Practical Considerations in Managing Variable Annuities

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					Practical Considerations in Managing
          Variable Annuities
  Stochastic Modeling Symposium
               Toronto
                  By
            Thomas S.Y. Ho
             Sang Bin Lee
            Yoon Seok Choi

             April 3-4 2006
         Problem Statement
• Variable annuities (GMIB) have embedded
  capital market options
  – Equity options (equity put option)
  – Interest rate options (annuity value)
• Pricing of the options
  – Capital market method: Cost of replications of
    the options
  – Identifying the embedded options
• Practical considerations
                                                     2
      Outline of the Presentation
• The valuation model
• Pathwise immunization and LPS
• Practical considerations
  –   Computational efficiency
  –   Equity/rate correlations and cost of guarantees
  –   Product risks
  –   Cost benefit analysis
  –   Suboptimal exercise of options
  –   Alternative to equity funds


                                                        3
           A Model of GMIB
• Invest in equity/bond funds
• End of the accumulation period, a choice:
  the account value or a fixed annuity
• The guarantee is a put option to the
  insurer
• If the fund is an equity fund, it is an equity
  put option, where the “strike price” is a
  bond, subject to interest rate risks
                                                   4
     Capital Market Approach:
     Equity/Interest Rate Model
• 2-factor interest rate model
• Recombining lattice, orthogonal yield
  curve movements
• Fit the term structure of interest rates and
  volatilities (arbitrage-free)
• Combining lognormal and normal behavior
• Equity returns are lognormal with the
  instantaneous rate of return = short rate
                                             5
    Interest Rate Model
•




                          6
Variable Annuity and GMIB Models

• Policyholders annuitize when the annuity
  value exceeds the account value
• GMIB is the policyholder’s equity put
  option with a stochastic strike price based
  on the annuity value.
• Can we represent the GMIB as a portfolio
  of equity options and bond options?

                                                7
    GMIB model: Capital Market
       Valuation Approach




• dV = dS – fVdt
• Max[V(T), B(T, T*)]
• GMIB value = average of the pathwise
  values

                                         8
 Decomposition using Linear Path
        Space (LPS)
• The path space is a representation of all the
  possible scenarios
• The recombining lattice offers a “co-ordinate
  system” to represent the possible scenarios
• Structured sampling of the path space
  provides equivalent classes of the possible
  scenarios, and the lattice framework enables
  us to measure the size of the classes

                                                  9
Linear Path Space




                    10
 GMIB Value and the Correlations

• X-axis: correlation of
  equity returns with             -0.5   0.0    0.5
  steepening movement
• Y-axis: correlation      -0.5   7.49   11.98 14.90
  with the parallel
  movement                 0.0    10.87 14.69 17.46
• Negative correlation
  implies natural          0.5    13.13 16.79 19.59
  hedging

                                                       11
  GMIB Value and Correlations
• The correlation with the
  steepening effect has
  greater impact on the
  GMIB value
• The worst scenario: yield
  curve steepens, rates and
  equity value fall
• Negative correlation with
  the steepening movement
  lowers the probability of
  the worst scenario
                                12
       GMIB & Account Value
• Increase account
  value: GMIB value
  falls
• Impact of the
  correlation on GMIB
  is significant,
  particularly when the
  account value is high


                              13
    Effectiveness of the Pathwise
             Immunization
• GMIB pathwise
  values are plotted
  against the hedging
  portfolio pathwise
  values
• The fit is 99% R-
  square



                                    14
  Decomposition Solution: Pathwise
      Immunization Portfolio
    A           B               C               D                E                F            G              H              I
 Hedging                                    Regression                                                                     Dollar
              Strike        Fair Value                      t-statistics     Dollar value     Delta        Duration
Instrument                                   Coeffs.                                                                      Duration
  Cash                  -            1.00           3.32              4.97             3.32        0.00            0.00           0.00

