Risk measures for insurance and finance

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Risk measures for insurance and finance

Definitions, properties and some applications

Christian Y. Robert

ISFA - Université Lyon 1

French-Vietnamese thematic school

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1 Introduction

1.1 Risks and risk management

⊲ Definitions of risk given in the Oxford Dictionary

- the possibility that something unpleasant or unwelcome will happen;

- a person or thing regarded as a threat or likely source of danger;

- a possibility of harm or damage against which something is insured;

- a person or thing regarded as likely to turn out well or badly in a particular context or respect;

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⊲ The main types of risks encountered in the financial industry are:

- the market risk: the risk of a change in the values of financial positions due to changes in the value of underlying components on which that positions depends, such as stock and bond prices, exchanges rates, commodity prices, etc...

- the credit risk: the risk of not receiving promises repayments on investments such as loans and bonds, because of the “default” of the borrower.

- the operational risk: the risk of losses resulting from inadequate or failed internal processes, people and systems, or from external events.

- the model risk: the risk associated with using a mis—specified (inappropriate) model for measuring risk.

- the liquidity risk: the risk for the lack of marketability of an investment that can not be bought or sold quickly enough to prevent or minimize a loss.

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⊲ The main types of risks encountered in the insurance industry are:

- the market risk, the credit risk, the operational risk, the model risk, the liquidity risk,

- the underwriting risk: the risks inherent in insurance policies that have been sold:

◦ the risk that premiums will not be sufficient to cover future incurred losses and that losses and loss adjustment expenses’ current reserves are not sufficient although the distributions of losses have been well assessed;

◦ the risk that may arise from an inaccurate assessment of the risks entailed in writing an insurance policy, or from factors that are not under the insurer’s con- trol (changes in patterns of natural catastrophes, changes in demographic tables underlying long-dated life products, changes in customer behaviour,...)

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⊲ A risk management process should involve the following steps:

- Establishing context: This includes an understanding (from an internal and an external viewpoint) of the current conditions in which the organization operates.

- Identifying risks: This includes the documentation of the material threats to the organization’s achievement of its objectives and the representation of areas to the organization may exploit for competitive advantage.

- Analyzing/quantifying risks: This includes the calibration and, if possible, creation of probability distributions of outcomes for each risk.

- Integrating risks: This includes the aggregation of all risk distributions, reflecting correlations and portfolio effects, and the formulation of the results in terms of impact on the organization’s key performance metrics.

- Assessing/prioritizing risks: This includes the determination of the contribution of

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- Treating/exploiting Risks: This includes the development of strategies for con- trolling and exploiting the various risks. There exist several response strategies for risks identified and analyzed, which may include:

◦ avoidance: exiting the activities giving rise to risk;

◦ reduction: taking action to reduce the likelihood or impact related to the risk;

◦ share or insure: transferring or sharing a portion of the risk, to finance it;

◦ accept: no action is taken, due to a cost/benefit decision;

◦ diversification: considering other similar and not too correlated risks to minimize the overall risk.

- Monitoring and reviewing: This includes the continual measurement and monitoring of the risk environment and the performance of the risk management strategies.

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Today, banks, insurance companies and other financial institutions face the same challenge: to collect and manage risks. But they have to balance diverse interests of internal and external stakeholders:

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1.2 Risks and risk measures

⊲ _Let _(Ω,_A_,P₎ be a probability space: Ω is the set of all possible outcomes (in economics often referred to as a state of nature), A is the σ-algebra, i.e. a set of subsets of Ω, called events, and P is the probability measure.

A one-period risky position (or simply risk) is defined as a random variable, i.e. a function on the probability space (Ω,A,P), characterized by its distribution function F(x) = P_(X _≤ _x).

⊲ A risk measure is a functional mapping a risk X to a real number Π(X), possibly infinite, representing the extra cash which has to be added toX to make it acceptable.

The idea is that quantifies the riskiness of X: large values of X tell us that X is

‘dangerous’.

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There are basically two ways to define a particular risk measure:

− In the first case, a set of properties to be satisfied by the risk measure is given and a risk measure is said to be “appropriate” in the situation under consideration if its characterizing properties are.

Axiomatization can then be used to justify a risk measure but also to criticize it.

− In the second case, a paradigm for decision under uncertainty can be selected to explain how decision-makers choose between uncertain prospects.

The risk measure is then obtained by an equivalence principle: the decision-maker is indifferent between the cash-flow Π(X) −X (corresponding to the case where X is covered) and 0 (X is not covered).

Note that this selection also amounts to opting for a set of properties.

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⊲ Hints for further reading:

− Some books:

◦ Quantitative risk management: concepts, techniques and tools. McNeil, A.J., R. Frey and P. Embrechts (2004). Princeton Series in Finance.

◦ Actuarial theory for dependent risks: measures, orders and models. M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas (2005). Wiley.

− Some papers:

◦ Risk measures and comonotonicity: a review. J. Dhaene, S. Vanduffel, Q.Tang, M. Goovaerts, R. Kaas, D. Vyncke (2006). Stochastic Models, 22, 573-606.

