## Annuity Baffles Area Man!

For a chronological index of my path to the CFA, click here.

Before I get to my main topic, let me just quickly note that I found some more HP 12C resources online at Hewlett-Packard’s site.

(1) The 12 C user’s guide is there in PDF. Not exciting unless you’ve lost yours…
(2) That same page also has some supplementary finance problems. Click on “Solutions Handbook”.
(3) Most exciting is this page. It has a bunch of training modules in PDF for CFA-style problems.

Now that we have that out of the way…

There are few things more frustrating than telling someone you’re studying a particular subject and then they don’t understand why you can’t solve a supposedly simple problem in that area.

Like coming home from college to visit your parents and your dad doesn’t understand why you can’t fix the intercom system that was hit by lightning or his broken VCR since you’re majoring in electrical engineering… ๐ฟ

Amazingly enough, the exact day I finished the Time Value of Money (TVM) reading in the CFA study guide, I got an email from a relative asking me a question about their annuity…

Got a mathematical question/dilemma for you please to think about or work out or whatever when you have the time.

We have an annuity, paid \$20,000 for it 5 years ago, worth either \$26826.79 right now as a one time payment in cash or redeemable for \$594.69 per month for 60 months.

(a) Which one should we pick?
(b) What would be our rate of return if we do the payout over 60 months?

Now part (a) is easy. Doesn’t matter how much they put into the annuity, we just have to compute the present value of the 60 payments, compare it to the lump-sum, and pick the one with greater value.*

* Actually it’s my understand that in the real world it’s not quite this simple since money in an annuity grows tax-deferred. But I haven’t gotten to that CFA chapter yet…

I’ll spare the details since this is TVM 101 kind of stuff. You do have to make an assumption about monthly interest rate. I picked 0.8% because this compounds to 10% annually – what an S&P 500 index fund averaged over the last 10 years.

The 60-payments turned out to be the better deal…

I approached part (b) with much confidence & chest-pounding. I drew the cash flow diagram. I reflected the initial principal & future payments to the same point in time on said diagram.

I equated the two, and prepared to solve for r, the monthly rate of return…

I failed! I may just be rusty on my algebra after years of letting computers solve problems for me, but I just couldn’t isolate r on one side of the equation or find a simple way to determine this polynomial’s roots.

I did estimate that the monthly return was about 0.65% – but that was just by plugging various values of r into the above equation until I got both sides to equal approximately the same number.

Eventually I gave up on an exact solution and decided to start my next chapter in the CFA study guides instead – Discounted Cash Flow (DCF). Lo and behold just a few pages into the DCF chapter I come across the concept of Internal Rate of Return (IRR). And the practice problems look very similar to the annuity interest rate problem I couldn’t solve.

The book presents some simplified IRR problems that can be solved through algebra but then says “…for most real-life problems, financial analysts use software, spreadsheets, or financial calculators to solve for IRR…”.

So… those professional analysts can’t solve for r either! ๐

I quickly consult my 12C manual to find out how to make it do IRR. Victory will soon be mine! I plug in the cash flows. I hit the IRR button and wait while it does its magic…

Too Complicated for 1981

…and it turns out that the 12C can’t solve this problem either because it spans 120 time periods and the 12C maxes out at 80. Sheesh!

Again I put the problem on the backburner and continue chugging through my CFA material. A week later I recall what the study guide said about IRR…“…for most real-life problems, financial analysts use software, spreadsheets, or financial calculators to solve for IRR…”

I fire up Excel and search Excel Help for IRR. Sure enough there’s a function with the same name. And sure enough…

IRRational Exuberance

I take Excel’s computed monthly interest rate of 0.65611%, plug it into my original equation from a couple weeks back, and – yes – both sides of the equation are equivalent. ๐

A couple weeks to solve one real-world annuity problem. It’s a good thing I like this stuff!

Update: I finally passed all the CFA exams and wrote an eBook about the program. If you’re interested, click here.