Respuesta :
To answer this question, we can assume some different possibilities for the answer, since it is incomplete (or with not clear options):
a. [tex] \\ \frac{PMT}{r}[/tex]
b. [tex] \\ PMT*\frac{(1+r)^{n}-1}{r}*(1 + r)[/tex]
c. [tex] \\ PMT*\frac{(1+r)^{n} - 1}{r}[/tex]
Answer:
a. [tex] \\ PV_{perpetuity}=\frac{PMT}{r}[/tex]
Step-by-step explanation:
The present value of a perpetuity is an amount of money needed to invest today to have a perpetuity, or an annuity paid for life, considering an interest rate of r.
PMT is a finance term for payment and r 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:
[tex] \\ PV(r) = \frac{PMT_{0}}{(1+r)^{0}} + \frac{PMT_{1}}{(1+r)^{1}} + \frac{PMT_{2}}{(1+r)^{2}}+\dotsc+\frac{PMT_{n}}{(1+r)^{n}}[/tex]
Where n represents the number of periods for the investment.
On the other hand, an annuity, given a present value PV, is defined by:
[tex] \\ PMT= A = PV*(1+r)^{n}*(\frac{r}{(1+r)^{n}-1})[/tex]
Solving this equation for PV (present value) to define the present value of an annuity, we have:
[tex] \\ PV = \frac{(1+r)^{n}-1}{(r*(1+r)^{n})}*PMT[/tex]
But the question is asking for an annuity paid for life (theoretically, for infinite periods of time); then, if we calculate the limit for the previous equation when n tends to infinity, we find that:
[tex] \\ lim_{n\to\infty} \frac{(1+r)^{n}-1}{(r*(1+r)^{n})}*PMT[/tex]
[tex] \\ (lim_{n\to\infty} \frac{(1+r)^{n}}{r*(1+r)^{n}} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT[/tex]
[tex] \\ (lim_{n\to\infty} \frac{(1+r)^{n}}{(1+r)^{n}}*\frac{1}{r} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT[/tex]
[tex] \\ (lim_{n\to\infty} 1*\frac{1}{r} - lim_{n\to\infty} \frac{1}{r*(1+r)^{n}})*PMT[/tex]
The second term of the previous expression tends to 0 (zero) when n tends to infinity, then:
[tex] \\ (lim_{n\to\infty} 1*\frac{1}{r})*PMT[/tex]
[tex] \\ (1*\frac{1}{r})*PMT[/tex]
[tex] \\ \frac{PMT}{r}[/tex] or
[tex] \\ PV_{perpetuity}=\frac{PMT}{r}[/tex]
This expression represents that, with an interest of r, if we make an investment of PMT today, then we will have an annuity of [tex] \\ \frac{PMT}{r}[/tex] for life, because in each period PMT would be the same again due to the interest rate (r).
