revisiting the risk-neutral approach to optimal policyholder behavior · pdf...

Revisiting the Risk-Neutral Approach to Optimal

Policyholder Behavior: A Study of Withdrawal

Guarantees in Variable Annuities

Thorsten Moenig and Daniel Bauer∗

Working Paper, December 2014

Abstract

Policyholder exercise behavior presents an important risk factor for pricing Variable Annuities.

However, approaches presented in the literature – building on value-maximizing strategies akin

to pricing American options – do not square with observed price and exercise patterns for

popular withdrawal guarantees. We show that including taxes into the valuation closes this

gap between theory and practice. In particular, we develop a subjective risk-neutral valuation

methodology that takes into consideration differences in the tax structure between investment

opportunities. We demonstrate that accounting for tax advantages significantly affects the

value of the guarantees and produces results that are in line with empirical patterns.

JEL classification: G22, D14, C61

Keywords: variable annuities, optimal policyholder behavior, risk-neutral valuation with taxes

∗Moenig, [email protected], Department of Mathematics, University of St. Thomas, 2115 Summit Ave,OSS 201, St. Paul, MN 55105, (651)962-5521; Bauer, [email protected], Department of Risk Management andInsurance, Georgia State University, 35 Broad Street, 11th Floor, Atlanta, GA 30303, (404)413-7490. We are thankfulfor helpful comments from participants at the 2012 ARIA Annual Meetings; the 47th Actuarial Research Conference;the 12th Symposium on Finance, Banking, and Insurance; the 2013 ASSA Meetings; the 2013 PARTY; the 2014International Congress of Actuaries; the 2014 SIAM FM Conference; as well as from seminar participants at GeorgiaState University, the University of Ulm, the University of Manitoba, Manhattan College, the University of St. Thomas,the University of Wisconsin-Madison, and the University of Minnesota. We are also indebted to Thierry Foucault,Glenn Harrison, Ajay Subramanian, Eric Ulm, and two anonymous referees for valuable input. Financial support fromthe Society of Actuaries under a CAE research grant is greatly appreciated. All remaining errors are ours.

REVISITING THE RISK-NEUTRAL APPROACH TO OPTIMAL POLICYHOLDER BEHAVIOR 1

1 Introduction

The market for Variable Annuities (VAs) is huge. Between 2011 and 2013, U.S. VA sales averaged

approximately $150 billion (bn) per year, with 76% of all new products containing optional features

in the form of so-called Guaranteed Living Benefits (GLBs).1 The total assets under management

pertaining to U.S. VAs well exceed 1.5 trillion dollars. Despite the size of the market, there is

a surprising disconnect between theory and practice in view of pricing some of the most popular

optional riders that depend on the policyholders’ exercise strategy, particularly for Guaranteed

Minimum Withdrawal Benefits (GMWBs). More precisely, an array of papers in the quantitative

finance and insurance literatures rely on (American and Bermudan) option pricing techniques to

solve for the fair fee to be charged for the guarantees,2 yet the derived fees considerably exceed

prevailing charges in the market.3 Researchers have shrugged off this deviation as a result of

sub-optimal or irrational policyholder behavior.

This paper demonstrates that this gap between theory and practice can be closed by taking into

account the very friction that arguably explains the rapid expansion of the VA market over the last

two decades: taxes (Milevsky and Panyagometh, 2001; Brown and Poterba, 2004). In particular,

we consider the policyholder’s subjective valuation of GMWBs in the presence of different in-

vestment opportunities with differing tax treatments. The key idea is that in a complete (pre-tax)

investment market, it is possible to replicate any given post-tax cash flow with a pre-tax cash flow

of some benchmark investments – irrespective of the tax treatment for the securities leading to

1See the corresponding fact sheets in the LIMRA (Life Insurance and Market Research Association) data bank.2See e.g. Milevsky and Salisbury (2006), Bauer et al. (2008), Chen and Forsyth (2008), Chen et al. (2008), Dai et

al. (2008), Blamont and Sagoo (2009), Piscopo (2010), Bacinello et al. (2011, 2014), Huang and Forsyth (2012), Peng

et al. (2012), Huang and Kwok (2014), among others.3For instance, Milevsky and Salisbury (2006) find an “underpricing of this feature [GMWBs] in a typically over-

priced VA market” (see also Dai et al. (2008), Blamont and Sagoo (2009), Piscopo (2010) for similar assertions); Bauer

et al. (2008) report considerable differences between market fees and the risk-neutral value based on their model; and

Chen et al. (2008) state that “only if several unrealistic modeling assumptions are made is it possible to obtain GMWB

fees in the same range as is normally charged.”


the former cash flow. We show that when accounting for taxes, a value-maximizing approach for

GMWBs yields pricing results that are in line with empirical patterns. We interpret our findings

as evidence that taxes considerably influence VA policyholder behavior, and that policyholders

roughly behave optimally or rationally with respect to the value of their contracts subject to taxa-

tion.

