Finance 101: Present Value of Prepetuities, Annuities

General:
Once we know the cash flow pattern we simply discount each payment by its appropriate discount factor

Perpetuity:
Consider an investment that pays a cash flow of CF in perpetuity. Then the present value of the cash flow is given as
PV = CF/r
r is the interest rate (also know as opportunity cost of capital)

Growth: Suppose the constant perpetuity payment grew at a per-period rate of g….
Cash flow at the end of 1 period = CF, CF2 = CF*(1+g), CF3 = CF*(1+g)*(1+g) and so on....
In this case the Present Value of Cash Flow is

PV = CF/(r-g)

Annuity:
Annuity is a payment or cash flows that are made for a particular period in future. For instance the 30 year mortage payment is a type of annuity for 30 years.
For instance
Consider a $4 annuity due in 4 periods with the interest rate of r per period. The present value in this case is given as

PV = $4/(1+r) + $4/(1+r)^2 + $4/(1+r)^3 + $4/(1+r)^4

In order to derive an equation for perpetuity we could consider the scenario where we split the above payments into 2 parts

1. Case A: Consider a $4 perpetual payment starting 1 year from today. Here the PV = $4/r
2. Case B: Consider a $4 perpetual payment starting in 5 year from today. Hence if you stand at year 4 it looks like a perpetual cash flow. Thus value of this payments at Year 4 is $4/r. We know that any cash flow in future can be discounted back to present value by using the discount factor for that period. Thus the present value of the perpetual cash flow in year 5 is given by Value of Cash flow at year 4 * Discount Factor for 4 years. PV = $4/r * 1/(1+r)^4

We could replicate our original payments of $4 per period of 4 period by subtracting the payments from Case A with payments from Cash B. In other words, we could obtain the present value of an Annuity by subtracing PV of Case A with PV of Case B.

PV of $4 for 4 period = $4/r * (1-1/(1+r)^4)

In general, PV of annuity with Cash Flow CF till period t is given as

PV = CF/r * (1-1/(1+r)^t)

Growth:

Suppose the constant annuity payment grew at a per-period rate of g…. Cash flow at the end of 1 period = CF, CF2 = CF*(1+g), CF3 = CF*(1+g)*(1+g) and so on.... In this case the Present Value of Cash Flow is

PV = CF/(r-g) * (1-((1+g)/(1+r))^t)