Efficient Valuation of Large Variable Annuity Portfolios ...valdez/Seoul2017effvaluation-KAS.pdf ·...

Efficient Valuation of Large Variable AnnuityPortfolios

Emiliano A. Valdez

joint work with Guojun Gan

University of Connecticut

Seminar Talk forThe Institute of Actuaries of Korea

Seoul, Korea

12 May 2017

Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 1 / 31

Outline of workThis presentation is based on a collection of work:

G. Gan and E.A. Valdez, Regression Modeling for the Valuation ofLarge Variable Annuity Portfolios, 2016, submitted to North AmericanActuarial Journal

G. Gan and E.A. Valdez, An Empirical Comparison of SomeExperimental Designs for the Valuation of Large Variable AnnuityPortfolios, 2016, Dependence Modeling

G. Gan and E.A. Valdez, Modeling Partial Greeks of Variable Annuitieswith Dependence, 2017, submitted in Insurance: Mathematics andEconomics

Just recently completed work on:G. Gan and E.A. Valdez, Valuation of Large Variable AnnuityPortfolios: Monte Carlo Simulation and Benchmark Datasets, 2017,submitted to ASTIN Bulletin

This collection of work tackles the issues related to efficient valuationof large variable annuity portfolios:

valuation of VA products present some computational challenges


What is a variable annuity?A variable annuity is a retirement product, offered by an insurancecompany, that gives you the option to select from a variety of investmentfunds and then pays you retirement income, the amount of which willdepend on the investment performance of funds you choose.

Policyholder

SeparateAccount

GeneralAccount

Premiums

Withdrawals/Payments

Charges

GuaranteePayments


Variable annuities come with guarantees

GMxB

GMDB GMLB

GMIB GMMB GMWB


Insurance companies have to make guarantee paymentsunder bad market conditions

Example (An immediate variable annuity with GMWB)

Total investment and initial benefits base: $100,000

Maximum annual withdrawal: $8,000

PolicyYear

INVReturn

FundBeforeWD

AnnualWD

FundAfterWD

RemainingBenefit

GuaranteeCF

1 -10% 90,000 8,000 82,000 92,000 02 10% 90,200 8,000 82,200 84,000 03 -30% 57,540 8,000 49,540 76,000 04 -30% 34,678 8,000 26,678 68,000 05 -10% 24,010 8,000 16,010 60,000 06 -10% 14,409 8,000 6,409 52,000 07 10% 7,050 8,000 0 44,000 9508 r 0 8,000 0 36,000 8,000...

......

......

......

12 r 0 8,000 0 4,000 8,00013 r 0 4,000 0 0 4,000


Dynamic hedging

Dynamic hedging is a popular approach to mitigate the financial risk, but

Dynamic hedging requires calculating the dollar Deltas of a portfolioof variable annuity policies within a short time interval

The value of the guarantees cannot be determined by closed-formformula

The Monte Carlo simulation model is time-consuming


Use of Monte Carlo method

Using the Monte Carlo method to value large variable annuity portfolios istime-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× 1010!

3.6× 1010 projections

200, 000 projections/second= 50 hours!


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 1/1/2014Maturity [15, 30] yearsAccount 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, respectivelyBase fee 200 bpsRider fee 20, 50, 60, 50, 50bps for DBRP,

DBRU, WB, WBSU, MB, respectivelyNumber of funds invested [1, 10]


Summary statistics of selected variablesCategorical Variables Category Count

gender Female 4071Male 5929

prodType DBRP 2028DBRU 2018

MB 1959WB 1991

WBSU 2004

Continuous Variables Minimum Mean Maximum

gmdbAmt 0 135116.88 986536.04gmwbAmt 0 7888.8 69403.72gmwbBalance 0 94151.68 991481.79gmmbAmt 0 54715.12 499925.4withdrawal 0 26348.89 418565.23FundValue1 0 33325.04 1030517.37FundValue2 0 43224.12 1094839.83FundValue3 0 28623.53 672927FundValue4 0 27479.09 547874.38FundValue5 0 24225.22 477843.32FundValue6 0 35305.36 819144.24FundValue7 0 28903.78 794470.82FundValue8 0 28745.1 726031.63FundValue9 0 27191.4 808213.6FundValue10 0 26666.22 709232.82age 34.36 49.38 64.37ttm 0.68 14.65 28.68


MetamodelingA metamodel, also a surrogate model, is a model of another model.Metamodeling has been applied to address the computationalproblems arising from valuation of variable annuity portfolios: anumber 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


Kriging has been used to build metamodels, but it assumesnormality

Fair Market Values (in thousands)

Frequency

0 50 100 200 300

0500

1000

2000


Use of GB2 distribution

GB2 provides a flexible family of distributions to model skewed data:

Z = Y + c (1)

f(z) =|a|

bB(p, q)

(zb

)ap−1 [1 +

(zb

)a]−p−q, z > 0, (2)

E[Z] =bB(p+ 1

a , q −1a

)B(p, q)

, −p < 1

a< q. (3)

We chose to incorporate covariates through the scale parameterb(zi) = exp(z′iβ).

MLE is used to estimate parameters and multi-stage optimizationapproach is used to find optimum parameters.


Some validation measures

PE =

∑ni=1(yi − yi)∑n

i=1 yi. (4)

R2 = 1−∑n

i=1(yi − yi)2∑ni=1(yi − y)2

, (5)

where y is the average fair market value given by

y =1

n

n∑i=1

yi.

