foundations of quantitative risk measurement€¦ · f hedging the liabilities using a replicating...

Foundations of Quantitative Risk Measurement

Chapter 6: Risk measures

Jan Dhaene and Daniel Linders

November, 2019

0 – Outline 2/41

1. Introduction

2. Risk measuresDefinitionCoherent risk measures

3. Value-at-Risk and Tail Value-at-Risk

4. The required solvency capital

5. The Haezendonck-Goovaerts risk measure

6. VaR, TVaR and comonotonicity

1 – The insurance business 3/41

Inverted production cycle:I Insured: pays a fixed premium.

I Insurer: accept the risk to pay the claim amounts related to possiblefuture events.

I Similar for insurer/reinsurer or reinsurer/reinsurer.

Solvency:I The insurer has to set a premium, such that he is able to pay all the

future benefits to the policy holders.

I When a claim has to be paid, the insurer should be solvent.

Example:I X and/or T can be random.

1 – Risks for an insurer 4/41

Underwriting risk:I The risk that the premiums are not sufficient to cover the obligations

of the insurer.

I Underwriting risk arises when the insurer underestimates the futurelosses.

Credit risk:I The risk that a counter party will not meet his obligations.

I The premiums are invested in bonds (and/or other assets), which candefault.

Market risk:I Exposure to the uncertain future market value of the investment

portfolio.

Other risks:I Liquidity risk, operational risk, . . .

1 – Solvency 5/41

Technical provisions

Technical provisions:I also called: actuarial reserves or best estimate;

I the amount the insurer has to hold in order to cover the expectedfuture claims;

I fair value of the future liabilities: the amount another party is willing topay to take over the insurance business.

Valuation = Market ConsistentI Mark-to-market:

F hedging the liabilities using a replicating portfolio gives a unique price.

I Mark-to-modelF Model-based approach to determine the expectation of the future

liabilities.

F Model risk: use a prudence margin,

1 – Solvency 6/41

What is solvency:I Solvency refers to the ability of the insurer to meet his obligations to

pay the present and future claims related to policyholders.

Solvency rules:I Imposed by the regulatory authority.

I Protecting policyholders.

Solvency Capital Requirements:I In addition to the technical provisions V, the insurer has to hold a

capital buffer.

I Minimal level of capital an insurer has to hold such that the insurer isvery likely to be able to meet his future obligations.

1 – Solvency II 7/41

http://www.solvency-2.com

2 – Risk measures 8/41

Definition

Definition (Risk measure)

Consider a set Λ of real-valued r.v.’s. The function ρ assigning a realnumber ρ [X] to a r.v. X ∈ Λ, is called a risk measure.

ρ is also called a risk measure with domain Λ.I In most cases, Λ is not specified and has to be taken ‘as broad as

possible’.

X represents a loss over a given reference periodI ρ [X] captures the ‘risk’ in a real number.

Example:I Standard deviation risk measure

I ρ [X] = Expected value + safety loading:

ρ [X] = µX + λσX, λ ≥ 0.

2 – Risk measures 9/41

References

Risk measures and premium principles:I Goovaerts, De Vylder & Haezendonck (1984), Kaas, Goovaerts,

Dhaene & Denuit (2008).

Risk measures and capital requirements:I Artzner (1999), Wirch & Hardy (2000).

2 – Properties 10/41

Distributional properties that a risk measure may satisfy

Law invariance:

X d= Y⇒ ρ [X] = ρ [Y] .

Preserving stop-loss order:

X sl Y⇒ ρ [X] ≤ ρ [Y] .

Comonotonic additivity:I For any comonotonic r.v.’s X, Y:

ρ [X + Y] = ρ [X] + ρ [Y] .

2 – Example: the stop-loss risk measure 11/41

Stop-loss premium with retention K ∈ R :I The risk measure ρ is defined as

ρ [X] = E[(X− K)+

].

ExerciseI Prove that ρ is law invariant.

I Prove that ρ preserves stochastic dominance and stop-loss order.

Prove also that ρ is not comonotonic additive.

2 – Properties 12/41

Distribution-free properties that a risk measure may satisfy

Monotonicity:

X ≤ Y⇒ ρ [X] ≤ ρ [Y] .

Positive homogeneity:I For any a > 0,

ρ [aX] = aρ [X] .

Translation invariance:I For any b ∈ R

ρ [X + b] = ρ [X] + b.

Subadditivity:

ρ [X + Y] ≤ ρ [X] + ρ [Y] .

2 – Coherent risk measures1 13/41

Definition (Coherent risk measure)

A risk measure is said to be coherent if it satisfies the properties

1. monotonicity;

2. positive homogeneity;

3. translation invariance;

4. subadditivity.

