Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the present value of $$\$ 40,000$$ due in 4 yr at the given rate of interest. \(8 \% /\) year compounded quarterly

Step-by-Step Solution

Answer

The present value of $$\$40,000$$ due in 4 years at an interest rate of $$8\%$$ per year compounded quarterly is approximately $$\$29,127.92$$.

Step 1: Understand the formula for compound interest

The formula for compound interest can be written as \(FV = PV(1 + i)^n\), where: - \(FV\) is the future value of the investment - \(PV\) is the present value or the amount invested today - \(i\) is the interest rate per compounding period - \(n\) is the number of compounding periods We need to find the present value (\(PV\)), given the future value (\(FV\)), interest rate, and the number of compounding periods.

Step 2: Convert the annual interest rate to a quarterly interest rate

Since the interest is compounded quarterly, we need to convert the annual interest rate to a quarterly interest rate. To do this, we'll divide the annual interest rate by the number of quarters per year: \[ i = \frac{Annual\, Rate}{4} = \frac{0.08}{4} = 0.02 \] So, the quarterly interest rate is $$2\%$$ or $$0.02$$.

Step 3: Find the number of compounding periods

Next, we'll determine how many quarters are in 4 years, as the interest is compounded quarterly: \[ n = \text{Years} \times \text{Quarters per Year} = 4 \times 4 = 16 \] So, there are 16 compounding periods.

Step 4: Rearrange the formula and solve for the present value

To find the present value (\(PV\)), we need to rearrange the formula for compound interest: \[ PV = \frac{FV}{(1 + i)^n} \] Now plug in the given future value, quarterly interest rate, and the number of compounding periods: \[ PV = \frac{40,000}{(1 + 0.02)^{16}} \]

Step 5: Calculate the present value

Now, we'll plug in the values into the formula and calculate the present value: \[ PV = \frac{40,000}{(1.02)^{16}} \approx \frac{40,000}{1.372785\ldots} \approx 29,127.92 \] Hence, the present value is approximately $$\$29,127.92$$.

Key Concepts

Compound Interest

When you invest money or take out a loan, compound interest is the concept that describes how the amount of interest earned or owed accrues not only on the initial principal but also on the accumulated interest over previous periods. It is the reason your savings grow over time and also why loan balances can increase when interest isn't paid down.

To illustrate the power of compound interest, consider this example: If you invest \(1,000 at an annual interest rate of 5% compounded yearly, after one year, you'll have \)1,050. In the second year, interest is calculated on the new total, \(1,050, not just the original \)1,000. Over time, this leads to exponential growth of your investment.

Key Formula:

The compound interest formula is expressed as: \(FV = PV(1 + i)^n\), where \(FV\) represents the future value of the investment, \(PV\) is the present value, \(i\) is the interest rate per period, and \(n\) is the number of compounding periods.

Future Value

The concept of future value (FV) is pivotal when assessing the expected growth of an investment or the final amount due on a loan. Future value tells you what your money can grow to over a given time frame and at a specified interest rate, assuming that all interest payments are reinvested at the same rate.

For instance, if you're planning to save for retirement or any long-term goal, understanding the FV of your current savings can help you determine how much more you need to save to meet your objectives. The marvel of compound interest significantly impacts the future value of investments, demonstrating that time can be as critical as the interest rate when it comes to financial growth.

Using the Future Value Formula:

To calculate future value using the formula \(FV = PV(1 + i)^n\), you'll need the present value, the interest rate per period, and the number of periods (compounding periods) over which the investment will grow.

Interest Rate

The interest rate is a percentage that denotes the cost of borrowing money, or the return on invested funds, over a specific period. It is a critical variable that affects the growth of investments and the cost of borrowing.

In our formula for calculating compound interest, \(i\) represents the interest rate per compounding period, which can vary based on how frequently the interest is compounded. If the interest is compounded more frequently, the effective interest rate becomes higher, resulting in more significant growth of the investment or debt over time.

Adjusting for Compounding Periods:

When dealing with different compounding frequencies, like quarterly compounding, it's important to adjust the annual rate to match the compounding periods. For example, an annual rate of 8% compounding quarterly is converted to 2% per quarter.

Compounding Periods

Compounding periods refer to the frequency with which interest is added to the principal balance of an investment or loan. Common compounding frequencies include yearly, semi-annually, quarterly, monthly, or even daily.

The number of compounding periods, represented by \(n\) in our formulas, directly influences how much your investment will grow or how much debt will accumulate. Generally, the more frequent the compounding periods, the higher the future value will be.

Calculating Compounding Periods:

To calculate the total number of compounding periods, multiply the number of years by the number of times interest is compounded per year. Using our initial example, interest compounded quarterly over four years results in \(n = 4 \times 4 = 16\) compounding periods.