Efficient Valuation of Large Variable Annuity Portfolios

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1 Efficient Valuation of Large Variable Annuity Portfolios Emiliano A. Valdez joint work with Guojun Gan University of Connecticut Seminar Talk at Wisconsin School of Business University of Wisconsin Madison, Wisconsin 31 March 2017 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

2 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) Seminar Talk - UW Madison 31 March / 31

3 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 - UW Madison 31 March / 31

4 Variable annuities come with guarantees GMxB GMDB GMLB GMIB GMMB GMWB Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

5 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, % 90,200 8,000 82,200 84, % 57,540 8,000 49,540 76, % 34,678 8,000 26,678 68, % 24,010 8,000 16,010 60, % 14,409 8,000 6,409 52, % 7,050 8, , r 0 8, ,000 8, r 0 8, ,000 8, r 0 4, ,000 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

6 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 - UW Madison 31 March / 31

7 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, = ! projections = 50 hours! 200, 000 projections/second Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

8 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, ] 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 - UW Madison 31 March / 31

9 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 gmwbamt gmwbbalance gmmbamt withdrawal FundValue FundValue FundValue FundValue FundValue FundValue FundValue FundValue FundValue FundValue age ttm Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

10 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 - UW Madison 31 March / 31

11 Kriging has been used to build metamodels, but it assumes normality Frequency Fair Market Values (in thousands) Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

12 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 - UW Madison 31 March / 31

13 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 - UW Madison 31 March / 31

14 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 (0.0018) (0.0120) (0.0123) (0.0258) R AAP E Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

15 FMV(Kriging) FMV(GB2) Scatter plots of the given fair market values and predicted fair market values when s = FMV(MC) FMV(MC) (a) (b) Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison March / 31

16 Kriging GB QQ plots of the given fair market values and predicted fair market values when s = Empirical Empirical (a) Gan/Valdez (U. of Connecticut) 0 (b) Seminar Talk - UW Madison 31 March / 31

17 The 95% confidence intervals of the parameters of the GB2 models Parameter Value Parameter Index Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

18 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 - UW Madison 31 March / 31

19 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 Parameter Estimation Prediction Total 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 - UW Madison 31 March / 31

20 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 - UW Madison 31 March / 31

21 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) R (0.25) (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 (0.66) (1.28) (59.28) (0.34) (1.73) Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

22 RS LDS DC LHS clhs (a) P E RS LDS DC LHS clhs (b) R RS LDS DC LHS clhs RS LDS DC LHS clhs (c) AAP E (d) Runtime Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

23 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 - UW Madison 31 March / 31

24 Summary statistics on the five indices Account Values of the five indices Variable Description Min Mean Max AV1 Account value of index AV2 Account value of index AV3 Account value of index AV4 Account value of index AV5 Account value of index 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 - UW Madison 31 March / 31

25 Scatter plots of dollar deltas - in thousands Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

26 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 σ2 2 + (µ 1 µ 2 ) 2 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

27 Numerical results when s = 320 Independence copula Delta1 Delta2 Delta3 Delta4 Delta5 PE MSE CCC t copula Delta1 Delta2 Delta3 Delta4 Delta5 PE MSE CCC Gumbel copula Delta1 Delta2 Delta3 Delta4 Delta5 PE MSE CCC Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

28 QQ plots: Monte Carlo vs independence copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

29 QQ plots: Monte Carlo vs t copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

30 QQ plots: Monte Carlo vs Gumbel copula when s = 320 Gan/Valdez (U. of Connecticut) Seminar Talk - UW Madison 31 March / 31

31 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 - UW Madison 31 March / 31

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