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

What This Is

Present Value (PV) is a fundamental concept in finance that helps investors and analysts determine the current worth of future cash flows. It's essential in finance because it allows us to compare the value of different investments, evaluate the feasibility of projects, and make informed decisions about capital allocation. For example, consider a $1,000 bond with a 5% coupon rate that matures in 5 years. If the market interest rate is 8%, the present value of the bond's future cash flows would be less than its face value.

Key Formulas & Symbols

• PV = FV / (1 + r)^n where FV = future value, PV = present value, r = periodic interest rate, n = number of periods.
• FV = PV × (1 + r)^n where FV = future value, PV = present 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.
• PV =? (CFt / (1 + r)^t) where PV = present value, CFt = cash flow at time t, r = periodic interest rate, t = time period.
• NPV =? (CFt / (1 + r)^t) where NPV = net present value, CFt = cash flow at time t, r = periodic interest rate, t = time period.
• IRR = r where IRR = internal rate of return, r = periodic interest rate.
• PVIF = 1 / (1 + r)^n where PVIF = present value interest factor, r = periodic interest rate, n = number of periods.

Step-by-Step Calculation

1. Identify the cash flows: Determine the future cash flows associated with the investment or project.
2. Choose a discount rate: Select a suitable discount rate (r) that reflects the risk-free rate, market rate, or cost of capital.
3. Determine the number of periods: Calculate the number of periods (n) over which the cash flows will occur.
4. Apply the present value formula: Use the PV formula to calculate the present value of each cash flow.
5. Sum the present values: Add up the present values of all cash flows to obtain the total present value.

Common Mistakes

• Mistake: Using an incorrect discount rate or cash flow estimate.
• Correction: Verify the accuracy of the discount rate and cash flow estimates, and consider using multiple scenarios to test sensitivity.
• Mistake: Failing to account for inflation or taxes in the discount rate.
• Correction: Adjust the discount rate to reflect the impact of inflation or taxes on the cash flows.
• Mistake: Confusing IRR and NPV ranking.
• Correction: Understand that IRR and NPV are different metrics that provide different insights into project value.

Exam / CFA Tips

• Tip: Be prepared to apply the present value formula to different types of cash flows, including annuities, perpetuities, and growing streams.
• Tip: Understand the distinction between nominal and effective interest rates.
• Tip: Be aware of the impact of compounding frequency on present value calculations.

Quick Practice Problem

A company is considering investing in a project with the following cash flows: $100,000 in year 1, $150,000 in year 2, and $200,000 in year 3. If the discount rate is 10%, what is the present value of the project?

Answer: $143,919.51

Explanation: Use the PV formula to calculate the present value of each cash flow, then sum the results.

Last-Minute Cram Sheet

• The present value formula assumes a constant discount rate and cash flows.
• The PVIF is used to calculate the present value of a single cash flow.
• The IRR is the discount rate that makes the NPV equal to zero.
• The NPV is the difference between the present value of cash inflows and outflows.
• The dividend discount model (DDM) requires g < r – otherwise the model explodes.
• The WACC is the weighted average cost of capital.
• The cost of equity is the return required by shareholders.
• The cost of debt is the interest rate on debt financing.
• The present value of a perpetuity is calculated using the formula PV = CF / r.