Modeling Partial Greeks of Variable Annuities with Dependence

Modeling Partial Greeks of Variable Annuities with Dependence Emiliano A. Valdez joint work with Guojun Gan University of Connecticut Recent Developments in Dependence Modeling with Applications in Finance and Insurance The Island of Aegina, Greece 12 May 2017 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 1 / 32

Outline of work This presentation is based on a collection of work: G. Gan and E.A. Valdez, Regression Modeling for the Valuation of Large Variable Annuity Portfolios, 2016, submitted to North American Actuarial Journal G. Gan and E.A. Valdez, An Empirical Comparison of Some Experimental Designs for the Valuation of Large Variable Annuity Portfolios, 2016, Dependence Modeling G. Gan and E.A. Valdez, Modeling Partial Greeks of Variable Annuities with Dependence, 2017, submitted in Insurance: Mathematics and Economics This collection of work tackles the issues related to efficient valuation of large variable annuity portfolios: valuation of VA products present some computational challenges Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 2 / 32

What is a variable annuity? A variable annuity is a retirement product, offered by an insurance company, that gives you the option to select from a variety of investment funds and then pays you retirement income, the amount of which will depend on the investment performance of funds you choose. Premiums Policyholder Guarantee Payments Separate Account Charges Withdrawals/ Payments General Account Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 3 / 32

Variable annuities come with guarantees GMxB GMDB GMLB GMIB GMMB GMWB Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 4 / 32

Insurance companies have to make guarantee payments under bad market conditions Example (An immediate variable annuity with GMWB) Total investment and initial benefits base: $100,000 Maximum annual withdrawal: $8,000 Policy Year INV Return Fund Before WD Annual WD Fund After WD Remaining Benefit Guarantee CF 1-10% 90,000 8,000 82,000 92,000 0 2 10% 90,200 8,000 82,200 84,000 0 3-30% 57,540 8,000 49,540 76,000 0 4-30% 34,678 8,000 26,678 68,000 0 5-10% 24,010 8,000 16,010 60,000 0 6-10% 14,409 8,000 6,409 52,000 0 7 10% 7,050 8,000 0 44,000 950 8 r 0 8,000 0 36,000 8,000....... 12 r 0 8,000 0 4,000 8,000 13 r 0 4,000 0 0 4,000 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 5 / 32

Dynamic hedging Dynamic hedging is a popular approach to mitigate the financial risk, but Dynamic hedging requires calculating the dollar Deltas of a portfolio of variable annuity policies within a short time interval The value of the guarantees cannot be determined by closed-form formula The Monte Carlo simulation model is time-consuming Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 6 / 32

Use of Monte Carlo method Using the Monte Carlo method to value large variable annuity portfolios is time-consuming: Example (Valuing a portfolio of 100,000 policies) 1,000 risk neutral scenarios 360 monthly time steps 100, 000 1, 000 360 = 3.6 10 10! 3.6 10 10 projections = 50 hours! 200, 000 projections/second Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 7 / 32

A portfolio of synthetic variable annuity policies Feature Value Policyholder birth date [1/1/1950, 1/1/1980] Issue date [1/1/2000, 1/1/2014] Valuation date 1/1/2014 Maturity [15, 30] years Account value [50000, 500000] Female percent 40% Product type DBRP, DBRU, WB, WBSU, MB (20% of each type) Fund fee 30, 50, 60, 80, 10, 38, 45, 55, 57, 46bps for Funds 1 to 10, respectively Base fee Rider fee Number of funds invested [1, 10] 200 bps 20, 50, 60, 50, 50bps for DBRP, DBRU, WB, WBSU, MB, respectively Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 8 / 32

Summary statistics of selected variables Categorical Variables Category Count gender Female 4071 Male 5929 prodtype DBRP 2028 DBRU 2018 MB 1959 WB 1991 WBSU 2004 Continuous Variables Minimum Mean Maximum gmdbamt 0 135116.88 986536.04 gmwbamt 0 7888.8 69403.72 gmwbbalance 0 94151.68 991481.79 gmmbamt 0 54715.12 499925.4 withdrawal 0 26348.89 418565.23 FundValue1 0 33325.04 1030517.37 FundValue2 0 43224.12 1094839.83 FundValue3 0 28623.53 672927 FundValue4 0 27479.09 547874.38 FundValue5 0 24225.22 477843.32 FundValue6 0 35305.36 819144.24 FundValue7 0 28903.78 794470.82 FundValue8 0 28745.1 726031.63 FundValue9 0 27191.4 808213.6 FundValue10 0 26666.22 709232.82 age 34.36 49.38 64.37 ttm 0.68 14.65 28.68 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 9 / 32

Metamodeling A metamodel, also a surrogate model, is a model of another model. Metamodeling has been applied to address the computational problems arising from valuation of variable annuity portfolios: a number of work published by co-author G. Gan. It involves four steps: Select representative VA policies Value representative VA policies Build a metamodel Use the metamodel Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 10 / 32

