Risk Insurance summary
Carine Wildeboer
April 2022
Contents
1 Chapter 1, Utility Theory and Insurance 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Utility functions and their properties . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Useful results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Implications for insurance business . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 The policyholder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 The insurance company . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.3 When is insurance possible? . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Stop-loss reinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Chapter 2, The Individual Risk Model 4
2.1 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Moment generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Other transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Mixed distributions and risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Mixed distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Mixed random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Law of iterated expectation or tower rule . . . . . . . . . . . . . . . . . . . 5
2.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 The rigid way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 The intuitive way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.3 The ultimate way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Chapter 3, Collective Risk Models 6
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Collective risk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Properties of compound Poisson distributions . . . . . . . . . . . . . . . . . . . . . 7
3.4 Individual versus collective risk model . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Advanced example: Maximum claim size . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Chapter 4, Ruin Theory 8
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Properties of the Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Characterization of the ruin process . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Lundberg’s exponential upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5.1 Ruin model with reinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5.2 Discrete-time ruin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1
,5 Chapter 5, Premium 10
5.1 Premium calculation from top-down: a case study . . . . . . . . . . . . . . . . . . 10
5.1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1.2 Setting premium with the ruin probability . . . . . . . . . . . . . . . . . . . 10
5.1.3 Including dividend payments . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1.4 Selecting initial investment . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.5 Dividing premium to individual policy . . . . . . . . . . . . . . . . . . . . . 11
5.2 Premium principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2.1 Premium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Coinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6 Chapter 6, Bonus-Malus Systems 12
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Example of bonus-malus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 Loimaranta efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.4 Hunger for bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Chapter 7, Ordering of Risks 13
7.1 Stochastic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.1.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.1.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.2 Thicker tailed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 Stop-loss order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3.3 Relations with other orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4 Exponential order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.3 Relations with other orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.5 Relation between ordering concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.6 Implications for the ordering concepts . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Chapter 8 15
9 Chapter 9, Generalized Linear Models in Insurance 15
9.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.2 Generalized linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.3 Poisson GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.4 Poisson GLM with exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.5 GLM estimation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2
, 1 Chapter 1, Utility Theory and Insurance
1.1 Introduction
St. Petersburg Paradox For price P, enter game. n trials, gain is 2n . Expected gain:
P ∞ n n
n=1 2 (1/2) = ∞. But, unless P is small only a few will enter game.
1.2 Utility functions and their properties
E[(u(w − X)]
• Property 1: Non-decreasing functions: u′ (w) ≥ 0. Marginal utility is non-negative.
• Property 2: Concave (risk-averse agents): u′′ (w) ≤ 0 or convex (risk-loving): u′′ (w) ≥ 0
Remark: E(u(w − X)) ≤ E(u(w − Y )) ⇐⇒
E(a ∗ u(w − X) + b) ≤ E(a ∗ u(w − Y ) + b)
1.2.1 Useful results
Risk aversion coefficient: r(w) of utility func. u(·) at wealth w is:
′′
(w)
r(w) = − uu′ (w)
Jensen’s inequality: If v(·) is convex: E(v(X)) ≥ v(E(X))
If v(·) is concave: E(v(X0) ≤ v(E(X)).
1.3 Implications for insurance business
Policyholders: risk averse, insurance company: risk averse or neutral.
1.3.1 The policyholder
Utility function u(·), is concave or linear and increasing. Buy insurance against loss X for premium
p. Then expected loss: E(X) = µ < ∞. If you buy, utility: u(w − P ). If you do not buy, utility:
E(u(w − X)). By Jensen:
E(u(w − X)) ≤ u(E(w − X)) = u(w − E(X)) = u(w − µ).
Max. premium acceptable: u(w − P + ) = E(u(w − X)) ⇒ P + ≥ µ
1.3.2 The insurance company
Utility function U (·), is concave or linear and increasing. P − : minimum premium company wants
to receive. By Jensen:
U (W ) = E(U (W + P − − X)) ≤ U (E(W + P − − X)) = U (W + P − − µ) ⇒ P − ≥ µ
1.3.3 When is insurance possible?
If P + ≥ P − ≥ µ
1.4 Stop-loss reinsurance
When claims are too big for an insurance company it transfers the risk to a reinsurance company.
