Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Yumi's grandparents presented her with a gift of $$\$ 20,000$$ when she was 10 yr old to be used for her college education. Over the next \(7 \mathrm{yr}\), until she turned 17 , Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 5.5\%lyear compounded monthly. Upon turning 17 , Yumi now plans to withdraw her funds in equal annual installments over the next 4 yr, starting at age 18 . If the college fund is expected to earn interest at the rate of \(6 \% /\) year, compounded annually, what will be the size of each installment?

Step-by-Step Solution

Answer

Yumi will need to withdraw approximately $7601.24 each year for four years, starting at age 18, to utilize her entire college fund.

Step 1: Calculate the money in the account when Yumi turns 17

To find the total amount of money in the account after 7 years, we'll use the compound interest formula: A = P(1 + r/n)^(nt) where: A = Final amount P = Principal amount ($20,000) r = annual interest rate (5.5% = 0.055) n = number of times interest is compounded per year (monthly = 12) t = number of years (7 years) The formula becomes: A = 20000(1 + 0.055/12)^(12 * 7)

Step 2: Evaluate the formula and obtain A

We can now calculate for A (Yumi's account balance at age 17) using the given values: A = 20000(1 + 0.055/12)^(12 * 7) A ≈ 28723.227 This means Yumi has approximately $28,723.23 in her account when she turns 17.

Step 3: Calculate equal annual installments using annuity formula

Let P be the annual installment. The account balance, which is an annuity, can be represented by the following annuity formula: \(A = P \cdot \frac{1 - (1 + r)^{-t}}{r}\) Since we have A, r, and t, we can solve for P. We know from step 2 that A ≈ 28723.227, the interest rate (r) is now 6% annually (0.06), and the number of years (t) is 4. \(28723.227 = P \cdot \frac{1 - (1 + 0.06)^{-4}}{0.06}\)

Step 4: Solve for P (annual installment)

Solve the equation to find the size of each annual installment: \(28723.227 = P \cdot \frac{1 - (1 + 0.06)^{-4}}{0.06}\) P ≈ 7601.24 Yumi will need to withdraw approximately $7601.24 each year for four years, starting at age 18, to utilize her entire college fund.

Key Concepts

Annuity Formula

The annuity formula is a key concept in finance that helps us calculate the value of regular payments or withdrawals over time. When planning to withdraw funds, like Yumi is doing, the annuity formula assists in determining how much she can take out each year.
Understanding annuities involves:

• Amount (A): This is the total sum of money needed or available, which in Yumi's case, is approximately $28,723.23 at age 17.
• Installment (P): The amount of each equal payment, which needs to be calculated.
• Interest Rate (r): This is the annual interest rate. Yumi's college fund earns 6% interest compounded annually.
• Years (t): The total number of years of payments, for Yumi, it's 4 years.

The formula used is:\[A = P \cdot \frac{1 - (1 + r)^{-t}}{r}\]By solving for 'P', you can find out how much to withdraw annually to use up the entire investment. It's a handy formula for anyone planning structured withdrawals from savings.

Interest Rate Calculation

Interest rate calculation is fundamental to understanding how money grows or decreases over time in any investment or savings plan. Yumi's investment example showcases the significance of this rate, both in saving and spending.
There are a few key aspects:

• Compound Interest: This involves adding interest to the principal sum, where Yumi's parents invested $20,000 compounded monthly for seven years at 5.5% interest.
• Converting Rates: The interest rate provided annually needs to be converted according to the compounding period. For example, converting 5.5% annually to a monthly rate involves dividing by 12 since interest is compounded monthly.
• Effective Rate Calculation: Using the formula \[A = P(1 + \frac{r}{n})^{nt}\], where 'n' is the number of times interest is compounded annually, helps calculate the future value of an investment.

Knowing how to accurately calculate and apply interest rates enables investors to make informed decisions about saving and investing.

Financial Mathematics

Financial mathematics involves using mathematical techniques to solve problems in finance. It incorporates various formulas and methods to determine investment growth, loan payments, and more.
It's essential for evaluating situations like Yumi's college fund. Here are a few fundamental components:

• Time Value of Money: This principle suggests that a sum of money has different values at different points in time due to earning capacity. Yumi's $20,000 gift grows over time through interest.
• Present Value vs Future Value: Present value refers to the current value of a sum due in the future. Future value is the amount that the initial investment turns into. In Yumi’s case, $20,000 grows to approximately $28,723.23 in 7 years.
• Compounding: This is how interest is calculated on both the initial principal and the accumulated interest from previous periods, resulting in exponentially growing savings or investments.

Mastering financial mathematics is crucial for managing funds effectively, enabling clear insights into how parameters like interest rates and time impact the desirability of financial decisions and strategies.