1Artzner et al. (1999)

2 – Upper expectation 14/41

Example of a coherent risk measure

Upper expectation2

I Π is a subset of all probability measures on the measurable space(Ω,F )

ρΠ [X] = sup EP [X] | P ∈ Π .

I Exercise: Prove that ρΠ is a coherent risk measure.

Interpretation:I The elements of Π are generalized scenarios.

I Expectation of X with respect to a worst-case scenario.

Upper expectations can also be considered as a G-expectations.3

2see Huber (1981)3see e.g. Peng (2007)


Law invariance and upper expectation

Define the risk measure ρ as:

ρ [X] = max E [X] , E [X | Z > z] ,

with P [Z ≤ z] > 0.

ρ is an upper expectation:

I P is the real-world probability measure.

I Q is a distorted probability measure:

Q [A] = P [A | Z > z] , for an event A.

I Then:ρ [X] = max

EP [X] , EQ [X]

.

ρ is not law invariant:

I For Y d= Z and Y and Z independent: ρ [Y] 6= ρ [Z] .


Risk and uncertainty

In Knight (1921), Frank Knight makes a distinction between risk anduncertainty.

Risk refers to the uncertainty about the realization of a loss X.I Risk is modeled by the cdf FX of X, which gives the probability of X

being in a given region.

I If the cdf FX is known, the expectation E[X] can be determined.

In practical situations, the cdf FX is not fully known.I We only have partial information: FX ∈ P .

I This induces an extra source of uncertainty.

A robust estimate for the expectation is the upper expectation

sup EF [X] | F ∈ P .

2 – Coherent risk measures and upper expectation 17/41

Theorem

Consider a risk measure ρ. The following statements are equivalent:

1. The risk measure ρ is coherent.

2. There exist a set Π of probability measures such that

ρ [X] = sup EP [X] | P ∈ Π .

A proof for the case where Ω is finite: Huber (1981).

The more general case is proven in Delbaen (2002).

3 – Value-at-Risk 18/41

Definition

Definition

For any p in (0, 1) the Value-at-Risk at level p is defined by

VaRp [X] = inf x ∈ R | FX (x) ≥ p .

Value-at-Risk = Quantile

I VaRp [X] = F−1X (p)

Interpretation:I Take p close to 1.

I Probability that X exceeds the threshold is small:

P[X > VaRp [X]

]= 1− FX

(VaRp [X]

)︸︷︷︸≥p

≤ 1− p.

3 – Properties of Value-at-Risk 19/41

Value-at-Risk is increasing in pI p < q⇒ VaRp [X] ≤ VaRq [X] .

Positive homogeneous and translation invariant:I For a > 0 and b ∈ R : VaRp [aX + b] = aVaRp [X] + b.

I Exercise: Prove these properties.

VaR is not a coherent risk measureI Exercise: give a counterexample.

3 – Solvency capital requirement 20/41

The VaR for avoiding bankruptcy

Solvency capital requirement:I X = future loss/liabilities.

I V = technical provision (fair valuation of X).

I K [X] = solvency capital

K [X] = VaRp [X]−V

I Extra buffer of capital for bad times4.

Interpretation:I K [X] =VaRp [X−V] .

I X−V = loss which is not covered by the technical provisions

P [X−V > K [X]] = P [X > K [X] + V] ≤ 1− p.

I Insolvency = technical provision + capital buffer are not sufficient tocover to loss:

F For p close to 1, the probability that the insurer is insolvent is small.4p should be sufficiently big, such that K [X] > 0.

3 – Example 21/41

VaR for normal and lognormal distributions

The cdf of a standard normal distribution = Φ.

Assume Y d= N

(µ, σ2) . Then

VaRp [Y] = µ + σΦ−1 (p) .

For a lognormal r.v. X d= eY :

VaRp [X] = eµ+σΦ−1(p).

3 – Problems with the Value-at-Risk 22/41

VaR and low frequency - high impact events

Consider the r.v. X:

P [X = −10000] = 0.999 and P [X = 10 000 000] = 0.001.

high probability of having a gain, but a low probability of having a bigloss.

The VaR does not detect there is an extreme risk:

VaR0.95[X] = −10000.

However, if something goes wrong, i.e. if the loss exceeds the VaR,the loss is enormous:

E [X | X > VaR0.95[X]] = 10 000 000.

3 – Value-at-risk 23/41

“I believe that VaR is the alibi that bankers will give shareholders(and the bailing-out taxpayer) to show documented due diligence, andwill express that their blow-up came from truly unforeseeablecircumstances and events with low probability not from taking largerisks that they didn’t understand. I maintain that VaR encouragesuntrained people to take misdirected risks with shareholders’, andultimately the taxpayers’, money.”

