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A company has borrowed to invest in a project. The loan calls for a payment of $17,384 every year for three years. The lender quoted A company a rate of 8.4% p.a. compounding monthly. How much has A company borrowed?

Question

A company has borrowed to invest in a project. The loan calls for a payment of $17,384 every year for three years. The lender quoted A company a rate of 8.4% p.a. compounding monthly. How much has A company borrowed?

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Solution

To solve this problem, we need to find the present value of the loan, which is the amount the company has borrowed. The present value (PV) is the total amount that a series of future payments is worth now.

Here are the steps to calculate the present value:

  1. Identify the variables from the problem:

    • The annual payment (PMT) is $17,384.
    • The number of periods (n) is 3 years.
    • The interest rate (r) is 8.4% per annum, compounded monthly. This means the monthly interest rate is 8.4%/12 = 0.7%.

  2. Use the formula for the present value of an annuity: PV = PMT * [(1 - (1 + r)^-n) / r]

  3. Substitute the values into the formula: PV = $17,384 * [(1 - (1 + 0.007)^-3) / 0.007]

  4. Calculate the present value: PV = 17,384∗[(1−(1.007)−3)/0.007]PV=17,384 * [(1 - (1.007)^-3) / 0.007] PV = 17,384 * [(1 - 0.978947) / 0.007] PV = 17,384∗[0.021053/0.007]PV=17,384 * [0.021053 / 0.007] PV = 17,384 * 3.0076 PV = $52,263.38

So, the company has borrowed approximately $52,263.38.

This problem has been solved

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