Stop-loss reinsurance: For a loss X the payment by the reinsurer to the insurer is:
(
X − d if X > d
(X − d)+ = max X − d, 0 =
0 if X ≤ d
3
Carine Wildeboer
April 2022
Contents
1 Chapter 1, Utility Theory and Insurance 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Utility functions and their properties . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Useful results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Implications for insurance business . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 The policyholder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 The insurance company . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.3 When is insurance possible? . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Stop-loss reinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Chapter 2, The Individual Risk Model 4
2.1 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Moment generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Other transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Mixed distributions and risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Mixed distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Mixed random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Law of iterated expectation or tower rule . . . . . . . . . . . . . . . . . . . 5
2.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 The rigid way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 The intuitive way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.3 The ultimate way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Chapter 3, Collective Risk Models 6
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Collective risk model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Properties of compound Poisson distributions . . . . . . . . . . . . . . . . . . . . . 7
3.4 Individual versus collective risk model . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Advanced example: Maximum claim size . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Chapter 4, Ruin Theory 8
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Properties of the Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Characterization of the ruin process . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Lundberg’s exponential upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5.1 Ruin model with reinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5.2 Discrete-time ruin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1
,5 Chapter 5, Premium 10
5.1 Premium calculation from top-down: a case study . . . . . . . . . . . . . . . . . . 10
5.1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1.2 Setting premium with the ruin probability . . . . . . . . . . . . . . . . . . . 10
5.1.3 Including dividend payments . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1.4 Selecting initial investment . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.5 Dividing premium to individual policy . . . . . . . . . . . . . . . . . . . . . 11
5.2 Premium principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2.1 Premium properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Coinsurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6 Chapter 6, Bonus-Malus Systems 12
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Example of bonus-malus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 Loimaranta efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.4 Hunger for bonus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Chapter 7, Ordering of Risks 13
7.1 Stochastic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.1.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.1.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.2 Thicker tailed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 Stop-loss order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3.3 Relations with other orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4 Exponential order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.2 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4.3 Relations with other orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.5 Relation between ordering concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.6 Implications for the ordering concepts . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Chapter 8 15
9 Chapter 9, Generalized Linear Models in Insurance 15
9.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.2 Generalized linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.3 Poisson GLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.4 Poisson GLM with exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.5 GLM estimation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2
, 1 Chapter 1, Utility Theory and Insurance
1.1 Introduction
St. Petersburg Paradox For price P, enter game. n trials, gain is 2n . Expected gain:
P ∞ n n
n=1 2 (1/2) = ∞. But, unless P is small only a few will enter game.
1.2 Utility functions and their properties
E[(u(w − X)]
• Property 1: Non-decreasing functions: u′ (w) ≥ 0. Marginal utility is non-negative.
• Property 2: Concave (risk-averse agents): u′′ (w) ≤ 0 or convex (risk-loving): u′′ (w) ≥ 0
Remark: E(u(w − X)) ≤ E(u(w − Y )) ⇐⇒
E(a ∗ u(w − X) + b) ≤ E(a ∗ u(w − Y ) + b)
1.2.1 Useful results
Risk aversion coefficient: r(w) of utility func. u(·) at wealth w is:
′′
(w)
r(w) = − uu′ (w)
Jensen’s inequality: If v(·) is convex: E(v(X)) ≥ v(E(X))
If v(·) is concave: E(v(X0) ≤ v(E(X)).
1.3 Implications for insurance business
Policyholders: risk averse, insurance company: risk averse or neutral.
1.3.1 The policyholder
Utility function u(·), is concave or linear and increasing. Buy insurance against loss X for premium
p. Then expected loss: E(X) = µ < ∞. If you buy, utility: u(w − P ). If you do not buy, utility:
E(u(w − X)). By Jensen:
E(u(w − X)) ≤ u(E(w − X)) = u(w − E(X)) = u(w − µ).
Max. premium acceptable: u(w − P + ) = E(u(w − X)) ⇒ P + ≥ µ
1.3.2 The insurance company
Utility function U (·), is concave or linear and increasing. P − : minimum premium company wants
to receive. By Jensen:
U (W ) = E(U (W + P − − X)) ≤ U (E(W + P − − X)) = U (W + P − − µ) ⇒ P − ≥ µ
1.3.3 When is insurance possible?
If P + ≥ P − ≥ µ
1.4 Stop-loss reinsurance
When claims are too big for an insurance company it transfers the risk to a reinsurance company.
Stop-loss reinsurance: For a loss X the payment by the reinsurer to the insurer is:
(
X − d if X > d
(X − d)+ = max X − d, 0 =
0 if X ≤ d
3
