Download Measures of Risk

CHAPTER 6
Measures of Risk
dddd, dddd financial position d risk dddd, dddddddddddd
ddddddddd.
d d d d, d d d d d d d d d d d d d d d d d d d d d d d d, i.e., d d
VT = VT (h̄) dddddd h̄ d final wealth, ddddddd
max VT (h̄).
h̄
ddddd, VT d random variable, ddddddddddddddddd h̄ dd
VT (h̄) dd random variable ddddd. dddddd. Let r = 0, T = 1, V0 = 0. The
bond price B0 = B1 = 1 and the stock price is given by 6.1
1/2
S1(w1) = 12
10
1/2
Figure 6.1. Stock price.
135
S1(w2) = 6
136
⎛
⎞
6. MEASURES OF RISK
0
⎜ −h ⎟
Consider h̄ = ⎝
⎠. Then
h0 /10
⎧
⎪
⎨ 1 h0 ,
0
h
5
S1 =
V1 (h̄) = h̄ · S 1 = −h0 +
⎪
10
⎩ − 2 h0 ,
5
if ω = ω1 ,
if ω = ω1 .
d ω1 d case dd, h0 dddd, dd ω2 d case dd, h0 ddddd. dddddd
dddddddd h̄ dddddd ω ∈ Ω dd, VT (h̄)(ω) dddddd.
ddddddddddddd?
dd random ddd, ddddddddddd, i.e., we consider
max E[VT (h̄)].
h̄
ddddddddddd:
(1) d d d d d d: d d d d d probability measure d expectation? d d d P,
equivalent martingale measure Q, dddddd probability measure? dddd
ddddddd.
(2) dddd: E[VT ]? ddddddddddddd? ddddddd utility
function: d d d d d d. d d d utility function d d d d d d d d d risk
neutral, risk averse, d risk lovingness. ddddddddddddd
max E[U (VT (h̄))],
h̄
dd U d utility function. ddd expectation utility optimization problem.
6.1. MONETARY MEASURE OF RISK
137
ddddddddddddddd, dddddddddd. ddddd? dddd.
Consider
V1 ≡ 1,
⎧
⎪
⎨ 2,
V̄1 =
⎪
⎩ 0,
with probability 1/2,
with probability 1/2,
We see that E[V1 ] = E[V̄1 ] = 1. ddddddddddddd, dd expectation dd
ddd, ddddddddddd mean dddd. ddddddd risk ddd. d
dddddddddddd risk. dddd risk dddd variance. ddddddd
dd mean ddddd, variance ddddd, dddd portfolio theory dddddd
mean-variance portfolio analysis.
dddddddddddddddd ρ ddddd financial position ddd.
6.1. Monetary measure of risk
Consider a measurable space (Ω, F), where Ω is a fixed set of scenarios.1
Notation 6.1. A financial position X : Ω −→ R is the discounted net worth at the
end of the trading period. Denote X a class of financial positions.
Definition 6.2. A mapping ρ : X −→ R is called a monetary measure of risk if for
all X, Y ∈ X ,
(i) (monotonicity) if X ≤ Y , then ρ(X) ≥ ρ(Y ).
(ii) (Cash invariance/translation invariance) if m ∈ R, then
ρ(X + m) = ρ(X) − m.
1We
do not assume that a probability measure is given on Ω
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6. MEASURES OF RISK
Remark 6.3.
(1) ρ(m) = ρ(0) − m for all m ∈ R.
(2) ρ(X + ρ(X)) = 0.
Remark 6.4. Sometimes, without loss of generality, we may assume that a given
monetary risk measure satisfies
(iii) (Normalization) ρ(0) = 0.
dddddddddddd, dddddddd.
Example 6.5. A most trivial example of monetary risk measure is given by
ρ(X) = −E[X].
It is easy to prove that ρ satisfies Definition 6.2 and Condition (iii).
Definition 6.6. On a probabilistic space (Ω, F, P), for some fixed λ ∈ (0, 1), we define
the Value at Risk at level λ as
VaRλ (X) := inf {m : P(X + m < 0) ≤ λ} .
VaRλ (X) dddddddddddd risk-free asset dddd m ddddd m + X
ddddddddd λ.
