Risk Measures and Optimal Portfolio Selection (with ... - CiteSeerX

Risk Measures and Optimal Portfolio Selection (with applications to elliptical distributions) Jan Dhaene§ Emiliano A. Valdez‡ Tom Hoedemakers§ §

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University of New South Wales, Sydney, Australia

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Katholieke Universiteit Leuven, Belgium

Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 1/278

Table of Contents: 1 2 3 4 5 6 7

Solvency Capital, Risk Measures and Comonotonicity..................................... pp. 3-48 Comonotonicity and Optimal Portfolio Selection............................................ pp. 49-88 Elliptical Distributions - An Introduction........................................................ pp. 89-103 Tail Conditional Expectations for Elliptical Distributions.............................. pp. 104-119 Bounds for Sums of Non-Independent Log-Elliptical Random Variables.... pp. 120-142 Capital Allocation and Elliptical Distributions.............................................. pp. 143-159 Convex Bounds for Scalar Products of Random Variables (With Applications to Loss Reserving and Life Annuities).................................................................... pp. 160-278


Lecture No. 1 Solvency Capital, Risk Measures and Comonotonicity

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Risk measures • Risk : random future loss.

• Risk Measure: mapping from the set of quantifiable risks to

the real line:

• Actuarial examples:

X → ρ(X).

◦ premium principles, ◦ technical provisions (liabilities), ◦ solvency capital requirements.

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◦ business risks, ◦ event risks.


Required vs. available capital • Required capital: required assets ρ(X) minus liabilities

L(X), to ensure that obligations can be met: K(X) = ρ(X) − L(X).

• Different kinds of capital:

◦ regulatory capital: you must have, ◦ rating agency capital: you are expected to have, ◦ economic capital: you should have,

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◦ available capital: you actually have.


Required vs. available capital • Parameters: ◦ default probability,

◦ time horizon, ◦ run-off vs. wind-up vs. going concern, ◦ valuation of liabilities: mark-to-model, ◦ valuation of assets: mark-to-market.

• Total balance sheet capital approach:

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ρ(X) = L(X) + K(X).


The quantile risk measure • Quantiles:

Qp (X) = inf {x ∈ R | FX (x) ≥ p} , 1

p ∈ (0, 1). FX (x)

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The quantile risk measure • Determining the required capital by

K(X) = Q0.99 (X) − L(X),

we have K(X) = inf {K | Pr [X > L(X) + K] ≤ 0.01} . • Qp (X) = F −1 (p) = V aRp (X). X • Meaningful when only concerned about ”frequency of

default” and not ”severity of default”.

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• Does not answer the question ”how bad is bad?”


Tail Value-at-Risk and Conditional Tail Expectation • Tail Value-at-Risk :

1 T V aRp (X) = 1−p

Z

1

Qq (X) dq, p

p ∈ (0, 1).

• Determining the required capital by

K(X) = T V aR0.99 (X) − L(X),

we define ”bad times” if X in ”cushion” [Q0.99 (X), T V aR0.99 (X)]. • Conditional Tail Expectation:

CT Ep (X) = E [X | X > Qp (X)] ,

p ∈ (0, 1) .

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• CT Ep = expectation of the top (1 − p)% losses. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 10/278

Relations between risk measures • Expected Shortfall:

ESFp (X) = E (X − Qp (X))+ ,

p ∈ (0, 1).

• ESFp (X) = expectation of shortfall in case required capital

K(X) = Qp (X) − L(X).

• Relations:

1 ESFp (X), 1−p 1 ESFp (X), CT Ep (X) = Qp (X) + 1 − FX (Qp (X))

T V aRp (X) = Qp (X) +

CT Ep (X) = T V aRFX (Qp (X)) (X).


Relations between risk measures • When FX is continuous:

CT Ep (X) = T V aRp (X). ESFp(X) 1

FX (x)

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Qp(X) TVaRp(X)


Normal random variables • Let X ∼ N µ, • Quantiles:

σ2

.

Qp (X) = µ + σ Φ−1 (p) .

where Φ denotes the standard normal distribution function. • Expected Shortfall: 0

ESFp (X) = σ Φ Φ

−1

• Conditional Tail Expectation:

(p) − σ Φ−1 (p) (1 − p) .

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CT Ep (X) = µ + σ

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Lognormal random variables • Let ln X ∼ N µ,

σ2

• Quantiles:

.

Qp (X) = eµ+σ

Φ−1 (p)

.

• Expected Shortfall:

ESFp (X) = e

µ+σ 2 /2

−eµ+σ

Φ σ−Φ

Φ−1 (p)

−1

(p)

(1 − p) .

• Conditional Tail Expectation:

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CT Ep (X) = e

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Φ−1 (p)


Risk measures and ordering of risks • Ordering of risks:

◦ Stochastic dominance:

X ≤st Y ⇔ FX (x) ≥ FY (x) for all x. ◦ Stop-loss order:

X ≤sl Y ⇔ E[(X − d)+ ] ≤ E[(Y − d)+ ] for all d. ◦ Convex order:

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X ≤cx Y ⇔ X ≤sl Y and E[X] = E[Y ].


Risk measures and ordering of risks • Stochastic dominance vs. ordered quantiles:

X ≤st Y ⇔ Qp (X) ≤ Qp (Y ) for all p ∈ (0, 1). • Stop-loss order vs. ordered TVaR’s:

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X ≤sl Y ⇔ T V aRp (X) ≤ T V aRp (Y ) for all p ∈ (0, 1).


Comonotonicity • A set S ⊂ Rn is comonotonic ⇔

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for all x and y in S either x ≤ y or x ≥ y holds.



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• A comonotonic set is a “thin” set.


Comonotonicity • A random vector (X1 , . . . , Xn ) is comonotonic ⇔

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d. d, Z > d, X and Y are comonotonic, but not perfectly correlated.


Comonotonic bounds for sums of dependent r.v.’s • Theorem: For any (X1 , X2 , . . . , Xn ) and any Λ, we have n X i=1

E [Xi | Λ] ≤cx

n X i=1

Xi ≤cx

n X

FX−1i (U ).

i=1

• Notations: P ◦ S= n ◦ ◦

i=1 Xi . Pn l S = i=1 E [Xi | Λ] = lower bound. Pn c S = i=1 FX−1i (U ) = comonotonic upper

bound.

• If all E [Xi | Λ] are % functions of Λ,

then S l is a comonotonic sum.


Risk measures and comonotonicity • Additivity of risk measures of comonotonic sums:

Qp ( T V aRp (

n X i=1 n X i=1

Xic ) = Xic ) =

n X i=1 n X

Qp (Xi ). T V aRp (Xi ).

i=1

• Sub-additivity of risk measures: Any risk measure that ◦ preserves stop-loss order ◦ is additive for comonotonic risks

is sub-additive: ρ(X + Y ) ≤ ρ(X) + ρ(Y ).

• Examples:

◦ TailVaRp is sub-additive. ◦ CTEp , Qp and ESFp are NOT sub-additive. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 22/278

Distortion risk measures • Expectation of a r.v.:

E[X] = −

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[1 − F¯X (x)] dx +

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with F¯X (x) = Pr[X > x].


Distortion risk measures 1 II

FX (x)

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with F¯X (x) = Pr[X > x]. • Distortion function:

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g : [0, 1] → [0, 1] is a distortion function ⇔ g is %, g(0) = 0 and g(1) = 1.


Distortion risk measures:

g(x) concave ⇒ g(x) ≥ x

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with F¯X (x) = Pr[X > x]. • Distortion function:

g : [0, 1] → [0, 1] is a distortion function ⇔ g is %, g(0) = 0 and g(1) = 1.

• Distortion risk measure:

ρg [X] = −

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1 − g F¯X (x)

dx +

∞ 0

¯ g FX (x) dx.

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ρg [X] = “distorted expectation” of X .

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Distortion risk measures:

g(x) ≥ x 1 II II'

FX (x)

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E[X] = I − (II+II') ρg [X] = (I+I') − II ≥ E[X]

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Examples of distortion risk measures • Expectation: X → E[X].

g(x) = x,

0 ≤ x ≤ 1.

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Examples of distortion risk measures • The quantile risk measure: X → Qp (X).

g(x) = I (x > 1 − p) ,

0 ≤ x ≤ 1.

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Examples of distortion risk measures • Tail Value-at-Risk : X → T V aRp (X).

g(x) = min

x ,1 , 1−p

0 ≤ x ≤ 1.

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g(x)

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1−p

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Examples of distortion risk measures • Conditional Tail Expectation: X → CT Ep (X).

is NOT a distortion risk measure.

• Expected Shortfall: X → ESFp (X).

is NOT a distortion risk measure.

• Stoch. dominance vs. ordered distortion risk measures:

X ≤st Y ⇔ ρg [X] ≤ ρg [Y ] for all distortion functions g.


The Wang transform risk measure • Problems with TVaR p : ◦ no incentive for taking actions that increase the

distribution function for outcomes smaller than Qp , ◦ accounts for the ESF ⇒ does not adjust for extreme low-frequency, high severity losses. • The Wang transform risk measure :

X → ρgp (X),

0 < p < 1,

with

gp (x) = Φ Φ

−1

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The Wang transform risk measure • Examples:

◦ if X is normal: ρgp (X) = Qp (X). ◦ if X is lognormal: ρgp (X) = Q Φ[Φ−1 (p)+ σ ] (X).

