Tilburg University
Master Program
Summary Risk & Regulation
Supervisor:
Author:
de Waegenaere, A
Rick Smeets
Melenberg, B
December 9, 2024
,Table of Contents
1 Hybrid Claims and Arbitrage Pricing 2
1.1 The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Types of Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Examples of Hybrid Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Market (In)completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The Fair Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Hedge-Based Valuation of Hybrid Claims 8
2.1 Mean-Variance Hedge-Based Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Two-Step Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Pricing and Diversification 13
3.1 Valuation at Portfolio Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Pricing at Claim Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Benefits from Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Risk Measures and Risk Margins 17
4.1 The Use of Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Distortion risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 Two Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Pricing Orthogonal Claims Using Risk Measures . . . . . . . . . . . . . . . . . . . . 22
5 Systemic Risk 23
5.1 Possible Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Systemic Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Multi-Period Hybrid Claim Valuation 29
6.1 The Valuation Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Disentangling the Claim Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Multi-Period Risk Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.1 The CoC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Risk Capital Allocation 36
7.1 Definition and Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.2 Desirable properties of capital allocations . . . . . . . . . . . . . . . . . . . . . . . . 37
7.3 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.4 Allocation via the Worst-Case Measure . . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Regulatory Impact 41
8.1 A Classical Monetary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.2 The Basic New Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1
,1 Hybrid Claims and Arbitrage Pricing
1.1 The Market Model
We consider a liquid and frictionless market with n + 1 traded assets that pay off at a given future
date T , such as stocks, futures, or options. The payoff is a random variable on the probability
space (Ω, G, P). Recall that Ω is the sample space (possible outcomes), G is a collection of events
(G ⊂ 2Ω ), and P is a probability measure.
The current price of a risky asset m ∈ {1, 2, . . . , n} is denoted y (m) , and the payoff at time T is a
random variable denoted as Y (m) . Hence, we denote vectors (in Rn ) as:
y = (y (0) , y (1) , . . . , y (n) )
Y = (Y (0) , Y (1) , . . . , Y (n) )
Note that asset 0 is a riskless asset, that is:
y (0) = 1 and Y (0) = er
where r is the (deterministic) continuously compounded interest rate.
A trading strategy θ = (θ(0) , θ(1) , . . . , θ(n) ) is a vector in Rn+1 , where θ(m) is the number of units
invested in asset m at time 0.
We denote Θ ⊂ Rn+1 as the set of admissible strategies. In a frictionless market, we have
Θ = Rn+1 because there are no transaction costs or other trading restrictions. Moreover, we
assume that the n + 1 assets are non-redundant, so there does not exist an investment strategy
θ ̸= 0 = (0, 0, . . . , 0) such that θ · Y = 0.
The market is arbitrage-free if and only if there is no investment strategy θ ∈ Θ such that
θ · y = 0, P(θ · Y ≥ 0) = 1, and P(θ · Y > 0) > 0
That is, the portfolio breaks even or makes profit with certainty and the portfolio has a strictly
positive chance of generating a strictly positive payoff.
1.1.1 Types of Claims
A claim is a random variable S on the probability space (Ω, G, P). We will distinguish three types
of claims:
1. A hedgeable claim is a claim that can be replicated by a trading strategy ν = (ν (0) , ν (1) , . . . , ν (n) )
∈ Θ such that
Xn
S =ν ·Y = ν (m) Y (m)
m=0
That is, any portfolio of traded assets such as stocks, bonds, options or futures.
2
, 2. An orthogonal claim is a claim that is P-independent of the vector of the traded risky
assets (Y (1) , Y (2) , . . . , Y (n) ). That is, any claim for which the payoff is independent of the
financial market. Specifically, regular insurance contracts with payoffs independent of the
financial market, such as an insurance contract that yields coverage in case of fire damage to
a house.
