2-2 Looking for Shortcuts-Perpetuities and Annuities

annuity

a fixed sum of money paid to someone each year, typically for the rest of their life (mortgage)

perpetuity

an annuity that has no end

How to Value Perpetuities

The annual rate of return on a perpetuity is equal to the promised annual payment divided by the present value:

$\text{return}=\frac{\text{cash flow}}{\text{present value}}$

The present value of a perpetuity can easily be found by flipping the formula:

$PV=\frac{\text{cash flow}}{\text{discount rate}}$

Sometimes you need to calculate the value of a perpetuity that starts payments several years from now. In three years the endowments will be worth:

$1 / r = $10 billion.

That is what it will be worth in the future. To find today's value we need to multiply by the three year discount factor 1 / (1 + r) ^ 3 = .751.

How to Value Annuities

1. Perpetual stream of $1 starting at the end of the first year.
2. Perpetual stream of $1 starting 4 years from now. Same payments as 1, but with a delay of 3 years.
3. Three-year annuity of $1.

Valuing Annuities Due

annuity due = stream of payments starting immediately

an annuity due is worth (1 + r) times the value of an ordinary annuity