VAs are unit-linked, tax-deferred savings plans usually entailing guaranteed payment lev-

els, upon death (Guaranteed Minimum Death Benefits, GMDBs) and/or survival until expiration

(GLBs). To finance these guarantees, most commonly insurers deduct an option fee at a constant

rate from the policyholder’s account value. The product menu of GLBs has expanded consider-

ably since the mid-1990s (Bauer et al., 2008; Ledlie et al., 2008). While initially only relatively

simple maturity guarantees (Guaranteed Minimum Accumulation Benefits, GMABs) and annuitiza-

tion guarantees (Guaranteed Minimum Income Benefits, GMIBs) were available, the most popular

guarantees over the last decade have been withdrawal guarantees in various forms. In particular,

a GMWB provides the policyholder with the right but not the obligation to withdraw the initial

investment over a certain period of time, while alive and irrespective of investment performance,

as long as annual withdrawals do not exceed a pre-specified amount. While GMWBs recently

only accounted for a relatively small fraction of new VA sales (approximately $6.5bn of VAs sold

between 2011 and 2013 in the U.S.), they presented the most popular guarantee in the mid-2000s

(e.g., in 2004 37% of all VAs included a GMWB, Sell, 2006).4

For evaluating and managing these long-term financial options, VA writers rely on option pric-

ing concepts and set up large-scale hedging programs that “rivaled small trading floors in invest-

ment banks” (Chopra et al., 2009). While these hedging programs proved essential in managing

embedded risks, VA providers experienced significant losses in the recent past – particularly in

the wake of the financial crisis – leading two major providers to accept TARP money and most

companies to “de-risk” their products (Chopra et al., 2009; Geneva Association, 2013). One key

4Recently, the largest fraction of VAs (approximately 55% of all U.S.-VAs sold in 2011-2013) were equipped with

related withdrawal guarantees, so-called Guaranteed Lifetime Withdrawal Benefits (GLWBs).


aspect in this context is the impact of policyholder behavior, i.e. how policyholders choose to

withdraw funds from their accounts, surrender their policies, and/or annuitize the remaining as-

set value. Understanding these choices is particularly relevant for newly introduced products or

under changing (regulatory or economic) conditions – such as the recent crisis – where historical

exercise probabilities are either unavailable or deceptive.5 However, results based on conventional

techniques familiar from pricing American and Bermudan options do not square well with market

observations (see Footnotes 2 and 3).

To reconcile empirical price and exercise patterns with theory to first order, we consider a

subjective risk-neutral valuation approach that accounts for differing tax treatments of different

investment opportunities. Similar to previous papers on GMWB pricing, we first introduce a model

for a simple VA plus GMWB that contains the basic features existing contracts share, even though

prevalent contract designs are typically more complex. We describe the evolution of the contract

over time in a Black-Scholes economy, including the policyholder’s options as well as the tax

treatment of the different cash flows, and then discuss our valuation approach. This valuation

is predicated on replicating all post-tax cash flows from the VA, which in the U.S. is subject to

deferred taxation on a last-in first-out basis, using ordinary mutual fund investments that are subject

to capital gains taxes. The approach leads to a non-linear equation for the subjective risk-neutral

value of the given future post-tax VA cash flow – rather than its (linear) expected discounted value

under the risk-neutral probability measure as is the usual case without taxation. We calibrate the

parameters to representative values for a typical U.S. VA contract and implement the policyholder’s

(subjective) value maximization problem using recursive dynamic programming.

