UNIVERSITY OF THE AEGEAN
DEPARTMENT OF FINANCIAL & MANAGEMENT ENGINEERING
THESIS
“An integrated optimization model for immunizing and matching pension funds bond portfolios”
GEORGIOS BOUZIANIS
Supervisor: Dr. P. Xidonas
Athens, June 2016
UNIVERSITY OF THE AEGEAN
DEPARTMENT OF FINANCIAL & MANAGEMENT ENGINEERING
THESIS
“An integrated optimization model for immunizing and matching pension funds bond portfolios”
GEORGIOS BOUZIANIS
Supervisor: Dr. Panos Xidonas
Approved by the 3-member committee in June 2016.
...... Dr. Panos Xidonas Dr. Vasilis Koutras Dr. Evangelos Vassiliou Adjunct Lecturer Lecturer Adjunct Lecturer
Athens, June 2016
GEORGIOS BOUZIANIS
Copyright© Georgios Bouzianis, 2016 All rights reserved.
Copying, storage and distribution of this paper for commercial purposes is forbidden. Reproduction, storage and distribution of this paper for research purposes, is allowed as long as the source is mentioned. The views and conclusions drawn in this document express only authors’ opinions and do not represent official views of the Department of Financial & Management Engineering.
ABSTRACT The primary purpose of this thesis is to develop two innovative bond portfolio optimization models, based on the portfolio dedication and immunization strategies. The first mathematical program minimizes the initial capital required for the creation of a bond portfolio, and is best suited to a defined liability-driven investment strategy. The second mathematical program operates under the uncertainty of term structure alterations, approached with Hull-White recombining trinomial lattice. In both cases, the exposure of transaction costs as well as the diversification and investment policy constraints regarding the portfolio structure, are strongly taken into account. In this sense, two mixed-integer linear programs are formulated. The validity of the proposed approach for the first model is verified through duration and convexity empirical testing. For the second model, it is verified through scenario evaluation in a well- diversified investment universe of bonds, including: US corporate bonds, European corporate bonds and sovereign bonds. Keywords: Bond portfolio optimization; Stochastic dynamic integrative asset and liability management strategy; Matching, immunization mathematical programming
PROLOGUE
I would like to express my profound gratitude to my supervisor, Dr. Panos Xidonas, for the assignment of this particular thesis and huge inspiration he provided me with. His excellent guidance, teaching ability and overall assistance were critical success factors for the completion of this work. I would also like to thank Professor Dimosthenis Drivaliaris, since he was the one who gave robust and mature shape in the mathematical skills and thinking potential of mine. His contribution to my progress was very important. My interaction with these two men changed the way I approach both science and life. I would also like to express my sincere thanks to the members of my thesis committee, Dr. Vasilis Koutras and Dr. Evangelos Vassiliou, for their useful comments and their overall contribution. Moreover, I would also like to thank all the people who contributed in their own unique way in supporting me during the time of my thesis elaboration. Finally, I dedicate this endeavor to my father Stavros, whose presence and input have been invaluable.
Georgios Bouzianis Athens, 2016
TABLE OF CONTENTS
CHAPTER 1 - INTRODUCTION ...... 11 1.1 The problem ...... 11 1.2 The Subject and the Aim of the Thesis ...... 19 1.3 The Contribution of the Thesis ...... 20 1.4 The Structure of the Thesis ...... 21
CHAPTER 2 - PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE ...... 24 2.1 Objective function ...... 25 2.2 Cashflow matching constraint ...... 26 2.3 Immunization constraints ...... 27 2.4 Investment policy adjustment ...... 32
CHAPTER 3 - STOCHASTIC SHORT RATE MODELS ...... 36 3.1 Vasicek Model ...... 37 3.2 Hull-White Model ...... 39 3.3 Dothan Model ...... 43 3.4 Black-Derman-Toy Model ...... 44 3.5 Cox Ingersoll Ross Model ...... 46
CHAPTER 4 - PROPOSED METHODOLOGY WITH EMBEDDED HULL - WHITE STOCHASTIC SHORT RATE MODEL ...... 50 4.1 Hull-White scenario structures ...... 51 4.2 General Methodological approach...... 55
CHAPTER 5 - EMPIRICAL TESTING ...... 59 5.1 Data and field of application...... 59 5.2 Results and discussion for the proposed methodology for small and parallel shifts in the term structure ...... 60 5.3 Results and discussion for the proposed methodology with embedded Hull-White stochastic short rate model ...... 65
CHAPTER 6 - CONCLUSIONS - PERSPECTIVES ...... 72
6.1 Conclusions ...... 72 6.2 Perspectives ...... 73
APPENDIXES A Programming code for the proposed methodology for small and parallel shifts in the term structure ...... 76 B Programming code for the proposed methodology with embedded Hull-White stochastic short rate model ...... 82
REFERENCES ...... 91
LIST OF FIGURES
CHAPTER 1 Figure 1.1.1 Depiction of a normal yield curve...... 15 Figure 1.1.2 Depiction of a flat yield curve...... 15 Figure 1.1.3 Depiction of an inverted yield curve...... 16 Figure 1.1.4 The u.s. bond market size is (in billions) according to the securities industry and financial markets association (SIFMA) as of Q4 2013 ...... 17 Figure 1.1.5 Outstanding u.s. bond market debt (in billions) from 1980 to 2014, according to sifma...... 17 Figure 1.1.6 Components of an alm system and basic procedure…………………………………………………………………………19
CHAPTER 2
Figure 2.2.1 A liability (lt) and the total inflows from mismatched portfolios (A and B), and from a perfectly matched portfolio (C)..27 Figure 2.3.1 The price-yield curve with approximations of duration and convexity...... 32
CHAPTER 4 Figure 4.1.1 Recombining trinomial Hull-White lattice, with five trading dates...... 51 Figure 4.1.2 A scenario structure derived from the trinomial lattice in figure 4.1.1...... 53 Figure 4.1.3 A scenario structure with common states up to t=3, derived from the trinomial lattice in figure 4.1.1...... 54
CHAPTER 5
Figure 5.2.1 Investment to the i-th bond of the portfolio and expected earnings...... 64 Figure 5.2.2 Bond portfolio incomes, necessary overflows and liabilities...... 64
Figure 5.3.1 Shifts in the term structure in the form of trinomial lattice, with Hull-White model...... 69
LIST OF TABLES
Table 5.2.1 Individual characteristics of the 70 bonds in the optimized portfolio ...... 61
Table 5.2.2 Monetary synthesis of the portfolio and evaluation of the earnings (in USD) from every incorporated bond...... 61
Table 5.2.3 Required overflows and overall inflows at any period t and corresponding liabilities (in USD)...... 62
Table 5.3.1 Individual characteristics of the 170 bonds in the optimized portfolio...... 66
Table 5.3.2 Monetary synthesis of the portfolio and evaluation of the earnings (in USD) from every incorporated bond...... 66
Table 5.3.3 Required overflows and overall inflows at any period t and state stv(t), and corresponding liabilities (in USD)...... 67
Table 5.3.4 Probabilities of the transitions in trinomial Hull-White lattice ...... 69
CHAPTER 1: `INTRODUCTION
CHAPTER 1. INTRODUCTION
1.1 The problem Financial intermediaries, insurance firms, pension funds and non- financial institutions all need to adopt a mechanism in order to cope efficiently with liability streams that stretch well in the future. The liabilities of a defined benefits pension fund may be considered as deterministic parameters, given the sufficient accuracy provided by average mortality rates (Zenios, 2005). For non-financial institutions planning acquisitions, expansions, or product development, a mechanism is required matching future cashflows known with certainty with the liability stream (Fabozzi, 2005; Zenios, 2005). Therefore it is important to develop an integrative asset and liability management strategy (ALM) so as to tackle this need effectively. In addition to the necessity for cashflow matching, large banks must protect their current net worth, whereas pension funds have the obligation of payments after a number of years. Naturally both these institutions are interested in protecting the future value of their portfolios and in this sense uncertain future interest rates are a risk factor (Macaulay, 1938). By using an immunization technique, large institutions can protect (immunize) their firm from exposure to interest rate fluctuations (Granito, 1984; Redington, 1952). A perfect immunization strategy establishes a virtually zero-risk profile in which interest rate movements have no impact on the value of a firm. The scientific field tackling with practical problems of day-to-day business is ―financial engineering systems‖. The financial engineering systems incorporate the required procedures for adapting existing instruments and techniques and creating new ones in order to provide the best results for the investors. In order to ensure the best of these results the application of mathematical modeling methods, statistical methods and computational techniques in finance are prerequisites. Investment strategies such as immunization and cashflow matching reside to asset and liability management strategies (ALM).
B ONDS A bond is a debt investment in which an investor lends money to an entity (typically corporate or governmental) for a defined period of time at a variable or fixed interest rate. Bonds are used by companies,
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CHAPTER 1: `INTRODUCTION municipalities, states and sovereign governments to raise money and finance a variety of projects and activities. Owners of bonds are officially called debt holders, or creditors of the issuer. Bonds are found in the product category of fixed-income securities and constitute one of the three main generic asset classes, along with stocks (equities) and cash equivalents. Many corporate and government bonds are publicly traded on exchanges, while others are traded only over the counter (OTC). In general, fixed-income securities are classified by their duration before maturity. These basic categories are: bills (debt securities maturing in less than one year), notes (debt securities maturing in one to ten years) and bonds (debt securities maturing in more than ten years). The basic types of bonds are:
Callable (Redeemable) bonds The bond issuer has the right but not the obligation to redeem his issue of bonds before its maturity. Issuers have to pay bond holders premium. There are two types of this bond: 1. American Callable bonds can be called by the issuer any time after the call protection. 2. European Callable bonds can be called by the issuer only on pre-specified dates. Convertible bonds Bondholder has the right but not the obligation, to change his bonds into different classes of securities. However, this privilege can be applied only in the case where the bondholder owns corporate bonds. Puttable bonds Bondholder has the right but not the obligation, to sell his bonds back to the issuer. The price and date of the bond sale are predefined. Investors can ask for the repayment of the bond. Municipal bonds Municipal bonds are issued by various cities having some low interest rates. They are tax free. Government bonds Government bonds are issued by governments in order to finance government spending. These bonds, in their majority are issued in the country's domestic currency. Even though this class of bonds is not as
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CHAPTER 1: `INTRODUCTION risky as the others, investors in government bonds must be aware of the risks associated to the country issuing them. The most common sources of risk in government bonds are: country risk, political risk, inflation risk, and interest rate risk. Investment-grade corporate bonds Corporate bonds are issued from several corporations in order to typically finance future operations. The backing for the bond usually depicts the payment ability of the company. Corporate bonds are considered higher risk than government bonds. Consequently this class of bonds has always higher interest rates than the rest classes, even for top-flight credit- quality companies. Foreign bonds A foreign bond is most often issued by a foreign firm to raise capital in a domestic market that would be most interested in purchasing the firm's debt. Bonds of this class are issued by foreign entity, in the domestic market's currency. Zero-coupon bonds Zero-coupon bonds do not pay interest (a coupon) but are traded at a deep discount, rendering profit at maturity when the bond is redeemed for its full face value.
Bonds are bought and traded mostly by institutions like central banks, sovereign wealth funds, pension funds, insurance companies, hedge funds, banks and investors. The main reason that insurance companies and pension funds invest on bond portfolios is because they have liabilities which essentially include fixed amounts payable on predetermined dates. Consequently, they buy the bonds to hedge this predetermined liability stream, and are sometimes even compelled by law to do so. Most individuals wanting to own bonds resort to bond funds. In the U.S., nearly 10% of all bonds outstanding are held directly by households. In general, bonds have less volatility (especially short and medium dated bonds) than that of equities (stocks). Investment in a bond portfolio is usually a safer choice compared to stocks (Puhle, 2008). However, bonds do suffer from less day-to-day volatility than stocks, and bonds' interest payments are sometimes higher than the general level of dividend payments. In a lot of cases, bonds tend to be liquid, providing the privilege
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CHAPTER 1: `INTRODUCTION to an institution to sell a large quantity of bonds without affecting the price much, which may be more difficult for equities. A great merit of bonds is that they provide a measure of legal protection: Assuming that a corporation goes bankrupt, under the law of many countries, it is the bondholders who are entitled to a partial refund, known as the recovery amount, while in this case the company's equity stock often ends up valueless. Fixed rate bonds are exposed to interest rate risk. There is a general rule that explains how bonds are affected by interest rates alterations: when the generally prevailing interest rates rise the market prices will decrease in value. Since the payments are fixed, a decrease in the market price of the bond corresponds to an increase in its yield. When the market price of bonds falls it reflects investors' ability to get a higher interest rate on their money elsewhere — perhaps by purchasing a newly issued bond that already features the newly higher interest rate. This does not affect the interest payments to the bondholder, so long-term investors who want a specific amount at the maturity date do not need to worry about price swings in their bonds and do not suffer from interest rate risk. Bonds are also subject to various other risks such as call and prepayment, credit, reinvestment, liquidity, event, exchange rate, volatility, inflation, sovereign and yield curve risk. Again, some of these will only affect certain classes of investors. In this study we focus on yield curve risk.
Y I E L D C U R V E A N D M A T U R I T Y D ATE The yield curve also known as ―term structure‖ is, by definition, the relationship between interest rates and time (Hull, 1989). Therefore, the plot of the yield curve is a two dimensional figure, determined by plotting the yields of all coupon bonds against their maturities. Since bonds with longer maturities combine higher yields, then the yield curve typically slopes upwards, even though there are special cases where it can be flat or even inverted. Normal Yield Curve As aforementioned, a normal yield curve is the natural relation between interest rates and maturities. The following graph illustrates the structure of interest rates over a period of time, which indicates that rates rise as maturities lengthen, i.e., short-term rates are lower than long-term rates. It is also known as ―positive and upward-sloping‖ yield.
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CHAPTER 1: `INTRODUCTION
Yield
Maturity
Figure 1.1.1: Depiction of a Normal Yield Curve.
Flat Yield Curve The flat yield curve is the special case where only a slight difference on short term and long term rates exists between bonds of the same credit quality. The flat yield curve illustrates the same yield for short maturity and long maturity bonds.
Yield
Maturity
Figure 1.1.2: Depiction of a Flat Yield Curve.
Inverted Yield Curve The inverted yield curve indicates the rare situation where the long term interest rates are lower than the short term interest rates. In that case the yield curve is negative because interest rates decrease as maturities lengthen. Subsequently, a negative yield curve is a two-dimensional
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CHAPTER 1: `INTRODUCTION graph, depicting the structure of interest rates over time, which indicates that as maturities lengthen interest rates fall, short-term rates exceed long-term ones.
Yield
Maturity
Figure 1.1.3: Depiction of an Inverted Yield Curve.