Example 6.7. Let X ∼ N (c, σ 2 ), then
VaRλ (X) = inf {m : P(X + m < 0) ≤ λ} = inf {m : P(σN + c + m < 0) ≤ λ}
c+m
c+m
= inf m : P N < −
≤ λ = inf m : Φ −
≤λ
σ
σ
c+m
−1
≤ Φ (λ) = inf m : m ≥ −c − σΦ−1 (λ)
= inf m : −
σ
= −c − σΦ−1 (λ) = σΦ−1 (1 − λ) − c,
where N ∼ N (0, 1) and Φ is the cumulative distribution function of N .
6.1. MONETARY MEASURE OF RISK
139
Lemma 6.8. VaRλ (X) is a monetary measure of risk.
Proof.
(i) If X ≤ Y , then for fixed m ∈ R, {X + m < 0} ⊇ {Y + m < 0}.
Thus,
VaRλ (X) = inf {m : P(X + m < 0) ≤ λ}
≥ inf {m : P(Y + m < 0) ≤ λ} = VaRλ (Y ).
(ii) For fixed m ∈ R,
VaRλ (X + m)
inf {m : P((X + m) + m < 0) ≤ λ}
=
k=m+m
=
inf {k − m : P(X + k < 0) ≤ λ}
=
inf {k : P(X + k < 0) ≤ λ} − m = VaRλ (X) − m.
k
k
(iii) Clearly, VaRλ (0) = 0.
Lemma 6.9. Any momentary measure of risk ρ is Lipschitz continuous with respect
to the supremum norm · ∞ ,i.e.,
|ρ(X) − ρ(Y )| ≤ X − Y ∞ .
Proof. Due to X ≤ Y + X − Y ∞ , we have
ρ(X) ≥ ρ (Y + X − Y ∞ ) = ρ(Y ) − X − Y ∞
by monotonicity and cash invariance. Reversing the roles of X and Y yields the assertion.
140
6. MEASURES OF RISK
6.2. Coherent and convex risk measures
Definition 6.10. A monetary risk measure ρ : X −→ R is called a convex measure of risk
if it satisfies
(iv) (Convexity) for 0 ≤ λ ≤ 1,
ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ).
Remark 6.11. If ρ is convex and normalized, then
ρ(λX) ≤ λρ(X),
for 0 ≤ λ ≤ 1,
ρ(λX) ≥ λρ(X),
for λ ≥ 1.
ddddddddddddddddd.
Definition 6.12. A convex measure of risk ρ is called a coherent risk measure if it
satisfies
(v) (Positive Homogeneity) If λ ≥ 0, then ρ(λX) = λρ(X).2
Remark 6.13.
(1) If a monetary measure of risk ρ is positive homogeneous, then
it is normalized (Condition (iii)), i.e., ρ(0) = 0.
(2) Under the assumption of positive homogeneity (Condition (v)), convexity is
equivalent to
(vi) (Subadditivity) ρ(X + Y ) ≤ ρ(X) + ρ(Y ).
Proof.
2ddd,
(1) ρ(0) = 0 comes directly from Condition (v) with λ = 0.
ddddddddd convex risk measure, dd coherent risk measure. dddddddd
dddd, dddddd nonlinear ddd.
6.2. COHERENT AND CONVEX RISK MEASURES
141
(2) Consider λ = 1/2, then
(v)
ρ(X + Y ) = 2ρ
=
1
1
X+ Y
2
2
(iv) 1
1
≤ 2
ρ(X) + ρ(Y )
2
2
ρ(X) + ρ(Y ).
Example 6.14. Consider the worst-case risk measure ρmax defined by
ρmax (X) = − inf X(ω)
for all X ∈ X .
ω∈Ω
The value ρmax is the least upper bound for the potential loss which can occur in any
scenario. Then
(i) If X ≤ Y , then inf X(ω) ≤ inf Y (ω). Thus,
ω∈Ω
ω∈Ω
ρmax (X) = − inf X(ω) ≥ inf Y (ω) = ρmax (Y ).