◦ Premium calculation and insurance pricing. R. Laeven, M. Goovaerts (2008).

Encyclopedia of Quantitative Risk Analysis and Assessment, Melnick, E. and Everitt, B. (eds). John Wiley & Sons Ltd, Chichester, UK, 1302-1314.

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2 Properties of risk measures

We will consider two situations to interpret the properties of the risk measure:

− A situation where the risk measure is used for calculating an actuarial premium

“Prem” (minimum amount that the insurer must raise from the insured in order that it is in the insurer’s interest to proceed with the contract).

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− A situation where the risk measure is used for determining provisions and capital requirements in order to avoid insolvency “Cap”.

X is then a possible loss or profit of some financial portfolio over a time horizon and we interpret Π(X) as the amount of capital that should be added as a buffer to this portfolio so that it becomes acceptable to an internal or external risk controller.

In such a case, X is the risk capital of the portfolio. X is a random variable with

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Properties to characterize a risk measure can be divided into four classes:

− rationality properties : these properties seem to be “rational”, in the sense that they are appropriate for almost people and they are not really questionable;

− additivity and homogeneity properties : these properties deal with sums of risks.

They describe the sensitivity of the risk measure with respect to risk aggregation.or scaling;

− comparison properties : these properties explain how risk measures preserve stochastic orders between risks;

− technical properties : these properties deal with technical requirements. They are usually necessary for obtaining mathematical proofs and are typically difficult to validate or to explain economically.

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2.1 Rationality properties

1. Non-excessive loading (or no-ripoff)

Π(X) ≤ max[X] = sup {x : P_{(X < x)} _< ₁_}_. This property can be taken in the following way:

− (Prem) “It is sensless to propose a premium larger than the maximal loss value”.

− (Cap) “It is useless to keep more capital than the maximal loss value”.

2. Non-negative loading

Π(X) ≥ E_[X_].

This property can be taken in the following way:

− (Prem) “The (technical) premium must be larger than the net premium, i.e. the mathematical expectation of the insured loss, otherwise ruin becomes certain (under the conditions of the law of large numbers)”.

− (Cap) “The capital (reserves/expected liabilities + required capital) must exceed

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3. No unjustified loading For any constant c,

Π(c) = c.

− (Prem) “A premium must be equal to the insured loss when the loss is certain”.

− (Cap) “For a degenerate risk c, the capital to be hold must not be different from the risk”.

4. Objectivity

For any risks X and Y with the same distribution, Π(X) = Π(Y ).

It means that Π(X) depends on X only through its distribution function F_X. This

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5. Translativity

For any constant c and for all risks X,

Π(X + c) = Π(X) + c.

− (Prem) “The underwriting costs must be added to the technical premium without modifying these costs”.

− (Cap) “Any increase in the liability by a deterministic amount c should result in the same increase in the capital”.

In particular

Π(X − Π(X)) = 0,

it means that, when we add Π(X) to the initial position −X, we obtain a ‘neutral’

position. Π(X) is the amount of safely invested capital that the holder of a risky

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2.2 Additivity and homogeneity properties

6. Subadditivity

For all risks X and Y (defined on the same probability space), Π(X + Y ) ≤ Π(X) + Π(Y ).

− (Prem) “The premium for insuring two risks must be smaller than the sum of the premiums raised by the insurer for the individual risks because the aggregated risk is reduced by diversification”.

− (Cap) “A merger does not create extra risk”.

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7. Comonotonic additivity

For all non-decreasing functions h and g,

Π(h(X) + g(X)) = Π(h(X)) + Π(g(X)).

− (Prem) “Insurers are not willing to give a reduction in the risk-load for a combined policy of comonotonic risks because they are bets on the same event and cannot act as a hedge against each other”.

− (Cap) “Putting comonotonic risks together never decreases the riskiness of the situation”.

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8. Positive homogeneity

For any positive constant c and for all risks X,

Π(cX) = cΠ(X).

− (Prem) “Multiplying all euro amounts of the claims by a constant exchange rate leads to multiplying the euro premium by the same exchange rate”.

− (Cap) “If the risk is multiplied by two, the capital must also be multiplied by two.”.

Positive homogeneity is closely related to comonotonic additivity. Indeed, assume that c is an integer, then comonotonic additivity implies the positive homogeneity.

However one can argue that the risk of a position is not always proportional to its size or scale and could be more for large positions (Π(cX) ≥ cΠ(X) for large c). It could

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2.3 Comparison properties

9. Monotonicity

For all risks X and Y (defined on the same probability space), P_(X _≤ _Y _{) = 1} _⇒ _Π(X₎ _≤ _Π(Y _).

− (Prem) “The amount of premium required for the loss X is smaller than the corresponding amount for Y when Y always exceeds X.”

− (Cap) “The amount of capital required for the loss X is smaller than the corresponding amount for Y when Y always exceeds X.”.

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10. Invariance by first-order stochastic dominance

D^EFINITION: Risk Y dominates risk X stochastically at first order, Y _DS₁ X if P_{(X > d)} _≤ P(Y > d), ∀d ∈ R_.

