Problem 35 - A Graphical Approach to College Algebra

Question

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Find the present value of an account that will be worth \(\$ 10,000\) in 5 years, if interest is compounded semiannually at \(3 \%\).

Step-by-Step Solution

Answer

The present value is approximately $8,616.77.

Step 1: Identify the Given Values

We are given that the future value \(A\) is \(\$10,000\), the number of years \(t\) is 5, the annual interest rate \(r\) is 3\% (or 0.03 as a decimal), and the number of compoundings per year \(n\) is 2 since the interest is compounded semiannually.

Step 2: Write Down the Formula for Present Value

The formula for the present value \(P\) is: \[P = A \left( 1 + \frac{r}{n} \right)^{-nt}\] We will substitute the given values into this formula.

Step 3: Substitute the Values into the Formula

Substitute \(A = 10,000\), \(r = 0.03\), \(n = 2\), and \(t = 5\) into the formula:\[P = 10000 \left( 1 + \frac{0.03}{2} \right)^{-2 \times 5}\] This simplifies the equation to:\[P = 10000 \left( 1 + 0.015 \right)^{-10}\]

Step 4: Simplify the Expression Inside the Parentheses

First, compute inside the parentheses:\[1 + 0.015 = 1.015\]

Step 5: Calculate the Exponent

Now calculate the exponent:\[1.015^{-10}\] First convert to positive exponent then compute:\[\frac{1}{1.015^{10}}\]

Step 6: Complete the Calculation

Calculate \(1.015^{10}\) using a calculator:\[1.015^{10} \approx 1.160968\]Now compute:\[\frac{1}{1.160968} \approx 0.861677\]Finally, multiply by 10,000 to find \(P\):\[P = 10000 \times 0.861677 \approx 8616.77\]

Key Concepts

Future Value

When dealing with investments or savings, the future value (FV) is essentially the amount of money that an investment will grow to over a given period when interest is applied. It's important to know how much your current savings will become in the future, thus why calculating the future value is essential. In contexts involving savings accounts or investments, this future value is what you expect to receive at the end of a certain amount of time. The future value equation getting discussed here is:

• \(A = P \left(1+\frac{r}{n}\right)^{nt} \)

Where:

• \(A\) denotes the future value.
• \(P\) is the principal, or present value.
• \(r\) represents the annual interest rate in decimal form.
• \(n\) is the number of compounding periods per year.
• \(t\) is the time in years.

Understanding this formula will allow you to predict the amount of money you will have in the future based on current funds under given conditions. This is incredibly useful for planning financial goals.

Compound Interest

Compound interest is a powerful financial concept where the interest earned on an initial deposit (or principal) also earns interest over time. This leads to exponential growth rather than linear. Here are some key points about compound interest:

• It involves earning "interest on interest", which over a long period, significantly increases the investment's total return.
• It makes use of regular compounding periods which can be annually, semi-annually, quarterly, or even monthly.To calculate compound interest:
  • Use the formula:\[A = P \left(1+\frac{r}{n}\right)^{nt} \]

As seen above, the equation shows us how we add the interest calculated for every period on top of the existing amount.Thus, compound interest accelerates the growth of an investment more than simple interest, which only applies to the original principal over time. Knowing how to calculate it can help you determine how your investment or savings will grow throughout its duration.

Exponentiation

Exponentiation in the context of finance typically refers to raising a number to a power, which is a crucial part of calculating the future value in compound interest. In the formula \(A = P \left(1+\frac{r}{n}\right)^{nt} \), exponentiation is used to account for the repeated application of the interest rate over multiple periods.Here's why it's key:

• The exponent, \(nt\), signifies applying the growth rate over a certain number of periods.
• In our example, \((1 + \frac{r}{n})\) is raised to \(10\) (since we compound semiannually for 5 years, \(n\times t = 2 \times 5\)).
• This process results in calculating how the principal grows, factoring in the compound nature of the interest.

Exponentiation makes compound interest powerful because it essentially means applying the interest multiplier several times.The outcome of this repeated multiplication leads to the compounding effect, where the amount grows faster over time, explaining the transformative power of compound interest. Knowing this helps in ensuring accurate calculation of future investment values.