SIMULATIONS STOCHASTIQUES ET APPLICATIONS EN GESTION … · Stochastic Simulation and Applications in Finance with MATLAB Programs (The Wiley Finance Series) “Because it adopts

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SIMULATIONS STOCHASTIQUES ET APPLICATIONS EN GESTION DES RISQUES

Van Son Lai, Ph.D., P. Eng. (Brit. Col.), CFADirecteur Fonds Conrad LeblancCodirecteur Laboratoire d’ingénierie financière de l’Université Laval (LABIFUL)

SÉMINAIRE DE L’ÉCOLE D’ACTUARIAT 13 février 2014

Plan de la présentation

Description des caractéristiques novatrices et distinctives de Stochastic Simulation and Applications in Finance with Matlab Programs » (Huynh, Lai et Soumaré, Wiley,2008, HLS), ayant contribué au succès du livre et en ayant fait une référence pour les examens de Fellowshipde la Society of Actuaries.

Applications au capital économique des portefeuilles d’assurance-crédit.- Les garanties financières- Risk-Based Capital and Credit Insurance Portfolios

Survol de la modélisation des risques de catastrophes naturelles et l’analyse des Cat Bonds.- Hedging Flood Losses in Quebec using CAT-Bonds- The Valuation of Catastrophe Bonds with Exposure to Currency Exchange Risk

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Marc Michaud, M. Sc. (ing. fin.), CFA Conseiller, Développement des produits d'épargne spécialisée

Gestion du patrimoine et Assurance de personnes, Mouvement Desjardins

« Comment décrire le fameux bouquin rouge HLS de Messieurs Huynh, Lai et Soumaré... Pour moi, il est MA première référence pour toute question d'ingénierie financière. C'est un livre qui m'a permis de grandir énormément comme ingénieur financier et qui me sert encore ÉNORMÉMENT!!! dans mon travail au quotidien chez Desjardins.

C'est véritablement un MUST pour toute personne qui désire en apprendre sur l'ingénierie financière tant les professionnels que les néophytes. Les références à des articles à la fin des chapitres permettent de pouvoir approfondir davantage des sujets plus poussés, mais également le livre incorpore des nouveaux concepts qui sont sortis dans des articles de recherche récents et avancés. »

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Marc Michaud, M. Sc. (ing. fin.), CFA Conseiller, Développement des produits d'épargne spécialisée

Gestion du patrimoine et Assurance de personnes, Mouvement Desjardins

« J'apprécie également le fait d’avoir les codes Matlab disponibles ce qui rend le livre encore plus accessible à une variété d'usagers. Ce livre fait véritablement le tour de la théorie mais également de l'application en pratique (c'est mon livre le plus utilisé dans ma bibliothèque tellement les pages sont gribouillés de détails et de réflexions) et doit nécessairement appartenir à l'élite des manuels en finance quantitative. »

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http://www.master272.com/ouvrages_finance.html

« Un ouvrage remarquable sur les méthodes de simulations en finance et en utilisant Matlab. Depuis sa parution en Français, il a été aussi traduit en anglais : la preuve qu'il comblait une lacune importante.»

HLS comme manuel de base dans des cours, i.e.,-Financial Derivatives & Stochastic Calculus, S. Rachev at Stony Brook University

-Quantitative Security Analysis, Axel KindUniversity of Basel

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Stochastic Simulation and Applications in Finance with MATLAB Programs (The Wiley Finance Series)

"This book provides a very useful set of tools for those who are

interested in the simulation method of asset pricing and its

implementation with MatLab. It is pitched at just the right level

for anyone who seeks to learn about this fascinating area of

finance. The collection of specific topics thoughtfully selected by

the authors, such as credit risk, loan guarantee and value-at-

risk, is an additional nice feature, making it a great source of

reference for researchers and practitioners. The book is a

valuable contribution to the fast growing area of quantitative

finance."

-Tan Wang, Sauder School of Business, UBC “This book is a good companion to text books on theory, so if

you want to get straight to the meat of implementing the

classical quantitative finance models here's the answer.” —Paul Wilmott, wilmott.com

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“This powerful book is a comprehensive guide for Monte Carlo methods in finance. Every quant knows that one of the biggest issues in finance is to well understand the mathematical framework in order to translate it in programming code. Look at the chapter on Quasi Monte Carlo or the paragraph on variance reduction techniques and you will see that Huu Tue Huynh, Van Son Lai and Issouf Soumaré have done a very good job in order to provide a bridge between the complex mathematics used in finance and the programming implementation.”

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“Because it adopts both theoretical and practical

point of views with a lot of applications, because it

treats about some sophisticated financial problems

(like Brownian bridges, jump processes, exotic

options pricing or Longstaff-Schwartz methods) and

because it is easy to understand, this handbook is

valuable for academics, students and financial

engineers who want to learn the computational

aspects of simulations in finance.” —Thierry Roncalli, Head of Investment Products

and Strategies, SGAM Alternative Investments & Professor of Finance, University of Evry

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À travers une approche intégrée, HLS permet aux lecteurs novices ou expert et chercheurs chevronnés d'appliquer le calcul stochastique et les techniques de simulation Monte Carlo (MC) et Quasi-MC aux problèmes de l'innovation financière et de développer et/ou d'adapter les modèles d’évaluation des titres contingents.

HLS initie les utilisateurs à la formulation des problèmes courants en finance et leur résolution tout en fournissant les programmes Matlab des différents travaux pratiques, ce qui permet au lecteur d'avoir des recettes spécifiques pour résoudre des problèmes de processus stochastiques en finance.

