Question

As shown in Table \(16.2, \$ 1,000\) invested at 10 percent compound interest will grow into \(\$ 1,331\) after three years. What is the present value of \(\$ 2,662\) in three years if it is discounted back to the present at a 10 percent compound interest rate?

Short answer

The present value is $2,000.

Step-by-step solution

Understanding the Problem

We need to find out what amount, when invested today at a 10% compound interest rate, will grow to $2,662 in 3 years.

Identifying the Formula

The present value (PV) formula for compound interest is given by \( PV = \frac{FV}{(1 + r)^n} \), where \( FV \) is the future value, \( r \) is the interest rate, and \( n \) is the number of years.

Plugging in the Values

We know that \( FV = 2,662 \), \( r = 0.10 \), and \( n = 3 \). Substitute these into the formula to get \( PV = \frac{2,662}{(1 + 0.10)^3} \).

Calculating the Denominator

Calculate \((1 + 0.10)^3\). This is \(1.10^3 = 1.331\).

Solving for Present Value

Now, compute \( PV = \frac{2,662}{1.331} \). This gives us a \( PV = 2,000 \).

Verifying the Calculation

Verify by recalculating if necessary to ensure the solution is accurate.

Key concepts

Compound Interest

Compound interest is a powerful concept in finance, which determines how money grows over time. Unlike simple interest, where interest is calculated only on the initial principal, compound interest accumulates on both the initial principal and the accumulated interest from previous periods. This means your money can grow exponentially.
For example, if you invest $1000 at a 10% compound interest rate, by the end of the first year, your investment grows to $1100. In the second year, the interest is calculated on $1100, not just your original $1000. This results in $1210 by the end of year two. This process continues, making compound interest a powerful tool for wealth accumulation over time.

• Compound interest = principal amount + interest on compound interest
• Results in exponential growth of money
• Vital for accurate financial forecasting

Future Value

The future value of an investment is the amount of money that an initial investment will grow to over a certain period of time, given a certain interest rate. It answers the question, 'How much will my investment be worth in the future?'
In the context of compound interest, the future value is calculated using the principal amount, the interest rate, and the number of compounding periods. For example, if you want to know how much $1000 will grow to after 3 years at 10% annual compound interest, you use a specific formula. The calculation involves raising the sum of 1 and the interest rate to the power of the number of years, then multiplying by the principal.

• Formula: \( FV = PV \times (1 + r)^n \)
• Key for long-term investment planning
• Depends on frequency of compounding

Interest Rate

The interest rate is the percentage charged on the total amount borrowed or earned. In the realm of investments, it's essential as it determines how much your money will grow over time. The interest rate can be annual, monthly, or even daily, depending on the terms of the investment or loan.
A 10% interest rate implies that for every $100 invested, you earn $10 in interest annually. However, with compound interest, this $10 will also earn interest in subsequent periods, increasing the total growth. When choosing investments, comparing interest rates can help you identify the best opportunities for growth.

• Higher interest rates lead to faster growth
• Can be nominal or effective depending on compounding
• Impact on loans and savings is significant

Financial Formulas

Financial formulas serve as essential tools for managing investments, understanding returns, and assessing risks in various scenarios. Specifically, for compound interest, the present value (PV) and future value (FV) formulas are crucial in determining the worth of money at different points in time.
The present value formula calculates how much a future sum of money is worth today. This is particularly useful for investors and financial analysts to determine if an investment is viable. It involves taking the future value of an investment and dividing it by the compound interest factor.
Examples of important formulas include:

• Present Value: \( PV = \frac{FV}{(1 + r)^n} \)
• Future Value: \( FV = PV \times (1 + r)^n \)
• Understanding these formulas is key to financial literacy and investment success.