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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