Growing Perpetuity

A growing perpetuity is a stream of cash flows that occur at regular intervals and grow at a constant rate forever.

We can find the present value by computing the amount you would need to deposit today to create the perpetuity yourself.

Suppose we want to create a perpetuity growing at 2%, so you invest $100 in a bank account that pays 5% interest.

• At the end of the year, you will have $105inthebank–youroriginal$100 plus $5 in interest.
• If you withdraw only $3,youwillhave$102 to reinvest – 2% more than the amount you had initially. This amount will then grow to \102 \times 1.05 = $107.10$inthefollowingyear,andyoucanwithdraw$$3 \times 1.02 = $3.06$,whichleavesuswith$$107.10-$3.06=$104.04$.
• Note that \102\times 1.02=$104.04$; both the amount you withdraw and the principal you reinvest grow by 2% each year.

In general, if we want to increase the amount we withdraw form the bank each year by $g$, then the principal in the bank will have to grow by the same factor $g$. Instead of reinvesting $P$ in the second year, we should reinvest $P(1+g)$. In order to increase our principal by $gP$, we need to leave $gP$ of the interest in the account.

The principal must grow at the same rate as the payments to sustain the perpetuity.

The present value can then be found with:

$$PV(\text{Growing Perpetuity})=\frac{C}{r-g}$$