Efficient Valuation of Large Variable Annuity Portfolios

Efficient Valuation of Large Variable Annuity Portfolios Emiliano A. Valdez joint work with Guojun Gan University of Connecticut Seminar Talk at Hanyang University Seoul, Korea 13 May 2017 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 1 / 34

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 Just recently completed work on: G. Gan and E.A. Valdez, Valuation of Large Variable Annuity Portfolios: Monte Carlo Simulation and Benchmark Datasets, 2017, submitted to ASTIN Bulletin 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) Seminar Talk - Hanyang University 13 May 2017 2 / 34

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) Seminar Talk - Hanyang University 13 May 2017 3 / 34

Variable annuities come with guarantees GMxB GMDB GMLB GMIB GMMB GMWB Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 4 / 34

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) Seminar Talk - Hanyang University 13 May 2017 5 / 34

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) Seminar Talk - Hanyang University 13 May 2017 6 / 34

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) Seminar Talk - Hanyang University 13 May 2017 7 / 34

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) Seminar Talk - Hanyang University 13 May 2017 8 / 34

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) Seminar Talk - Hanyang University 13 May 2017 9 / 34

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) Seminar Talk - Hanyang University 13 May 2017 10 / 34

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) Seminar Talk - Hanyang University 13 May 2017 11 / 34

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) Seminar Talk - Hanyang University 13 May 2017 12 / 34

Some validation measures P E = n i=1 (ŷ i y i ) n i=1 y. (4) i R 2 = 1 where y is the average fair market value given by y = 1 n n i=1 (ŷ i y i ) 2 n i=1 (y i y) 2, (5) n y i. i=1 AAP E = 1 n ŷ i y i n y i. (6) i=1 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 13 / 34

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) Seminar Talk - Hanyang University 13 May 2017 14 / 34

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) Seminar Talk - Hanyang University 200 250 13 May 2017 15 / 34

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) Seminar Talk - Hanyang University 13 May 2017 16 / 34

The 95% confidence intervals of the parameters of the GB2 models Parameter Value 20 10 0 0 10 20 Parameter Index Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 17 / 34

A list of parameters of the GB2 model Index Parameter Index Parameter 1 a 15 FundValue5 2 p 16 FundValue6 3 q 17 FundValue7 4 c 18 FundValue8 5 Intercept 19 FundValue9 6 gmdbamt 20 FundValue10 7 gmwbamt 21 age 8 gmwbbalance 22 ttm 9 gmmbamt 23 genderm 10 withdrawal 24 prodtypedbru 11 FundValue1 25 prodtypemb 12 FundValue2 26 prodtypewb 13 FundValue3 27 prodtypewbsu 14 FundValue4 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 18 / 34

Runtime 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 clhs 74.37 74.37 89.60 89.60 127.71 127.71 Parameter Estimation 2.65 0 4.83 0 17.90 0 Prediction 0.02 9.55 0.02 18.35 0.02 47.06 Total 77.04 83.92 94.46 107.95 145.63 174.77 The GB2 model is able to capture the skewness of the data better than the kriging model. The GB2 model is able to outperform the kriging model in term of computational speed. The GB2 model is able to produce comparably accurate predictions as the kriging model at the portfolio level. Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 19 / 34

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) Seminar Talk - Hanyang University 13 May 2017 20 / 34

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) Seminar Talk - Hanyang University 13 May 2017 21 / 34

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) Seminar Talk - Hanyang University 13 May 2017 22 / 34

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) Seminar Talk - Hanyang University 13 May 2017 23 / 34

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) Seminar Talk - Hanyang University 13 May 2017 24 / 34

Scatter plots of dollar deltas - in thousands Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 25 / 34

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) Seminar Talk - Hanyang University 13 May 2017 26 / 34

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) Seminar Talk - Hanyang University 13 May 2017 27 / 34

QQ plots: Monte Carlo vs independence copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 28 / 34

QQ plots: Monte Carlo vs t copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 29 / 34

QQ plots: Monte Carlo vs Gumbel copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 30 / 34

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) Seminar Talk - Hanyang University 13 May 2017 31 / 34

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) Seminar Talk - Hanyang University 13 May 2017 32 / 34

QQ plots: Monte Carlo vs t copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - Hanyang University 13 May 2017 33 / 34

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) Seminar Talk - Hanyang University 13 May 2017 34 / 34