Assuming that the value of the bonds in their trading bond portfolio falls, this will affect negatively professional investors such as banks, insurance companies, pension funds and asset managers, holding such bond portfolios, since the value of the portfolio will also fall. In such a case the holders of individual decreasing value bonds, will consider to "cash out". Interest rate risk could become a real problem (conversely, bonds market prices would increase if the prevailing interest rate were to drop, as it did from 2001 through 2003). The bond market is the financial market where participants can issue new debt (governments, corporations), known as the primary market, or buy and sell debt securities, known as the secondary market (Fabozzi, 1996). This is usually in the form of bonds, but it may include notes, bills, and so on. The bond market has largely been dominated by the United States, which accounts for about 44% of the market. As of 2009, the size of the worldwide bond market (total debt outstanding) is an estimated at $82.2 trillion, of which the size of the outstanding U.S. bond market debt was $31.2 trillion according to Bank for International Settlements (BIS), or alternatively $35.2 trillion as of Q2 2011 according to Securities Industry and Financial Markets Association (SIFMA).
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CHAPTER 1: `INTRODUCTION
$45,000 $40,000 $35,000 $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 $0
Figure 1.1.4: The U.S. bond market size is (in billions) according to the Securities Industry and Financial Markets Association (SIFMA) as of Q4 2013.
$45,000 $40,000 $35,000 $30,000 $25,000 $20,000 $15,000 $10,000 $5,000 $0
Figure 1.1.5: Outstanding U.S. Bond Market Debt (in billions) from 1980 to 2014, according to SIFMA.
In general, with the exception of U.S. Bond Market, the global bond market trend is particularly grown and offers investors and financial institutions substantial bargains, capable to develop effective asset and liability management strategies (ALM). The management of the firm’s balance sheet constitutes the core of enterprise-wide risk management (EWRM) for financial institutions.
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CHAPTER 1: `INTRODUCTION
The balance sheet reflects the risks of the environment from the asset side and most of the business risks from the liability side. The alignment of these risks is the goal of an ALM system. ALM starts from the seminal contributions of Harry M. Markowitz in the 1950s for asset allocation, and the subsequent extensions to include liabilities. In order for ALM to be effective in accomplishing the stated goals of EWRM additional information is required. ALM on the asset side focuses on market, credit, and liquidity risk, while on the liability side on volatilities of margins and costs. The major managerial activities in ALM are asset and equity allocations. The prices and the structure of the incorporated products are considered as inputs to the asset allocation phase and in determining hedging positions. In general ALM systems include several activities. The most important of those is to efficiently perform simulations with regard to the estimated earnings as well as balance sheet simulation. Additional tasks of ALM systems are sensitivity analysis, current and future valuation and dynamic balance sheet modeling. However, the implementation of these tasks requires information of specific components. The basic components of an ALM system whose quality defines the performance of ALM activities are: Data storage, Analysis tools, and reporting facilities (Zenios, 2005). Data storage provides data on the contractual obligations, market information such as prices and time series of financial data. Analysis tools are used in risk measurement and management. In this thesis, analysis tools are used more in the direction of valuation functionality in order to provide efficient simulations that can be extended well in the future so as for the institution’s balance sheet to be simulated dynamically across time. Dynamic analysis can either be passive, focusing on the time-evolution of the current balance sheet or active focusing on the analysis of explicit institutional ALM strategies.
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CHAPTER 1: `INTRODUCTION
Analysis tools Data storage Risk measurement Risk management Results Predetermined Reporting payouts Market valuation Active dynamic analysis Marketing Scenario analysis GAP assumptions Hedging Earnings-at risk VAR Risk factor Portfolio optimization scenarios Value-at-risk Earnings
New business RAROC
Figure 1.1.6: Components of an ALM system and basic procedure.
1.2 The Subject and the Aim of the Thesis The subject of this thesis is to develop a mathematical technique that focuses on the construction of a minimum- cost bond portfolio immunized against possible yield fluctuations and dedicated to matching the predetermined liability stream. Namely, a Mixed-Integer Linear Program (MILP) is solved, which minimizes the initial segregated required capital for accomplishing a matching-immunization strategy over a number of different type constraints, concerning also special investment policy statements. The corresponding programming code that allows the automated process of solving the above problem was formed, thus rendering it reachable to every financial and non-financial institution. In order to locate the optimal portfolio meeting the conditions of matching and immunization as mentioned above, the use of methods of applied mathematics and programming techniques is of the essence. In an effort to make this program more appealing, binary decision variables were added, as well as some special constraints regarding the bond ratings and maturities providing professional applied links to the pension fund and asset management industry. Also, a scenario-based optimization framework for solving the cashflow matching, immunization problem is proposed, where the liabilities’ time horizon coincides to the maturities of the available bonds. It faces the
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CHAPTER 1: `INTRODUCTION uncertainty of interest rates by applying standard short rate models for scenario generation within this framework. The optimal portfolio is attained by minimizing the initial cost required for this ALM strategy. This problem is also treated as Mixed-Integer Linear Program (MILP). The basic concept of this model is to optimize lending decisions dynamically under scenario uncertainty as new information arrives. Modeling the arrival of new information is possible with the use of scenario structures. Large-scale optimization models can then be formulated to optimize decisions on an event tree. These models were originally developed in the 1950s under the term stochastic programming and they are still widely used (Zenios, 2005).
1.3 The Contribution of the Thesis The benefit of investing in a matched-immunized portfolio of bonds for counterbalancing liabilities can be annuled given that the additional return of the incorporated portfolio is able to significantly reduce the amount of cash required for hedging the liability stream. Even though several models have been proposed approaching immunization problem, there has not yet been an official model to combine these two strategies. Two different mathematical models are hereby submitted, for matching and immunization strategies under the uncertainty of possible modifications in the term structure. The first model is a multiperiod static model: it will explicitly model rebalancing decisions at future time periods, under the assumption that the state of the world will shift with modest and limited changes during this period. However, since this setting is an unrealistic one, a new multiperiod stochastic model is presented. It enables dynamical evolution of both asset and liabilities through time, following a probability distribution. This model captures the stochastic nature of the problem. In the implementation of the multiperiod stochastic model hereby introduced, the short rates are not given, but are estimated with standard interest rate models for scenario generation. Hence, for the stochastic valuation of the assets and liabilities, the most suitable short rate model is embedded in the formulation. The validity of the proposed models is verified via empirical testing in a well-diversified investment universe of bonds including: US corporate bonds, European corporate bonds, Sovereign bonds, Zero-coupon bonds
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CHAPTER 1: `INTRODUCTION and Government bonds. This thesis embodies the simulation of a complete and well-defined capital management process with the use of real data so as to come up with the optimal bond portfolio complying to the investment policy of the decision maker and the ALM strategies of cashflow matching and yield immunization.
1.4 The Structure of the Thesis
CHAPTER 1 It is an introductory chapter revealing the main characteristics of the two problems which are presented and analyzed in detail in later chapters. Relevant conceptual information on the key components of these problems is included. The target, the contribution of this thesis and its structure are determined.
CHAPTER 2 The first integrative optimization model is put forward, which combines the ALM strategies of yield immunization and cashflow matching under the uncertainty of parallel and symmetric shifts in the term structure. It is explained how alterations in the term structure are possible to affect the price of assets and liabilities and how this problem can be managed. This model is a multiperiod static model and its functionality depends on how the yield curve will be changed across time. For bigger and non-parallel shifts in the term structure it is obligatory to adopt another mechanism for the valuation of the bond portfolio and the liabilities.
CHAPTER 3 In this chapter one can find the analysis of the case where the fluctuations in the term structure are not small and parallel. One of the oldest approaches is based on modeling the estimation of the instantaneous short interest rate. This is still quite popular for pricing interest rate derivatives and for risk management purposes, and represents the most commonly used type of dynamical stochastic model for interest rates. Therefore the aim of this chapter is to give information about these models and choose the most suitable with the integrated matching-immunization model of this thesis.
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CHAPTER 1: `INTRODUCTION
CHAPTER 4 The basic concepts of stochastic programming and dynamic models are distinguished. The proposed methodology of the second model is presented in detail and analyzed. It is about a stochastic multiperiod model that incorporates the stochastic evolution of the short interest rate. With this model the ALM strategy of matching-immunization is accomplished even in the case of larger and non-symmetric fluctuations in the term structure.
CHAPTER 5 In the fifth chapter of this thesis the suggested methodology for both models of chapter 2 and chapter 4, are implemented and applied on real data of bonds incl. US corporate bonds, European corporate bonds, Sovereign bonds, Zero-coupon bonds and Government bonds. In the beginning one can find the analysis of the specific features of the scope and then the application of the two models step by step. The results obtained by the application of the two models and their explanation are also presented in this chapter.
CHAPTER 6 It is the last chapter of the thesis where a brief description of the results and conclusions that arose from the proposed methodology, is presented, highlighting the usefulness of the suggested models.
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CHAPTER 2: PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
CHAPTER 2. PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
In this chapter we develop an optimization model for fixed income portfolios when the source of risk is the alterations of interest rates. The formulation of the model enables structuring portfolios that are dedicated to matching a predetermined liability stream while simultaneously have immunized returns against possible changes in the term structure (Fabozzi, 2007; Zenios, 2005). Financial institutions and non-financial institutions are in need to adopt a mechanism that matches fixed cashflows with the liability stream that stretch well in the future. In such cases the cashflows are uncertain and best represented with random variables. However, in the model promoted in this chapter the cashflows are the inflows from the coupons of the incorporated fixed rate bonds known in certitude. The liabilities of a defined benefits pension fund may be considered deterministic. However it is still argued on whether uncertainty plays a significant role in all of these cases. Nevertheless, this model deals only with the part of the problem known with certainty so it can be used extensively in practice. Even managers with perfect foresight on their liabilities have difficulties in developing an ALM strategy. Insurance companies search for a methodology to fund a guaranteed investment contract. In the upward slopping yield curve environment of the time insurers placed the proceeds from GIC sales into 10 to 30 years mortgages or public bond instruments. Industry capitalized on the large spread between the high rate on their long term assets and the credit rates on the shorter-term GIC contracts. Due to significant increase in rates in the early 1980s, insurance firms saw their liabilities maturing while their assets having 20 years remaining to maturity valued at a fraction of their original cost. Consequently, the concept of this model is the construction of a bond portfolio of minimum cost where the cashflows from the assets in the portfolio will match the liability at every period.
Paper submitted and is currently under review: Xidonas, P., Bouzianis, G., Hassapis, C., 2016. An integrated matching-immunization model for bond portfolio optimization. Computational Economics (Springer).
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CHAPTER 2: PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
2.1 Objective function The proposed model is structured based on total cashflow matching, the surplus reinvestment, and portfolio immunization. Total cashflow matching strategy can be accomplished with cashflow timings that coincide with the timing of the liability. It is therefore preferable to either purchase bonds maturing before the liability, or borrow cash. Immunization strategy can be achieved by matching the interest rate risk of an asset portfolio against the predetermined liability stream so as to ensure zero net market exposure. The initial basic idea of the proposed bond portfolio optimization model is to minimize the total required capital for satisfying the whole stream of liabilities in the future. Assuming that the synthesis of the portfolio is expressed in monetary terms, the underlying objective function of the problem is defined as follows:
nn b min f0 ( xi 0 p i 0 ) t i 0 ii11 nn bb (2.1.1) f0 ()() xi 0 p i 0 x i 0 p i 0 c i 0 ii11 n b f0 ( xi 0 p i 0 )(1 c i 0 ) i1
The continuous variables xbit represent the quantity to be bought from the i-th bond at period t, t=1,…,h. Also, pit denotes the price of the i-th bond at period t. The mathematical term xbi0pi0 denotes the initial monetary investment in the i-th bond. The summation of this term for all the incorporated bonds represents the total initial monetary investment in the bond portfolio composed of n different bonds.
The continuous variables ft, represent the necessary cash surplus at the end of every period t, while variable f0 represents the initial capital is used simultaneously with the purchase of the bonds.
The continuous variable tit represents the transaction costs from the investment in the i-th bond at period t of the h-year time investment horizon. The summation of this term for all the incorporated bonds at period 0 denotes the total transaction costs that the pension fund will be exposed. The general expression for total transaction costs at period 0 is:
b ti0=, x i 0 p i 0 c i 0 in [1,..., ] (2.1.2)
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CHAPTER 2: PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
where ci0 is the commission percentage for the i-th bond at period 0.
2.2 Cashflow matching constraint Total cashflow matching can be achieved by investing only in securities that their cashflow timings coincide with the payouts of the liabilities. In general, liability payouts may be due at any day of the month, while bond coupon payments and maturities are annual or semi-annual. There are cases where there are no securities maturing on the exact date to the liability. It is preferable to invest in a bond portfolio that matures before the liabilities and reinvest the principal received in short-term money markets. An ALM strategy that allows some shortfall or surplus around the liability payment day is known as symmetric cash matching. The cashflow matching constraint that describes the mathematical relation between the inflows, the outflows and the cash surplus at the end of every period t is:
n xi0 c it f t 1(1 r f ( t 1) ) f t = l t , i=1 th [1,..., ] (2.2.1)
The continuous variables xi0cit denote the inflows at period t from the investment in the i-th bond. The summation of this term for all the incorporated bonds represents the total inflows at period t from the investment in the bond portfolio. The variable rf(t) denotes the reinvestment rate at period t that cash surplus will be reinvested and lt denotes the corresponding liability at period t. The following figure depicts two different cases of cashflow matched portfolios. Portfolios A and B indicate the cases of mismatched bond portfolios where the earnings from the investment in the bonds and the surplus from the reinvestment do not coincide with the timing of the single payout. Portfolio C indicates the case of a perfectly matched bond portfolio with the liability timing. The model described in this chapter is exclusively dedicated in perfectly matched bond portfolios.
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CHAPTER 2: PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
Mismatched portfolio A Mismatched portfolio B
Σxi0Cit+1 + ft(1+rf(t)) Σxi0Cit-1 + ft-2(1+rf(t-2)) Σxi0Cit+1 + ft(1+rf(t)) Inflows Inflows
Time Time
t+1 t-1 t+1 Outflows Outflows
lt+ft lt+ft
Matched portfolio C
Σxi0Cit+1 + ft-1(1+rf(t-1))
Inflows Time Outflows
lt+ft Figure 2.2.1: A liability (lt) and the total inflows from mismatched portfolios (A and B), and from a perfectly matched portfolio (C).
The foundational framework for modern fixed income portfolio theory lays on portfolio dedication theory (Zenios, 2005; Fabozzi 2005) . In order to achieve total cashflow matching it is obligatory that the examined asset universe includes bonds with cashflow timings that coincide to the timing of liabilities. Ultimately, it is an asset and liability management strategy that includes cash surplus, reinvested in short-term deposits.
