Problem 36 - 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 \(\$ 25,000\) in 2.75 years, if interest is compounded quarterly at \(6 \%\).

Step-by-Step Solution

Answer

Present value is approximately $21,370.

Step 1: Identify the Variables

First, we need to identify the values given in the problem and what we are looking for. **Future Value (A)** is given as $25,000. **Interest rate (r)** is 6%, which we convert to a decimal as 0.06. **Time (t)** in years is 2.75. The number of times the interest is compounded per year (n) is quarterly, so n = 4.

Step 2: Set Up the Equation

We want to find the present value (P) using the given formula: \[ P = A \left(1+\frac{r}{n}\right)^{-nt} \]Insert the known values into the formula: \[ P = 25000 \left(1 + \frac{0.06}{4}\right)^{-4 \times 2.75} \]

Step 3: Calculate the Compound Interest Portion

Calculate the expression inside the parentheses first:\[ 1 + \frac{0.06}{4} = 1 + 0.015 = 1.015 \]Next, calculate the exponent portion:\[ -4 \times 2.75 = -11 \]

Step 4: Calculate the Present Value

Substitute back the values to find P:\[ P = 25000 \times (1.015)^{-11} \]Calculate the exponent:\[ (1.015)^{-11} \approx 0.8548 \]Multiply to find P:\[ P = 25000 \times 0.8548 \approx 21370 \]

Step 5: Present the Solution

The calculated present value is approximately \\(21,370. This is the amount you would need to invest today at a 6% annual interest rate compounded quarterly to reach a future value of \\)25,000 in 2.75 years.

Key Concepts

Present Value

The concept of present value is fundamental in finance and investing. It is the current worth of a future sum of money or stream of cash flows, given a specified rate of return. This calculation is based on the idea that money available now is worth more than the same amount in the future due to its earning potential.
To find the present value (\( P \)) of a future sum (\( A \)), you must discount it back to the present using the formula:

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

Where \( r \) is the interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the time in years.
Knowing the present value helps you understand how much you need to invest today to reach a desired financial goal in the future.

Future Value

The future value (\( A \)) is the value of a current asset at a specified date in the future, based on an assumed rate of growth. It's an important concept for finance professionals because it helps predict how much an investment made today will grow over time.
The basic formula for calculating future value when interest is compounded is:

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

This formula takes into account the present value (\( P \)), the interest rate (\( r \)), the number of compounding periods per year (\( n \)), and the time period in years (\( t \)).
Understanding future value can help with planning and decision-making for investments, savings goals, and managing debts. It provides a way to measure potential growth and financial outcomes over time.

Interest Rate

Interest rate is the percentage charged on a loan or paid on an investment for using money. It signifies the cost of borrowing or the gain from lending. A higher interest rate means more cost for borrowing and more gain for saving.
In the context of compound interest, the interest rate (\( r \)) can influence how quickly your investment grows. For most financial calculations, the interest rate is converted from a percentage to a decimal. For example, a 6% interest rate becomes 0.06.
Interest rates are vital for financial planning as they directly impact loan costs and investment returns. They are affected by economic factors, inflation rates, policies by central banks, and the creditworthiness of a borrower.

Quarterly Compounding

Quarterly compounding is a method of calculating interest whereby the compounding occurs four times a year. This is one type of compound interest calculation.
In this method, the annual interest rate is divided by four to get the quarterly rate, and compounding happens at that rate every three months. For instance, if a 6% annual interest rate is compounded quarterly, each quarter earns 1.5% interest.
The formula for compound interest with quarterly compounding is:

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

Here, \( n \) equals 4 because interest compounds four times a year. This approach typically results in higher returns compared to annual compounding, due to more frequent application of interest to the principal amount.