chapter 5

Introduction to Valuation

• Time value of money (TVM) is a core concept that denotes that the value of money changes over time due to potential earning capacity.
• A dollar today is worth more than a dollar received in the future due to inflation, interest rates, and opportunity costs.

Learning Objectives

• Calculate the future value (FV) of cash flows based on present investments.
• Determine the present value (PV) of future cash flows.
• Understand the return on an investment.
• Find out the time required for an investment to meet a specified value.

Future Value and Compounding

• Future Value (FV): The worth of an investment after a specified period, factoring in interest or growth.
• Example: If you invest $100 at an annual interest rate of 10%, after one year:
  • FV = $100  (1 + 0.10) = $110.
  • After two years: FV = $110  (1 + 0.10) = $121.
• Compounding: The process of earning interest on previously earned interest.
  • This results in compound interest, which is different from simple interest, as simple interest is only calculated on the original principal.

Future Value Formula

• The general formula for the future value of $1 invested for ( t ) periods at an interest rate of ( r ) is:
  • FV = 1 imes (1 + r)^t
• For example, for 5 years at 10%:
  • FV = 100 imes (1 + 0.10)^5 = 100 imes 1.61051 = 161.05
• This formula is crucial for evaluating long-term investments, with accuracy dependent on the interest rate assumption.

Present Value and Discounting

• Present Value (PV): The current worth of a future sum of money given a specified rate of return.
  • To find PV, future cash flows must be discounted back to present value.
  • Example: To find the present value of $1,000 expected in three years at a 15% discount rate, use:
  • Discount factor = 1/(1 + r)^t = 1/(1.15)^3
  • Calculating gives PV = 1,000 imes 0.65752 = 657.52.
• The present value interest factor can be used to further simplify calculations.

Discount Rate and Investment Evaluation

• The discount rate (r) is the interest rate used to determine the present value of future cash flows.
  • Inversely related: As the length of time until payment grows, present values decline.
• To find the implicit discount rate of an investment:
  • Set up the basic present value equation involving PV, FV, r, and t.
  • Use a financial calculator, algebraic methods, or future value tables for accuracy.

Financial Calculations Example

• For an investment costing $100 that doubles ($200) in 8 years:
  • Set parameters: PV = -100, FV = 200, and N = 8.
  • Solve for r, leading to a discount rate of approximately 9%.

Finding the Number of Periods

• If you wish to determine how long it will take to grow an investment from $25,000 to $50,000 at a 12% interest rate:
  • Use PV = 25,000, FV = 50,000, I/Y = 12% to find N.
  • Typically results in N ≈ 6.1163 years, demonstrating the concept of time value in compounded interest scenarios.

Summary of Key Equations

• Future Value: FV = PV imes (1 + r)^t
• Present Value: PV = FV imes 1/(1 + r)^t
• When discounting, the longer the timeframe, the lower the present value.
• Key Concepts: Compounding leads to higher future value as compared to simple interest calculations. The discounting process is essential for financial decision-making, allowing businesses to evaluate future cash flows accurately.