The results reveal that, when taking taxes into account, withdrawals are infrequent and are

5Relying on historical policyholder behavior has caught the industry off-guard on occasion. For instance, rising

interest rates in the 1970s led to the so-called disintermediation process, which caused substantial increases in sur-

renders and policy loans in the whole life market (Black and Skipper, 2000, p. 111). Similarly, in 2000, U.K.-based

Equitable Life – the world’s oldest life insurer – was closed to new business in part due to problems arising from a

misjudgment of policyholder behavior in guaranteed annuity options within individual pension policies (Boyle and

Hardy, 2003).


optimal primarily upon poor market performance or, more precisely, when the VA account has

fallen below the tax base. For instance, for our base case parameters, the policyholder withdraws on

average only between 3 and 11 percent of the initial investment over the contract period, depending

on the assumed excess return (Sharpe Ratio), and it is rarely optimal to surrender the policy. This is

in line with available information on dynamic functions describing GMWB policyholder behavior,

which are based on empirical exercise patterns and used by most companies (Society of Actuaries,

2008). These stipulate that withdrawals are prevalent only when the account value is low relative

to the guarantee, and that surrender rates are lower than in the absence of a GMWB (American

Academy of Actuaries, 2005). However, these findings are in stark contrast to our results in the

absence of taxes, which conform with the previous literature on GMWB pricing (Milevsky and

Salisbury, 2006; Chen et al., 2008, among others). More precisely, we find that without taxes,

withdrawing at least the guaranteed annual amount is optimal in most circumstances and complete

surrenders occur in 70 to 80 percent of all scenarios. As a consequence, in our base case, the fair

fee as a percentage of the policyholder’s account value reduces from more than 50 basis points

(bps) without taxes to less than 20 bps when accounting for the tax advantages – which is, again,

in line with typical fees for plain GMWB riders.

We analyze sensitivities to financial market parameters, tax rates, and contract features. While

the impact of the latter two aspects is relatively minor in a reasonable parameter range, the financial

market parameters considerably affect the fair fee rate. In particular, in a setting with a low interest

rate and high fund volatility, the fair fee rate increases to approximately 80 and 170 bps with and

without the consideration of taxes, respectively. This sensitivity lends force to VA providers’

efforts to “de-risk” investment choices in the presence of guarantees. However, the sensitivities

do not change the key observation: Taxation not only seems to be a major reason why consumers

purchase VAs but appears to also incentivize them not to withdraw prematurely. The intuition is

straightforward: In the absence of taxes, as the option moves far out-of-the-money, there is no

motivation for the policyholder to keep funds in the account that is subject to the option fee; with

taxation, on the other hand, withdrawals will be taxed as ordinary income and are subject to capital


gains taxes when invested outside whereas they grow tax-deferred inside the VA account, leading

policyholders to keep the funds in the VA.

In order to assess the empirical performance of the proposed approach, we analyze thirteen

GMWB products from seven different VA providers offered at different points in time. As already

indicated, real-world VA/GMWB contracts can be substantially more complex than the simple

versions analyzed in the literature and may contain a variety of features such as step-ups, ratchets,

qualifying periods, spousal continuation options, and/or fee forgiveness. However, the variety

of features across the different contracts we consider provides a comprehensive cross section of

the product space. We implement each of these policies one-by-one in an extended version of the

introduced framework and determine optimal withdrawal behavior as well as the values of different

cash flows. Our results show that, by and large, taking taxes into account explains policyholder

participation and aligns profit-and-loss expectations for the providers in view of the GMWB. More

precisely, the policies present a worthy investment in present value terms for the consumer and

surrender rates are low. In contrast, without taxes, value-maximizing consumers will not choose to

purchase the products in the first place and will, conditional on being endowed with a VA, surrender

the policies early in the contract term in most scenarios. Moreover, with taxes the present value of

collected fees for the GMWB rider roughly accords with the value of the corresponding liabilities.6

Related Literature and Organization of the Paper

As pointed out, a number of studies in the quantitative finance and insurance literatures apply

conventional option pricing techniques to VA plus GMWB pricing, however the resulting exercise

patterns and fee rates do not align with market observations (see Footnotes 2 and 3). There are vari-

ous potential reasons for this observation including: Market incompleteness, as the payoff depends

6Remaining discrepancies may be attributed to policyholders (rationally) deviating from a value-maximizing be-

havior, e.g. when facing liquidity constraints. For practical applications, it may be appropriate to augment the derived

exercise pattern by a deterministic surrender schedule as it is typical in insurance practice to capture such exogenous

factors.


on the policyholder’s survival and not all payoff profiles can be attained via existing securities;

trading restrictions, as there is no liquid secondary market for used VAs on which the policy could

be sold (or repurchased) at its fair value; and other frictions such as taxes or borrowing constraints.

