interval At,

XAt = R, - , , (11.9)

where / expresses the time of the measurement. Generally At is allowed to vary from 7 days to 301 days. A risk-sensitive measure of trading model performance can be derived from the utility function framework (Keeney and Raiffa, 1976). Let us assume that the variable , follows a Gaussian random walk with mean XAt and the risk aversion parameter a is constant with respect to XAt The resulting utility ( ,) of an observation is - exp(-aXAt), with an expectation value of ~ = u(XAt) ( 2 - /2), where is the varianceof XAt. The expected utility can be transformed back to the effective return, Xeg = - log(-u)/a where

Xeff = XAt--(11.10)

The risk term acx\t/2 can be regarded as a risk premium deducted from the original return where err is computed by

°i, = ;-r(*L-xL)

Unlike the Sharpe ratio, this measure is numerically stable and can differentiate between two trading models with a straight-line behavior ( ( = 0) by choosing the one with the better average return.7 The measure Xeff still depends on the size of the time interval It is hard to compare Xeg values for different intervals. The usual way to enable such a comparison is through the annualization factor, where AAt is the ratio of the number of At in a year divided by the number of Afs in the full sample

Xejf.ann.Ar = AAt Xeff = X - - , , (11.12)

where X is the annualized return and it is no longer dependent on At. The factor ? has a constant expectation, independent of At. This annualized measure still has a risk term associated with At and is insensitive to changes occurring with much longer or much shorter horizons. To achieve a measure that simultaneously considers a wide range of horizons, a weighted average of several Xeg ann is computed with n different time horizons Af,, and thus takes advantage of the fact that annualized Xejf.ann can be directly compared,

v J21=l wiXeff,ann.Atj .

Xeff = -=- (11.13)

7 An example for the limitation of the Sharpe ratio is its inability to distinguish between two straight line equity curves with different slopes.

where the weights w are chosen according to the relative importance of the time horizons Atj and may differ for trading models with different trading frequencies. Generally, a is set to a =0.1 when the returns are expressed as a percentage. If they are expressed in numbers, a would be equal 10. The risk term of Xeg is based on the volatility of the total return curve against time, where a steady, linear growth of the total return represents the zero volatility case. This volatility measure of the total return curve treats positive and negative deviations symmetrically, whereas foreign exchange dealers become more risk averse in the loss zone and hardly care about the clustering of positive profits.

11.3.2 Reff. An Asymmetric Effective Returns Measure

A measure that treats the negative and positive zones asymmetrically is defined to be Reg, (Miiller et al., 1993b; Dacorogna et al., 2001b) where Reff has a high risk aversion in the zone of negative returns and a low one in the zone of profits, whereas Xeg assumes constant risk aversion. A high risk aversion in the zone of negative returns means that the performance measure is dominated by the large drawdowns. The Reff has two risk aversion levels: a low one, a+, for positive AR, (profit intervals) and a high one, a , for negative AR, (drawdowns),

«= \a+ ! -! (1U4)

l a for AR, < 0

where a+ < a . The high value of a reflects the high risk aversion of typical market participants in the loss zone. Trading models may have some losses but, if the loss observations strongly vary in size, the risk of very large losses becomes un-acceptably high. On the side of the positive profit observations, a certain regularity of profits is also better than a strong variation in size. However, this distribution of positive returns is never as vital for the future of market participants as the distribution of losses (drawdowns). Therefore, a+ is smaller than a . and we assume that a+ = (X-/4 and a = 0.20. These values are under the assumption of the return measured as percentage. They have to be multiplied by 100 if the returns are not expressed as percentage figures.

The risk aversion a associated with the utility function u(AR) is defined in Keeney and Raiffa (1976) as follows:

d2«

a = WAjj ( 5)

d(AR)

The utility function is obtained by inserting Equation 11.14 in Equation 11.15 and integrating twice over AR:

u(AR)

for AR > 0

p " ( .16)

X JL for AR < 0

The utility function u(AR) is monotonically increasing and reaches its maximum 0 in the case AR -* oo (infinite profit). All other utility values are negative. (The absolute level of is not relevant; we could add and/or multiply all values with the same constant factor(s) without affecting the essence of the method.) The inverse formula computes a return value from its utility:

AR = AR(u) =

-log(-"+M) for >---

10g(l- --< - ) ,

--aZ- for « < - £

(11.17)

The more complicated nature of the new utility definition, Equation 11.16, makes deriving a formula for the mean utility quite difficult and offers no analytical solution. Moreover, the Reg is dominated by the drawdowns that are in the tail of the distribution, not in the center. The assumption of a Gaussian distribution, which may be acceptable for the distribution as a whole, is insufficient in the tails of the distribution, where the stop-loss, the leptokurtic nature of price changes, and the clustering of market conditions such as volatility cause very particular forms of the distribution.

Therefore, the use of explicit utilities is suggested in the Reg algorithm. The end results, Reg and the effective returns for the individual horizons, will however, be transformed back with the help of Equation 11.17 to a return figure directly comparable with the annualized return and Xeg. The utility of the jh observation for a given time interval At is

,,; = u(RtJ - RtJ-A,) (11.18)

The total utility is the sum of the utility for each observation

= N.- (11.19)

EjU v;

In this formula, Nj is the number of observed intervals of size Af that overlaps with the total sampling period of size T and the weight vj is the ratio of the amount of time during which the jh interval coincides with the sampling period over its interval size Af. This weight is generally equal to one except for the first observation(s), which can start before the sample starts, and the last one(s), which can end after the sample ends. To obtain a lower error in the evaluation of the mean utility, different regular series of overlapping intervals of size Af can be used. The use of overlapping intervals is especially important when the interval size Af is large compared to the full sample size T. Another argument for overlapping is the high, overproportional impact of drawdowns on Reg. The higher the overlap factor, the higher the precision in the coverage of the worst drawdowns.