An example of estimation in a financial setting

There’s been a social media uproar over the grading of an estimate problem. If you haven’t see it . . . be happy, I guess, but it got me thinking about how I use estimate in my work.

The answer isn’t simple, but estimation is one of the most important tools that I use. I can illustrate it with a non-work example that caught my eye last week.

I happened to see an article about pension plans in Illinois that included this graph:

Screen Shot 2015-10-16 at 11.46.13 AM

This graph caught my eye because the fact that the funding of the Teachers’ Pension plan had not improved since 2009 didn’t meet with my gut feeling of what should have happened during a time when the value of the stock markets roughly doubled. In fact, since the levels of the US stock markets are actually higher now that before the financial crisis, my guess would have been that the pension plans would have returned to pre-crisis funding levels.

Since my estimate of what should have happened was not even close to what actually happened, I decided to dig in to the numbers to see why.

The 2014 annual report of the Illinois Teachers’ Pension Plan is here:

Illinois Teacher’s Pension plan financial reports

Near the end of that report (on page 105 / 112 of the pdf) you’ll find this chart showing the money flowing in and out of the fund for the last 10 years:

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What I found interesting in this chart is from 2005 through 2014 roughly $39 billion in payments were made out of the plan (adding up the numbers in the “Total Deductions from Plan Net Position line).

I wondered what the estimated pension liability was at the end of 2004 figuring that with $39 billion of payments going out over the next 10 years, the figure had to have been pretty high.

I was surprised to learn from reading the 2004 report that the estimated liability of the pension at the end of 2004 was only about $51 billion.

Moving back to the 2014 report, I was surprised once again to learn that the estimated liability of the pension plan at the end of 2014 was just under $104 billion. The large growth of the pension liability over the last decade provides a partial explanation of why the funding levels haven’t recovered.

All these numbers lead to some estimation questions:

(1) Given the $39 billion of payments from the end of 2004 to the end of 2014, and the estimated liability of $104 billion at the end of 2014, does the liability estimate at the end of 2004 of $51 billion seem like a good estimate of the plan’s liability?

(2) What about the $104 billion liability estimate today – can we form any opinion as to whether or not that number is a good estimate of the liability at the end of 2014?

Well, on page 31 of 112 of the 2014 financial report there is a chart showing how the liability changes (or actually the under funding level) with different interest rate assumptions. The plan discounts future liabilities at a 7.5% discount rate. That means, roughly speaking, if they owe you $100 in a year, they’ll record a liability of $100 / 1.075 = $93, and if they owe you $100 in 10 years, they’ll record a liability of $100 / (1.075)^10 = $48.5.

The chart on page 31 of 112 shows that the pension plan is underfunded by about $61 billion assuming a 7.5% discount rate and by about $75 billion at a 6.5% discount rate. So, the liability of the plan increases from about $104 billion to $118 billion when you lower the discount rate from 7.5% to 6.5%.

You can get a feel for why the liability increases if you look back at the 10 year example I gave above. Using a 7.5% discount rate the $100 payment in 10 years was worth about $48.5, but using a 6.5% discount rate the $100 payment in 10 years is worth $100 / 1.065^10 = $53.3 – a larger liability.

One problem in trying to place a value on the plan’s liabilities is that long term interest rates are not 7.5% or 6.5% today, but are closer to 3%. A more difficult, but doable, estimation question is what is the estimated liability of the pension plan if we use a 3% discount rate instead of the 7.5% rate.

In my $100 in 10 years, the liability using a 3% discount rate is about $74.4. BUT, we don’t know when the payments are due, and the calculation of the liability involves many other estimates detailed in the financial report.

Still, though, there’s a reasonably good way to estimate the liability using different rates since we know what the value was using a 7.5% and a 6.5% discount rate. Actually this type of estimation is a pretty standard part of financial math. No need to go into that part here, though, other than to say it helps a little bit in thinking about whether or not the current $104 billion estimate of the pension liability is a good estimate or not.

Anyway, this was just an exercise that I went through myself after seeing the funding chart in the first article. Often in work settings (at least in my work setting) there’s no way to get exact answers, so doing your best to estimate numbers is really important. Equally (if not more) important is noticing things that don’t seem right – like the funding chart – and seeing if you can understand why it seems wrong.

All this has been a long way of saying that I’m glad they are talking about estimation in school.

A really intersting twitter conversation sparked by Kate Nowak

This tweet from Kate Nowak has been on my mind all day:

This part of the conversation on Twitter really got me thinking – in response to the question from Nowak “what is a hook to you?”:

That bit of the conversation caught my attention because I really liked Meyer’s definition, but at the same time I share Nowak’s concern about what happens if kids don’t have the tools to resolve the questions in their mind. However, despite sharing that concern, I occasionally show my kids ideas that are probably way over their head that I think they will find interesting.

My all time favorite example is the Numberphile -1/12 video.


I’ll never forget my younger son (who was in 2nd grade at the time) screaming at the computer screen “no no no no no.” I loved that something about the video bothered him – deeply and almost physically – even though it was hard for him to identify what had gone wrong.

You can see how much it bothers him in this video (at 6:50) from the project below:


After I read Jordan Ellenberg’s idea of “algebraic intimidation” in How not to be Wrong we talked about the series in this project:

Jordan Ellemberg’s “Algebraic Intimidation”

Ellenberg’s idea was a great way to explain to my younger son that it was ok not to believe the calculations he saw in the video.

Another fun project where the explanation was over the head of both kids was our look at the Chaos Game.

Computer Math and the Chaos Game

Around 2:17 in the video below we start down the path to an amazing result. That result – around 3:05 – was something that I hoped would plant the idea that computer math can be really fun even if the math was a little over their head.


More recently a Dan Anderson blog post got me thinking about how to use a little computer math and 3d printing to illustrate some math ideas usually reserved for advanced courses:

Those shapes allowed us to have a neat discussion about when some ideas in math work (finding the area of a triangle by approximating it with rectangles), when they don’t (finding the length of the hypotenuse using the same approximations), and how our old friend the -1/12 series actually played a role in the first part of the project 🙂

As with the first two projects, I’m happy to show them a strange result just to show that something interesting is happening. Even if they don’t yet have the tools to really understand what’s going on, I think it gets them thinking. As my older son says around 2:04 in the video below – “things get weird when you go to infinity.”


Here’s that project:

3d printing and Calculus concepts for kids

To circle back to Kate Nowak’s idea, I’m also a little dubious of trying to lead with ideas when kids don’t have the tools to fully understand them. Probably 95% of the math talks that I’ve had with my kids are not at all like the 3 projects I shared above. Every now and then, though, there’s been a more advanced idea that seemed like something they might find interesting. Sort of a hook to show that math has some amazing ideas that can be both fun and surprising.