Risk measures

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Risk measures

Proofs and additional remarks

Christian Y. Robert

ISFA - Université Lyon 1

October 2011

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⊲ COMONOTONIC RISKS D^EFINITIONS

1. A set S in R² is said to be comonotonic, if, for all (y₁, y₂) and (z₁, z₂) in this set, y_i < z_i for some i implies y_j ≤ z_j for j = i.

◦ Notice that a comonotonic set is a ‘thin’ set, in the sense that it is contained in a one-dimensional subset of R²_.

2. When the support of a random vector is a comonotonic set, the random vector itself and its joint distribution are called comonotonic.

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PROPOSITION :

1. A random vector (X, Y ) is comonotonic if and only if X and Y may be written as non-decreasing functions of the same random variable.

2. A random vector (X, Y ) is comonotonic if and only if

P_(X _≤ _{x, Y} _≤ _{y) = min}_{P_(X _≤ _x),P_(Y _≤ _y)}

for all x, y ∈ R_.

3. A random vector (X, Y ) is comonotonic if and only if (X, Y ) =_d (F_X⁻¹(U), F_Y⁻¹(U))

where F_X⁻¹ stands for the quantile function of X (see below), and U is a random variable that is uniformly distributed over the unit interval (0,1).

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⊲ COMPARISONS OF RISKS D^EFINITION

Let Y be a set of univariate distribution functions. The binary relation is a partial order on Y if for any elements X with df F_X, Y with df F_Y and Z with df F_Z in Y, the following properties hold :

(i) If XY and Y Z then XZ (transitivity).

(ii) XX (reflexivity).

(iii) If XY and Y X then X = Y (antisymmetry).

If, in addition, for any given pair X and Y of elements of either XY or Y X holds, then is said to be a total order.

Remark : We write XY but we actually mean F_XF_Y . In other words, when we say that a risk X is smaller than a risk Y for the stochastic order relation, we assert that this ordering holds for the respective dfs of these risks. Therefore, the joint distribution of X and Y is irrelevant ; only their marginal distributions are important.

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⊲ First-order stochastic dominance

P^ROP : Risk Y dominates risk X stochastically at first order if and only if there exist random variables X^′ =_d X and Y ^′ =_d Y such that P_(X^′ _≤ _Y ^′_{) = 1.}

Remark : If Y _DS1 X, then F_Y (d) ≤ F_X(d), ∀d ∈ R_.

If moreover F_X and F_Y are increasing df, we have F_Y⁻¹(F_Y (d) = d, F_X(X) =_d U where U is uniformly distributed over the unit interval (0,1) and F_Y⁻¹(U) =_d Y (see below).

Therefore X = X^′ and Y ^′ = F_Y⁻¹(F_X(X)) are desirable random variables since X ≤ Y ^′ a.s.

by the previous relation.

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PROP : Risk Y dominates risk X stochastically at first order if and only if E_[u(−X_)] _≥ E_[u(−Y _)]

for all non-decreasing functions u (such that the expectations exist).

PROOF : First note that, letting v(x) = −u(−x), the condition : E_[u(−X_)] _≥ E_[u(−Y _)]

for all non-decreasing functions u is equivalent to the condition : E_[v(X_)] _≤ E_[v(Y _)]

for all non-decreasing functions v.

The ⇐ part is obvious since F¯_X(z) = E_[I

{X>z]] and the function x → I

{X>z] is non-decreasing for any z. To get the converse implication, it suffices to invoke the previous proposition and to write

E_[v(X_{)] =} E_v(X^′₎ _≤ E_v(Y ^′₎ ₌ E_[v(Y _)]_.

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PROP : If X and Y have density propobability functions such that there exists a real number c and :

f_X (d) ≥ f_Y (d) for d ∈] − ∞, c) f_X (d) ≤ f_Y (d) for d ∈ [c,∞[.

then Y _DS₁ X.

P^ROOF : For x < c, we get F_X(x) =

_x

−∞f_X (u)du ≥

_x

−∞ f_Y (u) du = F_Y (x) For x > c, we get

F_X(x) = 1 −

_∞

x f_X (u)du ≥ 1 −

_∞

x f_Y (u)du = F_Y (x) and this concludes the proof.

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⊲ Second-order stochastic dominance

PROP : Risk Y dominates risk X stochastically at second order if and only if E_[u(−X_)] _≥ E_[u(−Y _)]

for all non-decreasing and concave function u (such that the expectations exist).

PROOF : First note that, letting v(x) = −u(−x), the condition : E_[u(−X_)] _≥ E_[u(−Y _)]

for all non-decreasing concave functions u is equivalent to the condition : E_[v(X_)] _≤ E_[v(Y _)]

for all non-decreasing convexe functions v.

The ⇐ implication is obvious since the function x → (x − t)₊ is convex for all t ∈ R. To get the converse, note that every continuous function v convex is the limit

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of an increasing sequence of functions : v_n(x) = α₁ + α₂x +

n j=0

β⁽ⁿ⁾_j

x − t⁽ⁿ⁾_j

+

with β⁽ⁿ⁾_j ≥ 0. It allows us to write E_[v_n_(X_{)] =} _α₁ ₊ _α₂E_[X_{] +}

n j=0

β⁽ⁿ⁾_j E

X − t⁽ⁿ⁾_j

+

≤ α₁ + α₂E_[Y _{] +}

n j=0

β⁽ⁿ⁾_j E

Y − t⁽ⁿ⁾_j

+

= E_[v_n_(Y _)]

for every n. Taking the limit (Monotone convergence theorem) yields E_[v(X_)] _≤ E_[v(Y _)].

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PROP : Risk Y dominates risk X stochastically at second order if and only if there exists a random variable D such that :

X + D =^d Y and E_[D|X_] _≥ ₀ _a.s.