2.3 Immunization constraints The property of portfolio immunization encloses three different constraints. The first one is the necessary condition for immunization. This condition requires that the present values of the portfolio cash inflows and of the liability outflow are equal. In order to understand this condition we start from a simple cashflow matching model. The cash surplus created at period 1 is given by:
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CHAPTER 2: PROPOSED METHODOLOGY FOR SMALL AND PARALLEL SHIFTS IN THE TERM STRUCTURE
n b (2.3.1) xii0 c 1 l 1= f 1 i=1
Moving from period 1 to 2 we get the second period constraint as follows:
nn bb (2.3.2) xi02 c i( x i 011 c i l )(1 r f (0,1) ) l 2 ii=1 =1
Repeating this process for each period until the end of the horizon h at reinvestment rates (1+rf(t,t+1)) we obtain the equation below:
n hh11 [xb c (1 r ) x b c (1 r ) ... x b c ] ii01 ftt (,1) ii 02 ftt (,1) iih 0 i=1 tt12 (2.3.3) hh11 l(1 r ) l (1 r ) ... l 1 f ( t , t 1) 2 f ( t , t 1) t tt12
Dividing both sides by the constant product of (1+rf(t,t+1)) for t=0,…,h-1 we obtain the relation that the present value of assets should be equal to the present value of the liabilities as follows:
n b xi00 p i pv l , i1 (2.3.4)
pvpl pv
The equation (2.3.4) arises from the classic cashflow matching model, but can be used properly without loss of generality as a constraint under the objective function (2.1.1), defined in section 2.1. However, it is not safe to say that (2.3.4) will endure when the rates change. In order to ensure that this condition will be satisfied, both sides of the equation must change evenly when the term structure changes. Duration is used so as to measure a first order estimate of the change of the present value of a stream of cashflows or a liability when term structure changes. The second constraint for the property of portfolio immunization is a first order sufficient condition that the duration of assets and liabilities match.
h Assume that the term structure rytm is subjected to a parallel shift tt1 h by a small amount Δrytm to rytm rytm so that: t t1 drytmt = drytm = Δrytm for each t=1,2,…,h. The price of a bond i under the given term structure is pi0, and the price sensitivity to the parallel shifts is given by the Fisher-Weil duration (Fisher and Weil, 1971) as:
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1 hi c t it , Di t1 piot1 (1 rytm it ) (2.3.5) in [1,..., ]
Duration is used to measure how sensitive a bond or a bond portfolio's price is to changes in interest rates. It expresses the weighted average time required for the complete payback of the capital and the interests on it. Note that, in the case of flat term structure, the Fisher –Weil duration reduces to modified duration (Weil, 1973).
The duration of a portfolio composed of n bonds with durations D1, D2,…,
Dn, is given by the following formula:
n (2.3.6) Dp w i D i i1 where wi= xbi0pi0/pvp is the percentage of the participation of the i-th bond in the portfolio with present value pvp. In order to prove the first order sufficient condition we start from the equation (2.3.4). Assume that S(rf(t)) is the difference between the present value of the bond portfolio and the present value of the liabilities. Then from equation (2.3.4) we conclude that:
n b (2.3.7) S() rf( t ) x i 0 p i 0 pv l i1
The investment in the bond portfolio will be sufficient to pay off the liability stream only if Sr(ft() ) 0 for every rrf()() t f t . This is possible only if the function Sr()ft() has (local) minimum at rf(t), i.e.
S( rf()() t ) S ( r f t ) 0 . Initially it is obligatory that: dS/0 drft() . The first derivative of the function Sr()ft() is:
ds n txb c tl i0 it t tt11 drf()()() ti1 t h(1 r f t ) t h (1 r f t ) tln txb c t i0 it (2.3.8) tt11 t h(1rrf()() t ) i 1 t h (1 f t )
Dl pv l D p pv p
From the equationdS/0 drft() , we have: Georgios Bouzianis [Page 29]
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Dl pv l D p pv p 0 (2.3.9) Dl pv l D p pv p
Using equation (2.3.4) we get the second constraint for the property of portfolio immunization:
DD (2.3.10) lp The duration of the bond portfolio and the duration of liabilities match. Nevertheless, duration matching constraint is used only for small and parallel shifts. In the case of larger changes, asset and liabilities will not be equal if the assets and liabilities have different price-yield relationships (Hull, 1993). It is necessary to add a second order condition so as to ensure that the asset price-yield curve bounds from above the liability price-yield curve. The third constraint for the property of portfolio immunization is a second order sufficient condition that the convexity of assets is bigger from the convexity of liabilities.
T Assume that the term structure rytm is subjected to a parallel shift by tt1 T a small amount Δrytm to rytm rytm so that: t t1 drytmt = drytm = Δrytm for all t=1,2,…,T. The price of a bond i under the given term structure is pi0, and the convexity of the bond is given by: 1 hi c (tt2 )it , Vi t2 piot1 (1 rytm i ) (2.3.11) in [1,..., ]
Convexity is a measure of the curvature in the relationship between bond prices and bond yields as the duration of a bond changes due to term structure alterations. Convexity is used as a risk-management tool, and helps to measure and manage the amount of market risk to which a portfolio of bonds is exposed.
The convexity of a portfolio composed of n bonds with convexities V1, V2,…,
Vn, is given by the following formula:
n (2.3.12) Vp wV i i i1
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For the second order sufficient condition that the convexity of assets is 22 bigger from the convexity of liabilities we have that d S/0 drft() , where the second derivative of the function Sr()ft() is:
ds2 n ()()t22 t xb c t t l i0 it t 2tt 2 2 drf()()() ti1 t h(1 r f t ) t h (1 r f t ) (2.3.13)
=Vp pv p V l pv l
22 From the inequalityd S/0 drft() , it follows that:
Vp pv p V l pv l 0 (2.3.14)
Vp pv p V l pv l
Using equation (2.3.4) we get the third constraint for the property of portfolio immunization:
VVpl (2.3.15)
The second-order Taylor series approximation of the price curve changes for small and parallel shifts in the term structure is written in terms of duration and convexity as follows: Vp p D p r ii() r 2 (2.3.16) i i i 2 The measures analytically explained in this section have a clear economic interpretation. They measure the sensitivity of a bond’s price to changes of the term structure only for small and parallel shifts. Hence, they measure only fixed-income market price risk but not shape risk. Duration in its various forms is the slope of the price-yield or price- interest rate curve at a given point. This slope provides a linear approximation to the curve. The most accurate representation can be obtained through the second-order Taylor series approximation to this curve.
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Bond Price
Yield
Figure 2.3.1: The price-yield curve with approximations of duration and convexity.
The total constraints of the model are h+3, where the first h constraints (2.2.1), indicate the property of cash flow matching and the additional three (2.3.4, 2.3.10 and 2.3.15) indicate the necessary condition for immunization, the first and the second order sufficient conditions respectively. It is very important to develop an integrative asset and liability management strategy which incorporates immunization properties for financial intermediaries, insurance firms, pension funds and non-financial institutions. However, immunization is a dynamic process. Once established, the portfolio must be periodically rebalanced in order to ensure that the duration is equal to the remainder of the horizon due to duration drift.
2.4 Investment policy adjustment
In order to give the portfolio selection program, a more realistic modeling of the decision situation, binary decision variables are inserted, bi=1 or bi=0 (Lenstra, 1983). Some lot sizes may be considered too small to be included in the portfolio. In this sense, we add constraints of the following form:
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b x xb b x , i li i0 i ui (2.4.1) in(1,..., ) where xli is the smallest allowable lot size and xui is the largest allowable lot size of the i-th bond, which may be a large number. It is obvious that for bi=1 we have xli ≤ xbi0 ≤ xui, and for bi=0 the constraint forces xbi0=0. Furthermore, a constraint allowing the direct determination of the number of incorporated bonds in the portfolio is added, thus addressing the diversification issue. The number of bonds in the portfolio must vary between Sl and Su as follows:
n (2.4.2) Sl b i S u i=1
Additionally a constraint with regard to the transaction costs for every incorporated bond is added, as follows: tt , i0 ui (2.4.3) in (1,..., )
where tui is the highest allowable transaction cost for the i-th bond.
Portfolios that consist a large number of small positions are not recommended for two main reasons. Firstly, an investment in many different positions implies larger transaction costs when an existing portfolio is revised. Secondly, operational costs are also higher due to monitoring requirements when managing large portfolios. With constraints (2.4.2) and (2.4.3), the program restricts the total number of assets that are included in the portfolio, while in the same time the transactions cost exposure is restricted within some maximum threshold. Additionally, constraints (2.4.1) and (2.4.3) can be adjusted with regard to the IPS. For example it may be significant to give more weight in specific sectors. Therefore the upper bound of (2.4.1) will be higher for the bonds that belong in these sectors. An additional investment constraint is proposed. The constraint pertains to the case that the treasury invests the budget in a portfolio of specific maturity. Consequently, the relation of the monetary composition of the optimized portfolio with the portfolio of specified maturity depends on policy. The constraint is defined as:
n b (2.4.4) Tu() x i00 p i T p i=1 Georgios Bouzianis [Page 33]
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Where Tp is the monetary placement in a portfolio composed of bonds with maturity more than Tx (Tx ≤ hi, i=1,…,n). Tu is the percentage of the budget that will be invested in bonds with specific maturity. Alternatively, the constraint (2.4.4) with regard to the policy can be expressed by setting a fixed maturity. This strategy will produce a new sample, comprised of bonds that comfort with the characteristics of risk and maturity that the DM concerns. The formulation of the model is mathematically structured as a Mixed Linear Integer Problem (MILP). The related hardware and software provides the solution to difficult mixed-integer programming problems for large asset universe in a few minutes. A complete investment policy statement regarding the bond portfolio’s structure, the level of diversification, the amount of transaction costs and the lot sizes, is precisely formulated.
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CHAPTER 3: STOCHASTIC SHORT RATE MODELS
CHAPTER 3. STOCHASTIC SHORT RATE MODELS In recent decades, we have witnessed a proliferation of new interest rate dependent securities, like bond futures, options on bonds, swaps, bonds with option features, etc., whose payoffs are strongly dependent on the interest rates (Clewlow and Strickland, 1998). It is noticeable that interest rates are used for discounting as well as for defining the payoff of the derivative. The values of these interest rate derivative products depend sensibly on the level of interest rates (Babbel, 1996; Fabozzi, 1998). In the construction of valuation models for these securities, it is crucial to incorporate the stochastic movement of interest rates (Brigo and Mercurio, 2006; Cairns 2004). Several approaches for pricing interest rate bonds have been proposed. The correct modeling of the stochastic behaviors of interest rates or more specifically, the term structure of the interest rate through time, is important for the construction of realistic and reliable valuation models for interest rate derivatives. The extension of the Black-Scholes valuation framework to bond options and other bond derivatives is doomed to be difficult because of the pull-to-par phenomenon, where the bond price converges to par at maturity, thus causing the instantaneous rate of return on the bond to be distributed with a diminishing variance through time. Since the information of the current term structure is available, an interest rate model should take the data on initial term structure as an input. Arbitrage exists if the theoretical bond prices solved from the model do not agree with the observed bond prices (Björk, 1998). In this chapter, we discuss no-arbitrage (arbitrage free) short rate models that contain time dependent parameter functions and the functions are determined in such a way that the current bond prices obtained from the model coincide with the observed market prices. The initial term structure may be prescribed as term structure of bond prices or forward rates. The interest rate market is where the price of rising capital is set. Bonds are traded securities and their prices are observed in the market. The bond price over a term depends crucially on the random fluctuations of the interest rate market. Naturally, interest rate, unlike bonds, cannot be traded. We only trade bonds and other instruments that depend on interest rates. Chan et al. (1992) conducted a comprehensive empirical analysis on the above list of one-factor short rate models. They found that the most
Georgios Bouzianis [Page 36]
CHAPTER 3: STOCHASTIC SHORT RATE MODELS successful models which capture the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the interest rate. The findings confirm the financial intuition that the volatility of the term structure is an important factor governing the value of contingent claims. The bond price when examined with alterations on interest rates depends on two basic factors. Initially it depends on the current time t and the time of maturity T for every bond. Therefore, the plot of B(t, T) is indeed a two- dimensional surface over varying values of t and T. For a given fixed t = t0, the plot of B(t, T) against T represents the whole spectrum of bond prices of different maturities at time t0. In this thesis, we address on-factor short rate models, Vasicek model (1977), Hull-White (1990 and 2000), Cox Ingersoll Ross model (1985), Dothan model (1978), Black-Derman-Toy model (1980).
3.1 Vasicek Model The Vasicek model was introduced by Oldrich Vasicek in 1977. Vasicek proposed the stochastic process for the instantaneous interest rate rt to be:
drt a() b r t dt dW t (3.1.1)
Where Wt is a Wiener process, modeling the random market risk factor, σ is the standard deviation parameter that determines the volatility of the interest rate, a(b-rt) is the drift factor that describes the expected change in the interest rate at that particular time, b is (long term mean value) the long run equilibrium value towards which the interest rate goes back and a is the speed of reversion. It gives the adjustment of speed and has to be positive in order to maintain stability on the long-term value.
When rt is below b, then the drift term becomes positive for positive a, generating a tendency for the interest rate to move upwards (toward equilibrium). To explain the mean reversion phenomenon, we argue that when interest rates increase, the economy slows down and that there is less demand for loans and a natural tendency for rates to fall. The reverse situation of rates dropping can be argued similarly in the reverse sense. Also, the reversion phenomena in interest rates do not violate the principle of market efficiency. Vasicek model was the first economics model to capture the value of mean reversion. Mean reversion is an important characteristic of the interest rate that is responsible to set it apart from other financial prices. Hence, Georgios Bouzianis [Page 37]
CHAPTER 3: STOCHASTIC SHORT RATE MODELS interest rates neither rise nor decrease indefinitely unlike stock prices. It moves within a limited range, showing a tendency to revert to a long run value. This model is used in order to create an interest rate tree with the corresponding probabilities. With the values of the possible interest rates of the lattice it follows the valuation of the bonds. By solving (3.2.1), for a,b and σ positive constants we get:
t r r eat b(1 e at ) e at e au dW tu0 0
Then
t :(),E r r a b E r du t t 0 u 0 (3.1.2) d ttab( ). dt
This is a linear ordinary differential equation (ODE). Now, by using the integrating factor eat we get that state variable is distributed normally with mean:
at at ttE r e r0 b(1 e ) (3.1.3)
And variance defined as:
t 22Var r E(), eat dW t t u 0 (3.1.4) 1 e2at 2 ( ). 2a The final formula for bond pricing after applying Vasicek model will be as follows:
BtTr( , ,tt ) exp( AtTr ( , ) DtT ( , )), (3.1.5) where
1 ea() T t A(,), t T a
2 2A(,) t T 2 D(,)( t T b )[(,)[ A t T t )] . 242
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The equation (3.1.5) is called an exponential affine bond price. The simple
Gaussian structure of rt leads to a closed form solution of the bond price. The main advantage of Vasicek Model is that it is analytically tractable and it has an explicit solution since the distribution of rt is normal. However, its fundamental drawback is that the short term interest rate ‘r’ can become negative with is a positive probability for this event to occur.
