designing and pricing guarantee options in defined contribution pension plans

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Introduction The Mathematics of Guarantee Options The Optimization Model Implementation and Results Conclusions Designing and pricing guarantee options in defined contribution pension plans Andrea ConsiglioMichele TumminelloStavros ZeniosUniversity of Palermo, IT University of Cyprus, CY June 2016 6th International Conference of the Financial Engineering and Banking Society 1 / 27

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  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Designing and pricing guarantee options in definedcontribution pension plans

    Andrea Consiglio Michele Tumminello Stavros Zenios

    University of Palermo, ITUniversity of Cyprus, CY

    June 20166th International Conference

    of the Financial Engineering and Banking Society

    1 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Outline

    1 Introduction

    2 The Mathematics of Guarantee Options

    3 The Optimization Model

    4 Implementation and ResultsThe effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    5 Conclusions

    2 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions

    Retirement plans are of two types:

    DB -Defined benefits plans shift the risks to the provider, be ita corporate employer or future taxpayersDC -Defined contributions plans pass the risks to retirees.

    The retirement income must be safe:

    DC politically acceptableEncourage participationIncrease savings

    Some type of guarantee is needed and the success ofDC hinges upon the design of appropriate guarantees.

    3 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions

    Retirement plans are of two types:

    DB -Defined benefits plans shift the risks to the provider, be ita corporate employer or future taxpayersDC -Defined contributions plans pass the risks to retirees.

    The retirement income must be safe:

    DC politically acceptableEncourage participationIncrease savings

    Some type of guarantee is needed and the success ofDC hinges upon the design of appropriate guarantees.

    3 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions

    Retirement plans are of two types:

    DB -Defined benefits plans shift the risks to the provider, be ita corporate employer or future taxpayersDC -Defined contributions plans pass the risks to retirees.

    The retirement income must be safe:

    DC politically acceptableEncourage participationIncrease savings

    Some type of guarantee is needed and the success ofDC hinges upon the design of appropriate guarantees.

    3 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions / 2

    Difficulty in designing the guarantee does not stop to thedefinition of its mechanism.

    Guarantees provisions are written on asset portfolio, and thesedecisions need to be optimised for their safety andperformance. (European Commission 2012)

    Given the complex interactions of financial, economic anddemographic risks, a guarantee may fail as much as a definedbenefit may be modified by government legislation.(World Bank 2000).

    4 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions / 2

    Difficulty in designing the guarantee does not stop to thedefinition of its mechanism.

    Guarantees provisions are written on asset portfolio, and thesedecisions need to be optimised for their safety andperformance. (European Commission 2012)

    Given the complex interactions of financial, economic anddemographic risks, a guarantee may fail as much as a definedbenefit may be modified by government legislation.(World Bank 2000).

    4 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Motivations and contributions / 3

    Our contributions:1 Modelling DC plans with alternative guarantee options2 Optimizing asset allocation to facilitate risk sharing

    5 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    DB vs DC pension plans in the OECD countries

    0

    20

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    100

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    Fran

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    Slov

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    Slov

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    Denm

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    Italy

    Austr

    alia

    (1)

    Mex

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    )

    Icelan

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    Unite

    d St

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    da (2

    )

    Germ

    any

    Finla

    nd

    Norw

    ay

    Switz

    erlan

    d

    Defined Contribution Defined Benefit / Hybrid Mixed

    6 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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 oncontribution, upon retirement

    Rung 3. Guaranteed return: equal to some industry averageupon retirement

    Rung 4. Guaranteed return: for each time period untilretirement

    Rung 5. Guaranteed income past retirement

    7 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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 oncontribution, upon retirement

    Rung 3. Guaranteed return: equal to some industry averageupon retirement

    Rung 4. Guaranteed return: for each time period untilretirement

    Rung 5. Guaranteed income past retirement

    7 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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 oncontribution, upon retirement

    Rung 3. Guaranteed return: equal to some industry averageupon retirement

    Rung 4. Guaranteed return: for each time period untilretirement

    Rung 5. Guaranteed income past retirement

    7 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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 oncontribution, upon retirement

    Rung 3. Guaranteed return: equal to some industry averageupon retirement

    Rung 4. Guaranteed return: for each time period untilretirement

    Rung 5. Guaranteed income past retirement

    7 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation 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 oncontribution, upon retirement

    Rung 3. Guaranteed return: equal to some industry averageupon retirement

    Rung 4. Guaranteed return: for each time period untilretirement

    Rung 5. Guaranteed income past retirement

    7 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The probabilistic structure

    Tt210

    NTNtN2N1N0

    8 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The basic minimum guarantee option

    Model the minimum guarantee provision as an option writtenon a reference fund, with value An at time T for each n NT ;

    We assume a closed fund with initial contribution L0 andregulatory equity requirement E0 = (1 )A0, < 1

    L0 = A0 and A0 = L0 + E0

    A0 is invested in a reference portfolio with proportions xj , andjJ xj = 1.

