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for the most part I understand this question, but i'm missing something. Any help would be appreciated.

  "Dipper has a 10 year increasing annuity immediate that pays 
  $100 at the end of the first year, $200 at the end of the second 
  year, ... , and $1000 at the end of the 10th year. He exchanges 
  the annuity for a perpetuity of equal value that pays X at the 
  end of each year. If the effective annual interest rate is 3%, 
  find the value of X."

What I did:

Increasing Annuity Immediate = Perpetuity

100(Is)10|.03 = X/i

100[((Annuity Due) - n)/.03] = X/.03

FV Annuity Due = (1+i)[(1+i^n)-1/i] = (1.03)[((1.03)^10 - 1)/.03]

FV Annuity Due = 11.807

100[((11.807) - 10/.03] = X/.03

6023 = X/.03

180.7 = X

user601206
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1 Answers1

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$$ 100\;(Ia)_{\overline{10}|3\%}=\frac{X}{3\%} $$ that is $$ X=3\cdot\frac{\ddot a_{\overline{10}|3\%}-10v^{10}}{3\%}=100\left(\ddot a_{\overline{10}|3\%}-10v^{10}\right)=134.52 $$ where $v=\frac{1}{1.03}$ and $\ddot a_{\overline{10}|3\%}=\frac{1-v^{10}}{1-v}=8.79$

alexjo
  • 14,976
  • Thank you! I feel like i am missing a really simple fundamental concept here. Can you explain why you used the present value instead of the accumulated value? – user601206 Oct 07 '18 at 04:10
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    Because you have an increasing annuity (now) that you want to exchange with a perpetuity (now). What you did is to accumulate a sum after 10 years (with an increasing annuity) and then to convert this sum into a perpetuity. – alexjo Oct 07 '18 at 08:50