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Properties of distortion risk measures • Additivity for comonotonic risks:

ρg [X1c + X2c + . . . + Xnc ] =

n X

ρg (Xi ).

i=1

• Positive homogeneity : for any a > 0,

ρg [aX] = aρg [X]. • Translation invariance:

ρg [X + b] = ρg [X] + b. • Monotonicity :

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0, b > 0)

1 Fβ (x) = β (a, b)

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x 0

ta−1 (1 − t)b−1 dt,

0 ≤ x ≤ 1.

• The Beta distortion risk measure:

X → ρFβ (X). ρFβ strictly preserves stop-loss order provided 0 < a ≤ 1, b ≥ 1 and a and b are not both equal to 1.

• A PH-transform risk measure: Wang (1995). T

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Sub-additivity of risk measures • Merging decreases the ‘insolvency risk’:

(X + Y − ρ [X] − ρ [Y ])+ ≤ (X − ρ [X])+ + (Y − ρ [Y ])+ ◦ Sub-additivity is allowed to some extent.

• Concave distortion risk measures are sub-additive:

ρg [X + Y ] ≤ ρg [X] + ρg [Y ] . ◦ Qp is not sub-additive, ◦ T V aRp is sub-additive. • Optimality of T V aRp :

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T V aRp (X) = min {ρg (X) | g is concave and ρg ≥ Qp } . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 38/278

Axiomatic characterization of risk measures • A risk measure is "Artzner-coherent” if it is sub-additive,

monotone, positive homogeneous and translation invariant. ◦ Qp is not ”coherent”. ◦ Concave distortion risk measures are ”coherent”.

• The Dutch risk measure:

ρ(X) = E [X] + E (X − E [X])+ . ρ(X) is coherent, but not comonotonic-additive ⇒ ρ(X) is NOT a distortion risk measure.

• Coherent or not?

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Markowitz (1959): “We might decide that in one context one basic set of principles is appropriate, while in another context a different set of principles should be used.” Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 39/278

Distortion risk measures for sums of dependent r.v.’s • Approximations for sums of dependent r.v.’s:

S=

Pn

i=1 Xi

Sl =

with given marginals, but unknown copula.

n X i=1

E [Xi | Λ] ≤cx S ≤cx

n X

FX−1i (U ) = S c

i=1

• Approximations for ρg [S]: (if all E [Xi | Λ] are % in Λ)

ρg [S c ] =

i=1 n X i=1

ρg [Xi ] , ρg [E (Xi | Λ)] .

l • If g is concave: ρg S ≤ ρg [S] ≤ ρg [S c ] . {

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h i ρg S l =

n X


Application: provisions for future payment obligations • Problem description ◦ Consider a payment obligation of 1 per year, due at

times 1, 2, ..., 20,

◦ Let e−Y (i) be the discount factor over [0, i]:

e−Y (i) ≡ e−(Y1 +Y2 +...+Yi ) . ◦ Assume the yearly returns Yj are i.i.d. and normal

distributed with parameters µ = 0.07 and σ = 0.1. ◦ The stochastic provision is defined by S=

20 X

e−(Y1 +Y2 +...+Yi ) .

i=1


Provisions for future payment obligations • Convex bounds for S =

P20

Let Λ = Then

i=1 Yi

P20

j=i

where S S

l

c

= =

n X i=1 n X

P20

−Y (i) e i=1 e−jµ and ri = corr [Λ, Y (i)] > 0.

S l ≤cx S ≤cx S c

e

−E[Y (i)]−ri σY (i) Φ−1 (U )+ 12 (1−ri2 )σY2 (i)

e

−E[Y (i)]+ σY (i) Φ−1 (U )

,

.

i=1


Provisions for future payment obligations • Provision (or total capital requirement) ◦ The provision for this series of future obligations is set

equal to ρg [S] ◦ Approximate ρg [S] by

ρg [S c ] =

i=1 n X i=1

ρg [Xi ] , ρg [E (Xi | Λ)] .

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Provisions for future payment obligations

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• The Quantile-provision principle: ρg [S] = Qp [S]

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Provisions for future payment obligations • The CTE-provision principle: ρg [S] =TVaRp [S]

TVARp [S l ] 17.24 18.45 20.03 21.22 23.98

‘TVARp [S]’ 17.26 18.50 20.10 21.30 24.19

TVARp [S c ] 18.61 20.14 22.16 23.69 27.29

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Theories of choice under risk • Expected utility theory :

◦ von Neumann & Morgenstern (1947). ◦ Prefer loss X over loss Y if

E [u(w − X)] ≥ E [u(w − Y )] , ◦ u(x) = utility of wealth-level x, % function of x. ◦ Risk aversion: u is concave.

• Yaari’s dual theory of choice under risk : ◦ Yaari (1987). ◦ Prefer loss X over loss Y if

ρf [w − X] ≥ ρf [w − Y ] ,

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◦ f (q) = distortion function. ◦ Risk aversion: f is convex. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 46/278

Compare theories of choice under risk • Transformed expected wealth levels:

E[w − X] = E[u(w − X)] = ρf [w − X] =

Z

Z

Z

1

Q1−q (w − X) dq,

0 1

u [Q1−q (w − X)] dq,

0 1 0

Q1−q (w − X) df (q).

• Ordering of risks:

◦ In both theories, stochastic dominance reflects

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common preferences of all decision makers. ◦ In both theories, stop-loss order reflects common preferences of all risk-averse decision makers. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 47/278

References (see www.kuleuven.ac.be/insurance) • Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002a). “The

concept of comonotonicity in actuarial science and finance: Theory” , Insurance: Mathematics & Economics, vol. 31(1), 3–33.

• Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002b). “The

concept of comonotonicity in actuarial science and finance: Applications”, Insurance: Mathematics & Economics, vol. 31(2), 133–161.

• Dhaene, Vanduffel, Tang, Goovaerts, Kaas, Vyncke (2003).

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“Capital requirements, risk measures and comonotonicity”


Lecture No. 2 Comonotonicity and Optimal Portfolio Selection

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Introduction • Strategic portfolio selection:

For a given savings and/or consumption pattern over a given time horizon, identify the best allocation of wealth among a basket of securities.

• The ’Terminal Wealth’ problem:

◦ Saving for retirement. ◦ A loan with an amortization fund with random return.

• The ’Reserving’ problem:

◦ The ’after retirement’ problem. ◦ Technical provisions.

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◦ Capital requirements.


Introduction • The ’Buy and Hold’ strategy :

◦ Keep the initial quantities constant. ◦ A static strategy.

• The ’Constant Mix’ strategy :

◦ Keep the initial proportions constant. ◦ A dynamic strategy.


Comonotonicity • Notations: ◦ U : uniformly distributed on (0, 1). ◦ X = (X1 , . . . , Xn ) .

◦ F −1 (p) = Qp [X] = VaRp [X]= inf {x ∈ R | FX (x) ≥ p} . X

• Comonotonicity of a random vector :

X is comonotonic ⇔ there exist non-decreasing functions f1 , . . . , fn and a r.v. Z such that d

X = [f1 (Z), . . . , fn (Z)] . • Comonotonicity: very strong positive dependency structure.

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• Comonotonic r.v.’s cannot be pooled.


Comonotonic bounds for sums of dependent r.v.’s • Theorem:

For any X and any Λ, we have n X i=1

E [Xi | Λ] ≤cx

n X i=1

Xi ≤cx

n X

FX−1i (U ).

i=1

• Notations: ◦ S= ◦ ◦

Pn

i=1 Xi . Pn l S = i=1 E [Xi | Λ] Pn c S = i=1 FX−1i (U )

= lower bound. = comonotonic upper bound.

• If all E [Xi | Λ] are increasing functions of Λ,

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Performance of the comonotonic approximations • Local comonotonicity:

Let B(τ ) be a standard Wiener process. The accumulated returns exp [µτ + σ B(τ )] , exp [µ (τ + ∆τ ) + σ B (τ + ∆τ )]

will be ’almost comonotonic’. • The continuous perpetuity :

S=

Z

∞ 0

exp [−µτ − σ B(τ )] dτ

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has a reciprocal Gamma distribution.


µ = 0.07 and σ = 0.1.

10

15

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25

30

35

Numerical illustration:

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Provisions for future liabilities • Problem description:

◦ α1 , . . . , αn : positive payments, due at times 1, . . . , n. ◦ R0 = initial provision established at time 0. ◦ Investment strategy : π (t) = (π1 , π2 , . . . , πm ). ◦ Provision at time j:

Rj (R0, π) = Rj−1 (R0, π) eYj (π) − αj

with R0 (R0, π) = R0 . ◦ What is the optimal investment strategy π ∗ ? ◦ Answer depends on ‘initial provision’ R0 and ‘probability level’ p.


The stochastic provision • Definition:

S (π) =

n X

αi e−(Y1 (π)+Y2 (π)+···+Yi (π)) .

i=1

• Relation:

Rn (R0, π) = (R0 − S (π)) e(Y1 (π)+···+Yn (π)) . • An investment strategy π is only acceptable if

Pr [Rn (R0, π) ≥ 0] is ”large enough”.

• Relation:

Pr [Rn (R0, π) ≥ 0] = Pr [S (π) ≤ R0 ] .

• PROBLEM: d.f. of S (π) too cumbersome to work with. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 78/278

Comonotonic approximations for S (π) • The comonotonic upper bound for S (π):

S (π) ≤cx S c (π) . • A comonotonic lower bound for S (π):

P P ◦ S l (π) = E S (π) n Yj (π) n αk e−k[µ(π)−σ 2 (π)] . k=j j=1

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◦ S l ≤cx S (π). ◦ S l (π) is a comonotonic sum.