ω∈Ω
ω∈Ω
(ii) Since inf (X(ω) + m) = inf X(ω) + m, we have
ω∈Ω
ω∈Ω
ρmax (X + m) = − inf (X(ω) + m) = − inf X(ω) − m = ρmax (X) − m.
ω∈Ω
ω∈Ω
(iii) Clearly, ρ(0) = 0.
(iv) Since
inf (λX(ω) + (1 − λ)Y (ω)) ≥ λ inf X(ω) + (1 − λ) inf Y (ω),
ω∈Ω
we see that ρmax is convex.
(v) holds clearly.
ω∈Ω
ω∈Ω
142
6. MEASURES OF RISK
Thus, we know that the worst-case risk measure ρmax is a coherent risk measure. Moreover,
ρmax is the most conservative measure of risk in the sense that any normalized monetary
risk measure ρ on X satisfies
ρ(X) ≤ ρ
inf X(ω) = ρ(0) − inf X(ω) = ρmax (X).
ω∈Ω
ω∈Ω
Note that ρmax can be represented in the form
ρmax (X) = sup EQ [−X],
Q∈M1
where M1 = M1 (Ω, F) denotes the class of all probability measures on (Ω, F)
Example 6.15. Value at Risk satisfies Condition (v) (positive homogeneity), but not
a convex measure.
(1) Claim: VaR satisfies the condition of positive homogeneity.
For given c > 0,
VaRλ (cX)
=
inf{m : P(cX + m < 0) < λ}
=
m
inf m : P X +
<0 <λ
c
k=m/c
=
c inf{k : P(X + k < 0) < λ} = c VaRλ (X).
(2) Claim: VaR is not a convex measure, we need only to find an example.
Consider an investment into two defaultable corporate bonds, each with return
r̃ ∈ (r, 1 + 2r), where r ≥ 0 is the return on a riskless investment. The discounted
net gain of an investment w > 0 in the ith bond is given by
⎧
⎪
⎨ −w,
i
X =
⎪
⎩ 1 · w(1 + r̃) − w = w(r̃ − r) ,
1+r
1+r
in case of default,
otherwise,
6.2. COHERENT AND CONVEX RISK MEASURES
143
for i = 1, 2. Suppose that the two bonds default independently of each other,
each of them with probability p ≤ λ, then
w(r̃ − r)
w(r̃ − r)
1
2
P X + −
<0 =P X + −
< 0 = p ≤ λ.
1+r
1+r
w(r̃ − r)
,
1+r
P X 1 + m < 0 = P X 2 + m < 0 = 1 > λ,
Since for any m < −
thus,
VaRλ (X 1 ) = VaRλ (X 2 ) = −
w(r̃ − r)
< 0.
1+r
Diversifying the portfolio by investing the amount w/2 into each of the two bonds
leads to the position Y = (X 1 + X 2 )/2, which is negative if at least on of the two
bonds defaults. Moreover, we have
P[Y < 0] = P[X 1 < 0] + P[X 2 < 0] − P[X 1 < 0, X 2 < 0] = 2p − p2 .
Consider λ ∈ [p, 2p − p2 ). Since
w
r̃ − r
P Y +
1−
< 0 = p2 < p ≤ λ,
2
1+r
w
r̃ − r
and for m <
1−
,
2
1+r
P [Y + m < 0] ≥ P[Y < 0] = 2p − p2 > λ,
we have
VaRλ (Y ) =
w 1 + 2r − r̃
·
>0
2
1+r
Thus,
VaRλ (Y ) = VaRλ
=
X1 + X2
2
>0>−
1
1
VaRλ (X1 ) + VaRλ (X2 ),
2
2
w(r̃ − r)
1+r
144
6. MEASURES OF RISK
which contradicts to the condition of convexity (Condition (iv)). This proves
that VaRλ is not a convex measure.
dddddddddd Value at Risk dddddd, ddddddddddddd
d?
Definition 6.16. The Average Value at Risk at level λ ∈ (0, 1) of X ∈ X is given by
1 λ
VaRγ (X) dγ.
AVaRλ (X) =
λ 0
Sometimes, the Average Value at Risk is also called the Conditional Value at Risk (CVaRλ (X))
or expected shortfall (ESλ (X)).
