Intro to Finance: Time Value of Money - Future Value, FV = PV(1 + r)^n

What This Is

Future Value (FV) is a fundamental concept in finance that calculates the value of a present amount (PV) at a future date, taking into account the periodic interest rate (r) and the number of periods (n). For example, if you invest $1,000 today at an 8% annual interest rate, the future value after 5 years would be $1,000 × (1 + 0.08)^5 = $1,469.30.

Key Formulas & Symbols

• FV = PV × (1 + r)^n where FV = future value, PV = present value, r = periodic interest rate, n = number of periods.
• PV = FV / (1 + r)^n where PV = present value, FV = future value, r = periodic interest rate, n = number of periods.
• r = (FV/PV)^(1/n) - 1 where r = periodic interest rate, FV = future value, PV = present value, n = number of periods.
• n = ln(FV/PV) / ln(1 + r) where n = number of periods, FV = future value, PV = present value, r = periodic interest rate.
• FV = PMT × (((1 + r)^n - 1) / r) where FV = future value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• PV = PMT × (((1 + r)^n - 1) / r) / (1 + r) where PV = present value, PMT = periodic payment, r = periodic interest rate, n = number of periods.
• r = (PMT/PV) × (1 + r)^n / ((1 + r)^n - 1) where r = periodic interest rate, PMT = periodic payment, PV = present value, n = number of periods.

Step-by-Step Calculation

1. Determine the present value (PV) of the investment or cash flow.
2. Determine the periodic interest rate (r) and the number of periods (n).
3. Use the formula FV = PV × (1 + r)^n to calculate the future value.
4. If the future value is known, use the formula PV = FV / (1 + r)^n to calculate the present value.
5. If the periodic payment (PMT) is known, use the formula FV = PMT × (((1 + r)^n - 1) / r) to calculate the future value.
6. If the periodic payment (PMT) is known, use the formula PV = PMT × (((1 + r)^n - 1) / r) / (1 + r) to calculate the present value.

Common Mistakes

• Mistake: Forgetting to account for compounding interest when calculating future value.
• Correction: Make sure to use the correct formula and account for compounding interest.
• Mistake: Using the wrong periodic interest rate or number of periods.
• Correction: Double-check the periodic interest rate and number of periods to ensure accuracy.
• Mistake: Confusing the formula for present value with the formula for future value.
• Correction: Make sure to use the correct formula for the present value or future value.

Exam / CFA Tips

• Tip: When calculating future value, make sure to use the correct formula and account for compounding interest.
• Tip: When given a choice between using the formula for present value or future value, choose the one that is most relevant to the problem.
• Tip: When dealing with periodic payments, use the formula FV = PMT × (((1 + r)^n - 1) / r) to calculate the future value.

Quick Practice Problem

Apple Inc. is expected to pay a dividend of $5 per share next year. If the required rate of return is 8%, what is the present value of the dividend?

Answer: $4.76 Explanation: Use the formula PV = FV / (1 + r)^n to calculate the present value.

Last-Minute Cram Sheet

• The formula for future value is FV = PV × (1 + r)^n.
• The formula for present value is PV = FV / (1 + r)^n.
• The periodic interest rate (r) should be expressed as a decimal.
• The number of periods (n) should be a positive integer.
• The formula for future value assumes compounding interest.
• The formula for present value assumes discounting.
• The required rate of return (r) should be greater than the periodic interest rate.
• The present value of a perpetual bond is equal to the periodic payment divided by the required rate of return.
• The future value of a perpetual bond is equal to the periodic payment divided by the periodic interest rate.