Optimal investment strategies • The Initial Provision: ◦ Definition:

R0 (π) = Eg [S (π)]

where S (π) is the Stochastic Provision. ◦ Eg [·] is a ‘distortion risk measure’. ◦ If g is concave, then Eg [·] is a ‘coherent’ risk measure.

• The optimal investment strategy : (π ∗ , R0∗ ) follows from

R0∗ = min Eg [S (π)]

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Reduced optimization problem • For σ (π 1 ) = σ (π 2 ) and µ (π 1 ) < µ (π 2 ) , we have that

S (π 2 ) ≤st S (π 1 ) . • Hence,

min Eg [S (π)] = min Eg [S (π σ )] . σ

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Minimizing the Initial Provision, for a given p • The p - quantile provision principle:

If investment strategy = π , then R0 (π) = Qp [S (π)] = inf {x | Pr [Rn (x, π) ≥ 0] ≥ p} . • Optimal strategy : (π ∗ , R0∗ ) follows from

R0∗ = min Qp [S (π)] . π

• Approximation: (π l , R0l ) follows from

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= min Qp S (π ) . σ

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Numerical illustration • Available assets: ◦ 1 riskfree asset with r = 0.03 ◦ 2 risky assets with

µ1 = 0.06, µ2 = 0.10,

and Corr • The tangency portfolio:

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Numerical illustration • Yearly consumptions: α1 = . . . = α40 = 1. • Stochastic provision:

S (π) =

40 X

e−(Y1 (π)+Y2 (π)+···+Yi (π)) .

i=1

• Optimal investment strategy :

R0∗ = min Qp [S (π)] . π

• Approximation:

R0l = min Qp [S (π σ )] .

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Numerical illustration 1.3

23

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minimal reserve

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0.4

optimal risky proportion

21

0.3 18

0.2 0.1

17 0.80

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Solid line (left scale): minimal initial provision R0l as a function of p. Dashed line (right scale): optimal proportion invested in the tangency portfolio.


Other optimization criteria • Minimizing the Initial Provision, given p:

R0∗ = min CTEp [S (π)] π

with CTEp [X] = E [X | X > Qp [X]] .

• Maximizing p for a given Initial Provision R0 :

p∗ = max Pr [Rn (R0 , π) > 0] .

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Generalizations • Investment restrictions: are taken into account by redefining

the set of efficient portfolios.

• Yaari’s dual theory : The ‘final wealth problem’ can be solved

for general distorted expectations.

• Distortion risk measures: The initial provision can be

defined in terms of general distortion risk measures.

• Stochastic sums: ‘How to avoid outliving your money?’

• Positive and negative payments: ‘The savings - retirement

problem’.

• Other distributions: Lévy-type or Elliptical-type distributions


Some references (www.kuleuven.ac.be/insurance) [1] Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002a). The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Mathematics & Economics, vol. 31(1), 3–33. [2] Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2002b). The concept of comonotonicity in actuarial science and finance: Applications. Insurance: Mathematics & Economics, vol. 31(2), 133–161. [3] Dhaene, Vanduffel, Goovaerts, Kaas, Vyncke (2004). Comonotonic approximations for optimal portfolio selection problems. (forthcoming)

[4] Dhaene, Vanduffel, Tang, Goovaerts, Kaas, Vyncke (2003). Risk measures and comonotonicity. (submitted)


Lecture No. 3 Elliptical Distributions - An Introduction

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Elliptical Distributions • This family coincides with the family of symmetric

distributions in the univariate case (e.g. normal, Student-t) and can be characterized using either: ◦ characteristic generator ◦ density generator

• References: ◦ Landsman and Valdez (2003) “Tail Conditional Expectations for Elliptical Distributions”, North American Actuarial Journal. ◦ Valdez and Dhaene (2004) “Bounds for Sums of Non-Independent Log-Elliptical Random Variables”, work in progress.

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• Normalizing constant: cn =

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where parameter

(x−µ)T Σ−1 (x−µ) 2kp

i−p

Γ(p) −n/2 (2πk ) p Γ(p−n/2)

• If p = (n + m) /2 where n, m are integers, and kp = m, we

get the traditional form of the multivariate Student t with density:

Γ n+m 2 p m (πm)n/2 Γ 2 |Σ|

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Generalized Student-t Distribution • Density: fX (x) = √ σ

1 2kp B(1/2,p−1/2)

B (·, ·) is the beta function.

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(x−µ)2 2kp σ 2

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, where

• For p > 3/2, usually kp = (2p − 3)/2 becaue it leads to the

important property that V ar (X) = σ 2 .

• For 1/2 < p ≤ 3/2, variance does not exist and kp = 1/2. • Note for example in the case where p = 1, we have

standard Cauchy distribution: "

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It is well-known that mean and variance for this distribution does not exist. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 99/278

Density Functions of GST - Figure 1

0.5 p = 2.5 p=5

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normal

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Multivariate Logistic Family e−u (1+e−u )2

• Density generator: g (u) = • Density:

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1 −1 (x − µ) exp − (x − µ) Σ 2 cn fX (x) = p io2 n h T |Σ| 1 + exp − 1 (x − µ) Σ−1 (x − µ) 2

• Normalizing constant:



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(−1)j−1 j 1−n/2 

5

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9

0 • Density:

n rh is o cn exp − (x − µ)T Σ−1 (x − µ) fX (x) = p 2 |Σ|

• Normalizing constant:

cn =

sΓ (n/2) (2π)n/2 Γ (n/2s)

rn/2s

• When r = s = 1, this reduces to multivariate normal. When √

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Lecture No. 4 Tail Conditional Expectations for Elliptical Distributions

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Introduction • Developing a standard framework for risk measurement is

becoming increasingly important.

• This paper is about a risk measure called tail conditional

expectations and their explicit forms for the family of elliptical distributions.

• This family coincides with the family of symmetric

distributions in the univariate case (e.g. normal, Student-t) and can be characterized using either:

◦ characteristic generator ◦ density generator


Introduction - continued • We introduce the notion of a cumulative generator which

plays a key role in computing tail conditional expectations.

• We extended the ideas into the multivariate framework

allowing us to decompose the total of the tail conditional expectations into its various constituents. ◦ decomposing the total into an allocation formula

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• Landsman and Valdez (2003)


Risk Measure • A risk measure ϑ is a mapping from the space of random

variables L to the set of real numbers: ϑ : X ∈ L → R.

• Some useful properties of a risk measure:

1. Monotonicity: X1 ≤ X2 with probability 1 =⇒ ϑ (X1 ) ≤ ϑ (X2 ) .

2. Homogeneity: ϑ (λX) = λϑ (X) for any non-negative λ. 3. Subadditivity: ϑ (X1 + X2 ) ≤ ϑ (X1 ) + ϑ (X2 ) .

4. Translation Invariance: ϑ (X + α) = ϑ (X) + α for any constant α. • Some consequences:

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ϑ (0) = 0; a ≤ X ≤ b =⇒ a ≤ ϑ (X) ≤ b; ϑ (X − ϑ (X)) = 0. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 107/278

The Tail Conditional Expectation • Notation: X : loss random variable; FX (x) : distribution

function; F X (x) = 1 − FX (x): tail function; xq : q -th quantile with F X (xq ) = 1 − q

• The tail conditional expectation (TCE) is

T CEX (xq ) = E (X |X > xq ) . • Other names used: tail-VAR, conditional VAR • Value-at-risk: xq = Qq (X)

• Expected Shortfall: E (X − xq ) = ESFq (X) + • Relationships:

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1 E (X − xq )+ T CEX (xq ) = xq +E (X − xq |X > xq ) = xq + 1−q Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 108/278

TCE for Univariate Elliptical

• Let X ∼ E1 µ, σ 2 , g so that density fX (x) =

where c is the normalizing constant.

c σg

h

1 2

i 2 x−µ σ

• Since X is elliptical distribution, the standardized random

variable Z = (X − µ) /σ will have a standard elliptical Rz 1 2 distribution function FZ (z) = c −∞ g 2 u du, with mean 0 R ∞ 2 1 2 2 and variance σZ = 2c 0 u g 2 u du = −ψ 0 (0), if they exist.

• Define the cumulative density generator:

G (x) = c

Z

x

g (u) du 0

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and denote G (x) = G (∞) − G (x) . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 109/278

- continued • The tail conditional expectation of X is

T CEX (xq ) = µ + λ · σ 2 1 1 2 1 1 2 G( 2 z q ) G( 2 z q ) where λ is λ = σF (x ) = σF (z ) and zq = (xq − µ) /σ. X

q

Z

q

• Moreover, if the variance of X exists, then

1 2 G σZ

1 2 2z

has

the sense of a density of another spherical random variable Z ∗ and λ has the form

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1 fZ ∗ (zq ) σZ2 . λ= σ F Z (zq ) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 110/278

Some Examples • Normal Distribution:

1 ϕ (zq ) λ= σ 1 − Φ (zq )

where ϕ (·) and Φ (·) denote respectively the density and distribution functions of a standard normal distribution. Notice that Z ∗ is simply the standard normal variable Z. • Student-t:

λ=

q

2p−5 2p−3

· fZ

q

2p−5 2p−3 zq ; p

F Z (zq ; p)

−1

only for the case where p > 5/2. Here, Z ∗ is simply a scaled GST with parameter p − 1. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 111/278

Examples - continued • Logistic:

# 1 ϕ (zq ) 1 1 σ λ= √ −1 2 2π + ϕ (zq ) F Z (zq ) "

which resembles that of a normal distribution, but with a correction factor. • Exponential Power:

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1 1 √ λ= F Z (zq ) 2Γ (1/(2s)) σ


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GST - Figure

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0

xq ) .