    9 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The basic minimum guarantee option

    Model the minimum guarantee provision as an option writtenon a reference fund, with value An at time T for each n NT ;We assume a closed fund with initial contribution L0 andregulatory equity requirement E0 = (1 )A0, < 1

    L0 = A0 and A0 = L0 + E0

    A0 is invested in a reference portfolio with proportions xj , andjJ xj = 1.

    9 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The basic minimum guarantee option

    Model the minimum guarantee provision as an option writtenon a reference fund, with value An at time T for each n NT ;We assume a closed fund with initial contribution L0 andregulatory equity requirement E0 = (1 )A0, < 1

    L0 = A0 and A0 = L0 + E0

    A0 is invested in a reference portfolio with proportions xj , andjJ xj = 1.

    9 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The dynamics of the asset and liability account

    Given a family of stochastic processes {Rt}tT defined as aJ-dimensional vector of returns, Rn

    (R1n , . . . ,R

    Jn

    )

    The asset account for each n N\{0} is

    An = Ap(n)eRAn

    whereRAn =

    jJ

    xjRjn

    The liability account is Ln = Lp(n) exp[g + max

    (RAn g , 0

    )]

    10 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The dynamics of the asset and liability account

    Given a family of stochastic processes {Rt}tT defined as aJ-dimensional vector of returns, Rn

    (R1n , . . . ,R

    Jn

    )The asset account for each n N\{0} is

    An = Ap(n)eRAn

    whereRAn =

    jJ

    xjRjn

    The liability account is Ln = Lp(n) exp[g + max

    (RAn g , 0

    )]

    10 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The dynamics of the asset and liability account

    Given a family of stochastic processes {Rt}tT defined as aJ-dimensional vector of returns, Rn

    (R1n , . . . ,R

    Jn

    )The asset account for each n N\{0} is

    An = Ap(n)eRAn

    whereRAn =

    jJ

    xjRjn

    The liability account is Ln = Lp(n) exp[g + max

    (RAn g , 0

    )]10 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The objective function

    We assume that shareholders cover shortfalls

    A rational strategy for the fund manager is to minimize theexpected value of shortfalls:

    = erTnNT

    qn max (Ln An, 0)]

    where the qn are the risk neutral probabilities.

    The cost of the guarantee is the cost of a put optionwritten on the value of the asset An with a stochasticstrike price Ln

    11 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The objective function

    We assume that shareholders cover shortfalls

    A rational strategy for the fund manager is to minimize theexpected value of shortfalls:

    = erTnNT

    qn max (Ln An, 0)]

    where the qn are the risk neutral probabilities.

    The cost of the guarantee is the cost of a put optionwritten on the value of the asset An with a stochasticstrike price Ln

    11 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The objective function

    We assume that shareholders cover shortfalls

    A rational strategy for the fund manager is to minimize theexpected value of shortfalls:

    = erTnNT

    qn max (Ln An, 0)]

    where the qn are the risk neutral probabilities.

    The cost of the guarantee is the cost of a put optionwritten on the value of the asset An with a stochasticstrike price Ln

    11 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Bilinear constraints

    Denote by wn and zn the final cumulative returns of the assetand liability accounts An and Ln.For all n NT , we have:

    wn =

    iP(n)

    RAi ,

    zn =

    iP(n)

    g + max(RAi g , 0

    ),

    Discontinuous nonlinear programming problem (DNLP)

    12 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Bilinear constraints

    Denote by wn and zn the final cumulative returns of the assetand liability accounts An and Ln.For all n NT , we have:

    wn =

    iP(n)

    RAi ,

    zn =

    iP(n)

    g + max(RAi g , 0

    ),

    Discontinuous nonlinear programming problem (DNLP)

    12 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Bilinear constraints / 2

    Introduce the set of equations to define the max operator:

    RAn g = +n n ,+n

    n = 0,

    +n , n 0.

    13 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Bilinear constraints / 3

    Similarly, for the max operator in the objective function:

    ln + zn wn = H+n Hn ,H+n H

    n = 0,

    H+n ,Hn 0,

    The cost of the guarantee becomes:

    (x1, x2, . . . , xJ) = erTA0

    nNT

    qnewn(eH

    +n 1

    ).

