frequency Figure 2-5 shows the change in the distribution of prices as the mean price (over shorter intervals) changes. This pattern indicates tliat a normal distribution is not appropriate for commodity prices, and that a log distribution would only apply to overall long-term distributions.

Choosing between Frequency Distribution and Standard Deviation

You should note that it is more likely that unreliable probabilities will result from using too little data than from the choice of method. For example, we might choose to look at the distribution of one month of daily data, about 23 daj-s; however, it is not much of a sample. The price or equity changes being measured might be completely different during the next month. Even the most recent five years of S&P data will not show a drop as large as October 1987.

FIGURE 2-5 Changing distribution at different price levels. A, B, and are increasing mean values of three shorter-term distributions.

Although we can identifj and measure skewness, it is difficult to get meaningful probabilities using a standard deviation taken on very distorted distributions. It is simpler to use a frequency dishibution for data with long tails on one side and truncated results on the other. To find the likelihood of retums using a frend sj-stem with a stop-loss, you can simply sort the data in ascending order using a ajreaddieet, then count from each end to find the extremes. You will notice that the largest I0°o of the profits cover a wide range, while the largest I0°o of the losses is clustered together.

A standard deviation is very helpful for giving some inihcation that a price move, larger than any we have seen in the data, is possible. Because it assumes a normally shaped curve, a large clustering of data toward one end will force the curve to extend fiirther. Although the usefiilness of the exact probabilities is questionable, there is no doubl that, given enough time, we will see price moves, profits, and losses that are larger than we have seen in the past.

Student t-test

Throughout the development and testing of a trading sj-stem we win want to know if the results we are seeing are as experted. The answer will keep referring back to the size of the sample and the amount of variance that is tj-pical of the data during this period. Readers are encouraged to refer to other sections in the book on sample error and chi-square test Another popular method for measuring whether the average price of the data is significantly different from zero, that is, if there is an underljing frend bias or if the pattem exhibits random qualities, is the shident t-test.

« of ptKC (haien

and where degrees of freedom = number of data items -1. The more frades in the sample, the more reliable the results The values of t needed to be significant can be found in Appendix 1, Table A1.2, "T-Distribution." The column headed ".10 gives the 90°o confidence level, ".05" is 95>o, and ".005" is 99.50o confidence.

If we separate data into two periods and compare the average of the two periods for consistency, we can decide whether the data has changed significantly. This is done with a 2-sample t-test:

where X. andSc arj± the averages of daia periCMls » <1 Z, r-, and Ui are «he variances of periods 1 and 2, and wi and rt arc rhc iiiiiiil x-i of data Icems in periods 1 aild 2. The tlcfrefsi tit bedum. Ofi ru.-<:ded to find the confidence levels in AJ-2. car, be caJculaced as:

JlL.JiL

The student t-test can also be used to compare the profits and losses generated by a trading sj-stem to show that the underljing sj-stem process is sound. Simply replace the data items by the average profit or loss of the sj-stem, the number of data items by the number

of trades, and calculate all other values using the profit/loss to get the student t-test value for the trading performance.

ST AM5ARDIZING RETURNS AND RISK

To compare one trading method with another, it is necessarj to standardize both the tests and the measurements used for evaluation. If one sj-stem has total returns of 50°o and the other of 250°o, we cannot decide which is best unless we know the duration of the test if the 50°o rdum was over 1 year and the 250°o return over 10 years, then the first one is best. Similarly, the rdum relative to the risk is crucial to performance as will be iHscussed in Chapter 21 ("Testing"). For now it is only important that retums and risk he annualized or standardized to make comparisons valid.

Calculatmg Retums

The calculation of rate of retum is essential for assessing performance as well as for many arbitrage situations. In its simplest form, the one-period rate of rdum R, or the holding period rate of retum is

where Po is the initial investment or startmg value, and P,l is the value of the investment after one period. In most cases, it is desirable to standardize the rdums by annualizing. This is particularly helpful when comparing two sets of test results, in which each covers a different time period. Although calculations on govemment instruments use a 360-dsy rate (based on 90-dsy quarters), a 365-dsy rate is common for most other pmposes. The following formulas show 365 dsj-s; however, 360 may be substituted.

The annualized rate of retum on a simple-interest basis for an investment over n dsj-s is

w-=(..r

The geometric mean is the basis for the compounded growth associated with mterest rates. If the initial mvestment is $1,000 (PO) and the ending value is $1,600 (P,) after 12 years (y = 12), there has been an increase of 60° o. The simple rate of retum is 5>o. but the compounded growth shows

Ending value - Startup value x (1 t Compounded retumj

The use of the standard deviation and compounded rate of return are combined to find the probability of a retum obeciive. In the following calculation,* the arithmetic mean of con-s retums is ln<l + Ky, and it is assumed that the retums are normally distributed.

where z=standardized vanable (can be looked up in Appendix Al) laigei value or raie<rfretum objenivc bnring imesunem value Rg - geometric average of periodic returns « = of periods

s=standard deviation of the logarithms of the quantities 1 plus the penodjc returns

Indexing Retums

The Federal Govemment has defined standards for calculating retums in the Futures Industry Commodity Trading Advisors (CTAs). This is simply an indexing of retums based on the current period percentage change in equity. It is the same process as creating any index, and it allows trading retums to be compared with, for exanple, the S&P Index or the Lehman Brothers Treasury Index, on equal footing. Readers should refer to the section later in this chapter, "Constmcting an Index."

Calculating Rid;

Although we would ahvaj-s like to think about retums, it is even more important to be able to assess risk. With that in mind, there are two tjpes of risk that are important for very different reasons. The first is catastrophic ride which will cause fetal losses or ruin. This is a complicated tjpe of risk, because it may be the result of a single price shock or a steadj deterioration of equity by being overleveraged. This form of risk will be discussed in detail later in the book.

Standard risk measurements are usefiil for comparing the performance of two sj-stems and for understanding how someone else might evaluate your own equity profile. The simplest estimate of risk is the variance of equity over a time interval commonly used by most inveshnent managers. To calculate the variance, it is first necessarj to find the mean retum, or the expected retum, on an inveshnent:

The most common measure of risk is variance, calculated by squaring the deviation of each retum from the mean, then multiplying each value by its associated probability

variance = <

The sum of these values is called the variance, and the square root of the variance is called the standard deviation. This was given in another form in the early section "Dispersion and Skewness."

Mandard deviation=Vp,(«, Elfdf * ,%- { - -Em

IWs and ottier very clear explanaKons of returns ean be found in Peter L. Bernsteins Tbe Portable MBA in Investment (John Wley & Sons, New Yoik, 1995).