Present Value of Perpetuity
Sometimes annuities last forever or so investor pretend. On first mirror image, investor might think that a perpetual annuity would be infinite. But remember that $1 paid in 10 years is worth only pennies today (excessively forward normal inflation and the inherent time value of money) and $1 paid in 100 years is not worth picking up from the sidewalk.
The present value of a eternal stream of future payments eventually reaches a limit. And, it turns out that the formula for an infinite series of equal payments which are discounted by a constant discount rate, is simplicity itself:
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PV = Pym tn / i
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PV
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the present value (or initial principal)
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Pym tn
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the payment made at the end of each of an infinite number of periods
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i
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the discount rate for each period presumed equal throughout
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investor can convince yourself the formula is correct. Imagine investor investor has to pay $2,000 a year forever and the discount rate is 8%. All that is needed is $25,000 -- that is, $2,000/.08. Each year, the $25,000 will produce a $2,000 return (excessively forward an interest rate of 8%), and this happy state of affairs will continue so long as the buffalo roam the plains and water gurgles in the brook and the sun rises in the east.
Although a perpetuity really exists only as a mathematical model,investor can approximate the value of a long-term stream of equal payments by treating it as an indefinite perpetuity. Our computational method can be employed even though investor might have doubts about payments in the distant future given the effect of discounting, they have a very little effect on our final computations.
A periodic amount awaiting payment indefinitely is known a perpetuity, in spite of the fact that few such instruments exist. The present value of a perpetuity can be computed by taking the limit of the above formula as n approaches infinity. There are some key presumptions:
> That it is not necessary to account for price inflation or in alternate manner, that the cost of inflation is comprised into the interest rate.
> That the likelihood of getting the payments is high or in alternate manner, that the default risk is incorporated into the interest rate.
> The present value(PV) of a growing perpetuity formula is the cash flow after the first period divided by the difference between the discount rate and the growth rate.
A growing perpetuity is a series of periodic payments that grow at a proportionate rate and are experienced for an infinite amount of time. An instance of when the present value of a growing perpetuity formula may be employed is commercial real estate. The rental cash flows could be believed indefinite and will grow over time.
It is significant to note that the discount rate must be higher than the growth rate when employing the present value of a growing perpetuity formula. This is due to the present value of a growing perpetuity formula is an infinite geometric series as illustrated in one of the following sections. In theory, if the growth rate is greater than the discount rate, the growing perpetuity would have an infinite value.
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