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Chapter 8: Problem 60

Give two examples that illustrate the difference between a compound interest problem involving future value and a compound interest problem involving present value.

Short Answer

Expert verified
Examples illustrating future value and present value in compound interest problems are computed as follows: The future value of a $5000 investment at 5% annual compound interest after 3 years will be $5788.13. On the other hand, to amass $6000 in a span of 4 years at the same annual compound interest rate of 5%, an initial investment of $4956.38 should be made.

Step by step solution

01

Example 1 - Future Value Compound Interest

Consider an investment of $5000 made today, which will compound annually at an interest rate of 5% for 3 years. To find the future value of this investment, use the compound interest formula:\[FV = PV \times (1 + r)^n\]where\(FV\) is future value\(PV\) is present value\(r\) is rate of interest \(n\) is time in yearsSubstituting the given values:\[FV = 5000 \times (1 + 0.05)^3 = $5788.13\]So, after 3 years, the initial $5000 will grown to $5788.13.
02

Example 2 - Present Value Compound Interest

Consider now that we have a goal of having $6000 in 4 years. We can find out how much we need to invest now in order to achieve this goal assuming same annual compound interest rate of 5%.We use the present value formula:\[PV = \frac{FV}{(1 + r)^n}\]where\(FV\) is future value\(PV\) is present value\(r\) is rate of interest \(n\) is time in yearsSubstituting the given values:\[PV = \frac{6000}{(1 + 0.05)^4} = $4956.38\]This implies that we need to invest $4956.38 now to have $6000 after 4 years.

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Most popular questions from this chapter

In Exercises 1-10, a. Find the value of each annuity. Round to the nearest dollar b. Find the interest.$$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \$ 1000 \text { at the end of } \\ \text { every three months } \end{array} & \begin{array}{l} 6.25 \% \text { compounded } \\ \text { quarterly } \end{array} & 6 \text { years } \\ \hline \end{array} $$

A bank bills its credit card holders on the first of each month for each itemized billing. The card provides a 20-day period in which to pay the bill before charging interest. If the card holder wants to buy an expensive gift for a September 30 wedding but can't pay for it until November 5 , explain how this can be done without adding an interest charge.

Solve for \(P\) : $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} . $$ What does the resulting formula describe?

a. Suppose that between the ages of 22 and 40 , you contribute \(\$ 3000\) per year to a \(401(\mathrm{k})\) and your employer contributes \(\$ 1500\) per year on your behalf. The interest rate is \(8.3 \%\) compounded annually. What is the value of the \(401(\mathrm{k})\), rounded to the nearest dollar, after 18 years? b. Suppose that after 18 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the \(401(\mathrm{k})\). How much money, to the nearest dollar, will you have in the plan when you reach age 65 ? c. What is the difference between the amount of money you will have accumulated in the \(401(\mathrm{k})\) and the amount you contributed to the plan?

Exercises 19 and 20 refer to the stock tables for Goodyear (the tire d. How many shares of this company's stock were traded company) and Dow Chemical given below. In each exercise, use yesterday? the stock table to answer the following questions. Where necessary, e. What were the high and low prices for a share yesterday? round dollar amounts to the nearest cent. f. What was the price at which a share last traded when the stock a. What were the high and low prices for a share for the past exchange closed yesterday? b. If you owned 700 shares of this stock last year, what dividend g. What was the change in price for a share of stock from the did you receive? h. Compute the company's annual earnings per share using c. What is the annual return for the dividends alone? How does Annual earnings per share this compare to a bank offering a \(3 \%\) interest rate? $$ =\frac{\text { Yesterday's closing price per share }}{P E \text { ratio }} . $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 56.75 & 37.95 & \begin{array}{c} \text { Dow } \\ \text { Chemical } \end{array} & \text { DOW } & 1.34 & 3.0 & 12 & 23997 & 44.75 & 44.35 & 44.69 & +0.16 \\ \hline \end{array} $$
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