Figuring the odds

COMMENT - Actually derivatives are securities whose value is derived from other securities. That would traditionally be all futures, warrants, and mainly options.

Figuring the odds
By Dr Guy Scott
Tue 16 Feb. 2010, 04:00 CAT

The financier Warren Buffet famously called them “financial weapons of mass destruction”; wildly successful currency speculator George Soros claimed that he did not trade in them because “we do not really understand how they work”. You can buy them now in Lusaka, from a highly respectable bank on Cairo Road. We are of course on to the subject of derivatives.

To make matters clear, I should here point out that when people talk about derivatives, especially in a worried way, they are talking about “custom-made” new-fangled derivatives. All sorts of “regular” financial instruments, including shares and insurance certificates, perhaps even a banknote, are strictly speaking derivatives; they derive their value as assets from underlying “fundamentals”. Both Buffet and Soros have a long history of dealing in such everyday derivatives, and have been accused of hypocrisy accordingly.

But, if we understand that their warnings refer to only certain types of instrument we perhaps have some chance of understanding their concern. This can be expressed as the worry that it is unavoidably difficult to compute the risk factors, and thence market value, of a new-fangled or one-off type of derivative, and if too much dependence is placed upon them as securities then big mistakes, big bubbles and bursts, will occur.

Some people think this is because the new stuff is too complicated to do calculations with but that is not really a problem if you are a big enough trader: you just hire a computer whiz or a mathematician (there is a book called Nuclear Physicists on Wall Street). No, the real problem exercising Warren and George has to do with elementary statistical theory; I will try and explain how in 1700 words, without any equations or graphs. If I succeed it may be some kind of record.

Every year I insure my car. The “actuarial value” of my insurance is less than what I pay for it; meaning that the “average” car-owning citizen pays out more in premiums than he/she claims in vehicle damage or loss. This excess of premium over actuarial value must be there otherwise the insurance company would not be able to operate, let alone at a profit. So why do I pay? Why do I part with more cash than the thing I am buying is worth? I do so because I wish to avoid the uncertainty, the nightmare scenario of a car theft or crash that keeps me awake at night; I am “risk averse” and I want to “hedge”, in the jargon, and am prepared to pay what it takes, essentially, for a good night’s sleep. Uncertainty is not equal to probability, especially since I am a damn good driver.

Now how does the insurance company know how much it is going to have to pay out in claims (and therefore how much it must charge in premiums)? Luckily, so long as it insures a sufficiently large number of cars, the past frequency of accidents and thefts gives a very good estimate of the mean probability per vehicle of accidents and thefts. This probability in turn is a good predictor of future frequency! In other words, for large numbers frequency and probability are very close together and can be used as estimates for each other.

This is not true when numbers are small however. An insurer who insured only ten vehicles (or ten houses or ten lives) could easily make a roaring profit one year and go bust the next. With small numbers, frequency and probability lose their intimate connection. But for an insurer who insures tens of thousands of cars, lives, houses or whatever, that intimate connection is there and he/she is assured of a financially stable if boring life.

Let me illustrate this difference between large and small by simulating on a computer (in Microsoft Excel) the outcomes of repeatedly tossing a coin various numbers of times. (If you do not know what a coin is ask your grandparents). Each “run” I toss the coin ten times (and do a trial of ten runs); this yields the following numbers of heads: 4, 6, 4, 7, 4, 3, 2, 6, 5, 4. The expected number of heads, based on a probability of 0.5 is exactly five each time – but this is the frequency on only one run! Frequency varies across the ten trials between 0.2 and 0.7. You cannot get a reliable estimate of the probability from observed frequency on a single small run.

Now if I do a simulation of coin tossing 100 times per run (with ten runs) I get the following numbers: 54, 49, 54, 52, 46 45, 51, 45, 42, 46. The frequency is running much closer to 0.5, staying comfortably between 0.4 and 0.6. If I toss the coin 1000 times the observed frequency hugs 0.5 very much more closely still.

