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Risk Measures
Birgit Rudloff
Princeton University
RTG Summer School 2013
Overview
Overview
Risk Measures
Primal Representation, Acceptance Sets
Dual Representation
Examples
Generalizations: Multivariate Risks
Risk Measures for Systemic Risk
Risk Measures in Markets with Transaction Costs
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Risk Measures
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1. Risk Measures
Basics
probability space (Ω, F, P )
sample space Ω
σ-algebra F
probability measure P
random variables (discounted payoff, profit-loss-profiles,
financial positions,...)
X : (Ω, F, P ) → R
values of X: real numbers (in USD, Euros or Chilean
Pesos...)
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1. Risk Measures
Risk Measures
mathematical model to quantify uncertainty
functional on Lp (Ω, F, P ), p ∈ [0, ∞] (or subspaces of
random variables)
ρ : Lp → R ∪ {+∞}
values: real numbers (in USD, Euro, even +∞ possible)
Interpretation
the larger ρ(X), the higher the risk
ρ(X) > 0: there is a risk
ρ(X) = 0: the position X is risk neutral
ρ(X) < 0: there is a chance (no risk)
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1. Risk Measures
There is not one absolutely objective risk measure! But, we can
ask the following:
Which properties a functional ρ should have to be a ”reasonable”
risk measure?
Which random variables are ”acceptable” for a risk measure ρ?
Which random variables are for all reasonable risk measures
acceptable?
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1.1 Coherent and Convex Risk Measures
Risk Measures
A function ρ : Lp → R ∪ {+∞} satisfying
(R0) Normalization
ρ(0) = 0,
(R1) Monotonicity
X1 , X2 ∈ Lp : X1 ≥ X2
=⇒
ρ(X1 ) ≤ ρ(X2 ),
(R2) Translation property
X ∈ Lp , c ∈ R
=⇒
ρ(X + c) = ρ(X) − c,
is called Risk Measure.
(R2) is sometimes called ”Cash invariance”.
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1.1 Coherent and Convex Risk Measures
Axioms of Convex Risk Measures
A function ρ : Lp → R ∪ {+∞} satisfying (R0), (R1), (R2) and
(R3) Convexity
X1 , X2 ∈ Lp , λ ∈ (0, 1)
=⇒
ρ (λX1 + (1 − λ)X2 ) ≤ λρ (X1 ) + (1 − λ)ρ (X2 ) .
it is called a Convex Risk Measure.
B Föllmer, Schied (02, 04): Stochastic Finance. Walter de Gruyter
B Frittelli, Gianin (02): Putting order in risk measure. Journal of Banking and Finance 26 (7)
Axioms of Coherent Risk Measures
A function ρ : Lp → R ∪ {+∞} satisfying (R0) - (R3) and
(R4) Positive homogeneity
X ∈ Lp , s > 0
=⇒
ρ (sX) = sρ (X) .
it is called a Coherent Risk Measure.
B Artzner, Delbaen, Eber, Heath (99): Coherent Measures of Risk. Mathematical Finance 9 (3)
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1.1 Coherent and Convex Risk Measures
Remarks
the inequality X1 ≥ X2 is to be understood P − a.s.,
X1 ≥ X2
:⇐⇒
P ({ω ∈ Ω : X1 (ω) < X2 (ω)}) = 0
(R2), ρ(X) ∈ R =⇒
ρ(X + ρ (X)) = ρ(X) − ρ(X) = 0.
i.e., ρ(X) can be seen as a capital requirement; when added
to X, it makes the overall position ”risk neutral”.
Is ρ(X) the smallest amount of capital with this property?
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1.2 Acceptance Sets
We call Aρ := {X ∈ Lp : ρ(X) ≤ 0} the acceptance set of the
risk measure ρ.
Lemma 1: Acceptance Set
Consider a function ρ : Lp → R ∪ {+∞}. It holds:
1
ρ monotone ⇒ Aρ + Lp+ ⊆ Aρ .