    14 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Bilinear constraints / 3

    Similarly, for the max operator in the objective function:

    ln + zn wn = H+n Hn ,H+n H

    n = 0,

    H+n ,Hn 0,

    The cost of the guarantee becomes:

    (x1, x2, . . . , xJ) = erTA0

    nNT

    qnewn(eH

    +n 1

    ).

    14 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Convex Put Option Model (CPOM)

    Minimizex1,...,xJ

    erTA0nNT

    qnewn(eH

    +n 1

    )(1)

    s.t.

    ln + zn wn = H+n Hn , n NT , (2)RAn g = +n n , n N\{0}, (3)

    zn = g T +

    iP(n)

    +i , n N\{0}, (4)

    wn =

    iP(n)

    RAi , n N\{0}, (5)

    RAn =jJ

    xjRjn, n N\{0}, (6)

    jJ

    xj = 1 (7)

    15 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Lemma

    Let us assume that x1 , x2 , . . . , 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 x1 , x2 , . . . , x

    J is an optimal portfolio choice for

    CPOM. Then, it exists a non empty subset of nodes B N suchthat n B we have

    +n n = 0.

    16 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Lemma

    Let us assume that x1 , x2 , . . . , 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 x1 , x2 , . . . , x

    J is an optimal portfolio choice for

    CPOM. Then, it exists a non empty subset of nodes B N suchthat n B we have

    +n n = 0.

    16 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Corollary

    Let x1 , x2 , . . . , x

    J be an optimal portfolio choice for the CPOM, if

    +k k > 0, for any k N , then it exists n NT such that

    k P(n) andHn > 0 or H

    n = H

    +n = 0.

    Theorem

    Let x1 , x2 , . . . , x

    J be an optimal portfolio choice for the CPOM,

    with optimal objective value . Let x1 , x2 , . . . , x

    J be an

    optimal portfolio choice of the NCPOM, with optimal objectivevalue . Then

    = .

    17 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Corollary

    Let x1 , x2 , . . . , x

    J be an optimal portfolio choice for the CPOM, if

    +k k > 0, for any k N , then it exists n NT such that

    k P(n) andHn > 0 or H

    n = H

    +n = 0.

    Theorem

    Let x1 , x2 , . . . , x

    J be an optimal portfolio choice for the CPOM,

    with optimal objective value . Let x1 , x2 , . . . , x

    J be an

    optimal portfolio choice of the NCPOM, with optimal objectivevalue . Then

    = .

    17 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 byEuropean countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 andBONDS-7-10). Salomon indices for corporate bond classes(CORP-FIN, CORP-ENE and CORP-INS). Morgan StanleyCapital 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 makingfor financial engineers, Blackwell-Wiley Finance, 2007)Simulate risk-neutral process of asset returns using standardMontecarlo approachModel implemented simulating fan of 1000 risk-neutral paths

    18 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 byEuropean countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 andBONDS-7-10). Salomon indices for corporate bond classes(CORP-FIN, CORP-ENE and CORP-INS). Morgan StanleyCapital 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 makingfor financial engineers, Blackwell-Wiley Finance, 2007)Simulate risk-neutral process of asset returns using standardMontecarlo approachModel implemented simulating fan of 1000 risk-neutral paths

    18 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 byEuropean countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 andBONDS-7-10). Salomon indices for corporate bond classes(CORP-FIN, CORP-ENE and CORP-INS). Morgan StanleyCapital 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 makingfor financial engineers, Blackwell-Wiley Finance, 2007)

    Simulate risk-neutral process of asset returns using standardMontecarlo approachModel implemented simulating fan of 1000 risk-neutral paths

    18 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 byEuropean countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 andBONDS-7-10). Salomon indices for corporate bond classes(CORP-FIN, CORP-ENE and CORP-INS). Morgan StanleyCapital 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 makingfor financial engineers, Blackwell-Wiley Finance, 2007)Simulate risk-neutral process of asset returns using standardMontecarlo approach

    Model implemented simulating fan of 1000 risk-neutral paths

    18 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 byEuropean countries (BONDS-1-3, BONDS-3-5, BONDS-5-7 andBONDS-7-10). Salomon indices for corporate bond classes(CORP-FIN, CORP-ENE and CORP-INS). Morgan StanleyCapital 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 makingfor financial engineers, Blackwell-Wiley Finance, 2007)Simulate risk-neutral process of asset returns using standardMontecarlo approachModel implemented simulating fan of 1000 risk-neutral paths