Remark 6.17. AVaRλ is a coherent risk measure. Clearly, (i), (ii), (iii), and (v)
hold, since VaRλ satisfies these conditions. Condition (iv) holds due to the representation
theory of AVaRλ , which will be shown in Section 7.4.
Example 6.18. Consider the entropic risk measure with parameter θ > 0 ρent (X) defined by
ρent (X) =
1
log E[e−θX ].
θ
Then ρent (X) is a convex risk measure, but not a coherent risk measure.
Proof.
(i) For X ≤ Y , we have
E[e−θX ] ≥ E[e−θY ],
which implies that ρent (X) ≥ ρent (Y ).
(ii) For constant m ∈ R,
1
1
log E[e−θ(X+m) ] = log e−θm · E[e−θX ]
θ
θ
1 −θX
=
] − θm = ρent (X) − m.
E[e
θ
ρent (X + m) =
6.3. ACCEPTANCE SETS
145
1
log E[e0 ] = 0.
θ
(iv) For λ ∈ (0, 1),
(iii) ρent (0) =
ρent (λX + (1 − λ)Y ) =
1
log E[e−θ(λX+(1−λ)Y ) ]
θ
≤ λρent (X) + (1 − λ)λρent (Y ).
(∗)
(∗): ddddddddd.
(v) Consider λ > 0 and
X=
⎧
⎪
⎨ 0,
with probability 1/2,
⎪
⎩ 2,
with probability 1/2.
Then
1 1 −2θ
1
1
−θX
log E[e
+ e
] = log
ρent (X) =
θ
θ
2 2
1
1
1 1 −2λθ
−θλX
.
] = log
ρent (λX) =
log E[e
+ e
θ
θ
2 2
Clearly, ρent (λX) = λρent (X).
6.3. Acceptance sets
Definition 6.19. A monetary measure of risk ρ induces the class
Aρ := {X ∈ X : ρ(X) ≤ 0}.
The class Aρ is called the acceptance set of ρ.
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6. MEASURES OF RISK
The financial position in Aρ is acceptable in the sense that they do not require additional capital. The following two propositions summarize the relations between monetary
measures of risk and their acceptance sets.
Proposition 6.20. Suppose that ρ is a monetary measure of risk with acceptance set
A := Aρ .
(1) ρ can be recovered from A, i.e.,
ρ(X) = inf{m ∈ R : m + X ∈ A}.
(6.1)
(2) ρ is convex risk measure if and only if A is convex.
(3) ρ is positive homogeneous if and only if A is a cone. In particular, ρ is coherent
if and only if A is a convex cone.
Proof.
(1) Due to cash invariance
ρ(X) = inf{m : ρ(X) ≤ m} = inf{m : ρ(X + m) ≤ 0}
= inf{m ∈ R : m + X ∈ A}.
(2) If ρ is a convex measure of risk, X, Y ∈ A, i.e., ρ(X), ρ(Y ) ≤ 0, then
ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) ≤ 0.
Hence, λX + (1 − λ)Y ∈ A, which implies that A is a convex set. The converse
will be shown later.
(3) If ρ is positive homogeneous and X ∈ A, then for λ ≥ 0,
ρ(λX) = λρ(X) ≥ 0.
Thus, λX ∈ A, which implies that A is a cone. Similar to (3), the converse will
be shown later.
6.3. ACCEPTANCE SETS
147
ddddd? dddddddd.
Proposition 6.21. Assume that A is a non-empty subset of X which satisfies
inf{m ∈ R : m ∈ A} > −∞
(6.2)
and
X ∈ A,
Y ∈ X,
Y ≥X
=⇒
Y ∈ A.
(6.3)
Then the functional ρA defined by
ρA (X) := inf{m ∈ R : m + X ∈ A}.
(6.4)
has the following properties:
(1) ρA is a monetary measure of risk.
(2) If A is a convex set, then ρA is a convex measure of risk.
(3) If A is a cone, then ρA is positive homogeneous. In particular, if A is a convex
cone, then ρA is a coherent measure of risk.
Proof.