This is akin to the situation for other financial decisions such as the exercise of executive stock

options (Carpenter, 1998; Detemple and Sundaresan, 1999) and more generally for decisions in

household finance (Campbell, 2006), e.g. in view of mortgage or pension choices.

The conventional approach in such situations is to solve life-cycle utility optimization problems

that incorporate the relevant decision variables,7 and several recent contributions follow this ap-

proach to analyze policyholder behavior in VAs (Gao and Ulm, 2012; Moenig, 2012; Steinorth and

Mitchell, 2012; Horneff et al., 2013). These models can then be used to analyze various attributes

of the guarantees. For instance, Horneff et al. (2013) show that a fair-priced VA plus GMWB can

enhance consumer welfare due to the guarantee paired with flexibility, particularly when consid-

ering tax advantages.8 However, capturing all relevant aspects and risk factors within a life-cycle

model is an ambitious task (Campbell, 2006), so that formulating a model for real-world GMWBs

with the variety of embedded features – as it is important for pricing – may not be feasible.9 In fact,

the (relative) simplicity of our approach, which provides the possibility to include the complexities

of existing products while still producing empirically viable results, is one of its key advantages.

In particular, it comes with the usual benefits of a risk-neutral approach in that it is independent of

preferences, wealth, and consumption decisions.

A few papers make similar points, namely that a risk-neutral approach reasonably predicts

exercise behavior to first order, in the context of mortgage options (Deng et al., 2000) or executive7See, e.g., Carpenter (1998), Detemple and Sundaresan (1999) in the context of executive stock options; Koijen

et al. (2009), Campbell and Cocco (2014) in the context of mortgages; and Chai et al. (2011), Koijen et al. (2011),

Hubener et al. (2014) in the context of pensions, among many others.8Several papers emphasize the importance of taxes in the context of financial planning, particularly the relevance

of tax-deferred accounts (Poterba, 2002; Dammon et al., 2004; Amromin et al., 2007).9Indeed, the motivation for this paper was the observation that a life-cycle model for a simple VA plus GMWB

including taxes, consumption, and outside savings opportunities delivers very similar results to the subjective risk-

neutral valuation approach presented here, see Moenig (2012).


stock options (Carpenter, 1998). In particular, Carpenter (1998) shows that for executive stock

options, a simple option pricing model in combination with an exogenous exercise probability

predicts “exercise times and payoffs just as well as an elaborate utility-maximizing model.” This

resonates with our approach – especially when augmented with an exogenous surrender schedule

(cf. Footnote 6). The key difference to our context is that a straight option pricing approach does

not deliver in the case of a VA plus GMWB; it needs to be modified to account for taxes.

Empirical studies on VA policyholder behavior are sparse. One exception is Knoller et al.

(2014), who analyze surrender behavior for a VA with a simple GMAB. For VA surrender data

from Japan, where according to the authors tax considerations are far less important than in the

U.S. market (Knoller et al., 2014, Footnote 2), they find that the value of the embedded guarantee

has by far the largest explanatory power. This is in line with our assertion that policyholders

exercise their options approximately optimally with regards to their (subjective) risk-neutral value.

Furthermore, as outlined above, our results are in line with available information on dynamic

functions that describe GMWB policyholder behavior and are based on empirical patterns.

This observation might be surprising in light of household decision making in general. Ev-

idence from the mortgage market indicates that some consumers behave “suboptimal[ly]” with

regards to embedded options (Campbell, 2012). Similarly, Koijen et al. (2014) show that, on av-

erage, households make significant mistakes in their (life and health) insurance decisions with an

associated welfare cost of 3.2 percent of total wealth, which is “an order of magnitude larger than

the welfare cost of under-diversification in stock and mutual fund portfolios.” Potential reasons

for this dissonance include that GMWB exercise may fall in the realm of financial (rather than

insurance) decisions possibly supported by a financial advisor – which is consistent with our ob-

servation that a value-maximizing approach yields viable results – and that participation in the VA

market is primarily limited to relatively wealthy individuals.10 It is well-established that wealthy

consumers are significantly more financially literate (Campbell, 2006; Lusardi and Mitchell, 2007)

10For instance, Brown and Poterba (2004) report that “[i]n the bottom half of the income distribution, [...] just over

2 percent of the population own Variable Annuities.”


and make better investment decisions (Agnew, 2006; Amromin et al., 2007).

