What is the future value of \$1 invested today at an annual interest rate of \(r\) for \(n\) periods, compounded once per period?

• A. \$1 \times (1 - r)^n$
• B. \$1 \times (1 + r \times n)$
• C. \$1 \div (1 + r)^n$
• D. \$1 \times (1 + r)^n$

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of future value. Future value (FV) is the value of a current amount of money after it has grown due to interest over a specified number of periods. Compounding means that interest is calculated on both the principal and any accumulated interest.
2. Step 2: Identify the formula for future value with compounding. The formula is: FV = P × (1 + r)^n, where P is the principal amount (initial investment), r is the annual interest rate, and n is the number of compounding periods.
3. Step 3: Analyze the given options. The correct formula for future value is \$1 \times (1 + r)^n$, as it accounts for compounding interest over n periods. The other options do not correctly represent the compounding process.
4. Step 4: Substitute the values into the formula. Replace P with \$1 (the initial investment), r with the given annual interest rate, and n with the number of periods to calculate the future value.
5. Step 5: Perform the calculation using the formula \(FV = 1 \times (1 + r)^n\). This will give the future value of the \$1 investment after n periods at the annual interest rate r.