Problem 54
Question
Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{n}\) 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.
Step-by-Step Solution
Verified Answer
The accumulated values for the scenarios are: a. compounded semiannually: approximately \$9287.66, b. compounded quarterly: approximately \$9321.78, c. compounded monthly: approximately \$9346.84, d. compounded continuously: approximately \$9360.48.
1Step 1: Identify Variables for Semiannual Compounding
From the problem, we know that the principle, P, is \$5000, interest rate, r, is 0.065 (6.5% converted to decimal), number of times it compounds in a year, n, is 2 (since it is semiannually), and number of years, t, is 10.
2Step 2: Use the Compound Interest Formula for Semiannual Compounding
By substituting these values into the compound interest formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\), we can calculate the accumulated amount, A.
3Step 3: Repeat Steps 1 and 2 for Quarterly, Monthly, and Continuous Compounding
We perform the same calculations but with different values for n. Quarterly compounding means n=4, monthly compounding means n=12. For continuous compounding, we utilize the formula \(A=Pe^{rt}\), where r is still the interest rate and t is still the number of years.
Key Concepts
Semiannual CompoundingQuarterly CompoundingMonthly CompoundingContinuous CompoundingAccumulated ValueInvestment Growth
Semiannual Compounding
When it comes to understanding how investments grow over time, the concept of semiannual compounding is pivotal. This term refers to the process where the interest on an investment is calculated and added back to the principal amount twice a year. Put simply, with semiannual compounding, your investment will earn interest every six months.
Using the formula for compound interest, \[A = P\left(1 + \frac{r}{n}\right)^{nt}\], where \(A\) represents the future value of the investment, \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time the money is invested in years, we can calculate the accumulated value of the investment after a certain period. Here, \(n=2\) for semiannual compounding.
To maximize comprehension, remember that as compounding occurs more frequently, interest is earned on the interest itself and thus the investment grows faster.
Using the formula for compound interest, \[A = P\left(1 + \frac{r}{n}\right)^{nt}\], where \(A\) represents the future value of the investment, \(P\) is the principal amount, \(r\) is the annual interest rate, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time the money is invested in years, we can calculate the accumulated value of the investment after a certain period. Here, \(n=2\) for semiannual compounding.
To maximize comprehension, remember that as compounding occurs more frequently, interest is earned on the interest itself and thus the investment grows faster.
Quarterly Compounding
Similar to semiannual compounding, quarterly compounding interest is added to the principal four times a year, meaning every quarter. So for quarterly compounding, \(n\) becomes 4.
Again using our principal formula, we calculate the accumulated amount after a given period by adjusting the formula to reflect the quarterly nature of the interest calculations: \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]. What's particularly intriguing about quarterly compounding is that with each passing quarter, the interest accumulates on a larger principal amount, which includes the interest from previous periods, accelerating the growth of your investment.
Again using our principal formula, we calculate the accumulated amount after a given period by adjusting the formula to reflect the quarterly nature of the interest calculations: \[A = P\left(1 + \frac{r}{n}\right)^{nt}\]. What's particularly intriguing about quarterly compounding is that with each passing quarter, the interest accumulates on a larger principal amount, which includes the interest from previous periods, accelerating the growth of your investment.
Monthly Compounding
When investments compound monthly, this means that interest is calculated and added to the principal balance 12 times per year, once at the end of each month. Therefore, in our compound interest formula, \(n\) would equal 12.
By substituting in the different value for \(n\), we get \[A = P\left(1 + \frac{r}{n}\right)^{nt}\] for monthly compounding. This frequency of compounding interest makes your investment grow even more swiftly than with quarterly or semiannual compounding. This is due to the more frequent application of interest to the principal, which increases the basis for each subsequent interest calculation.
By substituting in the different value for \(n\), we get \[A = P\left(1 + \frac{r}{n}\right)^{nt}\] for monthly compounding. This frequency of compounding interest makes your investment grow even more swiftly than with quarterly or semiannual compounding. This is due to the more frequent application of interest to the principal, which increases the basis for each subsequent interest calculation.
Continuous Compounding
Continuous compounding can be seen as the limit case of compounding frequency, where the number of compounding periods per year grows without bound. It represents a theoretical situation where the compounding occurs an infinite number of times instantaneously.
To handle this, we use a different formula: \[A = Pe^{rt}\], where \(e\) is the base of the natural logarithm (approximately 2.71828). This equation shows how an investment grows when it's subjected to the most intense form of compound interest. Continuous compounding is the mathematical ideal and shows the maximum possible accumulation for a given interest rate.
To handle this, we use a different formula: \[A = Pe^{rt}\], where \(e\) is the base of the natural logarithm (approximately 2.71828). This equation shows how an investment grows when it's subjected to the most intense form of compound interest. Continuous compounding is the mathematical ideal and shows the maximum possible accumulation for a given interest rate.
Accumulated Value
The term accumulated value is the total amount an investment is worth after interest is applied over a certain time period. It is the sum of the principal (the initial amount of money) and the interest earned.
Understanding accumulated value is critical to foresee the future worth of investments and savings. The higher the frequency of the compounding periods, the greater the accumulated value will be, given the same interest rate and investment period because the interest is being calculated on a gradually increasing principal over time.
Understanding accumulated value is critical to foresee the future worth of investments and savings. The higher the frequency of the compounding periods, the greater the accumulated value will be, given the same interest rate and investment period because the interest is being calculated on a gradually increasing principal over time.
Investment Growth
The investment growth refers to the increase in value of an investment due to earnings from interest over time. It's crucial for investors to understand how their money will grow, and the effects of different compounding frequencies on this growth.
It’s also significant to note that investment growth is affected not just by the frequency of compounding, but also by the rate of interest and the period of the investment. By utilizing compound interest formulas, investors can compare how their investments might perform under different compounding strategies and select the one that aligns best with their goals and timelines for their financial ambitions.
It’s also significant to note that investment growth is affected not just by the frequency of compounding, but also by the rate of interest and the period of the investment. By utilizing compound interest formulas, investors can compare how their investments might perform under different compounding strategies and select the one that aligns best with their goals and timelines for their financial ambitions.
Other exercises in this chapter
Problem 54
In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places.
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