3.2 Hull-White Model The Hull-White model was introduced by Hull and White in 1990. It is a model of future interest rate. It belongs to the class of no-arbitrage models that are able to fit today’s term structure of interest rate. The Hull-White model is varying the time parameter in Vasicek model. It allows the deviation of analytical formulae that is strength of its normal distribution. This special class of models take the following form:
drt(()())() at btrdt t tdW t (3.2.1) where Wt is a Wiener process, modeling the random market risk factor, σ is the standard deviation parameter that determines the volatility of the interest rate. With b = 0, the model can be considered an extension of the Vasicek model, and it becomes the continuous version of the HL model when α = 0. We consider this model when b and σ are constants and a has t- dependence. a is calculated from the initial yield curve describing the current term structure of interest rates.
t Let K a du and apply Ito’s lemma integration by parts to eKtrt, tu 0
KKKt t' t der()t e Krdt t t edr t (3.2.2)
By (3.2.1) and (3.2.2) we conclude that:
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KKKt t t der(t ) ebrdt t t e (( a t brd t t ) t t dW t ), KKKK ebrdtt ead t ebrd t e t dW , t t t t t t t t t (3.2.3) KKtt e at d t e t dW t ,
After integrating (3.2.3) we come to:
tt KKK et r r e u a du e u dW , t0 u u u 00 tt KKK r e t ( r ( e u a du e u dW )), (3.2.4) t0 u u u 00 tt KKKKK()() r et r e u t a du e u t dW . t0 u u u 00
In order to generalize the result for any t ≥ u, we have:
tt ()()()KKKKKK r et u r e t u a du e t u dW , (3.2.5) t0 u u u 00
By the initial assumption that b(t) and σ(t) are constants we get that:
tt r ebt r e b()() t u a du e b t u dW , (3.2.6) t0 u u u 00
tt r eb()()() t s r e b t u a du e b t u dW , (3.2.7) t s u u u ss
The final formula for bond pricing after applying Hull-White model will be as follows:
PST(, ) AST (, )exp( BSTrS (, )()), (3.2.8) where
1 exp( b ( T S )) BST(,), b P(0, T ) log( P (0, S )) 22 (exp( bT ) exp( aS )) (exp(2 aS ) 1) ASTBST( , ) exp( ( , ) ), P(0, S ) S 4 a3
A great advantage of this particular short rate model is that it can be fitted with the initial term structure of interest rate. For the calculation of Georgios Bouzianis [Page 40]
CHAPTER 3: STOCHASTIC SHORT RATE MODELS the initial prices of the zero coupon bond with maturity T and rt a stochastic process under a risk neutral measure Q, then the corresponding formula is:
T r ds s P(0, T ) EQ ( e 0 ), (3.2.9)
t In the above equation (3.2.9), define y() t r ds . y(t) is a sum of normal s 0 distributed random variables. Therefore we get that:
P(0, T ) EQ ( e y() T ). (3.2.10)
t By equations y() t r ds and (3.2.7) it follows that: s 0
t t s t s y(), t ebt r ds e b()() s u a duds e b s u dW ds 0 u u u (3.2.11) 0 0 0 0 0
It comes out that y(t) follows a normal distribution with mean:
re(bt 1) t ()(),t 0 ebs e bu a t u du (3.2.12) u b 0
And variance: t22 t e bt 1 v( t ) e22bs ( ). (3.2.13) 2b 2 b2 4 b 3 4 b 3
From the expression of the Laplace transform of Gaussian, the equation (3.2.10) can be written as:
1 P(0, T ) exp( EQ ( y ( T )) Var ( y ( T ))) 2 1 (3.2.14) exp( (t ) v ( t )) 2 re(bt 1) T 11T22 T e bT exp( 0 ebs e bu a ( T u ) du e 22 bs ( )) u 2 3 3 b0 2 2 b 2 b 4 b 4 b
Having set the formula for the initial price of the zero coupon bonds it is now easy to evaluate a(T) as follows:
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fT(0, ) 2 a( T ) bf (0, T ) (1 e2bT ). (3.2.15) Tb2
where f(0,T) is instantaneous forward rate, defined as follows: logPT (0, ) fT(0, ) (3.2.16) dT The generalized Hull-White model that we use for the construction of the trinomial short rate lattice is a model in which some function of the short rate obeys a Gaussian diffusion process of the following form (Hull-White, 2000): dfr()[() qt atfrdt ()()] stdz () (3.2.17) First, we set the current time to 0 and define a deterministic function g, which satisfies: dg[ q ( t ) a ( t ) g ( t )] dt (3.2.18) Define a new variable, x, that is: x(,)()() r t f r g t (3.2.19) The new variable obeys a much simpler diffusion process: dx a()() t xdt s t dz (3.2.20) Building a tree for f(r) involves four steps (Hull-White, 2000) The first step is to select the spacing of the tree nodes in the time dimension. The second step is to decide on the spacing of the nodes in the interest-rate dimension. The third step is to choose the branching process for x(r, t) through the grid of nodes. The final step involves the shifting of the tree by the value of g at each point in time. The main advantages of Hull and White Model is that compared to all the short rate models reviewed in this thesis it can be adjusted very accurately to the initial term structure and its implementation is relatively straight forward, based on a lattice structure (Hull and White, 1996; Hull and White, 1993). There is a wrong claim that the interest rates with this model can take negative values. Neglecting the stochastic term for a moment, the change in r can be negative if r is currently "large". The drawback of this model is that it is sensitive to its parameters; a small change in parameter values may result in a large change in bond price. The most suitable stochastic short rate model with the development of a matching-immunization ALM strategy, is Hull and White model. The fact Georgios Bouzianis [Page 42]
CHAPTER 3: STOCHASTIC SHORT RATE MODELS that bond prices can change significantly is an advantage for the purpose of this thesis because the results of the empirical testing verify the selective property of the proposed model that the final bond portfolio does not incorporate these bonds (Rama, 2010). Also Hull and White model (2000) can be efficiently used for the evaluation of coupon bonds with analytical Matlab functions.
3.3 Dothan Model Dothan introduced in 1978 a constant price of risk. It is a model of future interest rate (Pintoux and Privault, 2009). In Dothan model, the interest rate has a log-normal distribution which overcomes the drawback of negative values of the interest rate in Vasicek model. This special class of models take the following form:
drt ardt t rdW t t (3.3.1)
After integrating (3.3.1) it follows that:
tt r r ar ds r dW t0 s s s (3.3.2) 00
With regard to the log-normal distribution of rt., (3.3.2) transforms to: 1 lnr ln r at W 2 t . tt0 2 (3.3.3)
Then the equation for rt will be: 1 r rexp( at W 2 t ), tt0 2 1 r exp( a ( t u ) ( W W ) 2 ( t u )), (3.3.4) u t u 2 1 r exp(( a 2 )( t u ) ( W W )). u2 t u
This is the final mathematical expression for rt with Dothan Model.
Moreover, the conditional mean for rt is defined as follows:
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1 ()WW E[ r | F ] EQ [ r exp(( a 2 )( t u ))( e tu | F )], t u u2 u (3.3.5) a() t u reu
Also the variance of rt is defined as:
1 VarrFEr[ | ]Q [( (exp(( a 22 )( tu ))(exp( ( WW )) 1)) | F ], t u u2 t u u (3.3.6) 2 2a ( t u ) 2 ( t u ) ru e ( e 1).
Following the arbitrage pricing approach (arbitrage conditions), the governing equation for the bond price satisfies the following PDE for Dothan model:
BBB 1 2 ar 22 r rB 0, (3.3.7) t r2 r2
with final conditions:
B( T , T , r ) 1 , B ( t , T ,0) 1
The solution of the PDE (3.3.7) corresponds to the price of the bond with maturity T. Dothan model has the same advantage with BDT and CIR model, (the interest rate has a log-normal distribution) that overcomes the drawback of negative values in interest rates but the formula for bond evaluation is a very complex equation depended on three integrals.
3.4 Black-Derman-Toy Model The BDT model was introduced by Fischer Black, Emanuel Derman, and Bill Toy in 1990. It was first developed for in-house use by Goldman Sachs in the 1980s and was eventually published in the Financial Analysts Journal in 1990. The BDT model is a one-factor model; a single stochastic factor (the short rate) determines the future evolutions of all interest rate. In the model the
Georgios Bouzianis [Page 44]
CHAPTER 3: STOCHASTIC SHORT RATE MODELS short interest rate is log-normally distributed. The model does not have a closed-form solution for bond price. The original formulation of the Black-Derman-Toy model is in the form of a binomial tree. The continuous time equivalent of the model can be shown to be:
' ()t dln rt ( a ( t ) ln r t ) dt () t dW t . (3.4.1) ()t In this model, the changes in the short rate in the model are lognormally distributed, and the short rates are always non-negative. The first short a(t) is chosen so that the model fits the term structure of short rates and the second function σ(t) is chosen to fit the term structure of short rate volatilities. If σ(t) is a decreasing function then σ’(t) becomes less than zero. In this case the BDT model satisfies the mean reversion property. If σ(t) is an increasing function then σ’(t) becomes greater than zero. In this case, the BDT model will grow and it has no mean reversion effect. When the volatility function σ(t) is taken to be constant, the BDT model reduces to a lognormal version of the Ho-Lee model with σ’(t)=0. In this case the specified model becomes as follows:
dln rtt a ( t ) dt dW . (3.4.2)
By integrating the above equation (3.4.2) the formulation for the instantaneous short rate becomes as follows:
tt lnr ln r a ( s ) ds dW . ts0 00 (3.4.3) t lnr (() a s ds ) W . 0 t 0 Taking the base of natural logarithm both sides of the above equation (3.4.2) we get the final expression for the instantaneous short rate, in this particular case of the Black-Derman-Toy Model as follows:
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t r rexp( ( a ( s ) ds ) W ), tt0 0 t r exp( ( a ( s ) ds ) ( W W ), (3.4.4) u t u u t a() s ds u ()WW t u ru e e .
Even though Black-Derman-Toy model is one of the simplest short rate models for the prediction of future term structure in the form of lattices (Klose and Li, 2003), it does not have a closed mathematical form for the evaluation while short rates change through time. Therefore, it is not considered as a suitable model for the ALM strategy of this thesis. The main advantages of Black Derman Toy Model is that it does not allow negative values in the interest rates it predicts and it can be directly calibrated to an input term structure of discount bonds and cap volatilities. However, the fundamental property of mean reversion that is incorporated in BDT model does not hold for constant volatility and there is not explicit formulation for pricing interest rate derivatives.
3.5 Cox Ingersoll Ross Model The Cox Ingersoll Ross model was introduced in 1985 by Cox, Ingersoll and Ross as an extension of the Vasicek Model. The CIR bond pricing model assumes that the risk natural process for the Short interest rate is given by:
drt a(). b r t dt r t dW t (3.5.1)
The drift factor, a(b-rt), is exactly the same as in the Vasicek model. It ensures mean reversion of the interest rate towards the long run value b, with speed of adjustment governed by the strictly positive parameter a.
The standard deviation factor, rt avoids the possibility of negative interest rates for all positive values of a and b. An interest rate of zero is also precluded if the condition 2ab≥σ2 is met.
When the rate rt is close to zero, the standard deviation takes very small prices, which dampens the effect of the random shock on the rate.
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Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards equilibrium). After integrating the stochastic differential equation (3.5.1) we get that:
tt r r a(). b r ds r dW (3.5.2) t u s s s uu
In the case where μ=0, then from (3.5.2) we obtain the following mathematical relationship:
tt r r a(). b r ds r dW (3.5.3) t0 s s s 00
By (3.5.3), we get the unconditional mean and variance of rt as follows:
t EQQ( r ) r a ( bt E ( r ) ds ), (3.5.4) ts0 0
After solving the ordinary differential equation Φ(t)+aΦ(t)=ab, where t (t ) r a ( bt ( s ) ds ), we get the final formulation for the mean 0 0 defined as follows:
Q at E()(). rt b r0 b e (3.5.5)
The formulation for the variance of rt is defined as follows: 2 Var( r ) (1 eat )[ r e at ( b / 2)(1 e at )]. (3.5.6) t 0 The instantaneous short rate dynamics corresponds to a continuous time first-order autoregressive process where the randomly moving interest rate is elastically pulled toward a central location or long term value, b, which leads to mean reversion property. There is no explicit form for the solution to the CIR model. It is known that the model has unique positive solution. If the interest rate reaches zero, then it can subsequently become positive. Generally, when the rate is low (close to zero), then the standard deviation also becomes close to zero.
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CHAPTER 3: STOCHASTIC SHORT RATE MODELS
The price of a discount bond is a function of current interest rate, time and maturity. Under the free-arbitrage conditions B(t,T,r) satisfies the following PDE:
BBB 1 2 a( b r ) 2 r rB 0, (3.5.7) t r2 dr2 with final condition B(T,T,r)=1. The main advantage of the CIR model over the Vasicek model is that the short rate is guaranteed to remain non-negative. Unlike the Vasicek model, the CIR model is not Gaussian and is therefore considerably more difficult to analyze. For the evaluation of a zero coupon bond with CIR model the corresponding formula is defined as:
P(, t T ) A (, t T )exp( B (, t T )), rt (3.5.8) where
22 2ab 2( a2 2 2 ) e ((a 2 ) a ) T /2 A( t , T ) [ log( )], 2 22 ((a2 2 2 ) a )( e (aT 2 ) 1) 2( a 2 2 2 )
22 2(e(aT 2 ) 1) B(,). t T 22 ((a2 2 2 ) a )( e (aT 2 ) 1) 2( a 2 2 2 )
The main advantages of Cox Ingersoll Ross Model is that: it does not allow negative values in the interest rates it predicts; the conditional volatility depends on the level of the short rate; it is based on a coherent statistical model for behavior of interest rates; it has an explicit formulation for term structure and bond pricing. On the other hand, the particular model is not analytically tractable. The statistical model is in the simplest form allowing perfect correlation among all bond prices.
