Risk Optimization for Hybrid Pension Plans with ARMA and GARCH Investment Returns

101 Pages·2016·1.25 MB·English

Risk Optimization for Hybrid Pension Plans with
ARMA and GARCH Investment Returns
Ashwag ALzahrani
A Project
for The Department of
Mathematics and Statistics
Presented in Partial Fulfillment of the Requirements
for the Degree of Master of Science (Mathematics) at
Concordia University
Montr´eal, Qu´ebec, Canada
May 2016
(cid:13)c Ashwag ALzahrani, 2016
CONCORDIA UNIVERSITY
School of Graduate Studies
This is to certify that the project prepared
By: Ashwag ALzahrani
Entitled: Risk Optimization for Hybrid Pension Plans with ARMA and
GARCH Investment Returns
and submitted in partial fulfillment of the requirements for the degree of
Master of Science (Mathematics)
complies with the regulations of the University and meets the accepted standards with
respect to originality and quality.
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Abstract
The use of time series models in general, and conditional processes, in particular
for modelling returns on investment, have been considered recently for pension plan
funding. In this project, ARMA and GARCH models are applied to the rate of return
of a hybrid pension plan. The first and second moments of the fund, contributions,
and benefits are derived under both models. The aggregate risk and the optimal
spread period of amortization are studied under different risk measures; Value at Risk,
Coefficient of Variation, and Variance. All evaluations are done over finite as well
as infinite time horizons. Finally, numerical illustrations under different investment
strategies as well as different valuation interest Rates are proposed under GARCH
model.
Keywords: Hybrid Pension Designs, Funding Methods, Risk Sharing, Yield Rate,
ARMA(1,1), GARCH(1,1), Aggregate Risk, VaR, TVaR.
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In memory of my parents
Febuary 2013
April 2014
v
“Nothing is impossible, the word itself says ‘I’m possible’ ”
Audrey Hepburn
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Acknowledgments
I would like to express my deep gratitude and appreciation to my deceased parents;
Saeed ALzahrani and Fatimah ALzahrani; who supported my dreams to complete
my Masters degree. Without their encouragement I would have not made it this
far. Idedicatemyworktothemalthoughtheyarenotheretoseethefruitoftheirlabor.
I also would like to acknowledge my supervisor Jos´e Garrido for the support and
valuable advice that he has provided over the past few years. The amount of knowledge
that he shared with me during my reaserch work were more than useful, and it will
remain a constant encouragement for years to come.
I also thank Prof. Ricardas Zitikis and Dr. Maciej Augustyniak for helping me in
this project. Thank you, Omar Alzahrani, for providing the information that was the
foundation for my work. The advice that he provided was of great value that assisted
me in completing this project. All thanks goes out to friends and family that provided
support to me through out the years while in Canada.
Thanks for the monetary support that was provided by the Goverment of Saudi
Arabia through Ministry of Education as well as the Saudi Cultural Bureau.
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Contents
List of Tables xi
List of Figures xii
1 Introduction 1
1.1 Defined Benefit (DB) Plan . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Defined Contribution (DC) Plan . . . . . . . . . . . . . . . . . . . . . 1
1.3 Hybrid Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Hybrid Pension Designs . . . . . . . . . . . . . . . . . . . . . 2
1.4 Types of Risks In Pension Plans . . . . . . . . . . . . . . . . . . . . . 4
1.5 Actuarial Funding Methods . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Two Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.1 Pension Design In Saudi Arabia . . . . . . . . . . . . . . . . . 8
1.6.2 Pension Plan In Canada . . . . . . . . . . . . . . . . . . . . . 13
2 Risk Measurement 19
2.1 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Value at Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Conditional Value at Risk (CVaR) . . . . . . . . . . . . . . . . . . . . 22
3 Model Structure and Assumptions 24
3.1 Risk Sharing in a Hybrid Pension Plan . . . . . . . . . . . . . . . . . 24
3.2 Modeling a Hybrid Plan . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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3.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Models for the Return on Investment 29
4.1 Rate of Return i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
t
4.2 Autoregressive Moving Average (ARMA) Model . . . . . . . . . . . 30
4.2.1 Marginal Moments under ARMA(1,1) . . . . . . . . . . . . . 32
4.3 Generalized Autoregressive Conditional
Heteroskedasticity (GARCH) . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1 Conditional Moments of the GARCH(1,1) Model . . . . . . 41
4.3.2 Marginal Moments of the GARCH(1,1) Model . . . . . . . . 42
4.4 Long-Memory Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Autoregressive Fractionally Integrated Moving Average Model
(ARFIMA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.2 Fractionally Integrated Generalized Autoregressive
Conditional Heteroscedastic (FIGARCH) . . . . . . . . . . . 46
5 Risk Measurement under Hybrid Pension Plan 49
5.1 Moments of the Fund, Contribution, and Benefit Levels . . . . . . . . 50
5.1.1 Moments of the Fund Level under ARMA(1,1) . . . . . . . . 51
5.1.2 Moments of the Fund Level under GARCH(1,1) . . . . . . . 55
5.1.3 Comments on the Moments of Cash Flows . . . . . . . . . . . 57
5.2 Aggregate Risk and Optimal Spread Parameter . . . . . . . . . . . . 59
5.2.1 Analyzing the Optimal Spread Parameter . . . . . . . . . . . 60
5.2.2 Aggregate Risk under Variance, Coefficient of Variation, and
Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.3 Comments on the Aggregate Risk and Spread Parameter . . . 64
5.3 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.1 Varying Spread Periods . . . . . . . . . . . . . . . . . . . . . . 66
5.3.2 Varying Risk Measures and Spread Periods . . . . . . . . . . . 70
5.3.3 Varying Investment Strategies and Spread Periods . . . . . . . 73
5.3.4 Varying Valuation Interest Rates and Spread Periods . . . . . 75
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5.3.5 Varying Valuation Time and Spread Periods . . . . . . . . . . 78
6 Conclusion 79
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Bibliography 82
A Derivations 88
A.1 The Derivation of Equation (4.2) . . . . . . . . . . . . . . . . . . . . 88
A.2 The Derivation of Equation (4.3) . . . . . . . . . . . . . . . . . . . . 88
A.3 The Derivation of Equation (4.4) . . . . . . . . . . . . . . . . . . . . 89
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List of Tables
5.1 The Conditions of Existence the Limiting Value of the First Moment
of the Fund under Different Models of i . . . . . . . . . . . . . . . . 57
t
5.2 The Conditions of Existence the Limiting Value of the Second Moment
of the Fund under Different Models of i . . . . . . . . . . . . . . . . 58
t
5.3 Difference Between F and its Stationary Status AL under Different
∞
Model of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
t
5.4 AggR under Different i Models and the Optimal Spread Parameter . 64
t
5.5 Hybrid Plan Parameters for different investment strategy . . . . . . . 66
5.6 The First Moments of the Cash Flows C , B , and F , and When
∞ ∞ ∞
X = .5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.7 The Coefficient of Variation of the Cash Flows C , B , and F , and
∞ ∞ ∞
When X = .5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.8 The Fund under Different Risk Measures . . . . . . . . . . . . . . . . 72
5.9 Hybrid Plan Parameters for Different Valuation Rate . . . . . . . . . 77
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