Eternal annuity

A perpetual annuity (also perpetuity ) is an annuity that can be paid from the interest income of a fixed-income investment without changing the amount of the invested capital . Since the capital is retained, the return ( r ) is therefore achieved “ forever ”.

One example is the console loan issued in 1648 by the Dutch Hoogheemraadschap van de Lekdijk Bovendams , which still pays interest in the 21st century.

calculation

{\ displaystyle r = K \ cdot p}

${\ displaystyle r}$ is the pension amount to be paid repeatedly (in arrears), the initial capital and the discount rate. ${\ displaystyle K}$ ${\ displaystyle p}$

Derivation

The formula of the excess pension present value comes from the pension calculation .

Alternative derivation

The perpetual annuity assumption is that there are constant payments annually for an infinitely long period of time . These are therefore independent of the period under consideration . In order to calculate the earnings value of these constant payments, it is therefore necessary to discount the payments using the interest rate . ${\ displaystyle r}$ ${\ displaystyle t}$ ${\ displaystyle K}$ ${\ displaystyle p}$

It turns out

${\ displaystyle {\ begin {aligned} K = \ sum _ {t = 1} ^ {\ infty} {\ frac {r} {(1 + p) ^ {t}}} = r \ left [\ sum _ {t = 1} ^ {\ infty} {\ frac {1} {(1 + p) ^ {t}}} + 1-1 \ right] = r \ left [\ sum _ {t = 0} ^ { \ infty} \ left ({\ frac {1} {1 + p}} \ right) ^ {t} -1 \ right] \ end {aligned}}}$

Substitution: ${\ displaystyle \ quad q = {1 \ over 1 + p} \ quad {\ text {with}} | q | <1}$

{\ displaystyle {\ begin {aligned} = r \ left [\ sum _ {t = 0} ^ {\ infty} q ^ {t} -1 \ right] = r \ left [{\ frac {1} {1 -q}} - 1 \ right] = r \ left [{\ frac {q} {1-q}} \ right] \ end {aligned}}}

Consider : ${\ displaystyle {\ frac {q} {1-q}}}$

{\ displaystyle {\ begin {aligned} {\ frac {q} {1-q}} = {\ frac {\ frac {1} {1 + p}} {1 - {\ frac {1} {1 + p }}}} = {\ frac {1} {1 + p-1}} = {\ frac {1} {p}} \ end {aligned}}}

Resubstitution:

{\ displaystyle {\ begin {aligned} & = r \ cdot {\ frac {1} {p}} = {\ frac {r} {p}} \ quad. \ end {aligned}}}

The use of the formula of the geometric series must be taken into account in the derivation.

Application example

The “perpetual annuity” method is suitable for making decisions about “renting or selling”. If, for example, from the seller's point of view, the selling price K of a property including ancillary sales costs is lower than the quotient of the expected annual net rental income (rent excluding maintenance costs, taxes, etc.) and the calculated interest rate, rental is advantageous.

variants

In the case of perpetual pensions, a distinction is made between pensions paid in advance and pensions paid in arrears. In addition, a continuous perpetual annuity can be represented as an improper integral .

Eternal rising and falling pensions

Of course, there is also the concept of rising or falling annuity for the perpetual annuity. This is based on the consideration of safeguarding the value of the periodic interest payments (inflation). Thus, from a perpetually increasing pension, an annual amount increased by the rate of increase can be withdrawn without affecting the capital and preventing annual increases. In this case the formula is

{\ displaystyle r = K \ cdot (pg)}

r in turn denotes the periodic pension payment in arrears, K the initial capital, p the interest rate and g the periodic growth rate .

It should be noted here that the growth rate can also have a negative sign . The “slope” then becomes negative and in this case it is a falling pension.

Application example for a perpetual, increasing pension

A typical application example can be found in the final storage of radioactive waste. There are annual costs that have to be paid forever. However, the inflation rate must be taken into account. So we define a realistic growth rate (for example, 3%.), And can now calculate the necessary capital stock, the one required to complete all in the future payments lying, the annually by the inflation rate - in the formula by g represents - Increase to be able to cover.

Eternal Loan

Counterpart of "perpetuity" are (in Germany rather ungebräuchlichen) "eternal bonds" ( English perpetuals ), which reversed only serves the current interest , d. H. a must be paid, the loan debt itself, however, remains unpaid .

Individual evidence

↑ Berk, Jonathan and DeMarzo, Peter: Corporate Finance . 4th edition. Pearson, 2017, ISBN 978-1-292-16016-0 , pp. 144 (English).

↑ Arne Storn: Please be patient ! ; DIE ZEIT No. 15/2015, April 9, 2015 , last accessed August 20, 2016.

[1] Berk, Jonathan and DeMarzo, Peter: Corporate Finance . 4th edition. Pearson, 2017, ISBN 978-1-292-16016-0 , pp. 144 (English).