Annuity-Immediate Problem with Varying Payment (ASM FM Study Manual 10th Edition, Practice Exam 2 P.679 Q1) The question asks:
'A 35-year annuity immediate pays $1.05^{35}$ in the first year, $1.05^{34}$ in the second year, etc., until 1.05 is paid in the 35th year. The PV of this annuity at 5% effective is X. Determine X.'
I understand the answer = $1.05^{34} + 1.05^{32} + ... + 1.05^{-32} + 1.05^{-34}$.
However I don't get why it can be simplified to [(s-angle-34)+(a-angle-36)]/(a-angle-2)?
Please help! Thanks in advance!
 A: To be honest, don't get too hung up on trying to find equivalent ways to write the cash flow in terms of actuarial notation.  I would go about it this way:  We know that the series is geometric with common ratio $1/(1.05)^2 = v^2$, where $v$ is the annual present value discount factor.  We also know that this series begins at $(1.05)^{34} = v^{-34}$ and has a term of $35$ years.  So $$PV = \frac{v^{-34}(1-(v^2)^{35})}{1-v^2}.$$  And at this point, you can already do the calculation.  Now if you really had your heart set on writing it in actuarial notation, we would do some algebra:  $$\begin{align*} PV &= \frac{v^{-34} - v^{36}}{1-v^2} \\ &= \frac{(1+i)^{34} - 1 + 1 - v^{36}}{i} \cdot \frac{i}{1-v^2} \\ &= \frac{s_{\overline{34}\rceil} + a_{\overline{36}\rceil} }{a_{\overline{2}\rceil}}, \end{align*}$$ as claimed.  Note that we used a little trickery, subtracting and adding 1 in the numerator, and then multiplying/dividing by $i$ in the denominator, to get things to match up with the formulas we know for $a$ and $s$.  The reason why I don't recommend this approach is for two reasons:


*

*It can really mess you up.  The more algebra you need to do, the more likely you are to make a mistake somewhere.  If the problem doesn't require you to do it, you could spend all this extra effort to get something "elegant" only to find out it doesn't match the direct calculation.

*It wastes time.  You could have gotten the answer from the very first expression.  Again, unless the answer choices were written in actuarial notation (which is a possibility for the exam!), then once you get a numerical value, you should check it against the choices.


Now, doing this extra work could be a useful tool on the exam if you had extra time left over to do it.  You could use it as a way to check the validity of your calculations.  But the point here is to never make more effort to solve a question than is absolutely required.  Remember the basic formulas, but also don't rely on those formulas too much:  use your common sense.