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CHAPTER 4. PROPOSED METHODOLOGY WITH EMBEDDED HULL - WHITE STOCHASTIC SHORT RATE MODEL The main purpose of this chapter is to extend the model of chapter 2 in order to capture both the stochastic nature of the interest rate and the cases of bigger and no symmetric shifts in the term structure. There is a variety of different short rate models that describe the stochastic evolution of the interest rate. The most suitable model with the general aim of this thesis is the generalized Hull-White model. The first mathematical model that is presented in the second chapter of the thesis is a multiperiod static model. Even though it is suitable for matching and immunization ALM strategies it is obligatory that the shifts in the term structure are small, parallel and well defined during the investment horizon. It is already mentioned that this requirement is not totally realistic. The second mathematical model that is presented in this chapter is α multiperiod stochastic model. This model enables dynamical evolution of both asset and liabilities through time, and is suitable to incorporate dynamic trading strategies. It is based on stochastic dedication for the optimization of dynamic strategies of short term lending and stochastic linear programming that allows the formulation of stochastic evaluation of the short term structure. The planning horizon starts at t=0 and extends to t=T. The bond portfolio is constructed at t=0 under all the different scenarios of bond prices that are presented with Hull-White lattice structures (Hull and White, 1994). Decisions with regard to the necessary overflow can be made at any trading date of the planning horizon. The discrete scenario setting of this model presupposes that there do not exist events between the trading dates. The trading dates except t=0 are consisted of a finite number of possible states of the economy. As a new trading day arrives, every single state of this date stores unique information about the short rate, bond prices, and liabilities values. The fact that the Hull-White interest rate lattice is trinomial is translated as that there are exactly three possible transitions during the next time period when starting from any random state of the tree. These three transitions are: ‖up‖, ‖middle‖ and ‖down‖. Further information on linear scenario structures and their usefulness in the development of a stochastic
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matching-immunization asset and liability management strategy is given in the next sections of this chapter.
4.1 Hull-White scenario structures At every single trading date of the planning horizon there are exactly three possible states, because of the implementation of Hull-White model. The states are indexed by s and the index set of possible sates at trading
date t is denoted by Σt = { stv | v=1,2,..,St}. Every single state has a specific probability to occur defined by Hull- White. However in Hull-White short rate lattice not all the transitions have the same probabilities. At every trading date only one possible state will be finally reached. In order to understand better the linear scenario structures, suppose a five years planning horizon. The next figure illustrates the changes of the
short rate (drt) with Hull-White trinomial lattice.
4 4 s24 s3 s4 s54
s12 s23 s33 s43 s53
s22 s32 s42 s52 s11 s00
s21 s31 s41 s51
s10
0 0 s2 s3 s40 s50
t=0 t=1 t=2 t=3 t=4 t=5 t=T
Figure 4.1.1: Recombining trinomial Hull-White lattice, with five trading dates.
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Notice that, in figure 4.1.1 not every state in Σt can be reached from every state in Σt-1. For example in Σ2, the state s24 can be reached only from the state s12 of Σ1, while s22 can be reached from all the states of Σ1. Therefore it is important to create the right sets of states that are possible to occur in a transition from a trading day to the next, given the present state.
Consequently, given a random state stv(t), at a random trading date t<Τ, the possible states that can be reached from state stv(t), at the trading date t+1, are called the successor states of this state and their set is denoted by stv(t)+. Obviously the set stv(t)+ ε Σt+1. Conversely, the states at trading day t-
1>0 that can reach the state stv(t) are called predecessor states of this state and their set is denoted by stv(t)-. Obviously the set stv(t)- ε Σt-1.
Additionally, the first state s00 does not have predecessor states and it is defined as the root node. Also, the states in ΣT do not have successor states. In order to understand the sets stv(t)+ and stv(t)- consider the state s32 of the trinomial lattice in figure 4.1.1. The successor states of s32 are: s41, s42 and s43while the predecessor states of s32 is the set Σ2. Therefore for s32, s32+= { s41, s42, s43 } and s32- = { s20, s21, s22, s23, s24 } = Σ2. In general, the possible transitions between nodes as time evolves are donated by order pairs of states with regard to their corresponding trading days. Specifically, given the present state of the economy stv(t), only one of the successor states will occur in the next trading date and is denoted by st+1v(t+1). Hence, each transition can be defined by the pair (stv(t), st+1v(t+1)).
A sequence of possible states with regard to the sets stv(t)+ for every random state from t=0 to t=T (s0v(0),s1v(1),…,sTv(T)), is defined as a path or scenario and is denoted by l. Suppose that the present state of economy at trading day t is stv(t). A scenario l visits only the successor states of stv(t) at the next trading day. The state that will be visited by scenario l at the trading day t is denoted by nt(l). As it is easily verified by figure 4.1.1 there are many different paths-scenarios. In the proposed methodology of this chapter each scenario has the same last trading day which is t=T. Each scenario l ε Ω has a unique probability to occur which is denoted by pl, such that pl≥0 and
ΣleΩpl=1. In general the sample set Ω grows as the planning horizon also increases. It is obvious that in the cases where a pension fund builds a matching-
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immunization ALM strategy with very long time horizon, the transitions between the states of different trading dates increase significantly. The next figure illustrates the case of three different scenarios of the short rate trinomial lattice in figure 4.1.1.
4 4 4 s24 s3 s4 s5
2 3 3 3 s1 s2 s3 s4 s53
s11 2 s32 s42 s52 s00 s2
1 1 1 1 s10 s2 s3 s4 s5
s20 0 s30 s40 s5
t=0 t=1 t=2 t=3 t=4 t=5 t=T
Figure 4.1.2: A scenario structure derived from the trinomial lattice in figure 4.1.1.
The first scenario that is highlighted with the red line is: n0(1)=s00,
n1(1)=s12, n2(1)=s23, n3(1)=s32, n4(1)=s41, n5(1)=s52. The second scenario that
is highlighted with the blue line is: n0(2)=s00, n1(2)=s11, n2(2)=s23, n3(2)=s34,
n4(2)=s43, n5(2)=s52. The third scenario that is highlighted with the green
line is: n0(3)=s00, n1(3)=s10, n2(3)=s22, n3(3)=s33, n4(3)=s42, n5(3)=s51.
Obviously all scenarios of this trinomial lattice start from s00 and finish in
a state of Σ5. Moreover, different scenarios may have common states, up to a certain trading date in the future. The next figure illustrates the case of three scenarios with common states until t=3. It is very important to mention that even if the scenarios of a linear scenario structure present common
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states up to a trading day, there is still no ambiguity that the scenarios are independent.
4 4 4 s24 s3 s4 s5
53 3 s s12 s2 s33 s43
1 2 s1 2 s5 2 s32 s4 s00 s2
0 s21 s31 s41 s51 s1
s20 s30 s40 s50
t=0 t=1 t=2 t=3 t=4 t=5 t=T
Figure 4.1.3: A scenario structure with common states up to t=3, derived from the trinomial lattice in figure 4.1.1.
In figure 4.1.3, there are three scenarios with common states: s00, s12, s23
and s32 and at t=4 these scenarios follow different states up to the last trading date. However, all the possible scenarios of figure 4.1.1 are independent from each other, except of their common initial state which is the root node. In the proposed methodology of the next section, the changes in the term structure of the short rate are not supposed to be small and well defined as in the case of the proposed model in chapter 2. Changes in the short rate have a stochastic evolution which is described with the implementation of Hull-White model. Every different state of the Hull- White short term trinomial lattice in figure 4.1.1 describes a different possible change in the interest rate. With regard to this lattice it is possible to evaluate the prices of the incorporated bond at every different trading day. Since their prices depend
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CHAPTER 4: PROPOSED METHODOLOGY WITH EMBEDDED HULL- WHITE STOCHASTIC SHORT RATE MODEL on the changes drt, then the evolution in the price of every bond can also be represented with a scenario structure. With this information available it is easy to create a stochastic dynamic matching-immunization ALM strategy.
4.2 General Methodological approach The model suggested in this section follows closely the formulation of the proposed model in chapter 2. The model is structured based on total cashflow matching with surplus reinvestment and portfolio immunization. The core difference is that at every different trading day, the incorporated bonds change their values with regard to the changes in the term structure approached by Hull-White model. The problem is still defined as the minimization of the segregated initial capital is required for the construction of a matched-immunized portfolio. However, the number of the decision variables of the problem grows significantly, especially for a very long investment horizon. The objective function of the problem is defined as follows:
nn 0 b min f0 ( xi 0 p i 0 ) t i 0 ii11 nn 0 bb (4.2.1) f0 ()() xi 0 p i 0 x i 0 p i 0 c i 0 ii11 n 0 b f0 ( xi 0 p i 0 )(1 c i 0 ) i1
Also, pi0 denotes the expected present value of the i-th bond at period t, and is defined as follows:
ll pi00 l p p i (4.2.2) where pl is the probability of scenario l ε Ω to occur, and pi0l is the present value of the i-th bond under scenario l.
b The mathematical term xpii00 denotes the initial monetary investment in the i-th bond under all different scenarios for the evolution of bond’s price. It is obvious that the summation of this term for each i ε [1, n], is equal with the total initial monetary investment in the bond portfolio.
The continuous variables ftv(t), represent the necessary cash surplus at the end of every trading date t and state s. The variable f00 variable represents the initial capital is used simultaneously with the purchase of the bonds.
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The constraints of the problem are the same except the first and second order conditions for immunization. There are cases where there are not securities maturing on the exact date with the liability. The cashflow matching constraint that describes the mathematical relation between the inflows, the outflows and the cash surplus at the end of every period t is:
n vtvt()() vtvt (1)(1) vt (1) vtvt ()() xciit0 prf tt prf tt 1 1 (1 r ft ( 1) ) = prl tt , i=1 v ( t ) Σttv( t 1) v ( t ) v ( t ) Σ
n v()() t v t v () t v (1)(1) t v t vt( 1) xi0 c it pr t( f t l t ) pr t 1 f t 1 (1 r f (t1) ) = 0, (4.2.3) i=1 vt( ) Σt v( t 1) v ( t )
t [1,..., T ], v ( t ) t
Where prtv(t) is the probability from the Hull-White lattice for state stv(t) to occur. The rest variables that participate in equation (4.2.3) are the same with those in equation (2.2.1). However in equation (4.2.3) these variables are not only time dependent as in (2.2.1), but their values change with regard to different states stv(t) for each trading day t. Additionally, the coupons of the bonds are fixed, therefore there is no dependency with the different states of every single trading day. The coupon of the i-th bond at the trading day t is denoted by cit and is the same for every state stv(t) at trading day t. The second constraint with regard to the property of portfolio immunization requires that the expected present values of the portfolio cash inflows and of the liability outflow under all scenarios are equal.
pvpl pv n (4.2.4) pi00 x i pv l i1 where pvl is the expected present value of the liabilities under all scenarios. Nevertheless, this condition may be very expensive to satisfy for all scenarios and in the case of a long planning horizon where the total number of states increases significantly, it is hard to find a solution satisfying this constraint. In such cases, it is preferable to treat immunization as a single problem with different objective function.
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Moreover the constraints for the investment policy adjustment are the same with those that presented in the model of chapter 2. The proposed model of this chapter is a multiperiod stochastic model. In order to evaluate the incorporated interest-sensitive financial instruments, Hull-White tree method is applied in the model. In fact it is a trinomial lattice that describes the possible shifts in the short rate at each trading day of the planning horizon. With regard to this linear scenario structure, the set of the incorporated variables can also be represented with a similar tree structure. Moreover, the corresponding programming code of the model was developed, using Matlab functions for the construction of Hull-White tree and the evaluation of the coupon rate bonds. The application for both proposed models is presented in the next chapter of the thesis.
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CHAPTER 5: EMPIRICAL TESTING
CHAPTER 5. EMPIRICAL TESTING 5.1 Data and field of application
The recommended methodological approaches of the matching- immunization models presented in Chapter 2 and 4, have been applied on data concerning investment grade corporate US bonds, sovereign bonds and European corporate bonds. The study period includes the record of the closing prices, coupons percentages and maturity days of the incorporated bonds at 28/12/2015. The US corporate bond index consists of 5616 bonds, the sovereign bond index consists of 1091 bonds and the EUR corporate bond index consists of 1990 bonds. The population considered in the study consists of bonds covering a broad spectrum of different sector activities, with professional applied links with the pension fund and asset management industry, taking into account the major issue of the diversification effect. The types of data that are employed in the application are also available for the analysts and investors in other countries. The obtained evidence for the first model imply that if there is an optimal portfolio for a specific assets and liabilities universe under the prism of matching and immunization theory, the initial segregated capital, plus the necessary overflows of all periods will be less than the liabilities. While for the second model, the stochastic approach of the short rate provides a much more realistic matching-immunization ALM strategy with scenarios that reflect the reality of the market. Bloomberg US Corporate Bond Index
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Bloomberg Global Developed Sovereign Bond Index
Bloomberg EUR Investment Grade Corporate Bond Index
5.2 Results and discussion for the proposed methodology for small and parallel shifts on term structure. The number of bonds that can be selected is limited between 70 and 80, and the lot size of every bond between 4000 and 15000. Equal commission for every bond at 1% is considered, and the desired transaction cost upper limit at 20000. Table 5.2.1 contains the individual characteristics of the incorporated bonds. Table 5.2.2 contains the capital invested and the earnings for every bond. Table 5.2.3 contains the required overflows, the overall inflows and corresponding liabilities at any period t.
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A sample of 3500 bonds that includes US corporate bonds, Sovereign
bonds and European corporate bonds was used. After applying 24 ≥ hi, i=1,…,n, as an initial condition the final sample consisted of 2431 bonds with maturity less than 24 years is obtained. This sample consists of bonds that mature before the liability. Therefore borrowing cash is exempted from the model. The optimal portfolio is composed of 70 bonds. The results are presented in Tables 5.2.1, 5.2.2 and 5.2.3.
Ticker Coupon% Maturity Currency Price Sector Yield% Duration Convexity GE 6.750 03/15/32 USD 131.449 Financials 3.839 10.852 149.588 GS 7.500 02/15/19 USD 114.566 Financials 4.643 2.794 9.389 C 4.450 09/29/27 USD 99.380 Financials 3.507 9.309 101.515 ABIBB 7.750 01/15/19 USD 115.785 Consumer Staples 5.463 2.695 8.778 CMCSA 6.950 08/15/37 USD 132.689 Communications 4.028 12.900 222.732 WFC 4.300 07/22/27 USD 101.996 Financials 3.685 9.150 98.387 TWX 7.700 05/01/32 USD 125.832 Communications 4.526 10.488 141.551 PEMEX 6.625 06/15/35 USD 89.365 Energy 6.504 11.240 168.579 VZ 4.400 11/01/34 USD 93.314 Communications 3.782 13.173 214.583 T 4.500 05/15/35 USD 93.290 Communications 3.980 13.298 220.365 … … … … … … … … … RWE 6.625 01/31/19 EUR 128.348 Utilities 4.160 2.787 9.345 GE 6.000 01/15/19 EUR 128.257 Financials 4.133 2.767 9.196 ENGIFP 6.875 01/24/19 EUR 131.563 Utilities 4.316 2.758 9.177 SRGIM 5.000 01/18/19 EUR 124.615 Utilities 3.591 2.818 9.471 KPN 7.500 02/04/19 EUR 132.340 Communications 4.378 2.765 9.246 PEMEX 2.750 04/21/27 EUR 81.242 Energy 2.899 9.737 106.224 ETLFP 5.000 01/14/19 EUR 123.502 Communications 3.678 2.806 9.397 PEMEX 3.750 04/16/26 EUR 92.320 Energy 3.314 8.615 84.683 MS 6.500 12/28/18 EUR 129.184 Financials 5.031 2.779 9.013 VATFAL 6.750 01/31/19 EUR 129.600 Utilities 4.187 2.783 9.322
Table 5.2.1: Individual characteristics of the 70 bonds in the optimized portfolio
Ticker Investment on i-th bond Earnings from i-th bond Transaction costs GE 394347.000 624000.000 3943.470 GS 343698.000 367500.000 3436.980 C 1490700.000 2234250.000 14907.000 ABIBB 347355.000 369750.000 3473.550 CMCSA 1523284.816 2823533.982 15232.848 WFC 305988.000 441900.000 3059.880 TWX 1887480.000 3348000.000 18874.800 PEMEX 1340475.000 3388125.000 13404.750 VZ 279942.000 537600.000 2799.420 T 279870.000 556500.000 2798.700 … … … … RWE 385042.608 359625.000 3850.43 Georgios Bouzianis [Page 61]
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GE 384769.505 354000.000 3847.70 ENGIFP 394690.061 361875.000 3946.90 SRGIM 373845.377 345000.000 3738.45 KPN 397019.664 367500.000 3970.20 PEMEX 243726.509 390750.000 2437.27 ETLFP 370505.621 345000.000 3705.06 PEMEX 276959.549 412500.000 2769.60 MS 387553.183 358500.000 3875.53 VATFAL 388800.245 360750.000 3888.00 Table 5.2.2: Monetary synthesis of the portfolio and evaluation of the earnings (in USD) from every incorporated bond.
