## 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.