Find i (the rate per period) and n (the number of periods) for the following annuity. Annual deposits of $3,200 are made for 10 years into an annuity that pays 5.45% compounded annually. (Type an integer or a decimal.)
Find i (the rate per period) and n (the number of periods) for the following annuity. Annual deposits of $3,200 are made for 10 years into an annuity that pays 5.45% compounded annually. (Type an integer or a decimal.)
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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5.6
Transcribed Image Text:### Educational Resource: Understanding Annuities with Compounded Interest
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#### Problem Statement
**Objective: Find \( i \) (the rate per period) and \( n \) (the number of periods) for the following annuity.**
- **Scenario:** Annual deposits of \$3,200 are made for 10 years into an annuity that pays 5.45% compounded annually.
#### Solution Steps
1. **Identify the Regular Deposit Amount**:
- Annual Deposit (A): \$3,200
2. **Determine the Number of Periods**:
- Number of Years (T): 10 years
- Since the deposits are made annually, the number of periods (\( n \)) is equal to 10.
3. **Calculate the Periodic Interest Rate**:
- Annual Compounded Interest Rate (r): 5.45%
Since the interest is compounded annually, the rate per period (\( i \)) is the same as the annual rate. Thus,
\( i = 5.45\% \) or \( 0.0545 \) in decimal form.
#### Final Computations
- \( i = 0.0545 \) (interest rate per period when expressed as a decimal).
- \( n = 10 \) (number of periods).
### Summary Box for an Annuity Deposit
- Initial Annual Deposit: \$3,200
- Total Number of Deposits: 10
- Annual Compounded Interest Rate: 5.45%
Through this task, students learn how to find the periodic interest rate \( i \) and the number of periods \( n \) in the context of annuities, with considerations for compounded interest rates.
#### Interactive Element:
- **User Input:** \( i = \underline{\quad \quad} \) (Type an integer or a decimal.)
By accurately inputting the periodic interest rate and number of periods, students solidify their understanding of how annuity calculations are affected by compounding interest principles.
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### Visualization
While there are no graphs or diagrams in this example, students are encouraged to visualize the process through a timeline or chart that shows the series of deposits and compounded growth over the 10-year period.
**Note:** Ensure that the interactive segment for entering values \( i \) and \( n \) is clear and user-friendly on the educational platform.
![**Find i (the rate per period) and n (the number of periods) for the following annuity:**
Monthly deposits of $325 are made for 7 years into an annuity that pays 7.5% compounded monthly.
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\[ i = \ \text{(Type an integer or decimal rounded to four decimal places as needed.)} \]
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This problem involves calculating the rate per period and the number of periods for an annuity, given the rate is compounded monthly. To solve this, use the following steps:
1. **Determine the Rate per Period (i):**
- The annual interest rate is 7.5%.
- Since the interest is compounded monthly, divide the annual rate by 12.
- \( i = \frac{7.5\%}{12} = \frac{7.5}{100 \times 12} = 0.00625 \)
- Rounded to four decimal places, \( i = 0.0063 \)
2. **Determine the Number of Periods (n):**
- The period of annuity payments is 7 years.
- Since payments are made monthly, multiply the number of years by 12.
- \( n = 7 \times 12 = 84 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0335d89c-1e38-44bf-8e2e-69dc241551c2%2F42a26726-980b-4351-9120-a44bec0c1f4d%2F5tc1qj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find i (the rate per period) and n (the number of periods) for the following annuity:**
Monthly deposits of $325 are made for 7 years into an annuity that pays 7.5% compounded monthly.
---
\[ i = \ \text{(Type an integer or decimal rounded to four decimal places as needed.)} \]
---
This problem involves calculating the rate per period and the number of periods for an annuity, given the rate is compounded monthly. To solve this, use the following steps:
1. **Determine the Rate per Period (i):**
- The annual interest rate is 7.5%.
- Since the interest is compounded monthly, divide the annual rate by 12.
- \( i = \frac{7.5\%}{12} = \frac{7.5}{100 \times 12} = 0.00625 \)
- Rounded to four decimal places, \( i = 0.0063 \)
2. **Determine the Number of Periods (n):**
- The period of annuity payments is 7 years.
- Since payments are made monthly, multiply the number of years by 12.
- \( n = 7 \times 12 = 84 \)
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