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Quoique HLS se veut un livre d'initiation et pédagogique, nous introduisons plusieurs techniques utiles et récentes de simulations telles que:

Quadratic Resampling Technique, Barraquand (1995), Technique de programmation dynamique et

l'agrégation des états stratifiés (Stratified State Aggregation) de Barraquand et Martineau (1996) et Méthode des moindres carrés (Least-Squares Method) de Longstaff et Schwartz (2000) pour évaluer les options américaines,

Méthode d'extraction de la volatilité (MEV) proposée par Cvitanic, Goukasian et Zapatero (2002) pour estimer les coefficients de sensitivité des options (les grecques).

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Les modèles de structure à terme de Heath, Jarrow et Morton (1992) et le modèle de marché (Market Model) de Brace, Gatarek et Musiela(2001) en ce qui à trait aux dérivés de taux d'intérêt.

Une exposition extensive sur la valorisation des titres corporatifs et du risque de crédit basé sur l'approche structurelle de base de Merton (1974).

Des cas de gestion de risque de portefeuilles de garanties financières inspirés des publications des auteurs.

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Gestion des risques et Valeur à Risque (VaR) et VaR et Analyse en Composantes Principales (ACP).En bref, on trouve dans HLS: Pédagogie, rigueur, pertinence et application.Initie aux concepts de base tout en prenant en compte des problèmes actuels. Permet son accessibilité à un plus grand nombre de lecteurs sans pour autant sacrifier la rigueur mathématique tout en minimisant les concepts abstraits et jargons non nécessaires.

Corporate Finance & ERM Exam: Foundations of CFE Spring 2014

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3. Topic: Stochastic Modelling

Learning Objective

The candidate will understand how and when to apply various stochastic techniques to situations which have uncertain financial outcomes.

Learning Outcomes

The Candidate will be able to:

a) Explain the mathematical foundation of stochastic simulation.

b) Assess the appropriateness of a given stochastic simulation technique to quantify various market risk exposures.

c) Recommend the use of techniques to reduce the computational demand when applying stochastic methodology.

d) Assess the strengths and weaknesses of the calibration techniques for a given stochastic model.

e) Interpret the results of a given application of stochastic modelling and the impact of the chosen calibration process used.

f) Explain the differences and implications of the use of P-measure and Q-measure for risk assessment.

g) Explain the benefits and limitations of Value-at-Risk, Incremental Value-at-Risk, Component Value-at-Risk, and Expected Shortfall as tail risk measures.

Resources

• Stochastic Simulation and Applications in Finance, Huynh, Huu Tue, et. al.

o Ch. 1,(background)

o Ch.2 Introduction to Random Variables

o Ch. 3 Random Sequences

o Ch. 4 Introduction to Computer Simulation of Random Variables

o Ch. 5 Foundation of Monte Carlo,

o Ch. 6 Fundamental of Quasi Monte Carlo Method,

o Ch. 7 Introduction to Random Processes 7.1.1, 7.1.2, 7.3.1-7.3.3,

o Ch. 8 Solution of Stochastic differential equation,

o Ch. 9 General Approach of Valuation Technique, (Background)

o Ch. 10 Pricing Options using Monte Carlo Simulations only 10.1,

o Ch. 11 Term Structure of Interest Rate and Interest Rate Derivatives,

o Ch. 14 Risk Management and VaR

o Ch. 15 VaR and Principal Component Analysis

• Measures of Market Risk, Dowd, Kevin, 2nd Edition

o Ch. 2 Measures of Financial Risk,

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4. Topic: Advanced Risk Assessment Techniques

Learning Objective

The candidate will understand how to critique the appropriateness of advanced risk assessment methods for a given situation.

Learning Outcomes


Pricing

a) Apply and interpret the results of equilibrium pricing and no-arbitrage pricing theory to risk valuation.

Cost of Capital Methods

b) Compare and Contrast the methods that quantify the cost of capital within a risk valuation framework.

Concept of Tail Correlation and Copula

c) Describe the limitations of modelling dependences in risk variables.

d) Apply techniques to estimate tail correlation for long dated liabilities.

Operational Risk

e) Explain how to quantify risk when there is limited data

Resources

• Measuring Market Risk, Kevin Dowd, 2nd Edition

o Ch. 5 forecasting volatilities covariances and correlation including appendix – Modelling Dependence: Correlations and Copula.


o Ch. 4.6 MCMC o Ch. 15 VaR and Principal Component Analysis

• F-107-13: A Market Cost of Capital Approach to Market Value Margins • F-109-13: Application of Coherent Risk Measures to Capital Requirements in Insurance

• F-110-3: Chapters 4 and 5 of Financial Economics, Panjer, Harry, et. al.

• A Risk Management Tool for Long Liabilities: The Static Control Model, 2009 Enterprise Risk Management Monograph

• A Practical Concept of Tail Correlation 2008 Enterprise Risk Management Monograph • Measuring Operational Risk Interdependencies using Interpretative Structural Modeling, 2007 ERM

Symposium, Concurrent Sessions 3

http://www.soa.org/library/monographs/other-monographs/2009/april/mono-2009-m-as09-1-manistre.aspx

http://www.soa.org/library/monographs/other-monographs/2008/april/mono-2008-m-as08-1-manistre.aspx

http://www.ermsymposium.org/2007/handouts.php

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5. Topic: Financial Risk Management

Learning Objective

The candidate will understand how to identify and recommend appropriate risk assessment and monitoring techniques for financial risk management.

