Designing and pricing guarantee options in de fined contribution pension plans

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The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. For DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio of assets and we develop a discrete model that optimally selects the reference portfolio to minimise the cost of a guarantee. While the relation DB-DC is typically viewed as a binary one, the model can be used to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signalling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.
Transcript
  • 1. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Designing and pricing guarantee options in defined contribution pension plans Andrea Consiglio† Michele Tumminello† Stavros Zenios‡ †University of Palermo, IT ‡University of Cyprus, CY June 2016 6th International Conference of the Financial Engineering and Banking Society 1 / 27
  • 2. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Outline 1 Introduction 2 The Mathematics of Guarantee Options 3 The Optimization Model 4 Implementation and Results The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing 5 Conclusions 2 / 27
  • 3. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions Retirement plans are of two types: 3 / 27
  • 4. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions Retirement plans are of two types: DB -Defined benefits plans shift the risks to the provider, be it a corporate employer or future taxpayers DC -Defined contributions plans pass the risks to retirees. 3 / 27
  • 5. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions Retirement plans are of two types: DB -Defined benefits plans shift the risks to the provider, be it a corporate employer or future taxpayers DC -Defined contributions plans pass the risks to retirees. The retirement income must be “safe”: DC politically acceptable Encourage participation Increase savings Some type of guarantee is needed and the success of DC hinges upon the design of appropriate guarantees. 3 / 27
  • 6. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions / 2 Difficulty in designing the guarantee does not stop to the definition of its mechanism. Guarantees provisions are written on asset portfolio, and these decisions need to be “optimised for their safety and performance”. (European Commission 2012) 4 / 27
  • 7. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions / 2 Difficulty in designing the guarantee does not stop to the definition of its mechanism. Guarantees provisions are written on asset portfolio, and these decisions need to be “optimised for their safety and performance”. (European Commission 2012) Given the complex interactions of financial, economic and demographic risks, a guarantee may fail as much as a “defined benefit” may be modified by government legislation. (World Bank 2000). 4 / 27
  • 8. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions / 3 Our contributions: 1 Modelling DC plans with alternative guarantee options 5 / 27
  • 9. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Motivations and contributions / 3 Our contributions: 1 Modelling DC plans with alternative guarantee options 2 Optimizing asset allocation to facilitate risk sharing 5 / 27
  • 10. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions DB vs DC pension plans in the OECD countries 0 20 40 60 80 100 C hileR epublicEstonia FranceG reeceH ungary PolandR epublicSloveniaD enm ark Italystralia (1)M exicoaland (1)Iceland SpainStates (2) Turkey Israel Koreabourg (3)Portugal anada (2)G erm anyFinlandN orwayitzerland Defined Contribution Defined Benefit / Hybrid Mixed 6 / 27
  • 11. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Type of guarantees Rung 1. Money-safe accounts: guarantee the contribution, (nominal or real value) upon retirement 7 / 27
  • 12. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Type of guarantees Rung 1. Money-safe accounts: guarantee the contribution, (nominal or real value) upon retirement Rung 2. Guaranteed return: guarantee fixed rate of return on contribution, upon retirement 7 / 27
  • 13. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Type of guarantees Rung 1. Money-safe accounts: guarantee the contribution, (nominal or real value) upon retirement Rung 2. Guaranteed return: guarantee fixed rate of return on contribution, upon retirement Rung 3. Guaranteed return: equal to some industry average upon retirement 7 / 27
  • 14. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Type of guarantees Rung 1. Money-safe accounts: guarantee the contribution, (nominal or real value) upon retirement Rung 2. Guaranteed return: guarantee fixed rate of return on contribution, upon retirement Rung 3. Guaranteed return: equal to some industry average upon retirement Rung 4. Guaranteed return: for each time period until retirement 7 / 27
  • 15. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Type of guarantees Rung 1. Money-safe accounts: guarantee the contribution, (nominal or real value) upon retirement Rung 2. Guaranteed return: guarantee fixed rate of return on contribution, upon retirement Rung 3. Guaranteed return: equal to some industry average upon retirement Rung 4. Guaranteed return: for each time period until retirement Rung 5. Guaranteed income past retirement 7 / 27
