Efficient valuation of variable annuity portfolios with dynamic programming

• Published date: 01 December 2021
• Author: Thorsten Moenig
• Date: 01 December 2021
• DOI: http://doi.org/10.1111/jori.12355

J Risk Insur. 2021;88:1023–1055. wileyonlinelibrary.com/journal/jori

|

1023

Received: 26 March 2020

|

Revised: 22 May 2021

|

Accepted: 1 June 2021

DOI: 10.1111/jori.12355

ORIGINAL ARTICLE

Efficient valuation of variable annuity

portfolios with dynamic programming

Thorsten Moenig

Department of Risk, Insurance, and

Healthcare Management, Temple

University, Philadelphia,

Pennsylvania, USA

Correspondence

Thorsten Moenig, Department of Risk,

Insurance, and Healthcare Management,

Temple University, Philadelphia, PA

19122, USA.

Email: moenig@temple.edu

Abstract

The valuation of variable annuity portfolios presents

major challenges for US life insurers. Recent studies

propose machine learning and metamodeling techni-

ques based on selecting a few representative guaran-

tees. However, these methods face a critical trade‐off

between speed and accuracy. In contrast, I propose a

recursive dynamic programming approach and demon-

strate its ability to value a large and highly hetero-

geneous variable annuity portfolio with a high degree

of accuracy and within a few seconds—even under

stochastic interest rates and volatility—since the heavy

computational burden can be fully front‐loaded (in a

one‐time effort at the guarantee's pricing stage). This

makes the dynamic programming approach ideally

suited for all variable annuity applications, including

the computation of reserves and capital requirements

and to determine the insurer's hedging position.

Moreover, dynamic programming can naturally in-

corporate optimal policyholder behavior into the in-

surer's valuation.

KEYWORDS

guaranteed minimum benefits, hedging, portfolio valuation,

recursive dynamic programming, variable annuities

JEL CLASSIFICATION

C61, G22

© 2021 American Risk and Insurance Association

1|INTRODUCTION

Variable annuities (VAs) are equity‐linked personal savings and investment vehicles. Through

a unique combination of investment flexibility, long‐term financial guarantees and tax benefits,

VAs have become an increasingly important part of US household retirement planning. By the

end of 2019, they accounted for around $2.0 trillion dollars in net assets and constituted the

largest category of liabilities for US life insurers (Koijen & Yogo, 2018).

The liabilities stem from the guarantees that the insurers provide—mostly as optional riders

for a separate fee—to make the product more attractive as a retirement savings vehicle. While

their payout structure resembles financial put options, VA guarantees tend to be substantially

more complex and have much longer maturities (10–25 years, and possibly longer) than tra-

ditional exchange‐traded options. Some VA guarantees promise a specified minimum payout

upon either the policyholder's death (so‐called GMDBs) or her survival to the contract's ma-

turity (so‐called GMABs). There are also withdrawal guarantees which either ensure that the

policyholder can withdraw the initial investment over time in small annual installments

(GMWBs) or guarantee a certain withdrawal amount each year for the remainder of the pol-

icyholder's life (GLWBs).

For a typical VA provider, a portfolio of in‐force policies contains a wide range of these

guarantee types with different contract specifications and policyholder characteristics.

1

This

makes a timely and accurate valuation of a large VA portfolio quite challenging. Nonetheless,

insurers must conduct such a valuation on a regular basis for the purpose of financial reporting,

hedging, reserving, and capital calculations. The insurer could, of course, value each in‐force

policy individually at the valuation time and aggregate these values to the portfolio level. In

fact, much of the academic research in this area has focused on the pricing and hedging of a

single guarantee (see e.g., Bauer et al., 2008; Milevsky & Posner, 2001; Milevsky & Salisbury,

2006, among many others). However, the direct valuation of each policy is sufficiently time

consuming to make this approach infeasible for large VA portfolios.

Recent studies have attempted to remedy this issue by developing valuation methods spe-

cifically for VA portfolios, typically using machine learning methods (Hejazi & Jackson, 2016;

Xu et al., 2018)ormetamodeling techniques that rely on the selection of a small number of

representative contracts that are valued in detail (e.g., using simulation), and then extending

the values to the portfolio using spatial interpolation or regression methods based on the

policies' characteristics (see e.g., Feng et al., 2020; Gan, 2013; Gan & Lin, 2015; Gan & Valdez,

2018; Hejazi et al., 2017; Liu & Tan, 2020).