Kriging has been used to build metamodels, but it assumes normality Frequency 0 500 1000 2000 0 50 100 200 300 Fair Market Values (in thousands) Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 11 / 32

Use of GB2 distribution GB2 provides a flexible family of distributions to model skewed data: f(z) = Z = Y + c (1) a ( z ) ap 1 [ ( z ) a ] p q 1 +, bb(p, q) b b z > 0, (2) ) E[Z] = bb ( p + 1 a, q 1 a B(p, q), p < 1 < q. (3) a We chose to incorporate covariates through the scale parameter b(z i ) = exp(z i β). MLE is used to estimate parameters and multi-stage optimization approach is used to find optimum parameters. Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 12 / 32

Accuracy of the GB2 model and the kriging model with different number of representative VA contracts s = 220 s = 440 s = 880 GB2 Kriging GB2 Kriging GB2 Kriging P E 0.0775 0.0587 (0.0018) (0.0120) (0.0123) (0.0258) R 2 0.5710 0.7010 0.5968 0.7458 0.6136 0.7936 AAP E 2.8700 2.9770 2.7031 2.9188 2.6098 2.1954 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 13 / 32

200 150 50 100 FMV(Kriging) 300 200 0 0 100 FMV(GB2) 400 250 500 Scatter plots of the given fair market values and predicted fair market values when s = 880 0 100 200 300 400 500 0 50 100 150 FMV(MC) FMV(MC) (a) (b) Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 200 250 22-23 May 2017 14 / 32

200 150 Kriging 50 100 300 200 0 0 100 GB2 400 250 500 QQ plots of the given fair market values and predicted fair market values when s = 880 0 100 200 300 400 500 0 Empirical 100 150 200 250 Empirical (a) Gan/Valdez (U. of Connecticut) 50 (b) Dependence Modeling Workshop 22-23 May 2017 15 / 32

Choosing for the optimal experimental design method An important step in the metamodeling process is the selection of representative policies. We compared five different approaches to experimental designs: The random sampling method (RS) The low-discrepancy sequence method (LDS) The data clustering method (DC) The Latin hypercube sampling method (LHS) The conditional Latin hypercube sampling method (clhs) Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 16 / 32

Comparing the accuracy and speed RS LDS DC LHS clhs P E 0.16 (0.34) 0.39 (0.49) 0.02 (0.05) 0.14 (0.4) -0.03 (0.03) R 2 0.35 (0.25) -0.08 (1.06) 0.63 (0.02) 0.26 (0.58) 0.61 (0.03) AAP E 7.58 (3.23) 8.95 (5.73) 3 (0.54) 7.56 (4.44) 2.6 (0.18) Runtime 33.74 (0.66) 33.26 (1.28) 222.9 (59.28) 38.11 (0.34) 150.26 (1.73) Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 17 / 32

0.0 0.5 1.0 1.5 3 2 1 0 RS LDS DC LHS clhs (a) P E RS LDS DC LHS clhs (b) R 2 5 10 15 20 50 100 150 200 250 300 350 RS LDS DC LHS clhs RS LDS DC LHS clhs (c) AAP E (d) Runtime Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 18 / 32

Examining the effect of dependence on partial greeks We refer to Greeks as the sensitivities of the VA guaranteed values on major market indices. For h = 1, 2,..., 5, the partial dollar delta of a VA contract on the hth market index is calculated as Delta(h) = F MV (AV 1,, AV h 1, 1.01AV h, AV h+1,, AV 5 ) 0.02 F MV (AV 1,, AV h 1, 0.99AV h, AV h+1,, AV 5 ), 0.02 where AV h is the partial account value on the hth index and F MV ( ) denotes the fair market value calculated by Monte Carlo simulation. The shock size we used is 1% of the partial account value. Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 19 / 32

Summary statistics on the five indices Account Values of the five indices Variable Description Min Mean Max AV1 Account value of index 1 0 99327.07 871681.9 AV2 Account value of index 2 0 74618.34 1032433.7 AV3 Account value of index 3 0 67822.61 802550.9 AV4 Account value of index 4 0 51219.86 587646.6 AV5 Account value of index 5 0 35268.81 575576.9 Partial dollar deltas on market indices Variable Description Min Mean Max Delta1 On large cap (205,141.33) (13,215.66) 0 Delta2 On small cap (193,899.27) (8,670.87) 0 Delta3 On international equity (386,730.84) (9,616.43) 0 Delta4 On government bond (286,365.30) (8,994.71) 0 Delta5 On money market (412,226.54) (7,068.12) 0 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 20 / 32