The property of invariance by first order stochastic dominance holds if for all risks X and Y ,

X _DS1 Y ⇒ Π(X) ≤ Π(Y ).

− (Prem/Cap) “If, for any amount d, the probability that a loss is larger than d is bigger for risk Y than for risk X, then the amount of premium/capital required for X is smaller than the corresponding amount for Y ”.

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PROPOSITION:

1. Risk Y dominates risk X stochastically at first order if and only if E_[u(₋_X_)] _≥ E_[u(₋_Y _)]

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

2. If X and Y have density propobability functions such that there exists a real number c ≥ 0 and:

f_X (d) ≥ f_Y (d) for d ∈] − ∞, c[

f_X (d) ≤ f_Y (d) for d ∈ [c,∞[.

then Y _DS₁ X.

3. Risk Y dominates risk X stochastically at first order if and only if there exist

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11. Invariance by second-order stochastic dominance

D^EFINITION: Risk Y dominates risk X stochastically at second order, Y _DS₂ X if E_[(X ₋ _d)₊_] _≤ E_[(Y ₋ _d)₊_] _∀_d _∈ R_.

The property of invariance by second order stochastic dominance holds if for all risks X and Y ,

X _DS2 Y ⇒ Π(X) ≤ Π(Y ).

− (Prem/Cap) “If, for any amount d, the stop-loss premium/shortfall risk with deductible d is bigger for risk Y than for risk X, then the amount of premium/capital required for X is smaller than the corresponding amount for Y ”.

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PROPOSITION:

1. Risk Y dominates risk X stochastically at second order if and only if E_[u(₋_X_)] _≥ E_[u(₋_Y _)]

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

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

X + D =^d Y and E_[D_|_Y _] _≥ ₀ _a.s.

3. If E_[X_] _≤ E_[Y _] and if there exists a real number c ≥ 0 such that F_X (d) ≤ F_Y (d) for d ∈] − ∞, d)

F_X (d) ≥ F_Y (d) for d ∈ [c,∞[ then Y _DS₂ X.

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2.4 Technical properties

12. Convexity

For all risks X and Y (defined on the same probability space), for any constant α ∈ [0, 1],

Π(αX + (1 − α)Y ) ≤ αΠ(X) + (1 − α)Π(Y ) whatever the dependency structure of X and Y .

13. Iterativitivity

For all risks X and Y (defined on the same probability space),

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14. Convergence in distribution

If (X_n) converges in distribution to X and if max[X_n] → max[X], then Π(X_n) → Π(X).

15. Stability by mixing

Let X^′, X₁ and X₂ be risks. If Π(X₁) = Π(X₂), then

Π(pF_X₁ + (1 − p)F_X_′) = Π(pF_X₂ + (1 − p)F_X_′) for any p ∈ [0,1].

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2.5 Definitions and relations between the properties

DEFINITIONS:

1. A risk measure is a monetary risk measure if it satisfies the following properties 1) Translativity, 2) Monotonicity.

2. A risk measure is a convex (or weakly coherent) risk measure if it satisfies the following properties 1) Translativity, 2) Monotonicity, 3) Convexity.

3. A risk measure is a coherent risk measure if it satisfies the following properties 1) Translativity, 2) Subadditivity, 3) Positive homogeneity, 4) Monotonicity.

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The properties of risk measures are not independent on each other as it is shown in the following proposition.

PROPOSITION:

1. Π satisfies the properties 1) Monotonicity, 2) Objectivity, iff it satisfies the Invari- ance by first-order stochastic dominance property.

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

3. Assume that Π is a risk measure that satisfies the Positive homogeneity property.

Π satisfies the Convexity property iff it satisfies the Subadditivity property.

4. Assume that Π is a risk measure that satisfies the Convexity property and Π(0) = 0. Π satisfies the Positive homogeneity property iff it satisfies the Subadditivity property.

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5. If Π satisfies the Monotonicity property and the “No unjustified loading” property, then it satisfies the “Non-excessive loading” property.

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

7. If Π satisfies the properties 1) Objectivity, 2) “No unjustified loading”, 3) Con- vexity, 4) Convergence in distribution, then it satisfies the “Non-negative loading”

property.

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

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

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⊲ Properties and families of risk measures

T^HEOREM: Π satisfies the properties 1) No unjustified loading, 2) Comonotonic additivity, 3) For all risks X and Y , E_[X_|_{X > F}⁻¹

X (h)] ≤ E_[Y _|_{Y > F}⁻¹

Y (h)]

∀h ∈ (0,1) ⇒ Π(X) ≤ Π(Y ), 4) Convergence in distribution, iff there exists a non-decreasing function H : [0,1] → [0,1] such that

Π(X) = H(0)E_[X_{] +}

₁ 0

E_[X_|_{X > F}⁻¹_(h)]dH_{(h) + (1} ₋ _H_{(1)) max[X}_].

THEOREM: Π satisfies for all risks S with bounded support the properties 1) No unjustified loading, 2) Invariance by second-order stochastic dominance, 3) Stability by mixing, iif there exist a continuous, non-decreasing and convexe function f such that

Π(S) = f⁻¹E_[f_(S_)]

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3 Families of risk measures

3.1 VaR, TVaR and other associated measures

DEFINITION: The Value at Risk is defined as the quantile of level α ∈ (0, 1) V aR[X;α] = inf{x ∈ R _: _F_(x) _≥ _α_} ₌ _F⁻¹_(α).