Earnings Necessary Total bond Reinvestment from Liabilities overflows earnings rate Time reinvestments 0 0.000 - - - 5.00% 1 0.000 0.000 2500000.000 2500000 5.20% 2 100000.000 0.000 2500000.000 2400000 5.10% 3 4369676.444 5100.000 6964476.444 2700000 4.90% 4 4572487.254 214114.146 2179957.311 2200000 4.80% 5 3476496.440 219479.388 2179957.311 3500000 5.00% 6 3123325.581 173824.822 2179957.311 2700000 5.40% 7 3087381.417 168659.581 3407889.558 3600000 5.32% 8 386474.951 164248.691 3032374.937 5900000 5.35% 9 1700035.355 20676.410 2692999.937 1400000 5.28% 10 362087.183 89761.867 5671099.937 7100000 5.20% 11 2731721.344 18828.533 6750515.958 4400000 5.14% 12 0.000 140410.477 2326229.146 5200000 5.08% 13 0.000 0.000 2900000.000 2900000 4.96% 14 211387.802 0.000 3911387.802 3700000 4.94% 15 0.000 10442.557 2978127.363 3200000 4.88% 16 30367.168 0.000 8030367.168 8000000 4.74% 17 0.000 1439.404 2368150.915 2400000 4.75% 18 0.000 0.000 3600000.000 3600000 4.82% 19 0.000 0.000 3100000.000 3100000 4.89% 20 0.000 0.000 2000000.000 2000000 4.95% 21 0.000 0.000 4800000.000 4800000 5.10% 22 213878.998 0.000 1913878.998 1700000 5.05% 23 0.000 10800.889 975213.173 1200000 5.00% 24 0.000 0.000 1000000.000 1000000 -
Total overflows Initial capital Total capital Total liabilities
24365319.935 47230068.114 71595388.050 81200000
Table 5.2.3: Required overflows and overall inflows at any period t and corresponding liabilities (in USD). In order to prove the accuracy of the proposed model theory, empirical evidence is used. More specifically, an empirical study was developed, implementing the model using actual market data. The existing historical data of the corporate US bonds, Sovereign bonds and European corporate
Georgios Bouzianis [Page 62]
CHAPTER 5: EMPIRICAL TESTING bonds are used in order to estimate the performance of the model (backtesting). The model proposed contains variables that change values through time, such as the yields, the coupons, the durations and the prices of the incorporated bonds. Consequently, in order to efficiently construct a portfolio that meets the conditions of the stated strategies, we backtest the price changes of the bonds for the predetermined investment horizon using duration and convexity as Fabozzi (1999) and Fabozzi (2005). The results offer statistics that can be used to gauge the effectiveness of the strategy. In Table 5.2.1 it is verified that the optimized portfolio consists of bonds of a variety of different sectors as well as and different types of bonds. Premium, discount and par bonds are included. The initial samples also contain extremely evaluated bonds which were not selected by the program. Moreover, there are bonds included with significantly different convexity and duration prices. In general, bonds that offer the same duration and yield but some exhibit greater convexity, will be affected differently by changes in interest rates. Bonds with greater convexity have a higher price than bonds with a lower convexity, regardless of whether interest rates rise or fall. The majority of the bonds that were selected from the program exhibit great values of convexity. This fact underlines the importance of the first and second order sufficient conditions of the portfolio immunization theory. The optimal portfolios will appear small changes in the case of larger changes in the term structure due to their synthesis. Consequently, the risk of the portfolio value to not meet the liability target value when liabilities become due is minimized. In Table 5.2.2 it is verified that the required investment in every bond of Table 5.2.1 is significantly less than the estimated earnings from each bond. This is one of the greatest advantages of fixed income strategies pointing out the return of the investment in a bond portfolio. Furthermore, the corresponding transaction costs for every incorporated bond is nonessential comparing with the earnings at each period t. The next figure illustrates the earnings compared with the investment to every single bond of the final portfolio.
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Figure 5.2.1: Investment to the i-th bond of the portfolio and expected earnings.
As is already mentioned previously, the sum of the investments in every individual bond, while simultaneously concerning the corresponding transaction costs, provides the information on the total capital required in bond investments. From the results of Table 5.2.3 it is verified how prosperous the adoption of an integrative asset and liability management strategy may be in the long run for the pension fund. The combination of the selected bond portfolio and the overflows manage to hedge the liability stream with less total capital than the total amount of liabilities. The next figure illustrates the fixed incomes from the investment in the bond portfolio and the necessary overflows at every trading day, compared to the predetermined liability stream.
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Figure 5.2.2: Bond portfolio incomes, necessary overflows and liabilities.
The basic inflows of the pension fund at any period t, were previously categorized through the segregation of the initial capital. There are the inflows from the reinvestments of the required overflows and the fixed incomes from the coupons of the bonds. The usefulness of investing in a matched-immunized portfolio of bonds for counterbalancing liabilities can be cleared out considering the fact that the additional return of the incorporated portfolio can considerably reduce the amount of cash required for this purpose. An additional notice is the confirmation of the selective property of the matching-immunization program as the number of included bonds stays to the minimum accepted. The program includes fewer assets in general but it allows higher lot sizes, accomplishing higher returns.
5.3 Results and discussion for the proposed methodology with embedded Hull-White stochastic short rate model The number of bonds which can be selected is limited to between 70 and 200, and the lot size of every bond between 700 and 1000. Equal commission for every bond at 1% is considered, and the desired transaction cost upper limit at 20000. For the implementation of Hull-White model, the valuation day is considered at 12/28/2015 with a planning horizon of five years. The initial term structure of the short rate is considered to be 4.1%, 3.9%, 3.8%, 4% and 4.4%, AlphaCurve = 0.1 and VolCurve = 0.01. Table 5.3.1 contains the individual characteristics of the incorporated bonds and their expected present value under all 81 possible scenarios. Table 5.3.2 contains the capital invested and the earnings for every bond and Table 5.3.3 contains the required overflows, the overall inflows and the corresponding liabilities at any different state stv(t) of every trading day t of the planning horizon. A sample of 3500 bonds that includes US corporate bonds, Sovereign bonds and European corporate bonds was used. After applying 5 ≥ hi, i=1,…,n, as an initial condition the final sample consisted of 1916 bonds with maturity less than 5 years is obtained. This sample consists of bonds that mature before the liability. Therefore borrowing cash is once again
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exempted from the model. The optimal portfolio is composed by 170 bonds. The results are presented in Tables 5.3.1, 5.3.2 and 5.3.3.
Expected price Ticker Coupon% Currency Sector under all scenarios ABBV 1.750 USD Health Care 96.1525 INTC 1.350 USD Technology 95.2128 ORCL 1.200 USD Technology 95.3144 T 1.400 USD Communications 95.3867 CVX 1.104 USD Energy 94.8172 C 1.850 USD Financials 96.2460 TOYOTA 1.250 USD Consumer Discretionary 95.4620 KO 1.875 USD Consumer Staples 89.4066 WSTP 1.500 USD Financials 95.5703 BDX 1.800 USD Health Care 96.0549 F 1.724 USD Consumer Discretionary 95.9569 USB 1.375 USD Financials 95.8181 COST 1.125 USD Consumer Staples 94.7917 CVX 1.345 USD Energy 95.3814 RY 1.400 USD Financials 95.6694 … … … … … NOVNVX 0.750 EUR Health Care 80.3080 ATLIM 1.125 EUR Industrials 82.2903 WESAU 1.250 EUR Consumer Staples 83.2027 PLD 1.375 EUR Financials 87.4346 NESNVX 2.125 EUR Consumer Staples 87.8722 AIFP 2.125 EUR Materials 87.5883 HEIANA 2.000 EUR Consumer Staples 88.5535 EFFP 1.750 EUR Health Care 87.3519 PEP 1.750 EUR Consumer Staples 87.1811 RENAUL 2.250 EUR Consumer Discretionary 89.7878 BSHBOS 1.875 EUR Consumer Discretionary 89.2563 SCMNVX 1.875 EUR Communications 86.6324 RBOSGR 1.625 EUR Financials 86.3411 ENEXIS 1.875 EUR Utilities 89.2563 ICADFP 2.250 EUR Financials 89.6487 Table 5.3.1: Individual characteristics of the 170 bonds in the optimized portfolio.
Investment on i-th Earnings from i-th Ticker Transaction costs bond bond ABBV 96152.452 101750.000 961.525 INTC 95212.758 101350.000 952.128 ORCL 95314.411 101200.000 953.144 T 95386.661 101400.000 953.867 CVX 94817.250 101104.000 948.172 C 96245.973 101850.000 962.460 TOYOTA 95462.044 101250.000 954.620 KO 89406.626 107500.000 894.066 WSTP 95570.256 101500.000 955.703 BDX 96054.881 101800.000 960.549 F 95956.896 101724.000 959.569 USB 95818.113 101375.000 958.181 COST 94791.697 101125.000 947.917 Georgios Bouzianis [Page 66]
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CVX 95381.359 101345.000 953.814 RY 95669.409 101400.000 956.694 … … … … NOVNVX 80308.003 103750.000 803.080 ATLIM 82290.300 105625.000 822.903 WESAU 83202.694 106250.000 832.027 PLD 87434.625 105500.000 874.346 NESNVX 87872.153 110625.000 878.722 AIFP 87588.339 110625.000 875.883 HEIANA 88553.528 110000.000 885.535 EFFP 87351.879 108750.000 873.519 PEP 87181.053 108750.000 871.811 RENAUL 89787.798 111250.000 897.878 BSHBOS 89256.258 107500.000 892.563 SCMNVX 86632.420 109375.000 866.324 RBOSGR 86341.085 108125.000 863.411 ENEXIS 89256.258 107500.000 892.563 ICADFP 89648.717 111250.000 896.487 Table 5.3.2: Monetary synthesis of the portfolio and evaluation of the earnings (in USD) from every incorporated bond.
State Liability Necessary overflow Bond portfolio - - 0.000 incomes-
s00 4073618.432 1260579.539 5334197.971
s10 4528959.685 487954.220 190733.190
s11 634934.081 827985.521 190733.190
s12 4566879.281 1459265.702 190733.190
s20 3161796.231 667177.493 190733.190
s21 487702.025 0.000 190733.190
s22 1392491.094 519524.400 190733.190
s23 2734407.596 589558.916 190733.190
s24 4787534.177 0.000 190733.190
s30 4824442.676 0.000 3395546.038
s31 788065.408 1854027.910 3395546.038
s32 4852963.909 0.000 3395546.038
s33 4785834.741 0.000 3395546.038
s34 2426878.244 806283.418 3395546.038
s40 4001402.344 4799192.736 8714844.000
s41 709431.693 14474720.981 8714844.000
s42 2108806.413 4944711.009 8714844.000
s43 4578677.626 4312164.010 8714844.000
s44 3961036.648 6757800.758 8714844.000
Table 5.3.3: Required overflows and overall inflows at any period t and state stv(t), and corresponding liabilities (in USD).
Georgios Bouzianis [Page 67]
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In Table 5.3.1 it is verified that the optimized portfolio consists of bonds of a variety of different sectors, but in their majority, they are discount bonds with low coupon payments. Moreover, there were scenarios in which the bonds of the initial sample were extremely evaluated through trading dates, due to big shifts in the short rate term structure. However, in the final synthesis of the bond portfolio of the program there are no bonds of this category. Moreover, in the initial constraints of this problem, the range of the allowed selected bonds is bigger than the range for the case of small and parallel shifts on term structure, and the lot size is significantly lower. In the cases where it is allowed to build a small bond portfolio with few different bond placements and large lot sizes two basic problems may occur: Either there is no solution satisfying the constraints of the problem, or the total cost for the construction of the bond portfolio that meets the stated conditions of matching-immunization, exceeds the liabilities. It is cleared out, that the investment policy adjustment has a significant role in the efficiency of the proposed model. Additionally, from Table 5.3.2 it is verified that the required investment in every bond of Table 5.3.1, is less than the estimated earnings from each bond, but the profit margin from every bond is less than in the case of small and parallel shifts in the term structure. However, the size of the bond portfolio for the case of the stochastic approach is much bigger than in the case of the static model of Chapter 2. Furthermore the corresponding transaction costs for every incorporated bond is nonessential compared to the earnings at each period t. From the results of Tables 5.3.3 and 5.3.2 it is verified that the adoption of an integrative stochastic asset and liability management strategy is a more realistic approach. The capital required in order to efficiently hedge the liability stream is in most cases less than the liabilities. However, there are some scenarios in which the total initial capital is equal, or marginally bigger than the total payouts. This fact is attributed to the incorporation of real market variables and their stochastic evolution, as well as the mean reversion property of the short rate. The next figure illustrates the shifts in the term structure of the short rates with the implementation of Hull-White model. Additionally, two random scenarios of the tree in the form of a table have been selected.
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Figure 5.3.1: Shifts in the term structure in the form of trinomial lattice, with Hull-White model.
The next table contains the probabilities of the possible transitions at every trading day t and state stv(t).
Table 5.3.4: Probabilities of the transitions in trinomial Hull-White lattice.
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The importance of an integrative asset and liability management strategy incorporating immunization properties for financial intermediaries, insurance firms, pension funds and non-financial institutions is given. Bearing in mind that immunization is a dynamic process, the portfolio, once established, cannot remain constant but needs to be periodically rebalanced in order to ensure that the duration is equal to the remainder of the horizon due to duration drift. The synthesis of the final portfolio of the programs highlights the importance of diversification. Both programs neutralize the exposure in possible yield alterations and simultaneously offer sector and currency diversification.
