Risk of Ruin

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John Ansbacher

02 Jul, 2007

in Money Management

Optimal Risk Size Using the Kelly Criterion
Traders often ask how much they should risk on each transaction in order to ensure the highest possible returns and the lowest probability of depleting their account to zero. The problem is that the more you risk, the higher your potential yield, but this is coupled with greater potential for significant loss.

In allocating capital to a trading or betting strategy a trader can significantly improve his performance by ensuring that he risks the optimal amount per transaction.

What Is The Kelly Criterion?
The Kelly Criterion is a formula for calculating the percentage of a trader’s total risk capital that he should stake given his probability of winning and the odds offered on the bet.

The formula was originally published by J.L Kelly in 1956 and published in the Bell System Technical Journal. It was devised for minimising noise over long-distance telephone lines but it’s application to the speculator, whose market information may contain errors (‘noise’), was quickly realised.

There are a number of advantages to the Kelly Criterion including:

  1. Assuming (and one must be careful here) that money and bet sizes are continuously divisible, use of the formula leads to a zero risk of ruin.
  2. It enables the trader to adjust his risk size for every transaction in order to allocate more capital to higher probability events than to lower. Thus capitalising on the best bets available.
  3. It is guaranteed to maximise profit potential and minimise ruin given the odds and the probability of winning.

Calculating Bet Size
Now for a little bit of simple maths, to explain and demonstrate the formula.

Anyone familiar with Microsoft Excel should be able to do the calculation in a spreadsheet (I have pasted the Excel formula for the example below in a footnote to save you time).

 f* = bp - q / b

f* is the percentage of the available capital to risk

b represents the odds received on the bet (for example, 2/1 would be represented as 2)

p is the probability of winning the bet. This may require some guess work but more experienced traders are often good at figuring their percentage probability of being right. It just takes practice.

q is the probability of losing the bet (100% - p). So, if you have a 70% chance of winning, q would be 30%.

I recently placed a EUR/USD bet and considered that my probability of winning was at very least 30% and the odds quoted on the site were 3/1   =((3*30%)-70%)/3

I punched the numbers into my Kelly Criterion spreadsheet which told me that I ought to risk 6.67% of my available capital on this bet. If my estimation of the minimum probability of winning was right (and I am always very conservative in my guesses so I usually do well at this) then this was a great bet.

Even by winning such a bet just 30% of the time, one could make substantial profits over the course of many bets.

Risk Is A Matter Of Taste
The Kelly Criterion is mathematically optimal for picking risk size per bet but it does have a drawback that many traders like to adjust for.

A feature of it is that it provides a ‘mirror’ probability of losing a given percentage of your capital. When a series of bets are placed using the Criterion it gives the trader a 50% chance of at some time reducing his capital to 50% of the starting amount, a 40% chance or reducing it to 40%, a 10% chance of reducing it to 10% and so on. While he will probably make the losses back if he uses the model correctly this is just too much risk for many traders.

The solution to this is to calculate the risk amount per bet using the Kelly Criterion and then to divide it by a number depending on your risk appetite. The reduction in bet size does not decrease the long run returns linearly. For example, betting at 50% of the suggested amount will still pay the trader 75% of the returns that risking the full Kelly bet amount would have done but with far less volatility.

Why And Where To Use The Kelly Criterion
The Kelly Criterion provides the mathematically optimal balance of risk versus reward. Some traders are more concerned with risk and others with reward.

Betting more than the Kelly percentage is never optimal and will probably lead to long term depletion of capital and many traders are most comfortable at 50% or even 33.33% of the Kelly bet size. I am a fairly aggressive trader so choose to bet 100% of the Kelly size relative to the funds in each of my trading accounts.

If you have a tougher time determining the probability of success for each bet you make then Kelly betting is not for you. It is predicated on the ability to pick 'good bets'. You may however be interested in considering risk diversification and bet size in light of a model which does not require such knowledge. If so, see my article here: Beating The Risk Of Ruin.

Kelly betting is best deployed in trading scenarios where the stake and payout are known in advance. My interest in Kelly bet sizes is that the model is perfect for fixed odds financial betting and lends itself perfectly to Binarys, One Touches, No Touches, Time Switches and their respective ranges.

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while I appreciate author's good intentions (and knowledge) there are some objections. Kelly and shown risk calculating may be suitable for fixed-odds betting, but trading is unfortunately more complicated environment and cannot be 'mathematicized'. For example, one may have less than 50% succesfull trades and still be ahead (because he will gain more when right than lose when he is wrong). Also Kelly in trading is somewhat elusive because it is exact science, while trading is highly unstable environment where rules slightly change all the time. I experimented with Kelly alot and found no use of it on stock/forex mkt.

Another, risk of ruin calculation is also something suited for fixed-odds environment. In trading, if one uses fixed percentage MM, every loss will be smaller in terms of money so risk of ruin is theoretically non-existent. When we add that experienced trader rarely faces more than 8 losses in a row, our chances are even better. Trading cannot be painted or comprehended by using black-white only formulas.

Jul 05, 2007

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