Perpetuity Immediate Present Value Question?

Question

>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: $a_{\\infty\\neg i}=\\cfrac{X}{i}\ ag{1}$ $\ ext{Brian's share}=0.4\\cfrac{X}{i}\ ag{2}$I don't know where to go from here. Can anyone please help or explain? Thank you.

Answer 1

I do not know what language you are speaking in. If you presented the problem as a word problem. The steps are pretty straight forward. Let's start here. we can look at the perpetuity as a stream of X dollars per year. Without loss of generality we can let X= 1.

The value of one dollar in perpetuity is P= 1/i. Let A = The Value of an annuity that pays $1 for n years discounted at i.. Let B = Brian's share, C = Colleen's share and K = Jeff's Share.

B+C+K = 1. B= .4 so C +K = .6. Colleen's Annuity starts paying in n+1 years.

Let d= (1/(1+i)) which is the discount factor. Colleen's Annuity is worth dⁿA. Colleen's share, C is therefore equal to C=dⁿ(A/P) note That (A/P)=Brians share or .4 so C= .4dⁿ

Jeff's Perpetuity is worth d²ⁿP.. so his share, K = d²ⁿ

Boy this is fun!!!! Recognize 1 = B+C+K

We have K = 1-B-C = 1-.4(1+dⁿ) = 1- .4(1+ (1/(1+i))ⁿ

So Jeff's residual is a declining function of both i and n..

I would be very happy to help you with the mechanics of any cash flow analysis. I have bee doing this stuff professionally for decades

We can deal with this several ways