P^ROOF : The ⇐ implication is derived by using the conditional Jensen’s inequality E_[(Y ₋ _d)₊_{] =} E_[(X ₊ _D ₋ _d)₊_{] =} E_X_[E_[ _(X ₊ _D ₋ _d)⁺

+|X

≥ E_X_[(X ₊ E_[D|X_] ₋ _d)₊ _≥ E_[(X ₋ _d)₊_].

The other implication is difficult to prove.

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PROP : If E_[X_] _≤ E_[Y _] and if there exists a real number c such that F_X (d) ≤ F_Y (d) for d ∈] − ∞, c)

F_X (d) ≥ F_Y (d) for d ∈ [c,∞[

then Y _DS₂ X. PROOF : Note that

E_{[ (X} ₋ _d)

+] =

_∞

d (x − d)dF_X (x)

= −[(x − d)(1 − F_X (x))]^∞_d +

_∞ d

F¯_X(x)dx =

_∞ d

F¯_X(x)dx π^′_X(d) = −(1 − F_X(d)) = −F¯_X(d).

Moreover lim_d→∞E_{[ (X} ₋ _d)

+] = 0 and since E_{[ (X} ₋ _d)

+] +d = E[max(X, d)]

d→−∞lim

E_{[ (X} ₋ _d)

+] + d = E_[X_].

Let us consider the function φ(d) = π_Y (d) − π_X(d). We have lim_d→−∞ φ(d) = E_[Y _] ₋ E_[X_] _≥ _0, _lim_d→∞ _{φ(d) = 0} _and _φ^′_{(d) =} _F_Y _(d) ₋ _F_X_(d).

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PROP : If Y _DS₁ X, then Y _DS₂ X. P^ROOF : Y _DS₁ X if and only if

E_[u(−X_)] _≥ E_[u(−Y _)]

for all non-decreasing function u (such that the expectations exist). Hence the inequality holds if u is a non-decreasing and concave function. So it is clear that Y _DS₂ X.

Note that two risks may be stochastically comparable at second order but not at first order.

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⊲ PROPERTIES OF RISK MEASURES

P^ROP : Π satisfies the properties 1) Monotonicity, 2) Objectivity, iff it satisfies the Invariance by first-order stochastic dominance property.

PROOF : The ⇐ implication is derived by noting that if P_(X _≤ _Y _{) = 1} _then P_{(X > d)} _≤ P_{(Y > d)} _{for all} _d _∈ R and hence the Monotonicity property is satisfied. Moreover if X =_d Y then X _DS₁ Y and Y _DS₁ X. By the first-order stochastic dominance property, we deduce that Π(X) = Π(Y ).

The ⇒ part is proven by noting that X _DS₁ Y if and only if there exist random variables X^′ =_d X and Y ^′ =_d Y such that P_(X^′ _≤ _Y ^′) = 1. By Monotonicity property Π(X^′) ≤ Π(Y ^′) and by the Objectivity property Π(X) ≤ Π(Y ).

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PROP : If Π satisfies the Invariance by second-order stochastic dominance property, then it satisfies the Invariance by first-order stochastic dominance property.

PROOF : The proposition is proven by noting that, if Y _DS1 X, then Y _DS2 X. Indeed, assume that X _DS₁ Y , then Y _DS₂ X and by the Invariance by second- order stochastic dominance property, we deduce that Π(X) ≤ Π(Y ). Therefore we have shown that

X _DS₁ Y ⇒ Π(X) ≤ Π(Y ).

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PROP : Assume that Π is a risk measure that satisfies the Positive homogeneity property. Π satisfies the Convexity property iff it satisfies the Subadditivity property.

PROOF : Assume that for any positive constant c and for all risks X, Π(cX) = cΠ(X).

i) ⇒ part : take α = 1/2 in the Convexity property, 1

2Π(X + Y ) = Π

1

2X + 1 2Y

≤ 1

2Π(X) + 1

2Π(Y ) to derive the Subadditivity property.

ii) ⇐ part : for α ∈ [0,1],

Π(αX + (1 − α)Y ) ≤ Π(αX) + Π((1 − α)Y ) = αΠ(X) + (1 − α)Π(Y ).

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PROP : Assume that Π is a risk measure that satisfies the Convexity property and Π(0) = 0. Π satisfies the Positive homogeneity property iff it satisfies the Subaddi- tivity property.

PROOF : If Π is a risk measure that satisfies the Convexity property, then t → Π(tX)/t is a non-decreasing function for t > 0, since by taking 0 < t₁ < t₂, and α = t₁/t₂,

Π(t₁X) = Π(αt₂X + (1 − α) × 0) ≤ αΠ(t₂X) + (1 − α)Π(0) = t₁

t₂Π(t₂X).

i) ⇒ part : obvious since

Π(X + Y ) = 2Π

1

2X + 1 2Y

≤ Π(X) + Π(Y ).

ii) ⇐ part : for k ∈ N _and _k _≥ _2, _Π(kX₎ _≤ _kΠ(X_{), i.e.} _Π(kX_)/k _≤ _Π(X_{). But} since Π is a risk measure that satisfies the Convexity property, t → Π(tX)/t is a non-decreasing and therefore it must be constant for t > 0.

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PROP : If Π satisfies the Monotonicity property and the “No unjustified loading”

property, then it satisfies the “Non-excessive loading” property.

PROOF : Since P_(X _≤ _max[X]) = 1, we deduce that

Π(X) ≤ Π(max[X]) = max[X] by the “No unjustified loading” property.

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PROP : If Π satisfies the properties 1) “Non-excessive loading”, 2) Convexity, then it satisfies the Monotonicity property.

PROOF : For α ∈ (0,1)

Π(αX) = Π

αY + (1 − α) α

(1 − α)(X − Y )

≤ αΠ(Y ) + (1 − α)Π

α

(1 − α)(X − Y )

.

If P_(X _≤ _Y ) = 1, then max[X − Y ] ≤ 0, Π α(1 − α)⁻¹(X − Y ) ≤ 0 and Π(αX) ≤ αΠ(Y ).

Let α ր 1 to conclude.

