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Chapter 8: Problem 29

To offer scholarships to children of employees, a company invests \(\$ 10,000\) at the end of every three months in an annuity that pays \(10.5 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of 10 years? b. Find the interest.

Short Answer

Expert verified
a) The company will have about $457,819.05 in scholarship funds at the end of ten years. b) The interest earned will be approximately $57,819.05.

Step by step solution

01

Clarify the Variables

Identify and understand the values given: - The periodic investment (PMT) is $10,000, which is made every three months (quarterly). - The interest rate per period (i) is 10.5%, but since it's compounded quarterly, we need to divide it by 100 and then by 4 to convert it into a rate per quarter, giving us \(0.105/4 = 0.02625\) or 2.625%.- The number of periods (n) is 10 years, but since the investment is made quarterly, we multiply it by 4 to get the number of investment periods, resulting in \(10*4 = 40\) periods.
02

Find the Future Value of the Annuity (Part a)

Applying the formula for the future value of ordinary annuity: \( FV = PMT * [(1 + i)^n - 1] / i \)Substitute values as follows: \( FV = 10,000 * [(1 + 0.02625)^{40} - 1] / 0.02625 \)which simplifies to\( FV = 10,000 * [2.2080404 - 1] / 0.02625 \).So the future value becomes approximately $457,819.05.
03

Calculate the Interest Earned (Part b)

The interest earned can be found by subtracting the total amount invested from the future value. The total amount invested is the periodic investment amount multiplied by the number of periods: \(10,000 * 40 = $400,000\). Subtracting this from the future value gives the total interest: $457,819.05 - $400,000 = $57,819.05.

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Most popular questions from this chapter

In Exercises 31-34, round up to the nearest dollar. You would like to have \(\$ 3500\) in four years for a special vacation following college graduation by making deposits at the end of every six months in an annuity that pays \(5 \%\) compounded semiannually. a. How much should you deposit at the end of every six months? b. How much of the \(\$ 3500\) comes from deposits and how much comes from interest?

What is a down payment?

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ Round answers to the nearest dollar. Suppose that you borrow \(\$ 10,000\) for four years at \(8 \%\) toward the purchase of a car. Find the monthly payments and the total interest for the loan.

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In Exercises 11-14, use the formula $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} $$ Round all computations to the nearest dollar. Suppose that you drive 40,000 miles per year and gas averages \(\$ 4\) per gallon. a. What will you save in annual fuel expenses by owning a hybrid car averaging 40 miles per gallon rather than an SUV averaging 16 miles per gallon? b. If you deposit your monthly fuel savings at the end of each month into an annuity that pays \(5.2 \%\) compounded monthly, how much will you have saved at the end of six years?
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