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… ๐Ÿ‘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! ๐Ÿ˜‰

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…

Excel IRR

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. ๐Ÿ™‚

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.

16 thoughts on “Annuity Baffles Area Man!”

  1. if i am following the question right, the person can have $26,826 now or an annuity of 594.69/m for 60 months. so what you really want to do is solve for the yeild of the annuity.

    $26,826 = 594.69/(1+r) + 594.69/(1+r)^2 + 594.69/(1+r)^3 +…..594.69/(1+r)^60

    not sure about HP calculator but witht he BAII Plus you simply use the time value of money feature and solve for interest rate (I/Y).

    PV= -26,826 N=60, PMT=594.69, FV=0 solve for I/Y. to avoid a calulator error you must make sure you make the PV negative.

    I/Y = .9874%. this is the monthly yeild of the annuity, assuming monthly compounding the effective annual yeild (EAY) is an approx 12.7%

    so the choice is take the $26826 now and invest it somewhere else or take the annuity that is gaurunteeing 12.7% effective annual yeild. since it seems to be risk free you would definitely want the annuity.

  2. also, i do not know much about the HP 12C, but the TI BA II plus is the calcualtor that was suggested in college for finance classes. I also can not recall anyone having the HP 12C at the exams. It seems everyone uses the BA II plus. It might be more user friendly or more advanced. I am not sure. I was a little confused why you were using 120 time periods. I know the BA II plus can solve for much more than that. 360 time periods is necessary for a simple 30 yr mortgage calulations. I would think both calculators could compute that. another way to do the problem with the BAII is to use the Cash flow feature. intial out lay (CFo) -26826, CF1 594.69 you can use frequency (FO1) of 60 instead of punching in all 60 CF’s. Then you just Compute (CPT) for IRR.

  3. Thanks for the comments Mick.

    The interest rate I was trying to compute was from start to finish if the relative were to take the monthly payments. So t=0 on the cash flow diagram starts with the -$20K initial investment, then 5 years (60 months) of 0 cash flow before the $594.69 payments start, which proceed for another 5 years.

    I did do the same computation you did above in order to advise them between the lump-sum or the 60 payments and came to the same conclusion you did. They’d probably be hard pressed to take the lump-sum and achieve 12.7% risk free!

    Oh man, I hope I haven’t chosen the wrong calculator. The 12C has the same sort of short cuts where you can enter a single cash flow and then tell it that it occurs N number of times. I used these shortcuts for the 120 months but when I told it to solve for IRR the calculator returned an error code number which, according to the 12C manual, you can get if your try to do IRR with n>80.

    Oddly, you can use n>80 on other computations, just not IRR (or so it seems). In order to compute PV of perpetuities I enter it as I would an annuity, but with very large n (like 2000).

    Thanks for the head’s up. I might indeed have bought the wrong calculator…

  4. Hey Lumilog,

    How did you type that math equation so nicely? I’ve seen professors use a similar application to write notes for class and since I’m trying to keep my notes digital for the CFA (in word), I was wondering if you can explain how you created it?



  5. Hi Ryan,

    In MS Word I did Insert->Object->Microsoft Equation which gives you the nice equation editor. Then I did a screenshot, cropped it, and converted to JPEG for the blog.


  6. Oh ok i see what you were doing. yeah the BAII Plus had no problem. CFo=-20,000, CF1=0 FO1=60, CF2=594.69, FO2=60 CMT IRR=0.649.

  7. this is what i did…

    Part (a) my IRR was 6.049%
    – $20,000 * (1+x)^5 = $26,826.79
    – x = 6.049285%

    Part (b) my IRR was 8.06796% and monthly was 0.6487% (i realized i added another period in my irr calculation or it would have been the same as yours)
    – BA II Plus, I was able to do the IRRs without any trouble. (calc steps below)
    1) [CF] -20,000 [ENTER] = sets inital cash outflow
    2) [down arrow] 0 [ENTER] – sets cashflow 1
    3) [down arrow] 60 [ENTER] – sets frequency of cashflow 1
    4) [down arrow] 594.69 [ENTER] – sets cashflow 2
    5) [down arrow] 60 [ENTER] – sets frequency of cashflow 2
    6) [IRR] [CPT] – computes IRR

    After computing IRR, I got 0.64867912 as monthly (once again please note that I used t0 – t120 in my calculation (121 periods).

