Risk Management Theory

Nobody wins all the time – just ask Leeds United – but losing needn’t lead to disaster. Clem Chambers examines risk management in trading and gambling, from maths-based strategies to portfolio theory. It’s a truism to say that if you thought you were wrong about a horse or the direction of the FTSE, you wouldn’t make the bet. But inevitably there are times you will be wrong, and will consequently make a loss. The key to making a profit on balance is to employ good risk management, thereby avoiding ‘gambler’s ruin’ and optimising returns.

Stop losses are often proffered as the first line of defence in risk management – supposedly ensuring that a trader is profitable by guaranteeing losses don’t eat into his capital. However it’s never that simple. In a vacuum, a stop loss system is at best a neutral strategy but in practise, due to costs, a slightly negative one. There is no free money to be made in the market and a robotic stop loss system cannot add to a trader’s returns unless he is already trading so badly that he is doomed anyway.

Doing the random walk. Oi!

All that follows is underpinned by the idea that markets are usually highly random. Randomness isn’t a term many traders or investors like to hear – how can a trader predict the market when it’s random? However, a look at stock price histories will show bell curves of distribution again and again. Bell curves are the signature of randomness and plot the spread of results created by a process. The bell curve represents a statistical evenness suggesting that the process is generating random outcomes, similar to dice being rolled on a tabletop.

One example, taken from my records, is a graph of five years’ worth of Dow moves between the opening of business and the previous night’s close. This bell-curved distribution is the footprint of what physicists call ‘the random walk’. To view this graph, click the ‘How To’ button.

However, despite this randomness, by reading and taking advantage of short-term trends, it is still clearly possible to make a profit in line with the risks you are taking. If it wasn’t, you would put your money in the bank and wait for retirement. The key to reaping return while avoiding disaster is risk management. Avoiding disaster sounds like a good idea, yet many traders simply don’t. This is often because they either don’t understand the risks or the full, potential ramifications of taking them.

At this point I’d like to introduce the martingale. This is a technique that involves increasing stake size as a run of losses extends. Classically, the martingale comes into play with gambling games like blackjack or roulette, where a stake is doubled after every sequential loss. When a win is made, the player gets back his losses and earns a ‘one-unit’ win; the stake size is then reset. In theory, with infinite money you’ll always end up winning in the end.

The martingale and its many permutations are popular systems for trading. However the model is flawed as, in reality, you are liable to run out of money before your luck turns. Bad luck simply happens with greater regularity than the capital required to fund the operation. I’ve actually witnessed red come up 23 times successively on a roulette table. If I’d been running a martingale, it would have cost me £4 million to win back my £1 – and even if I had £4 million, the table limit would defeat me.

Another classic example of the perils of misunderstanding risk management is the early 20th Century phenomenon of the ‘bucket shop’.

These were stock market day trading systems run by unlicensed trading establishments. Their trading rules were simple. You could, for example, buy a £100 share on margin for £2 (50 times leverage). This £2 also acted as a stop loss. If the share went down £2, the position was closed and you lost your margin money. If the share went to £110 or higher, you would make the amount by which the price had surpassed £100 (£14 if it reached £114, £50 for £150, etc).

A fixed downside with an infinite upside looks like too good an offer to refuse, but volatility and probability are formidable foes in this set-up. Even if the odds of this trade are slightly in your favour, on a random basis, a price will at some point go below £100 around nine times out of ten. So with a mere £2 margin, you can get stopped out very easily. If a price has a normal daily range of four points even in a sideways trend, you are going to be closed out every time. It takes a very strong jump for a stock not to hit £98 at some point in the future of the proceedings. One touch however and you’re out. This is an example of gambler’s ruin in microcosm, and it is essential to avoid it.

The view from Kelly’s eye
So if stop losses by themselves are not the answer, what is? Visualise a roulette table and imagine the wheel has 35 reds on it and one black, yet the payout is still even money. On this table we could make lots of money very quickly indeed, yet we can still lose everything. If we have 100 £1 chips and play £1 at a time we’ll make £1, 35 times out of 36. That’s pleasant but hardly optimal – after all, we could get very rich very quickly at this table. Place £2 down and we would clearly make twice as much per go. We could place £4 or £10 down at each turn and increase our profit each time. However, the higher your stake, the higher the probability of a run of bad luck taking all your money.

