Present values of two perpetuities

Question

I have two perpetuities:

1. Pays $1$ at the beginning of every year
2. Pays $1.8$ at the beginning of every odd year

I have to show which one is more profitable if $i=0.05$. I calculated present value of each of them. For the first one we have: $$PV_1=\frac{1}{d}=21$$ For the second one: $$PV_2=\sum_{k=0}^\infty 1.8\cdot v^{2k+1}=1.8\cdot\frac{v}{1-v^2}\approx18.45$$ I'm not sure about the second one though. Is that correct? I see this as the cash flow of $1.8$ for odd $t$ and $0$ for even $t$.

Answer 1

The value for $PV_1$ is correct.

The beginning of an odd year coincides with the end of an even year, so the correctly discounted cash flow in the second case is, with $v = (1.05)^{-1}$,

$$PV_2 = \sum_{k=0}^\infty 1.8 \cdot v^{2k} = \frac{1.8}{1- v^2} \approx 19.36 $$