## Perpetuities and Growth Perpetuities

Perpetuities are annuity payments that never end. There are two types of perpetuities, which are normal perpetuities or consols and growth perpetuities.

### What is a perpetuity?

A perpetuity is an annuity payment that never ends.  In other words, the darn thing keeps paying out forever.  However, the payout never changes either.

### What is formula for a perpetuity?

The Formula for a perpetuity is dividend one divided by rate.

Dividend or payment / Rate

### Sample Problem for a Perpetuity:

Your great great grandmother is retiring at the end of this year. She would like to ensure that her son receive annual payments of \$10,000 a year forever and ever, starting when she retires next year. If she can earn 8.0 percent annually, how much does your grandmother need to invest to create the desired payments to her son?

PV = 10,000 / .08

PV = 125,000

### What is a growing perpetuity?

A growing perpetuity is an annuity payment that never ends either.  Except the damn thing keeps growing and growing and growing... at the designated growth rate.

### What is formula for a growing perpetuity?

The formula for a growing perpetuity is dividend 1 or the next dividend paid divided by the difference between the expected return and expected growth rate.

Present Value of Perpetuity = Dividend 1 / (Rate - Growth)

### Sample growing perpetuity problem:

A person is evaluating a growing perpetuity investment option from a large financial services firm in the US.  The investment guarantees an initial payment of \$10,000 at the end of this year and following payments that will increase at a rate of 5 percent annually.  If you use a 10 percent discount rate for investments such as this, what is the present value of the growing perpetuity?

PV = 11,000 / (.1 - .05)

PV = 11,000 / .05

PV =220,000

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