Learning Outcomes


a) Evaluate the methods and processes for measuring and monitoring market risk positions.

b) Describe the types of models and the sources of model risk.

c) Assess the methods and process for quantifying and managing model risk within a financial institution.

d) Design an appropriate stress-testing process and evaluate its limitations for a given risk position.

e) Interpret the results of back-testing.

Resources

• Measures of Market Risk, Dowd, Kevin , 2nd Edition

o Ch. 2 Measures of Financial Risk

o Ch. 3 Estimating Market Risk Measures and Introduction and Overview

o Ch. 4 Non-parametric Approaches (Including Appendix 1 – 4)

o Ch. 6&7 Parametric Models

o Ch. 10 Option Risk measures

o Ch. 12 Mapping Positions to Risk Factors

o Ch. 13 Stress Testing Risk

o Ch. 15 Back Testing Risk

o Ch. 16 Model Risk

• Variable Annuities A Global Perspective, Kalberer, Tigran and Ravindran, Kannoo

o Ch. 8, 9, 15, 16 & 17


o Ch. 14 Risk Management and VaR

• F-108-13: The Known, the Unknown, and the Unknowable in Financial Risk Management: Measurement and Theory Advancing Practice, Diebold, et.al., Ch. 3

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Les garanties de prêts financières

Van Son Lai, Ph.D., P.Eng.(Brit.Col.), CFA

FAI, Université Laval

Québec, Canada

Introduction

1. Importance des garanties financières

•Garanties de prêts

-publiques, privées, implicites, explicites,omniprésentes (Merton & Bodie, 1992, «On the Management of Financial Guarantees», FinancialManagement).

•Risque de crédit/défaut, rehaussement de crédit, atout dans innovations financières et gestion des risques.

•Analyses des garanties : Cadre conceptuel pour l’analyse du risque de crédit.

Introduction (suite) 2. Brève revue de la littérature

•La plupart des études sur les garanties de prêts sont sur celles offertes par les gouvernements, publiques, « sans-défaut » (Merton, 1977, «An Analytic Derivation of the Cost of Deposit Insurance and Loan Guarantees : An Application of Modern Option Pricing Theory», Journal of Banking and Finance, Jones & Mason, 1980, «The Valuation of Loan Guarantees», Journal of Banking and Finance, Sosin, 1980, «On the Valuation of Federal Loan Guarantees to Corporations», Journal of Finance) et autres.

Brève revue de la littérature

•Assurance-dépôt : Merton (1977), Ronn &Verma (1986, « Pricing Risk-Adjusted DepositInsurance : An Option-Based Model», Journal ofFinance), Lai (1996, «The Effects of Variations inLaxity (or Strictness) of Closure Rule on theValuation of Deposit Insurance», The FinancialReview), Lai & Warywoda (1997, «Coinsuranceand Private-Public Partnership in DepositInsurance», Research in Finance) parmi tantd’autres.


• Garanties privées/vulnérables/risquées dans un contexte

d’un prêt avec un garant représentatif : Lai (1992, «An Analysis of Private Loan Guarantees», Journal of Financial Services Research), Lai et Gendron (1994, «On Financial Guarantees Insurance Under Stochastic Interest Rates», Geneva Papers on Risk and Insurance Theory), Lai, (1995, «On the Valuation of Private Loan Guarantees : Wealth Effects and Government Insurance Protection», Research in Finance), Lai et Yu (1999, «An Accurate Analysis of Vulnerable Loan Guarantees», Research in Finance), Chang, Lai & Yu (2002, «Credit Enhancement and Loan Default Risk Premia», Canadian Journal of Administrative Sciences).


En contexte de portefeuille de garanties d’un seul endosseur : Gendron, Lai & Soumaré (2002, «An Analysis of Private Loan Guarantee Portfolios», Research in International Business and Finance, Financial Risk and Financial Risk Management), d’un gros prêt garanti par plusieurs garants : Chang, Chung & Yu (2006, «Loan Guarantee Portfolios and Joint Loan Guarantees with Stochastic Interest Rates», The Quarterly Review of Economics and Finance.

1. Recherche en contexte de portefeuille de garanties

- Identité fondamentale (dans le sens fonctionnel et de

pricing, Merton & Bodie (1992)).

Dette risquée ≡ Dette sans-défaut – Valeur de la

garantie.

"This identity strictly applies only if the guarantee

itself is default-free and it covers the whole loan"

(Merton and Bodie (1992)).

→ Dans Gendron, Lai & Soumaré (2002) on répond aux questions suivantes:

Q 1. Comment la structure de la covariance

influence le pricing des garanties?

Q 2. Combien de garanties constitue un portefeuille

de garanties bien diversifié?

Q 3. Trade-off entre la diversification sectorielle et

la diversification par taille/nombre?



• R 1. Le risque du portefeuille augmente avec des

corrélations élevées (positives et négatives) entre les

firmes emprunteuses et l’assureur.

• R 2. Entre 10 à 15 garanties produisent un portefeuille

de garanties bien diversifié.

• R 3. Dans le scénario avec leviers élevés, la

diversification par nombre semble être plus efficace

que la diversification par secteurs.


→ Extensions :

Structure à terme des taux d’intérêt

Volatilités stochastiques

Violation de la règle de subordination

(Violation of Absolute Priority Rule)

Diversification temporelle

Valeur à risque

Capital à risque versus capital réglementaire.


→ Méthodologie :

Spécifier les caractéristiques stochastiques du processus des sous-jacents, autrement dit

Dynamiques des sous-jacents par des équations différentielles stochastiques (EDS)

-Sous-jacents: valeur des actifs, taux d’intérêt,

risque/volatilité, pertes, etc.

