Problem 53
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 \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.
Step-by-Step Solution
Verified Answer
a. The accumulated value for semiannually compounding is approximately $14,137.07. b. The accumulated value for quarterly compounding is approximately $14,192.91. c. The accumulated value for monthly compounding is approximately $14,229.10. d. The accumulated value for continuous compounding is approximately $14,244.17.
1Step 1: Compute for semiannual compounding
Use the formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), substituting \(P = $10,000\), \(r = 0.055\), \(n = 2\) (since semiannually means twice a year), and \(t = 5\). So, \(A = $10,000 \left(1 + \frac{0.055}{2}\right)^{2*5}\). Calculate this expression to find the accumulated value.
2Step 2: Compute for quarterly compounding
Use the compound interest formula again, but this time with \(n = 4\) (since quarterly means four times a year). So, \(A = $10,000 \left(1 + \frac{0.055}{4}\right)^{4*5}\). Calculate to find the accumulated value.
3Step 3: Compute for monthly compounding
Once again apply the same formula, but now with \(n = 12\) (since monthly means twelve times a year). So, \(A = $10,000 \left(1 + \frac{0.055}{12}\right)^{12*5}\). Calculate to find the accumulated value.
4Step 4: Compute for continuous compounding
For continuous compounding, use the formula \(A = Pe^{rt}\) instead. Substitute \(P = $10,000\), \(r = 0.055\), and \(t = 5\). So, \(A = $10,000*e^{0.055*5}\). Calculate to find the accumulated value.
Key Concepts
Semiannual CompoundingQuarterly CompoundingMonthly CompoundingContinuous Compounding
Semiannual Compounding
In semiannual compounding, interest is calculated and added to the principal twice a year. This means after every six months, the interest earned gets reinvested.
To figure out the accumulated value using semiannual compounding, you can use the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \(P\) is the principal amount, \(r\) is the interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
For an investment of \$10,000 at a rate of 5.5% for 5 years, compounding occurs twice a year, so \(n = 2\). Substitute these values into the formula to get \[ A = 10000 \left(1 + \frac{0.055}{2}\right)^{2 \times 5} \].
This method of compounding will give you an exact calculation of how much your investment grows when interest is added every half year.
To figure out the accumulated value using semiannual compounding, you can use the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where \(P\) is the principal amount, \(r\) is the interest rate, \(n\) is the number of compounding periods per year, and \(t\) is the time in years.
For an investment of \$10,000 at a rate of 5.5% for 5 years, compounding occurs twice a year, so \(n = 2\). Substitute these values into the formula to get \[ A = 10000 \left(1 + \frac{0.055}{2}\right)^{2 \times 5} \].
This method of compounding will give you an exact calculation of how much your investment grows when interest is added every half year.
Quarterly Compounding
Quarterly compounding means the interest is calculated and added to the principal four times a year. This happens every three months.
Like semiannual compounding, the formula used is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]. However, for quarterly compounding, \(n = 4\).
In the example of the \$10,000 investment at an interest rate of 5.5% over 5 years, the calculation becomes \[ A = 10000 \left(1 + \frac{0.055}{4}\right)^{4 \times 5} \]. Each quarter, as the interest adds up, it will be reinvested, increasing the amount of interest earned by the next period.
This compounding frequency will typically result in a higher accumulated value than semiannual compounding because the interest is calculated more frequently.
Like semiannual compounding, the formula used is \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]. However, for quarterly compounding, \(n = 4\).
In the example of the \$10,000 investment at an interest rate of 5.5% over 5 years, the calculation becomes \[ A = 10000 \left(1 + \frac{0.055}{4}\right)^{4 \times 5} \]. Each quarter, as the interest adds up, it will be reinvested, increasing the amount of interest earned by the next period.
This compounding frequency will typically result in a higher accumulated value than semiannual compounding because the interest is calculated more frequently.
Monthly Compounding
With monthly compounding, interest is added to the principal every month. This frequency allows for more compounding periods in a year, usually resulting in a higher accumulated value compared to semiannual and quarterly compounding.
Again, the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] is used, with \(n = 12\) since there are 12 months in a year.
For the \$10,000 investment at 5.5% interest over 5 years, you would compute \[ A = 10000 \left(1 + \frac{0.055}{12}\right)^{12 \times 5} \].
This frequent compounding means every month, the interest is calculated and added to the current principal, allowing for a compounding effect that builds up more interest over time.
Again, the formula \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] is used, with \(n = 12\) since there are 12 months in a year.
For the \$10,000 investment at 5.5% interest over 5 years, you would compute \[ A = 10000 \left(1 + \frac{0.055}{12}\right)^{12 \times 5} \].
This frequent compounding means every month, the interest is calculated and added to the current principal, allowing for a compounding effect that builds up more interest over time.
Continuous Compounding
Continuous compounding is a concept in which interest is continually added to the principal at every possible moment. This leads to the highest possible compounding effect.
The formula used for continuous compounding is slightly different from other compounding methods: \[ A = Pe^{rt} \] where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
For an initial investment of \$10,000 at an interest rate of 5.5% over 5 years, the calculation would be \[ A = 10000 \times e^{0.055 \times 5} \].
This scenario showcases the power of continuous compounding. Since interest is added in an ongoing manner without discrete intervals, it results in the highest accumulated value among all compounding methods.
The formula used for continuous compounding is slightly different from other compounding methods: \[ A = Pe^{rt} \] where \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
For an initial investment of \$10,000 at an interest rate of 5.5% over 5 years, the calculation would be \[ A = 10000 \times e^{0.055 \times 5} \].
This scenario showcases the power of continuous compounding. Since interest is added in an ongoing manner without discrete intervals, it results in the highest accumulated value among all compounding methods.
Other exercises in this chapter
Problem 53
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