**INSIDE THE NUMBERS**

Most professional traders and hedge fund investors know the monthly mean and standard deviation of their returns. These numbers are needed as inputs to the risk of ruin equation. The mean and standard deviation may be measured per month, per day, per week, per year or per trade. It is important to measure both the mean and standard deviation over the same time period. Do not mix the annual mean return with the standard deviation per trade.

Markets usually have fat-tailed distributions. This means that extreme events happen more often than you’d see in a random data set. The fat tail is often caused by variation in the standard deviation. The risk of ruin formulae are sensitive to standard deviation. It is dangerous to underestimate standard deviation because that will underestimate risk of ruin. An increase in standard deviation makes large losses more likely.

Monthly numbers work well in this formula. Few traders have enough years of data to use annual numbers. Typical trading systems make several trades in a month, so the monthly returns often have a distribution reasonably close to a normal bell curve.

For example, suppose a trader has a mean return of 6% per month, with a standard deviation of 13% (see “Risk of loss,” above). The risk of losing one standard deviation, or 13% of the account, is given by this formula:

R = e^(-2*a/d)

Where:

R =

risk of losing one standard

deviation

e =

2.71828, the base of the

natural logarithm

a =

average, or mean return,

in this case 6%, or 0.06. a must be a positive number, or else R is defined as 100%.

d =

standard deviation of returns,

in this case 13%, or 0.13. d must be a positive number between 0 and 1.

Plugging in the numbers, we find that

R = e^(-(2*0.06)/0.13)

R = e^-0.923

R = 0.397

There is a 39.7% chance of losing 13% of the account from any point on the equity curve. This trader will spend 39.7% of his time at least 13% below a prior maximum equity high.

Now suppose this same trader wants to calculate the probability of losing half of his account. From Cox and Miller, the equation is:

R = e^(-(2*a*z)/(d*d))

Where:

R =

risk of losing z fraction

of the account

e =

2.71828, the base of the

natural logarithm

z =

The fraction of the account that might be lost, in this case 50%,

or 0.50

a =

average, or mean return,

in this case 6%, or 0.06

d =

standard deviation of returns,

in this case 13%, or 0.13

Plugging in the numbers, we find that

R = e^(-(2*0.06*0.50)/(0.13*0.13))

R = e^-3.550

R = 0.029

There is a 2.9% chance of losing 50% of the account from any point on the equity curve. This trader will spend 2.9% of his time at least 50% below a prior maximum equity high.

This equation assumes that the trader cannot reduce the trade size as the account value falls. The monthly standard deviation remains at 13% of the original account size and the average return remains at 6% of the original account size.