2
ρ convex ⇒ Aρ convex.
3
ρ positively homogeneous ⇒ Aρ is a cone.
4
ρ closed ⇒ Aρ is closed.
A function f : Lp → R ∪ {±∞} is called closed (or lower
semicontinuous) if
epi f := {(v, r) ∈ Lp × R : f (v) ≤ r} ⊆ Lp × R is closed in
Lp × R.
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1.2 Acceptance Sets
On the other hand: Can we construct a risk measure from a
given set of ”acceptable positions” A ⊆ Lp ?
Lemma 2: Acceptance Set
Consider a set A ⊆ Lp . Define
ρA (X) := inf{t ∈ R : X + t ∈ A}.
(1)
It holds:
1
inf{t ∈ R : t ∈ A} = 0 ⇒ ρA (0) = 0.
2
A + Lp+ ⊆ A ⇒ ρA monotone.
3
ρA satisfies the translation property.
4
A is convex ⇒ ρA is convex.
5
A is a cone ⇒ ρA is positively homogeneous.
6
A is a closed ⇒ ρA is closed.
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1.3 Lsc Risk Measure and Closed Acceptance Sets
A set A ⊆ Lp satisfying properties in 1 and 2 in lemma 2 is
called an acceptance set.
Lemma 3
There is a one-to-one relationship between lower semicontinuous
risk measures and closed acceptance sets via
1
A = {X ∈ Lp : ρ (X) ≤ 0} and
2
ρ (X) = inf {t ∈ R : X + t ∈ A}.
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1.4 Dual Representation of Convex Risk Measures
Dual Representation
A function ρ : Lp → R ∪ {+∞} is a lower semicontinuous,
convex risk measure if and only if there exists a representation
of the form
n
o
ρ(X) = sup E Q [−X] − α(Q) ,
(2)
Q∈Q
q
where Q := { prob. measures Q : dQ
dP ∈ L } and
dQ
α(Q) = supX∈Aρ E Q [−X] = ρ∗ (− dP ) is called the penalty
function.
For a lsc coherent risk measure we have
ρ(X) = sup E Q [−X],
(3)
Q∈Q̄
for some Q̄ ⊆ Q.
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1.5 Examples
Counterexamples
Variance σ 2 (X) is not monotone, not translative!
Value at Risk V aRα (X) at level α ∈ (0, 1) is not convex!
V aRα (X) := inf{x ∈ R : P {X + x ≤ 0} ≤ α} = −xα .
Exercise: Consider two defaultable corporate bonds with
face value $500, 000. Payoff X1 , X2 (r = 10%, default prob.
0.8%, independent). Calculate V aRα (X1 ), V aRα (X2 ) and
V aRα ( 21 (X1 + X2 )) for α = 1%!
upper α-quantile
xα := inf{x ∈ R : P {X ≤ x} > α} = sup{x ∈ R : P {X ≤ x} ≤ α}.
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Risk Measures
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1.5 Examples
Examples of Coherent Risk Measures
Conditional Value at Risk
1
CV aRα (X) := − E[X1{X≤xα } ] + xα (α − P [X ≤ xα ])
α
Zα
1
=
V aRγ (X)dγ
α
0
also Average Vaue at Risk (AV aRα ) or Expected Shortfall
(ESα ) at level α ∈ (0, 1).
Expected loss
ρ(X) := E[−X]
”Worst Case” risk measure
ρmax (X) := ess.sup(−X)
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1.5 Examples
Example of a Convex Risk Measure
Consider the exponential utility function u(x) = 1 − e−αx and
the set of acceptable positions whenever the expected utility is
nonnegative:
A := {X ∈ L∞ | E[u(X)] ≥ 0}.
The set A is convex, satisfies inf{t ∈ R : t ∈ A} = 0 and
A + L∞
+ ⊆ A. Thus, ρA (as defined in (1)) is a convex risk
measure on L∞ .
It is called the entropic risk measure.
