A few fun Perfect Bracket stats questions for students

ESPN had 18.8 million entries in their bracket challenge for the NCAA men’s basketball tournament. There were also several other bracket contests going, too. Below are a couple of fun bracket-related questions for students learning about statistics:

(1) Perfect Brackets after the first round:

The ESPN contest went from 18.8 million entries to 164 perfect brackets at the end of the first round of games:

So . . .

(i) If you were running a contest that had 100,000 entries instead of 18.8 million, how many perfect brackets would you expect to have in your contest?

(ii) What do you think the probability of having 0 in your contest would be?

(iii) How about exactly 1?

(2) Perfect brackets after Michigan St. beat Miami

The number of perfect brackets in the ESPN contest fell from 952 to 513

Prior to the Michigan St win you had 5 perfect brackets left in your contest. Given what happened in the ESPN contest how many do you think you’ll have after the Michigan St. win?

What do you think the probability is that you will have 0?

How about 1?

(3) The USC vs SMU game was the 22nd game of the tournament

You had 241 perfect brackets going in and 22 after USC won.

In the ESPN contest 81.6% of the entries picked SMU to win and 18.4% picked USC.

(i) Suppose you have a coin that flips heads 18.4% of the time. If you flip it 241 times what is the probability that you will have 22 or fewer heads? (probably best to use a computer for this one . . . )

(ii) Do you think having 22 brackets left rather than 44 (which would roughly be 18.1%) was random chance or was there another factor in the reduction?

(4)  Expected upsets

I’ll make up the numbers for purposes of this problem, but you can get the actual numbers here if you want:

Some statistics for the ESPN bracket tournament

Suppose that the 18.8 million entries have selected the winners of the games this way:

Teams 1 – 5:  95% of the entrants guessed these teams would win their 1st round game

Teams 6 – 10: 90% of the entrants guessed these teams would win their 1st round game

Teams 11 – 15:  85% of the entrants guessed these teams would win their 1st round game

Teams 16 – 20:  80% of the entrants guessed these teams would win their 1st round game

Teams 21 – 25:  70% of the entrants guessed these teams would win their 1st round game

How many games out of these 25 do you expect the team that was not favored by the ESPN contestants to win?  Why?

 

 

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A risk pricing formula for simple situations

I was reading Nassim Taleb’s most recent paper this morning:

https://twitter.com/nntaleb/status/718456742141673472

and this short passage caught my attention:

Screen Shot 2016-04-14 at 4.16.52 PM

It reminded me of a simple risk pricing idea I had (and used) 15 years ago that nearly got me laughed out of the room. The idea isn’t applicable to general situations – just pretty simple ones in which you have low-probability, insurance-like risk. So a situation something like:

(i) The event you trying to price has a low probability -> I’ll say 1/k, and

(ii) you have a known (and binary) payout if the event happens, and

(iii) you keep whatever premium you collect if the event doesn’t happen.

The original source of the conversations was providing insurance for Pepsi’s “Play for a Billion” contest in which there was a 1 in 1000 chance that someone would win $1 billion.

Most of the traditional ways that people think about insurance risk fall flat here. Do you want a 10% more than the “loss cost”? So, an extra $1 million to take a 1 in 1,000 shot a losing $1 billion? Maybe 50% more?

My idea was to imagine taking the bet over time and look at the probability that you would collect enough money to pay one loss before the first loss actually happened. The math behind this idea makes for a nice Calculus 1 example. Assume for simplicity that your payout will be $1.

Probability of loss -> 1/k

Premium collected each time -> x / k, so you are getting “x” times the loss cost.

It will thus take you k / x trials to collect enough money to pay a single loss. The probability of not paying out in the first k / x trials is:

(1 - 1/k)^{(k/x)} = ( (1 - 1/k)^k )^{(1/x)} \approx e^{(-1/x)} since k is large.

This math gives us a very simple formula which is independent of the probability (as long as the probability is low). If, say, we want a 75% chance of collecting a $1 before paying out the $1 we have to charge 3.5x the loss cost since:

e^{(1/x)} = 0.75, and x = - \ln{ 0.75 } \approx 3.5

For a 95% change you’d have to charge about 9.5x the loss cost.

Anyway, as I said this approach was laughed at by just about everyone on the other side of the table. I’m glad to see serious discussion about time averages rising to the top now!

An interest rate question extending Kate Nowak’s rate post

Earlier in the week Kate Nowak wrote a neat post about rates. The perspective in the post (in my words) is coming from writing curriculum materials for 6th grade math:

Here’s an alternate perspective on the same (or at least similar) issue that I encountered at work this week.

