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Chapter 8: Problem 20

How much money should be deposited today in an account that earns \(10.5 \%\) compounded monthly so that it will accumulate to \(\$ 22,000\) in four years?

Short Answer

Expert verified
The initial deposit required to accumulate \$22000 after 4 years with an interest rate of 10.5% compounded monthly is approximately \$16267.45.

Step by step solution

01

Understand compound interest formula

The formula for compound interest is \(A = P(1+\frac{r}{n})^{nt}\), which gives the final amount accumulated after a given period, considering that the interest earned is compounded.
02

Fill given data into the formula

We know the final amount \(A = \$22000\), the interest rate \(r = 10.5\% = 0.105\), compounding frequency \(n = 12\) times a year (once per month), time period \(t = 4\) years. The aim is to find the principal amount, \(P\). Therefore, replace the known values into the formula.
03

Solve for the principal amount

The equation from step 2 is now as follows: \(22000 = P(1+0.105/12)^(12*4)\). This must be rearranged and solved for \(P\), giving \(P = \frac{22000}{(1+0.105/12)^{12*4}}\). Now calculate the value of \(P\).

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Most popular questions from this chapter

Describe what happens to the portions of payments going to principal and interest over the life of an installment loan.

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? $$ =\frac{\text { Yesterday's closing price per share }}{P E \text { ratio }} . $$ $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 73.25 & 45.44 & \text { Goodyear } & \text { GT } & 1.20 & 2.2 & 17 & 5915 & 56.38 & 54.38 & 55.50 & +1.25 \\ \hline \end{array} $$

Make Sense? In Exercises 47-53, determine whether each statement makes sense or does not make sense, and explain your reasoning. By putting \(\$ 10\) at the end of each month into an annuity that pays \(3.5 \%\) compounded monthly, I'll be able to retire comfortably in just 30 years.

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