    18 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    Minimum guarantee rate (g)

    Min

    imum

    gua

    rant

    ee c

    ost

    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

    alpha0.7 0.8 1

    19 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    delta

    Min

    imum

    gua

    rant

    ee c

    ost

    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

    alpha0.7 1

    20 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    Minimum guarantee rate (g)

    Min

    imum

    gua

    rant

    ee c

    ost

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0

    0.01

    0.02

    0.03

    0.04

    0.05

    alpha0.7 0.8 0.9 1

    21 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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

    g0 0.01 0.03 0.05

    22 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    Years

    Year

    ly r

    etur

    ns (

    %)

    40

    20

    0

    20

    40

    1995 2000 2005 2010

    10Year TBond 3Month TBill

    Portfolio

    1995 2000 2005 2010

    40

    20

    0

    20

    40S&P500

    23 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    Minimum guarantee rate (g)

    Min

    imum

    gua

    rant

    ee c

    ost

    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

    alpha0.7 0.8 0.9 1

    24 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 = 1 (zero equity) we have cost 0.84.

    Case 1. The pension fund bears the risk of the guarantee. Thebeneficiary will pay 1.84 and will get a return only on 1 euro.

    Case 2. Risk sharing between the beneficiary and a thirdparty. For example, the cost of sharing consist in paying 0.3 ofequity ( = 0.7), and investing it in the reference portfolio.The cost of the guarantee is 0.09.

    25 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 = 1 (zero equity) we have cost 0.84.

    Case 1. The pension fund bears the risk of the guarantee. Thebeneficiary will pay 1.84 and will get a return only on 1 euro.

    Case 2. Risk sharing between the beneficiary and a thirdparty. For example, the cost of sharing consist in paying 0.3 ofequity ( = 0.7), and investing it in the reference portfolio.The cost of the guarantee is 0.09.

    25 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    The effect of policy parameters on the cost of the guaranteePortfolio 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 = 1 (zero equity) we have cost 0.84.

    Case 1. The pension fund bears the risk of the guarantee. Thebeneficiary will pay 1.84 and will get a return only on 1 euro.

    Case 2. Risk sharing between the beneficiary and a thirdparty. For example, the cost of sharing consist in paying 0.3 ofequity ( = 0.7), and investing it in the reference portfolio.The cost of the guarantee is 0.09.

    25 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Concluding remarks

    A general and computationally tractable model for pricing thecost of alternative embedded guarantee options in DC pensionfunds.

    Model to determine the asset allocation choice that is optimalfor a given guarantee, in that it minimizes the cost of theguarantee.

    Can be used to benchmark existing portfolios by applying it totest portfolios of State and local government pension funds.

    Can be used to calculate risk premia for risk sharing.

    26 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Concluding remarks

    A general and computationally tractable model for pricing thecost of alternative embedded guarantee options in DC pensionfunds.

    Model to determine the asset allocation choice that is optimalfor a given guarantee, in that it minimizes the cost of theguarantee.

    Can be used to benchmark existing portfolios by applying it totest portfolios of State and local government pension funds.

    Can be used to calculate risk premia for risk sharing.

    26 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Concluding remarks

    A general and computationally tractable model for pricing thecost of alternative embedded guarantee options in DC pensionfunds.

    Model to determine the asset allocation choice that is optimalfor a given guarantee, in that it minimizes the cost of theguarantee.

    Can be used to benchmark existing portfolios by applying it totest portfolios of State and local government pension funds.

    Can be used to calculate risk premia for risk sharing.

    26 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Concluding remarks

    A general and computationally tractable model for pricing thecost of alternative embedded guarantee options in DC pensionfunds.

    Model to determine the asset allocation choice that is optimalfor a given guarantee, in that it minimizes the cost of theguarantee.

    Can be used to benchmark existing portfolios by applying it totest portfolios of State and local government pension funds.

    Can be used to calculate risk premia for risk sharing.

    26 / 27

  • IntroductionThe Mathematics of Guarantee Options

    The Optimization ModelImplementation and Results

    Conclusions

    Reference

    A. Consiglio, M. Tumminlello and S.A. Zenios, Designing andpricing guarantee options in defined contributions pension plans,Insurance: Mathematics and Economics, 65:267279, 2015.

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    IntroductionThe Mathematics of Guarantee OptionsThe Optimization ModelImplementation and ResultsThe effect of policy parameters on the cost of the guaranteePortfolio composition, moral hazard and risk sharing

    Conclusions