There is a lot of current interest in various “crypto currencies” such as Bitcoin, but that does not mean there have not been previous combined ledger and token recording systems. Others have noticed the relevance of Crawfurd v The Royal Bank (the case where money became money), and we are going to write about this yet again.
Very roughly: a Bitcoin is a cryptographic secret that is considered to have some value. Bitcoins are individual data tokens, and duplication is prevented through a distributed shared ledger (called the blockchain). As interesting as this is, we want to point out notional value existing both in ledgers and as possessed tokens has quite a long precedent.
This helps us remember that important questions about Bitcoins (such as: are they a currency or a commodity?) will be determined by regulators, courts, and legislators. It will not be a simple inevitable consequence of some detail of implementation as this has never been the case for other forms of value (gold, coins, bank notes, stocks certificates, or bank account balances).
Value has often been recorded in combinations of ledgers and tokens, so many of these issues have been seen before (though they have never been as simple as one would hope). Historically the rules that apply to such systems are subtle, and not completely driven by whether the system primarily resides in ledgers or primarily resides portable tokens. So we shouldn’t expect determinations involving Bitcoin to be simple either.
What the Sharpe ratio does is: give you a dimensionless score to compare similar investments that may vary both in riskiness and returns without needing to know the investor’s risk tolerance. It does this by separating the task of valuing an investment (which can be made independent of the investor’s risk tolerance) from the task of allocating/valuing a portfolio (which must depend on the investor’s preferences).
Having worked in finance I am a public fan of the Sharpe ratio. I have written about this here and here.
One thing I have often forgotten (driving some bad analyses) is: the Sharpe ratio isn’t appropriate for models of repeated events that already have linked mean and variance (such as Poisson or Binomial models) or situations where the variance is very small (with respect to the mean or expectation). These are common situations in a number of large scale online advertising problems (such as modeling the response rate to online advertisements or email campaigns).
von Neumann and Morgenstern’s “Theory of Games and Economic Behavior” is the famous basis for game theory. One of the central accomplishments is the rigorous proof that comparative “preference methods” over fairly complicated “event spaces” are no more expressive than numeric (real number valued) utilities. That is: for a very wide class of event spaces and comparison functions “>” there is a utility function u() such that:
a > b (“>” representing the arbitrary comparison or preference for the event space) if and only if u(a) > u(b) (this time “>” representing the standard order on the reals).
However, an active reading of sections 1 through 3 and even the 2nd edition’s axiomatic appendix shows that the concept of “events” (what preferences and utilities are defined over) is deliberately left undefined. There is math and objects and spaces, but not all of them are explicitly defined in term of known structures (are they points in R^n, sets, multi-sets, sums over sets or what?). The word “event” is used early in the book and not in the index. Axiomatic treatments often rely on intentionally leaving ground-concepts undefined, but we are going to work a concrete example through von Neumann and Morgenstern to try and illustrate a bit more of the required intuition and deep nature of their formal notions of events and utility. I also will illustrate how, at least in discussion, von Neuman and Morgenstern may have held on to a naive “single outcome” intuition of events and a naive “direct dollars” intuition of utility despite erecting a theory carefully designed to support much more structure. This is possible because they never have to calculate in the general event space: they prove access to the preference allows them to construct the utility funciton u() and then work over the real numbers. Sections 1 through 3 are designed to eliminate the need for a theory of preference or utility and allow von Neuman and Morgenstern to work with real numbers (while achieving full generality). They never need to make the translations explicit, because soon after showing the translations are possible they assume they have already been applied. Continue reading Working an example of von Neumann and Morgenstern utility
This is an elementary mathematical finance article. This means if you know some math (linear algebra, differential calculus) you can find a quick solution to a simple finance question. The topic was inspired by a recent article in The American Mathematical Monthly (Volume 117, Number 1 January 2010, pp. 3-26): “Find Good Bets in the Lottery, and Why You Shouldn’t Take Them” by Aaron Abrams and Skip Garibaldi which said optimal asset allocation is now an undergraduate exercise. That may well be, but there are a lot of people with very deep mathematical backgrounds that have yet to have seen this. We will fill in the details here. The style is terse, but the content should be about what you would expect from one day of lecture in a mathematical finance course.
There is plenty of blame to go around from the current global financial crisis. But, I would like to point out that it is not “all the quants’ fault.” We are all now, unfortunately, sitting in the middle of a high quality (and extremely expensive) lesson in financial mathematics. I would hate for some of the truly important points to be lost to paying too much attention to some of the shiny details.