3. A hybrid claim is a claim that is neither hedgeable nor orthogonal. That is, any claim
that has both financial risk and risk that is independent of the financial market. Specifically,
insurance contracts with payoffs dependent on the financial market are hybrid claims.
From now on, C denotes the set of all claims, C h denotes the subset of all hedgeable claims, and C ⊥
denotes the subset of all orthogonal claims. Note that the risk-free claims S = a for some a ∈ R
are the only claims that are orthogonal but hedgeable.
1.1.2 Examples of Hybrid Claims
Examples of hybrid claims are life insurance contracts with benefits that depend on a stock market
index. This means the amount the policy pays out can vary depending on how well the stock market
performs. Policyholders can often choose which stock market index or investment portfolio they
want their policy linked to.
There exist various types of contracts. The simplest type of these policies pays out based on the
value of a predefined stock portfolio at a specified future time T , but only if the policyholder is still
alive. More complex policies include a return guarantee. This is similar to having an embedded
put option, which ensures that the policyholder receives at least a minimum amount regardless of
stock market performance.
Example 1.1. (Pure endowment contracts)
Let Ti be the random remaining lifetime of an insured, T a deterministic future date, and Y the
value of financial assets at date T . Then, a pure endowment contract pays off f (Y ) on date T if
the insured is still alive, for some given function f : Rn+1 → R. The payoff on date T is
S = 1{Ti ≥T } × f (Y )
This contract is a hybrid claim that has elements of an insurance contract (payment depending on
survival) and a financial contract (payment depending on financial market performance).
A catastrophe-linked bond (CAT bond) is a bond whose principal is reduced when a catastrophe
occurs. By issuing these bonds, insurance risks associated with catastrophes are converted into a
form that investors can buy and sell. This process is known as securitization of insurance risk.
Insurance companies are also exposed to financial risk. They often take out reinsurance to help
cover large potential losses. This means that in the event of a significant claim, the financial burden
is shared between the insurer and the reinsurer. However, if the equity market performs well, an
insurance company with large claims may not need as much compensation from the reinsurer.
3
Master Program
Summary Risk & Regulation
Supervisor:
Author:
de Waegenaere, A
Rick Smeets
Melenberg, B
December 9, 2024
,Table of Contents
1 Hybrid Claims and Arbitrage Pricing 2
1.1 The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Types of Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Examples of Hybrid Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Market (In)completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The Fair Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Hedge-Based Valuation of Hybrid Claims 8
2.1 Mean-Variance Hedge-Based Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Two-Step Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Pricing and Diversification 13
3.1 Valuation at Portfolio Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Pricing at Claim Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Benefits from Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Risk Measures and Risk Margins 17
4.1 The Use of Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Distortion risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 Two Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Pricing Orthogonal Claims Using Risk Measures . . . . . . . . . . . . . . . . . . . . 22
5 Systemic Risk 23
5.1 Possible Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Systemic Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Multi-Period Hybrid Claim Valuation 29
6.1 The Valuation Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Disentangling the Claim Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Multi-Period Risk Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.1 The CoC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Risk Capital Allocation 36
7.1 Definition and Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.2 Desirable properties of capital allocations . . . . . . . . . . . . . . . . . . . . . . . . 37
7.3 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.4 Allocation via the Worst-Case Measure . . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Regulatory Impact 41
8.1 A Classical Monetary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8.2 The Basic New Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1
,1 Hybrid Claims and Arbitrage Pricing
1.1 The Market Model
We consider a liquid and frictionless market with n + 1 traded assets that pay off at a given future
date T , such as stocks, futures, or options. The payoff is a random variable on the probability
space (Ω, G, P). Recall that Ω is the sample space (possible outcomes), G is a collection of events
(G ⊂ 2Ω ), and P is a probability measure.
The current price of a risky asset m ∈ {1, 2, . . . , n} is denoted y (m) , and the payoff at time T is a
random variable denoted as Y (m) . Hence, we denote vectors (in Rn ) as:
y = (y (0) , y (1) , . . . , y (n) )
Y = (Y (0) , Y (1) , . . . , Y (n) )
Note that asset 0 is a riskless asset, that is:
y (0) = 1 and Y (0) = er
where r is the (deterministic) continuously compounded interest rate.