———- Nassim Taleb 1997.

“The risk-taking model that emboldened Wall Street to trade withimpunity is broken and everyone is coming to the realization that noalgorithm can substitute for old-fashioned due diligence. VaR failed todetect the scope of the market’s collapse. The past months haveexposed the flaws of a financial measure based on historical prices.”

———- Financial Reporter Christine Harper, January 2008.

3 – Tail Value-at-risk 24/41

Definition

For any p in (0, 1) the Tail Value-at-Risk at level p is defined by

TVaRp [X] =1

1− p

∫ 1

pVaRq [X] dq.

Interpretation:I TVaR is an average of VaR’s.

I Exercise: prove that VaRp [X] ≤ TVaRp [X] .

What is bad?I Probability that X exceeds VaRp [X] is small, but not 0.

I If X exceeds VaRp [X] , what will be the loss?

I Bad times are those when X ∈[VaRp [X] , TVaRp [X]

].

3 – Example 25/41

TVaR as a cushion for extreme losses

Consider the loss X :

P [X = 0] = 0.6,P [X = 100] = 0.37,

P [X = 10000] = 0.02,P [X = 100000] = 0.01.

VaR0.95 [X] = 100.I Probabillity of a loss bigger than 100 is ‘only’ 5%.

I What can we expect if the loss will exceed VaR0.95 [X] ?

TVaR0.95 [X] = 24040.I If the loss will exceed VaR0.95 [X] , then it will be, on average, equal to

TVaR0.95 [X] .

3 – Tail Value-at-Risk 26/41

Properties of TVaR

Tail Value-at-Risk is increasing in pI p < q⇒ TVaRp [X] ≤ TVaRq [X] .

Positive homogeneous and translation invariant:I For a > 0 and b ∈ R : TVaRp [aX + b] = aTVaRp [X] + b.

I Exercise: Prove these properties.

TVaR is a coherent risk measure.I We give a prove in the next chapter.

3 – Solvency 27/41

Solvency capital requirement:I X is a future loss/liability.

I V = technical provision

I K [X] = solvency capital

K [X] = TVaRp [X]−V.

Interpretation:I K [X] = TVaRp [X−V] ≥ VaRp [X−V] .

I X−V = loss which is not covered by the technical provisions

P[X−V > VaRp [X−V]

]≤ 1− p.

I First buffer = VaRp [X−V] .

I Second buffer = TVaRp [X−V] .

3 – Other risk measures 28/41

Conditional Tail Expectation:I For any p ∈ (0, 1) and any r.v. X, the Conditional Tail Expectation at

level p isCTEp [X] = E

[X | X > VaRp [X]

].

I VaRp [X] ≤ CTEp [X] .

Expected Shortfall:I For any p ∈ (0, 1) and any r.v. X, the Expected Shortfall at level p is

ESFp [X] = E[(

X− VaRp [X])+

].

I Solvency capital = Value-at-Risk + Expected Shortfall.

3 – Relation between VaR, TVaR, CTE and ESF 29/41

Theorem

For p ∈ (0, 1) , we have that

TVaRp [X] = F−1X (p) +

11− p

ESFp [X] ,

CTEp [X] = F−1X (p) +

1

1− FX

[F−1

X (p)]ESFp [X] ,

CTEp [X] = TVaRFX[F−1X (p)] [X] .

Remarks:

I In general, FX

[F−1

X (p)]

is not always equal to p.

I If FX is a continuous distribution function:

CTEp [X] = TVaRp [X] .

3 – The normal distribution 30/41

X is normal distributed with mean µX and variance σ2X.

Value-at-Risk:I For p ∈ (0, 1)

VaRp [X] = µX + σXΦ−1 (p) .

Stop-loss premium:

E[(X− K)+

]= σXφ

(K− µX

σX

)− (K− µX)

[1−Φ

(K− µX

σX

)].

Expected Shortfall:

ESFp [X] = E[(

X− VaRp [X])+

]= σXφ

(Φ−1 (p)

)− σXΦ−1 (p) (1− p) .

3 – The normal distribution 31/41

Conditional Tail Expectation:I For p ∈ (0, 1)

CTEp [X] = F−1X (p) +

1

1− FX

[F−1

X (p)]ESFp [X]

= µX + σXφ(Φ−1 (p)

)1− p

.

I TVaRp [X] = CTEp [X] .