Our findings also indicate that VA fees are appropriate from the demand side, i.e. the subjective

risk-neutral value is close to (and mostly slightly exceeds) the invested amount for the considered

empirical products. This seems to be in contrast to evidence from the market for structured retail

financial products that indicates overpricing (Henderson and Pearson, 2011; Célérier and Vallée,

2014). An explanation for this discrepancy is that tax deferral within VAs creates a wedge between

the demand and the supply side, and indeed our calculations imply that the collected fees consid-

erably exceed the value of the providers’ total liabilities. In other words, preferred tax treatment of

VAs may make it possible to sustain high base fees, potentially leading VA providers to innovate

by offering optional riders at cost in order to best serve consumers’ objectives and thereby increas-

ing their share of the investment market. However, it is also conceivable that, as in Carline and

Manso (2011), VA providers continue to innovate for the purpose of obfuscation by increasing the

level of complexity of GMWBs and by introducing new guarantees. The question of what drives

product innovation in the VA market, therefore, is a very interesting direction for future research

but is beyond the scope of the current paper.

The remainder of the paper is structured as follows: The next section presents our model for

a simple VA plus GMWB contract and our valuation approach. Section 3 provides results for a

calibrated version of the model. Section 4 discusses extensions of the model to capture preva-

lent GMWB features in the VA market and presents valuation results for empirical VA/GMWB

products. Finally, Section 5 concludes.

2 Model and Valuation Approach

The U.S. market features a large variety of available VA products. The policies differ by how the

premiums are collected; policyholder investment opportunities, including whether the policyholder

can reallocate funds after underwriting; and guarantee specifics, for instance what type of guaran-

tees are included, how the guarantees are designed, etc. In what follows, we first present a (simple)


representative example VA contract including a GMWB rider. We then briefly describe the tax

treatment of VAs in the U.S., outline our valuation approach, and discuss the implementation and

calibration of the model.

2.1 VAs with GMWB in a Black-Scholes Economy

We consider an x-year old individual that just (time t = 0) purchased a VA with finite integer

maturity T against a single up-front premium P . The premium is invested in a risky asset (typically

a mutual fund) – denoted by (St)t≥0 – which evolves according to a geometric Brownian motion:

dStSt

= µ dt+ σ dZt , S0 > 0 ,

where µ, σ > 0, and (Zt)t>0 is a standard Brownian motion.11 The market also contains a risk-free

asset, which accumulates continuously at an annual rate of r.

The insurer will return the policyholder’s concurrent account value, denoted by Xt, at the end

of the policyholder’s year of death or at maturity T , whichever comes first. Relying on standard

actuarial notation, we denote by qy the probability that an individual aged y dies in the following

year, and by py = 1 − qy the corresponding survival probability. We assume that all cash flows

as well as all relevant decisions come into effect at policy anniversary dates, t = 1, . . . , T . At

these dates, the policyholder has the right to make withdrawals from the account, in which case the

account value will be updated as follows:

X+t =

(X−t − wt

)+, (1)

where wt denotes the amount withdrawn at time t. Here we use the notations X−/+t to denote the

time-t VA account values immediately before and after the withdrawal is made, respectively, and

(a)+ = maxa, 0 .

11As usual in this context, underlying our considerations is a complete filtered probability space (Ω, F,P,F =

(Ft)t≥0), where F satisfies the usual conditions and P denotes the physical probability measure.


The contract contains a return-of-premium GMWB rider, which grants the policyholder the

right but not the obligation to withdraw the initial premium P free of charge and independent

of investment performance, as long as annual withdrawals do not exceed the guaranteed annual

amount gWt . Withdrawals in excess of either gWt or the remaining guaranteed aggregate withdrawal

amount, henceforth called the benefits base Gt, carry a (partial) surrender charge of st ≥ 0 as a

percentage of the excess withdrawal amount.12

The initial benefits base is G1 = P . We model adjustments to the benefits base in case of a

withdrawal prior to maturity based on the following popular specification (Bauer et al., 2008): If

the withdrawal does not exceed the guaranteed annual amount gWt , the benefits base will simply be

reduced by the withdrawal amount; otherwise, the benefits base will be the lesser of that amount

and a pro-rata adjustment according to the VA account value. Hence:

Gt+1 =

(Gt − wt)+ , if wt ≤ gWt ,(

min

Gt − wt , Gt ×

X+t

X−t

)+

, if wt > gWt .(2)

To finance the guarantee, the insurer continuously deducts an option fee at constant rate φ ≥ 0

from the policyholder’s account value. As a result, between policy anniversary dates, the VA

account value evolves according to:

X−t+1 = X+t × e−φ × eσγ+µ−

12σ2

, (3)

where the annual net return γ follows a standard normal distribution under P.