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PROP : If Π satisfies the properties 1) Objectivity, 2) “No unjustified loading”, 3) Convexity, 4) Convergence in distribution, then it satisfies the “Non-negative loading”

property.

P^ROOF : Let X, X₁, X₂... be iid random variables. First note that by the convexity property

Π

1

2X₁ + 1 2X₂

≤ 1

2Π (X₁) + 1

2Π (X₂) = Π (X).

Let X¯_k = (X₁ + ... + X_k)/k. We show by induction and the convexity property that, for any k ∈ N_, _Π X¯_2k ≤ Π (X) since

X¯_2k+2 = k

k + 1X¯_2k + 1 k + 1

1

2X_2k+1 + 1

2X_2k+2

. Now by the law of large numbers X¯_k →P_,d E_[X_] _and

Π X¯_2k → Π (E_[X_{]) =} E_[X_] and hence Π (X) ≥ E_[X_].

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PROP : If Π satisfies the properties 1) Objectivity, 2) Monotonicity, 3) Convexity, 4) if (X_n) converges in distribution to X then Π(X_n) → Π(X), then the risk measure does not depend on risks.

PROOF : Fix two numbers a < b. Define X₀ = b and for n = 1,2,3, ... let X_n = a + 2ⁿ(b − a)I

U∈[0,2⁻ⁿ)

X_n^′ = a + 2ⁿ(b − a)I

U∈[2⁻ⁿ,2⁻ⁿ⁺¹)

where U is a random variable that is uniformly distributed over the unit interval (0,1). Thus X_n =_d X_n^′ ,

X_n = 1

2 X_n+1 + X_n+1^′ and X_n →_d a. Convexity and law-invariance imply

Π(X_n) = Π

1

2 X_n+1 + X_n+1^′ ≤ 1

2Π(X_n+1) + 1

2Π(X_n+1^′ ) = Π(X_n+1).

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Thus n → Π(X_n) is an increasing sequence. Therefore monotonicity and convergence in distribution of the risk measure imply

Π(a) = lim

n→∞Π(X_n) ≥ Π(X₀) = Π(b) ≥ Π(a).

Thus Π(a) = Π(b) = π and by monotonicity we get for any X with a ≤ X ≤ b that Π(X) = π.

As a and b are arbitrary, this implies that Π is constant on the set of all bounded random variables.

Finally approximate an unbounded random variable Y by the sequence Y_n = (Y ∧ n)∨ −n of bounded random variables to extend to unbounded random variables too.

Remark : X_n →_d a but max[X_n] →_d ∞ = a!

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PROP : If Π satisfies the properties 1) Objectivity, 2) Comonotonic additivity, 3) Invariance by second-order stochastic dominance, then it satisfies the Subadditivity property.

P^ROOF : Let X and Y be two random variables, and U be a random variable that is uniformly distributed over the unit interval (0,1). Set

X^c = F_X⁻¹(U) and Y ^c = F_Y⁻¹(U).

For d = d₁ + d₂, we have

(x + y − d)₊ = ((x − d₁) + (y − d₂))₊ ≤ ((x − d₁)₊ + (y − d₂)₊)₊

= (x − d₁)₊ + (y − d₂)₊. Let us now choose

d₁ = F_X⁻¹c(F_X^c_+Y ^c(d)) and d₂ = F⁻¹

Y ^d (F_X^c_+Y ^c(d)) and note that, for any d where F_X^c_+Y ^c is increasing,

d₁ + d₂ = (F_X⁻¹c + F_Y⁻¹c )(F_X^c_+Y ^c(d)) = F_X⁻¹c+Y ^c(F_X^c_+Y ^c(d)) = d.

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It follows that

E_[(X ₊ _Y ₋ _d)₊_]

≤ E_[(X ₋ _d₁₎₊_{] +} E_[(Y ₋ _d₂₎₊_]

= E_[(X^c ₋ _F⁻¹

X^c(F_X^c_+Y ^c(d)))₊] + E_[(Y ^c ₋ _F⁻¹

Y ^d (F_X^c_+Y ^c(d)))₊]

= E_[(F⁻¹

X (U) − F_X⁻¹_c(F_X^c_+Y ^c(d)))₊] + E_[(F⁻¹

Y (U) − F⁻¹

Y ^d (F_X^c_+Y ^c(d)))₊]

= E_[(F⁻¹

X + F_Y⁻¹)(U) − (F_X⁻¹_c + F⁻¹

Y ^d )(F_X^c_+Y ^c(d)))₊]

= E_[(X^c ₊ _Y ^c ₋ _d)₊_] and then

X + Y _DS₂ X^c + Y ^c.

If the risk measure Π is invariant by second-order stochastic dominance and is additive for comonotonic risks, then

Π(X + Y ) ≤ Π(X^c + Y ^c) = Π(X) + Π(Y ), which proves the stated result.

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⊲ _VaR

There are basically two ways to define a generalized inverse for a distribution function.

DEFINITION

Given a df F, we define the inverse functions F⁻¹ and F⁻¹⁺ of F as F⁻¹(α) = inf{x ∈ R _: _F_(x) _≥ _α} _{= sup{x} _∈ R _: _F_(x) _{< α}}

and

F⁻¹⁺(α) = inf{x ∈ R _: _F_(x) _{> α}} _{= sup{x} _∈ R _: _F_(x) _≤ _α}

for α ∈ [0,1], where, by convention, inf ∅ _{= +∞} _and _sup∅ ₌ _−∞.

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One can check that :

1. F⁻¹ and F⁻¹⁺ are both non-decreasing (they are continuous everywhere, except on an at most countable set of points) ;

2. F⁻¹ is left-continuous while F⁻¹⁺ is right-continuous ;

3. F⁻¹(α) = F⁻¹⁺(α) if, and only if, α does not correspond to a ‘flat part’ of F or equivalently, if, and only if, F⁻¹ is continuous at α.

