A payment of $11,050 is due in 1 year, $19,500 is due in 5 years, and $8,550 is due in 6 years. What single equivalent payment made today would replace the three original payments? Assume that money earns 5.50% compounded monthly. Round to the nearest cent
A payment of $11,050 is due in 1 year, $19,500 is due in 5 years, and $8,550 is due in 6 years. What single equivalent payment made today would replace the three original payments? Assume that money earns 5.50% compounded monthly. Round to the nearest cent
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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![**Present Value of Multiple Payments with Monthly Compounding**
A payment of $11,050 is due in 1 year, $19,500 is due in 5 years, and $8,550 is due in 6 years. What single equivalent payment made today would replace the three original payments? Assume that money earns 5.50% compounded monthly.
**Steps to Solve:**
1. Calculate the present value (PV) of each payment using the formula for present value with monthly compounding:
\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \(FV\) is the future value (the payment amount),
- \(r\) is the annual interest rate (as a decimal),
- \(n\) is the number of compounding periods per year,
- \(t\) is the time in years.
2. Sum the present values of all three payments to find the total present value.
**Given Data:**
- Annual interest rate (\(r\)): 5.50% or 0.055
- Compounding periods per year (\(n\)): 12
- Payment 1: $11,050, due in 1 year (\(t\) = 1)
- Payment 2: $19,500, due in 5 years (\(t\) = 5)
- Payment 3: $8,550, due in 6 years (\(t\) = 6)
**Calculations:**
*For Payment 1:*
\[ PV_1 = \frac{11,050}{(1 + \frac{0.055}{12})^{12 \times 1}} \]
*For Payment 2:*
\[ PV_2 = \frac{19,500}{(1 + \frac{0.055}{12})^{12 \times 5}} \]
*For Payment 3:*
\[ PV_3 = \frac{8,550}{(1 + \frac{0.055}{12})^{12 \times 6}} \]
**Graphical Explanation:**
- A graph could illustrate the relationship between the future values and their respective present values over the given time periods.
- It would show how each original payment is discounted back to its present value using the given interest rate.
**Conclusion:**
Add the present](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F816cc47c-f059-4914-940d-8196a7c7af6b%2F6cf0fb42-3c0c-4827-8772-378ae9951c89%2Fz4o4j2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Present Value of Multiple Payments with Monthly Compounding**
A payment of $11,050 is due in 1 year, $19,500 is due in 5 years, and $8,550 is due in 6 years. What single equivalent payment made today would replace the three original payments? Assume that money earns 5.50% compounded monthly.
**Steps to Solve:**
1. Calculate the present value (PV) of each payment using the formula for present value with monthly compounding:
\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \(FV\) is the future value (the payment amount),
- \(r\) is the annual interest rate (as a decimal),
- \(n\) is the number of compounding periods per year,
- \(t\) is the time in years.
2. Sum the present values of all three payments to find the total present value.
**Given Data:**
- Annual interest rate (\(r\)): 5.50% or 0.055
- Compounding periods per year (\(n\)): 12
- Payment 1: $11,050, due in 1 year (\(t\) = 1)
- Payment 2: $19,500, due in 5 years (\(t\) = 5)
- Payment 3: $8,550, due in 6 years (\(t\) = 6)
**Calculations:**
*For Payment 1:*
\[ PV_1 = \frac{11,050}{(1 + \frac{0.055}{12})^{12 \times 1}} \]
*For Payment 2:*
\[ PV_2 = \frac{19,500}{(1 + \frac{0.055}{12})^{12 \times 5}} \]
*For Payment 3:*
\[ PV_3 = \frac{8,550}{(1 + \frac{0.055}{12})^{12 \times 6}} \]
**Graphical Explanation:**
- A graph could illustrate the relationship between the future values and their respective present values over the given time periods.
- It would show how each original payment is discounted back to its present value using the given interest rate.
**Conclusion:**
Add the present
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