• Instead, we denote this as T CEX |S (xq ) = E (Xk |S > xq ), k

the contribution to the total risk attributable to risk k .

• It can be interpreted as follows: in case of a disaster as

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measured by an amount at least as large as the quantile of the total loss distirbution, this refers to the average amount that would be due to the presence of risk k . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 116/278

Theorem on Risk Decomposition • Let X = (X1 , X2 , ..., Xn )T ∼ En (µ, Σ, gn ) such that

condition

R∞ 0

g1 (x)dx < ∞ holds and let S = X1 + · · · + Xn .

• Then the contribution of risk Xk , 1 ≤ k ≤ n, to the total TCE

T CEXk |S (xq ) = µk + λS · σk σS ρk,S ,

for k = 1, 2, ..., n, where ρk,S

σk,S = σk σS

1 2 G 12 zS, q σ and λS = SF (z ) . Z

S,q

• Notice that if we take the sum of T CEX |S (xq ), we have k n X

T CEXk |S (xq ) = µS + λS

k=1

n X k=1

σk σS ρk,S = µS + λS · σS2 | {z } σk,S


Multivariate Normal Case • Panjer (2002) demonstrated that in the case of a

multivariate normal random vector i.e. X ∼ Nn (µ, Σ), we have   1 xq −µ ϕ σS σ−k  σS  2 E (Xk |S > xq ) = µk +  ,  σk 1 + ρk,−k xq −µ σk 1 − Φ σS

where they have used the negative subscript −k to refer to the sum of all the risks excluding the k th risk, that is, S−k = S − Xk .

• Therefore, according to this notation, we have

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σk,−k σ−k σk,−k σk,S Cov (Xk , S − Xk ) σ−k ρk,−k = = = = 2 −1. 2 2 σk σk σ−k σk σk σk σk Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 118/278

Multivariate Normal - continued • Thus, our formula for risk decomposition becomes

  1 ϕ xqσ−µ  σS S  E (Xk |S > xq ) = µk +   σk σS ρk,S xq −µ 1 − Φ σS

which gives the case of multivariate normal.

• This confirms the formula above for risk decomposition

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which holds for multivariate elliptical distributions including multivariate normal distributions.


Lecture No. 5 Bounds for Sums of Non-Independent Log-Elliptical Random Variables

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Emiliano A. Valdez


Introduction • This paper is about finding bounds for sums of

non-independent log-elliptical random variables.

• Extends the ideas developed in ◦ “The Concept of Comonotonicity in Actuarial Science

and Finance: Theory” IME, Dhaene, et al. ◦ “The Concept of Comonotonicity in Actuarial Science and Finance: Applications” IME, Dhaene, et al. • These papers considered bounds for the log-normal

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random variables.


Outline of Talk • Comonotonicity • Convex Upper and Lower Bounds • Elliptical, Spherical, and Log-Elliptical Distributions • Extension to Log-Elliptical Distributions

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• The Results for Log-Normal Distributions


Sums of Dependent Random Variables • Consider an insurance portfolio X = (X1 , X2 , · · · , Xn )T

◦ Xi : claim amount of policy i at the end of the period. ◦ Assumption: all Xi are i.i.d.

• Introduction of stochastic financial aspects in actuarial

models reveals the necessity of determining distributions of sums of dependent random variables.

• Assumption that the Xi are mutually independent ◦ is often approximately, ◦ leads to easier mathematics,

◦ but is sometimes violated.


- continued • Individual risks Xi may be influenced by the same

economic/physical environment:

◦ catastrophes (storms, explosions, etc.) cause an

accumulation of claims; weather conditions in automobile; fire insurance; pension fund; and lifetimes of a couple.

◦ ◦ ◦ ◦

• The independence assumption probably underestimates:

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Ordering of Random Variables • Upper and lower tails ◦ E (X − d) = surface above the d.f., from d on. +

◦ E (d − X) = surface below the d.f., from −∞ to d. + • Convex order: X ≤cx Y

◦ ⇔ the upper tails as well as the lower tails of Y eclipse

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the respective tails of X . • → Extreme values are more likely to occur for Y than for X. ◦ ⇔ E (X) = E (Y ) and E [u (−X)] ≥ E [u (−Y )] for all non-decreasing concave functions u. • → Common preferences of risk averse decision makers between rv’s with equal means. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 125/278

- continued • Sufficient condition: ◦ E (X) = E (Y ) and the d.f.’s only cross once, (finally,

FY ≤ F X ) ◦ ⇒ X ≤cx Y.

• Convexity order and moments: ◦ X ≤cx Y ⇒ E (X) = E (Y ) ◦ X ≤cx Y ⇒ V ar (X) ≤ V ar (Y ).

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d ◦ X ≤cx Y and V ar (X) = V ar (Y ) ⇒ X = Y.


Comonotonicity • Suppose X has joint d.f. F . Well-known Frechet bounds:

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≤ FX (x) ≤ min [F1 (x1 ) , ..., Fn (xn )] .

• Hoeffding (1940) and Frechet (1951). • X is comonotonic if its joint distribution is the Frechet upper

bound:

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FX (x) = min [F1 (x1 ) , ..., Fn (xn )] .


Comonotonicity - continued • Comonotonicity is very strong positive dependency

structure.

• Comonotonic rv’s are not able to compensate each other,

they cannot be used as ”hedge” against each other.

• Quantiles, distribution functions, and tails of sums of

comonotonic random variables follow immediately from the respective quantities of the marginals.

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◦ Notation: (X1c , · · · , Xnc ).


Bounds for Sums • COMONOTONIC UPPER BOUND: ◦ Define the comonotonic vector corresponding to X by

Xc = (X1c , ..., Xnc )T where Xkc = Fk−1 (U ) . ◦ Sum: S c = X1c + · · · + Xnc .

• IMPROVED UPPER BOUND:

◦ Define the random vector corresponding to X by

Xu = (X1u , ..., Xnu )T where Xku = FX−1k |Λ (U ).

◦ Sum: S u = X1u + · · · + Xnu .

• LOWER BOUND:

◦ Define the vector corresponding to X by

Xl

=

where Xkl = E (Xk |Λ ) .

· + Xnl .

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◦ Sum:

T l l X1 , ..., Xn S l = X1l + · ·


Bounds for Sums - continued • We have the following bounds:

S l ≤cx S ≤cx S u ≤cx S c • Proofs can be found in: ◦ Tchen (1980) ◦ Dhaene, Wang, Young & Goovaerts (1997) ◦ Müller (1997); and

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Class of Elliptical Distributions • Y v En (µ, Σ,φ) if c.f. can be expressed as T

T

ϕY (t) = exp(it µ) · φ t Σt

for some scalar function φ and where Σ is given by Σ = AAT for some matrix A(n × m). i h • Density: fY (y) = √cn gn (y − µ)T Σ−1 (y − µ) , for some |Σ|

function gn (·) called density generator.

• Normalizing constant: cn =

Γ(n/2) π n/2

R ∞

−1

. R ∞ n/2−1 gn (z)dz < ∞ guarantees gn as density Condition 0 z generator. 0

z n/2−1 g

n (z)dz

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• Kelker (1970); Fang, et al. (1990). Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 131/278

Some Properties • Mean: E (Y) = µ. • Covariance: Cov (Y) = −φ0 (0) Σ. • Y ∼E n (µ, Σ, φ) , iff for any

1), bT Y

b(n ×

∼E1

bT µ, bT Σb,φ

.

• Marginals are also elliptical with the same characteristic

generator:,

Yk ∼

E1 µk ,σk2 , φ

.

• For any matrix B (m × n), any vector c (m × 1) and any

random vector Y ∼ En (µ, Σ, φ), we have that T

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BY + c ∼ Em Bµ + c, BΣB , φ . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 132/278

Independence and Elliptical • Any multivariate elliptical distribution with mutually

independent components must necessarily be multivariate normal, see Kelker (1970).

• Let Y ∼En (µ, Σ, φ) with mutually independent components

Yk . Assume that the expectations and variances of the Y k exist and that var (Yk ) > 0. Then it follows that Y is multivariate normal.

• Thus, it follows that the joint distribution of mutually

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independent elliptical random variables is not elliptical, unless all the marginals are normal.


Spherical Distributions • Z is spherical with c.g. φ if Z ∼En (0n , In ,φ). • Notation: Sn (φ) for En (0n , In ,φ).

• Z ∼Sn (φ) iff E exp itT Z

=φ

tT t

.

d • Suppose m-dim vector Y is such that Y = µ + AZ, for some

µ(n × 1), some matrix A(n × m) and some m-dim elliptical vector Z ∼Sm (φ). Then Y ∼En (µ, Σ,φ) where Σ = AAT .

• Z ∼Sn (φ) iff for any n-dim vector a,

aT Z √ ∼S1 (φ) . aT a • Any component Zi of Z has a S1 (φ) distribution.

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• Density: fZ (z) = cg zT z . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 134/278

Conditional Distributions • Conditional distributions of bivariate Normal is again

Normal.

• GENERALIZATION OF RESULT TO ELLIPTICAL: ◦ Let Y ∼ En (µ, Σ, φ) with d.g. gn (·). Define Y and Λ to

be linear combinations of Y, i.e. Y = αT Y and Λ = β T Y, for some αT = (α1 , α2 , . . . , αn ) and β T = (β1 , β2 , . . . , βn ). Then, (Y, Λ) ∼ E2 µ(Y,Λ) , Σ(Y,Λ) , φ .