    Annualized will be (1 + 0.64867912%)^12 – 1 = 8.067961%

    With 120 periods my results will be the same as yours so therefore I was wondering if I am suppose to use 121 periods or 120…I’m still not clear which one to use. Any clarification would be great! Thanks!


  8. Thanks Mick & DJ. I’m leaning closer and closer to getting a BAII. I see that Schweser also recommends the BAII…

    DJ – my cash flow diagram begins with the initial payment at t=0 with the first payment happening 5 years later at t=60, and the last payment at t=119 for a total of 60 payments.


  9. Hi everyone,

    I just also read through this and the first formula in the CFA “Quantitative Investement Analysis (2007) is:

    FV= PV(1 +R)

    Perhaps this can help. BTW I’m a young father with a baby seeing what you guys are up to (I’m not writing the exam).

    I am curious about these calculators though. I haven’t used any of these and are they worth it? I heard that some of these are available on the web? Have you all chosen the same one or are there one or two top contenders?

  10. Hi Ray,

    As far as the exam goes, you are only allowed to bring in one of two calculator models as specified here.

    I picked the HP b/c I used a similar model to get me through engineering courses in college. However my understanding is that I’ve picked incorrectly – most seem to prefer the TI for the CFA exams.

    Personally I’d ditch these calculators and bring a laptop with Matlab to the test if they’d let me. But alas…


  11. Yes, I have the same feelings about the antiquation calculation of the calculator methodology and it does give me pause or at least raises reservations about the currency/applicability of the exam. In terms of design, I like the look of the HP but the cool digital videos regarding the TI I suppose make it a strong contender. I wonder if anyone here has investigated the online paratextual course cribs for the CFA. I found this one: which looks interesting and another comparsion here: Somebody, preferably a CFA candidate should do a cost value analysis here and publish it online!

  12. Hi, I went back home and tried the calculation on my old HP12C model. My model has a blue “Nf” printed under Fv that enables one to repeat the previously entered CFn for less than 99 times. Therefore, after storing -26826.79 into CF0 and 594.69 into CFn, you just punch 60 into Nf, then click IRR, HP12C can get the correct IRR just fine. Which model of HP12C do you use? Mine is the classic model I think. You can find the instructions for Nf in the section called “additional financial functions in the handbook.

    I love the cool look and RPN methodology it adopts, and it performs so beautifully! I have never used an advanced calculator before (I am a biology major). Though I think it’s a little inadequate in statistical functions, I doubt TA model is stronger in that…

  13. You used to have an equation here about a perpetuity and how to calculate it with 12,000 a year or something do you have that infp still? I’m trying to figure one out for myself.. thanks ๐Ÿ™‚

  14. Hi Spencer,

    Hmmm.. I haven’t heavily editing this post. Maybe it was:

    In either case, the general formula for valuing a perpetuity with a fixed payment is

    Value = Payment / DiscountRate

    So arbitrarily picking a 10% annual discount rate, a $12,000 perpetuity is worth $120,000 today.

    Or said another way, if you put $120,000 into an investment vehicle today, providing 10% annual interest, it would give you $12,000 annually forever.

    Hope that helps?

  15. No it was something along the lines of if you invest X amount maybe 900 a month lets say for X years, say 30 years at an assumed realistic rate, say 6 % or even sp 500 10%, you would end up getting about 10,000 a month forever starting at the end of the 30 years.. sorry for taxing your memory ๐Ÿ™‚

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