Principles of Finance/Section 1/Chapter 3/Applications of Time Value of Money/Perpetuities

A "perpetuity" is a theoretical bond which makes payment during its entire life. One realizable example of this is a company issued stock which always pays dividends each year. The lifetime is defined from the time of issuance until buy back or potential bankruptcy.

A perpetuity, such as preferred stock, can be valued by simply dividing the payment by the applicable discount rate.

PV_Perpetuity = ${\frac {C}{r}}$

Example:

XYZ Corp. preferred stock pays a $2 dividend every year. This dividend is expected to remain constant for the forseeable future. If investors are requiring a 10% return, what is the stock selling for?

$PV={\frac {\$2}{.10}}=\$20$

Therefore, XYZ Co. stock should sell for $20 per share.

Growing Perpetuities

A growing perpetuity is one whose payments increase at a certain rate forever. They can be valued by the following formula, where C₁ is the payment during the upcoming payment period, r is the discount rate, and g is the growth rate:

PV_{Growing Perpetuity} = ${\frac {C}{r-g}}$

Example:

Madeline and Thurgood Johnson wish to set up a trust fund for their grandson which will begin paying $5,000 next year. They wish to have the payments grow at 5% per year to keep pace with inflation. If the current discount rate is 8%, what should they pay for the perpetuity?

PV_{Growing Perpetuity} = ${\frac {\$5,000}{.08-.05}}=\$166,666.67$