◦ Also, given Λ = λ, Y has a univariate elliptical

distribution:

Y |Λ = λ ∼ E1

µY+ r (Y, Λ) σσYΛ (λ − µΛ ) , 1 − r (Y, Λ)2 σY2 , φa

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char. gen. φa (·) depending on a = (λ − µΛ )2 /σΛ2 . Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 135/278

Log-Elliptical Distributions • X is multivariate log-elliptical with parameters µ and Σ if

log X is elliptical:

log X ∼ En (µ, Σ, φ) . • Notation: log X ∼ En (µ, Σ, φ) as X ∼ LEn (µ, Σ, φ) . • When µ = 0n and Σ = In , we write X ∼ LSn (φ) . • If Y ∼ En (µ, Σ, φ) and X = exp (Y), then

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is given by fZ ∗ (x) =

fZ (x)eσx φ(−σ 2 ) .

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Non-Independent Log-Elliptical Risks • Payments: α1 , ..., αn • Rates of return: Yi (i − 1, i) , i = 1, 2, . . . , n. • Define Y (i) = Y1 + · · · + Yi , the sum of the first i elements

of Y.

◦ Xi = exp [−Y (i)]. • Present Value: P

n i=1 αi exp [− (Y1

S=

+ · · · + Yi )] =

Pn

i=1 Xi

• Assume return vector Y ∼ En (µ, Σ, φ) with parameters µ

and Σ.

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- continued • We know Y (i) ∼ E1

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i X

µk and σ 2 (i) =

k=1

• Conditioning rv: Λ =

with i X i X

σkl .

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Pn

i=1 βi Yi

• Using the property of elliptical, Λ ∼ E1 µΛ ,σ 2 , φ where Λ

µΛ =

n X

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i=1

2 βi µi and σΛ =

n X


The Bounds • COMONOTONIC UPPER BOUND: Pn −1 c

S =

i=1 αi exp

−µ (i) + σ (i) FZ (U )

• IMPROVED UPPER BOUND: n X

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i=1

q αi exp −µ (i) − ri σ (i) FZ−1 (U ) + 1 − ri2 σ (i) FZ−1 (V )

• LOWER BOUND: n X

l

S =

αi e[−µ(i)−ri

i=1

σ(i) FZ−1 (U )]

2

· φa −σ (i) 1 −

ri2

where U and V are mutually indep. U (0, 1) rv’s, Z ∼ S1 (φ), i k=1

σ(i)σΛ

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Sums of Log-Normal RV’s In the case of log-Normal, we have the results from Dhaene, et al. (2002):

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S

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αi e−E[Y (i)]−ri αi e

σY (i) Φ

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(U )+

−E[Y (i)]+σY (i) Φ−1 (U )

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Lecture No. 6 Capital Allocation and Elliptical Distributions

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Emiliano A. Valdez


Introduction • Why do we need to allocate capital? ◦ Redistribute capital cost equitably ◦ Division of capital provides division of risks across

business units ◦ Allocation of expenses, prioritizing capital budgeting projects ◦ Fair assessment of manager performance • This paper examines what constitutes a fair allocation and

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studies Wang’s allocation within this fair allocation principle and then extends to class of elliptical distributions.


Fair Allocation • Let XT = (X1 , X2 , ..., Xn ) denote the vector of losses. • Define an allocation A to be a mapping A : XT → Rn such

= (K1 , K2 , ..., Kn )T where that A Pn i=1 Ki = K = ρ (Z) . XT

• Each component Ki of allocation is viewed as the i-th line

of business contribution to total capital.

• Because allocation must also reflect the fact that each line

operates in the presence of other lines, the notation A (Xi |X1 , ..., Xn ) = Ki

is well-suited for this purpose.

Pn

i=1 Ki

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= ρ (Z) is sometimes called the “full allocation” requirement.

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• Notice also that the requirement


What constitutes a fair allocation? • Let N = {1, 2, ..., n} be the set of the first n positive integers.

An allocation A is said to be a fair allocation if:

◦ No Undercut: For any subset M ⊆ N , we have

P

i∈M

A (Xi |X1 , ..., Xn ) ≤ ρ

P

i∈M

Xi .

◦ Symmetry: Let N ∗ = N − {i1 , i2 } . If M ⊂ N ∗ (strict

subset) with |M | = m, XTm = (Xj1 , ..., Xjm ) and if T T A Xi1 Xm , Xi1 , Xi2 = A Xi2 Xm , Xi1 , Xi2 for every M ⊂ N ∗ , then we must have Ki1 = Ki2 . ◦ Consistency: For any subset M ⊆ N with |M | = m, let T Xn−m = Xj1 , ..., Xjn−m for all jk ∈ N − M where k = 1, ..., n − m. Then we have ! X X X A (Xi |X1 , ..., Xn ) = A Xi Xi , XTn−m . i∈M

i∈M

i∈M


Relative Allocation • This gives the allocation to the i-th line of business as

ρ (Xi ) . A(Xi |X1 , · · · , Xn ) = ρ(Z) ρ (X1 ) + · · · + ρ (Xn ) • Simple and appealing, but not a fair allocation.

• Consider 3 indep. risks X1 , X2 , and X3 with mean

E (Xi ) = 0 and variances V ar (Xi ) = σ 2 (Xi ) for i = 1, 2, 3. Define the risk measure ρ (Xi ) = Fi−1 (1 − α) · σ (Xi ) .

• Now suppose a life company has four lines each facing risks

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X1 , −X1 , X2 , and X3 so that total risk is Z = X2 + X3 . Consider the subset M consisting of the risks {X1 , −X1 , X2 } and observe that P −1 ρ X ) = F = ρ(X i 2 2 (1 − α) · σ (X2 ). i∈M


Relative Allocation - continued • Because

X

i∈M

A (Xi |X1 , −X1 , X2 , X3 )

ρ (X1 ) + ρ(−X1 ) + ρ(X2 ) , = ρ(X2 + X3 ) ρ (X1 ) + ρ(−X1 ) + ρ(X2 ) + ρ(X3 )

the “no undercut” cannot be satisfied unless the risks have symmetric distributions. • The “consistency” property is also not satisfied because

! X X Xi Xi , X 3 = A (X2 |X2 , X3 )

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ρ(X2 ) = ρ(X2 + X3 ) ρ(X2 ) + ρ(X3 )


Relative Allocation - continued • Hence

X

i∈M

A (Xi |X1 , −X1 , X2 , X3 ) 6= A

! X X Xi Xi , X 3 .

i∈M

i∈M

• However, it can be shown that the “symmetry” property is

satisfied for this allocation formula. Consider for example the case where A (X1 |X1 , −X1 , X2 ) = A (X2 |X1 , −X1 , X2 ) and it is straightforward to show that in this case ρ (X1 ) = ρ (X2 ) so that A (X1 |X1 , −X1 , X2 , X3 ) = A (X2 |X1 , −X1 , X2 , X3 )

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and symmetry is satisfied. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 149/278

Covariance-Based Allocation • The allocation formula is based on

A(Xi |X1 , · · · , Xn ) = λi ρ(Z) where λT = (λ1 , ..., λn ) denotes a vector of weights that add up to one so that full allocation is satisfied.

• To determine these weights λi , we minimize the following

quadratic loss function h i E ((X − µ) − λ (Z − µZ ))T W ((X − µ) − λ (Z − µZ ))

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E [(Xi − µi ) (Z − µZ )] Cov (Xi , Z) h i λi = , for i = 1, 2, ..., n. = 2 V ar (Z) E (Z − µZ ) ^

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where the weight-matrix W is assumed to be positive definite. Differentiating with respect to λ and equating to zero yields


Is the Covariance Principle Fair? • The capital allocation formula based on the covariance

principle satisfies the three properties of a fair allocation: ◦ no undercut, ◦ symmetry, and

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◦ consistency.


Wang’s Capital Decomposition Formula • Preserving the notation used by Wang (2002), denote the

expectation of Xi,Q by

E[X · exp (λZ)] Hλ [Xi , Z] = E (Xi,Q ) = E[exp (λZ)]

and the expectation of the aggregate loss ZQ by E[Z · exp (λZ)] . Hλ [Z, Z] = E (ZQ ) = E[exp (λZ)]

This exactly gives the Esscher transform of Z . • Price of a random payment Xi traded in the market is

Hλ [Xi , Z] so that one can think of the difference ρ (Xi ) = E (Xi,Q ) − E (Xi ) = Hλ [Xi , Z] − E (Xi ) as the risk premium. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 152/278

- continued • For the aggregate payment Z , its risk premium is given by

ρ (Z) = ρ (

Pn

i=1 Xi )

= Hλ [Z, Z] − E (Z) .

• It is rather straightforward to show ρ (Xi ) =

and ρ (Z) =

Cov(Xi ,exp(λZ)) E[exp(λZ)]

Cov(Z,exp(λZ)) E[exp(λZ)] .

• Wang proposes computing the allocation of capital to

individual business unit i based on the following formula: Ki = Hλ [Xi , Z] − E (Xi ) .

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company as a whole, the parameter λ can be computed using K = Hλ [Z, Z] − E (Z) . Pn • For i = 1, 2, ..., n, it can readily be shown that K = i=1 Ki . ¢

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• Assuming an aggregate capital of K for the insurance


Multivariate Normal • If X1 , ..., Xn follow a multivariate normal, we have that

Wang’s allocation method reduces to the covariance method.

