Find the term of the ordinary annuity in years and months (from 0 to 11 months). Payment Interval months Future Value Periodic Rent $7928 $349 The term is year(s) and month(s). Interest Rate 5% Conversion Period semi-annually
Find the term of the ordinary annuity in years and months (from 0 to 11 months). Payment Interval months Future Value Periodic Rent $7928 $349 The term is year(s) and month(s). Interest Rate 5% Conversion Period semi-annually
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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![### Determining the Term of an Ordinary Annuity
To find the term of the ordinary annuity in years and months (from 0 to 11 months), we are given the following information:
- **Future Value (FV):** $7,928
- **Periodic Rent (PMT):** $349
- **Payment Interval:** 6 months
- **Interest Rate (APR):** 5%
- **Conversion Period:** Semi-annually (twice a year)
### Calculation Steps:
1. **Identify Given Values:**
- \( FV = 7928 \)
- \( PMT = 349 \)
- Payment Interval = 6 months
- Interest Rate = 5% per annum (compounded semi-annually)
2. **Use Formula for Future Value of an Ordinary Annuity:**
The formula for the future value of an ordinary annuity is:
\[
FV = PMT \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = Future Value
- \( PMT \) = Periodic Payment (Rent)
- \( r \) = Periodic Interest Rate
- \( n \) = Total Number of Payments
3. **Convert Annual Interest Rate to Periodic Rate:**
- Since the payments are made semi-annually and the interest is compounded semi-annually, the periodic interest rate, \( r \), is:
\[
r = \frac{5\%}{2} = 2.5\% = 0.025
\]
4. **Substitute the Known Values:**
We rearrange the formula to solve for \( n \) (the number of payments):
\[
7928 = 349 \times \frac{(1 + 0.025)^n - 1}{0.025}
\]
5. **Calculate the Number of Payments (n):**
Rearranging to solve for \( (1 + 0.025)^n \):
\[
\frac{7928 \times 0.025}{349} + 1 = (1 + 0.025)^n
\]
Simplify:
\[
\frac{198.2}{349} + 1 = (1.025)^n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62ca1c58-ac8a-4380-bd93-744ecdfbeb4a%2Fdd8b4162-346c-4a8a-be1f-826c3e058b3d%2Fi5mvnur_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Term of an Ordinary Annuity
To find the term of the ordinary annuity in years and months (from 0 to 11 months), we are given the following information:
- **Future Value (FV):** $7,928
- **Periodic Rent (PMT):** $349
- **Payment Interval:** 6 months
- **Interest Rate (APR):** 5%
- **Conversion Period:** Semi-annually (twice a year)
### Calculation Steps:
1. **Identify Given Values:**
- \( FV = 7928 \)
- \( PMT = 349 \)
- Payment Interval = 6 months
- Interest Rate = 5% per annum (compounded semi-annually)
2. **Use Formula for Future Value of an Ordinary Annuity:**
The formula for the future value of an ordinary annuity is:
\[
FV = PMT \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- \( FV \) = Future Value
- \( PMT \) = Periodic Payment (Rent)
- \( r \) = Periodic Interest Rate
- \( n \) = Total Number of Payments
3. **Convert Annual Interest Rate to Periodic Rate:**
- Since the payments are made semi-annually and the interest is compounded semi-annually, the periodic interest rate, \( r \), is:
\[
r = \frac{5\%}{2} = 2.5\% = 0.025
\]
4. **Substitute the Known Values:**
We rearrange the formula to solve for \( n \) (the number of payments):
\[
7928 = 349 \times \frac{(1 + 0.025)^n - 1}{0.025}
\]
5. **Calculate the Number of Payments (n):**
Rearranging to solve for \( (1 + 0.025)^n \):
\[
\frac{7928 \times 0.025}{349} + 1 = (1 + 0.025)^n
\]
Simplify:
\[
\frac{198.2}{349} + 1 = (1.025)^n
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