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Chapter 2: Problem 23

Future value for various compounding periods Find the amount to which \(\$ 500\) will grow under each of these conditions: a. 12 percent compounded annually for 5 years. b. 12 percent compounded semiannually for 5 years. c. 12 percent compounded quarterly for 5 years. d. 12 percent compounded monthly for 5 years. e. 12 percent compounded daily for 5 years. f. Why does the observed pattern of FVs occur?

Short Answer

Expert verified
The future values are: a) $881.60, b) $895.42, c) $903.92, d) $911.49, e) $914.05. Increasing compounding frequency increases the future value due to more frequent interest application.

Step by step solution

01

Understanding the Formula

To calculate the future value with different compounding periods, use the formula: \( FV = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \), where \( P \) is the principal (initial investment), \( r \) is the annual interest rate, \( n \) is the number of times the interest is compounded per year, and \( t \) is the time in years.
02

Calculate FV for Annual Compounding

For annual compounding, \( n = 1 \). Plug in the values: \( P = 500 \), \( r = 0.12 \), \( n = 1 \), and \( t = 5 \). Calculate as follows: \[ FV = 500 \times \left(1 + \frac{0.12}{1}\right)^{1 \times 5} = 500 \times (1.12)^5 \approx 881.60 \]
03

Calculate FV for Semiannual Compounding

For semiannual compounding, \( n = 2 \). Plug in the values: \[ FV = 500 \times \left(1 + \frac{0.12}{2}\right)^{2 \times 5} = 500 \times (1.06)^{10} \approx 895.42 \]
04

Calculate FV for Quarterly Compounding

For quarterly compounding, \( n = 4 \). Plug in the values: \[ FV = 500 \times \left(1 + \frac{0.12}{4}\right)^{4 \times 5} = 500 \times (1.03)^{20} \approx 903.92 \]
05

Calculate FV for Monthly Compounding

For monthly compounding, \( n = 12 \). Plug in the values: \[ FV = 500 \times \left(1 + \frac{0.12}{12}\right)^{12 \times 5} = 500 \times \left(1.01\right)^{60} \approx 911.49 \]
06

Calculate FV for Daily Compounding

For daily compounding, \( n = 365 \). Plug in the values:\[ FV = 500 \times \left(1 + \frac{0.12}{365}\right)^{365 \times 5} \approx 500 \times (1.000329)^{1825} \approx 914.05 \]
07

Observing the Pattern

The future value increases as the compounding frequency increases due to more frequent application of interest, allowing the interest to be calculated on an increasingly larger principal amount with each compounding period.