Suppose I ask you to play the following game:

(1) You pay me $2 today.
(2) I’ll then select an integer from 1 to 10 at random (uniformly)
(3) At the end of year 1 you pay me $1, and if my random number was 1 I’ll pay you $10 and the game stops. If my number wasn’t 1 we’ll meet again next year.
(4) In general, at the end of year n, you’ll pay me $1 and if the random number I picked was n, the game stops.

The interest rate question relating to this games is this: What is your expected rate of return for playing my little game?

Here are two different ways to think about it:

(1) Internal rate of return

You’ll see an expected set of cash flows that look something like this:

Screen Shot 2016-01-23 at 11.09.40 AM

The “internal rate of return” on those cash flows is about 12%, so you might say (and I think that many people would be quite comfortable saying) that your expected rate of return playing my game is about 12%.

(2) Accounting for the costs and the investment returns differently

One possible objection to the internal rate of return calculation is that your cash outflows are really part of your investment in the game and so are quite different than the investment return. In fact, to play the game all the way through, in addition to the $2, you need to be sure that you have access to $10 over time to play.

So, you might prefer to discount your cash outflows at a less risky rate – I’ve picked 4% just for example purposes – and discount the inflows (the investment returns) at a risky rate to measure your return. That calculation looks something like this:

Screen Shot 2016-01-23 at 11.14.43 AM

Using this method the expected investment return you’ll get for paying $2 to play my games is more like 8% per annum.

So, what is the correct way to think about the rate of return for playing my game?

I think the rate of return question here is pretty interesting to think about and gives a real life example of the things that Nowak is thinking about writing 6th grade curriculum.

A neat probability paper Christopher Long shared with me

A longer than usual, and more fun than usual sequence started on Twitter yesterday with this tweet:

I was excited about it because we have some non-transitive “Grime” dice and the kids still play around with them. Here’s a little project we did when we got them:

Non-Transitive Grime Dice

Because of the original twitter post, Christopher Long ended up sharing this neat (and fairly accessible) probability paper:

Long wrote several more (maybe 10) tweets following this one which are all pretty great follow ups to the paper.

What especially caught my eye in the paper was section 5 on duration calculations. I’ve spent lot of time in the last couple of years thinking about how quantities like duration and internal rate of return change when you have both positive and negative cash flows. Originally my interest in these ideas came from noticing that many people in the financial area like to talk about internal rate of return (“IRR”) even when IRR seemed to be to be a pretty dubious way to analyze the particular financial product.

The interest in duration calculations followed a similar path when people were marketing some “short duration” products that didn’t seem to have short durations at all (to me).

The connection with fractional coins and dice seems fascinating. Although I haven’t really settled some of the ideas in my mind, I’m happy to have stumbled into this interesting connection to a problem that was already on my mind.

Pretty amazing power that Twitter has 🙂

The most important piece of math I learned in 2015

My introduction to finance came back in graduate school when a friend working at an option shop in NY asked me to explain Edgar Peters’s Fractal Market Analysis to him. This is probably a pretty non-standard first introduction to finance, but it was a lucky one for me – I found the presentation to be compelling. Two other books that helped me refine my thinking on finance were both by Mandelbrot – The (Mis) Behavior of Markets and Fractals and Scaling in Finance.

My only bit of background in finance that would be considered “standard” by the business school crowd came from reading Zvi Bodie’s paper “On the Risk of Stocks in the Long Run”. This paper asks a simple, but quite educationally useful, question about option pricing via Black-Scholes – what is the price a European put maturing in the future with a strike price equal to the current price of the asset brought forward to the maturity date at the risk free rate? The answer was a surprise to me (though, of course, I’d not studied much option pricing, so anything would have been a surprise, I suppose) and understanding that result proved to be useful many years later when the friendly Wall St. folks showed up wanting to write 15 and 20 year puts with us.

With that all as background, this ~30 minute talk by Nassim Taleb has the most important piece of math that I learned in 2015:

 

These screen shots in particular show two parts that have stuck with me since the moment I saw them. The first shows roughly how much data you need to be able to talk sensibly about something governed by a power law distributions, and the second makes the point that if you try to estimate the mean of a power law distribution ( with 1 < \alpha < 2) from a data set, you will likely underestimate it.

Screen Shot 2016-01-01 at 7.01.09 AM

Screen Shot 2016-01-01 at 7.08.26 AM

If you are analyzing anything that lives in (to borrow Taleb’s word) extremistan, you have to understand the ideas in Taleb’s talk. If you don’t, the slides I highlighted show two easy traps that you’ll fall into – and there many are others, too.