(1) (a) Monotonicity: If X ≤ Y , by (6.3) we know that for m + X ∈ A,
m + Y ∈ A. This implies that
{m : m + X ∈ A} ⊆ {m : m + Y ∈ A}.
Thus, ρA (X) ≥ ρA (Y ).
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6. MEASURES OF RISK
(b) Cash invariance: If m ∈ R,
ρA (X) = inf{k ∈ R : k + X ∈ A}
= inf{k ∈ R : k + (m + X) ∈ A} + m
= ρA (X + m) + m.
(2) Suppose that X1 , X2 ∈ X , m1 , m2 ∈ R such that m1 + X1 , m2 + X2 ∈ A. Since
A is convex,
λ(m1 + X1 ) + (1 − λ)(m2 + X2 ) = (λm1 + (1 − λ)m2 ) + (λX1 + (1 − λ)X2 ) ∈ A
for λ ∈ (0, 1). Hence
ρA (λX1 + (1 − λ)X2 ) ≤ λm1 + (1 − λ)m2
for all m1 , m2 such that m1 +X1 , m2 +X2 ∈ A. Taking infimum on the right-hand
side, we get the convexity condition
ρA (λX1 + (1 − λ)X2 ) ≤ λρA (X1 ) + (1 − λ)ρA (X2 ).
(3) (a) Suppose that A is a cone, i.e. if Z ∈ A, cZ ∈ A for c ≥ 0. Thus,
ρA (λX)
=
inf{m ∈ R : m + λX ∈ A}
=
m
inf m ∈ R :
+X ∈A
λ
=
λ inf {m ∈ R : m + X ∈ A} = λ ρA (X),
m =m/λ
which implies that ρA is positive homogeneous.
(b) If A is a convex cone, then we know that ρA satisfies the Conditions (iv)
and (v), which implies that ρA is a coherent risk measure.
6.3. ACCEPTANCE SETS
Proof of Proposition 6.20 (-continuous).
149
(3) Due to (6.1) and (6.4) and
the second assertion in Proposition 6.21 we know that if A is convex, ρ is a convex
risk measure.
(4) Similarly to (3), due to the third assertion in Proposition 6.21 we see that if A
is a cone, then ρ is positive homogeneous.
In the following examples, we take X as the linear space of all bounded measurable
functions on some measurable space (Ω, F).
Example 6.22.
(1) Consider the worse-case risk measure ρmax (see Example 6.14).
The acceptable set of ρmax is given by
Aρmax = {X ∈ X : ρmax (X) ≤ 0} = {X ∈ X : inf X(ω) ≥ 0}
ω∈Ω
= {X ∈ X : X(ω) ≥ 0 for all ω ∈ Ω},
which is clearly a convex cone. By Proposition 6.20, we see that ρmax is a coherent
risk measure.
(2) Consider a utility function u on R, constant c ∈ R, and a probability measure
Q ∈ M1 . Take a class
A := {X ∈ X : EQ [u(X)] ≥ u(c)}
as a set of acceptable positions. Then
(a) Clearly, (6.2) is satisfied.
(b) If X ∈ A, Y ∈ X , Y ≥ X, then
EQ [u(Y )] ≥ EQ [u(X)] ≥ u(c),
which means Y ∈ A.
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6. MEASURES OF RISK
(c) If X, Y ∈ A, λ ∈ (0, 1), then
EQ [u(λX + (1 − λ)Y )] ≥ λEQ [u(X)] + (1 − λ)EQ [u(Y )]
≥ λu(c) + (1 − λ)u(c) = u(c),
i.e., λX + (1 − λ)Y ∈ A. This implies that A is a convex set.
By Proposition 6.21 we may use the set A to induce a convex measure of risk ρA .
(3) Fixed a probability measure P on (Ω, F). Consider a one-period model with
interest rate r = 0. Then for an asset with payoff X̃ ∈ L2 = L2 (Ω, F, P),
variance = σ 2 (X̃) = 0 at time 1, price = π(X̃) at time 0, define
X = X̃ − π(X̃)
the corresponding discounted net worth. The Sharpe ratio is defined as
E[X]
E[X̃] − π(X̃)
.
