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Chapter 3: Problem 55

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{n t}\) to solve.\( Round answers to the nearest cent. Suppose that you have \)\$ 12,000\( to invest. Which investment yields the greater return over 3 years: \)7 \%\( compounded monthly or \)6.85 \%$ compounded continuously?

Short Answer

Expert verified
The answer will be the name of the investment exercise that yields a higher return. This can only be determined by plugging values into the formulas and calculating the results.

Step by step solution

01

Setup the equations

The compound interest formula for interest that is compounded periodically is given by: \(A=P(1+\frac{r}{n})^{nt}\). Where A represents the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money). r is the annual interest rate (in decimal). n is the number of times that interest is compounded per year, and t is the time the money is invested for in years. In the case of the investment yielding 7% compounded monthly: r = 7/100 = 0.07 (converting into decimal), n = 12 (for 12 months in a year), P = $12,000, and t = 3. Substituting these values in the formula we get: \(A_1 = 12000(1 + \frac{0.07}{12})^{12*3}\). The compound interest continuously compounded is given by: \(A=P e^{rt}\). Here, e is the base of natural logarithm, approximated as 2.71828. For the investment yielding 6.85% compounded continuously, r = 6.85/100 = 0.0685, P = $12000, and t = 3. So, we substitute into the continuous compound interest formula to get: \(A_2 = 12000 e^{0.0685*3}\)
02

Calculate the future investment value

Substitute the known values into the formulas and calculate using a calculator to get the future values of the investment after 3 years for both cases. These values will be the total amount of money accumulated after 3 years, including interest. Use rounding rules to round off the answer to the nearest cent. Investment returns equals A - P, i.e the future value minus the initial investment.
03

Comparison of returns

Once the values of \(A_1\) and \(A_2\) are known, you compare these values to determine which investment results in a higher return.

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Key Concepts

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

Continuous Compounding
Continuous compounding is a fascinating concept in the world of finance. It refers to the process where interest is calculated and added to the principal an infinite number of times within a specific period, effectively compounding at every moment. This method of interest calculation uses the formula: \[ A = Pe^{rt} \]where:
  • A is the amount of money accumulated after time t, including interest
  • P is the principal amount
  • e is Euler's number, approximately 2.71828
  • r is the annual interest rate in decimal form
  • t is the time the money is invested for in years
Using continuous compounding typically results in a slightly higher return compared to periodic compounding, particularly over long periods. This is because interest accrual happens more frequently, allowing the investment to grow faster.
Periodic Compounding
Unlike continuous compounding, periodic compounding occurs at specific intervals such as annually, semi-annually, quarterly, or monthly. The compound interest formula for periodic compounding is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where:
  • A is the future value of the investment
  • P is the principal amount
  • r is the annual interest rate as a decimal
  • n is the number of compounding periods per year
  • t is the time in years
For example, if you choose to compound monthly, n would be 12. The more frequently interest is compounded, the higher the future value will be. This illustrates how the frequency of compounding affects the growth of an investment. Compounding monthly may provide more benefit than yearly compounding, but it won't quite match the returns of continuous compounding.
Investment Comparison
When deciding between two investment options, it's crucial to compare them under the same conditions to find which one yields a higher return. To do this, you must calculate the future value for each investment using appropriate compounding methods. For instance, if you compare a 7% return compounded monthly against a 6.85% return compounded continuously, you can directly plug in these values into their respective formulas to arrive at:- For monthly compounding: \( A_1 = 12000(1 + \frac{0.07}{12})^{12 \times 3} \)- For continuous compounding:\( A_2 = 12000 e^{0.0685 \times 3} \)Once calculated, compare the future values (\(A_1\) and \(A_2\)) to see which one gives you the better return on your investment.
Interest Calculations
Calculating interest accurately is fundamental to understanding how investments grow over time. It involves determining how much interest is earned on an initial investment over a period, based on the compounding method used. Interest calculations for both periodic and continuous compounding scenarios follow specific formulas:- For periodic compounding:\( A - P = P \left( 1 + \frac{r}{n} \right)^{nt} - P \)- For continuous compounding:\( A - P = Pe^{rt} - P \)These equations help investors compute the accrued interest by subtracting the initial principal from the future value of the investment. Calculating interest allows investors to plan better and choose the most profitable investment options. It's integral to making informed financial decisions and maximizing potential returns.

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