Note that for all x ∈ R and for all α ∈ (0, 1)

V aR[X;α] ≤ x ⇔ α ≤ F(x).

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PROPERTIES:

1. VaR satisfies the “Non-excessive loading” property.

2. VaR does not satisfy the “Non-negative loading” property.

3. VaR satisfies the “No unjustified loading” property 4. VaR satisfies the Objectivity property.

5. VaR satisfies the Translativity property.

6. VaR does not satisfy the Subadditivity property.

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7. VaR satisfies the Comonotonic additivity property.

8. VaR satisfies the Positive homogeneity property.

9. VaR satisfies the Monotonicity property.

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

11. VaR does not satisfy the Invariance by second-order stochastic dominance property.

12. VaR does not satisfy the Convexity property.

13. VaR does not satisfy the Iterativity property.

14. VaR satisfies the Convergence in distribution.

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PROPOSITION:

V aR[X;α] = arg min

π (E_[(X ₋ _π)₊_{] + (1} ₋ _α)π₎_.

Consider a portfolio with loss X. The regulator wants the solvency capital requirement related to the loss X to be large enough to ensure that the shortfall (measure by E_[(X ₋ _π)₊]) is sufficiently small. On the other hand, holding capital has a cost which is measure by (1 − α)π.

The capital requirement is determined as the solution to a minimization problem which balances the two conflicting criteria of low residual risk and low cost of capital:

VaR appears as an optimal capital requirement.

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LOCAL RISK ANALYSIS:

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

∂

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

and

∂²

∂γ²V aR[X + γY ;α] = ∂

∂sV_[Y _|_X ₊ _γY ₌ _s]

_{s=V aR[X}_+γY_;α]

+ V_[Y _|_X ₊ _γY ₌ _s] ^∂

∂sf_X_+γY(s)

s=V aR[X+γY;α]

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DEFINITIONS:

1. The Tail Value at Risk is defined as the arithmetic average of the VaRs of X from α on

T V aR[X;α] = 1 1 − α

₁

α V aR[X;ξ]dξ.

2. The Conditional Tail Expectation is defined as

CT E[X;α] = E_[X_|X > V aR[X;α]] . 3. The Conditional VaR is defined as

CV aR[X; α] = E_[X ₋ _{V aR[X}_;_α]_|X > V aR[X;α]]

= CT E[X;α] − V aR[X;α].

4. The expected Shortfall is defined as

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PROPOSITION:

For any α ∈ (0,1)

T V aR[X;α] = V aR[X;α] + 1

1 − αES[X;α]

CT E[X;α] = V aR[X;α] + 1

1 − F(V aR[X;α])ES[X;α]

CV aR[X;α] = 1

1 − F(V aR[X;α])ES[X;α]

(if F has a positive pdf, T V aR[X; α] = CT E[X;α]).

For any α ∈ (0,1)

min(E_[(X ₋ _π₎ _{] + (1} ₋ _{α)π) = (1} ₋ _{α)T V aR[X}_;_α].

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PROPERTIES:

1. TVaR satisfies the “Non-excessive loading” property.

2. TVaR satisfies the “Non-negative loading” property.

3. TVaR satisfies the “No unjustified loading” property.

4. TVaR satisfies the Objectivity property.

5. TVaR satisfies the Translativity property.

6. TVaR satisfies the Subadditivity property.

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8. TVaR satisfies the Positive homogeneity property.

9. TVaR satisfies the Monotonicity property.

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

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

12. TVaR satisfies the Convexity property.

13. TVaR does not satisfy the Iterativity property.

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

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LOCAL RISK ANALYSIS:

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

∂

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

and

∂²

∂γ²T V aR[X + γY ;α] = 1

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

×f_X_+γY (V aR[X + γY ;α]).

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3.2 Risk measures based on expected utility theory

Consider a decision-maker who has to choose between two uncertain future incomes modelled by the rvs R₁ and R₂.

He does not base his decisions on simply comparing the expectations of the incomes under consideration because the level of the expected income is not equal to his wellness itself.

A decision-maker bases his preferences on the ‘expected utility hypothesis’ if there exists a real-valued function u which represents the decision-maker’s utility-of-wealth, such that R₁ is preferred over R₂, if

E_[u(R₁_)] _≥ E_[u(R₂_)].

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A decision-maker is said to be risk-averse if he always prefers a certain income to a risky income with the same expectation.

Let p ∈ [0,1] and

R₁ = px + (1 − p)y, R₂ =

x with probability p

y with probability 1 − p , then

E_[u(R₁_)] _≥ E_[u(R₂_)] _⇔ _u(px _{+ (1} ₋ _p)y) _≥ _{pu(x) + (1} ₋ _p)u(y)

which means that u is concave.

A decision-maker is said to be risk-neutral if it is indifferent between a certain income

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AXIOMS OF EXPECTED UTILITY THEORY:

◦ A^XIOM EU1: If R₁ and R₂ are identically distributed then the decision-maker considers R₁ and R₂ as equivalent.

