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Chapter 8: Problem 17

In Exercises 17-20, round answers up to the nearest cent. How much money should be deposited today in an account that earns \(6 \%\) compounded semiannually so that it will accumulate to \(\$ 10,000\) in three years?

Short Answer

Expert verified
The initial amount to be deposited in the account, rounded up to the nearest cent, will be the solution obtained after following the above steps.

Step by step solution

01

Identify and write down the given values

The given values are: \( A = \$10000 \) (the amount the account will accumulate to), \( r = 6\% \) or \( 0.06 \) in decimal form, \( n = 2 \) (interest is compounded semiannually), and \( t = 3 \) years.
02

Plug the given values into the compound interest formula

So the formula becomes \( 10000 = P(1 + 0.06/2)^(2*3) \)
03

Solve the equation for P

To find P, the initial deposit needed, you'll have to divide both sides of the equation by \( (1 + 0.06/2)^(2*3) \). Do the calculation to get the value of P.
04

Round up to the nearest cent

After solving for P, make sure to round your answer up to the nearest cent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Time Value of Money
The time value of money is a fundamental concept in financial mathematics that highlights the principle that a sum of money in hand today is worth more than the same sum at a future date due to its potential earning capacity. This principle is predicated on the opportunity to earn interest, which means that money available today can be invested to earn additional income over time.

Essentially, this concept provides the basis for understanding the value of investments, loans, and savings. Calculating how much money one should deposit today to obtain a future amount involves discounting the future amount by the interest rate over the period. This ensures that you consider the earning capacity of the initial investment and understand its true value over time鈥攁 crucial decision-making factor for both individuals and businesses planning their finances.
Financial Mathematics
Financial mathematics is the application of mathematical methods to financial problems. It involves formulas and calculations to analyze and solve issues related to money, investments, and other financial instruments. One of the key formulae in financial mathematics is the compound interest formula, which calculates the amount of money that an investment will grow to over a certain period, at a given interest rate and compounding frequency.

In the context of the exercise, financial mathematics helps determine how much money needs to be deposited (the present value) to achieve a desired future sum with semiannual compounding. Using the formula, you can solve various types of problems, from saving for retirement to calculating loan payments. Understanding these formulas and concepts is essential for making informed financial decisions and optimizing the growth of investments over time.
Semiannual Compounding
Semiannual compounding refers to the process where interest is calculated and added to the principal balance of an investment or loan twice a year. Unlike simple interest, which is calculated only on the principal amount, compound interest is calculated on the principal and the interest that has been added previously.

With semiannual compounding, the annual interest rate is divided by 2 to find the semiannual rate, and the number of years is multiplied by 2 to determine the number of compounding periods. The result is a faster accumulation of interest because the investment grows at each compounding period. It's crucial for students to understand that the more frequently interest is compounded, the more interest will be earned on an investment over time. Therefore, semiannual compounding can significantly affect the growth of your investment compared to annual compounding, emphasizing the importance of compounding frequency in financial planning and investment strategies.

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Most popular questions from this chapter

The cost of a home is financed with a \(\$ 120,00030\)-year fixed-rate mortgage at \(4.5 \%\). a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage. Round entries to the nearest cent.$$ \begin{array}{|c|c|c|c|} \hline \begin{array}{l} \text { Payment } \\ \text { Number } \end{array} & \text { Interest } & \text { Principal } & \text { Loan Balance } \\ \hline 1 & & & \\ \hline 2 & & & \\ \hline 3 & & & \\ \hline \end{array} $$

What is meant by the value of an annuity?

Here are additional formulas that you will use to solve some of the remaining exercises. Be sure you understand what each formula describes and the meaning of the variables in the formulas. Here are two ways of investing 30,000 for 20 years $$ \begin{array}{|l|l|l|} \hline \text { Lump-Sum Deposit } & \text { Rate } & \text { Time } \\ \hline \$ 30,000 & \begin{array}{l} 5 \% \text { compounded } \\ \text { annually } \end{array} & 20 \text { years } \\ \hline \text { Periodic Deposit } & \text { Rate } & \text { Time } \\ \hline \begin{array}{l} \text { \$1500 at the end of } \\ \text { each year } \end{array} & \begin{array}{l} 5 \% \text { compounded } \\ \text { annually } \end{array} & \text { 20 years } \\ \hline \end{array} $$ a. After 20 years, how much more will you have from the lump-sum investment than from the annuity? b. After 20 years, how much more interest will have been earned from the lump- sum investment than from the annuity?

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

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