Ch 5: Introduction to Valuation: The Time Value of Money

Time and Money

• Would you rather receive $\$$100 today or $\$$50 today and $\$$50 next year?
• Why?
  • Inflation/purchasing power
  • Uncertainty/risk
  • Taxes
  • Utility
  • Opportunity costs (could using the money for something else)
• These forces work together in the marketplace to yield an interest rate.
• The interaction of time and an interest rate allows us to evaluate cash flows at any point in time.
• The important thing: comparing apples to apples.

Definitions

• Present Value (PV) - beginning point on time line
• Future Value (FV) - end point on time line
• Interest Rate (r) - conversion (exchange) rate between money at PV and money at FV
  • Discount rate
  • Cost of capital
  • Opportunity cost of capital
  • Required rate of return

Example

• PV = 1000
• r = 5%
• FV = ? in one year
• In the future, we receive our initial principal plus interest (compensation for delaying use of capital).
• FV = 1000 * (1.05) = 1000 * (1 + .05) = 1000 + 50
• FV = 1050

Example (cont.)

• PV = 1000
• r = 5%
• FV = ? in two years
• FV = 1000 * (1.05) * (1.05) = 1000 * (1.05)$^2$ = 1102.50
• FV = 1102.50 = 1000 + 1000 * .05 + 1000 * .05 + 50 * .05
• Simple vs. compound interest

General Formula

• $FV = PV(1+r)^t$
• $PV = FV/(1+r)^t$ (discounting)
  • How does PV vary with t?
  • How does PV vary with r?
• $r = (FV/PV)^{(1/t)} - 1$
• $t = \ln{(FV/PV)}/\ln{(1+r)}$

Practice Problem

• You have $10,000 to invest for five years.
• How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return?
• How long will it take your $10,000 to double in value if it earns 5% annually?
• What annual rate has been earned if $\$$1,000 grows into $\$$4,000 in 20 years?
• Now use a financial calculator.