A perpetuity is simply a stream of payments that will never end. The present value of a perpetuity is very easy to calculate. Assuming that the payments are for a fixed amount paid at regular intervals, it is:
c ÷ r
Where c is the amount of the payment, and,
r is the discount rate.
If the payments grow at a fixed rate (i.e. each payment is a fixed percentage greater than the last), then the present value is:
c ÷ (r - g)
where g is the growth rate.
So if £1,000 is to be paid in the first year, and the payment will grow five percent each year (so £1,050 the second year, £1102.5 in the third year, etc.), and the discount rate 10%, then the present value is:
£1,000 ÷ (0.10 - 0.05) = £1,000 ÷ .0.5 = £20,000
Securities that actually offer perpetuities are fairly rare: most bonds have a maturity date. Exceptions include some gilts, although no more of these seem to be being issued.