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CHAPTER 6: CONCLUSIONS-PERSPECTIVES
CHAPTER 6. CONCLUSIONS – PERSPECTIVES
6.1 Conclusions
T H E P R O B L E M Nowadays financial intermediaries, insurance firms, pension funds and non-financial institutions, have several liabilities and obligations to payout. Financial engineering systems offer mechanisms than match the obligation stream with incomes from investment to fixed income instruments. The adoption of such a mechanism, reduces the capital required in order to efficiently cover the liabilities due to the return of the investment. Therefore it is important to develop a long term integrative asset and liability management strategy (ALM). However a significant factor that must be incorporated in strategies like this is the uncertainty of future interest rate. Shifts in the term structure of the interest rate can affect the future value of the investors’ portfolios, as well as the amount of their liabilities. Consequently, it is a necessity to adopt a yield immunization technique in which interest rate movements have no impact on the value of a firm. Therefore, the main goal is the creation of an integrative asset and liability management strategy that satisfies the properties of cashflow matching and yield immunization.
P ROPOSED METHODOLOGY Within this thesis, two bond portfolio optimization programs were developed. Both programs are based on portfolio dedication (cashflow matching) and portfolio immunization strategies. The operation of both programs lies on the minimization of the total initial capital required for the construction of a least cost bond portfolio that satisfies these strategies. Additionally, special factors such as transaction costs, logical decisions (binary variables) and diversification factor are properly incorporated in both models. Also, investment policy constraints providing flexibility to the models are taken into account. The first model is a multiperiod static model that operates under the assumption that the shifts in the term structure will be small, parallel and well defined, approached with duration and convexity empirical testing. The second model is a multiperiod stochastic model that enables larger and no symmetric shifts in the term structure. Also, the evaluation of both Georgios Bouzianis [Page 72]
CHAPTER 6: CONCLUSIONS-PERSPECTIVES asset and liabilities is dynamical and relational with the stochastic evolution of the short rate, approached with Hull-White short rate trinomial lattice.
A PPLICATION OF THE SU GGESTED METHODOLOGY Both models where tested in a well-diversified investment universe of bonds that includes: US corporate bonds, European corporate bonds and Sovereign bonds. The validation process of the results indicates that both models offer completely satisfactory conclusions that were discussed in Chapter 5. For solving the Mixed-Integer linear Programming formulations, the optimization toolbox of matlab is used and in particular the function intlinprog. For the implementation of Hull-White model and linear scenario structure, functions intevset, hwvolspec, hwtimespec and hwtree are used. The full programing code for both models is presented in the appendixes. The implementation of the programs was made on a laptop with an Intel (R) i7-5500U CPU @ 2.4 GHz processor. Normally we have reported solution time under 30 minutes for asset universes consisted of 2000 to 3000 bonds while for bigger sets of assets multiple minutes of running time may be required. However, for asset universes consisted of 7000 bonds and more, matlab runs out of memory. An important reason for required time increase is the involvement of binary variables.
6.2 Perspectives Both proposed methodologies contain only continuous variables that change their prices relatively with the time horizon. Both models can be improved by embedding more decision variables with regard to the matching – immunization strategy. For instance, borrowing variables can be incorporated in the model. However both suggested methodologies refer mostly to pension funds. Hence, it is wise to except borrowing from the model and construct portfolios with securities that mature before the liabilities. Additionally, it is possible to turn the problem into a multiobjective linear problem by adding several objective functions with regards to different investment preferences. Both models can operate with the same set of constraints under a new objective function that will be a utility function. Even though it seems appealing, both models become very complicated and it is hard to apply in real cases for managerial purposes.
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CHAPTER 6: CONCLUSIONS-PERSPECTIVES
Moreover, both proposed models can be applied in real cases and can be adjusted in a variety of different investment policies. The results of both models can be directed in the desired investment direction of the DM (Decision Maker), by adjusting the constraints of both models with regard to the corresponding IPS.
Finally, this thesis is dedicated in the yield curve risk, when adopting an integrative asset and liability management strategy. It would be very interesting to incorporate more risk factors in the model, such as credit, liquidity, event, exchange rate, inflation, and sovereign risk. In that case the problem would be a multifactor problem corresponded to the construction of a liability matched portfolio, immunized not only against yield fluctuations. However the complexity of the model would also increase dramatically and will not offer easy managerial solutions.
Georgios Bouzianis [Page 74]
APPENDIXES
A PROGRAMMING CODE FOR THE PROPOSE D METHODOLOGY F O R S M A L L AND PARALLEL SHIFTS I N THE TERM STRUCTURE .
%% Obtaining Data c=xlsread('Data_ab',4,'b2:b3499');%Bond coupon P=xlsread('Data_ab',4,'e2:e3499');%Bond price euro2dollar=1.0968; %Currency at 28/12/2015 for i=2023:3498 P(i)=P(i)*euro2dollar; end M=xlsread('Data_ab',4,'p2:p3499'); [num1,Tick]=xlsread('Data_ab',4,'a2:a3499'); %Bond ticker [num2,Base]=xlsread('Data_ab',4,'d2:d3499'); %Bond monetary base [num3,sector]=xlsread('Data_ab',4,'i2:i3499'); %Bond sector data=readtable('dates_ab.xlsx'); t=data(:,1); settle='12/28/15'; Md=table2array(t); L=xlsread('MI1',1,'B21:B44'); %Liability stream Rytm=xlsread('MI',1,'a20:a44'); %Initial term structure d=0.01; FV=100; %Face value H=24; %Planning horizon j=1;
%Excluding bonds with maturity less than three years and more than %the 24years of the planning horizon for i=1:numel(M) if M(i)>=3 && M(i)<=H B(j)=P(i); T(j)=M(i); C(j)=c(i); com(j)=0.01; Md(j)=Md(i); Tick(j)=Tick(i); Base(j)=Base(i); sector(j)=sector(i); index(j)=i; j=j+1; end end
%Calculating the yield for every incorporated bond of the sample. for i=1:numel(C) Georgios Bouzianis [Page 76]
R(i)=bndyield(B(i),C(i),settle,Md(i)); end
%Investment policy adjustment (IPS) Su=80; %Maximum number of different bonds in the portfolio Sd=70; %Minimum number of different bonds in portfolio Nu=15000; %Maximum portion can be bought from every different bond Nd=3000; %Minimum portion can be bought from every different bond Tu=20000; %Maximum numerical exposure in transaction costs
%% Bond Price,Duration calculation and payoff matrix %Payoff matrix for i=1:numel(C) for t=1:T(i) if t end end end %Di calculation Di=Sum(t*wti) for i=1:numel(C) [Dmod(i),YD(i),D(i)]= bnddury(R(i)/100, C(i)/100, settle, Md(i)); [YearConv(i), Conv(i)] = bndconvy(R(i)/100, C(i)/100, settle, Md(i)); end % DL,PL and Ql calculation ( Duration, Present value and Convexity of %the liability stream) for i=1:numel(L) PV_L(i)=L(i)/(1+Rytm(i)+d)^i; end PL=sum(PV_L(:)); sumLconv=0; for i=1:numel(PV_L) Lw(i)=(PV_L(i)/PL)*i; sumLconv=sumLconv+((i^2+i)*L(i))/(1+Rytm(i)+d)^i; end DL=sum(Lw(:)); convL=sumLconv/PL; %% Problem Setting % Obj Creation obj(1,1)=1; for i=2:H+1 obj(i,1)=0; end Georgios Bouzianis [Page 77] j=1; for i=H+2:H+1+numel(B) obj(i,1)=(1+com(j))*B(j); j=j+1; end for i=H+2+numel(B):H+1+2*numel(B) obj(i,1)=0; end %Binary Variables intcon=(H+2+numel(B)):(H+1+2*numel(B)); %linear Equality Constraints %Construction of the tables Aeq and Beq for i=1:H+3 k=0; z=0; q=0; p=0; for j=1:(H+2*numel(B)+1) if i<=H if j<=H+1 if j==i Aeq(i,j)=(1+Rytm(i)+d); elseif j==(i+1) Aeq(i,j)=-1; else Aeq(i,j)=0; end elseif j<=H+1+numel(B) k=k+1; Aeq(i,j)=cf(i,k); else Aeq(i,j)=0; end Beq(i,1)=L(i); elseif i>H && i<=H+1 if j>H+1 && j<=H+1+numel(B) z=z+1; Aeq(i,j)=(YD(z)*B(z))/PL; else Aeq(i,j)=0; end Beq(i,1)=DL; elseif i>H+1 && i<=H+2 if j>H+1 && j<=H+1+numel(B) q=q+1; Georgios Bouzianis [Page 78] Aeq(i,j)=B(q); else Aeq(i,j)=0; end Beq(i,1)=PL; elseif i>H+2 && i<=H+3 if j>H+1 && j<=H+1+numel(B) p=p+1; Aeq(i,j)=(YearConv(p)*B(p))/PL; else Aeq(i,j)=0; end Beq(i,1)=convL; end end end %linear inequality constraints %Constructions of the tables Aineq and Bineq for i=1:numel(B) for j=1:numel(B) if i==j T1(i,j)=1; T2(i,j)=-Nu; T3(i,j)=-1; T4(i,j)=Nd; T5(i,j)=com(i)*B(i); else T1(i,j)=0; T2(i,j)=0; T3(i,j)=0; T4(i,j)=0; T5(i,j)=0; end end end T6=zeros(numel(B)); T7=zeros(2,numel(B)); T8=ones(1,numel(B)); T9=-T8; T11=[T8;T9]; T10=zeros((3*numel(B)+2),H+1); T12=[T1 T2;T3 T4;T5 T6;T7 T11]; Aineq=[T10 T12]; for i=1:2*numel(B) Bineq(i)=0; end Georgios Bouzianis [Page 79] for i=2*numel(B)+1:3*numel(B) Bineq(i)=Tu; end Bineq(3*numel(B)+1)=Su; Bineq(3*numel(B)+2)=-Sd; Bineq=transpose(Bineq); %Bound constraints lb=zeros(H+1+2*numel(B),1); for i=1:H+1+2*numel(B) if i<=H ub(i,1)=Inf; elseif i==H+1 ub(i,1)=0; elseif i>H+1 && i<=H+1+numel(B) ub(i,1)=Inf; else ub(i,1)=1; end end %% Problem Solution x=intlinprog(obj,intcon,Aineq,Bineq,Aeq,Beq,lb,ub); %Solution Table for i=1:numel(B) N(i)=x(H+1+i,1); b(i)=x(H+1+numel(B)+i); end for i=1:numel(B) Budg(i)=B(i)*N(i); Tr(i)=com(i)*B(i)*N(i); end Solution=sum(Budg)+sum(Tr)+x(1); %Total initial capital is required Total_N=sum(b); %Total different incorporated bonds Total_Tr=sum(Tr); %Total portfolio trasanction costs F=x(2:25); %Total earnings at every trading day t of the planning horizon for i=1:H port_incomes(i)=0; for j=1:numel(C) port_incomes(i)=port_incomes(i)+N(j)*cf(i,j); end end Georgios Bouzianis [Page 80] %Final Portfolio synthesis and characteristics j=1; l=1; for i=1:numel(b) if floor(b(i))==1 || round(b(i))==1 Budg_port(j)=Budg(i); % Investment in every incorporated bond C_port(j)=C(i); %Coupon N_port(j)=N(i); %Portion bought Md_port(j)=Md(i); %Maturity day Tr_port(j)=Tr(i); %Transaction cost exposure B_port(j)=B(i); %Price R_port(j)=R(i); %Yield D_port(j)=YD(i); %Duration Tick_port(j)=Tick(i); %Ticker Base_port(j)=Base(i); %Base sector_port(j)=sector(i); %Sector Conv_port(j)=YearConv(i); %Convexity index_port(j)=index(i); %Index from initial sample j=j+1; for k=1:H cf_port(k,l)=cf(k,i); %Payoff matrix end l=l+1; end end %Earnings from every bond in the portfolio for