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How much should you invest in a stock?



THE science of numbers has developed quite a bit since June 1955, when a new quiz show called The $64,000 Question made its debut on American television. The show was a hit. It captured as much as 85 per cent of the viewing audience and spawned dozens of copycat shows.

There was even betting on which contestants would win. But the problem was the show was produced in New York and aired live on the East Coast of the US. The telecast, however, was delayed by three hours on the West Coast. A gambler took advantage of the time difference by finding out by phone who the winners were, and then placing his bets before the West Coast airing.

John Kelly, a physicist at Bell Labs, heard about the scam on the news. After some pondering, he was convinced that a gambler with ‘inside information’ could use some of the equations developed by his colleague – Claude Shannon – to achieve the highest return on his capital. Mr Shannon, who created information theory after having realised that computers could express numbers, words, pictures, audio and video as strings of digital 0s and 1s, had developed formulas to deal with the signal noise of long-distance telephone transmission. Mr Kelly saw that the equations could be applied to the problem of a gambler who has inside information, say, about a horse race, and who is trying to determine his optimum bet size.

A gambler gets a tip on a race’s outcome. He could bet everything he’s got on the horse that’s supposed to win. But if the gambler adopts this approach, he will lose everything should the information turn out to be wrong. Alternatively, he could play it safe and bet a minimal amount on each tip. This squanders the considerable advantage the inside tips supply.

In Kelly’s analysis, the smart gambler should be interested in ‘compound return’ on capital. That is, to optimise the long-term growth rate of one’s capital, the gambler – the theory also applies to investors – should vary the wager as a proportion to his overall capital depending on the probability of bet being a winning one, and on the payout of the winning bet.

The Kelly formula or Kelly criterion has become a popular money-management formula for investors, hedge fund managers and economists.

Uncertain events

As the saying goes, the only certainties in life are death and taxes. For everything else, we form some kind of expectations of the outcome based on our experience, or what have been documented by others.

In finance, the expected value is used to account for the uncertain outcome of, say, a project or an investment.

Project A, has 30 per cent chance of making a profit of 20 per cent, 40 per cent of a profit of 12 per cent and 30 per cent of making a loss of 15 per cent. So the expected return is 6.3 per cent (30%x20% + 40% x12% + 30% x -15%).

If there are numerous investment opportunities with that kind of probabilistic outcome, then over a long period of time, the investor will earn an average of 6.3 per cent return per investment, as indicated by the expected return.

However, some investments or gambles are such that there is a one in a million chance of getting a $2.5 million payoff for a $1 bet. But for the other 999,999 times, you lose 100 per cent of your wager.

The expected return is a good 150 per cent. But if one were to bet a significant sum of one’s wealth on this kind of gamble, it is almost a certainty that one would be bankrupted before the big payoff comes along.

According to Kelly’s formula, two numbers should decide how much capital one should allocate to bet on an uncertain event: the probability of the bet being a winning one, and the payout. The formula is this: (Probability of bet being a winning one x (expected rate of return +2) -1) / (expected rate of return + 1).

So if you think that an investment has a 51 per cent chance of returning 20 per cent, then according to the formula, you should put 10 per cent of your capital in that investment. But if the probability of the investment yielding 20 per cent drops to 45 per cent or less, then you should not make any bet at all.

Meanwhile, a stock with half a chance of returning 30 per cent, the wager size should be 11.5 per cent of your portfolio.

This method forces an investor to seriously and thoroughly analyse the potential of a stock. And when he or she comes across a stock whose potential they think is severely unappreciated by the market, then they should have the conviction to commit a sizeable bet on it.

The prerequisite for this kind of approach is that the investor must have deep understanding of a stock and the industry it operates in, and have knowledge of how companies are valued by the market.

In a way, the world’s most successful investor, Warren Buffett, also subscribes to this strategy. He advocates focus investing, and to bet big on high probability events.

There have been studies done that, using Kelly’s formula one can minimise the expected time to reach a fixed fortune.

From the table, it appears that the probability of a favourable outcome carries a much bigger weight in determining how much one should put into an investment.


The Kelly formula was developed to solve similar problems in gambling where the outcome is either win or lose. And it assumes a 100 per cent loss when the outcome is unfavourable. In the stock market, one rarely lose 100 per cent of one’s investment in a single trade.

Also, a financial investor cannot completely rely on the number suggested by the Kelly formula as it does not take into consideration the possibility of a few available investment options.

In gambling, using Kelly’s formula can produce a rather volatile result. There is a one-third chance of halving the bankroll before it is doubled. According to some literature, a popular alternative is to bet only half the amount suggested, which gives three-quarters of the investment return with much less volatility.

Where money accumulates at 9.06 per cent compound interest with full bets, it still accumulates at 7.5 per cent for half-bets.

And as mentioned, this kind of strategy is applicable to those who know their ways around the stock market. But even for experts, putting 30 per cent of one’s portfolio in a stock with a 60 per cent probability of a 35 per cent return seems rather big a bet. The numbers should at best be used as a rough guide.

And for those who don’t have the time or the inclination to carry out in-depth studies of stocks, a diversified approach is perhaps safer.

The writer is a CFA charterholder. She can be reached at hooiling@sph.com.sg

Source: Business Times 24 Nov 07


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