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LEMMA : For all x ∈ R and for all α ∈ (0,1)

(i) F⁻¹(α) ≤ x ⇔ α ≤ F(x)

(ii) F⁻¹⁺(α) ≥ x ⇔ α ≥ F(x−) = P_{(X < x)}

PROOF : We only prove (i) ; (ii) can be proven in a similar way. The ‘⇒’ part is proven if we can show that

α > F(x) ⇒ F⁻¹(α) > x

Assume that α > F(x). Then there exists an ǫ > 0 such that α > F(x + ǫ). From the sup-definition of V aR[X; α], we find that x + ǫ ≤ F⁻¹(α), which implies that

F⁻¹(α) > x.

We now prove the ‘⇐’ part. If α ≤ F(x) then we find that α ≤ F(x + ǫ) for all ǫ > 0. From the inf-definition of F⁻¹(α), we can conclude that F⁻¹(α) ≤ x + ǫ for all ǫ > 0. Taking the limit for ǫ ↓ 0, we obtain F⁻¹(α) ≤ x.

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PROPOSITION : Let X be an rv. For any 0 < α < 1, the following equalities hold : (i) If t is non-decreasing and continuous then F_t(X⁻¹₎(α) = t F_X⁻¹(α).

(ii) If t is non-decreasing and continuous then F_t(X)⁻¹⁺(α) = t F_X⁻¹⁺(α).

PROOF : We only prove (i) ; (ii) can be proven in a similar way. By application of the previous lemma, we find that the following equivalences hold for all real x :

F_t(X⁻¹₎(α) ≤ x ⇔ α ≤ F_t(X₎(x) ⇔ α ≤ F_X(t⁻¹⁺(x))

⇔ F_X⁻¹(α) ≤ t⁻¹⁺(x) ⇔ t F_X⁻¹(α) ≤ x

Note that the above proof only holds if t⁻¹⁺ is finite. But one can verify that the equivalences also hold if t⁻¹⁺(x) = ±∞.

Remark : The continuity assumption put on the function t can be relaxed as follows : in (i) it is enough for t to be left-continuous, whereas in (ii) it is enough for t to be right-continuous.

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PROPOSITION :

(i) If an rv X has a continuous df F, then F(X) ∼ U ni(0,1).

(ii) Let X be an rv with df F, not necessarily continuous. If U ∼ U ni(0,1), then X =_d F⁻¹(U) =_d F⁻¹⁺(U).

P^ROOF :

(i) For all 0 < u < 1,

P_(F_(X₎ _≥ _{u) =} P_(X _≥ _F⁻¹_{(u)) = 1} ₋ _F_(F⁻¹_{(u)) = 1} ₋ _u from which we conclude that F(X) ∼ U ni(0,1).

(ii) We see from the lemma that

P_(F⁻¹_(U₎ _≤ _{x) =} P_(U _≤ _F_{(x)) =} _F_(x).

The other statement has a similar proof.

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PROP : VaR satisfies the “Non-excessive loading” property.

PROOF : Since X ≤ max[X] we have that V aR[X;α] ≤ max[X] whatever α, so that VaR is indeed no-ripoff.

P^ROP : VaR does not satisfy the “Non-negative loading” property.

PROOF : Let us define α^∗ = F(E_[X]). It is clear that VaR does not exceed the expected loss X for probability levels less than α^∗.

P^ROP : VaR satisfies the “No unjustified loading” property.

P^ROOF : It is easy to see that for any probability level α > 0, V aR[c;α] = c.

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PROP : VaR satisfies the Objectivity property.

PROOF :This is a direct consequence of the definition of VaR, since it only depends on the df of X.

P^ROP : VaR satisfies the Translativity property.

PROOF : VaR possesses the very convenient stability property that the VaR of a non- decreasing function t of some rv X is obtained by applying the same function to the initial VaR. Let us consider the function t : x → x +c, we deduce that VaR has the translativity property.

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PROP : VaR fails to be subadditive.

i) A counter-example

Let us consider two independent risks with unit Pareto distribution X ∼ P ar(1,1) and Y ∼ P ar (1,1), i.e.

P (X > t) = P (Y > t) = 1

1 + t, t > 0.

On the one hand,

V aR [X;α] = V aR [Y ;α] = 1

1 − α − 1.

On the other hand, one can show that

P (X + Y ≤ t) = 1 − 2

2 + t + 2ln (1 + t) (2 + t)² .

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Since

P (X + Y ≤ 2V aR [X;α]) = α − (1 − α)² 2 ln

1 + α 1 − α

< α, we get

V aR[X;α] + V aR[Y ;α] < V aR[X + Y ;α]

and, in such a case, diversification will lead to more risk being reported.

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ii) Elliptical distributions et subadditivity of VaR D^EFINITION :

1. A random vector X = (X₁, . . . , X_n) has spherical distribution, if for every ortho- gonal map U ∈ R^n×n _(i.e. _U^′_U ₌ _UU^′ ₌ _I_d_),

U X =^d X.

◦ The multivariate standard Gaussian distribution is a spherical distribution since f_UX(x) = f_X(U⁻¹x) = 1

(2π)^n/2 exp

−1

2(U⁻¹x)^′(U⁻¹x)

= 1

(2π)^n/2 exp

−1

2x^′UU⁻¹x)

= f_X(x)

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PROP : The following are equivalent.

(i) X is spherical

(ii) There exists a function ψ such that, for all t ∈ R^d E_[e^it^′^X_{] =} _ψ(t²_).

(iii) For every a ∈ R^d

a^′X =_d aX₁.

(iv) X =_d RS where S is uniformly distributed on the unit sphere Sⁿ⁻¹ = {t ∈ R^d _: _t² _{= 1}} _and _R _≥ ₀ is a radial random variable, independent of S.

ψ is called the characteristic generator of the spherical distribution and we write X ∈ S_n(ψ).

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2. A random vector X = (X₁, . . . , X_n) has an elliptical distribution (X ∈ E(µ, A, ψ)) if there exist µ ∈ Rⁿ_, _A _∈ R^n×d _and _Y _∈ _S_d_(ψ) _{such that}

X =^d µ + AY.

