# Reference for elementary and “cool” statistics or financial math

I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to do business :-)

The closest I could come up with (using ideas of a friend) is some game theoretic stuff like price of anarchy (a related and possibly with much simpler mathematics is The Economics of Caste and of the Rat Race and Other Woeful Tales by Akerlof), Arrow's Impossibility Theorem, and possibly some Prospect theory.

Does anyone have any idea about elementary interesting math related to Statistics or Finance?

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I recommend against game theory and prospect theory. This is a high school student who probably won't be used to distinguishing between normative models and descriptive models, and the result might be that he picks up jargon without understanding anything useful. That could be practice for business, but not satisfying for you, I hope.

A fundamental question in financial math is how to evaluate a security which pays you $1/year forever, a "perpetuity." I would start there. You only need high school algebra to approach this. You can add inflation and risk, and then get into basic statistical analysis, options pricing (use simple discrete models with no calculus), etc. - I sort of disagree and will argue that this article at hss.caltech.edu/~camerer/Ec101/ProspectTheory.pdf by Kahneman and Tversky is more or less accessible to a high school student. Well, on a second thought it seems that to properly appreciate the article one has to know about expected utility theory, so I back off. – auniket Jan 13 '10 at 3:08 I vehemently disagree with the idea of teaching prospect theory, and I say that as someone who majored in behavioral economics in college before becoming a mathematician. Prospect theory is not a normative theory. It is a descriptive theory designed to explain anomalies. Do you think it's better to teach this student how to be irrational in a supposedly popular way, or how to evaluate risks in a consistent fashion? Should we teach common calculation errors before teaching people how to calculate things correctly? – Douglas Zare Jan 13 '10 at 3:31 I contend that you have a valid point. Thanks a lot for your arguments, which I clearly have not thought about. I would not try prospect theory, but would give the article suggested below by Michael a try. – auniket Jan 13 '10 at 15:36 Robert Aumanns theorem on agreeing-to-disagree (the original article can be found here) is a fun result that can be motivated by simple puzzles. Moreover, it is the basis of the no-trade-theorems in finance that show that rational expectations rule out speculative trading. A leisurely introduction to this, Aumann's theorem, and related issues can be found here. This article also motivates these issues in terms of a puzzle for children. I think it should be possible to select some topics that are (at some level) understandable by someone in high school. Moreover, this is a topic that raises questions and gives no answers, so you will not indoctrinate your student. - Thanks! It seems very intriguing. – auniket Jan 12 '10 at 22:54 You can prove that there will be no speculative trading... and then see new story after news story about huge industries of speculative trading. That's because the assumptions turn out not to be valid. Don't look for something interesting. If something is mathematically interesting to you as a mathematician, it's going to be beyond most high school students, particularly those aiming for business. That article is too interesting to be a good choice. – Douglas Zare Jan 13 '10 at 3:45 You might consider looking at the binomial asset pricing model. A good-looking book is here. - "Efficient Algorithms for Universal Portfolios" http://jmlr.csail.mit.edu/papers/volume3/kalai02a/kalai02a.pdf An algorithm for investing in a portfolio of stocks that guarantees you do almost as well as the best stock in hindsight. Very cool and not super complicated. - How long do you think it would take to explain that paper to a high school student? It has quantifiers, vectors, products using$\Pi\$ notation, integrals over higher dimensional simplices, n-cubes, etc. This high school student is a prospective business student who probably doesn't understand the basic risk versus reward tradeoff model of the stock market, not a math major in college. –  Douglas Zare Jan 12 '10 at 5:19
On the first look, it does seem to be a bit too technical for a high school student. I will use it if I want to scare him some time :-) –  auniket Jan 13 '10 at 2:59

At the high school level, perhaps they would be interested in the Monty Hall Problem -- that kept me stumped for a while.

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The Monty Hall puzzle confuses the vast majority of people who encounter it. Unfortunately, even the majority of people who think they understand it don't. For example, they overlook the assumptions about whether the host could simply show you that you lose when your initial choice is wrong. I think mathematics should be a source of clarity before confusion, and that it would take a tremendous amount of work to make this a good choice. –  Douglas Zare Jan 21 '10 at 16:42
When I was an undergraduate TA for an introduction to proofs class, we gave the students a puzzle: Why does this inductive proof fail? Claim: Any n numbers are all equal. Proof: True for n=1. Suppose it's true for n and consider a collection of n+1. The first n are equal, and the last n are equal, so they are all equal. By induction, this holds for all n. Neat, right? We thought they would find the flaw in the proof. Instead, many rejected induction, and mathematics. One otherwise promising student dropped the class and then avoided me. Fun puzzles may be bad lessons. –  Douglas Zare Jan 21 '10 at 16:51
@Douglas: Could you please elaborate on "they overlook the assumptions about whether the host could simply show you that you lose when your initial choice is wrong"? I thought the host was supposed to open a door regardless of the player's initial choice. Or is it a variant of the "original" Monty Hall puzzle (I admit I do not know what is the original version). –  auniket Jan 23 '10 at 0:10
In the actual game show, there was no requirement that the host had to offer to let you switch. The host knew whether your choice was right, and could decide to offer the chance to switch only when your choice was right, in which case switching would never help. Even if the host is required to offer you a chance to switch, there is some ambiguity about which door he opens, and you may get different information if the host prefers to one door versus another. Most solutions, and analogies to other problems, overlook these necessary assumptions for coming to the usual conclusion. –  Douglas Zare Jan 23 '10 at 20:31
@Douglas: Thanks, I overlooked these assumptions too. –  auniket Jan 29 '10 at 22:41

One topic that I would cover is the St. Petersburg Game (or paradox). Daniel Bernoulli's solution is the basis for expected utility theory and the rivalry between himself and his cousin Nicolas just makes it more fun. (The problem is well-suited to a visual description as well.)

Good luck.

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That sounds like it might be a good, approachable topic to cover. Make sure the student knows that there are some infinite series with finite sums first. While this is not needed to see that there is a paradox, it can lead people to misunderstand its nature. –  Douglas Zare Jan 21 '10 at 20:18
This seems very good to me as well. Let's see how the subject responds :) –  auniket Jan 22 '10 at 23:46

You might try a simplified version the CAPM with an application regarding the choice of a utility based portfolio on the (Markowitz) Efficient Frontier of market portfolios. It uses concepts of arbitrages, utility functions, portfolio of assets with gaussian yields and all this is quite illuminating.

http://en.wikipedia.org/wiki/Capital_asset_pricing_model

Hope it helps

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