Question

As shown in Table \(18.2,1,000\)dollars invested at 10 percent compound interest will grow into \(1,331\)dollars after three years. What is the present value of \(2,662\)dollars in three years if it is discounted back to the present at a 10 percent compound interest rate? (Hint: \(2,662\)dollars is twice as much as \(1,331.\)dollars)

Short answer

The present value of 2,662 dollars in three years at a 10% interest rate is 2,000 dollars.

Step-by-step solution

Understand Compound Interest

The problem uses compound interest, which means that interest is calculated not only on the initial principal but also on the accumulated interest from previous periods. The formula for compound interest is given by \( FV = PV \times (1 + r)^n \), where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the interest rate, and \( n \) is the number of periods.

Recognize Given Values

In the problem, we are given the future value (\( FV = 2,662 \) dollars), the interest rate (\( r = 10\% \) or 0.10), and the time duration (\( n = 3 \) years). Our task is to find the present value (\( PV \)).

Use the Present Value Formula

To find the present value, we need to rearrange the compound interest formula to solve for \( PV \). This can be done by using the equation \( PV = \frac{FV}{(1 + r)^n} \).

Calculate the Denominator

First, calculate \((1 + r)^n = (1 + 0.10)^3 = 1.1^3\). Calculate 1.1 times itself three times: \(1.1 \times 1.1 \times 1.1 = 1.331\).

Insert Values into the Formula

Now that we've calculated the denominator, substitute the known values into the formula: \( PV = \frac{2,662}{1.331} \).

Compute the Present Value

Complete the division: \( PV = \frac{2,662}{1.331} \approx 2,000 \). Thus, the present value of \( 2,662 \) dollars in three years at a 10% discount rate is 2,000 dollars.

Key concepts

Compound Interest

Compound interest is a critical concept in finance and investing. It refers to the process where interest is calculated not only on the initial principal amount but also on the accumulated interest from prior periods. This means your money can grow exponentially over time, as each period builds on the last.

To calculate compound interest, you use the formula:

• \[ FV = PV \times (1 + r)^n \]
• Where \( FV \) is the future value, \( PV \) is the present value, \( r \) is the interest rate per period, and \( n \) is the number of periods.

This formula shows how your investment grows over each compounding period. The process captures the power of compounding, leading to accelerated growth of savings or investments, which is why it’s so admired in the realm of finance.

In practical terms, the longer the time span and the higher the interest rate, the more exponential the growth due to compounding.

Future Value

The future value is the amount an investment is expected to grow to, at a specified interest rate, over a particular period. This is essentially what all investors aim for - understanding future value helps in planning financial goals.

With the help of the formula for compound interest, future value can be calculated easily by rearranging that formula to focus on the future aspect, like so:

• Use \( FV = PV \times (1 + r)^n \) to find out how much a present investment will grow after a certain number of years at a given interest rate.
• This equation helps you foresee how much money you will have in the future if it is invested wisely.

This is the essence of understanding how investments could grow over time, allowing you to make informed decisions about where to place your money.

Interest Rate

The interest rate is the percentage at which interest is charged or earned over a specified period of time. This rate is crucial as it directly influences how much you will earn on your investments or pay on loans.

There are different types of interest rates, such as fixed or variable, and each has its implications.

• A fixed interest rate remains the same throughout the period, providing predictability.
• A variable or floating interest rate can change, which affects the amount of interest to be paid or earned over time.

In our case, the problem uses a fixed interest rate of 10% per annum, which affects the calculation of both the future and present value. Understanding how different interest rates work can help in making better financial decisions regarding borrowing and investing.

Time Value of Money

The time value of money is a foundational financial principle that suggests money today is worth more than the same amount in the future. This concept takes into account the potential earning capacity of money.

Essentially, it underlines the idea that \(1\) dollar now has greater value compared to \(1\) dollar later because the current dollar can earn interest over time.

• This is reflected in calculations involving present and future values.
• By understanding the time value of money, individuals and businesses can determine the true value of investments and the cost of future cash flows.

When using formulas to find present value, like in our example, reducing a future amount to its equivalent worth today, considering the interest rate, embodies this concept perfectly. Employing the formula \( PV = \frac{FV}{(1 + r)^n} \) helps portray how the time value of money works in real-world finance.