Decision-makers only take into account the probability distribution of the random income and are not influenced by other details of the income or by other aspects of the state of the world.

◦ AXIOM EU2: The preferences of the decision-maker generate a complete weak order, that is to say, it is reflexive, transitive and connected.

◦ AXIOM EU3: The preferences of the decision-maker are continuous with respect to the Wasserstein distance d_W, defined as

_∞

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If R₁,R˜₁, R₂ and R˜₂ are four random incomes such that R₁ is preferred over R₂ and if there exists ǫ > 0 such that

maxd_W(R₁,R˜₁), d_W(R₂,R˜₂) < ǫ then R˜₁ is preferred over R˜₂.

This axiom means R₁ is preferred over R₂, the same ordering holds for pairs R˜₁ and R˜₂ of risky prospects sufficiently close to R₁ and R₂.

◦ A^XIOM EU4: If F_R₁(t) ≤ F_R₂(t) ∀t, then R₁ is preferred over R₂.

◦ AXIOM EU5 (independence): Assume that R₁ is preferred over R₂. Let R˜₁ =

R₁ with probability p

S with probability 1 − p R˜₂ =

R₂ with probability p

S with probability 1 − p where S is an income independent of both R₁ and R₂, then R˜₁ is preferred over R˜₂ for any p ∈ [0,1].

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DEFINITION:

Consider an insurance company with initial wealth R and with an increasing and concave utility function u.

The company covers a risk X and sets its price for coverage Π(X) as the solution of the following indifference equation

E_[u(R ₋ _X _{+ Π(X}_{))] =} _u(R).

The premium Π(X) is fair in terms of utility: the right-hand side represents the utility of not issuing the contract; the left-hand side represents the expected utility of the insurer assuming the random financial loss X.

Putting R = 0, we get the so-called zero-utility principle proposed by Bühlmann in

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PROPERTIES OF ZERO-UTILITY PRINCIPLE:

1. Π(.) satisfies the “Non-excessive loading” property.

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

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

4. Π(.) satisfies the Objectivity property.

5. Π(.) satisfies the Translativity property.

6. Π(.) does not satisfy the Subadditivity property. Π(X) satisfies the Additivity for comonotonic risks property iff u(x) = −e⁻^αx or u(x) = x (up to a linear relation).

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8. Π(.) does not satisfy the Positive homogeneity property.

9. Π(.) satisfies the Monotonicity property.

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

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

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

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

14. Π(.) satisfies the Convergence in distribution property.

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3.3 Risk measure based on distorted expectation theory

Consider a decision-maker who has to choose between two uncertain future incomes modelled by the rvs R₁ and R₂. He does not base his decisions on simply comparing the expectations of the incomes under consideration because the level of the expected income is not equal to his wellness itself.

A decision-maker bases his preferences on the ‘distorted expectation hypothesis’ if there exists a non-increasing function g with g(0) = 0 et g(1) = 1, called a distortion function, such that R₁ is preferred over R₂ if H_g_[R₁_] _≥ H_g_[R₂_] _where

H_g_[R₁_{] =} ₋

₀

−∞

1 − g( ¯F_R₁(r))dr +

_∞

0 g( ¯F_R₁(r))dr.

Note that

E_[R₁_{] =} ₋

₀

1 − F¯_R (r)dr +

_∞

F¯_R (r)dr.

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Risk aversion and distorted probability distribution

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A decision-maker is said to be risk-averse if he always prefers a certain income to a risky income with the same expectation.

Let p ∈ [0,1], x > 0 and

R₁ = px, R₂ =

x with probability p

0 with probability 1 − p then

H_g_[R₁_] _≥ H_g_[R₂_] _⇔ _px _≥ _g(p)x _⇔ _p _≥ _g(p).

Note that, if g is convexe then the decision-maker is risk-averse.

A decision-maker is said to be risk-neutral if it is indifferent between a certain income

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AXIOMS FOR DISTORTED EXPECTATION THEORY (YAARI’S THEORY):

◦ A^XIOMS ED1-4 = A^XIOMS EU1-4.

◦ A^XIOM ED5 (independence): Assume that R₁ is preferred over R₂. Let R˜₁ and R˜₂ be defined by

F¯⁻_˜ ¹

R₁ = pF¯_R⁻¹

1 + (1 − p) ¯F_S⁻¹ F¯⁻_˜ ¹

R₂ = pF¯_R⁻¹

2 + (1 − p) ¯F_S⁻¹ where S is an other income, then R˜₁ is preferred over R˜₂ for any p ∈ [0,1].

Remind that for A^XIOM EU5, R˜₁ and R˜₂ are defined by F¯_R_˜

1 = pF¯_R₁ + (1 − p) ¯F_S F¯_R_˜

2 = pF¯_R₂ + (1 − p) ¯F_S.