In contrast, this article proposes to apply the principle of recursive dynamic programming

(RDP) to the valuation of large VA portfolios. Here, the value of each policy in the VA portfolio

is approximated individually from a grid of predetermined values (among representative policy

characteristics and stochastic variables such as the current VA account value). Notably, the grid

is established at the product's design and pricing stage, and the “online”computation time at

each valuation date requires merely a linear interpolation per in‐force policy. As a result, the

valuation of a realistic VA portfolio with 100,000 heterogeneous policies can be done within

seconds, while competing methods require at least several minutes of computation time.

An in‐depth numerical analysis demonstrates that the RDP approach is also highly accu-

rate. For a sample VA portfolio of 100,000 in‐force policies with disperse contract

1

This includes heterogeneity in the policyholder's age and sex, potential step‐up features of the guarantee, the contract's years to maturity, the current

“moneyness”of the guarantee, and the underlying fund allocation, among others.

1024

|

MOENIG

characteristics, guarantee types (including GMDB, GMAB, GMWB, and GLWB) and features,

the RDP estimate is on average within 0.2% of the corresponding benchmark policy value (for

both fees and guarantees) in the Black–Scholes setting, and it falls within 0.6% in the worst case

when also modeling stochastic interest rates (Vasicek model) and stochastic equity volatility

(using regimes). At the portfolio level, the RDP estimation does even better, and it is also able to

estimate the portfolio delta with an accuracy of 0.1%. These error rates are at the same level or

(in most cases) below the errors reported by existing studies, even though I employ a more

complex financial market model, include stochastic policyholder behavior, value the policies

between (rather than only on) anniversary dates, and—to my knowledge—include a greater

level of heterogeneity in the sample portfolio than competing studies.

The real‐world nature of this sample portfolio and the extremely positive results of this

study demonstrate that the RDP approach is feasible in any realistic industrial context. In

particular, with its combination of speed and accuracy, the RDP approach can be readily

applied to all aspects of insurance operations that require a market‐consistent valuation of a

firm's VA portfolio(s). This includes the calculation of GAAP reserves (especially under the

new accounting standard ASU 2018‐12, commonly abbreviated as LDTI), finding the portfolio

Greeks for the purpose of updating the hedging position, and computing statutory reserves and

capital requirements under the new regulatory standard VM‐21. This standard entails sto-

chastic projections of the insurer's liabilities from the VA portfolio, which requires the frequent

calculation of its hedging position along each path. For that, the relevant portfolio Greeks have

to be determined thousands of times under differing conditions. Here, the computational

advantage of the RDP approach over machine learning methods and simulation techniques is

particularly impactful.

RDP has a long history of solving complex optimization problems related to sequential

decision making, particularly under uncertainty, and is widely used in economics and finance,

among other fields. The approach was developed by Richard E. Bellman in the 1940s and

entails the recursive computation of dynamic value functions that depend on time and on

additional “state variables.”In the case of this study where the value function represents the

value of a VA guarantee (or the expected present value of all future fee payments), state

variables may include the VA account value, the guaranteed amount, and (stochastic) financial

market parameters such as equity volatility and interest rate. The state variables are re-

presented by a discrete grid with a finite number of nodes and the value function is computed

for all grid points, starting at some maturity date Tand then proceeding recursively to times

T−

1

,

T−

2

,…,

0

. Hereby, the value function at each time‐

t

grid point (for

t

T=0,…, −

1

)is

defined as the value (or utility, etc.) gained over period

tt

[

,+1

)

, plus the expected (discounted)

continuation value from time

t

+

1

onward. The latter is captured on the grid by the

time‐t

(

+1

)

value function—which was computed in the prior time‐step calculations and can

be interpolated as needed. For a more general and more detailed introduction to dynamic

programming, I refer the interested reader to Bertsekas (1995) and Cooper and Cooper (2016).

At the pricing stage, the insurer can implement the grid computations for a representative

set of policy (holder) characteristics, which include the guarantee type and specifications, the

time to maturity of the contract, and the policyholder's age at inception and sex. These grid

calculations can become very computationally intensive, due to the curse of dimensionality that

makes the RDP computation time increase exponentially with every additional state variable

(and with every dimension of characteristics). However, I find that even when incorporating

stochastic interest rates and volatility regimes into the model, the RDP approach is still easily

manageable with the resources of even a small insurance company. And importantly, this grid

MOENIG

|

1025