Distribution of returns generated by stochastic exposure 75

directional bets. A good illustration is provided by the Bankers Trust 1994

annual report reproduced in Chew (1996: 210). On the one hand, the statistical

signiﬁcance of the trading proﬁts reported in both papers, Jorion (2001) and

Berkowitz and O’Brien (2001), tend to be extremely high as measured by the

T-statistics which roughly vary between 3.8 and 22. On the other hand, a pure

directional bet is considered as very successful when the T-statistic reaches 2.

This can be seen by studying the performance of alternative investments over

long periods of time (see Managed Account Reports

1

). Although interesting,

studies on banking proﬁts can be difﬁcult to interpret. Indeed, a too high VaR

estimate may reﬂect a change of management policy rather than a method-

ology issue. Implementing corrective actions when directional losses start to

develop is not uncommon.

Within asset management, there is a growing literature which modelizes

the effect of stochastic weights within portfolios. Directional trading rules are

typical examples of strategies affecting the distribution of return (Acar and

Satchell, 1998). Extension of this work and its relevancy to hedge fund man-

agement has been investigated by Lundin and Satchell (2000). Generalization

to active fund management and relative returns has been recently formulated

in Hwang and Satchell (2001).

The purpose of this chapter is to highlight the direct effect of uncertain

exposures on Value at Risk calculations both theoretically and empirically.

Whereas corporations’ cash ﬂows are difﬁcult to analyse for conﬁdentiality

reasons and banking proﬁts are unﬁltered, we have chosen to concentrate

our examples on popular strategies used by active investors and directional

traders. They encompass both discrete and continuous distributions of cash

ﬂows. The market timing ability is best rendered by the performance returns,

which are the result of the by-product between the stochastic exposure and

the market returns. Deans (2000) provides numerous examples of proﬁt and

loss calculation for backtesting. He especially recommends the use of proﬁt

and loss histograms to detect if the distribution is approximately normal,

skewed, fat-tailed or if it has other particular features. This is why section 3.2

discusses the distribution of performance returns when the exposure is stochas-

tic. Section 3.3 quantiﬁes the implications for Value at Risk calculations.

Section 3.4 illustrates trading returns in the futures markets. Section 3.5 sum-

marizes our ﬁndings and proposes new avenues for future research.

3.2 DISTRIBUTION OF PERFORMANCE RETURNS

It is clear that no money manager or trader has control over the market

returns denoted X. The best a trader can do is to time his entry and exit

1

http://www.marhedge.com/

76 Performance Measurement in Finance

in the market via his exposure, labelled B (long, squared or short). In other

words, the directional views are captured by the changing weights over the

period. The market timing ability is best rendered by the performance returns

Z = BX. Our goal here is not to forecast the excess returns generated by

active timers but rather to quantify the risk taken by active money managers

under the random walk assumption. Then we will assume no forecasting

ability, which implies that active timing either based on discretion or trading

rules cannot generate proﬁts above and beyond the buy-and-hold returns. In

statistical terms, this means that there is independence between the exposure

B and the forthcoming returns X. Despite violating the inner purpose of using

a forecasting strategy, the random walk assumption is nevertheless useful in

giving us a proxy for VaR calculations. Indeed, it is critical for performance

returns to include several different contributions other than those related to

market risk measurement, namely leverage and timing.

We are interested in establishing the distribution of the performance returns

resulting from the product of two independent random variables. B,the

stochastic exposure, follows either a discrete or continuous distribution. X,

the market returns, is supposed in this section to follow a normal distribution

with mean μ

x

and volatility σ

x

.

3.2.1 Discrete exposure

Let’s suppose that the exposure B =

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0 with probability p

0

b

1

with probability p

1

b

2

with probability p

2

.

.

.

b

n

with probability p

n

with

n

i=0

p

i

= 1,p

i

≥ 0,i= 0,...,n and b

i

= 0,i= 1,...,n

In this case, the performance returns Z = BX satisfy:

Prob[Z<z] =

n

i=1

p

i

z − b

i

μ

x

|b

i

|σ

x

if z<0

Prob[Z = 0] = p

0

Prob[Z<z] = p

0

+

n

i=1

p

i

z − b

i

μ

x

|b

i

|σ

x

if z>0

where is the cumulative function of a normal distribution N(0, 1).

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