Palma wants to establish a fund for her granddaughter's college education. What lump sum must she deposit in an account that pays an annual interest rate of 6%, compounded monthly, if she wants to have $10,000 in 10 years?
Palma wants to establish a fund for her granddaughter's college education. What lump sum must she deposit in an account that pays an annual interest rate of 6%, compounded monthly, if she wants to have $10,000 in 10 years?
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
Related questions
Question
100%
![**Problem Statement:**
Palma wants to establish a fund for her granddaughter's college education. What lump sum must she deposit in an account that pays an annual interest rate of 6%, compounded monthly, if she wants to have $10,000 in 10 years?
**Solution Details:**
To determine the required lump sum deposit, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given:
- \( A = 10,000 \)
- \( r = 0.06 \)
- \( n = 12 \) (monthly compounding)
- \( t = 10 \)
We need to solve for \( P \) (the principal amount).
Rearranging the formula to solve for \( P \):
\[ P = \frac{A} {\left(1 + \frac{r}{n}\right)^{nt}} \]
Plugging in the given values:
\[ P = \frac{10,000}{\left(1 + \frac{0.06}{12}\right)^{12 \times 10}} \]
\[ P = \frac{10,000}{\left(1 + 0.005\right)^{120}} \]
\[ P = \frac{10,000}{\left(1.005\right)^{120}} \]
By calculating the compound factor:
\[ (1.005)^{120} \approx 1.8194 \]
So:
\[ P = \frac{10,000}{1.8194} \]
\[ P \approx 5496.86 \]
Therefore, Palma needs to deposit approximately $5,496.86 in the account today to have $10,000 in 10 years, given the conditions mentioned.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8311114-a7cb-49e3-b44e-1e05de4bc29a%2F2218b546-912a-4cba-91df-d9b6ebb92dc3%2F6beaz4.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Palma wants to establish a fund for her granddaughter's college education. What lump sum must she deposit in an account that pays an annual interest rate of 6%, compounded monthly, if she wants to have $10,000 in 10 years?
**Solution Details:**
To determine the required lump sum deposit, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given:
- \( A = 10,000 \)
- \( r = 0.06 \)
- \( n = 12 \) (monthly compounding)
- \( t = 10 \)
We need to solve for \( P \) (the principal amount).
Rearranging the formula to solve for \( P \):
\[ P = \frac{A} {\left(1 + \frac{r}{n}\right)^{nt}} \]
Plugging in the given values:
\[ P = \frac{10,000}{\left(1 + \frac{0.06}{12}\right)^{12 \times 10}} \]
\[ P = \frac{10,000}{\left(1 + 0.005\right)^{120}} \]
\[ P = \frac{10,000}{\left(1.005\right)^{120}} \]
By calculating the compound factor:
\[ (1.005)^{120} \approx 1.8194 \]
So:
\[ P = \frac{10,000}{1.8194} \]
\[ P \approx 5496.86 \]
Therefore, Palma needs to deposit approximately $5,496.86 in the account today to have $10,000 in 10 years, given the conditions mentioned.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, finance and related others by exploring similar questions and additional content below.Recommended textbooks for you
Essentials Of Investments
Finance
ISBN:
9781260013924
Author:
Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:
Mcgraw-hill Education,
Essentials Of Investments
Finance
ISBN:
9781260013924
Author:
Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:
Mcgraw-hill Education,
Foundations Of Finance
Finance
ISBN:
9780134897264
Author:
KEOWN, Arthur J., Martin, John D., PETTY, J. William
Publisher:
Pearson,
Fundamentals of Financial Management (MindTap Cou…
Finance
ISBN:
9781337395250
Author:
Eugene F. Brigham, Joel F. Houston
Publisher:
Cengage Learning
Corporate Finance (The Mcgraw-hill/Irwin Series i…
Finance
ISBN:
9780077861759
Author:
Stephen A. Ross Franco Modigliani Professor of Financial Economics Professor, Randolph W Westerfield Robert R. Dockson Deans Chair in Bus. Admin., Jeffrey Jaffe, Bradford D Jordan Professor
Publisher:
McGraw-Hill Education