In addition to this base case specification, in our numerical analysis, we consider common

modifications of the fee structure, modifications of the adjustment to the benefits base, and a chang-

ing asset allocation over time. More precisely, we analyze a product for which the fees amount to

12This contract design can be easily extended to include other popular guarantees such as GMDBs and GMABs,

possibly at the cost of a larger state space unless the benefits bases coincide. Similarly, it is also straightforward to

incorporate e.g. step-up or ratchet features (see Section 4). We refer to Bauer et al. (2008) for details.


a proportion of the benefits base, so that the account evolution changes from (3) to:

X−t+1 =(X+t − φ×Gt+1

)+× eσγ+µ−

12σ2

. (3’)

For the adjustment to the benefits base in the case of an excess withdrawal, we consider two

alternative specifications. In the first, the benefits base is reduced to the lesser of the account value

and the previous benefits base minus the withdrawal amount:

Gt+1 =


minGt − wt , X+

t

)+

, if wt > gWt .

(2’)

In the second specification, the benefits base is first reduced by the guaranteed annual amount, and

then adjusted on a pro-rata basis for the excess withdrawal:

Gt+1 =


Gt − gWt)+× X+

t(X−t − gWt

)+

, if wt > gWt .(2”)

Finally, for a changing asset allocation over time, we obtain the same law of motion but the asset

parameters in (3) now depend on the policy year t. In particular, we analyze a specification where

the proportion of equity in the mutual fund – and thus µt and σt – decrease over time.

2.2 Tax Treatment of VAs

We model taxation of income and investment returns based on current U.S. regulation, albeit with

a few necessary simplifications. We assume that all contributions to the VA are post-tax and

non-qualified. As such, taxes will only be due on future investment gains, not the initial invest-

ment (principal) itself. We further assume a constant (marginal) income tax rate τ and a constant


(marginal) tax rate κ on capital gains from investments outside of the VA.13

When withdrawn, earnings from the VA are taxed as ordinary income. More precisely, with-

drawals are taxed on a last-in first-out basis, meaning that earnings are withdrawn before the prin-

cipal. The so-called tax base, denoted by Ht, represents the amount that may still be withdrawn

from the VA tax-free at or after time t. If the account value X−t exceeds the tax base, any with-

drawal up to(X−t −Ht

)+

will be taxed at rate τ and will not affect the tax base. Withdrawals in

excess of(X−t −Ht

)+

, as reductions of principal, are not subject to taxes but reduce the tax base.

Hence, starting with H1 = P , the tax base evolves as:

Ht+1 = Ht −(wt −

(X−t −Ht

)+

)+. (4)

Moreover, any withdrawal prior to age 59.5 carries an early withdrawal penalty of sg (typically

10% of the amount withdrawn).

Upon death (in year t), the beneficiaries receive the concurrent account value as a lump-sum

payout net of taxes:

Xt − τ ×(X−t −Ht

)+. (5)

In our base case specification, we assume that upon survival to maturity, the policyholder takes

out the remaining account value as a lump sum resulting in the payoff (5) at time T . In addition,

we analyze two alternative specifications in which the policyholder annuitizes 50% or 100% of the

account value, respectively. Upon annuitization in level installments, the annual tax-free amount is

calculated as the benefits base over a given life expectancy estimate. Annuity payments in excess

of the annual free amount are taxed as ordinary income (IRS, 2003).

13A constant income tax rate is a mild assumption as holders of VAs typically are relatively wealthy, so that brackets

over which the applicable marginal income tax rate is constant are fairly large. Moreover, we want to avoid withdrawal

behavior being affected unpredictably by “fragile” tax advantages. For capital gains, we abstract from differences in

the taxation of investment income (e.g., dividends or coupon payments) and consider a constant effective rate.