• Some straightforward calculation yields the results:

E ZeλZ

E Xi eλZ

2 2 λ σZ = exp λµZ + · (µ + λσZ2 ) 2 2 2 λ σZ = exp λµZ + · (µi + λσi,Z ) 2

• Then it follows that K = λσ 2 and Ki = λσi,Z which is clearly Z

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equivalent to the covariance method.


Some Notation and Assumptions • Suppose X ∼En (µ, Σ,gn ) and e = (1, 1, ..., 1)T . • Assume density generator gn exists. • Define

Z = X1 + · · · + X n =

n X

Xk = eT X

k=1

which is the sum of elliptical risks. We know that Z ∼ E1 eT µ, eT Σe, g1 . P • Denote by µZ = eT µ = n µj and j=1 P σZ2 = eT Σe = ni,j=1 σij . • Define the tail generator by

∞ 1 2 u 2

cn gn (x) dx.

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Tn (u) =

Z


Two Useful Lemmas • Let X ∼ En (µ, Σ, gn ) . Then for 1 ≤ i ≤ n, the vector

Xi,Z = (Xi , Z)T has an elliptical distribution with the same generator, i.e., Xi,Z ∼ E2 µi,Z , Σi,Z , g2 , where ! P T 2 σi σi,Z µi,Z = µi , nj=1 µj , Σi,Z = , and 2 σi,Z σZ Pn Pn 2 2 σi = σii , σi,Z = j=1 σij , σZ = j,k=1 σjk .

• Let X ∼En (µ, Σ,gn ) and assume condition for existence of

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density generator holds. Let T be the tail generator as defined above and associated with Z . Then Z exp (λµZ ) ∞ Hλ [Xi , Z] = µi +λρi,Z σi σZ Tn (w) exp (λσZ w) dw MZ (λ) −∞ Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 156/278

Wang’s Allocation for Elliptical • Let X ∼En (µ, Σ,gn ) and assume conditions for existence of

density generator and |ψ 0 (0)| < ∞ hold. Then Wang’s capital allocation formula can be expressed as Ki = −λψ 0 (0) ρi,Z σi σZ

• The result immediately follows from the previous lemma:

Ki = Hλ [Xi , Z] − E (Xi ) Z ∞ 1 exp [λ (µZ + σZ w)] Tn (w) dw = λρi,Z σi σZ MZ (λ) −∞ 0 1 −ψ (0) MZ (λ) = λρi,Z σi σZ MZ (λ)

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ü ý

ø

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= −λψ 0 (0) ρi,Z σi σZ .


Panjer’s Example • Panjer (2002) example to illustrate the capital allocation. • Insurer has 10 lines of business is faced with risks

represented by vector XT = (X1 , ..., X10 ) where each Xi represents the P.V. of losses over a specified time horizon.

b , (in millions-squared) • The estimated covariance structure, Σ

is given by [see paper for variance matrix] and the bT (in millions) is given by estimated mean vector µ

(25.69, 37.84, 0.85, 12.70, 0.15, 24.05, 14.41, 4.49, 4.39, 9.56) .

• The resulting allocation KT is given by

(2.72, 12.55, 0.08, 1.92, 0.37, 6.27, 2.51, −0.70, −0.30, 1.89) ,

"

!

expressed in millions, with total capital equal to 27.31 million. Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 158/278

Graph of Allocation

10 9 8

Line of Business

7 6 5 4 3 2 1 0%

10%

20%

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50%

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Lecture No. 7 Convex Bounds for Scalar Products of Random Variables (With Applications to Loss Reserving and Life Annuities)

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i) + . . . + I(Tx(N0 ) >i) e−Y (i)

X

αi Ni e−Y (i) ,

i=1

=

X

bω−xc

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Three types of life annuities • Consider a portfolio of N0 homogeneous life annuity

contracts. From the Law of Large Numbers for sufficiently large N0 :     bω−xc bω−xc bω−xc X X X N i −Y (i) −Y (i)  ≈ N0  αi Ni e = N0  αi e αi i px e−Y (i)  N0 i=1

i=1

i=1

=⇒ in the case of large portfolios of life annuities it suffices to compute risk measures of an ‘average’ portfolio: Sxaverage

=

bω−xc

X

αi i px e−Y (i)

i=1

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= E[Sxpolicy |Y (1), · · · , Y (bω − xc)] Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 209/278

The Gompertz-Makeham law • Force of mortality at age ξ :

µξ = α + βcξ

-α > 0: constant component → capturing accident hazard -βcξ : variable component → capturing the hazard of aging (β > 0, c > 1) • Survival probability:

Z t px = Pr(Tx > t) = exp −

x+t

µξ dξ x

t cx+t −cx

=sg

,

β log c

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where s = exp(−α) and g = exp −


The Gompertz-Makeham law • Denote by Tx0 the future lifetime of (x) from the Gompertz

family with force of mortality µ0ξ = βcξ

d • Tx = min(Tx0 , E/α) and E ∼ exp(1)

Pr(min(Tx0 , E/α) > t)

= Pr(Tx0 > t) Pr(E > αt) Z x+t µ0ξ dξ e−αt = exp −

= exp −

x Z x+t x

= Pr(Tx > t).

µξ dξ

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Simulation from Makeham’s law : (a) Generate G from the Gompertz’s law by the inversion method (b) Generate E from the exp(1) distribution (c) Retain T = min(G, E/α) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 211/278

Personal finance problem • Suppose that (x) disposes a lump sum L.

What is the amount that (x) can yearly consume to be almost sure (i.e. sure with a sufficiently high probability e.g. p = 99%) that the money will not be run out before death? A solution to the latter problem is crucial to determine the fair value of future liabilities and the solvency margin.

• Notice that the presented methodology is appropriate not

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only in the case of large portfolios when the limiting distribution can be used on the basis of the law of large numbers but also for portfolios of average size (e.g. 1000-5000) which are typical for the life annuity business.


Single life annuity: CDF upper bound SLAcx • Xi = I(T >i) ∼ Bern(i px ) ⇒ F −1 (p) = x Xi

SLAcx =

∞ X

FX−1i (U )Fα−1 −Y (i) (V ) = ie

i=1

FSLAcx (y) =

bω−xc

X k=1

-conditional quantiles:

-conditional df:

k X

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1 for p > i qx 0 for p ≤ i qx .

(U )c bFT−1 x X

Fα−1 −Y (i) (V ) ie

i=1

k| qx FSLAcx |Kx =k (y)

−1 FSLA c |K =k (p) x x

=

k X

αi e

−µi +sign(αi )σi Φ−1 (p)

i=1

αi exp −µi + sign(αi )σi Φ

−1

(FSLAcx |Kx =k (y)) = y


Single life annuity: SL upper bound SLAcx E[(SLAcx

c − d)+ ] = EKx E (SLAx − d)+ |Kx =

bω−xc

X k=1

k| qx

k X i=1

E[(αi e−Y (i) − dk,i )+ ] ,

with dk,i = αi exp −µi + sign(αi )σi Φ−1 (FS˜c (d)) k

E[(SLAcx − d)+ ] = bω−xc

X

k=1

k| qx

k X

αi e

i=1

−µi +

σi2 2

h

Φ sign(αi )σi − Φ

−1

(FS˜c (d)) − d 1 − FS˜c (d) k

i

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not (SLAcx |Kx =k = S˜kc )


Single life annuity: CV lower bound SLAlx • Γ = Tx ⇒ E[I(T >i) |Tx ] = I(T >i) x x • Λ?

a)

Pbω−xc

−µi + 12 σi2

= i=1 αi i px e Y (i) → first order approximation to the PV of the limiting portfolio

Λ(a)

b) Λ(M ) := Λj0 with j0 = arg max{Var(SLAl,j x ), j = 1, . . . , bω − xc} j

-SLAl,j x -Λ j =

P Kx

−Y (i) |Λ ] j i=1 E[αi e 1 2 −µi + 2 σi α e Y (i) i i=1

= Pj

αi e

i=1

=

−µi + 12 σi2 (1−ri2 )−σi ri Φ−1 (V )

SLAlx |Kx =k

k X


Single life annuity: CDF lower bound SLAlx SLAlx =

Kx X i=1

FSLAlx (y) =

E[αi e−Y (i) |Λ]

bω−xc

X k=1

-conditional quantiles:

−1 FSLA l |K =k (p) x x

=

k X

αi e

−µi + 12 σi2 (1−ri2 )+σi ri Φ−1 (p)

i=1

1 αi exp −µi + σi2 (1 − ri2 ) + σi ri Φ−1 (FSLAlx |Kx =k (y)) 2 i=1

=y

-conditional df:

k X

k| qx FSLAlx |Kx =k (y)


Single life annuity: SL lower bound SLAlx E[(SLAlx − d)+ ] = EKx E SLAlx − d |Kx +

bω−xc

X

=

k| qx

i=1

k=1

with dk,i = αi exp −µi +

1 2 2 σi (1

E[(SLAlx − d)+ ] = bω−xc

X

k=1

k| qx

k X i=1

h

−

ri2 )

E

E[αi e−Y (i) |Λ] − dk,i

+ σ i ri

Φ−1 (F

Φ ri σi − Φ−1 FS˜l (d) k

! +

SLAlx |Kx =k (d))

i

− d 1 − FS˜l (d) k

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k X


Single life annuity: an alternative approximation • Take as conditioning variable:

