Question

A loan of $100,000 is made today. The borrower will make equal repayments of $3568 per month with the first payment being exactly one month from today. The interest being charged on this loan is constant (but unknown).

For the following two scenarios, calculate the interest rate being charged on this loan, expressed as a nominal annual rate in percentage:

(a) The loan is fully repaid exactly after 33 monthly repayments, i.e., the loan outstanding immediately after 33 repayments is exactly 0.

Answer 1

The annual interest rate is 6.12%.

Step-by-step explanation:

This is a compound interest problem

The compound interest formula is given by:

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

In this problem, we have that:

To find the interest rate, we first have to find the value of A, that is, the amount paid. The total amound paid was $3568 paid monthly for 33 months, so:

P is the value of the loan, so

r is the interest rate, the value we have to find.

We have to find the annual interest rate, so .

This value was paid in 33 months. However, the unit of t is years. So

Applying the formula:

To find r

The annual interest rate is 6.12%.

Answer 2

The annual interest rate is 6.12%.

Explanation:

This is a compound interest problem

The compound interest formula is given by:

`A = P(1 + (r)/(n))^(nt)`

In which A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

In this problem, we have that:

To find the interest rate, we first have to find the value of A, that is, the amount paid. The total amound paid was $3568 paid monthly for 33 months, so:

`A = 33*3,568 = 117,744`

P is the value of the loan, so
`P = 100,000`

r is the interest rate, the value we have to find.

We have to find the annual interest rate, so
`n = 1`.

This value was paid in 33 months. However, the unit of t is years. So
`t = (33)/(12) = 2.75`

Applying the formula:

`A = P(1 + (r)/(n))^(nt)`

`117,744 = 100,000(1 + r)^(2.75)`

`(1 + r)^(2.75) = 1.17744`

To find r

`\sqrt[2.75]{(1 + r)^(2.75)} = \sqrt[2.75]{1.17744}`

`1 + r = 1.0612`

`r = 0.0612`

The annual interest rate is 6.12%.