A perpetuity-immediate pays $X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian's share of the present value of the original perpetuity is 40%, and Jeff's share is K. Calculate K.
The perpetuity immediate present value is:
$a_{\infty\neg i}=\cfrac{X}{i}\tag{1}$
$\text{Brian's share}=0.4\cfrac{X}{i}\tag{2}$
I don't know where to go from here. Can anyone please help or explain? Thank you.
Best Answer
Hint 1: You are correct about Brian's share, but there is another expression that can give the value of Brian's payments, which is the present value of an $n$ year annuity immediate that pays $X$ per year. Set this equal to the $0.4X/i$. You will need this equation after Hint 2.
Hint 2: Jeff gets every payment after the first $2n$ payments. So, he gets a perpetuity that starts $2n$ years later, with payments of $X$. The present value of this is then $$\frac{X}{i} v^{2n}.$$ You need to figure out this as a percent of $\frac{X}{i}$, so in other words you need to know the value of $v^{2n}$. Use your equation from Hint 1.