Λ Kx =

Kx X

αi e

−µi + 12 σi2

Y (i)

i=1

• The lower bound is then given by bω−xc

X k=1

k| qx

k X

αi e

2 −µi + 12 σi2 (1−ri,k )−σi ri,k Φ−1 (Uk )

,

i=1

with ◦ correlations: ri,k = √

Cov(Y (i),Λk ) √ Var[Y (i)] Var[Λk ]

◦ {Uk }k=1,...,bω−xc ∼ Unif(0, 1) ⇒ multidimensional lower -

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26

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bound


Single life annuity: an alternative approximation A new approximation based upon this lower bound: ◦ SLAcl x =

bω−xc

X

k| qx

k=1

k X

αi e

2 −µi + 12 σi2 (1−ri,k )−σi ri,k Φ−1 (U )

i=1

• The “comonotonic upper bound of the lower bound”

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X

αi Ni e−Y (i) ,

i=1

=

X

bω−xc i=1

=

N0 X

SLA(j) x

j=1

=

N0 bω−xc X X j=1 i=1

αi I(Tx(j) >i) e−Y (i)

Ni ∼ binomial(N0 , i px ) ⇒ difficult to deal with !

n

x

s

n

r

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r

t

s

l

s

qr

no

lm

p

−→ Normal Power Approximation (NPA) Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 223/278

Homogeneous portfolio of life annuities: NPA ˜i Approximate the distribution of Ni by the NPA N s ! 3 6(x − µNi ) 9 FN˜i (x) = Φ − + + +1 2 γ Ni γ Ni σNi γ Ni

with µ Ni

= E[Ni ] = N0 i px

2 σN i

= Var[Ni ] = N0 i pxi qx

γ Ni

=

E[Ni − µNi ]3 1 − 2 i px =√ 3 σNi N0 i pxi qx

˜i : The p-th quantile of N

{

{

y

~

{|

yz

}

FN−1 ˜i (p)

= µ Ni + σ Ni Φ

−1

γNi σNi −1 2 (Φ (p)) − 1 (p) + 6


homogeneous portfolio of life annuities: convex bounds • The upper bound is straightforward, from

P LAcx |U =u

=

bω−xc

X i=1

−µi +sign(αi )σi Φ αi FN−1 (u)e ˜

−1

(V )

i

• Conditioning variables of the lower bound ◦ Γ = Ni0 → the number of policies-in-force in the year i0 E[Ni |Ni0 = n0 ] E[Ni |Ni0 = n0 ]

= (Bayes)

=

i−i0 px+i0 n0 N0 X

k=n0

=

k−n0 N0 −k i0 −i qx+i i qx N0 −n0 i0 q x

for i < i0

Pr(Ni0 = n0 |Ni = k)Pr(Ni = k) Pr(Ni0 = n0 )

N0 X N0 − n 0 k−n0 k i px k − n0

k=n0

k

for i ≥ i0


homogeneous portfolio of life annuities: convex bounds • Conditioning variables of the lower bound ◦ Γ : take for simplicity Γ = N1 ⇒ E[Ni |N1 ] = ◦ Λ=

Pbω−xc i=1

α i i px e

−µi + 12 σi2

i−1 px+1 N1

Y (i)

• The lower bound is then straightforward, from

P LAlx |U =u

=

bω−xc

X i=1

−1 −µi + 12 σi2 (1−ri2 )−σi ri Φ−1 (V ) αi i−1 px+1 FN˜ (u)e 1

• Moments based approximation P LAm x

~ ]] + Var[E[P LAx |Y ~ ]] Var[P LAx ] = E[Var[P LAx |Y ~ ]] + N 2 Var[E[SLAx |Y ~ ]] = N0 E[Var[SLAx |Y 0

~ ]] = N0 Var[SLAx ] + (N02 − N0 )Var[E[SLAx |Y Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 226/278

A numerical illustration: Quantiles -Return process: Black & Scholes model µ = 0.05, σ = 0.1 -Mortality process: Makeham’s model MR, 65 years, males -Portfolio: 1000 policies -Payments: αi = 1 ∀i c P LAl65 P LAm P LA 65 65 MC (s.e.)

0.995 20209

20250

22620

20242 (22.09)

0.975 17252

17272

18722

17276 (8.80)

0.95

15937

15951

17029

15947 (8.15)

0.90

14565

14574

15290

14568 (5.08)

0.75

12574

12577

12821

12577 (3.90)

¢

¬

§

¢

¦

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§«

ª

©

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¨

§

§

¥¦

¢£

¡

¤

p


5000

10000

15000

20000

A numerical illustration: QQ-plot

6000

8000

10000

12000

14000

16000

18000

20000

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´

¯

³

³

´¸

·

¶

³

µ

´

´

²³

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±

c QQ-plot of the quantiles of P LAl65 (◦), P LAm 65 (4) and P LA65 () versus those of ‘P LA65 ’ (MC).


d

c P LAl65 P LAm 65 P LA65 MC (s.e.)

0

11094

11094

11094

11098 (2.11)

5000

6094

6094

6095

6098 (2.10)

10000 1608

1610

1793

1611 (1.95)

15000 153.7

155.3

278.4

155.3 (1.78)

20000 10.23

10.57

36.02

10.67 (1.26)

25000 0.680

0.734

4.816

0.743 (0.09)

30000 0.051

0.059

0.711

0.036 (0.02)

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A numerical illustration: Stop-loss premia


Part II Applications

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Loss Reserving


Loss Reserving: general framework • Stochastic liability payments: Li ≥ 0 at times i = 1, 2, . . . , n (modified by certain forces that influence the liability over time) • Lt = Lt−1 RLt , t = 1, . . . , i i i ◦ Lt : amount of liability expressed in money values of i

time t ◦ RLt = 1 + rLt ◦ rLt : inflation of claim costs over interval (t − 1, t]

• At = At−1 RAt ◦ At : holding of assets of value At at time t ◦ RAt = 1 + rAt • Assume RXt (X = A, L) follows CAPM:

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rXt = rF t + βX ∆t + Xt


Loss Reserving: general framework • rXt = rF t + βX ∆t + Xt ◦ ∆t = rM t − rF t (distribution independent of t) ◦ rF t : risk-free rate in period t

◦ rM t : periodic increase in value of the economy wide

portfolio of assets ◦ βX : CAPM beta associated with X ◦ Xt ∼ i.i.d. and E[Xt ] = 0 and Var(Xt ) := ω 2 X ◦ At , Lt , ∆t independent

2) • Assume RXt ∼ i.i.d logN(µX , σ 2 ) and L0s ∼ logN(ν0s , τ0s X

• L0s and RXt independent ∀s, t, X

• ρ = Corr(log RAt , log RLt ) and κ(rs) = Corr(log L0r , log L0s )

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¯ X = E[RXt ] = exp(µX + 1 σ 2 ) • R 2 X Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 232/278

Discounted loss reserve Discounted loss reserve: V

= =

n X i=1 n X

Vi =

n X

−1 Ltt RA (t)

i=1

−1 L0t RL (t)RA (t)

i=1

• RX (i) = RX1 + · · · + RXi ∼ logN(iµX , iσ 2 ) X

=⇒ Vi ∼ logN(α(i) , δ 2(i) )

◦ α(i) = ν0i + i(µL − µA )

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õ

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õù

ø

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ô

ö

õ

î

õ

óô

ðñ

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ò

◦ δ 2(i) = τ 2 + i(σ 2 + σ 2 − 2ρσL σA ) 0i L A


Loss Reserving: general framework Three relevant values of the loss reserve: Pn 0 • i=1 E[Li ] : CAPM-based economic value of the liability

• E[V ] : expected value of the discounted liability cash-flows

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ý

û

ýþ

ûü

ÿ

• F −1 (p) : 100p% confidence loss reserve V


Convex bounds discounted loss reserve V =

n X i=1

V l :=

n X i=1

l

Qp [V ] = c

Qp [V ] =

n X not Vi : = e Zi i=1

E[Vi |Λ] ≤cx V ≤cx V c :=

n X i=1 n X

e e

E[Zi ]+σZi Φ−1 (p)

,

c

E[V ] = E[V ] = E[V ] =

n X

FV−1 (U ) i

i=1

2 E[Zi ]+ 12 (1−ri2 )σZ +ri σZi Φ−1 (p) i

,

p ∈ (0, 1)

p ∈ (0, 1)

i=1

l

n X

e

2 E[Zi ]+ 12 σZ i

i=1


Convex bounds discounted loss reserve Λ=

Pn

i=1 βi Zi

with βi = exp(E[Zi ] + 12 σZ2 i )

  ! i  R ¯ 2 1/2  ¯ L 1 + (β 2 σ 2 + ω 2 )/R A M A A E[Zi ] = ν0i + log ¯2 ¯ A 1 + (β 2 σ 2 + ω 2 )/R  R  L M L L

2 Var(Zi ) = σZ2 i = τ0i + iˆ σ2

The variability of the discounting structure not

2 − 2ρσ σ is given by σ ˆ 2 = σL2 + σA L A

[1 +

2 σ2 (βA M

2 )/R ¯ 2 ][1 + (β 2 σ 2 + ωA A L M 2 /R ¯AR ¯ L ]2 [1 + βA βL σM

+

2 2 ¯ ωL )/RL ]

!

log


Convex bounds discounted loss reserve The correlation between Zi and Λ is given by Pn

σ ˆ 2 min(i, k)

η (i,k)

+ Cov(Zi , Λ) k=1 βk qP ri = = Pn n σZ s σΛ σZ i σ 2 min(k, l) + η (k,l) ) k=1 l=1 βk βl (ˆ

with

η

(k,s)

= Cov

log L0k , log L0s

= κ(ks) τ0k τ0s

$

.

