher the monthly payments.

It will take 65.29 months to pay off the loan.

The loan is payoff when the present value of the monthly payments is equal to the amount borrowed. We can use the following formula to compute the monthly payment for a loan with principal {eq}P {/eq}, monthly interest rate {eq}r{/eq} and number of monthly payments {eq}T{/eq}:

• {eq}\displaystyle \frac{Pr}{1 - (1 + r)^{-T}} {/eq}

In this question, the amount borrowed is 933,000, the effective monthly payment is 9.6% / 12 = 0.8%, and the monthly payment is 18.4*1000 = 18,400. Applying the formula, we have:

• {eq}\displaystyle \frac{9330,000*0.8\%}{1 - (1 + 0.8\%)^{-T}} = 18,400 {/eq}
• {eq}1 - (1 + 0.8\%)^{-T} = 0.4056521739 {/eq}
• {eq}(1 + 0.8\%)^{-T} = 0.5943478261 {/eq}
• {eq}T = 65.29 {/eq}