To answer this question, we can assume some different possibilities for the answer, since it is incomplete (or with not clear options):
a.
b.
c.
Answer:
a.
Step-by-step explanation:
The present value of a <em>perpetuity</em> is an <em>amount of money needed to invest today</em> to have a perpetuity, or an annuity paid for life, considering an interest rate of <em>r</em>.
PMT is a finance term for <em>payment</em> and <em>r </em>is the interest rate (roughly, an important quantity that defines how much it can be obtained for an investment).
In general, the present value can be mathematically defined as:
Where <em>n</em> represents the number of periods for the investment.
On the other hand, an annuity, given a present value <em>PV</em>, is defined by:
Solving this equation for <em>PV</em> (present value) to define the present value of an annuity, we have:
But the question is asking for an annuity paid for life (theoretically, for infinite periods of time); then, if we calculate the <em>limit</em> for the previous equation when <em>n</em> tends to <em>infinity</em>, we find that:
The second term of the previous expression tends to 0 (zero) when <em>n</em> tends to <em>infinity</em>, then:
or
This expression represents that, with an interest of <em>r</em>, if we make an investment of PMT today, then we will have an annuity of for life, because in each period PMT would be the same again due to the interest rate (r).