I have a question with regards to calculating annuity due in terms of annuity immediate.
I thought that one way to do so is annuity due for n periods = annuity immediate n-1 periods + 1 (where the payment is 1). However, I did this for a part of a problem that is basically Present value is 500,000 of a 10 year annuity due of X. Interest rate is 9% annual effective. Calculate X. Why doesn’t the following work??
PV of annuity due of 10 years of payment 1 = annuity immediate of 9 years of payment 1 + 1 Then divide 500,000 by the right hand side of this equation?
When I do this, I get PV = 7 from the calculator, so payment is 500,000/7. But when I do another method, I get a lower answer, which is the right answer according to the manual. But I don’t understand why they should be different.
The other method is PV of annuity due of n years = (1 + i) * PV annuity immediate of n years. This gives the correct answer. >.<
Sorry if I’ve misunderstood something fundamental and thank you in advance for the help!
You cannot have different results, because
$$ \ddot a_{\overline{n} |i} =a_{\overline{n-1} |i}+i=\frac{1-(1+i)^{-(n-1)}}{i} +1=\frac{1+i-(1+i)^{-n+1}}{i}=(1+i) \frac{1-(1+i)^{-n}}{i}=(1+i)a_{\overline{n} |i} $$