Misleading Elvish Statistics

by chenicus

Ponzi the magical elf offers two investments that last three days and average the same daily return.  One triples its value each day (for 3 days), and one double, triples, and then quadruples.  Supposing you believe Ponzi, which do you pick?  Well 3x3x3=27, 2x3x4=24.  So despite having the same average daily return, the two investments differ in outcomes. Go ahead and take it to an extreme: average(6,3,0)=3, 6*3*0=0.  Who cares what the average return is if Ponzi runs off with your money in the end?  Now we can go into finance-specific resolutions [1], but consider: barring uncertainty and elf-trust issues, what actually mattered?  Cash money, dollar dollar bills, how much you end up with.  I feel that as much of statistical misuse [2] is caused by misunderstood tools as misunderstood purpose.

[1] For instance, average log returns are not vulnerable to this specific problem  [http://quantivity.wordpress.com/2011/02/21/why-log-returns/] as the transformation preserves the features we want (one-to-one with total return).  I like my example because the risk-return tradeoff in finance is typically presented in a way that can seduce you (or at least me) into thinking they are two separate entities.  Risk in the sense of the variance or spread in periodic relative returns directly affects the bottom line independent of your probabilistic philosophy.  Even the popular Sharpe ratio will not save you, as it has a sort-of wink-and-nod Normal dist approach to risk (e.g. ignoring skew).  For instance, take a risk-free return rate of 0, on year with -100% return, and N years of R>0 annual return.  Then the Sharpe ratio is (N*R-1)/(sqrt(N+1)*(1+R)).  Set N to be 15, R to be 1 and you have nice Sharpe ratio of 1.75, and have basically described Bernie Madoff’s portfolio.

[2] Check out  [https://en.wikipedia.org/wiki/List_of_paradoxes#Statistics], specifically the ones based on fallacious reasoning.  What is amazing to me is that Paul Erdos fell prey to the Monty Hall problem [http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf]. As an aside, check out the Erdos-Bacon-Sabbath [http://erdosbaconsabbath.com/] number.