Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The managers of a pension fund have invested $$\$1.5$$ million in U.S. government certificates of deposit that pay interest at the rate of \(5.5 \%\) /year compounded semiannually over a period of 10 yr. At the end of this period, how much will the investment be worth?

Step-by-Step Solution

Answer

At the end of the 10-year period, the investment will be worth approximately $2,632,500.

Step 1: Calculate the interest rate per period

We need to find the interest rate per period as it's compounded semi-annually. Hence, divide the annual interest rate (5.5%) by the number of compounding periods in a year (2): \(i = 5.5\% / 2 = 2.75\%\)

Step 2: Determine the number of periods

Now let's determine the total number of periods, which is the number of years (10) multiplied by the number of compounding periods in a year (2): \(n = 10 * 2 = 20\)

Step 3: Apply the compound interest formula to find the future value

We can now use the compound interest formula to find the future value of the investment: \(FV = P(1 + i)^n\) Where: - \(FV\) is the future value of the investment - \(P\) is the principal, in our case \(1.5\) million - \(i\) is the interest rate per period, we found it to be \(2.75\%\) - \(n\) is the total number of periods, we found it to be \(20\) Before plugging in the values, convert the interest rate from percentage to decimal by dividing it by 100: \(i = 2.75\%/100 = 0.0275\) Now, plug in the values into the formula: \(FV = 1,500,000 (1 + 0.0275)^{20}\)

Step 4: Calculate the future value

Finally, calculate the future value of the investment: \(FV = 1,500,000 (1.0275)^{20}\) \(FV \approx 1,500,000 * 1.755\) \(FV \approx 2,632,500\) At the end of the 10-year period, the investment will be worth approximately $2,632,500.

Key Concepts

Future Value

In the realm of finance, the term "future value" refers to the amount of money an investment will grow to over a period of time. This concept hinges on the principle of compound interest, which means that your investment does not just earn interest on the initial amount, but also on the interest accumulated in previous periods. To imagine this, think of snowballing: as the snowball rolls, it picks up more snow, just like how your initial investment accumulates more interest.
Future value is crucial for investors as it gives them an idea of what their current investments will be worth in the future, allowing them to make more informed financial decisions.The basic formula to compute future value when compound interest is involved is:- \(FV = P(1 + i)^n\)where:

• \(FV\) is the future value
• \(P\) is the principal amount (the initial amount or investment)
• \(i\) is the interest rate per period
• \(n\) is the total number of compounding periods

By understanding and calculating the future value, you can plan your investments better and have clearer expectations about financial growth.

Interest Rate per Period

The concept of "interest rate per period" is pivotal in the calculation of compound interest, particularly when interest is compounded more than once a year. In our example, the annual interest rate is given as 5.5%, and it's important to convert this annual rate into an interest rate per period since the interest compounds semiannually.The annual interest rate of 5.5% is divided by the number of compounding periods in a year, which is 2 for semiannual compounding. So, the calculation is as follows:- \(i = \frac{5.5\%}{2} = 2.75\%\)This results in an interest rate per period of 2.75%. It is critical to convert this percentage into a decimal for calculations in the compound interest formula:- \(i = \frac{2.75\%}{100} = 0.0275\)Understanding the interest rate per period is fundamental, as it signifies the portion of the annual interest that applies to each compounding period. Thus, it enables more accurate calculations for the growth of investments over multiple periods.

Compounding Periods

Compounding periods justify how often interest is calculated and added to the principal within a specified timeframe. This concept affects how interest accumulates over time. The more frequent the compounding, the higher the amount of compounded interest. In the given example, we have semiannual compounding, meaning the interest is calculated and added twice a year. Compounding more frequently leads to slightly higher future values compared to less frequent compounding because each time interest is calculated, it is added to the total and then used to calculate future interest. To find the total number of compounding periods, simply multiply the number of years by the number of compounding periods per year. In this case, it's calculated as:- \(n = 10 \text{ years} \times 2 \text{ (semiannual compounding)} = 20 \text{ periods}\)Understanding compounding periods helps investors know how interest is applied, ultimately affecting the future value of their investment. This knowledge is crucial in maximizing returns.