Present Value of Growing Perpetuities and Annuities

To find the present value of a stream of cash flows, the fundamental method is to add up the present values of each individual cash flow in that stream.
Present Value (PV) of a Cash Flow Stream: The total value is calculated by applying the discount factor to each specific payment and summing them: $PV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}$

For certain types of level cash flows, convenient mathematical shortcuts can be derived to simplify the calculation of present value.
Perpetuity
- Definition: A perpetuity is a stream of level cash payments (where the cash flow remains the same every period) that never ends.
- Formula: $PV = \frac{C}{r}$
- Variables:
  - $C$ : The constant cash payment starting exactly one period from the current time ( $t=1$ ).
  - $r$ : The interest rate per period.
Annuity
- Definition: An annuity is a level stream of cash flows (where the cash flow amount is identical per period) occurring at regular intervals with a finite maturity.
- Formula: $PV = \frac{C}{r} \left[ 1 - \frac{1}{(1+r)^t} \right]$
- Variables:
  - $C$ : The constant cash payment.
  - $r$ : The interest rate per period.
  - $t$ : The total number of periods until maturity.

Growing Perpetuity
- Definition: A series of cash flows that continue indefinitely and grow at a constant rate ( $g$ ) each period.
- Formula: $PV = \frac{C}{r-g}$
- Mathematical Constraint: This formula holds as long as the growth rate is less than the interest rate ( $g < r$ ).
- Implications: If the growth rate ( $g$ ) were equal to or higher than the interest rate ( $r$ ), the present value would not be a finite sum, as the growing payments would overwhelm the effect of the discount rate.
Growing Annuity
- Definition: A finite level stream of cash flows that increases by a constant growth rate ( $g$ ) each period for a defined length of time ( $t$ ).
- Formula: Based on the calculation logic provided by Jeroen Ligterink, the formula is: $PV = \frac{C}{r-g} \left( 1 - \frac{(1+g)^t}{(1+r)^t} \right)$

For the following examples, assume an interest rate of $r = 0.05$ .

Example 1: Perpetuity
- Scenario: A perpetuity of $50$ each year.
- Calculation: $PV = \frac{50}{0.05}$
- Solution: $1000$
Example 2: Fixed Annuity
- Scenario: An annuity of $50$ for exactly $30$ years.
- Calculation: $PV = \frac{50}{0.05} \left[ 1 - \frac{1}{(1+0.05)^{30}} \right]$
- Solution: $768.62$
Example 3: Growing Perpetuity
- Scenario: A growing perpetuity where the cash flow one year from now is $50$ and it grows with a constant rate of $2\%$ per year forever.
- Variables: $C = 50$ , $r = 0.05$ , $g = 0.02$
- Calculation: $PV = \frac{50}{0.05 - 0.02}$
- Solution: $1666.67$
Example 4: Growing Annuity
- Scenario: A growing annuity where the cash flow one year from now is $50$ and it grows with a constant rate of $2\%$ per year for a period of $20$ years.
- Variables: $C = 50$ , $r = 0.05$ , $g = 0.02$ , $t = 20$
- Calculation: $PV = \frac{50}{0.05 - 0.02} \left( 1 - \frac{(1+0.02)^{20}}{(1+0.05)^{20}} \right)$
- Solution: $733,27$