Instead of independence being postulated for convex combinations which are formed along the probability axis, independence is postulated in Yaari’s theory for convex

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COMPARISON WITH EXPECTED UTILITY THEORY:

Let R be an income. In expected utility theory, the attractiveness of R is evaluated with the help of

E_{[u(R)] =}

₁

0 u(V aR[R;α])dα whereas in distorted expectation theory, it is replaced by

H_g_{[R] =}

₁

0 V aR[R; 1 − α]dg(α) =

₁

0 V aR[R;α]dg(1 − α).

These two equations highlight the difference between the two approaches: under the expected utility hypothesis, the possible amounts of fortune V aR[R;α] are adjusted by a utility function, while under the distorted expectation hypothesis, the tail probabilities are adjusted.

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WANG’S RISK MEASURES:

Consider an insurance company with initial wealth R and with a non-decreasing and convexe distortion measure g.

The company covers a risk X and sets its price for coverage Π(X) as the solution of the following indifference equation

H_g_[R ₋ _X _{+ Π(X}_{)] =} H_g_[R].

The premium Π(X) is fair in terms of distorted expectation: the right-hand side represents the distorted expectation of not issuing the contract; the left-hand side represents the distorted expectation of the insurer assuming the random financial loss X. The solution is then given by

Π(X) = H_g_¯_[X_]

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PROPERTIES:

6. Π(.) satisfies the Subadditivity iff.¯g is a concave function.

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8. Π(.) satisfies the Positive homogeneity property.

12. Π(.) satisfies the Convexity property.

13. Π(.) does not satisfy the Iterativity property.

14. Π(.) satisfies the Convergence in distribution property.

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EXAMPLES: 1. If

¯

g(p) = I_[p _≥ ₁ ₋ _α], then

Π(X) = H_g_¯_[X_{] =} _{V aR[X}_; _α].

This function is not concave and then the V aR is not subadditive.

2. If

¯

g(p) = min

p

1 − α,1 , then

Π(X) = H_g_¯_[X_{] =} _{T V aR[X}_;_α].

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3. Dual power: g¯ is defined as

¯

g(p) = 1 − (1 − p)^ξ, ξ ≥ 1.

If X ≥ 0 and ξ ∈ N_{, then} _Π(X_{) =} E_[max(X₁, . . . , X_ξ)] where X₁, . . . , X_ξ are iid random variables with the same distribution as X.

4. Proportional hazard: g¯ is defined as

¯

g(p) = p^1/ξ, ξ ≥ 1.

Let X^∗ be the random variable whose distribution is given by F¯^∗(.) = ( ¯F(.))^1/ξ (i.e. such that f^∗/F¯^∗ = ξ⁻¹f /F¯), then Π(X) = E_[X∗].

5. Normal transform: g¯ is defined as

¯

g(p) = Φ(Φ⁻¹(1 − p) + Φ⁻¹(p)),

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An important characterization of the T V aR is the following one:

T V aR[X;α] = min

¯ g

H_g_¯_[X_]_|_g_¯ is concave and H_g_¯_[X_] _≥ _{V aR[X}_;_α] _.

The T V aR can be seen as the distorted expectation risk measure with a concave distortion function that provides the mimimum capital larger than the V aR.

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3.4 Premium calculation principles

The price of insurance is the monetary value for which two parties agree to exchange risk and “certainty”.

There are two commonly encountered situations in which the price of insurance is subject of consideration:

- when an individual agent (for example, a household), bearing an insurable risk, buys insurance from an insurer at an agreed premium;

- when insurance portfolios (that is, a collection of insurance contracts) are traded in the financial industry (e.g., being transferred from an insurer to another insurer or from an insurer to the financial market (securitization)).

Pricing in the former situation is usually referred to as premium calculation while pricing in the latter situation is usually referred to as insurance pricing.

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1. The expected value principle DEFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) = (1 + β)E_[X_], _β _≥ _0.

If β = 0, Π(X) (= E_[X_]) is called the net premium. The net premium can be justified as follows: for the risk S_n = ⁿ_i=1 X_i of a large homogeneous portfolio, with X_i iid and E_[X_i_{] =} _{µ <} _∞, we have on behalf of the law of large numbers, for any ǫ > 0,

nlim→∞P ₍_|_S_n ₋ _nµ_| > nǫ) = 0.

Hence, in case the net premium µ is charged for each risk transfer X_i, the premium

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However, it follows from the central limit theorem (see also ruin theory) that if no loading is applied, ruin will occur with probability 1

nlim→∞P_(S_n ₋ nµ > ǫ) = 1.

Therefore a positive safety loading is required to avoid the ruin

nlim→∞P_(S_n ₋ _{n(1 +} β)µ > ǫ) = 0.

In this spirit, the loading factor β should be determined by setting sufficiently pro- tective solvency margins.

The main drawback of this principle is that the loading margin increases with the expected value and does not depend on the uncertainty of the risk X.

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For light tailed claims, one can use Chernoff’s Lemma to determine the loading factor.

Chernoff’s Lemma : Let X₁, X₂,.. , X_n be iid positive random variables such that E_[e^tX1] < ∞ for some t > 0. Let S_i = X_i − (1 + β) E_[X_i_{]. Then}

P



 n

i=1

S_i ≥ 0



 ≤ ρⁿ and lim

n→∞

1

n log P



 n

i=1

S_i ≥ 0



 = logρ, where

ρ = inf

t M_S(t) < 1, M_S(t) = E_[e^tSⁱ_{] = exp (}₋_{t(1 +} _β)E_[X_i_])_M_X_(t).

