Question from How to Pass

[kme]

This problem has been driving me nuts. I'm probably making a silly mistake.

At an effective rate of interest of 5%, the present value of a perpetuity-immediate is the same as the accumulated value of an n-year annuity-immediate. Find the present value of an n-year annuity-due.

Answer: 10.5

[Gandalf]

You should know the value of a perpetuity immediate at i = .05.
Call it X.

Then write the formula for the n-year immediate annuity, set it equal to X.
Multiply through by i (which equals 0.05).
Solve for (1+i)^n.

That gives you v^n.

Then you have everything you need to get the value of the annuity due. Get d from i; plug in the values of v^n and d.

Comment: n = 14.2067. Non-integral n is somewhat strange.

[Gandalf]

Polls are still open for almost 30 minutes, (or 24 hours and 30 minutes, if the mole is to be trusted). I trail by one.

[kme]

x=1/.05=20

20=((1+.05)^n)-1)/.05

1=((1.05)^n)-1

2=(1+i)^n=1.05^n

d=.05/1.05=.04762

AV(annuity due)=(((1+i)^n)-1)/d
=(2-1)/.04762
=1/.04762
=21

[Gandalf]

Originally Posted by kme

At an effective rate of interest of 5%, the present value of a perpetuity-immediate is the same as the accumulated value of an n-year annuity-immediate. Find the present value of an n-year annuity-due.

You are getting the accumulated value of the annuity due.

[kme]

ACK! that's been my silly mistake all along!

[D]

given a_inf = s_n, i=0.05

=> a_inf = 1/i = [(1+i)^n -1 ]/i = s_n
=>1 = (1+i)^n -1
=> (1+i)^n = 2
=>v^n = 1/2

now a_n(due) = (1+i)*a_n

=(1+i)*(1-v^n)/i
=(1.05)*(1-1/2)/0.05
=10.5

[Svak]

Easy to read solution: