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Perpetuity
A perpetuity or a perpetual bond is a special type of bond that yields a fixed amount of cash flow annually for ever in future. The present value of a perpetual cash flow (i.e. payment stream) of $ 100 per annum is given by the following infinite summation:

PV = 100/(1 + r) + 100/(1 + r)2 + 100/(1 + r)3 + 100/(1 + r)4 + ….

PV = 100{1/(1 + r) + 1/(1 + r)2 + 1/(1 + r)3 + 1/(1 + r)4 + …..}                         (1)


The above present value of the stream of cash flows paid annually can be easily obtained by using the following formula which gives the present discounted value of cash flow of $ 1 per annum forever.

PV of $ 1 payable per annum for ever = 1/r, where r is rate of interest.

Thus, Present value (PV) of the perpetuity of $ 100 = 100/r

If rate of interest is 5 percent, then PV of a perpetual bond with $ 100 as annual cash flow

= 100/0.5 = $ 2000

If r = 10 percent, then

PV of above bond = 100/0.10 = $ 1000.


Thus the lower the rate of interest, the higher the present value of a perpetual bond.

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