Yumi's grandparents presented her with a gift of $18,000 when she was 8 years old to be used for her college education. Over the next 9 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 3.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 4%/year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.) $ 5805.62 Need Help? Read It Watch It
Yumi's grandparents presented her with a gift of $18,000 when she was 8 years old to be used for her college education. Over the next 9 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 3.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 4%/year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.) $ 5805.62 Need Help? Read It Watch It
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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![**Educational Content: Calculating Future Value and Withdrawals**
**Problem Scenario:**
Yumi's grandparents presented her with a gift of $18,000 when she was 8 years old to be used for her college education. Over the next 9 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 3.5% per year, compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 4% per year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.)
**Solution:**
Based on the conditions stated, we need to determine the size of each annual installment Yumi can withdraw from age 18 over the next 4 years.
**Calculation Steps:**
1. **Calculate the Future Value (FV) of the initial investment:**
- Principal = $18,000
- Interest Rate = 3.5% per year, compounded monthly
- Time = 9 years
2. **Use the formula for compound interest:**
\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]
- \(P\) = principal amount ($18,000)
- \(r\) = annual interest rate (3.5% or 0.035)
- \(n\) = number of compounding periods per year (12)
- \(t\) = time in years (9)
3. **Find the amount available at age 17 and calculate equal withdrawals for 4 years at 4% annual interest, compounded annually.**
**Solution Explanation:**
This problem involves understanding both compound interest and annuities. First, we calculate the future value of Yumi’s account at age 17 using the initial conditions. Then, we solve for the annuity, which represents equal annual withdrawals over four years.
Remember: The assumption that no further interest accrues after turning 17 ensures simplicity in calculating the withdrawal amounts directly based on the accrued value at age 17.
**Interactive Components:**
- **Need Help](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8719c815-bc4c-45f2-8134-fbe23974a9e7%2Ff6942956-9864-4207-b31c-3222587ba295%2Foznjsn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Calculating Future Value and Withdrawals**
**Problem Scenario:**
Yumi's grandparents presented her with a gift of $18,000 when she was 8 years old to be used for her college education. Over the next 9 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 3.5% per year, compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 4% per year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.)
**Solution:**
Based on the conditions stated, we need to determine the size of each annual installment Yumi can withdraw from age 18 over the next 4 years.
**Calculation Steps:**
1. **Calculate the Future Value (FV) of the initial investment:**
- Principal = $18,000
- Interest Rate = 3.5% per year, compounded monthly
- Time = 9 years
2. **Use the formula for compound interest:**
\[
FV = P \left(1 + \frac{r}{n}\right)^{nt}
\]
- \(P\) = principal amount ($18,000)
- \(r\) = annual interest rate (3.5% or 0.035)
- \(n\) = number of compounding periods per year (12)
- \(t\) = time in years (9)
3. **Find the amount available at age 17 and calculate equal withdrawals for 4 years at 4% annual interest, compounded annually.**
**Solution Explanation:**
This problem involves understanding both compound interest and annuities. First, we calculate the future value of Yumi’s account at age 17 using the initial conditions. Then, we solve for the annuity, which represents equal annual withdrawals over four years.
Remember: The assumption that no further interest accrues after turning 17 ensures simplicity in calculating the withdrawal amounts directly based on the accrued value at age 17.
**Interactive Components:**
- **Need Help
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