  • 16. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The probabilistic structure Tt210 NTNtN2N1N0 8 / 27
  • 17. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The basic minimum guarantee option Model the minimum guarantee provision as an option written on a reference fund, with value An at time T for each n ∈ NT ; 9 / 27
  • 18. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The basic minimum guarantee option Model the minimum guarantee provision as an option written on a reference fund, with value An at time T for each n ∈ NT ; We assume a closed fund with initial contribution L0 and regulatory equity requirement E0 = (1 − α)A0, α < 1 L0 = αA0 and A0 = L0 + E0 9 / 27
  • 19. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The basic minimum guarantee option Model the minimum guarantee provision as an option written on a reference fund, with value An at time T for each n ∈ NT ; We assume a closed fund with initial contribution L0 and regulatory equity requirement E0 = (1 − α)A0, α < 1 L0 = αA0 and A0 = L0 + E0 A0 is invested in a reference portfolio with proportions xj , and j∈J xj = 1. 9 / 27
  • 20. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The dynamics of the asset and liability account Given a family of stochastic processes {Rt}t∈T defined as a J-dimensional vector of returns, Rn ≡ R1 n , . . . , RJ n 10 / 27
  • 21. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The dynamics of the asset and liability account Given a family of stochastic processes {Rt}t∈T defined as a J-dimensional vector of returns, Rn ≡ R1 n , . . . , RJ n The asset account for each n ∈ N{0} is An = Ap(n)eRA n where RA n = j∈J xj Rj n 10 / 27
  • 22. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The dynamics of the asset and liability account Given a family of stochastic processes {Rt}t∈T defined as a J-dimensional vector of returns, Rn ≡ R1 n , . . . , RJ n The asset account for each n ∈ N{0} is An = Ap(n)eRA n where RA n = j∈J xj Rj n The liability account is Ln = Lp(n) exp g + max δRA n − g, 0 10 / 27
  • 23. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The objective function We assume that shareholders cover shortfalls 11 / 27
  • 24. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The objective function We assume that shareholders cover shortfalls A rational strategy for the fund manager is to minimize the expected value of shortfalls: Γ = e−rT n∈NT qn max (Ln − An, 0)] where the qn are the risk neutral probabilities. 11 / 27
  • 25. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The objective function We assume that shareholders cover shortfalls A rational strategy for the fund manager is to minimize the expected value of shortfalls: Γ = e−rT n∈NT qn max (Ln − An, 0)] where the qn are the risk neutral probabilities. The cost of the guarantee is the cost of a put option written on the value of the asset An with a stochastic strike price Ln 11 / 27
  • 26. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Bilinear constraints Denote by wn and zn the final cumulative returns of the asset and liability accounts An and Ln. For all n ∈ NT , we have: wn = i∈P(n) RA i , zn = i∈P(n) g + max δRA i − g, 0 , 12 / 27
  • 27. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Bilinear constraints Denote by wn and zn the final cumulative returns of the asset and liability accounts An and Ln. For all n ∈ NT , we have: wn = i∈P(n) RA i , zn = i∈P(n) g + max δRA i − g, 0 , Discontinuous nonlinear programming problem (DNLP) 12 / 27
  • 28. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Bilinear constraints / 2 Introduce the set of equations to define the max operator: δRA n − g = ε+ n − ε− n , ε+ n ε− n = 0, ε+ n , ε− n ≥ 0. 13 / 27
  • 29. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Bilinear constraints / 3 Similarly, for the max operator in the objective function: ln α + zn − wn = H+ n − H− n , H+ n H− n = 0, H+ n , H− n ≥ 0, 14 / 27
  • 30. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Bilinear constraints / 3 Similarly, for the max operator in the objective function: ln α + zn − wn = H+ n − H− n , H+ n H− n = 0, H+ n , H− n ≥ 0, The cost of the guarantee becomes: Γ(x1, x2, . . . , xJ) = e−rT A0 n∈NT qnewn eH+ n − 1 . 14 / 27
  • 31. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Convex Put Option Model (CPOM) Minimize x1,...,xJ e−rT A0 n∈NT qnewn eH+ n − 1 (1) s.t. ln α + zn − wn = H+ n − H− n , n ∈ NT , (2) δRA n − g = ε+ n − ε− n , n ∈ N{0}, (3) zn = g T + i∈P(n) ε+ i , n ∈ N{0}, (4) wn = i∈P(n) RA i , n ∈ N{0}, (5) RA n = j∈J xj Rj n, n ∈ N{0}, (6) j∈J xj = 1 (7) 15 / 27
  • 32. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Lemma Let us assume that x∗ 1 , x∗ 2 , . . . , x∗ J is an optimal portfolio choice for the CPOM. Then, H+ n H− n = 0, for all n ∈ NT . 16 / 27
  • 33. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Lemma Let us assume that x∗ 1 , x∗ 2 , . . . , x∗ J is an optimal portfolio choice for the CPOM. Then, H+ n H− n = 0, for all n ∈ NT . Lemma Let us assume that x∗ 1 , x∗ 2 , . . . , x∗ J is an optimal portfolio choice for CPOM. Then, it exists a non empty subset of nodes B ⊂ N such that ∀n ∈ B we have ε+ n ε− n = 0. 16 / 27