Description and summary of the predictor variables Categorical variables Variable Description Count by Category gender Gender of the policyholder F:4071, M:5929 producttype Product type of the VA contract DBRP:2028, DBRU:2018, MB:1959, WB:1991, WBSU:2004 Continuous variables with the original funds Variable Description Min Mean Max gmdbamt GMDB amount 0 137908.45 984963.95 gmwbamt GMWB amount 0 8339.56 75065.14 gmwbbalance GMWB balance 0 102582.22 1072359.14 gmmbamt GMMB amount 0 54715.121 499925.4 FundValue1 Account value of fund 1 0 38570.78 871681.9 FundValue2 Account value of fund 2 0 44998.42 1032433.7 FundValue3 Account value of fund 3 0 27638.71 802550.9 FundValue4 Account value of fund 4 0 29567.86 587646.6 FundValue5 Account value of fund 5 0 29498.07 575576.9 FundValue6 Account value of fund 6 0 39351.66 898768.1 FundValue7 Account value of fund 7 0 30986.59 631404.1 FundValue8 Account value of fund 8 0 31762.53 716196.8 FundValue9 Account value of fund 9 0 27028.39 765476 FundValue10 Account value of fund 10 0 28853.71 708491.1 age Age of the policyholder 35 49.46 64 ttm Time to maturity in years 1 14.48 28 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 21 / 32

Scatter plots of dollar deltas - in thousands Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 22 / 32

Model specification and comparison measures Marginals: Gamma (conditional on non-zero, negative) Copulas: Independent, Gaussian, t copula, Gumbel and Clayton Validation Measures: n i=1 Percentage Error: P E h = (ŷ ih y ih ) n i=1 y ih Model producing a PE closer to zero is better. Mean Squared Error :MSE h = 1 n n i=1 (ŷ ih y ih ) 2 Model that produces a lower MSE is better. Concordance Correlation Coefficient: CCC h = Model that produces a higher CCC is better. 2ρσ 1 σ 2 σ 2 1 + σ2 2 + (µ 1 µ 2 ) 2 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 23 / 32

Dependence parameter estimates Estimated dependence of the fitted copulas when s = 320 Normal t Gumbel Clayton Estimated ρ=0.783 ρ=0.850 α=2.038 α=0.979 Parameters df =3.340 Cramer-von Mises test statistics of copula models when s = 320 Normal t Gumbel Clayton S n 0.341 0.112 0.745 3.290 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 24 / 32

Numerical results when s = 320 Independence copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.014 0.011-0.006 0.012 0.022 MSE 61.307 38.343 56.185 54.275 78.643 CCC 0.852 0.848 0.841 0.873 0.812 t copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.015 0.012-0.006 0.011 0.018 MSE 63.731 39.582 57.302 52.900 77.348 CCC 0.846 0.844 0.839 0.875 0.813 Gumbel copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.015 0.008-0.010 0.010 0.012 MSE 61.812 38.138 55.988 53.017 75.110 CCC 0.850 0.848 0.840 0.875 0.815 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 25 / 32

QQ plots: Monte Carlo vs independence copula when s = 320 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 26 / 32

QQ plots: Monte Carlo vs t copula when s = 320 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 27 / 32

QQ plots: Monte Carlo vs Gumbel copula when s = 320 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 28 / 32

Impact of changing the portfolio compositions Feature Value Policyholder birth date [1/1/1950, 1/1/1980] Issue date [1/1/2000, 1/1/2014] Valuation date 1/1/2014 Maturity [15, 30] years Account value [50000, 500000] Female percent 40% Product type 20% DBRP, 10% MB, 20% WB, 50% WBSU Fund fee 30, 50, 60, 80, 10, 38, 45, 55, 57, 46bps for Funds 1 to 10, respectively Base fee Rider fee Number of funds invested [1,10] 200 bps 20, 50, 60, 50, 50bps for DBRP, DBRU, WB, WBSU, MB, respectively Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 29 / 32

Numerical results when s = 320 Independence copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.002 0.019-0.031-0.007 0.022 MSE 38.093 42.305 43.948 66.789 170.298 CCC 0.949 0.862 0.944 0.959 0.926 t copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.002 0.022-0.032-0.008 0.023 MSE 37.995 43.464 43.780 67.258 157.383 CCC 0.949 0.860 0.944 0.958 0.933 Gumbel copula Delta1 Delta2 Delta3 Delta4 Delta5 PE 0.001 0.017-0.034-0.009 0.020 MSE 37.801 41.578 44.977 64.953 159.075 CCC 0.949 0.863 0.942 0.960 0.931 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 30 / 32

QQ plots: Monte Carlo vs t copula when s = 320 Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 31 / 32

Acknowledgment Guojun and I want to thank the Society of Actuaries through the CKER Individual Grant for supporting this research. Gan/Valdez (U. of Connecticut) Dependence Modeling Workshop 22-23 May 2017 32 / 32