Valeur du titre contingent =

EQ (Valeur Présente des Payoffs à T)

EQ (.) = Espérance sous la probabilité risque neutralisé Q

Risk-Based Capital and Credit Insurance PortfoliosFinancial Markets, Institutions & Instruments (2010)

Van Son LAI(Joint with Issouf SOUMARÉ)

Faculty of Business Administration, Laval University, Canada

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Main providers of financial guarantees

International organizations World Bank MIGA Other development banks

Export Credit Agencies US Ex-Im Bank Japan NEXI China Sinosure UK ECGD EDC Canada COFACE France

Banks and credit insurance companies

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Guarantees used to facilitate access to funds at improved terms (source: World Bank)

Debt Maturity Interest Spread

Colombia

Philippines

10

15

7

6.5%

5%

2.5%

3%

with Guarantee

without Guarantee

5

Côte

d’Ivoire1

12

3%

2.75%

Uganda 0

16 3.1%

8%

1Bangladesh3%

2%14

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Purpose of the paper

The aim of this paper is to study the impact of loan maturities, assets’ risks and time-varying assets correlations on pricing portfolios of financial guarantees

We propose an extension of Merton (1974)’s model that incorporates time-varying correlations using a one factor Gaussian copula model

Furthermore, the pricing of the guarantee takes into account the expected losses as well as the unexpected losses (CreditVaR)

We also consider a risk averse insurer with disutility proportional to the square of large deviation from expectation

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Overview of the contribution

Constant correlations lead to undervaluation of the guarantee contract since it understates the losses

There is an optimal average maturity for the loans composing the portfolio

The insurance premium increases with the risk level of the projects composing the guarantee portfolio

However, the risk based capital contribution from the guarantor does not necessarily increase since it has already been accounted for in the premium

The insurer’s asset risk affects its insuring capacity even if an appropriate risk-based capital has been set to cover losses in the portfolio

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The model

The model captures different types of risks inherent to portfolios of financial guarantees by allowing stochasticity of the following : Cash flows of the policyholders Net-assets of the insurer Cash flows volatilities Short-term risk free interest rate Correlations.

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Cash flows available for repayment

Portfolio of N financial guarantee contracts Each contract consists of insuring one debt Vi denotes the cash flows available for loan repayment

by the policyholder i (i = 1,…,N)

)()())()(()(

)(tdZtdtttr

tV

tdViii

i

iσδ +−=

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Insurer's net assets

The total insurance premium raised and the economic capital brought by the insurer’s equityholders are deposited in an account (reserve) denoted by W

W is the total net-assets available to the insurance company to support credit insurance claims

)()())()(()(

)(tdZtdtttr

tW

tdWWWW σδ +−=

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Stochastic volatilities & short-term interest rate

)()())(()( 22tdZtdtttd

kkkkkkk σσφσβνσ +−=

)()())(()( tdZtrdttrtdr rrσθκ +−=

Stochastic volatilities of the cash flows

Stochastic risk-free interest rate

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Stochastic correlations

The risk source of each variable k is defined as linear combination of two independent Wiener processes X(common component) and Yk (idiosyncratic component)

where

),()(1)()()( 2tdYttdXttdZ kkkk ρρ −+=

1 and , ,otherwise

X)(for )( 2

≤≥

≤

= k

L

k

H

kL

k

H

k

k

tXt ρρρ

ρ

ρρ

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Debts covenants

The debt i has coupon rate ci, face value Fi and mature at Ti

Default before the maturity of the debt is triggered by missing on a coupon payment or hitting the barrier

The insurer seizes the policyholder’s remaining cash flows at the first default date, and then proceeds to the payment on the guaranteed debt

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Claims valuation

)()( such that min ii Ttg

iii eFtVt−

≤=τ

∫

+∫

=

−−

∫ i

dssr

ii

T dssr

i FedtFceED

iT

i

t

ττ

τ

ττ

)()(

)(Price of the default-free coupon bond

)()0( 00

)(

0

)(

0,

∫

+∫

=

−−

∫ ii

dssr

ii

dssr

NGi VedtFceED

i

i

t

τ

τ

τ Price of the non-guaranteed risky debt

−∫

+=

−

)()(),(min)0()0( 0

)(

0,, iiiii

dssr

NGiGi VDWeEDD

i

τττ

τ

Price of the guaranteed risky debt

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Illustration of the expected loss & guarantee

Di,NG(0)

Non-guaranteed risky debt

Di,G(0)

guaranteedrisky debt

Di(0)

Default-free debt

Guarantee (Gi)

Loss (Li)

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Premium & utility of the insurer

Expected vs. unexpected losses EL : Expected Loss MPL : Maximum Probable Loss at 99% confidence level UL = MPL – EL : Unexpected Loss (or CreditVaR)

Portfolio total premium raised

Utility specification

)(][Pr2

][Pr TOCemiumVark

EGemiumEU −−−=

444 3444 21

43421

oncompensati rsShareholde

capital EconomicpriceFair 1

)(PrPr RELMPLELemiumemiumN

i

i ×−+==∑=

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Constant vs. stochastic correlation regimes

