Efficient Valuation of Large Variable Annuity Portfolios Emiliano A. Valdez joint work with Guojun Gan University of Connecticut Seminar Talk for The 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
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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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 2 / 31
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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 3 / 31
Variable annuities come with guarantees
GMxB
GMDB GMLB
GMIB GMMB GMWB
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 4 / 31
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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 5 / 31
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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 6 / 31
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!
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 7 / 31
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]
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 8 / 31
Summary statistics of selected variablesCategorical Variables Category Count
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 9 / 31
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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 10 / 31
Kriging has been used to build metamodels, but it assumesnormality
Fair Market Values (in thousands)
Frequency
0 50 100 200 300
0500
1000
2000
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 11 / 31
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.
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 12 / 31
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)
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 13 / 31
Accuracy of the GB2 model and the kriging model withdifferent number of representative VA contracts
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.
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 19 / 31
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)
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 20 / 31
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.
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 23 / 31
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
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 24 / 31
Scatter plots of dollar deltas - in thousands
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 25 / 31
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.
Gan/Valdez (U. of Connecticut) Seminar Talk - The Institute of Actuaries of Korea 12 May 2017 26 / 31