Suppose the interest rate is 6.9% APR with monthly compounding. What is the present value of an annuity that pays $110 every three months for five years? (Note: Be careful not to round any intermediate steps less than six decimal places.) The present value of the annuity is $ (Round to the nearest cent.)
Suppose the interest rate is 6.9% APR with monthly compounding. What is the present value of an annuity that pays $110 every three months for five years? (Note: Be careful not to round any intermediate steps less than six decimal places.) The present value of the annuity is $ (Round to the nearest cent.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Calculating the Present Value of an Annuity
#### Problem Statement:
Suppose the interest rate is 6.9% APR with monthly compounding. What is the present value of an annuity that pays $110 every three months for five years? *(Note: Be careful not to round any intermediate steps less than six decimal places.)*
---
The present value of the annuity is \$\_\_\_\_. *(Round to the nearest cent.)*
---
#### Explanation:
To find the present value of an annuity, you need to discount each payment back to the present value using the given interest rate and compounding frequency. This involves the following steps:
1. **Determine the periodic interest rate**: Since the interest is compounded monthly, divide the annual percentage rate (APR) by the number of months in a year.
2. **Convert the payment frequency**: Payments are made every three months, so you need to adjust the number of periods accordingly.
3. **Use the present value of annuity formula**:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \(P\) is the payment amount ($110)
- \(r\) is the periodic interest rate
- \(n\) is the total number of payments
Be certain to carry out intermediary calculations with precision to six decimal places to avoid rounding errors. Finally, round the present value to the nearest cent.
This concept is crucial for financial planning and investment decisions, helping you understand the time value of money and making informed choices about annuities and other financial instruments.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeb1fcba-4a85-4c8e-822f-1910eb414763%2Fd654a8d9-2ba3-40f1-bb49-764340c2b061%2Fq0xxt9e_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Present Value of an Annuity
#### Problem Statement:
Suppose the interest rate is 6.9% APR with monthly compounding. What is the present value of an annuity that pays $110 every three months for five years? *(Note: Be careful not to round any intermediate steps less than six decimal places.)*
---
The present value of the annuity is \$\_\_\_\_. *(Round to the nearest cent.)*
---
#### Explanation:
To find the present value of an annuity, you need to discount each payment back to the present value using the given interest rate and compounding frequency. This involves the following steps:
1. **Determine the periodic interest rate**: Since the interest is compounded monthly, divide the annual percentage rate (APR) by the number of months in a year.
2. **Convert the payment frequency**: Payments are made every three months, so you need to adjust the number of periods accordingly.
3. **Use the present value of annuity formula**:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \(P\) is the payment amount ($110)
- \(r\) is the periodic interest rate
- \(n\) is the total number of payments
Be certain to carry out intermediary calculations with precision to six decimal places to avoid rounding errors. Finally, round the present value to the nearest cent.
This concept is crucial for financial planning and investment decisions, helping you understand the time value of money and making informed choices about annuities and other financial instruments.
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