Risk Management and Value Creation in ... - Arabictrader.com

Capital Structure in Banks 217
Theoretical Concerns Schröder was one of the first 390 to point out that VaRbased
risk measures have a serious shortcoming in that they lack a sound
theoretical foundation. 391 We can define the so-called lower partial moment
(LPM) of a distribution as:
t
n
LPMn() t =
∫
( t −X) f ( X)
dX
(5.39) 392
where t = Target (minimum) return
X = Realized return
n = Moment of the distribution
f = Probability density function of the returns X
−∞
The moment n of the distribution determines (theoretically) the type of
the utility function used. 393 For instance, for n = 0, 394 this approach 395 assumes
a risk-neutral investor who is only interested in the probability of
falling short of the target minimum return t, ignoring the extent (or severity)
of this event when it occurs. For n = 1, we consider a risk-averse investor
who is interested in both the probability and the extent (severity) of the
actual return falling short of the target return t. It therefore calculates
the expected value of the shortfall. 396 For n = 2, the result is similar to the
semivariance and is, therefore, also often called target semivariance. 397
Replacing the target return t with VaR α
, the value at risk at the (1 – α)
confidence level, it is easy to show, because:
– VaR
∫
LPM0 (– VaR ) = f ( X) dX = F(– VaR ) = %,
α α
α
– ∞
that
–VaR α
= F -1 (LPM 0
) (5.40)
where F
F -1
= Cumulative probability function
= The inverse of the cumulative probability function
390 At least in discussing this in the context of VaR approaches.
391 See Schröder (1996), pp. 1–2. Obviously, Markowitz (1959) already points into
the same direction.
392 See Fishburn (1977), p. 116.
393 See Wittrock (1995), p. 43.
394 Obviously, this is the least restrictive shortfall risk measure possible.
395 As can be easily seen, LPM 0
reduces to the (cumulative) probability distribution.
396 In the above methodology, we would call this expected loss.
397 See Copeland and Weston (1988), p. 152. This measure is also suitable for riskaverse
investors.