3 – Exercise 32/41

Consider the r.v. X with distribution function

FX (x) =

x , if 0 ≤ x < 0.85,

0.85 , if 0.85 ≤ x < 0.9,0.95 , if 0.9 ≤ x < 0.95

x , if 0.95 ≤ x ≤ 1.

Value-at-Risk:I Prove that the value-at-risk is given by:

Varp [X] =

p , if 0 < p ≤ 0.85,0.9 , if 0.85 < p ≤ 0.95p , if 0.95 < p ≤ 1.

.

Conditional Tail Expectation:I Take p = 0.9. Prove that

CTEp [X] = 0.975.

4 – The required solvency capital 33/41

Regulators view:I V = technical provisions of a future loss X.

I Solvency capital: ρ [X]−V.

I ϕ : risk measure to calculate the shortfall risk.

I Cost of insolvency:ϕ[(X− ρ [X])+

]Investors view:I Holding an extra amount of capital requires a return equal to ε :

(ρ [X]−V) ε.

Cost function:I For ε ∈ (0, 1) :

C (X, ρ [X]) = ϕ[(X− ρ [X])+

]+ (ρ [X]−V) ε.

I Find the optimal capital requirement K such that the cost functionC (X, K) is minimal.

4 – Optimal capital requirement 34/41

Theorem

Take 0 < ε < 1. The cost function C (x, ρ [X]) defined by

C (X, ρ [X]) = E[(X− ρ [X])+

]+ (ρ [X]−V) ε,

reaches its minumum if

ρ [X] = VaR1−ε [X] ,

4 – TVaR as a minimum 35/41

Minimal value of the cost function:

C (X, VaR1−ε [X]) = ε (TVaR1−ε [X]−V) .

Take ε = 1− p. For any K :

C (X, K) ≥ (1− p)(

TVaR [X]p −V)

.

So we find:

TVaRp [X] ≤ K +1

1− pE[(X− K)+

].

Note that we have an equality if K = VaRp[X].

4 – TVaR as a minimum 36/41

Theorem

TVaR is a minimum:

TVaRp [X] = minK

K +

11− p

E[(X− K)+

].

5 – The Haezendonck-Goovaerts risk measure 37/41

Consider an insurer, facing the loss X ∈ (0, max[X]).

Total available capital (including a solvency buffer)= V.

Capital with extra risk buffer = ρ.

The random variable Z is defined as:

Z =(X−V)+

ρ−V.

I If X ≤ V, there is no extra capital needed and Z = 0.

I If Z ≤ 1, the extra buffer ρ−V absorbs the unanticipated losses.

I If Z > 1, the extra buffer ρ−V is not sufficient to cover realized loss.


We force E[Z] to be equal to 1− α, where α ∈ (0, 1).

The solution is then denoted by ρ[X, V]:

E

[(X−V)+

ρ[X, V]−V

]= 1− α.

We can then write:

ρ[X, V] = V +1

1− αE[(X−V)+

].

We determine the capital level V, such that ρ[X, V] is minimal.


The linear Haezendonck-Goovaerts risk measure ρ[X] is definedas:

ρ[X] = infV∈[0,max[X]]

ρ[X, V].

We can write:

ρ[X] = infV∈[0,max[X]]

V +

11− α

E[(X−V)+

]We find that:

ρ[X] = TVaRα[X].

6 – Risk measures and comonotonic sums 40/41

VaR, TVaR, ESF are additive for comonotonic risks

Theorem

Consider a comonotonic random vector (Xc1, Xc

2, . . . , Xcn) and let

Sc = ∑ni=1 wiXc

i , where wi are positive weigth factors. Then we have for allp ∈ (0, 1) that

VaRp [Sc] =n

∑i=1

wiVaRp [Xi] ,

TVaRp [Sc] =n

∑i=1

wiTVaRp [Xi] ,

ESFp [Sc] =n

∑i=1

wiESFp [Xi] .

Exercise: prove this theorem.

6 – CTE and comonotonic risks 41/41

Continuous distribution functions:I Assume that the cdf of Xc

i are continuous.

I Then we have that CTEp [Xi] = TVaRp [Xi] .

I Sc has also a continuous distribution function: CTEp [Sc] = TVaRp [Sc] .

I CTE is additive for comonotonic risks.

In general: CTE is not additive for comonotonic risksI counter example: FY is uniform over [0, 1].

I FX is

FX (x) =

x , if 0 ≤ x < 0.85,

0.85 , if 0.85 ≤ x < 0.9,0.95 , if 0.9 ≤ x < 0.95

x , if 0.95 ≤ x ≤ 1.

I For p = 0.9 : CTEp [Sc] < CTEp [X] +CTEp [Y] .

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