Problem 55 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The parents of a child have just come into a large inheritance and wish to establish a trust fund for her college education. If they estimate that they will need $$\$ 100,000$$ in \(13 \mathrm{yr}\), how much should they set aside in the trust now if they can invest the money at \(8 \frac{1}{2} \% /\) year compounded (a) annually, (b) semiannually, and (c) quarterly?

Step-by-Step Solution

Answer

The parents should set aside approximately \(\$42,517.40\) if the interest is compounded annually, \(\$41,930.16\) if it's compounded semiannually, and \(\$41,719.45\) if it's compounded quarterly.

Step 1: (Step 1: Write down the compound interest formula)

(The compound interest formula is given by the equation: \[A = P\left(1 + \frac{r}{n}\right)^{nt}\] Where: - \(A\) is the future value - \(P\) is the present value (initial investment) - \(r\) is the annual interest rate (in decimal form) - \(n\) is the number of times the interest is compounded per year - \(t\) is the number of years We need to find the initial investment, \(P\), in present value for different compounding frequencies.)

Step 2: (Step 2: Convert the interest rate to decimal form)

(The given interest rate is \(8\frac{1}{2}\%\). To convert it to a decimal, we need to divide it by 100: \[r = \frac{8\frac{1}{2}}{100} = \frac{17}{200} = 0.085\])

Step 3: (Step 3: Solve for the present value (P) for each compounding frequency)

(We will use the given future value, \(A=\$100,000\), the annual interest rate \(r=0.085\), the time \(t=13\) years, with different values of \(n\) for each case.) (a) Annually (\(n=1\)): We rearrange the compound interest formula to get the present value, \(P\): \[P = \frac{A}{(1 + \frac{r}{n})^{nt}}\] Then, plug in the values: \[P = \frac{100,000}{(1 + \frac{0.085}{1})^{1\times13}} = \frac{100,000}{(1 + 0.085)^{13}} \approx \$42,517.40\] (b) Semiannually (\(n=2\)): \[P = \frac{100,000}{(1 + \frac{0.085}{2})^{2\times13}} = \frac{100,000}{(1 + 0.0425)^{26}} \approx \$41,930.16\] (c) Quarterly (\(n=4\)): \[P = \frac{100,000}{(1 + \frac{0.085}{4})^{4\times13}} = \frac{100,000}{(1 + 0.02125)^{52}} \approx \$41,719.45\] The required present value of the trust fund for each compounding frequency is approximately: (a) Annually: \(\$42,517.40\) (b) Semiannually: \(\$41,930.16\) (c) Quarterly: \(\$41,719.45\)

Key Concepts

Present Value

Understanding the concept of Present Value is key when dealing with compound interest problems. Present Value (\( P \)) tells us how much money we need to invest today in order to achieve a specified Future Value at a certain interest rate.
This is crucial in financial planning, such as setting up a trust fund for a child's education. Two main factors affect Present Value:

• Interest Rate: Generally, higher interest rates decrease the present value because money can grow faster.
• Time: The longer the investment duration, the smaller the amount needed today, as there is more time for compounding.

In the given exercise, the parents must decide the present amount to invest (\( P \)) to grow it to $100,000 over 13 years at an 8.5% annual interest rate compounded differently. The calculation is done by rearranging the compound interest formula to solve for \( P \).

Future Value

Future Value (\( A \)) refers to the amount of money an investment will grow to over a period of time at a specified interest rate. It accounts for the original investment, interest applied, and compounding over time.
In financial contexts, understanding Future Value helps anticipate how much an investment will yield, facilitating better decision-making.

• Future Value relies on initial investment, interest rate, compounding frequency, and time.
• For example, the Future Value of $100,000 needed for college education is determined with a duration of 13 years in the exercise.

Calculating the Present Value requires knowing the Future Value you aim for. For investors, knowing the desired Future Value helps backtrack to the appropriate present investment amount.

Compounding Frequency

Compounding Frequency refers to how often interest is calculated and added to the balance. The more frequently interest is compounded, the more the investment grows.
Higher compounding frequency increases Future Value and reduces Present Value. Common frequencies include:

• Annually: Interest calculated once a year.
• Semiannually: Interest calculated twice a year.
• Quarterly: Interest calculated four times a year.

In our exercise, varying the compounding frequency changes the present value required to achieve $100,000. The more frequent the compounding, the less you need to invest today, thanks to faster accumulation of interest.

Interest Rate Conversion

Interest Rate Conversion is transforming the nominal interest rate to match the compounding frequency. This step ensures appropriate calculation of interest over different periods. It's crucial when calculating compound interest.

• Convert percentages to decimals for ease of calculation. For instance, 8.5% becomes 0.085.
• Adjust the nominal rate by the compounding frequency. For semiannual compounding, split the annual rate into two.

In the exercise, we convert an 8.5% annual rate to different compounding frequencies for precise calculations. Understanding Interest Rate Conversion prevents misjudging the potential returns on investments.