Magicians And Mathematicians

Quantitative finance and risk management are not just about the numbers. Numbers play a part, but so does the human side of the business. When analyzing risk it is important to be able to think creatively about scenarios. Unfortunately the training that most quants get seems to actively discourage creativity.

Some of the following appeared on the BBC website in December 2008.

We’ve learned the hard way how important it is to measure and manage risk. Despite the thousands of mathematics and science PhDs working in risk management nowadays we seem to be at greater financial and economic risk than ever before. To show you one important side of banking I’d like you to follow me in an exercise with parallels in risk management.

You are in the audience at a small, intimate theatre, watching a magic show. The magician hands a pack of cards to a random member of the audience, asks him to check that it’s an ordinary pack, and would he please give it a shuffle. The magician turns to another member of the audience and asks her to name a card at random. “Ace of Hearts,” she says. The magician covers his eyes, reaches out to the pack of cards, and after some fumbling around he pulls out a card. The question to you is what is the probability of the card being the Ace of Hearts?

Think about this question while I talk a bit about risk management. Feel free to interrupt me as soon as you have an answer. Oh, you already have an answer? What is that, one in fifty two, you say? On the grounds that there are 52 cards in an ordinary pack. It certainly is one answer. But aren’t you missing something, possibly crucial, in the question? Ponder a bit more.

One aspect of risk management is that of ‘scenario analysis.’ Risk managers in banks have to consider possible future scenarios and the effects they will have on their bank’s portfolio. Assign probabilities to each event and you can estimate the distribution of future profit and loss. Not unlike our exercise with the cards. Of course, this is only as useful as the number of scenarios you can think of.

You have another answer for me already? You’d forgotten that it was a magician pulling out the card. Well, yes, I can see that might make a difference. So your answer is now that it will be almost 100% that the card will be the Ace of Hearts, the magician is hardly going to get this trick wrong. Are you right? Well, think just a while longer while I tell you more about risk and its management.

Sometimes the impact of a scenario is quite easy to estimate. For example, if interest rates rise by 1% then the bank’s portfolio will fall in value by so many hundreds of millions. But estimating the probability of that interest rate rise in the first case might be quite tricky. And more complex scenarios might not even be considered. What about the effects of combining rising interest rates, rising mortgage defaults and falling house prices in America? Hmm, it’s rather looking like that scenario didn’t get the appreciation it deserved.

Back to our magician friend. Are those the only two possible answers? Either one in 52 or 100%? Suppose that you had billions of dollars of hedge fund money riding on the outcome of this magic trick would you feel so confident in your answers? When I ask this question of finance people I usually get either the one in 52 answer or the 100%. Some will completely ignore the word ‘magician,’ hence the first answer. Some will say “I’m supposed to give the maths answer, aren’t I? But because he’s a magician he will certainly pick the Ace of Hearts.” This is usually accompanied by an aren’t-I-clever smile! Rather frighteningly, some people trained in the higher mathematics of risk management still don’t see the second answer even after being told.

This is really a question about whether modern risk managers are capable of thinking beyond maths and formulas. Do they appreciate the human side of finance, the herding behaviour of people, the unintended consequences, what I think of as all the fun stuff. And this is a nice question because it very quickly sorts out different types of thinkers.

There is no correct answer to our magician problem. The exercise is to think of as many possibilities as you can. For example when I first heard this question an obvious answer to me was zero. There is no chance that the card is the Ace of Hearts. This trick is too simple for any professional magician. Maybe the trick is a small part of a larger effect, getting this part ‘wrong’ is designed to make a later feat more impressive…the Ace of Hearts is later found inside someone’s pocket. Or maybe on the card are written the winning lottery numbers that are drawn randomly 15 minutes later on live TV. Or maybe the magician was Tommy Cooper. Or it was all the magician’s performance-anxiety dream the night before. When I ask non mathematicians this is the sort of answer I get.

The answer one in 52 is almost the answer least likely to be correct! Magicians only rarely rely on probability. Clue: How many times did Houdini die during his Water Tiorture trick? (Unless the magician was using an ordinary deck of cards, was aiming to pull out a different card but accidentally pulled out the Ace of Hearts instead! Accidentally not making the intended ‘mistake.’)

A member of didn’t believe me when I said how many people get stuck on the one in 52 answer, and can’t see the 100% answer, never mind the more interesting answers. He wrote “I can’t believe anyone (who has a masters/phd anyway) would actually say 1/52, and not consider that this is not…a random pick?” So he asked some of his colleagues the question, and his experience was the same as mine. He wrote “Ok I tried this question in the office (a maths postgraduate dept), the first guy took a fair bit of convincing that it wasn’t 1/52 !, then the next person (a hardcore pure mathematician) declared it an un-interesting problem, once he realised that there was essentially a human element to the problem! Maybe you have a point!” Does that not send shivers down your spine, it does mine.

Once you start thinking outside the box of mathematical theories the possibilities are endless. And although a knowledge of advanced mathematics is important in modern finance I do rather miss the days when banking was populated by managers with degrees in History and who’d been leaders of the school debating team. A lot of mathematics is no substitute for a little bit of commonsense and an open mind.

How can we get quants and risk managers to think beyond the mathematics? I’m afraid I don’t think we can, the way the majority of them are currently educated.