5 10 15 20 25 30 35 40100

200

300

400

500

600

700

800

900

1000

1100 Expected Loss

T

EL

ρH=ρ

L

ρH=0.80

5 10 15 20 25 30 35 40400

600

800

1000

1200

1400

1600

1800

2000

2200 Maximum Probable Loss

T

MP

Lρ

H=ρL

ρH=0.80

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5 10 15 20 25 30 35 40200

400

600

800

1000

1200

1400

1600 Insurance Premium

T

Pre

miu

m

ρH=ρ

L

ρH=0.80

5 10 15 20 25 30 3560

65

70

75

80

85

90 Utility of the Guarantor

T

Util

ityρ

H=ρL

ρH=0.80

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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1500

600

700

800

900

1000

1100

1200

1300

1400

1500 Insurance Premium

σ

Pre

miu

m

ρH=ρ

L

ρH=0.80

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11050

1100

1150

1200

1250

1300 Excess Capital brought by Insurer

σ

Cap

ital

ρH=ρ

L

ρH=0.80

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Conclusion

This paper provides a portfolio framework to analyse the inter-temporal decision choice of loan guarantees

Although the model is based on the structural approach of Merton (1974), it is richer with important features such as stochastic assets correlations and barrier based default triggering à la Black & Cox (1976)

By not taking into account the changes in correlations with the economic conditions, undervalues the guarantee policy and requires more economic capital from the insurer’s equity holders, especially for portfolios with longer maturity loans.

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Hedging Flood Losses in Quebec using CAT-Bonds

Alexandre Têtu, Van Son Lai, IssoufSoumaré & Michel Gendron

Alexandre Têtu, M.Sc. (ing. fin.)

Financial Markets Analyst

1Laboratoire d’ingénierie financière de l’UL (www.fsa.ulaval.ca/LABIFUL/)

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Motivation

In Canada, floods are natural disasters with the highest frequency of occurrence

Floods are uninsurable for homeowners in Canada

To eliminate big unexpected expenditures caused by catastrophic floods, the government wants flood risk to become insurable for homeowners

One way the government could control the expenses related to its financial aid program, without altering it, would be to transfer part of the risk to capital markets

Whether the risk is transferred to capital markets or to insurance companies, the first step is quantifying the risk.


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Transferring risk to capital markets:Quantifying the risk/costs

1- Develop step by step framework that can be used to model catastrophe risk and value hedging costs

2- Apply it to flood risk in Quebec to adress the followingquestions: How to model the risk of occurrences of flood in Quebec?

How to value CAT-Bonds issued to transfer floods risk to capital markets?

How does the part of the risk beared by the federal governmentimpact these CAT-Bonds?

Looking back, what kind of return would the investors have received?


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CAT-Bonds – Brief explanation

It’s a securitization of disaster risks. Catastrophe bonds are issued by insurers (re-) or by governments

who wish to transfer some of their exposure to the risk of occurrence of a natural disaster


5

Literature review

Literature on CAT-Bonds valuation models contains different valuation approaches.(Dassios & Jang (2003), Jarrow (2010), Perrakis & Boloorforoosh (2013), among others)

No-arbitrage approach (albeit incomplete markets in the case of disaster risk)

Equilibrium model (dependant of the utility function used)

Actuarial approach used to evaluate the risk premium of past issues

Laboratoire d’ingénierie financière de l’UL (www.fsa.ulaval.ca/LABIFUL/)5

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Literature review

Most literature on modeling disaster risk use the Poisson process (PP) as the appropriate stochastic process. Cummins and Geman (1995) PP with constant jump amplitude.

Baryshnikov, Mayo and Taylor (1999) double compound PP.

Louberge, Kellezi and Gilli (1999) PP with log-normal distribution for jump amplitude.

Lin, Chang and Powers (2009) reflected the likelihood of an increase in frequency of occurrence due to global warming using a double stochastic PP.

Hainaut (2010) proposed a process that takes into account the seasonal effect of some disasters. The intensity of the PP is the sum of a Ornstein-Uhlenbeckprocess and a periodic function.

In most literature, the loss distribution is modeled as independent of the occurrence process.


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Literature review

Risk Premium Puzzle:

Bantwal and Kunreuther (1999) : CAT-Bonds have higher risk premium compared to equivalent rated bonds. Among causing factors: ignorance of investors towards this new type of securitized instrument and aversion to risk.

Cummins (2008) : Risk premium of issues between 2001 and 2007 ranges from 2 to 6 times the expected loss.

Froot (2001) : For the same expected loss, risk premium is higher for disaster with lower probability of occurrence of disaster but larger severity of damage.

Froot (2008) : Risk premium is greatly affected by the recent occurrence of disasters (Whether it is of the same type or different type than the one covered by the CAT-bond)


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Loss data

Data comes from: Financial Assistance Program of the Government of Quebec (Canada)

Database contains all floods for which costs exceeded 300,000 $ for the provincial government.

These costs represents only a portion of what would be covered floods were insurable since the program only covers essential goods.

Data supplied by the Department of Public Safety of the Government of Quebec.

Database provides monthly data from March 1992 to June 2011.


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Loss data


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Catastrophe modeling – Occurrence process

Poisson process used to model the occurrence of floods. Intensity λ of the PP can be constant (HPP), time-dependant

or stochastic (NHPP) Homogeneous Poisson process (HPP) :

Non-homogeneous Poisson process (NHPP): Include the effects of seasonality

In Quebec, historical floods during the spring/summer are far more often then during winter.

Modeled using a periodic (sinusoidal) function with a period of 12 months.

Include the effects of global warming

Calibration to historical data:

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Catastrophe modeling – Occurrence process

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Catastrophe modeling – Loss distribution

Loss distribution independant of occurrence process Search distribution that best fits historical data Mean excess function of the historical data is increasing :

Distribution must have fatter tails than the exponential distribution

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Catastrophe modeling – Loss distribution

Loss distribution : 300,000$ + Fitted distribution to (historical data –300,000$).