◦ It follows that any random vector with components that are linear combinations of the components of an elliptical distribution is again an elliptical distribution with the same characteristic generator.

◦ The Gaussian and the Student distributions are examples of elliptical distributions.

◦ Any multivariate elliptical distribution with mutually independent components and finite variance must necessarily be multivariate normal.

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PROPOSITION : Let X ∈ E(µ, A, ψ) and M = {L : L = λ₀ + λ^′X}. For any L₁, L₂ ∈ M, α ≥ 0.5

V aR[L₁ + L₂;α] ≤ V aR[L₁;α] + V aR[L₂;α].

PROOF : Let L₁ = λ_0,1 + λ^′₁X and L₂ = λ_0,2 + λ^′₂X. We have V aR[L₁ + L₂;α]

= λ_0,1 + λ_0,2 + V aR[(λ₁ + λ₂)^′X;α]

= λ_0,1 + λ_0,2 + V aR[(λ₁ + λ₂)^′µ + (λ₁ + λ₂)^′AY₁;α]

= λ_0,1 + λ_0,2 + (λ₁ + λ₂)^′µ + (λ₁ + λ₂)^′A V aR[Y₁;α]

If α ≥ 0.5, then V aR[Y₁;α] ≥ 0 and V aR[L₁ + L₂;α]

≤ λ_0,1 + λ_0,2 + (λ₁ + λ₂)^′µ + λ^′₁A + λ^′₂A V aR[Y₁;α]

= V aR[L₁;α] + V aR[L₂;α].

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PROP : VaR satisfies the Comonotonic additivity property.

P^ROOF : For all non-decreasing (left-continuous) functions h and g,

V aR[h(X) + g(X);α] = V aR[(h + g)(X);α] = (h + g)(V aR[X;α])

= h(V aR[X;α]) + g(V aR[X;α])

= V aR[h(X); α] + V aR[g(X);α]

PROP : VaR satisfies the Positive homogeneity property.

P^ROOF : Let us consider the function t : x → λx with λ > 0, we deduce that VaR has the translativity property.

P^ROP : VaR satisfies the Monotonicity property.

PROOF : Clearly, if P_(X _≤ _Y _{) = 1} _{holds then} _F_X_(x) _≥ _F_Y _(x) is true for any x.

Therefore, V aR[X;α] ≤ V aR[Y ;α] holds in such a case for any probability level α (by symmetry with respect to the main diagonal).

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PROP : VaR satisfies the Invariance by first-order stochastic dominance property.

PROOF : It is easy to show that

X _DS₁ Y ⇔ P_(Y _≤ _d) _≤ P_(X _≤ _d) _∀d _∈ R

⇔ V aR[X;α] ≤ V aR[Y ;α] ∀α ∈ (0,1).

P^ROP : VaR does not satisfy the Invariance by second-order stochastic dominance property.

PROOF : Since X _DS₂ Y _X _DS1 _Y , the Invariance by second-order stochastic dominance property may not be satisfied.

P^ROP : VaR does not satisfy the Convexity property.

P^ROOF : VaR is not subadditive and satisfies the Positive homogeneity property.

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PROP : VaR does not satisfy the Iterativity property.

PROOF : Let

X Y

∼ N

µ_X µ_Y

,

σ²_X ρσ_Xσ_Y ρσ_Xσ_Y σ²_Y

The conditional distribution of X given Y = y is X|Y = y ∼ N

µ_X + ρσ_X

σ_Y (y − µ_Y ), σ²_X(1 − ρ²)

.

Hence

V aR[X;α] = µ_X + σ_XΦ⁻¹(α) V aR[X|Y ;α] = µ_X + ρσ_X

σ_Y (Y − µ_Y ) + σ_X

(1 − ρ²)Φ⁻¹(α) V aR[V aR[X|Y ;α];α] = µ_X + σ_X (1 − ρ²) + ρ

Φ⁻¹(α) and we deduce that V aR[V aR[X|Y ;α];α] = V aR[X;α] only if ρ = 0.

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PROP : VaR satisfies the Convergence in distribution.

PROOF : It is well known that the weak convergence of the dfs ensures the same type of convergence for the quantile functions.

P^ROP : VaR does not satisfy the Stability by mixing property.

PROOF : Consider for example the case where X =_d 1

2δ₀ + 1

2N (0,1). It is easily seen that

V aR[X;α] = 1

2Φ⁻¹(α)

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PROP : Let (X, Y ) be a random vector with pdf f(., .), then

∂

∂γV aR[X + γY ;α] = E_[Y _|X ₊ _γY ₌ _{V aR[X} ₊ _γY _;_α]]

PROOF : It may be found for example in Gouriéroux C., J.P. Laurent and O. Scaillet (2000). Sensitivity Analysis of Values at Risk. Journal of Empirical Finance, 7, 225- 245.

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∂²

∂γ²V aR[X + γY ;α] = ∂

∂sV_[Y _|X ₊ _γY ₌ _s]

_{s=V aR[X}_+γY_;α]

+ V_[Y _|X ₊ _γY ₌ _s] ^∂

∂sf_X_+γY(s)

s=V aR[X+γY;α]

PROOF : It may be found for example in Gouriéroux C., J.P. Laurent and O. Scaillet (2000). Sensitivity Analysis of Values at Risk. Journal of Empirical Finance, 7, 225- 245.

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⊲ TVaR and associated risk measures P^ROP : For any α ∈ (0,1)

T V aR[X;α] = V aR[X;α] + 1

1 − αES[X;α]

CT E[X;α] = V aR[X;α] + 1

1 − F(V aR[X;α])ES[X;α]

CV aR[X;α] = 1

1 − F(V aR[X;α])ES[X;α]

PROOF : The first expression follows from ES[X;α] =

₁

0 (V aR[X;ξ] − V aR[X;α])₊ dξ

=

₁

α V aR[X;ξ]dξ − V aR[X;α](1 − α).

