So I've currently been learning about perpetuities in my actuarial math class and have come to a conceptual roadblock in the topic of infinite annuities. More specifically finding a present value for something that is infinite in size doesn't make sense to me. How could we calculate some value now that is expected to go infinity later on? As well as what is the significance of this value, (why must it be that and not something smaller or greater)? In traditional present value questions, it's a very logical calculation, but I can't wrap my head around the infinite case.
Also, would it be logical to assume that the accumulated value of a perpetuity is infinite? As in infinite payments accumulated at any interest rate is infinite in size? Thanks!
The present value of a perpetuity is finite since each of the elements of the sum, is the product of the cashflow and the discount factor which is $\dfrac{1}{(1+r)^t}<1$ . Then, from a direct application of geometric series, the sum with infinite elements converges. In other words, the stream of cashflows in not something that is infinite in size. The number of elements in the sum is infinite while none of the elements in the sum is an infinite element.
Why is it not something smaller or greater? Future cashflows are random variables as well as the values of $(1+r)$. For the sake of simplicity, ususally one assumes $r$ is constant through time but the true value of $r$ of the future is uncertain. In practice, the estimates of these numbers vary from person to person (analyst, etc), so a present value can be smaller or greater from person to person. In other words, a present value is no observable, but it is only estimable.
Would it be logical to assume that the accumulated value of a perpetuity is infinite? Would you pay an infinite ammount of money for an asset? Is it logical that the price of an asset today equals an infinite ammount of money. My personal opinion is that it is not plausible.