A trading strategy θ = (θ(0) , θ(1) , . . . , θ(n) ) is a vector in Rn+1 , where θ(m) is the number of units
invested in asset m at time 0.
We denote Θ ⊂ Rn+1 as the set of admissible strategies. In a frictionless market, we have
Θ = Rn+1 because there are no transaction costs or other trading restrictions. Moreover, we
assume that the n + 1 assets are non-redundant, so there does not exist an investment strategy
θ ̸= 0 = (0, 0, . . . , 0) such that θ · Y = 0.
The market is arbitrage-free if and only if there is no investment strategy θ ∈ Θ such that
θ · y = 0, P(θ · Y ≥ 0) = 1, and P(θ · Y > 0) > 0
That is, the portfolio breaks even or makes profit with certainty and the portfolio has a strictly
positive chance of generating a strictly positive payoff.
1.1.1 Types of Claims
A claim is a random variable S on the probability space (Ω, G, P). We will distinguish three types
of claims:
1. A hedgeable claim is a claim that can be replicated by a trading strategy ν = (ν (0) , ν (1) , . . . , ν (n) )
∈ Θ such that
Xn
S =ν ·Y = ν (m) Y (m)
m=0
That is, any portfolio of traded assets such as stocks, bonds, options or futures.
2
, 2. An orthogonal claim is a claim that is P-independent of the vector of the traded risky
assets (Y (1) , Y (2) , . . . , Y (n) ). That is, any claim for which the payoff is independent of the
financial market. Specifically, regular insurance contracts with payoffs independent of the
financial market, such as an insurance contract that yields coverage in case of fire damage to
a house.
3. A hybrid claim is a claim that is neither hedgeable nor orthogonal. That is, any claim
that has both financial risk and risk that is independent of the financial market. Specifically,
insurance contracts with payoffs dependent on the financial market are hybrid claims.
From now on, C denotes the set of all claims, C h denotes the subset of all hedgeable claims, and C ⊥
denotes the subset of all orthogonal claims. Note that the risk-free claims S = a for some a ∈ R
are the only claims that are orthogonal but hedgeable.
1.1.2 Examples of Hybrid Claims
Examples of hybrid claims are life insurance contracts with benefits that depend on a stock market
index. This means the amount the policy pays out can vary depending on how well the stock market
performs. Policyholders can often choose which stock market index or investment portfolio they
want their policy linked to.
There exist various types of contracts. The simplest type of these policies pays out based on the
value of a predefined stock portfolio at a specified future time T , but only if the policyholder is still
alive. More complex policies include a return guarantee. This is similar to having an embedded
put option, which ensures that the policyholder receives at least a minimum amount regardless of
stock market performance.
Example 1.1. (Pure endowment contracts)
Let Ti be the random remaining lifetime of an insured, T a deterministic future date, and Y the
value of financial assets at date T . Then, a pure endowment contract pays off f (Y ) on date T if
the insured is still alive, for some given function f : Rn+1 → R. The payoff on date T is
S = 1{Ti ≥T } × f (Y )
This contract is a hybrid claim that has elements of an insurance contract (payment depending on
survival) and a financial contract (payment depending on financial market performance).
A catastrophe-linked bond (CAT bond) is a bond whose principal is reduced when a catastrophe
occurs. By issuing these bonds, insurance risks associated with catastrophes are converted into a
form that investors can buy and sell. This process is known as securitization of insurance risk.
Insurance companies are also exposed to financial risk. They often take out reinsurance to help
cover large potential losses. This means that in the event of a significant claim, the financial burden
is shared between the insurer and the reinsurer. However, if the equity market performs well, an
insurance company with large claims may not need as much compensation from the reinsurer.
3
