magine you are sitting at a blackjack table. You have been counting the number of aces and picture cards that have so far been dealt from the pack and this has enabled you to determine that the odds of you winning the next hand are better than 50%. You expect to have an unlimited number of similar opportunities like this since the casino does not know what you are up to. What percentage of your chips should you bet on each hand to maximise your long-term return?

This is the situation in which Ed Thorp, a physics graduate teaching mathematics at the Massachusetts Institute of Technology, found himself in 1961 having pioneered the technique of card-counting. He used a result from work on information theory developed by Claude Shannon, a genius whose work laid the foundations for the Internet. However, it is relatively easy to derive this result from first principles. After n hands our geometric mean return is given by

{(1w)f¥(1+aw)s}1/n

where s = number of successful outcomes

f = number of failures

– = s + f

a = ‘odds’ offered by the house (for blackjack a=1)

and w = is the percentage of our bankroll

Now if p is the probability of winning each hand then as n tends towards infinity, s and f tend toward n.p and n(1p) respectively and the geometric mean g(w) therefore tends towards

g(w) =(1w)(1p)¥(1+aw)p

It is left as an exercise to the reader to prove that this is maximised for a value of w equal to

where h=1/(1+a) ,the probability of winning implied by the house odds. A crude proof is provided in the box opposite.

This is more usually expressed as ‘edge/odds’ and is referred to as the ‘Kelly bet’ after a gun-toting physicist from Texas who first published the formula in a paper entitled ‘Information Theory and Gambling’ in 1956.

It can immediately be seen that, as you would expect, w is positive only if p>h, ie you only bet if the odds are in your favour. Further, if p=1 then w=1, ie if you are absolutely certain of winning then you should bet your entire bankroll.

More importantly, the formula shows that there is an optimum size of bet and that betting even a small amount either side of that amount can produce significantly lower returns, as can be seen in figure 1. This also shows that the expected return approaches zero as you tend towards betting your entire bankroll. Another key feature is that, unlike other betting systems such as Martingale whereby you bet a certain amount and then double it each time you lose, those using the Kelly formula can never lose their entire stake.

The same principles can be used for an event with more than two discrete outcomes, for example a football match where you can bet on win, lose, or draw. In this case, provided the odds on one outcome are sufficiently favourable, the geometric mean will be maximised by also betting on another outcome, even if the odds on that particular outcome are not in your favour. This is because of the reduction in volatility of your returns.

In the 1970s the Kelly formula came under stinging criticism from Paul Samuelson, a leading economist of his generation. Joining Samuelson in the criticism was Samuelson’s most brilliant protégé, Robert C Merton. It might seem odd to non-economists that someone should criticise what is a mathematically correct formula. However, the criticism was aimed not at the formula itself but at the premise on which it was based that one’s objective should be to maximise geometric mean returns in the longer term. Samuelson was quick to point out the often-used quote from John Maynard Keynes ‘in the long run we are all dead’. He argued that one should think in terms of maximising utility. The argument between the Kelly protagonists and Samuelson’s economist was heated and in one article Samuelson explained why he believed the premise to be flawed entirely in words of one syllable! One particular criticism of the approach of maximising geometric returns was that the consequent returns were too volatile. From any point in time, there is always a 1 in n chance that you will at some point in the future be reduced to 1/n of your bankroll. For this reason, many gamblers using the formula tend to bet ‘1/2 Kelly’. At the other extreme the formula was criticised for producing too small a bet. For example, if you were placing a single bet with hugely favourable odds then to maximise your return you should bet your entire bankroll.

Princeton-Newport

In 1964 Ed Thorp turned his attention to the biggest casino of all, the stockmarket. Using probability distributions of random walks, he deduced that warrants were overpriced and therefore saw that there was money to be made by selling warrants short. To reduce risk, he purchased an equivalent amount of the underlying stocks, a technique now known as ‘delta hedging’. By 1967 Thorp had devised a version of what are now called the Black-Scholes pricing formulae for options. Having proved this worked, he later set up a hedge fund partnership called Convertible Hedge Associates, later to be renamed Princeton-Newport Partners.

At about the same time, Merton was working on the same problem, building on work done by Myron Scholes and Fisher Black. The Black-Scholes formula was published in 1973 and quickly made the market in derivatives more efficient, making it more difficult for Thorp to find arbitrage opportunities. However, Princeton-Newport were adept at finding other opportunities. In the early 1970s, Thorp’s founding partner of Princeton-Newport, Jay Regan, came up with the idea of detaching coupons from new treasury bonds (which were then pieces of paper) and selling those pieces of paper separately. Not only did this offer investors the choice of income or deferred capital, it took advantage of a loophole in the tax law at the time. The loophole was closed in 1982 and at the same time paper bond certificates were replaced by electronic bookkeeping.

In 1981 the US telecoms giant AT&T was broken up into eight pieces and it was possible for investors to buy shares of the ‘Baby Bells’ and the new AT&T before they were officially issued. Thorp calculated that the shares of the old AT&T were slightly cheaper than the equivalent amounts of new companies. Princeton-Newport hedged by buying the old AT&T and selling short the new shares, borrowing to fund the entire capital of the fund by about six times and in doing so making the largest ever trade in the history of the New York Stock Exchange.

In 1982 S&P futures began trading. Thorp calculated that the futures were overvalued and went short, hedging by buying selected sets of the S&P500 to provide protection while minimising trading costs. For four months the profits rolled in, until the prices for S&P contracts reduced as other traders started using computers.

Throughout the life of the fund, Thorp and his associates used the principles underlying the Kelly formula to determine the size of their positions, and borrowing large amounts when there was a near certainty of success.

Between 1969 and 1988, Princeton-Newport Partners achieved an average return of 15.1% pa after fees, compared to 8.8% pa from the S&P500 over the same period, and with much less volatility.

LTCM

Long Term Capital Management (LTCM) started in March 1994 and was the first fund to raise a billion dollars. With Merton and Scholes on board, what could go wrong? Like Princeton-Newport the fund made money from arbitrage opportunities identified using complex financial models. In 1994 and 1995 the fund earned 43% and 41% respectively after fees. However, to achieve these returns LTCM used a huge amount of leverage about 30 times.

In August 1998, Russia defaulted on their treasury bonds and devalued the ruble, leading to huge losses for LTCM. It was the beginning of the end. In a 2003 issue of Wilmott magazine, Thorp linked the LTCM collapse to Merton and Scholes’s intellectual critique of the Kelly system: ‘I could see that they didn’t understand how it controlled the danger of extreme risk and the danger of fat-tailed distributions’, Thorp said. ‘It came back to haunt them in a grand way.’

I rarely meet actuaries who invest their own money directly in the stockmarket, let alone any that use spread-betting to hedge their returns. However, for any that do then the lesson here is that over-betting can seriously damage your wealth.

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