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The second and third expression follow from

ES[X;α] = E_[X ₋ _{V aR[X}_;α]|X > V aR[X;α]]P(X > V aR[X; α]).

Remark :

1. If F has a positive probability distribution function, for any α ∈ (0,1), T V aR[X;α] = CT E[X;α].

2. For any α ∈ (0,1)

minπ (E_[(X ₋ _π)₊_{] + (1} ₋ _α)π)

= E_[(X ₋ _{V aR[X}_;_α])₊_{] + (1} ₋ _{α)V aR[X}_;_α]

= (1 − α)T V aR[X; α].

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EXAMPLES :

1. Consider a random variable X ∼ N(µ, σ²) which is normally distributed with mean µ and variance σ². We have

V aR[X;α] = µ + σΦ⁻¹(α) T V aR[X;α] = µ + σϕ(Φ⁻¹(α))

1 − α CT E[X;α] = µ + σϕ(Φ⁻¹(α))

1 − α CV aR[X;α] = σ

ϕ(Φ⁻¹(α))

1 − α − Φ⁻¹(α)

ES[X;α] = σϕ(Φ⁻¹(α)) − σΦ⁻¹(α)(1 − α)

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2. Consider a random variable that is lognormally distributed, i.e. ln X ∼ N(µ, σ²).

We have

V aR[X;α] = e^µ+σΦ⁻¹^(α)

T V aR[X;α] = e^µ+σ²^/2Φ(σ − Φ⁻¹(α)) 1 − α

CT E[X;α] = e^µ+σ²^/2Φ(σ − Φ⁻¹(α)) 1 − α

CV aR[X;α] = ^µ+σ²^/2Φ(σ − Φ⁻¹(α))

1 − α − e^µ+σΦ⁻¹^(α)

ES[X;α] = e^µ+σ²^/2Φ(σ − Φ⁻¹(α)) − e^µ+σΦ⁻¹^(α)(1 − α)

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PROP : TVaR satisfies the “Non-excessive loading” property.

PROOF : This comes from the fact that VaR is known to be no-ripoff, so that T V aR[X;α] = 1

1 − α

₁

α V aR[X;ξ]dξ ≤ 1 1 − α

₁

α max[X]dξ.

PROP : TVaR satisfies the “Non-negative loading” property.

PROOF : This is again an immediate consequence of the corresponding properties for VaRs, since

T V aR[c;α] = 1 1 − α

₁

α V aR[c;ξ]dξ = 1 1 − α

₁

α cdξ = c.

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PROP : TVaR satisfies the “No unjustified loading” property.

PROOF : If U ∼ Uni(0,1) then

E_[X_{] =} E_[F⁻¹_(U_{)] =}

₁

0 F⁻¹(u)du = T V aR[X; 0].

The claimed property will hold if we are able to show that TVaR is non-decreasing in the probability level. We clearly have that

T V aR[X;α] = 1 1 − α

E_[X_] ₋

_α

0 V aR[X;ξ]dξ

. Therefore, we can write

d

dαT V aR[X;α] = 1

1 − α (T V aR[X;α] − V aR[X;α]). Since α → V aR[X;α] is non-decreasing,

T V aR[X;α] = 1 1 − α

₁

α V aR[X;ξ]dξ ≤ 1 1 − α

₁

α V aR[X;α]dξ = V aR[X;α]

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which gives

d

dαT V aR[X;α] ≥ 0.

We conclude

T V aR[X;α] ≥ T V aR[X; 0] = E_[X_]

so that TVaR induces a non-negative loading whatever the probability level α.

PROP : TVaR satisfies the Objectivity property.

P^ROOF : Knowing α → T V aR[X;α] is equivalent knowing α → V aR[X;α] since by definition

T V aR[X;α] = 1 1 − α

₁

α V aR[X;ξ]dξ.

and

V aR[X;α] = T V aR[X;α] − (1 − α) d

dαT V aR[X;α].

Hence TVaR satisfies the Objectivity property.

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PROP : TVaR satisfies the Translativity property.

PROOF : This is immediate from the corresponding properties of the VaRs T V aR[X + c;α] = 1

1 − α

₁

α V aR[X + c;ξ]dξ

= 1

1 − α

₁

α V aR[X;ξ]dξ + c = T V aR[X;α] + c PROP : TVaR satisfies the Subadditivity property.

PROOF : First note that

T V aR[X;α] = min

π

π + 1

(1 − α)E_[(X ₋ _π)₊_]

.

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We thus have for any 0 < λ < 1 that T V aR[λX + (1 − λ)Y ;α]

≤

π + 1

(1 − α)E_[(λX _{+ (1} ₋ _λ)Y ₋ _π)₊_]

π=λV aR[X;α]+(1−λ)V aR[Y ;α]

= λV aR[X;α] + (1 − λ)V aR[Y ;α]

+ 1

(1 − α)E_[(λX _{+ (1} ₋ _λ)Y ₋ _{(λV aR[X}_;_{α] + (1} ₋ _{λ)V aR[Y} _;_α]))₊

≤ λV aR[X;α] + (1 − λ)V aR[Y ;α]

+ λ

(1 − α)E_[(X ₋ _{V aR[X}_;_α])₊_{] +} ⁽¹ ⁻ ^λ)

(1 − α)E_[(Y ₋ _{V aR[Y} _;_α])₊_]

= λT V aR[X;α] + (1 − λ)T V aR[Y ;α].

Hence the TVaR is convexe and since it is positive homogeneous, it is subadditive.

P^ROP : TVaR satisfies the Comonotonic additivity property.

P^ROOF : This is immediate from the corresponding properties of the VaRs.

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PROP : TVaR satisfies the Positive homogeneity property.

PROOF : This is immediate from the corresponding properties of the VaRs.

P^ROP : TVaR satisfies the Monotonicity property.

P^ROOF : This is immediate from the corresponding properties of the VaRs.