When the fitted distribution deviates from the true distribution in the tails, the Anderson-Darling (A2) test is the most powerful statistic to use.

Three distributions have the lowest values for the test statistics: Log-normal, Generalized Pareto and Burr distributions.

Using additional statistics (KS, W2) and comparing the distributions mean excess function with the historical function, we chose the Pareto generalized.

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Maximum probable loss (MPL)

The maximum probable loss (MPL) is defined as the amount that is equal to or exceeds the amount of damages for a certain percentage of trials.

The exceedance probability is probability that annual losses are greater than this amount

15

CAT-Bonds Valuation model


16

CAT-Bonds Valuation model

Based on Jarrow (2010) no-arbitrage model.

Modifications to allow multiple events before remainingcapital is returned to investors.

CAT-Bond model validated through comparison withMonte-Carlo simulations (PP & Gamma distribution)

Can’t be used in the case of Quebec floods becausethe density function of a sum of the i.i.d. GeneralizedPareto variables has no analytical solution. Using Monte Carlo simulations for floods in Quebec.


17

Hedging Quebec Floods loss – CAT-Bonds Characteristics

The coupons are paid monthly and the annual coupon rate is c

The principal amount at issuance is A0 = $ 500 million

The maturity of the bond is 1 year (to allow a comparison with the historical losses on an annual basis)

The bonds are issued on December 1st of each year.

We use the zero-coupon yield curves data from the Bank of Canada. (For the methodology of the zero-coupon yield curve, we refer the interested reader to Bolder, Johnson and Metzler (2004)).

We use the PP and loss distribution defined before.

When a catastrophic event occurs, we assume that the loss amount can be immediately assessed and the principal is reduced at the end of the period.


18

Hedging Quebec Floods loss – Costs evaluation

Under the risk neutral probability, the value of the catastrophe bond at issuance is given by:

We use Monte Carlo simulations to obtain the adjusted principal amounts An.

Then, using the zero-coupon yield curve, we determine the coupon rate which allows us to issue the CAT-Bond at par.

The costs of an issue is the present value of the coupon spread paid on the CAT-Bond relative to an equivalent risk-free bond. It is given by:


19

Historical analysis – actuarial costs

We compare the annual losses that have occurred with the costs that the government would have born if there was a hedging strategy via a CAT-Bond issue of one year maturity.


20

Historical analysis – risk premium

We compare the impact of different levels of risk premium. Lane & Mahul (2008) found an average long-term trend towards a risk

premium ratio of 2.69.


21

Historical analysis – risk premium & trigger level


We chart the costs for different trigger levels but with a the long-term risk premium of 2.69.

Evidently, losses must be significantly larger for the insurance hedge to be viable.

22

Historical analysis – Optimal deductible


Next, we plot the average annual costs over the study period as a function of the deductible level with a risk premium ratio of 2.69.

23

Historical analysis – Federal aid inclusion


Under the current provincial flood financial aid program, the federal government has to defray part of the costs when a disaster exceeds a certain amount per capita of the province.

The federal and provincial governments could share the costs of the CAT-Bonds hedge.

The provincial government could include the federal aid and issue CAT-Bonds to hedge only the part not covered by the federal aid.

The risk premium being usually less for small scale events, the inclusion of the federal aid would decrease the risk premium required by the market.

24

Historical analysis – Federal aid inclusion


We chart the costs for complete hedging (risk premium=2.69) with and without including the federal aid.

Evidently, expected losses and hedging costs are lower when federal aid is taken into account

25

Attractiveness for investors


The principal benefit to an investor that invests a portion of its portfolio in CAT-Bonds is diversification. Returns offered by disaster risk markets have little or no correlations with traditional securities (stocks and bonds) markets. (Litzenberger et al. (1996), Constantin (2011))

For the past 19 years, these bonds would have offered performance that, except for the year 2011, seem to be uncorrelated with the S&P/TSX.

Average annual returns of catastrophe bonds during these 19 years was higher than that of the S&P/TSX over the same period. The highest average annual return is obtained with zero deductible and with federal aid included.

26



27



We can see that CAT-Bonds with the Federal aid included not onlygives higher returns, but also higher Sharpe Ratios.

28

Hedging costs – looking forward


We use the model developed above to estimate the costs of issuance of catastrophe bonds allowing the government of Quebec to hedge against future floods losses.

We assume a risk premium of 2.69 and that the term structure of interest rates stays the same.

Costs increase very fast because of upward trend in the occurrence of catastrophe. As the time pass, if the number of occurrence of floods stabilize, future costs will be lower.

29

Conclusion


We have developed a methodology to model the risk of flood losses

Although our data have some shortcomings, we were able to estimate the function parameters that best fits our data for :

a non-homogeneous Poisson process with upward trend and seasonal effect.

a loss distribution independent of the Poisson process.

Applied to the Quebec Government flood financial aid program, this methodology allowed us to estimate the historical and future costs of hedging flood losses.

We have shown that these instruments could be attractive for investors that wishes to diversify and enhance the yield of their Canadian portfolio.