=
σ(X)
σ(X̃)
Suppose that we find the position X is acceptable if the Sharpe ratio is bounded
from below by some constant c > 0, i.e.,
Ac := {X ∈ L2 : E[X] ≥ c · σ(X)}.
Note that Ac is a convex cone. However, Ac does not satisfy (6.3). Hence we
cannot apply Proposition 6.21 to conclude that the function defined by (6.4) is a
coherent measure of risk. Moreover, consider the corresponding function defined
6.3. ACCEPTANCE SETS
151
by (6.4) on L2
ρc (X) = inf{m ∈ R : m + X ∈ Ac }
= inf{m ∈ R : E[X + m] ≥ c · σ(X + m)}
= inf{m ∈ R : E[X] + m ≥ c · σ(X)}
= E[−X] + c · σ(X).
Then we have
(a) (Cash invariance) If m ∈ R, then
ρc (X + m) = E[−X − m] + c · σ(X + m)
= E[−X] + c · σ(X) − m = ρc (X) − m.
(b) (Convexity) Since σ(·) is a convex functional on L2 , ρc is convex.
(c) (Positive homogeneity) If λ ≥ 0, then
ρc (λX) = E[−λX] + c · σ(λX) = λ E[−X] + cλ · σ(X)
= λ (E[−X] + c · σ(X)) = λ ρc (X).
But ρc is not a monetary risk measure, since it is not monotonic.
(4) The acceptance set of the entropic risk measure is the set of payoffs with positive
expected utility, i.e.,
Aρent = {X ∈ Lp : E[e−θX ] ≤ 1}.
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6. MEASURES OF RISK
6.4. Robust representation of coherent risk measure
Notation 6.23. Denote
M1 := M1 (Ω, F) =
M1,f := M1,f (Ω, F) =
the set of all probability measures on (Ω, F).
the set of all finitely additive set functions
Q : F −→ [0, 1] which is normalized to Q(Ω) = 1.
Clearly, we have
M1 M1,f .
Proposition 6.24. ρ : X −→ R is a coherent measure of risk if and only if there
exists Q ⊆ M1,f such that
ρ(X) = sup EQ [−X]
for all X ∈ X .
Q∈Q
Example 6.25.
(1) As in Example 6.5,
ρ(X) = E[−X] = sup EQ [−X],
Q∈Q
where Q = {P}.
(2) The worst-case risk measure ρmax :
ρmax = − inf X(ω) = sup(−X(ω)) = sup EQ [−X] = sup EQ [−X],
ω∈Ω
ω∈Ω
Q∈M1
Q∈Q
where Q = M1
(3) The Average Value at Risk AVaRλ (X) has the representation
AVaRλ (X) = max EQ [−X],
Q∈Qλ
for allX ∈ X ,
6.5. ROBUST REPRESENTATION OF CONVEX RISK MEASURES
153
where Qλ is the set of all probability measures Q P whose density dQ/dP3 is
P-a.s. bounded by 1/λ, i.e.,
Qλ =
1
dQ
≤
QP:
dP
λ
P-a.s. .
6.5. Robust representation of convex risk measures
Lemma 6.26. Let α : M1,f −→ R ∪ {∞} be a functional satisfying
inf
Q∈M1,f
α(Q) ∈ R.
Then
ρ(X) = max (EQ [−X] − α(Q))
Q∈M1,f
(6.5)
is a convex measure of risk.
Notation 6.27. The function α in (6.5) is called the penalty function for ρ in M1,f .
Theorem 6.28. Any convex risk measure ρ on X is of the form
ρ(X) = max (EQ [−X] − αmin (Q))
Q∈M1,f
where the penalty function αmin is given by
αmin (Q) = sup EQ [−X]
X∈Aρ
for Q ∈ M1,f . Moreover, αmin is the minimal penalty function satisfying (6.5).
3Q
P ddddd dQ/dP dd density, ddd Chapter 9 ddddd.
154
6. MEASURES OF RISK
Remark 6.29. The robust representation of a coherent risk measure is a special case
of the robust representation of a convex risk measure, since it corresponds to the penalty
⎧
⎪
⎪
⎨0
function
α(Q) =
⎪
⎪
⎩+∞
if Q ∈ Q,
if Q ∈ Q.