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Chapter 3: Problem 54

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{n t}\) to solve.\( Round answers to the nearest cent. Find the accumulated value of an investment of \)\$ 5000\( for 10 years at an interest rate of \)6.5 \%$ if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Short Answer

Expert verified
The accumulated values after 10 years for different modes of compounding are: For semiannual compounding, $9382.99. For quarterly compounding, $9413.14. For monthly compounding, $9431.71. For continuous compounding, $9452.13.

Step by step solution

01

Calculate for Semiannual Compounding

For semiannual compounding, \(n = 2\). Substituting \(P = 5000\), \(r = 0.065\), \(n = 2\), and \(t = 10\) into the formula, we get: \(A = 5000 \left(1 + \frac{0.065}{2}\right)^{2*10}\). After simplifying, the calculated accumulated value is approximately $9382.99.
02

Calculate for Quarterly Compounding

For quarterly compounding, \(n = 4\). Substituting \(P = 5000\), \(r = 0.065\), \(n = 4\), and \(t = 10\) into the formula, we get: \(A = 5000 \left(1 + \frac{0.065}{4}\right)^{4*10}\). After simplifying, the calculated accumulated value is approximately $9413.14.
03

Calculate for Monthly Compounding

For monthly compounding, \(n = 12\). Substituting \(P = 5000\), \(r = 0.065\), \(n = 12\), and \(t = 10\) into the formula, we get: \(A = 5000 \left(1 + \frac{0.065}{12}\right)^{12*10}\). After simplifying, the calculated accumulated value is approximately $9431.71.
04

Calculate for Continuous Compounding

For continuous compounding, we use the formula \(A = Pe^{rt}\). Substituting \(P = 5000\), \(r = 0.065\), and \(t = 10\) into the formula, we get: \(A = 5000e^{0.065*10}\). After simplifying, the calculated accumulated value is approximately $9452.13.

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Key Concepts

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

Understanding Semiannual Compounding
When it comes to increasing the value of an investment over time, compounding frequency plays a pivotal role. With semiannual compounding, the interest on an investment is calculated and added to the principal twice a year. To comprehend this better, let's use the example of a \(5000 investment over 10 years at 6.5% interest.

In the semiannual compounding formula, we see the variable 'n' represents the number of times interest is compounded per year. For semiannual, this is simply 2. Plugging in the values into the compound interest formula \(A=P\left(1+\frac{r}{n}\right)^{nt}\), the math unfolds to provide a future value of the investment. In our example, that results in an accumulated value of approximately \)9382.99.
When you compare this to annual compounding, the added frequency allows for interest to be earned on interest more often, effectively 'snowballing' the growth of the investment over time.
Quarterly Compounding Explained
Quarterly compounding takes the principle of semiannual compounding and increases the frequency to four times per year. Each quarter, interest earned is added to the principal, which in turn earns more interest.

Using the same formula, but substituting 'n' with 4 for quarterly compounding, we find that the \(5000 investment grows slightly more compared to semiannual compounding, due to the increased compounding events. Here, the accumulated value after 10 years at 6.5% interest is roughly \)9413.14, illustrating the beneficial impact of higher compounding frequency on the growth of an investment.
Monthly Compounding and Its Impact
Taking the frequency a notch higher, monthly compounding occurs when interest is calculated and added to the principal 12 times a year. Each month, as the interest is compounded, it contributes to an even larger base for the next month's interest calculation.

For our \(5000 example, using 'n' equal to 12, the compound interest formula reveals an accumulated value of about \)9431.71 after 10 years at a 6.5% interest rate. This increase from quarterly to monthly compounding underscores the concept that the more frequent the compounding, the more wealth is generated over time.
Continuous Compounding: Maximizing Growth
In the realm of compounding, continuous compounding represents the theoretical limit of compounding frequency, where interest is compounded an infinite number of times. This approach uses a unique formula, \(A = Pe^{rt}\), where 'e' is the mathematical constant approximately equal to 2.71828.

Applying this to our ongoing example, with a \(5000 investment at 6.5% interest over 10 years, continuous compounding yields an accumulated value of about \)9452.13. This demonstrates the profound effect of compounding as frequently as possible, accentuating the potential for exponential growth in one's investments.

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