i=1:numel(C_port) port_bond_incomes(i)=0; for j=1:H port_bond_incomes(i)=port_bond_incomes(i)+N_port(i)*cf_port(j,i); end end Georgios Bouzianis [Page 81] B PROGRAMING CODE FOR THE PROPOSED METHODOLOGY WITH EMB E D D E D H U L L - W H I T E STOCHASTIC SHORT RAT E M O D E L . %% Obtaining Data M=xlsread('Data_ab',4,'p2:p3499'); %Bond maturity [num1,Tick2]=xlsread('Data_ab',4,'a2:a3499'); %Bond ticker [num2,Base2]=xlsread('Data_ab',4,'d2:d3499'); %Bond base [num3,sector2]=xlsread('Data_ab',4,'i2:i3499'); %Bond sector Rates=xlsread('MI1',1,'a22:a26'); %Initial term structure data=readtable('dates_ab.xlsx'); t=data(:,1); Maturity2=table2array(t); %Bond maturity day CouponRate2 =xlsread('Data_ab',4,'b2:b3499'); %Bond coupon j=1; H=5; %Planning horizon FV=100;%Face value %Excluding bonds with maturity more than one year and less than the five years %of the planning horizon for i=1:numel(M) if M(i)>=1 && M(i)<=H T(j)=M(i); Maturity(j)=Maturity2(i); CouponRate(j)=CouponRate2(i); com(j)=0.01; Tick(j)=Tick2(i); Base(j)=Base2(i); sector(j)=sector2(i); index(j)=i; j=j+1; end end %Implementation of Hull-White model and construction of the Hull-White trinomial lattice ValuationDate = '12/28/15'; StartDates = ValuationDate; VolDates = ['12/28/16';'12/28/17';'12/28/18';'12/28/19';'12/28/20']; VolCurve = 0.01; AlphaDates = '12/28/20'; AlphaCurve = 0.1; Compounding=-1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', VolDates,'Rates', Rates); Georgios Bouzianis [Page 82] HWVolSpec = hwvolspec(ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); TimeSpec = hwtimespec(ValuationDate, VolDates, Compounding); HWTree = hwtree(HWVolSpec, RateSpec, TimeSpec); treeviewer(HWTree) Settle = '12/28/15'; % Display the instrument portfolio [Price,PriceTree] = bondbyhw(HWTree, CouponRate/100, Settle, Maturity); treeviewer(PriceTree) %Investment policy adjustment (IPS) Su=300; %Maximum number of different bonds in the portfolio Sd=70; %Minimum number of different bonds in portfolio Nu=1000; %Maximum portion can be bought from every different bond Nd=700; %Minimum portion can be bought from every different bond Tu=20000; %Maximum numerical exposure in transaction costs %% Payoff matix for i=1:numel(CouponRate) for t=1:T(i) if t end end end %% Present value of liabilities for every state for i=1:H if i==1 s=1; elseif i==2 s=3; elseif i==3 s=5; elseif i==4 s=5; elseif i==5 s=5; Georgios Bouzianis [Page 83] end for j=1:s L(i,j)=5000000*rand; PV_L(i,j)=L(i,j)/(1+Rates(i)+(HWTree.FwdTree{i}(j)/100)); end end %% Obtaining scenarios (l) m=1; while m<=81 count=0; for i=1:H if i==1 y = 1; elseif i==2 || i==3 y = datasample(y:y+2,1,'Replace',false); else if y==1 y = datasample(y:y+2,1,'Replace',false); elseif y==5 y = datasample(y-2:y,1,'Replace',false); else y = datasample(y-1:y+1,1,'Replace',false); end end scenarios(m,i)=y; end for i=1:size(scenarios,1) for w=1:size(scenarios,1) if i~=w if scenarios(i,:)==scenarios(w,:) count=count+1; end end end end if count==0 m=m+1; else m=m; end end %% Present value of liabilities for every scenario for i=1:size(scenarios,1) PVL_l(i,1)=0; SUM_l(i,1)=0; Georgios Bouzianis [Page 84] for j=1:H PVL_l(i,1)=PVL_l(i)+PV_L(j,scenarios(i,j)); SUM_l(i,1)=SUM_l(i)+L(j,scenarios(i,j)); end end PVL_l_A=0; for i=1:81 PVL_l_A=PVL_l_A+(1/81)*PVL_l(i,1); end %% Problem Setting % Obj Creation obj(1,1)=1; for i=2:20 obj(i,1)=0; end j=1; for i=21:20+numel(CouponRate) obj(i,1)=(1+com(j))*PriceTree.PTree{1}(j); j=j+1; end for i=21+numel(CouponRate):20+2*numel(CouponRate) obj(i,1)=0; end %Binary Variables intcon=(21+numel(CouponRate)):(20+2*numel(CouponRate)); %linear Equality Constraints %Construction of the tables Aeq and Beq Aeq1=zeros(15,20); Aeq1(1,1)=(1+0.04); Aeq1(1,2)=-1; Aeq1(2,2)=1+Rates(1)+HWTree.FwdTree{1}(1); Aeq1(2,3)=-1*HWTree.Probs{1}(1); Aeq1(2,4)=-1*HWTree.Probs{1}(2); Aeq1(2,5)=-1*HWTree.Probs{1}(3); Aeq1(3,3)=1+Rates(2)+HWTree.FwdTree{2}(1); Aeq1(3,6)=-1*HWTree.Probs{2}(1,1); Aeq1(3,7)=-1*HWTree.Probs{2}(2,1); Aeq1(3,8)=-1*HWTree.Probs{2}(3,1); Aeq1(4,4)=1+Rates(2)+HWTree.FwdTree{2}(2); Aeq1(4,7)=-1*HWTree.Probs{2}(1,2); Aeq1(4,8)=-1*HWTree.Probs{2}(2,2); Georgios Bouzianis [Page 85] Aeq1(4,9)=-1*HWTree.Probs{2}(3,2); Aeq1(5,5)=1+Rates(2)+HWTree.FwdTree{2}(3); Aeq1(5,8)=-1*HWTree.Probs{2}(1,3); Aeq1(5,9)=-1*HWTree.Probs{2}(2,3); Aeq1(5,10)=-1*HWTree.Probs{2}(3,3); Aeq1(6,6)=1+Rates(3)+HWTree.FwdTree{3}(1); Aeq1(6,11)=-1*HWTree.Probs{3}(1,1); Aeq1(6,12)=-1*HWTree.Probs{3}(2,1); Aeq1(6,13)=-1*HWTree.Probs{3}(3,1); Aeq1(7,7)=1+Rates(3)+HWTree.FwdTree{3}(2); Aeq1(7,11)=-1*HWTree.Probs{3}(1,2); Aeq1(7,12)=-1*HWTree.Probs{3}(2,2); Aeq1(7,13)=-1*HWTree.Probs{3}(3,2); Aeq1(8,8)=1+Rates(3)+HWTree.FwdTree{3}(3); Aeq1(8,12)=-1*HWTree.Probs{3}(1,3); Aeq1(8,13)=-1*HWTree.Probs{3}(2,3); Aeq1(8,14)=-1*HWTree.Probs{3}(3,3); Aeq1(9,9)=1+Rates(3)+HWTree.FwdTree{3}(4); Aeq1(9,13)=-1*HWTree.Probs{3}(1,4); Aeq1(9,14)=-1*HWTree.Probs{3}(2,4); Aeq1(9,15)=-1*HWTree.Probs{3}(3,4); Aeq1(10,10)=1+Rates(3)+HWTree.FwdTree{3}(5); Aeq1(10,13)=-1*HWTree.Probs{3}(1,5); Aeq1(10,14)=-1*HWTree.Probs{3}(2,5); Aeq1(10,15)=-1*HWTree.Probs{3}(3,5); Aeq1(11,11)=1+Rates(4)+HWTree.FwdTree{4}(1); Aeq1(11,16)=-1*HWTree.Probs{4}(1,1); Aeq1(11,17)=-1*HWTree.Probs{4}(2,1); Aeq1(11,18)=-1*HWTree.Probs{4}(3,1); Aeq1(12,12)=1+Rates(4)+HWTree.FwdTree{4}(2); Aeq1(12,16)=-1*HWTree.Probs{4}(1,2); Aeq1(12,17)=-1*HWTree.Probs{4}(2,2); Aeq1(12,18)=-1*HWTree.Probs{4}(3,2); Aeq1(13,13)=1+Rates(4)+HWTree.FwdTree{4}(3); Aeq1(13,17)=-1*HWTree.Probs{4}(1,3); Aeq1(13,18)=-1*HWTree.Probs{4}(2,3); Aeq1(13,19)=-1*HWTree.Probs{4}(3,3); Aeq1(14,14)=1+Rates(4)+HWTree.FwdTree{4}(4); Aeq1(14,18)=-1*HWTree.Probs{4}(1,4); Aeq1(14,19)=-1*HWTree.Probs{4}(2,4); Aeq1(14,20)=-1*HWTree.Probs{4}(3,4); Aeq1(15,15)=1+Rates(4)+HWTree.FwdTree{4}(5); Aeq1(15,18)=-1*HWTree.Probs{4}(1,5); Aeq1(15,19)=-1*HWTree.Probs{4}(2,5); Aeq1(15,20)=-1*HWTree.Probs{4}(3,5); Georgios Bouzianis [Page 86] Aeq2(1,1:numel(CouponRate))=cf(1,1:numel(CouponRate)); Aeq2(2,1:numel(CouponRate))=cf(2,1:numel(CouponRate)); Aeq2(3,1:numel(CouponRate))=cf(3,1:numel(CouponRate)); Aeq2(4,1:numel(CouponRate))=cf(3,1:numel(CouponRate)); Aeq2(5,1:numel(CouponRate))=cf(3,1:numel(CouponRate)); Aeq2(6,1:numel(CouponRate))=cf(4,1:numel(CouponRate)); Aeq2(7,1:numel(CouponRate))=cf(4,1:numel(CouponRate)); Aeq2(8,1:numel(CouponRate))=cf(4,1:numel(CouponRate)); Aeq2(9,1:numel(CouponRate))=cf(4,1:numel(CouponRate)); Aeq2(10,1:numel(CouponRate))=cf(4,1:numel(CouponRate)); Aeq2(11,1:numel(CouponRate))=cf(5,1:numel(CouponRate)); Aeq2(12,1:numel(CouponRate))=cf(5,1:numel(CouponRate)); Aeq2(13,1:numel(CouponRate))=cf(5,1:numel(CouponRate)); Aeq2(14,1:numel(CouponRate))=cf(5,1:numel(CouponRate)); Aeq2(15,1:numel(CouponRate))=cf(5,1:numel(CouponRate)); Aeq3=zeros(15,numel(CouponRate)); Aeq=[Aeq1 Aeq2 Aeq3]; q=1; for j=1:(20+2*numel(CouponRate)) if j>20 && j<=20+numel(CouponRate) Aeq(16,j)=PriceTree.PTree{1}(q); q=q+1; else Aeq(16,j)=0; end end Beq(1,1)=L(1,1); Beq(2,1)=HWTree.Probs{1}(1)*L(2,1)+HWTree.Probs{1}(2)*L(2,2)+HWTree .Probs{1}(3)*L(2,3); Beq(3,1)=HWTree.Probs{2}(1,1)*L(3,1)+HWTree.Probs{2}(2,1)*L(3,2)+HW Tree.Probs{2}(3,1)*L(3,3); Beq(4,1)=HWTree.Probs{2}(1,2)*L(3,2)+HWTree.Probs{2}(2,2)*L(3,3)+HW Tree.Probs{2}(3,2)*L(3,4); Beq(5,1)=HWTree.Probs{2}(1,3)*L(3,3)+HWTree.Probs{2}(2,3)*L(3,4)+HW Tree.Probs{2}(3,3)*L(3,5); Beq(6,1)=HWTree.Probs{3}(1,1)*L(4,1)+HWTree.Probs{3}(2,1)*L(4,2)+HW Tree.Probs{3}(3,1)*L(4,3); Beq(7,1)=HWTree.Probs{3}(1,2)*L(4,1)+HWTree.Probs{3}(2,2)*L(4,2)+HW Tree.Probs{3}(3,2)*L(4,3); Beq(8,1)=HWTree.Probs{3}(1,3)*L(4,2)+HWTree.Probs{3}(2,3)*L(4,3)+HW Tree.Probs{3}(3,3)*L(4,4); Beq(9,1)=HWTree.Probs{3}(1,4)*L(4,3)+HWTree.Probs{3}(2,4)*L(4,4)+HW Tree.Probs{3}(3,4)*L(4,5); Beq(10,1)=HWTree.Probs{3}(1,5)*L(4,3)+HWTree.Probs{3}(2,5)*L(4,4)+H WTree.Probs{3}(3,5)*L(4,5); Georgios Bouzianis [Page 87] Beq(11,1)=HWTree.Probs{4}(1,1)*L(5,1)+HWTree.Probs{4}(2,1)*L(5,2)+H WTree.Probs{4}(3,1)*L(5,3); Beq(12,1)=HWTree.Probs{4}(1,2)*L(5,1)+HWTree.Probs{4}(2,2)*L(5,2)+H WTree.Probs{4}(3,2)*L(5,3); Beq(13,1)=HWTree.Probs{4}(1,3)*L(5,2)+HWTree.Probs{4}(2,3)*L(5,3)+H WTree.Probs{4}(3,3)*L(5,4); Beq(14,1)=HWTree.Probs{4}(1,4)*L(5,3)+HWTree.Probs{4}(2,4)*L(5,4)+H WTree.Probs{4}(3,4)*L(5,5); Beq(15,1)=HWTree.Probs{4}(1,5)*L(5,3)+HWTree.Probs{4}(2,5)*L(5,4)+H WTree.Probs{4}(3,5)*L(5,5); Beq(16,1)=PVL_l_A; %linear inequality constraints %Constructions of the tables Aineq and Bineq for i=1:numel(CouponRate) for j=1:numel(CouponRate) if i==j T1(i,j)=1; T2(i,j)=-Nu; T3(i,j)=-1; T4(i,j)=Nd; T5(i,j)=com(i)*PriceTree.PTree{1}(i); else T1(i,j)=0; T2(i,j)=0; T3(i,j)=0; T4(i,j)=0; T5(i,j)=0; end end end T6=zeros(numel(CouponRate)); T7=zeros(2,numel(CouponRate)); T8=ones(1,numel(CouponRate)); T9=-T8; T11=[T8;T9]; T10=zeros((3*numel(CouponRate)+2),20); T12=[T1 T2;T3 T4;T5 T6;T7 T11]; Aineq=[T10 T12]; for i=1:2*numel(CouponRate) Bineq(i)=0; end for i=2*numel(CouponRate)+1:3*numel(CouponRate) Bineq(i)=Tu; end Georgios Bouzianis [Page 88] Bineq(3*numel(CouponRate)+1)=Su; Bineq(3*numel(CouponRate)+2)=-Sd; Bineq=transpose(Bineq); %Bound constraints lb=zeros(20+2*numel(CouponRate),1); for i=1:20+2*numel(CouponRate) if i<=15 ub(i,1)=Inf; elseif i>=16 && i<=20 ub(i,1)=Inf; elseif i>=21 && i<=20+numel(CouponRate) ub(i,1)=Inf; else ub(i,1)=1; end end %% Problem Solution x=intlinprog(obj,intcon,Aineq,Bineq,Aeq,Beq,lb,ub); %Solution Table for i=1:numel(CouponRate) N(i)=x(20+i,1); b(i)=x(20+numel(CouponRate)+i); end for i=1:numel(CouponRate) Budg(i)=PriceTree.PTree{1}(i)*N(i); Tr(i)=com(i)*PriceTree.PTree{1}(i)*N(i); end Solution=sum(Budg)+sum(Tr)+x(1); %Total initial capital is required Total_N=sum(b); %Total different incorporated bonds Total_Tr=sum(Tr); %Total portfolio trasanction costs %Total earnings at every trading day t of the planning horizon for i=1:H port_incomes(i)=0; for j=1:numel(CouponRate) port_incomes(i)=port_incomes(i)+N(j)*cf(i,j); end end %Final Portfolio synthesis and characteristics j=1; l=1; for i=1:numel(b) Georgios Bouzianis [Page 89] if floor(b(i))==1 || round(b(i))==1 Budg_port(j)=Budg(i); C_port(j)=CouponRate(i); N_port(j)=N(i); Md_port(j)=Maturity(i); Tr_port(j)=Tr(i); B_port(j)=PriceTree.PTree{1}(i); Tick_port(j)=Tick(i); Base_port(j)=Base(i); sector_port(j)=sector(i); index_port(j)=index(i); j=j+1; for k=1:H cf_port(k,l)=cf(k,i); end l=l+1; end end for i=1:numel(C_port) port_bond_incomes(i)=0; for j=1:H port_bond_incomes(i)=port_bond_incomes(i)+N_port(i)*cf_port(j,i); end end for i=1:size(scenarios,1) SUM_f(i,1)=0; for j=1:H if j==1 SUM_f(i,1)=SUM_f(i)+x(2,1); elseif j==2 SUM_f(i,1)=SUM_f(i)+x(2+scenarios(i,j),1); elseif j==3 SUM_f(i,1)=SUM_f(i)+x(5+scenarios(i,j),1); elseif j==4 SUM_f(i,1)=SUM_f(i)+x(10+scenarios(i,j),1); elseif j==5 SUM_f(i,1)=SUM_f(i)+x(15+scenarios(i,j),1); end end end success=0; for i=1:81 if Solution <= SUM_l(i,1) success=success+1; 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