An incredible article about data science

If you or your students are interested in understanding ways that math can be applied to problems outside of academic / school settings, this recent article from “I Quant NY” is an absolute must read. Hat tip to Patrick Honner for pointing it out to me:

So much of what is important in mathematical problem solving is on full display in the piece – noticing, wondering, basic number sense, and tons and tons of persistence.

oh, and no equations more complicated than calculating a 20% tip.

I’d guess that students ranging in age from middle school to graduate school can get something – and probably quite a lot – out of this article. The analysis, methods, and conclusions shared in the article provide such valuable lessons that I honestly can’t think of a better starting point to understand what quantitative analysis can bring to the table in the mythical “real world.”

If you want one little sound bite / takeaway, let it be this passage:

When Doing Data Science, Look at Your Raw Data. If there is one thing I have learned doing data science, it is to always look closely at your raw data in addition to your aggregate statistics. It would not have been possible to figure this out without looking at a subset of individual rides.”

Bravo I Quant NY!!

An interesting math example coming from the world of finance

Saw this article tonight on Bloomberg:

The Best and Worst Investments of 2014

What caught my eye was the description of the best performing bond fund (about 2/3 of the way down the page):

“Vanguard’s index fund invests in U.S. government bonds that don’t mature for 20 to 30 years. They did well in 2014, reflecting expectations that inflation will remain low for quite a while. If you’d invested $10,000 in VEDTX on Jan. 1, it would be worth $14,506 today.”

Here’s the math that students interested finance (and, of course, anyone else) might find interesting:

According to Bloomberg data, the 30 year US Government bond had a yield of 3.969% on December 31, 2013, and a yield of 2.752% on December 31, 2014. So, throughout 2014 the yields of long-dated US government bonds – the exact type of bonds that the Vanguard fund invests in – were quite low. In that sort of yield environment how could a fund that invests in US government bonds have had a return of 45%?

Answering this question will help you gain a little bit of knowledge about how bond prices move with interest rates.

Let’s start with a simple example. Say that on December 31, 2013 someone had offered to pay you a 3.969% per year return on your initial investment which would result in you receiving $1,000 in 30 years. Just to be clear, you’ll receive no interest payments on this investment – the only time cash changes hands is on day 1 when you hand over the initial investment, and at year 30 when you are handed back $1,000. How much would you have to give that person today to make this investment (ignoring taxes and all sorts of other potential complications – just to keep things easy)?

One way to get to the answer to this question is to take the final $1,000 payment and divide it by the 1.03969% return that you are promised each year for 30 years. That math tells you the initial investment would be $1,000 / (1.03969)^30 = $311.09.

Now, one year later these long-dated yields have gone down to 2.752%. How much is the $1,000 you receive in 29 years worth at that rate now? We can do the same math: $1,000 / (1.02752)^29 = 455.07. Wow – what a difference!

In just one year the so-called “present value” of the future payment has gone from $311 to $455 – a gain of 46.3%! That gain is due to combination of interest rates falling by 1.25% and the $1,000 payment being far off in the future so that the change in interest rates compounds for many years.

I’m cheating a little in the above calculation by using the wrong rates. For a single payment in the future I should be using so-called “zero coupon” rates, but I’m just trying to illustrate where these high returns that the Vanguard fund achieved can come from.

If we look at the 30 year coupon bond we can see a similar, though not exactly the same, move in prices. Again according to Bloomberg, the price of the official 30 year US government bone on December 31, 2013 was about $96 and that bond paid an annual coupon of $3.75. Roughly speaking that means you could have bought this bond for $96 on 12/31/2013, then received $3.75 every year for the next 30 years and then received $100 at the end of 30 years.

One year later the price of the exact same bond was about $120. So, at the end of the first year you could have sold the bond for $120 and kept all of the $3.75 in interest that you received during the year. You would have paid $96 and received $123.75 in interest and sales proceeds. That’s a return of about 29% on your $96 investment! Again, that return comes from the combination of falling interest rates and the fact that the maturity of the bond is so far in the future.

So, understanding the math on bonds helps us see how it is at least plausible that a US government bond fund could have 46% returns in a year when interest rates were quite low by historical standards. It may seem surprising that a seemingly small movement in interest rates – that is rates falling from 4% to 2.75% – could produce a 46% return, but that more or less exactly how the math works out. It was a lucky year to be taking bets on the values of long-dated bonds!