AAPE =1

n

n∑i=1

∣∣∣∣ yi − yiyi

∣∣∣∣ . (6)


Accuracy of the GB2 model and the kriging model withdifferent number of representative VA contracts

s = 220 s = 440 s = 880

GB2 Kriging GB2 Kriging GB2 Kriging

PE 0.0775 0.0587 (0.0018) (0.0120) (0.0123) (0.0258)R2 0.5710 0.7010 0.5968 0.7458 0.6136 0.7936

AAPE 2.8700 2.9770 2.7031 2.9188 2.6098 2.1954


Scatter plots of the given fair market values and predictedfair market values when s = 880

0 100 200 300 400 500

0100

200

300

400

500

FMV(MC)

FMV(GB2)

(a)

0 50 100 150 200 2500

50100

150

200

250

FMV(MC)

FMV(Kriging)

(b)


QQ plots of the given fair market values and predicted fairmarket values when s = 880

0 100 200 300 400 500

0100

200

300

400

500

Empirical

GB2

(a)

0 50 100 150 200 2500

50100

150

200

250

Empirical

Kriging

(b)


The 95% confidence intervals of the parameters of theGB2 models

0

10

20

0 10 20Parameter Index

Par

amet

er V

alue


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


Runtime of the GB2 model and the kriging model withdifferent number of representative VA contracts

s = 220 s = 440 s = 880

GB2 Kriging GB2 Kriging GB2 Kriging

cLHS 74.37 74.37 89.60 89.60 127.71 127.71Parameter Estimation 2.65 0 4.83 0 17.90 0

Prediction 0.02 9.55 0.02 18.35 0.02 47.06Total 77.04 83.92 94.46 107.95 145.63 174.77

The GB2 model is able to capture the skewness of the data betterthan the kriging model.

The GB2 model is able to outperform the kriging model in term ofcomputational speed.

The GB2 model is able to produce comparably accurate predictions asthe kriging model at the portfolio level.


Choosing for the optimal experimental design method

An important step in the metamodeling process is the selection ofrepresentative policies. We compared five different approaches toexperimental 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)


Comparing the accuracy and speed

RS LDS DC LHS cLHS

PE 0.16 (0.34) 0.39 (0.49) 0.02 (0.05) 0.14 (0.4) -0.03 (0.03)R2 0.35 (0.25) -0.08 (1.06) 0.63 (0.02) 0.26 (0.58) 0.61 (0.03)AAPE 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)


●

●

●

RS LDS DC LHS cLHS

0.0

0.5

1.0

1.5

(a) PE

●

●

●

●

●

RS LDS DC LHS cLHS

−3

−2

−1

0

(b) R2

●

●

●●

RS LDS DC LHS cLHS

510

1520

(c) AAPE

●

●

●

RS LDS DC LHS cLHS

5010

015

020

025

030

035

0

(d) Runtime

Figure: Box plots of the performance measures from 10 runs of the experimentaldesign methods when s = 880.


Examining the effect of dependence on partial greeks

We refer to ”Greeks” as the sensitivities of the VA guaranteed values onmajor market indices.

For h = 1, 2, . . . , 5, the partial dollar delta of a VA contract on the hthmarket index is calculated as

Delta(h) =FMV (AV1, · · · , AVh−1, 1.01AVh, AVh+1, · · · , AV5)

0.02

−FMV (AV1, · · · , AVh−1, 0.99AVh, AVh+1, · · · , AV5)0.02

,

where AVh is the partial account value on the hth index and FMV (·)denotes the fair market value calculated by Monte Carlo simulation. Theshock size we used is 1% of the partial account value.


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 871681.9AV2 Account value of index 2 0 74618.34 1032433.7AV3 Account value of index 3 0 67822.61 802550.9AV4 Account value of index 4 0 51219.86 587646.6AV5 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) 0Delta2 On small cap (193,899.27) (8,670.87) 0Delta3 On international equity (386,730.84) (9,616.43) 0Delta4 On government bond (286,365.30) (8,994.71) 0Delta5 On money market (412,226.54) (7,068.12) 0


Scatter plots of dollar deltas - in thousands


Model specification and comparison measures

Marginals: Gamma (conditional on non-zero, negative)

Copulas: Independent, Gaussian, t copula, Gumbel and Clayton

Validation Measures:

Percentage Error: PEh =

∑ni=1(yih − yih)∑n

i=1 yihModel producing a PE closer to zero is better.

Mean Squared Error :MSEh = 1n

∑ni=1(yih − yih)2

Model that produces a lower MSE is better.

Concordance Correlation Coefficient: CCCh =2ρσ1σ2

σ21 + σ22 + (µ1 − µ2)2Model that produces a higher CCC is better.


Numerical results when s = 320

Independence copula

Delta1 Delta2 Delta3 Delta4 Delta5

PE 0.014 0.011 -0.006 0.012 0.022MSE 61.307 38.343 56.185 54.275 78.643CCC 0.852 0.848 0.841 0.873 0.812

t copula


PE 0.015 0.012 -0.006 0.011 0.018MSE 63.731 39.582 57.302 52.900 77.348CCC 0.846 0.844 0.839 0.875 0.813

Gumbel copula


PE 0.015 0.008 -0.010 0.010 0.012MSE 61.812 38.138 55.988 53.017 75.110CCC 0.850 0.848 0.840 0.875 0.815


QQ plots: Monte Carlo vs independence copula whens = 320


QQ plots: Monte Carlo vs t copula when s = 320


QQ plots: Monte Carlo vs Gumbel copula when s = 320


Acknowledgment

Guojun and I want to thank the Society of Actuaries through theCKER Individual Grant for supporting this research.


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