11.18: Perpetuities- No-Growth Perpetuities
A perpetuity is a series of equal cash flows that arrive in equal intervals and are never-ending, a kind of forever annuity. We cannot evaluate its FV since it would be infinite , as would be its nominal value. However, we can figure the PV.
The PV of a perpetuity is simply the (fixed) payment divided by the interest rate. For example, if the cash flows are $100 and the discount rate is 10%, the PV would be:
PV = $100 ÷ .10 = $1,000
The general formula would thus be:
PV = CF ÷ i
This works because of the mathematical “law of limits. ” Simply put, as the cash flows grow more distant, the respective PVs of these cash flows approach zero. Adding increasingly distant cash flows will have an infinitesimal impact on the outcome, the present value. Thus, the sum can be calculated as mention ed. Let u s examine this further.
For example, a $100 annuity for five years at 10% has a PV of $379.08. A ten-year annuity has a PV of $614.46. An annuity of 25 years has a PV of $907.70 and a 50-year annuity would be $991.48. As time goes on, the PV of each additional year ’s cash flow becomes very, very small (i.e., time increases, PV decreases) so that adding such infinitesimal amounts provides no material addition to the aggregate. In the end, you will note that the above present values, when aggregated, will eventually approach, and theoretically equal $1,000.
It is interesting to note that where perpetuities are concerned the present value does not depend on compounding frequency (“p”) . This appears to violate one of our inviolable rules of TVM! For example, the present value of a $1 ,2 00 perpetuity paid monthly at 12% p.a. will be the same as a $1,200 annual perpetuity:
PV = ( $1,200 ) ÷ (0 .12 ) = ( $1,200 /4) ÷ (0 .12 /4) = $10,000
T he frequency of perpetuity payments does not influence its future value, which will still be infinity ! It is also interesting to note that, because a perpetuity is forever, both the nominal and future values of the perpetuity are the same – infinity !
This formula is applicable to preferred stock whose dividends are fixed.