Present value of perpetuity with one extra payment

Question

Consider the perpetuity that pays $3$ by the end of the 2nd year and then every $4$ years. I need to calculate the present value of it when $i=0.05$. So the second payment will be at 6th year. I thought about doing it by just discounting the first one and then express the rest of the discounts by a geometric series: $$PV=3\cdot v^2+\sum_{k=1}^\infty 3\cdot v^{4k+2}=3\cdot \left( v^2+\frac{v^6}{1-v^4}\right)\approx15.348.$$ Is that correct?

Answer 1

Your answer is correct. However, the perpetuity formula works for the second term.

$$PV = 3v^2 + v^2\cdot \frac{3}{1.05^4-1} = 15.348$$

The second term is the perpetuity formula discounted the first two years. Saves from using the infinite sum.

Answer 2

This is just a deferred perpetuity-due with deferral period of $2$ years, and periodic payments of $4$ years: $$\require{enclose} PV = 3v^2 \ddot a_{\enclose{actuarial}{\infty} j} = 3\frac{1+j}{(1+i)^2 j}$$ where $v = 1/(1+i)$ is the annual present value discount factor, and $j = (1+i)^4 - 1$ is the $4$-year effective interest rate. Therefore, the present value is $$PV = \frac{3(1+i)^4}{(1+i)^2 ((1+i)^4 - 1)} = \frac{3(1+i)^2}{(1+i)^4 - 1} = \frac{3(1.05)^2}{(1.05)^4-1} \approx 15.3476.$$