OUTLINEFACTS

PREVIOUS RESEARCHCONTRIBUTION

VALUATION MODELMONTE CARLO SIMULATION

FINDINGSCONCLUSION

THE VALUATION OF CAT BONDS WITHEXPOSURE TO CURRENCY EXCHANGE RISK

Van Son Lai, Ph.D., P.Eng. (BC), CFALaval University

(Joint with Mathieu Parcollet & Bernard F. Lamond)

86th ARIA Annual Meeting,August 4–7, 2013, Washington DC, USA

1/27 Van Son Lai, Ph.D., P.Eng. (BC), CFA, Laval University The Valuation of CAT Bonds & Currency Exchange Risk

OUTLINEFACTS



FINDINGSCONCLUSION

FACTS

I Catastrophe Bonds (Cat Bonds) payoffs are tied to occurrence ofnatural disasters such as hurricanes and earthquakes.

I Property insurance claims of approximately $60 billion in the lastfive years (Canter et al [1996]) have caused great concern to theinsurance industry and resulted in the insolvency of a number offirms.

I The CAT bond we are interested in is described by specifying theregion, type of events, type of insured properties etc.

I Cat Bonds offer insurers the ability to hedge events that couldotherwise leave them insolvent.

I Cat Bonds offer investors a unique opportunity to enhance theirportfolios with an assets that provides an attractive return that isuncorrelated with the markets.


OUTLINEFACTS



FINDINGSCONCLUSION

I Cat bonds have been severely tested by both natural and financialdisasters, on each occasion they have performed robustly.

I Attractive yields with low sensitivity to traditional markets.

I Cat bond yields -at typically 5-15% above LIBOR and significantlyless volatility (at around 3%) that traditional asset classes.

I Due to liquidity, Cat bonds was mainly issued in US Dollars (USD).

I Almost USD 7 billion in Cat bonds was issued in 2012.

I Exposure to Currency Exchange Risk.

I Question : How to deal with Currency Exchange Risk ?


OUTLINEFACTS



FINDINGSCONCLUSION

I To deal with Exchange Risk, Garman & Kohlhagen [1983] extendBlack-Scholes’ formula to options on currency, where the foreigninterest rate plays the same role as a dividend rate.

I Grabbe [1983] developed a model in which the prices of domesticand foreign obligations are assumed to follow geometric Brownianmotions and obtained a closed-form formula.

I Amin & Jarrow [1991] tackled the same problem using theframework of Heath, Jarrow & Morton [1992].

I Hilliard et al [1991] provided a generalization of the previous modelsfor stochastic domestic and foreign interest rates correlated with theexchange rate.


OUTLINEFACTS



FINDINGSCONCLUSION

I Hilliard et al [1991] showed that the model where interest rates arestochastic yields better performance than that with constant ratesfor estimating the price of options on currencies.

I Hakala & Wystup [2002] adapted Heston’s volatility model toexchange rate options.

I Haastrecht et al [2009] and Grzelak & Oosterle [2010] considered amodel with four factors with stochastic rates and volatilities,

I But these authors did not find a closed-form expression for theoptions on currency.


OUTLINEFACTS



FINDINGSCONCLUSION

CONTRIBUTION

I Cat bond new model taking into account sponsor’s exposure tocurrency risk and risk of occurrence of catastrophic events.

I New Natural Risk Index with jumps process.

I Extension of Poncet & Vaugirard [2001]’s model including a hedgingcost for the current risk, jump-diffusion process for catastrophicevents, and three dimensional stochastic process for exchange rate,domestic and foreign interest rates.

I Derivation of a semi-closed form for Cat bond’s price undermartingale measure.

I Joshi & Leung [2007]’s extension approach for three factors MonteCarlo simulation.


OUTLINEFACTS



FINDINGSCONCLUSION

VALUATION MODEL

Let St the exchange rate governed by geometric Brownian motions, rf ,and rd the foreign and domestic interest rates each according to themodel of Vasicek, described as follows :

dSt/St = (rd − rf )dt + σSdWSt (1)

drd = κd(θd − rd)dt + σddWdt (2)

drf = κf (θf − rf )dt + σf dWft (3)

where κd and κf are the mean reversion rates, θd and θf are thelong-term means, σS , σd and σf are the instantaneous volatilities, andW S

t , W dt and W f

t are three Brownian motions with correlation matrix

Γ =

1 ρSd ρSf. 1 ρdf. . 1

.


OUTLINEFACTS



FINDINGSCONCLUSION

In the model of Vasicek, the price at t of the zero-coupon domestic bond(here American, that is denominated in USD) earning $1 at T is

Pd(t,T ) = exp A(τ)− B(τ)rd , (4)

where

B(τ) =1− e−κdτ

κd

and

A(τ) =

(θd −

σ2d

2κd

)(B(τ)− τ)− σ2

d

4κdB(τ)2,

with time to maturity τ = T − t.


OUTLINEFACTS



FINDINGSCONCLUSION

ν2(τ) the conditional variance of the forward exchange rate F (t, T )From Hilliard et al [1991]

ν2(τ) = σ2Sτ +

τ

3(σ2

d + σ2f − 2σdf ) + τ 2(σSd − σSf ) (5)

and the price of the call option on the foreign currency in this model is

C (t,T ) = Pd(t,T )[F (t,T )N(d1)− KN(d2)], (6)

N(·) is the cumulative distribution function of the standard normaldistribution,

d1 =ln (F (t, T )/K ) + 1

2ν2(τ)

ν(τ)

andd2 = d1 − ν(τ).

The spot and forward exchange rates are linked by the interest rate parity,

F (t,T ) =Pd(t,T )

Pf (t,T )St . (7)


OUTLINEFACTS



FINDINGSCONCLUSION

As Vaugirard [2002], combine with Merton [1976] jump diffusion process,the loss index follows

dIt/It− = µdt + σdWt + (Y − 1)dNt , (8)

Nt is a standard Poisson process of constant intensity λN , and Y is alog-normal random variable, such that at a jump epoch tn,

It+n = YIt−n .