One of the core difficulties with the “custom-made” derivatives, it seems to me, is a problem of small numbers. Small numbers (some derivatives are designed for only one customer) make it difficult to work out the underlying probabilities. Also, the magical effect of large numbers in insurance, where people paying premiums provide the money for those making claims at any one time, is not there. The derivatives currently being “custom made” in Zambia are in some respects like insurance policies.

For example, a producer of, let us say, sweet potatoes (I think this is an imaginary example as it should be) is worried that the export price for his commodity when sold south of the Zambezi is (a) likely to fluctuate in Rand terms and (b) in kwacha terms may be adversely affected by appreciation of the Zambian (ZMK) relative to the Rand. So he negotiates a contract (which may be in more than one part) with the derivative supplier (“the bank”) under which the bank guarantees a floor price in kwacha; in exchange the bank sets a ceiling price, above which it takes the excess. This is called a “collar”. What is relevant is the fact that, while there is a long history of insuring against all kinds of eventualities in Zambia (including even hail and drought) there is no history of insurance against negative market conditions (unless you count Government’s loss-making interventions in the maize market).

Regular insurers have steered clear of commodity markets for good reason. Only some farmers are hit by hail in any one season, droughts are similarly localised, but a depressed market affects everyone: there are no “lucky” punters to pay out the “unlucky” ones. And there is no simple frequency counting to determine probabilities: but it is probabilities that determine the value of the derivative to the bank, or to someone to whom the bank sells it as a security.

Note that problems with underlying value of a derivative on the securities markets do not affect its value to the customer, who entered into the contract on the basis of evading risk and getting a good night’s sleep. Hedging is good and the actuarial value is not his concern (so long as the bank does not go bust). Nor would derivatives much affect the stability of the banking system in Zambia; unless they came to be a substitute for more easily valued assets, as happened in the world banking system before the crash.

So far as I can establish the types of hedging done in Zambia currently include: FX forwards, cross currency swaps (CCS), Interest Rate Swaps (IRS), FX Options, call (buy), puts (sells), collars (buy and sell). Though some of it sounds like gobbledegook it probably makes sense: you just need to find someone in the know to explain it to you if you are having sleep problems.

Plan B, if you are trying to discover the probability of something, is to work it out from first principles. In the case of the physical tossing of a physical coin, you do not have to observe frequencies nor be a red hot engineer to figure out that, give or take some tiny effects due to distortion or wear and tear of the coin the odds of a head or a tail are equal i.e. the probability of either is 0.5. Though more complex, clever people can look at the underlying factors driving commodity prices, and that of sweet potatoes in our example, and come to some estimate, all factors considered, of how prices might move. This is chancy, however, given the complex dynamics of commodity markets and the role of externalities (such as recession). And even simple mechanical devices can spring surprises as the following true story illustrates!

Joseph Jagger was an English engineer of the 19th Century who had an intuition that roulette wheels, if made to a poor standard, would not produce equal probabilities of the little ball ending up in each of the 38 slots in the wheel. His conjecture appears to have concerned the inaccuracies in the machining of the “frets” – the little metal plates that separate the slots. If some stood higher than others they would cause bias in the destination of the ball.

In 1873 he hired six clerks to clandestinely record all the numbers that came up from the six roulette wheels in the main casino in Monte Carlo over a period of weeks. Upon analysis he discovered from the frequencies that one of the wheels showed a distinct bias towards nine numbers: 7, 8, 9, 17, 18, 19, 22, 28, and 29. He immediately moved to Monte Carlo and started betting heavily on these numbers. It took the management some time to notice what was going on and he made money heavily.

Management eventually twigged and moved the wheels around at night, causing Jagger to lose money until he rediscovered his biased table. In the end the management took to moving the frets around each day; finally confounding Jagger. So he took his winnings – about USD5 million in today’s money – and invested it wisely. Other punters who had emulated Jagger’s bets also took large sums home with them. Although others have laid claim to the title, Jagger (who is supposedly an ancestor of Mick the Rolling Stone) was the genuine Man Who Broke the Bank at Monte Carlo. And gave a lesson in very practical statistics to us all.