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1.5 Examples
Example of a Coherent Risk Measure
Superhedging price (NA-price bounds) for discounted payoff X
p(X) =
sup
E Q [X],
Q∈EM M
where EMM is the set of equivalent martingale measures.
Thus, ρ(X) := p(−X) is a coherent risk measure!
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Risk Measures
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2. Generalizations: Multivariate Risks
Multivariate Risks
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Risk Measures
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2. Generalizations: Multivariate Risks
X ∈ Lpd . Why important?
e.g. systemic risk: d banks, interlinked, want capital
requirement for each bank, not separately, but taking the
interconnectedness into account
e.g. markets with transaction costs: d assets, Xi payoff in
asset i = 1, ..., d, cannot sum up over Xi bc. of transaction
costs!
Collection of all initial portfolio vectors, that make X
acceptable:
R(X) = {u ∈ Rd : X + u ∈ A}
for some acceptance set A ⊂ Lpd .
Usually initial capital just in a few ’eligible’ assets: subspace
M ⊆ Rd , typically dim M = m << d
R(X) = {u ∈ M : X + u ∈ A}
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1. Risk measures under transaction costs
M ⊆ Rd
M+ = M ∩ Rd+
Set-Valued Risk Measure
A set-valued function
R : Lpd (FT ) → P(M, M+ ) = {D ⊆ M : D = D + M+ } is a risk
measure if
1
Finite at zero: ∅ =
6 R(0) 6= M
2
M translative: R(X + m) = R(X) − m for any m ∈ M
3
Monotone: if X − Y ∈ Lpd (FT )+ then R(X) ⊇ R(Y )
normalized if R(X) = R(X) + R(X).
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Risk Measures
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1. Risk measures under transaction costs
Primal Representation
Risk measures and acceptance sets are one-to-one via
R(X) = {u ∈ M : X + u ∈ A}
and
A = {X ∈ Lpd (FT ) : 0 ∈ R(X)}.
finite at zero
monotone
market compatible
Birgit Rudloff
R
A
∅=
6 R(0) 6= M
M ∩ A 6= ∅
M ∩ (Lpd \A) 6= ∅
Y − X ∈ Lpd (FT )+
⇒ R(Y ) ⊇ R(X)
convex
positively homogeneous
subadditive
closed images
lsc
R(X) = R(X) + K M
Risk Measures
A + Lpd (FT )+ ⊆ A
convex
cone
A+A⊆A
directionally closed
closed
A + Lpd (K M ) ⊆ A
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1. Risk measures under transaction costs
Let G(M, M+ ) = {D ⊆ M : D = cl co(D + M+ )}.
Dual Representation, 1 ≤ p ≤ ∞
A function R : Lpd (FT ) → G(M, M+ ) is a closed coherent
conditional risk measure if and only if there is a nonempty
q
set WR
⊆ W q such that
R(X) =
\
(E Q [−X] + G (w)) ∩ M.
q
(Q,w)∈WR
Q vector probability measure with components Qi
q
Q
Q1
Qd
T
i
(i=1,...,d), dQ
dQ ∈ L and E [X] = (E [X1 ], ..., E [Xd ]) .
w ∈ (M+ )+
G(w) = {v ∈ Rd : E[wT v] ≥ 0}.
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1. Risk measures under transaction costs
Wq =
n
(Q, w) ∈ Md (P) × (M+ )+ \ (M )⊥ :
dQi
q
∈ L (FT )+ ∀i = 1, ..., d .
wi
dP
Md (P) vector probability measures with components
dQi
dQ
∈ L1 .
Proof of dual representation: Set-valued convex analysis.
analog for convex set-valued risk measures
static set-valued risk measures:
B Jouini, Touzi, Meddeb (2004),
Hamel, Rudloff (2008), Hamel, Heyde (2010), Hamel, Heyde, Rudloff
(2011)
dynamic set-valued risk measures:
B Feinstein, Rudloff (2013a,b),
BenTahar, Lepinette (2013)
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