)

$

(

(

)-

,

+

(

*

)

"

)

'(

%$"#

&

Note that if the liability cash-flows are independent 2 I η (k,s) = τ0k (k=s) and I(.) the indicator function.


Moment matching vs. convex bounds Security margin for confidence level p (Taylor, 2004): not

SMp [V ] = (Qp [V ]/E[V ]) − 1 LN ] − SM [V M C ] SMp [V l ] − SMp [V M C ] not SMp [V p LB = = ×100% and LN ×100%, M C M C SMp [V ] SMp [V ] not

(MC: Monte Carlo simulation - LN: lognormal moment matching) Stochastic liability cash-flow structure: (n = 30) -ν0i = −4.46 for i = 1, . . . , 30   5% s ≤ 5; 10% 5 < i ≤ 15; τ0i =  20% 25 < i ≤ 28; 25% 28 < i ≤ 30 P30

15%

15 < i ≤ 25 .

1

6

1

5

; 5

6:

9

8

5

7

6

/

6

45

12

/0

3

- i=1 E[Li ] = 100% and E[L0i ] = 35.51% - ωL = 10% and ωA = 5%


Moment matching vs. convex bounds

p = 0.975

σM = 0.05

σM = 0.10

σM = 0.15

σM = 0.20

σM = 0.25

σM = 0.30

LB

−0.19%

−0.15%

−0.23%

−0.16%

−0.11%

−0.17%

0.4390

0.5250

0.6528

0.8103

0.9924

1.1970

LN MC

−1.95%

−1.56%

p = 0.995

p = 0.975

p = 0.95

p = 0.90

p = 0.80

p = 0.70

p = 0.60

LB

−0.93%

−0.04%

−0.02%

−0.18%

−0.03%

−0.6%

+0.86%

4.4521

2.2264

1.4998

0.8814

0.3508

0.0761

(37.63)

(2.99)

(7.44)

(2.79)

(0.78)

(0.27)

−0.1069

−3.94%

(0.29)

+3.78%

(0.41)

+7.22%

(0.69)

+11.29%

(1.22)

+19.68%

(3.78)

+53.46%

−15.50% (0.08)

>

H

C

>

B

B

CG

F

E

B

D

C

?

−3.92%

s.e.(×105 )

LN

0)


Statistical analysis • Check the model assumptions!

- Gauss-Markov conditions of a regression model - Normality for inference

• Goodness-of-Fit

- (Adjusted) coefficient of determination and AIC/BIC - Residual plots - Plot of the observed values vs. the fitted values

• Estimation of the parameters by maximum likelihood

methods ~ˆ 0 (Z ~ˆ ~ − Xβ) ~ − Xβ) ˆ 2 = n1 (Z -σ ˆ ~ - β~ = (X0 X)−1 X0 Z ~ˆ 0 (Z ~ˆ → unbiased ~ − Xβ) ~ − Xβ) Remark: σ ˜ 2 = 1 (Z n−p

À

Ê

Å

À

Ä

Ä

ÅÉ

È

Ç

Ä

Æ

Å

¾

Å

ÃÄ

ÀÁ

¾¿

Â

estimator of σ 2


Lognormal models • The mean of the IBNR reserve equals W =

t X

t X

0 −1 0 1 2 ~ e(Rβ)ij + 2 σ (1+(R(X X) R )ij )

i=2 j=t+2−i

• The unique UMVUE of the mean of the IBNR reserve is given by t t n − p SS X X ~ˆ ij z (Rβ) ˆ ; e , WU = 0 F 1 2 4 i=2 j=t+2−i

where 0 F1 (α; z) denotes the hypergeometric function. • The MLE of the mean of the IBNR reserve: ˆM = W

t X

t X

e

2 0 −1 0 ~ˆ ij + 1 σ (Rβ) R )ij ) 2 ˆ (1+(R(X X)

Í

×

Ò

Í

Ñ

Ñ

ÒÖ

Õ

Ô

Ñ

Ó

Ò

Ë

Ò

ÐÑ

ÍÎ

ËÌ

Ï

i=2 j=t+2−i


Lognormal models ˆ M , but with • Verrall (1991) has considered an estimator similar to W ˜2: σ ˆ 2 replaced with σ ˆV = W

t X

t X

e

2 0 −1 0 ~ˆ ij + 1 σ (Rβ) R )ij ) 2 ˜ (1+(R(X X)

i=2 j=t+2−i

• Doray (1996) has considered the following simple estimator estimator t t X X 2 ~ˆ ij + 1 σ (Rβ) ˆ 2˜ e WD = i=2 j=t+2−i

⇒ Now we have the order relation ˆD < W ˆV , Û < W W

Ú

ä

ß

Ú

Þ

Þ

ßã

â

á

Þ

à

ß

Ø

ß

ÝÞ

ÚÛ

ØÙ

Ü

ˆ D ] < E[W ˆV ] ˆ U ] < E[W which implies that W = E[W Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 251/278

Generalized Linear Models 1. Random component f (yij ; θij , φ) = exp {[yij θij − b(θij )] /a(φ) + c(yij , φ)}

• f (.) belongs to the exponential family • a(.), b(.) en c(., .) are known functions: a(φ) = φ/wij • E[Yij ] = µij = b0 (θij ) and Var[Yij ] = b00 (θij )a(φ) = V (µij )a(φ)

2. Systematic Component

~ ij = β1 Rij,1 + · · · + βp Rij,p , ηij = (Xβ)

i, j = 1, . . . , t

3. Link function ηij = g(µij )

ç

ñ

ì

ç

ë

ë

ìð

ï

î

ë

í

ì

å

ì

êë

çè

åæ

é

g is a monotone, differentiable function Samos 2004 Workshop, Risk Measures and Optimal Portfolio Selection, Dhaene/Valdez/Hoedemakers – p. 252/278

Generalized Linear Models: link function • Canonical link → when g(µij ) = θij

~ ,→ sufficient statistic in ~η (when ~η = θ~) given by R0 Y

• Logarithmic link → multiplicative parametric structure +

positive fitted values

Density ÿ

Distribution

√1 exp σ 2π

y

ÿ

νy µ

ν

y −3/2 √ exp 2πσ 2

Variance

link θ(µ)

function µ(θ)

function V (µ)

σ2

µ

θ

1

1

log(µ)

eθ

µ

exp − νy µ −(y−µ)2 2yσ 2 µ2

1 y

1 ν

1/µ

−1/θ

µ2

σ2

1/µ2

(−2θ)−1/2

µ3

ô

þ

ù

ô

ø

ø

ùý

ü

û

ø

ú

ù

ò

ù

÷ø

ôõ

òó

ö

IG(µ, σ 2 )

1 Γ(ν)

Mean

ÿ

e−µ µy!

Poisson(µ) Gamma(µ, ν)

(y−µ)2 − 2σ2

ÿ

N(µ, σ 2 )

Canonical

φ


Generalized Linear Models • Estimation of the parameters by maximum likelihood

methods (using iteratively reweighted least squares)

• Suppose:

  

response is always positive ⇒ no particular distr. data are invariably skew to the right   variance increases with mean

• Quasi-likelihood (Wedderburn, 1974) estimation allows us

to model the response variable in a regression context without specifying its distribution. We need only to specify the link and variance functions to estimate the regression coefficients.

• If all the data are positive (greater than 0), identical

parameter estimates are obtained using full or quasi-likelihood.


Generalized Linear Models 1. Over-dispersed Poisson model: The incremental claims Yij are distributed as independent over-dispersed Poisson random variables, with Var[Yij ] = φE[Yij ] ,→ not only suitable for data consisting exclusively of positive integers ⇓ quasi-likelihood approach

2. Gamma model:

Var[Yij ] = φ (E[Yij ])2


Generalized Linear Models Log-normal model: Zij = log(Yij ) ∼ N (µij , σ 2 ) ⇓ ηij = µij and φ = σ 2 ,→ limitation: incremental claim amounts must be positive

'

"

!

!

"&

%

$

!

"#

"

!

1 2 ηij =g(µij ) ˆ ˆ îj ) ←→ Yij = µ îj = g −1 (ˆ ηij ) Yij = exp(ηîj + σ 2


Quasi-likelihood equations When using a logarithmic link function, the quasi-likelihood equations are given by t+1−i X

eηij =

j=1

t+1−j X i=1

t+1−i X

Yij

1 ≤ i ≤ t;

Yij

1 ≤ j ≤ t.

j=1

eηij =

t+1−j X i=1

*

/

*

.

.

4

⇒ Work without GLM-software and without the log-link /3

2

1

.

/0

(

/

-.

*+

()

,

,→ The sum of the incremental claims in every row and column has to be non-negative ⇒ problems when modelling incurred data with a large number of negative incremental claims in the later stages of development, which is the result of overestimates of case reserves in the first development years.


~ˆ Distribution of µ ~ˆ ∼ M N (Rβ, ~ Σ(Rβ)) ~ˆ (asymptotically) with • Rβ

ˆ a } = R(X0 WX)−1 R0 ~ - Σ(Rβ) = Σa = {σij - W = diag{w11 , · · · , wt1 } with wij = Var[Yij ]−1 (dµij /dηij )2

• The function g −1 (η11 , · · · , ηtt ) has a nonzero differential

~ , where ψij = dµij /dηij ~ = (ψ11 , · · · , ψtt )0 at (Rβ) ψ

• Delta method:

h

7

A

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