Question

A loan is repaid with payments of $2226 made at the end of each month for 12 years. If interest on the loan is 5.2%, compounded semi-annually, what is the initial value of the loan? Enter to the nearest cent (two decimals). Do not use $ signs or commas.

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Murray

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46 Answers

To find the initial value of the loan, we can use the formula for the present value of an ordinary annuity:

PV = \frac{P}{r} \left(1 - \frac{1}{(1+r)^n}\right)

where:

- PV is the present value or initial value of the loan

- P is the payment amount made at the end of each period (monthly)

- r is the interest rate per period (semi-annually)

- n is the total number of periods (monthly payments for 12 years = 12 * 12 = 144 periods)

First, let's convert the interest rate from annual to semi-annually:

r = \frac{5.2}{2} = 2.6\% = 0.026

Now we can calculate the initial value of the loan using the given values:

PV = \frac{2226}{0.026} \left(1 - \frac{1}{(1+0.026)^{144}}\right)

Simplifying the equation:

PV = \frac{2226}{0.026} \left(1 - \frac{1}{(1.026)^{144}}\right)

Calculating the present value:

PV=\frac{2226}{0.026}\left(1-\frac{1}{(1.026)^{144}}\right)\approx83.475

Therefore, the initial value of the loan is approximately 83.475

where:

- PV is the present value or initial value of the loan

- P is the payment amount made at the end of each period (monthly)

- r is the interest rate per period (semi-annually)

- n is the total number of periods (monthly payments for 12 years = 12 * 12 = 144 periods)

First, let's convert the interest rate from annual to semi-annually:

Now we can calculate the initial value of the loan using the given values:

Simplifying the equation:

Calculating the present value:

Therefore, the initial value of the loan is approximately 83.475

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