  • 34. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Corollary Let x∗ 1 , x∗ 2 , . . . , x∗ J be an optimal portfolio choice for the CPOM, if ε+ k ε− k > 0, for any k ∈ N, then it exists n ∈ NT such that k ∈ P(n) and H− n > 0 or H− n = H+ n = 0. 17 / 27
  • 35. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Corollary Let x∗ 1 , x∗ 2 , . . . , x∗ J be an optimal portfolio choice for the CPOM, if ε+ k ε− k > 0, for any k ∈ N, then it exists n ∈ NT such that k ∈ P(n) and H− n > 0 or H− n = H+ n = 0. Theorem Let x∗ 1 , x∗ 2 , . . . , x∗ J be an optimal portfolio choice for the CPOM, with optimal objective value Γ∗. Let x∗∗ 1 , x∗∗ 2 , . . . , x∗∗ J be an optimal portfolio choice of the NCPOM, with optimal objective value Γ∗∗. Then Γ∗ = Γ∗∗ . 17 / 27
  • 36. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Experiments setup Experiments for T = 30 yrs and J = 12 financial asset indices. 18 / 27
  • 37. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Experiments setup Experiments for T = 30 yrs and J = 12 financial asset indices. J.P. Morgan aggregate indices of sovereign bonds issued by European countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 and BONDS-7-10). Salomon indices for corporate bond classes (CORP-FIN, CORP-ENE and CORP-INS). Morgan Stanley Capital International Global for stock market indices (STOCKS-EMU, STOCKS-EX-EMU, STOCKS-PAC, STOCKS-EMER, and STOCKS-NA) 18 / 27
  • 38. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Experiments setup Experiments for T = 30 yrs and J = 12 financial asset indices. J.P. Morgan aggregate indices of sovereign bonds issued by European countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 and BONDS-7-10). Salomon indices for corporate bond classes (CORP-FIN, CORP-ENE and CORP-INS). Morgan Stanley Capital International Global for stock market indices (STOCKS-EMU, STOCKS-EX-EMU, STOCKS-PAC, STOCKS-EMER, and STOCKS-NA) Data from FINLIB (Zenios, Practical Financial Optimization. Decision making for financial engineers, Blackwell-Wiley Finance, 2007) 18 / 27
  • 39. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Experiments setup Experiments for T = 30 yrs and J = 12 financial asset indices. J.P. Morgan aggregate indices of sovereign bonds issued by European countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 and BONDS-7-10). Salomon indices for corporate bond classes (CORP-FIN, CORP-ENE and CORP-INS). Morgan Stanley Capital International Global for stock market indices (STOCKS-EMU, STOCKS-EX-EMU, STOCKS-PAC, STOCKS-EMER, and STOCKS-NA) Data from FINLIB (Zenios, Practical Financial Optimization. Decision making for financial engineers, Blackwell-Wiley Finance, 2007) Simulate risk-neutral process of asset returns using standard Montecarlo approach 18 / 27
  • 40. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Experiments setup Experiments for T = 30 yrs and J = 12 financial asset indices. J.P. Morgan aggregate indices of sovereign bonds issued by European countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 and BONDS-7-10). Salomon indices for corporate bond classes (CORP-FIN, CORP-ENE and CORP-INS). Morgan Stanley Capital International Global for stock market indices (STOCKS-EMU, STOCKS-EX-EMU, STOCKS-PAC, STOCKS-EMER, and STOCKS-NA) Data from FINLIB (Zenios, Practical Financial Optimization. Decision making for financial engineers, Blackwell-Wiley Finance, 2007) Simulate risk-neutral process of asset returns using standard Montecarlo approach Model implemented simulating fan of 1000 risk-neutral paths 18 / 27
  • 41. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Minimum guarantee rate (g) Minimumguaranteecost 0.0 0.5 1.0 1.5 2.0 2.5 0 0.01 0.02 0.03 0.04 0.05 0.7 0 0.01 0.02 0.03 0.04 0.05 0.8 0 0.01 0.02 0.03 0.04 0.05 1 alpha 0.7 0.8 1 19 / 27
  • 42. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing delta Minimumguaranteecost 0.0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 alpha 0.7 1 20 / 27
  • 43. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Minimum guarantee rate (g) Minimumguaranteecost 0.0 0.5 1.0 1.5 2.0 2.5 0 0.01 0.02 0.03 0.04 0.05 alpha 0.7 0.8 0.9 1 21 / 27
  • 44. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Portfolio percentages BONDS_1_3 CORP_FIN CORP_INS STOCKS_EMER STOCKS_EMU STOCKS_PAC 0.0 0.2 0.4 0.6 0.8 0.7 0.0 0.2 0.4 0.6 0.8 0.9 g 0 0.01 0.03 0.05 22 / 27
  • 45. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Years Yearlyreturns(%) −40 −20 0 20 40 1995 2000 2005 2010 10−Year T−Bond 3−Month T−Bill Portfolio 1995 2000 2005 2010 −40 −20 0 20 40 S&P500 23 / 27
  • 46. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Minimum guarantee rate (g) Minimumguaranteecost 0 1 2 3 4 5 0 0.01 0.02 0.03 0.04 0.05 Benchmark portfolio 0 0.01 0.02 0.03 0.04 0.05 Optimal reference portfolio alpha 0.7 0.8 0.9 1 24 / 27
  • 47. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Risk Sharing Cost of guarantee can be used to set risk sharing premia. E.g., for g=3% and α = 1 (zero equity) we have cost 0.84. 25 / 27
  • 48. Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions The effect of policy parameters on the cost of the guarantee Portfolio composition, moral hazard and risk sharing Risk Sharing Cost of guarantee can be used to set risk sharing premia. E.g., for g=3% and α = 1 (zero equity) we have cost 0.84. Case 1. The pension fun
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