Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A city has $$\$ 2.5$$ million worth of school bonds that are due in \(20 \mathrm{yr}\) and has established a sinking fund to retire this debt. If the fund earns interest at the rate of \(7 \%\) /year compounded annually, what amount must be deposited annually in this fund?

Step-by-Step Solution

Answer

The city must deposit approximately $$45,238.51$$ annually into the sinking fund to retire the \(2.5\) million debt in 20 years with an interest rate of 7% per year compounded annually.

Step 1: Identify the given values and the formula required to solve the problem

In this case, we are given: - Future Value (FV) = $2,500,000 - Time (t) = 20 years - Interest rate (r) = 7% per year, compounded annually We need to find the annual deposit amount, which we can represent as P. The formula for the future value of an annuity is: \[FV = P \times \frac{(1+r)^t-1}{r}\] Where: - FV is the future value of the annuity - P is the annuity payment (annual deposit) - r is the interest rate - t is the time period in years

Step 2: Convert the interest rate to a decimal

Before plugging the values into the formula, let's convert the interest rate from percentage to decimal: r = 7% = 0.07

Step 3: Rearrange the formula to solve for P

To find the annual deposit amount (P), we need to rearrange the annuity formula: \[P = \frac{FV \times r}{(1+r)^t-1}\]

Step 4: Plug in the given values and solve for P

Now, we can plug in the given values into the formula: \[P = \frac{2,500,000 \times 0.07}{(1+0.07)^{20}-1}\] Using a calculator, we get: \[P = \frac{2,500,000 \times 0.07}{(1.07)^{20}-1}\] \[P = \frac{175,000}{3.8693}\] \[P \approx 45,238.51\]

Step 5: State the final answer

The city must deposit approximately $$45,238.51$$ annually into the sinking fund to retire the \(2.5\) million debt in 20 years with an interest rate of 7% per year compounded annually.

Key Concepts

Future Value of Annuity

The future value of an annuity is the total value of a series of regular payments at some point in the future, taking into account interest earned over time. In simpler terms, it's what you will have in your account after investing a certain amount of money regularly over a period of time.
To calculate the future value of an annuity, we use the following formula:\[FV = P \times \frac{(1+r)^t-1}{r}\]

• FV represents the future value, or what you will have after the investment period.
• P is the annuity payment, which is the amount you deposit regularly.
• r stands for the interest rate expressed in decimal form.
• t is the time period over which the deposits are made.

In the context of the sinking fund for the school bonds, the future value is the amount needed to retire the debt ($2.5 million) after 20 years of regular annual deposits.
Understanding how future value works helps individuals and organizations plan for future financial needs by knowing how much to save regularly to achieve a financial goal.

Compounded Interest

Compounded interest refers to earning interest not only on your initial investment but also on the interest that accumulates each year. It's a powerful concept that can significantly increase the value of an investment over time.
This concept is illustrated by the interest rate of 7% per year compounded annually in the city’s sinking fund. This means that each year, the interest is calculated on the total amount (initial deposits plus the previously earned interest), and this new total becomes the starting point for calculating interest the next year.To visualize this:

• Year 1: Interest is calculated on the initial deposit.
• Year 2: Interest is calculated on the sum of the initial deposit and the interest earned in Year 1.
• This process continues, compounding over the years.

The formula used in the problem \[P = \frac{FV \times r}{(1+r)^t-1}\]emphasizes using compounded interest to figure out the required deposit. Compounded interest accelerates the growth of your investments, and comprehending its mechanics can guide better financial decisions.

Debt Retirement

Debt retirement involves planning and saving adequate funds to pay off existing debt by a specified period. For entities like cities, which often issue bonds, ensuring that they can retire (or pay back) these debts when they become due is crucial.
A sinking fund is typically used for this purpose. It is a special account established to save regularly for repaying debts. In the original exercise, the city uses a sinking fund method where they set aside a specific amount every year so that by the end of 20 years, they can fully repay the $2.5 million school bond.
The process involves:

• Determining future financial needs, in this case, $2.5 million.
• Calculating how much needs to be deposited regularly, considering an annual return (7% compounded interest).
• Setting aside this amount every year to ensure the total required is accumulated by the end of the term.

Effectively planning for debt retirement, especially using a sinking fund, can help avoid financial shortages when repayments are due and contributes to solid financial management. Understanding these concepts is valuable in both personal and organizational financial planning.