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Key Concepts

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

Compounding Periods
Compounding periods refer to the frequency with which the interest is applied to the principal investment. Each compounding period affects how the investment grows over time.
Common compounding periods include annual, semiannual, quarterly, monthly, and daily compounding.
For example, if you invest $500 with an annual interest rate of 12%, the interest can be calculated annually, semiannually, or even daily, each altering the end value due to periodic interest application.
  • Annually: Interest is compounded once a year.
  • Semiannually: Compounding occurs twice a year.
  • Quarterly: Interest is compounded four times a year.
  • Monthly: Compounding happens each month, totaling twelve times a year.
  • Daily: Interest is compounded every day, significantly increasing the total number of compounding periods.
Annual Interest Rate
The annual interest rate is the percentage of the principal sum that the investment earns in a year. This rate is vital for calculating the future value of an investment.
The annual interest rate is typically expressed as a percentage, like 12%.
In the formula to find Future Value (FV), the annual interest rate is crucial as it determines the amount of interest generated each year:
- If you have a principal of $500, and the annual interest rate is 12%, then each year, theoretically, your investment will earn $60 if compounded annually.
However, when compounded more frequently, the interest earned each period is calculated as a portion of this annual rate, effectively increasing the overall growth due to more frequent application.
Investment Growth Calculations
Investment growth calculations help determine the future value of an investment based on initial principal, annual interest rate, and specified compounding periods using the formula:\[ FV = P \times \left(1 + \frac{r}{n}\right)^{n \times t} \]Where:
  • \( FV \) is the future value
  • \( P \) is the principal amount
  • \( r \) is the annual interest rate
  • \( n \) is the number of compounding periods per year
  • \( t \) is the number of years
For example, when compounding is semiannual, values are adjusted for this frequency:
- If \( P = 500 \), \( r = 0.12 \), \( n = 2 \), and \( t = 5 \), the calculation becomes \[ 500 \times \left(1 + \frac{0.12}{2}\right)^{2 \times 5} \approx 895.42 \], demonstrating how changes in \( n \) affect the growth.
Compounding Frequency Effect
The compounding frequency effect illustrates how often interest is applied and affects the ultimate growth of an investment. The more frequently interest is compounded, the higher the investment's future value.
Compounding frequency increases the growth rate as it regularly adds accrued interest to the principal, boosting the base for subsequent interest calculations.
  • For a 12% interest rate compounded annually, the growth over 5 years might yield a future value of $881.60.
  • With semiannual compounding, this grows to $895.42.
  • Quarterly compounding further increases to $903.92.
  • Monthly compounding results in $911.49.
  • Daily compounding pushes it to approximately $914.05.
As this pattern shows, increasing compounding frequency results in greater future values, making frequent compounding a powerful tool for investment growth.

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Most popular questions from this chapter

Growth rates Shalit Corporation's 2005 sales were \(\$ 12\) million. Its 2000 sales were \(\$ 6\) million. a. At what rate have sales been growing? b. Suppose someone made this statement: "Sales doubled in 5 years. This represents a growth of 100 percent in 5 years, so, dividing 100 percent by \(5,\) we find the growth rate to be 20 percent per year." Is the statement correct?

Loan amortization and EAR You want to buy a car, and a local bank will lend you \(\$ 20,000 .\) The loan would be fully amortized over 5 years \((60\) months), and the nominal interest rate would be 12 percent, with interest paid monthly. What would be the monthly loan payment? What would be the loan's EAR?

Future value of an annuity Your client is 40 years old, and she wants to begin saving for retirement, with the first payment to come one year from now. She can save \(\$ 5,000\) per year, and you advise her to invest it in the stock market, which you expect to provide an average return of 9 percent in the future. a. If she follows your advice, how much money would she have at \(65 ?\) b. How much would she have at \(70 ?\) c. If she expects to live for 20 years in retirement if she retires at 65 and for 15 years at \(70,\) and her investments contintue to earn the same rate, how much could she withdraw at the end of each year after retirement at each retirement age?

\(\mathrm{PV}\) and a lawsuit settlement It is now December \(31,2005,\) and a jury just found in favor of a woman who sued the city for injuries sustained in a January 2004 accident. She requested recovery of lost wages, plus \(\$ 100,000\) for pain and suffering, plus \(\$ 20,000\) for her legal expenses. Her doctor testified that she has been unable to work since the accident and that she will not be able to work in the future. She is now \(62,\) and the jury decided that she would have worked for another 3 years. She was scheduled to have earned \(\$ 34,000\) in \(2004,\) and her employer testified that she would probably have received raises of 3 percent per year. The actual payment will be made on December 31 , \(2006 .\) The judge stipulated that all dollar amounts are to be adjusted to a present value basis on December \(31,2006,\) using a 7 percent annual interest rate, using compound, not simple, interest. Furthermore, he stipulated that the pain and suffering and legal expenses should be based on a December, \(31,2005,\) date. How large a check must the city write on December \(31,2006 ?\)

Amortization schedule a. Set up an amortization schedule for a \(\$ 25,000\) loan to be repaid in equal installments at the end of each of the next 3 years. The interest rate is 10 percent, compounded annually. b. What percentage of the payment represents interest and what percentage represents principal for each of the 3 years? Why do these percentages change over time?
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