Problem 114
Question
A principal of \(\$ 5000\) is invested at 6\(\%\) interest. Find the amount after 10 years if the interest is compounded (a) annually, (b) semi-annually, (c) quarterly, (d) monthly, and (e) daily.
Step-by-Step Solution
Verified Answer
The total amounts after 10 years are approximately: (a) \$8,982.58 for annual compounding, (b) \$9,059.15 for semi-annual compounding, (c) \$9,094.03 for quarterly compounding, (d) \$9,118.41 for monthly compounding, and (e) \$9,140.66 for daily compounding.
1Step 1: Calculate for Annual Compounding
Calculate using the formula \(A=P\left(1+\frac{r}{n}\right)^{nt}\), where \(P=5000\), \(r=0.06\), \(n=1\) (as it is compounded once a year), \(t=10\).\nThus, the amount after 10 years is \(A=5000\left(1+\frac{0.06}{1}\right)^{1*10}=5000*(1.06)^{10}\) dollars.
2Step 2: Calculate for Semi-Annual Compounding
Calculate again using the formula with \(n=2\) (since the interest is compounded twice a year).\nThis yields us \(A=5000\left(1+\frac{0.06}{2}\right)^{2*10}=5000*(1.03)^{20}\) dollars.
3Step 3: Calculate for Quarterly Compounding
Now, let's compute for \(n=4\) (quarterly compounding).Hence, \(A=5000\left(1+\frac{0.06}{4}\right)^{4*10}=5000*(1.015)^{40}\) dollars.
4Step 4: Calculate for Monthly Compounding
For monthly compounding, we should set \(n=12.\) This gives us an amount after 10 years of \(A=5000\left(1+\frac{0.06}{12}\right)^{12*10}=5000*(1.005)^{120}\) dollars.
5Step 5: Calculate for Daily Compounding
Finally for daily compounding, set \(n=365.\) The calculation then becomes \(A=5000\left(1+\frac{0.06}{365}\right)^{365*10}=5000*(1.00016438)^{3650}\) dollars. Remember that the more frequent the compounding, the greater the amount due to interest.
Key Concepts
Annual CompoundingSemi-Annual CompoundingQuarterly CompoundingMonthly CompoundingDaily Compounding
Annual Compounding
When an investment is subject to annual compounding, the interest is calculated and added to the principal once a year. For the given exercise, where we have a principal amount of $5000 invested at an interest rate of 6% over 10 years, the formula \(A=P\left(1+\frac{r}{n}\right)^{nt}\) simplifies greatly. \(
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\)Since it's compounded annually, \(n=1\), thus our formula becomes \(A=5000\left(1+\frac{0.06}{1}\right)^{1*10}=5000*(1.06)^{10}\). By solving this, we find the future value of the investment after 10 years due to the effect of interest compounded once per year.
\) \(
\)Since it's compounded annually, \(n=1\), thus our formula becomes \(A=5000\left(1+\frac{0.06}{1}\right)^{1*10}=5000*(1.06)^{10}\). By solving this, we find the future value of the investment after 10 years due to the effect of interest compounded once per year.
Semi-Annual Compounding
Semi-annual compounding involves calculating and adding interest to the principal twice a year. Here, the term 'semi-annual' refers to the two times that interest is compounded within a year. In our exercise, with a 6% annual interest rate, the rate per period becomes 3% (\( \frac{6\%}{2} \)), and there are 20 compounding periods over 10 years (2 per year). \(
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\)Our modified formula for semi-annual compounding is \( A=5000\left(1+\frac{0.06}{2}\right)^{2*10}=5000*(1.03)^{20}\). This results in a slightly higher amount than with annual compounding, demonstrating the benefit of more frequent compounding periods.
\) \(
\)Our modified formula for semi-annual compounding is \( A=5000\left(1+\frac{0.06}{2}\right)^{2*10}=5000*(1.03)^{20}\). This results in a slightly higher amount than with annual compounding, demonstrating the benefit of more frequent compounding periods.
Quarterly Compounding
Quarterly compounding means that interest is compounded four times a year. The quarter refers to one-fourth of a year. This increase in frequency typically results in a higher return compared to annual or semi-annual compounding for the same interest rate and principal. \(
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\)In our exercise, with the number of periods \(n=4\), the formula adapts to \( A=5000\left(1+\frac{0.06}{4}\right)^{4*10}=5000*(1.015)^{40}\). So, every quarter, the interest rate applied is 1.5% (\( \frac{6\%}{4} \)), which when calculated over 40 periods (10 years), provides us the amount with interest compounded quarterly.
\) \(
\)In our exercise, with the number of periods \(n=4\), the formula adapts to \( A=5000\left(1+\frac{0.06}{4}\right)^{4*10}=5000*(1.015)^{40}\). So, every quarter, the interest rate applied is 1.5% (\( \frac{6\%}{4} \)), which when calculated over 40 periods (10 years), provides us the amount with interest compounded quarterly.
Monthly Compounding
When it comes to monthly compounding, interest is added to the principal 12 times within the year. This higher frequency allows interest to be calculated on the interest that was previously earned, thereby earning 'interest on interest' more often. \(
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\)Applying this to our scenario with \(n=12\), the formula becomes \( A=5000\left(1+\frac{0.06}{12}\right)^{12*10}=5000*(1.005)^{120}\). Here, each month the investment grows by an effective rate of 0.5% (\( \frac{6\%}{12} \) per month), leading to 120 compounding periods over the decade.
\) \(
\)Applying this to our scenario with \(n=12\), the formula becomes \( A=5000\left(1+\frac{0.06}{12}\right)^{12*10}=5000*(1.005)^{120}\). Here, each month the investment grows by an effective rate of 0.5% (\( \frac{6\%}{12} \) per month), leading to 120 compounding periods over the decade.
Daily Compounding
Daily compounding is the process where interest is calculated and added to the principal balance every day. It represents the greatest frequency included in our exercise, with compounding occurring 365 times each year. \(
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\)The daily compounding calculation in our exercise is described by the formula \( A=5000\left(1+\frac{0.06}{365}\right)^{365*10}=5000*(1.00016438)^{3650}\). This results in an even higher amount over the 10-year period as compared to monthly, quarterly, semi-annual, or annual compounding, illustrating how the compounding effect becomes more powerful as the frequency of compounding increases.
\) \(
\)The daily compounding calculation in our exercise is described by the formula \( A=5000\left(1+\frac{0.06}{365}\right)^{365*10}=5000*(1.00016438)^{3650}\). This results in an even higher amount over the 10-year period as compared to monthly, quarterly, semi-annual, or annual compounding, illustrating how the compounding effect becomes more powerful as the frequency of compounding increases.
Other exercises in this chapter
Problem 113
The sum of the first \( 20 \) terms of an arithmetic sequence with a common difference of \( 3 \) is \( 650 \). Find the first term.
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In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{i=1}^{\infty} 5 \left(\dfrac{1}{2} \right)^i \) Fourth partial sum
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The sum of the first \( n \) terms of anarithmetic sequence with first term \( a_1 \) and common difference \( d \) is \( S_n \) Determine the sum if each term
View solution Problem 114
In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\dfrac{1}{3} \right)^i \) Fifth partial sum
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