d. At what discount rate would the present value of 15 annual payments of $100, with the first payment right now, be 0? e. How many annual payments of $100, with the first payment right now, would it take to be worth more than $1,000, if the discount rate is 0.05? f. What is the value of 15 annual payments which begin at $100 one year from now and increase at 2% per year thereafter, if the discount rate is 0.05?
d. At what discount rate would the present value of 15 annual payments of $100, with the first payment right now, be 0? e. How many annual payments of $100, with the first payment right now, would it take to be worth more than $1,000, if the discount rate is 0.05? f. What is the value of 15 annual payments which begin at $100 one year from now and increase at 2% per year thereafter, if the discount rate is 0.05?
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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![## Annuity Formulae Problem
### Questions:
d. **At what discount rate would the present value of 15 annual payments of $100, with the first payment right now, be 0?**
To find the discount rate where the present value (PV) becomes zero, use the formula for the present value of an annuity:
\[ \text{PV} = P \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \]
Set PV to 0 and solve for the rate \( r \).
e. **How many annual payments of $100, with the first payment right now, would it take to be worth more than $1,000, if the discount rate is 0.05?**
Determine the smallest number \( n \) such that:
\[ 100 \times \left(\frac{1 - (1 + 0.05)^{-n}}{0.05}\right) > 1000 \]
f. **What is the value of 15 annual payments which begin at $100 one year from now and increase at 2% per year thereafter, if the discount rate is 0.05?**
For increasing payments, use the formula for a growing annuity:
\[ \text{PV} = P \times \left(\frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}\right) \]
Where \( g \) is the growth rate of payments (0.02), and solve for the present value with \( r = 0.05 \) over 15 years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ea22554-2b6e-4908-9457-383afcfa1888%2Fe77fb023-68d3-4500-b6f1-ee624e86731b%2Fckrrm4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Annuity Formulae Problem
### Questions:
d. **At what discount rate would the present value of 15 annual payments of $100, with the first payment right now, be 0?**
To find the discount rate where the present value (PV) becomes zero, use the formula for the present value of an annuity:
\[ \text{PV} = P \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \]
Set PV to 0 and solve for the rate \( r \).
e. **How many annual payments of $100, with the first payment right now, would it take to be worth more than $1,000, if the discount rate is 0.05?**
Determine the smallest number \( n \) such that:
\[ 100 \times \left(\frac{1 - (1 + 0.05)^{-n}}{0.05}\right) > 1000 \]
f. **What is the value of 15 annual payments which begin at $100 one year from now and increase at 2% per year thereafter, if the discount rate is 0.05?**
For increasing payments, use the formula for a growing annuity:
\[ \text{PV} = P \times \left(\frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}\right) \]
Where \( g \) is the growth rate of payments (0.02), and solve for the present value with \( r = 0.05 \) over 15 years.
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