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View Full Version : Kellison 9.8 #22 and #24 (Algebra Questions)

November 28th 2005, 03:53 AM
Sorry for double posting, I'm just really confused with these problems.

9.8 Duration
For 22 it asks you to show that lim i-->infinity d is equal to the point in time at which the first payment is made. (I think the answer here is 1)

For this one I expanded the numerator and denominator of the duration definition and attempted to take the limit as i --> infinity, I find this troublesome however because I left with 1(0)R1 + 2(0)R2 + ....... + N(0)RN all over (0)R1 + (0)R2 + ...... + (0)RN How do I find the limit...?

For 24 it asks to Find the duration of a common stock which pays dividends at the end of each year, if it is assumed that each dividend is 4% greater than the prior dividend and the effective rate of interest is 8% (Ans 27)

I am positive I set up the expression correctly, but I do not know how to derive a simpler expression..
I have duration = (v + 2v^2 (1.04) + 3v^3 (1.04)^2 + .....) / (v + v^2 (1.04) + v^3 (1.04)^2 ....) Somehow I need to incorporate the growing annuity formula (Ia)n with n=infinity.... I'm lost here

Any help is appreciated.

Tetrabit
December 13th 2005, 06:35 PM
#22: It should be possible to cancel a factor of v from the numerator & the denominator before taking the limit. Then it'll work out.

#24: Again if you cancel one factor of v from num & denom and call V=v(1.04), it becomes

1+2V+3V^2+.../1+V+V^2+...

and now the numerator is (by one of the standard formulae) 1/(1-V)^2 and the denominator is 1/(1-V) so the answe is 1/(1-V). Plugging in V=1.04/1.08 gives the answer 27.

HTH,
Tetra