Bond Call              10           57.20           -0.19         -10.20           -10.81          0.00           21.41     -231.43

Bond Call              20           50.90           -0.16         -10.25              -8.11        0.00           22.78     -184.87

Bond Call              30           44.61           -0.11         -10.33              -4.82        0.00           24.54     -118.25

Bond Call              40           38.31           -0.02            -6.51            -0.73        0.00           26.87       -19.57

Bond Call              50           32.01           0.14              9.20             4.41        0.00           30.13      132.72

Bond Call              60           25.71           0.42              9.71            10.84        0.00           34.98      379.22

Bond Call              80           13.94           0.33             12.35             4.62        0.00           50.85      234.92

Bond Call           100              5.42           0.28             12.95             1.54        0.00           69.42      107.12

Bond Call           120              1.53           0.17              8.69             0.26        0.00           87.96          22.57

Equity Put             40            0.14           1.56              4.04             0.22        -0.01          53.54          11.71

Equity Put          100              6.98           0.37              5.22             2.55        -0.16          32.86          83.86

Equity Put          150             22.55           0.51             14.88            11.40        -0.35          26.07      297.28

Replicating
                                                                                      14.69        -0.24          48.68      715.26
 Portfolio
  GMIB                                                                                14.69        -0.22          48.34      710.09

                                                                                                                                  15
        A Solution in Pricing
• Separate pricing from valuation
• Use the decomposition to identify the
  pricing strategy
• The solution is a combination of hedging
  and directional investment
• Product design to seek the optimal
  solution

                                             16
           Fixed Annuities
• Based in interest rate models
• Investment portfolio can be fixed income
• The embedded options would be interest
  rate options
• Decomposition based on yield curve
  scenarios


                                             17
    Computational Efficiency
• Extended LPS model
• Structured sampling approach
• Convergence




                                 18
Equity/rate correlations and cost of
            guarantees
• Correlations of equity returns to the yield
  curve movement can be incorporated
• This approach is important to GMIB
  valuation because it focuses on the worst
  scenario: steep yield curve, falling rates,
  and underperforming equity market



                                                19
             Product risks
• Lapse risk, mortality risk, partial
  withdrawal can be simulated as risk
  sources
• Distribution of the pathwise values would
  be higher
• Hedging would not less perfect resulting in
  lower hedged positions

                                            20
        Cost benefit analysis
• Pathwise values provide the present value
  and risk adjusted values by scenarios
• The distribution of the pathwise values
  post partial hedging provides a measure of
  the downside risks
• The market view can be incorporated in
  the risk and return tradeoff in the “optimal
  hedging”
                                             21
 Suboptimal exercise of options
• The exercise rule can be adjusted for
  inefficiency in exercising the options
• Valuation and hedging can be adjusted by the
  model results




                                                 22
    Alternative to equity funds
• Fixed income fund can be evaluated in the
  same framework
• Policyholders’ behavior can be simulated
  by the model resulting in the appropriate
  tradeoff of risk and returns




                                          23
Implications on the Product Design

• Depends on the insurer’s business model
• Market directions and balance sheet
  positions are relevant
• Correlations of the fund returns and the
  guarantees are important
• Deciding on the risk and return tradeoff


                                             24
             Conclusions
• Pricing of a product begins the
  determination of the building blocks of
  value of the guarantees
• Then determine the optimal combination of
  hedge cost and directional investment
• Business model and the portfolio structure
  must be taken into consideration

                                          25
                References
• Ho, Lee and Choi: “Practical Considerations in
  Managing Variable Annuities” Working Paper
  2006
• Ho and Mudavanhu: “Decomposing and
  Managing Multivariate Risks: the Case of
  Variable Annuities” Journal of Investment
  Management 2005
• Ho and Mudavanhu “Managing Stochastic
  Volatility Risk of Interest Rate Options: Key Rate
  Vega” working paper 2006
• Papers available at www.thomasho.com

                                                   26

				
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