If the insurer wants to bound its ruin probability by ε, i.e. P ⁿ_i=1 S_i ≥ 0 ≤ ε, then he may choose β such that ρⁿ(β) = ε.

Example : If S has an Exponential distribution with mean α, ρ(β) = e⁻^β (1 + β).

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PROPERTIES:

1. Π(.) does not satisfy the “Non-excessive loading” property only if β = 0.

3. Π(.) does not satisfy the “No unjustified loading” property.

5. Π(.) does not satisfy the Translativity property only if β = 0.

6. Π(.) satisfies the Subadditivity property.

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13. Π(.) does not satisfy the Iterativity property only if β = 0.

14. Π(.) does not satisfy the Convergence in distribution property.

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2. The variance principle D^EFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) = E_[X_{] +} _βV_ar _(X₎ _, _β _≥ _0.

One of the main drawbacks of this principle is that the loading margin is symmetric with respect to the expectation of X. The negative values of X − E_[X_] _{are taken} into account in the calculation of the loading (with too heavy weight) although they are not dangerous to the insurer.

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PROPERTIES:

1. Π(.) does not satisfy the “Non-excessive loading” only if β = 0.

6. Π(.) does not satisfy the la Subadditivity property.

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8. Π(.) does not satisfy the Positive homogeneity property (for n ≥ 2, nΠ (X/n) <

Π(X)).

9. Π(.) does not satisfy the Monotonicity property.

10. Π(.) does not satisfy the Invariance by first-order stochastic dominance property.

11. Π(.) does not satisfy the Invariance by second-order stochastic dominance property.

12. Π(.) does not satisfy the Convexity property.

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3. The standard deviation principle D^EFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) = E_[X_{] +} _βV_ar _(X_), _β _≥ _0.

This principle shares a large number of drawbacks with the variance principle.

But contrary to the variance principle, it is positive homegenous (hence there is no incitation for the insured to split up the risk in several parts) and it is subadditive.

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PROPERTIES:

1. Π(.) does not satisfy the “Non-excessive loading” only if β = 0.

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13. Π(.) does not satisfy the Iterativity property sauf si β = 0.

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4. The Dutch principle D^EFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) = E_[X_{] +} _βE_[(X ₋ _αE_[X_])₊_], _α _≥ _1, ₀ _{< β} _≤ _1.

The principle only takes into account for the laoding margin the positive values of X − αE_[X_].

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PROPERTIES:

1. Π(.) does not satisfy the “Non-excessive loading” property only if β = 0.

3. Π(.) satisfies the “No unjustified loading” only if α = 1.

5. Π(.) satisfies the Translativity only if α = 1.

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5. The mean value principle D^EFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) = f⁻¹E_[f_(X_)]

where f is an increasing and convexe function.

Examples:

1. If f (x) = e^αx (α > 0), the exponential principle is obtained Π(X) = 1

α lnE_e^αX_.

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2. If f (x) = x^β (β ≥ 1), the β-norm principle is obtained Π(X) = E_X^β^1/β

Π(X) increases with β, lim_β_→₁ Π(X) = E_[X_] _and _lim_β

→∞Π(X) = max [X].

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PROPERTIES:

2. Π(.) satisfies the “Non-negative loading” property (by Jensen’s inequality).

5. Π(.) satisfies the Translativity property iff f(x) = e^αx or f(x) = x (up to a linear relation).

6. Π(.) does not satisfy the Subadditivity property.

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13. Π(.) satisfies the Iterativity property.

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6. The Esscher principle D^EFINITION:

The insurance company accepts the risk X at the price Π(X) given by Π(X) =

E_Xe^αX

E_e^αX ^, ^α ^≥ ^0.

Π(X) may be interpreted as the expectation of the random variable X under the Esscher transformed probability measure G defined by

dG(x) = e^αxdF(x)

_∞

0 e^αxdF(x).

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PROPERTIES:

6. Π(.) does not satisfy the Subadditivity property.

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13. Π(.) does not satisfy the Iterativity property.

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4 Applications

4.1 Optimal reinsurance

Reinsurance is insurance that is purchased by an insurer (also sometimes called a

“cedant”) from another insurance company (reinsurer) as a means of risk management.

The reinsurer and the insurer enter into a reinsurance agreement which details the conditions upon which the reinsurer would pay the insurer’s losses (in terms of excess of loss or proportional to loss). The reinsurer is paid a reinsurance premium by the insurer.

The main reason for insurers to buy reinsurance is to transfer risk from the insurer

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Let X be the (aggregated) risk that has been accepted by the insurer. The insurance market where the reinsurance treaties may be bought imposes conditions on the way of sharing the risk X.