So, if we assume that this uncommonly gambler-friendly wheel will only be open for one night and we only have £100 to play, how do we optimise returns? We can be assured that if we do find a wrinkle in the market it won’t last forever, so extracting value will always be against the clock. The question for us is how to make as much as possible that evening.

The answer is not to put down the whole of your money on the red each time as although the chance of black coming up is low, it will come up. If it did we’d lose all our money and experience gambler’s ruin. So somewhere between £1 and £100 a go, there’s an optimal stake threshold, which maximises your return but avoids gambler’s ruin.

In equities, this stake limit is often said to be between 2.5% to 5% of your investment capital. In options trading it’s suggested that a trader needs capital of three times his maximum losing streak. The losing streak divided by the money lost would therefore dictate the position size.

Mathematically, the rule – known as Kelly’s Optimisation Model – is a lot more exact. This model is often used in gambling but is just as applicable to investment. It assumes that the size of a positive outcome is the same as for a negative outcome and as such is a useful outlet for stop losses.

If the probability of the trade being good is 52.5% (5.25 times out of 10) then the Kelly model says you should put: (2×0.525)-1 of your capital on the trade. That is 1.05-1 of your capital or 5%. If your chances were 90% then you would stake: (0.9×2)-1, which is 1.8-1 or 80% of your capital.

While it is interesting to know exactly where results are optimal, it’s more important to realise that if you stake more than the sum given by this equation, you are guaranteed to suffer gambler’s ruin. It might take time, but it’s a mathematical certainty.

If you were using Kelly’s Optimisation, the size of your positions would be right on the make-or-break line. This is quite a torrid place, so many gamblers simply back 50% of this amount, on the basis that they will still make plenty of money but avoid the ulcer. In the case of our funny roulette wheel, it only takes an hour or so to own the casino whether you back a half or a whole of the amount.

Any portfolio in a storm
Here, we’ve just avoided the disaster of compound events. The sister technique to this is portfolio management, where resources are spread in such a way that they are engaged in risk yet protected from what is known as ‘unsystemic risk’. An example of unsystemic risk would be that our wonky roulette table attracts the casino management’s attention and we end up wearing a concrete overcoat. Portfolio management prevents us suffering gambler’s ruin caused by a single disaster.

While Kelly’s optimisation applies to the allocation of resources for a single position, portfolio management describes the type of simultaneous positions to carry. With diversification, more money can be safely deployed and therefore a better return made.

Portfolio theory considers that a market has a systemic rate of return. This rate is, on average, proportional to the risk involved with the instrument invested in. For example, return on government bonds will be small as the risk is low; corporate bonds are riskier, so returns will be higher. As risk levels grow for any instrument, so does the chance of a big loss. The problem is picking the risky stock that won’t crash.

Portfolio theory’s solution to this is to buy a bundle, thereby capturing the average result, as winners will counterbalance losers. Advocates of the random walk state that this is the best you can do in any event as markets, like the weather, are impossible to predict.

Not putting all your eggs in one basket is not the most revolutionary of maxims, but depending on the volatility – and hence, risk – of an asset, the more varieties of them that are held, the more certain your money is of growing at the ‘systemic return’. In the case of equities, this is considered to be 30 or more stocks in a portfolio.

You could, for instance, pick a portfolio of tech stocks. In this way, you would receive the returns of the tech sector, without leaving yourself vulnerable to wipe-out on a single stock. Alternatively, you could capture returns on a whole market, or perhaps the returns of a collection of markets, or even a broad spread of different asset types. Portfolio theory enables you to slice and dice the risk return potential of all kinds of markets.

While creating a portfolio might feel like removing the thrill of risk, it can be used to increase returns, by providing a structure for holding extremely risky investments. To capture these returns, the portfolio must be wider. That way, a few huge winners will outweigh the large proportion of losses. This is not free money; you are being rewarded for taking risk and providing liquidity.