PROP : TVaR satisfies the Invariance by first-order stochastic dominance property.

PROOF : TVaR satisfies the Invariance by second-order stochastic dominance property and so it satisfies the Invariance by first-order stochastic dominance property..

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PROP : TVaR satisfies the Invariance by second-order stochastic dominance property.

PROP : For any random pair (X, Y ) we have that X _DS2 Y if and only if their respective T V aR’s are ordered :

X _DS₂ Y ⇔ T V aR[X;α] ≤ T V aR[Y ;α] ∀α ∈ (0,1)

PROOF : First we assume X _DS₂ Y and let α ∈ (0,1). Consider the function f(d) defined by

f(d) = (1 − α)π + E_[(X ₋ _d)₊_].

We have

T V aR[X;α] = f(V aR[X;α])

1 − α ≤ f(V aR[Y ;α]) 1 − α

= V aR[Y ;α] + 1

1 − αE_[(X ₋ _{V aR[Y} _;_α])₊_]

≤ V aR[Y ;α] + 1

1 − αE_[(Y ₋ _{V aR[Y} _;_α])₊_{] =} _{T V aR[Y} _;_α].

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To prove the other implication, we assume that the TVaR’s are ordered for all α ∈ (0,1). Note that for any random variable X, we have that

E_[(X ₋ _d)₊_{] =} E_[(F⁻¹

X (U) − d)₊]

=

₁

F_X(d) V aR[X;α]dα − d(1 − F_X(d)).

Hence, for d such that 0 < F_X(d) < 1, we find

E_[(X ₋ _d)₊] = (T V aR[X;F_X(d)] − d) (1 − F_X(d))

≤ (T V aR[Y ;F_X(d)] − d) (1 − F_X(d))

= E_[(Y ₋ _d)₊_{] +}

_F_Y_(d)

F_X(d) (V aR[Y ;α] − α)dα

Using the equivalence α ≤ F_Y (d) ⇔ d ≥ V aR[Y ;α], it is straightforward to prove that

_F_Y_(d)

F_X(d) (V aR[Y ;α] − α)dα ≤ 0.

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If F_X(d) = 1, we find E_[(X ₋ _d)₊_{] = 0} _≤ E_[(Y ₋ _d)₊_{]. Since} E_[X_] _≤ E_[Y _] Thus E_[(X ₋ _d)₊_{] =} E_[X_] _≤ E_[(Y ₋ _d)₊_] also holds for d such that F_X(d) = 0.

Hence, we have proven that X _DS2 Y .

PROP : TVaR satisfies the Convexity property.

PROOF : See the proof for the Subadditivity property.

PROP : TVaR does not satisfy Iterativity property.

PROOF : Consider the case where

X Y

∼ N

µ_X µ_Y

,

σ²_X ρσ_Xσ_Y ρσ_Xσ_Y σ²_Y

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PROP : TVaR satisfies the Convergence in distribution property if moreover E_[X_n_] _→ E_[X_n_].

P^ROOF : By using the Objectivity property.

PROP : TVaR does not satisfy the Stability by mixing property.

P^ROOF : Consider for example the case where X =_d 1

2δ₀ + 1

2N (0,1).

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Remark : Let X and x be such that F(x) > 0. For any event A such that P_{(A) =} F(x),

E_[X_|A] _≤ E_[X|X > x].

It suffices to write

≥ x + E_[X ₋ x|X > x, A] P(A|X > x)

= x + E_[X ₋ x|X > x, A] P(X > x|A)

≥ x + E_[X ₋ x|X > x, A] P(X > x|A) +E_[X ₋ _x|X _≤ _{x, A]}P_(X _≤ _x|A)

= E_[X_|A].

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It sheds a new light on CTE, which can be represented as a worst-case conditional expectation since

CT E[X;α] = sup E_[X_|A]|P_(A) _≥ F¯(V aR[X; α]) which reduces to

CT E[X;α] = sup{E_[X_|A]|P_(A) _≥ ₁ ₋ _α}

when F is continuous.

This result is closely related to the notion of scenario or stress testing : the CTE appears as the largest possible expected value of X under the set of all plausible scenarios (that is, those whose probabilities exceed 1 − α).

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∂

∂γT V aR[X + γY ;α] = E_[Y _|X ₊ _γY _≥ _{V aR[X} ₊ _γY _;_α]]

PROOF : See for example Scaillet, O., 2004. Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall. Mathematical Finance, 14 : 115-129.

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∂²

∂γ²T V aR[X + γY ;α] = 1

1 − αV_[Y _|X ₊ _γY ₌ _{V aR[X} ₊ _γY _;_α]]

×f_X_+γY (V aR[X + γY ;α]).

PROOF : See for example Scaillet, O., 2004. Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall. Mathematical Finance, 14 : 115-129.

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⊲ RISK MEASURES BASED ON EXPECTED UTILITY THEORY

Remark : u may be chosen such that u(0) = 0, u^′(0) = 1 and u^′′(0) = −a ≤ 0.

PROP : Π(.) satisfies the “Non-excessive loading” property (no ripoff).

PROOF : Because u is non-decreasing and X ≤ max[X] a.s., we have 0 = E_[u(Π(X₎ ₋ _X_)] _≥ _u(Π(X₎ ₋ _max[X_])

so that Π(X) ≤ max[X] holds, and the zero-utility premiums satisfy the no-ripoff condition.

PROP : Π(.) satisfies the “Non-negative loading” property.

PROOF : If u is concave then Jensen’s inequality ensures that 0 = E_[u(Π(X₎ ₋ _X_)] _≤ _u(Π(X₎ ₋ E_[X_])

so that Π(X) ≥ E_[X_] and the zero-utility premiums contain a non-negative loading.

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PROP : Π(.) satisfies the “No unjustified loading” property.

PROOF : Note that

0 = E_[u(Π(c) ₋ _{c)] =} _u(Π(c) ₋ _c).

Since u^′(0) > 0, we deduce that Π(c) = c.

