SSD CONSISTENT CRITERIA AND COHERENT RISK MEASURES

Condition (5) represents the required convexity properties to model diversification
advantages. Note that the second cumulative distribution functions
F 2 (X, r) are convex with respect to random variables X [13]. Hence,
taking two random variables Y ′ and Y ′′ both with the same distribution as
X one gets F 2 (αY ′ + (1 − α)Y ′′ , r) ≤ F 2 (X, r) for any 0 ≤ α ≤ 1 and any
r ∈ R. Thus, αY ′ + (1 − α)Y ′′ ≽ RS
X and convexity of ϱ(X) is necessary
to meet the requirement (5).
The concept of separation risk measures properties following Theorem 1
is applicable while considering general (arbitrary) distributions. It may be,
however, adjusted to some specific classes of distribution. In particular, we
will show that it remains valid for a class of symmetric distributions. Indeed,
a more subtle construction
F 1 (Z, r) =
{
F1 (X, r), r < µ(Y )
F 1 (X, r + 2(µ(X) − µ(Y ))), r ≥ µ(Y )
(7)
preserves symmetry of the distribution thus leading us to the following assertion.
For any symmetric random variables with X ≽ SSD
Y , there is a
symmetric random variable Z such that X ≽ F SD
Z ≽ RS
Y . One may also
notice that the ‘a.s.’ characteristic of the FSD relation may be, respectively,
enhanced for symmetric distributions.
It follows from the majorization theory [6, 10] that in the case of simple
lotteries constructed as random variables corresponding n-dimensional real
vectors (probability 1/n is assigned to each coordinate if they are different,
while probability k/n is assigned to the value of k coinciding coordinates) a
convex, positively homogeneous and translation invariant (dispersion type)
risk measure is SSD 1-safety consistent if and only if it is additionally expectation
bounded (3). We will demonstrate this for more general space
of lotteries. Hereafter, a lottery is a discrete random variable with a finite
number of steps.
Lemma 3 Lotteries X and Y satisfies X ≽ RS Y if and only if F −2 (X, p) ≥
F −2 (Y, p) for all p - cumulative probability of a step of F 1 (X, α) or F 1 (Y, α)
and F −2 (X, 1) = F −2 (Y, 1).
Proof. From quantile characterization of SSD we have inequality for all
p ∈ (0, 1). On the other hand, we can see that p - cumulative probability
of steps are sufficient. Let p 1 , p 2 , . . . , p m - cumulative probability of the
steps of F 1 (X, α) or F 1 (Y, α), and let c ∈ (p i , p i+1 ). Then, F −2 (X, c) =
F −2 (X, p i ) + (c − p i )F −1 (X, p i+1 ) = F −2 (X, p i+1 ) − (p i+1 − c)F −1 (X, p i+1 ).
Hence, F −2 (X, c) ≥ F −2 (Y, c) whenever F −2 (X, p i ) ≥ F −2 (Y, p i ) and
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