From Merton [1976], risk due to jump process is assumed to bediversifiable. The index It is not traded on the market ; a specific riskpremium is thus applied to it. Under the risk-adjsted Q-dynamics, Itbecomes

dIt/It− = (µ− λσ) dt + σdWt + (Y − 1)dNt , (9)

where λ is the risk premium due to the jumps.


OUTLINEFACTS



FINDINGSCONCLUSION

We define the instant η when the Cat bond is triggered as the first timewhen the index It reaches the trigger threshold H :

η = mint≥0

It ≥ H.

The promised cash flow of the Cat Bond at maturity T is

X = V1η>T + (1− ω)V1η<T = V (1− ω1η<T ) ,

1 is the indicator random variable.

I If the trigger is not hit before maturity, the face value V is entirelyreimbursed.

I If not, Cat Bond is triggered before maturity, it is reduced by aproportion ω.


OUTLINEFACTS



FINDINGSCONCLUSION

I To take into account the exchange risk, we must add the pay-off ofa hedge that would be setup by the sponsor who would not want toassume this risk.

I When the Cat bond is triggered, the guarantor takes a long positionon a call on his own currency to fix the exchange rate at T .

I The payoff of this call is (ST − K )1ST>K1η<T .

I The total payoffs is

XCAT = V − ω1η<T −ωV

K(ST − K )1ST>K1η<T . (10)


OUTLINEFACTS



FINDINGSCONCLUSION

General expression for the price of a Cat Bond

I Discount factorD(t,T ) = e−

∫ Tt

rd (u)du. (11)

We assume the underlying probability space (Ω,F ,Q) endowed withsome filtration Ft

I The price of the Cat Bond is then

PCat(t,T ) = EQ [D(t,T )XCAT|Ft ] (12)

Let QT the equivalent forward martingale measure for the date T .Using equation (10), after some algebraic manipulations and changeto forward martingale measure (Geman et al [1995]), one obtains

PCat(t,T ) = VPd(t,T )

1− ωEQT [1η<T |Ft ]

− ω

KEQT [(ST − K )1ST>K1η<T |Ft ] (13)


OUTLINEFACTS



FINDINGSCONCLUSION

To simplify equation (13) it is reasonable to assume that :

A1 : Catastrophe index in independent of the rest of the economy

A2 : The dynamics of It remain unchanged i.e.,

EQT [1η<T |Ft ] = EQ[1η<T |Ft ]

The price formula (13) becomes

PCat(t,T ) = VPd(t,T )

1− ω

(1 +

C (t,T )

KPd(t,T )

)Q(η < T )

(14)


OUTLINEFACTS



FINDINGSCONCLUSION

MONTE CARLO SIMULATION

I Joshi & Leung [2007]’s method adapted to Cat bond

I Decomposition :

PCat(t,T ) = Q(t1 > T )PCat(t,T )|no jump+Q(t1 ≤ T )PCat(t,T )|jump

I Waiting time between jumps follows an exponential distribution withparameter λN .So p = Q(t1 > T ) = e−λNT and Cat bond follows the explicitformula (14) and the trigger probability can be expressed as

Q(η < T ) = N(d1) + N(d2)

(I0H

)1−2γ/σ2

with γ = µ− λσ,

d1 =ln(I0/H) + (γ − σ2/2)T

σ√T

, d2 =ln(I0/H)− (γ − σ2/2)T

σ√T20/27 Van Son Lai, Ph.D., P.Eng. (BC), CFA, Laval University The Valuation of CAT Bonds & Currency Exchange Risk

OUTLINEFACTS



FINDINGSCONCLUSION

I Impact of the variance of the interest rates

σd Cat bond price σf Cat bond price0.02 722.14 0.02 722.170.03 722.07 0.03 722.070.05 721.98 0.05 721.83

0.075 721.97 0.075 721.480.1 722.09 0.1 721.08

0.15 722.74 0.15 720.19

Variance of domestic interest rate exhibits a positive relation with Catbond price.


OUTLINEFACTS



FINDINGSCONCLUSION

I Influence of the correlation structure

ρSd Cat bond price ρSf Cat bond price ρdf Cat bond price-0.8 722.84 -0.7 721.86 -0.8 722.08-0.5 722.65 -0.5 721.99 -0.5 722.07-0.2 722.46 -0.2 722.21 -0.2 722.06

0 722.34 0 722.36 0 722.060.2 722.23 0.2 722.51 0.2 722.060.5 722.07 0.5 722.73 0.5 722.07

0.75 721.94 0.8 722.97

I Confirmation of intuition provided by the equation forvariance of the forward rate


OUTLINEFACTS



FINDINGSCONCLUSION

I Influence of the initial level of the interest rates

rd(0) Cat bond price rf (0) Cat bond price0.08 735.52 0.08 723.010.09 728.78 0.09 722.56

0.1 722.07 0.1 722.070.11 715.37 0.11 721.530.12 708.7 0.12 720.96


OUTLINEFACTS



FINDINGSCONCLUSION

CONCLUSION

I Negative Impact of the exchange rate on the Cat bond price

I Negative Impact of the exchange rate variance on the Catbond price

I Variance of domestic interest rate exhibits a positive relationwith Cat bond’s price

I Negative Impact of correlation between Exchange rate andDomestic Interest rate on the Cat bond price

I Strong Negative Impact of the initial level of domesticinterest rate on the Cat bond’s price.


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