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Chapter 8: Problem 18

How much money should be deposited today in an account that earns \(7 \%\) compounded semiannually so that it will accumulate to \(\$ 12,000\) in four years?

Short Answer

Expert verified
The amount of money that should be deposited today to accumulate to \$12,000 in four years in an account that earns a 7% interest compounded semiannually is approximately \$9,366.40

Step by step solution

01

Understand the Concept

The present value \(P\) in a compound interest model is given by the formula \(P = F{(1 + r/n)}^{-nt}\) where:\n- \(F\) is the future value\n- \(r\) is the annual interest rate (decimal)\n- \(n\) is the number of times that interest is compounded per year\n- \(t\) is the time the money is invested for in years.
02

Substitute the Known Values

Given that:\n- The future value \(F = \$ 12,000\)\n- The annual interest rate \(r = 7\%\) or \(0.07\) when expressed as a decimal\n- Interest is compounded semiannually, so \(n = 2\)\n- The investment time period \(t = 4\) years, substitute these values into the present value formula.
03

Calculate the Present Value

Substitute all the known values into the formula: \(P = 12000{(1 + 0.07/2)}^{-2*4}\). Simplifying this leads to the calculation of the present value.

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Key Concepts

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

Present Value
The Present Value is a key concept in finance. It represents the amount of money that needs to be invested today to reach a specific future sum, taking into account a certain interest rate and compounding period.
In simple terms, it answers the question: "How much is a future sum worth today?" Understanding present value helps in planning effective investments and saving strategies.

To calculate the present value, we use the formula: \[P = \frac{F}{(1 + r/n)^{nt}}\]
  • \(P\) is the present value or the amount to be invested today.
  • \(F\) is the future value, or the amount you aim to reach in the future (in our case, $12,000).
  • \(r\) is the annual interest rate in decimal form (0.07 for 7%).
  • \(n\) is the frequency of compounding per year (2 for semiannual).
  • \(t\) is the time in years (4 years here).
This formula helps us determine how much we need to deposit now to ensure that we reach our future goal, given an interest rate and compounding frequency.
Future Value
The Future Value is what an investment is worth after one or more periods, considering any interest that has been accumulated over that period. It tells us how much an investment will grow over time.
Imagine you deposit a certain amount today. With the added interest over time, this will grow into a larger sum at a future date, which is called the future value.
To find the future value, we use the formula:\[F = P (1 + r/n)^{nt}\]
  • \(F\) stands for the future value.
  • \(P\) is the present value or initial investment.
  • \(r\) is the annual interest rate (as a decimal).
  • \(n\) represents how often the interest is compounded annually.
  • \(t\) is the number of years the money is invested.
By understanding the future value, investors and savers can better appreciate how money will grow given specific conditions. It's instrumental for long-term financial planning.
Interest Rate
The Interest Rate in the context of compound interest, is the percentage at which your money grows annually. It鈥檚 a vital component when evaluating how quickly investments appreciate.
With compound interest, the interest rate isn't merely applied to the initial investment but to the accumulated amount, which includes previously earned interest.
Interest rates are usually presented in percentage form but must be converted to a decimal when used in formulas.
  • For example, a 7% interest rate converts to 0.07 in formulas.
Compounding frequency matters. It refers to how many times the interest is applied per year, affecting the total return on an investment.
  • Annual compounding means once a year, semiannual means twice, quarterly four times, and so on.
Thus, the interest rate, along with the compounding frequency, influences the growth of your investment and helps in planning for future financial goals effectively.

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Most popular questions from this chapter

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 $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { 52-Week High } & \text { 52-Week Low } & \text { Stock } & \text { SYM } & \text { Div } & \text { Yld \% } & \text { PE } & \text { Vol 100s } & \text { Hi } & \text { Lo } & \text { Close } & \text { Net Chg } \\ \hline 73.25 & 45.44 & \text { Goodyear } & \text { GT } & 1.20 & 2.2 & 17 & 5915 & 56.38 & 54.38 & 55.50 & +1.25 \\ \hline \end{array} $$

a. Suppose that between the ages of 25 and 37 , you contribute \(\$ 3500\) per year to a \(401(\mathrm{k})\) and your employer matches this contribution dollar for dollar on your behalf. The interest rate is \(8.25 \%\) compounded annually. What is the value of the \(401(\mathrm{k})\), rounded to the nearest dollar, after 12 years? b. Suppose that after 12 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 3-4 involve credit cards that calculate interest using the average daily balance method. The monthly interest rate is \(1.2 \%\) of the average daily balance. Each exercise shows transactions that occurred during the June 1 -June 30 billing period. In each exercise, a. Find the average daily balance for the billing period. Round to the nearest cent. b. Find the interest to be paid on July 1, the next billing date. Round to the nearest cent. c. Find the balance due on July 1 . d. This credit card requires a \(\$ 30\) minimum monthly payment if the balance due at the end of the billing period is less than \(\$ 400\). Otherwise, the minimum monthly payment is \(\frac{1}{25}\) of the balance due at the end of the billing period, rounded up to the nearest whole dollar. What is the minimum monthly payment due by July 9? $$ \begin{array}{|l|c|} \hline \text { Transaction Description } & \text { Transaction Amount } \\ \hline \text { Previous balance, } \$ 2653.48 & \\ \hline \text { June } 1 \quad \text { Billing date } & \\ \hline \text { June } 6 \quad \text { Payment } & \$ 1000.00 \text { credit } \\\ \hline \text { June } 8 \quad \text { Charge: Gas } & \$ 36.25 \\ \hline \end{array} $$$$ \begin{array}{|ll|l|} \hline \text { June } 9 & \text { Charge: Groceries } & \$ 138.43 \\ \hline \text { June 17 } & \begin{array}{l} \text { Charge: Gas } \\ \text { Charge: Groceries } \end{array} & \$ 42.36 \\ \hline \text { June } 27 & \text { Charge: Clothing } & \$ 214.83 \\ \hline \text { June } 30 & \text { End of billing period } & \\ \hline \text { Payment } & \text { Due Date: July } 9 & \\ \hline \end{array} $$

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