Problem 20
Question
In Exercises 19 and \(20,\) find the amount of time required for a \(\$ 2000\) investment to double if the annual interest rate \(r\) is compounded (a) annually, (b) monthly, (c) quarterly, and (d) continuously. \(r=8.25 \%\)
Step-by-Step Solution
Verified Answer
The time it takes for the investment to double depends on the frequency of compounding. For annual compounding, it’ll take approximately 9 years, for monthly compounding it will take around 8.6 years, for quarterly compounding it will take about 8.7 years and for continuously compounding, the investment will double in approximately 8.4 years.
1Step 1: Calculate for annual compounding
The formula for annual compound interest is \(A=P(1+r)^n\) where: \(A\) is 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 years the money is invested forSince the money is to double, \(A=2P\). Substituting this into the formula, we get \(2= (1 + 0.0825)^n\). Solve for \(n\)
2Step 2: Calculate for monthly compounding
The formula for compounded interest is \(A = P(1+ r/n) ^{nt}\), with n now representing the number of times the interest is compounded per year. Since the money is to be doubled, we substitute \(A=2P\) into the formula to get \(2= (1 + 0.0825/12)^{12n}\). Solve for \(n\)
3Step 3: Calculate for quarterly compounding
We still use the formula \(A = P(1+ r/n) ^{nt}\) but with \(n=4\) because the interest is compounded quarterly. We substitute \(A=2P\) into the formula to get \(2= (1 + 0.0825/4)^{4n}\). Solve for \(n\)
4Step 4: Calculate for continuous compounding
For continuously compounded interest, the formula is \(A = Pe^{rt}\), where \(e\) is Euler's number (\(2.71828\)). Substituting \(A=2P\) into the formula, we get \(2=e^{0.0825n}\). Solve for \(n\)
Key Concepts
Annual Interest RateCompounding FrequencyPrincipal AmountContinuously Compounded Interest
Annual Interest Rate
The annual interest rate is the percentage increase in your investment over a year. For example, in this exercise, the annual interest rate is 8.25%. An annual interest rate is used to calculate the growth of your money when compounded in various timeframes like annually, monthly, or quarterly. To use the interest rate in formulas, you need to convert it into a decimal by dividing by 100, so 8.25% becomes 0.0825.
Understanding the annual interest rate is crucial because it directly impacts how much your investment grows. A higher interest rate means faster growth, while a lower rate prolongs the time needed for your investment to reach a desired amount such as doubling. When solving for time using different compounding methods, consider how often the interest is applied within the year, as this affects the computation involving the annual interest rate.
Understanding the annual interest rate is crucial because it directly impacts how much your investment grows. A higher interest rate means faster growth, while a lower rate prolongs the time needed for your investment to reach a desired amount such as doubling. When solving for time using different compounding methods, consider how often the interest is applied within the year, as this affects the computation involving the annual interest rate.
Compounding Frequency
Compounding frequency refers to how often the interest is added to the principal amount within a given period. The frequency could be annually, semi-annually, quarterly, monthly, daily, or continuously.
For example, if interest is compounded more frequently, such as quarterly or monthly, the investment will grow faster compared to annual compounding. Therefore, when solving for the time required for an investment to double, the compounding frequency will play a pivotal role.
- Annually: Interest is compounded once a year.
- Quarterly: Interest is compounded four times a year.
- Monthly: Interest is compounded twelve times a year.
- Continuously: Interest is compounded every moment, theoretically.
For example, if interest is compounded more frequently, such as quarterly or monthly, the investment will grow faster compared to annual compounding. Therefore, when solving for the time required for an investment to double, the compounding frequency will play a pivotal role.
Principal Amount
The principal amount is the initial sum of money invested or lent before the application of interest. In this exercise, the principal amount is $2000. The principal acts as the base upon which interest is calculated. Understanding its significance helps you see how investments grow over time.
When calculating compound interest, the principal amount remains the reference point for calculating growth. This starting figure is important because regardless of the compounding frequency, the initial sum remains unchanged during interest calculations.
In scenarios like this exercise, the principal serves as a constant across calculations for doubling the investment, be it annually, monthly, or any other compounding method. Starting with a solid principal often means more significant accumulation of interest, thereby achieving financial goals faster.
When calculating compound interest, the principal amount remains the reference point for calculating growth. This starting figure is important because regardless of the compounding frequency, the initial sum remains unchanged during interest calculations.
In scenarios like this exercise, the principal serves as a constant across calculations for doubling the investment, be it annually, monthly, or any other compounding method. Starting with a solid principal often means more significant accumulation of interest, thereby achieving financial goals faster.
Continuously Compounded Interest
Continuously compounded interest uses Euler's number (approximately 2.71828) to calculate interest that is constantly being added to the principal amount. The formula for this is expressed as:\[ A = Pe^{rt} \]Here, **\( P \)** is the principal, **\( e \)** is Euler's number, **\( r \)** is the annual interest rate, and **\( t \)** is the time in years. This formula portrays a situation where interest is compounded an infinite number of times in a year rather than a fixed number.
Continuously compounded interest is generally more beneficial than standard periodic compounding as it results in greater growth of an investment. In this exercise, using continuous compounding allows us to calculate how long it would take for a $2000 investment to double at an annual interest rate of 8.25%.
By solving the equation \( 2 = e^{0.0825t} \), you derive \( t \), which represents the time in years needed for the investment to double under continuous compounding. This method showcases the power of continuous growth, often seen in financial calculations for investment growth models.
Continuously compounded interest is generally more beneficial than standard periodic compounding as it results in greater growth of an investment. In this exercise, using continuous compounding allows us to calculate how long it would take for a $2000 investment to double at an annual interest rate of 8.25%.
By solving the equation \( 2 = e^{0.0825t} \), you derive \( t \), which represents the time in years needed for the investment to double under continuous compounding. This method showcases the power of continuous growth, often seen in financial calculations for investment growth models.
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