PROP : Π(.) satisfies the Objectivity property.

PROOF : If X and Y have the same distribution,

0 = E_[u(Π(X₎ ₋ _X_{)] =} E_[u(Π(Y ₎ ₋ _Y _{)] =} E_[u(Π(X₎ ₋ _Y _)]

and hence Π(X) = Π(Y ).

PROP : Π(.) satisfies the Translativity property.

P^ROOF : We have

0 = E_[u(Π(X ₊_c)₋_(X ₊_{c))] =} E_[u((Π(X ₊_c)₋_c)₋_X_{)] =} E_[u(Π(X₎₋_X_)]

and then Π(X + c) = Π(X) + c.

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PROP : Π(.) does not satisfy the Subadditivity property.

PROOF : Consider the case where

X Y

∼ N

0 0

,

1 ρ ρ 1

and u(x) = −e^−αx for α > 0. Then Π(X) = 1

α lnE_e^αX ₌ ^α

2 and Π(X + Y ) = (1 + ρ)α and therefore Π(X + Y ) = Π(X) + Π(Y ).

Note that Π(X) satisfies the Additivity for independent risks property iff u(x) =

−e^−αx or u(x) = x (up to a linear relation).

P^ROP : Π(.) does not satisfy the Comonotonic additivity property.

P^ROOF : Π(.) does not satisfy the Positive homogeneity property and therefore it does not satisfy the Comonotonic additivity property.

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PROP : Π(.) does not satisfy the Positive homogeneity property.

PROOF : Consider the case where X ∼ N (0,1). Then Π(λX) = λ 1

λα lnE_[e^λαX_{] =} _λ^αλ

2 = λ²Π(X).

P^ROP : Π(.) satisfies the Monotonicity property.

P^ROOF : Π(.) satisfies the Invariance by first-order stochastic dominance property and therefore it satisfies the Monotonicity property.

PROP : Π(.) satisfies the Invariance by first-order stochastic dominance property.

PROOF : Π(.) satisfies the Invariance by second-order stochastic dominance property and therefore it satisfies the Invariance by first-order stochastic dominance property.

PROP : Π(.) satisfies the Invariance by second-order stochastic dominance property.

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PROOF : Assume that X _DS₂ Y . We have

E_[u(Π(X₎ ₋ _X_)] _≥ E_[u(Π(X₎ ₋ _Y _)].

But

0 = E_[u(Π(X₎ ₋ _X_{)] =} E_[u(Π(Y ₎ ₋ _Y _)] _≥ E_[u(Π(X₎ ₋ _Y _)]

and it follows that Π(Y ) ≥ Π(X).

PROP : Π(.) satisfies the Convexity property if u^′′ < 0.

P^ROOF : Consider two risks X and Y and define

g(t;X, Y ) = Π(X + tV ) where V = Y − X.

Assume that g(t) = g(t;X, Y ) is convexe for all X and Y and for α ∈ (0,1). Then Π(αX + (1 − α)Y ) = Π(X + (1 − α)V ) = g((1 − α))

≤ αg(0) + (1 − α)g(1) = αΠ(X) + (1 − α)Π(Y ).

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It is now enough to show that

g^′′(0;X, Y ) ≥ 0 for all X and Y since

g^′′(0;X + tV, Y ) = (1 − t²)g^′′(t;X, Y ).

But

g^′′(0;X, Y ) = −E_[u^′′_(Π(X₎ ₋ _X_)(g^′′_(0;_{X, Y} ₎ ₋ _V ₎²_] E_[u^′_(Π(X₎ ₋ _X_)] ^.

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PROP : Π(.) satisfies the Iterativity property iff u(x) = −e^−αx or u(x) = x (up to a linear relation).

P^ROOF : The ‘⇐’ part is proven if we can show that Π(X) = Π(Π(X|Y )) for u(x) = x (obvious) and for u(x) = −e^−αx. But

Π (X) = 1

α lnE_[e^αX_{] =} ¹

α lnE_[E_[e^αX_|Y _]]

= 1

α lnE

e^α^α¹ ^ln^E^[e^αX^|Y ^]

= Π (Π(X|Y )).

The ‘⇒’ part is proven the following way. Let z > 0 and y₁, y₂ ∈ [0,1]. Define : Y = 1

2δ_y₁ + 1

2δ_y₂ and X|Y = y =_d X_y,z =_d (1 − y)δ₀ + yδ_z. On the other hand

X =_d 1

2X_y₁_,z + 1

2X_y₂_,z =_d X_q,z with q = 1

2(y₁ + y₂).

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We will use the following notation

π(y_i) = Π (X|Y = y_i) which satisfies

y_iu (π(y_i) − z) + (1 − y_i)u(π(y_i)) = 0.

Differentiating one time with respect to y_i

∂

∂y_i (y_iu (π(y_i) − z) + (1 − y_i)u(π(y_i))) = 0 and letting y_i tends to 0 leads to

u(−z) = −π^′(0).

By iterativity, we have that Π(X) = Π (Π (X|Y )) i.e. : 1

2u (π(q) − π(y₁)) + 1

2u (π(q) − π(y₂)) = 0

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Differentiate two times with respect to y_i

∂²

∂y_i²

1 2u

π

y₁ + y₂ 2

− π(y₁)

+ 1 2u

π

y₁ + y₂ 2

− π(y₂)

= 0.

And choosing y₁ = y₂ = y, we get

π^′′(y) + aπ^′(y)² = 0 with π(0) = 0, π(1) = z.

If a = 0, π(y) = y. If a > 0, π(y) = a⁻¹ log(1 − y + ye^az) and u(−z) =

−π^′(0) = −a⁻¹ (e^az − 1).

P^ROP : Π(.) satisfies the Convergence in distribution property if lim_n→∞ E_[u(−X_n_{)] =} E_[u(−X_)].

PROOF : Obvious.

PROP : Π(.) does not satisfy the Stability by mixing property.

P^ROOF : By choosing an appropriate counter-example.