For a function I : R⁺ _→ R⁺_, _I_(X₎ is the part of X that is transferred to the reinsurance. The function I must be continuous, non-negative, non-increasing and must inscrease . More formally, I must belong to the set:

I = I(.)|I(0) = 0; 0 ≤ I^′(s) ≤ 1

Some particularly important elements of I are:

- Quota Share Treaty Reinsurance I(s) = αs for α ∈ [0,1]

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We assume that the insurer has decided to use an amount P to his reinsurance program. Hence he may choose a treaty in the following subset:

IP = {I(.) ∈ I|Π_r(I(X)) = P}

where Π_r is the premium calculation principle used by the reinsurance company.

The insurer only considers the (net) risk, i.e. the random variable Z = X − I(X).

He chooses a risk measure Π and decides to minimize Π(Z) over the set I_P

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◦ The optimal reinsurance contract:

Assume that Π_r is the expected value principle and that Π satisfies the Invariance by second-order stochastic dominance property.

The set of treaties is then given by

I_P = {I(.) ∈ I|(1 + β)E_[I_(X_{)] =} _P_} where β is the loading coefficient.

PROPOSITION: Let I₁ and I₂ be two elements of I such that E_[I₁_(X_)] _≥ E_[I₂_(X_)]

(and Z₁ and Z₂ be the respective parts of the initial risk X that have been transfered to the reinsurer).

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THEOREM: The optimal reinsurance contract is given by I_d(s) = (s − d)⁺,

where d is such that (1 + β)E_[I_d_(X_{)] =} _P.

PROOF:

I

I₁

I₃ I₂

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Risk comparisons and risk aggregation

◦ First stochastic dominance PROPOSITION:

1. If X is a random variable independent on X₁ and X₁^′, and X₁ _DS1 X₁^′, then:

X₁ + X _DS1 X₁^′ + X.

2. If X₁, ..., X_n and X₁^′, ..., X_n^′ are independent random variables such that X_i _DS₁ X_i^′ for all i then:

n

X_i _DS₁

n

X_i^′.

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3. If X₁, ..., X_n and X₁^′, ..., X_n^′ are sequences of independent random variables such that X_i _DS1 X_i^′ for all i and if N and N^′ are integer valued random variables independent on (X_n) and X_n^′ such that N _DS1 N^′, then:

N

i=1

X_i _DS₁

N^′

i=1

X_i^′.

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◦ Second stochastic dominance

1. If X₁, ..., X_n and X₁^′, ..., X_n^′ are independent random variables such that X_i _DS₂ X_i^′ for all i then:

n

i=1

X_i _DS₂

n

i=1

X_i^′.

2.If X₁, ..., X_n and X₁^′, ..., X_n^′ are sequences of independent random variables such that X_i _DS₂ X_i^′ for all i and if N and N^′ are integer valued random variables independent on (X_n) and X_n^′ such that N _DS2 N^′, then:

N

i=1

X_i _DS₂

N^′

i=1

X_i^′.

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Let us consider the collectif risk model X = ^N_i=1 X_i, where the X_i are iid amounts of loss and N is a random integer valued random variable indepent of (X_i).

We assume that the reinsurance treaty is of the following form:

T(n, x₁, x₂, ...) =

n

j=1

I(x_j), with I ∈ I_P,

where n is the number of claims and P is such that (1 + β)E_[T_(X_{)] =} _P.

THEOREM: (Optimal per claim reinsurance) The optimal reinsurance contract is given by

T_d =

n

j=1

(x_j − d)⁺

where d is such E_(X ₋ _d)⁺ ₌ _{P/((1 +} _β)E_[N_]).

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4.2 Markowitz portfolio optimization

An investor wants to invest a proportion of his wealth W₀ in a portfolio of risky assets with the remainder in cash (or indeed may borrow money to fund his purchase of risky assets in which case there is a negative cash weighting).

He buys securities at date 0 and sell his overall portfolio at date 1. It is a one-period (static) model.

1. Model and assumptions i) Assets

The investor may find in the market n + 1 assets:

- a risk-free asset (cash) (i = 0), - n risky assets (i = 1, . . . , n),

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The asset returns are given by:

- the risk-free interest rate: r_f = (p_0,1−p_0,0)/p_0,0 > 0 (we assume p_0,0 = 1), - the risky asset returns r_i = (p_i,1 − p_i,0)/p_i,0.

The first two moments of the vector of risky asset returns are given by:

- E_{[r] =} _µ

(n×1), - V_{ar(r) =} E_rr′

− E_[r]E_r′

= E_(r_t ₋ _µ)(r_t ₋ _µ)′

= Ω _(n_×_n).

We assume that the investor have an accurate conception of these two moments of the distribution of returns and that Ω is strictly positive definite.

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ii) The market

We willl use the following notation:

- b is the amount in cash,

- a = (a₁, ..., a_n)^′ is the vector of amounts in risky assets.

The assumptions are:

- the investor can lend/buy and borrow unlimited amounts in cash and in risky assets: (b, a) ∈ R _× Rⁿ (in reality, every investor has a credit limit and short selling is rarely permitted);

- there are no taxes or transaction costs;

- all investors are price takers, i.e., their actions do not influence prices: in particular the vector of the initial price p₀ = p_0,1, ..., p_0,n does not depend on a

Risk measures for insurance and finance