2.3 Valuation Approach

Arbitrage pricing in the presence of taxation is not straightforward. As demonstrated by Ross

(1987), no universal pricing measure exists when tax rates vary for different agents but an agent’s

valuation of a given cash flow depends on the individual endowment and tax rates. Our valuation

approach is predicated on the idea that we can nevertheless identify a unique – albeit individual or

subjective – valuation if the pre-tax financial market for ordinary investments in stocks and bonds

is complete.14 Then, consistent with standard arbitrage pricing arguments, we define the time-zero

value of a given post-tax cash flow as the amount necessary to set up a pre-tax portfolio in stocks

and bonds that – after taxes – replicates this cash flow.15 This valuation rule is subjective in the

sense that it depends on the investor’s current position and tax rates. For instance, if the investor

has additional investments that offset tax responsibilities for the replicating portfolio, the relative

value will change. In what follows, we consider the basic case with no offsetting investments.

For the VA described in Section 2.1, at each policy anniversary date, the policyholder’s decision

is based on observing the concurrent state variables X−t , Gt, and Ht (denoting the VA account,

benefits base, and tax base, respectively). This decision entails choosing the withdrawal amount

wt that maximizes the value of the VA given by the Bellman equation:

Vt(X−t , Gt, Ht) = max

wt

wt − fee− pen− tax + V +

t

, (6)

where withdrawals are constrained by the guarantee values gWt and Gt in case of a withdrawal

within the limits of the GMWB and by the account value otherwise:

0 ≤ wt ≤ maxX−t , min

gWt , Gt

.

14Note that even in the absence of taxes, arbitrage pricing does not give a unique pricing rule if the market is

incomplete.15This approach is similar to that in Sibley (2002), although he only considers deterministic cash flows from

different tax-advantaged retirement accounts.


In Equation (6), “fee” denotes the excess withdrawal fee, “pen” is the early withdrawal penalty,

“tax” denotes (income) taxes associated with the withdrawal, and V +t is the value of the VA after

the withdrawal. According to the specifications outlined in the preceding subsections, we have for

the fees, penalties, and taxes:

fee = st ×(wt −mingWt , Gt

)+

;

pen = sg × (wt − fee)× 1x+t<59.5 ; and

tax = τ ×minwt − fee− pen, (X−t −Ht)+ .

Appendix A shows that under the described valuation approach, the subjective risk-neutral

value at time t of the future VA cash flows, V +t , is given via a non-linear, implicit equation:

V +t = EQ

t

[e−r

(qx+t bt+1 + px+t Vt+1

(X−t+1, Gt+1, Ht+1

))]+

κ

1− κ× EQ

t

[e−r

(qx+t bt+1 + px+t Vt+1

(X−t+1, Gt+1, Ht+1

)− V +

t

)+

].

(7)

Here, bt+1 is the bequest amount in case of death, occurring with probability qx+t:

bt+1 = X−t+1 − τ ×(X−t+1 −Ht+1

)+.

In case of survival (probability px+t), the policyholder receives the continuation value Vt+1, which

is again given by the Bellman equation (6) for t+1 < T and by the terminal condition for t+1 = T .

In our base case, this is simply the lump sum of the remaining account value:

VT (X−T , GT , HT ) = X−T − τ ×(X−T −HT

)+.

The updating equations of the benefits base and the tax base in (7) are given by (2) [or (2’)/(2”)]

and (4) in our base case [alternative] specification. For the account value, the evolution is described

by (1) and (3) [or (3’)], albeit projection is carried out under the (unique) risk-neutral measure

Q associated with the complete pre-tax market for stocks and bonds. As is well-known from


arbitrage/replication pricing, the dynamics can be specified by replacing the drift µ in Equations

(3) and (3’) by the risk-free rate r.

As a benchmark, we also include a basic risk-neutral valuation not considering taxes, which

is the approach used in previous studies on GMWB pricing (see the references in Footnote 2).

The corresponding equations follow analogously by setting the tax rates (κ, τ ) as well as the early

withdrawal penalty (sg) to zero.

2.4 Implementation and Calibration

Since the optimization problem cannot be solved analytically, we use recursive dynamic program-

ming to implement it numerically. More specifically, we discretize the state space, and – for each

point in the discretized state space – compute the (after-tax) value of the payoff at maturity (time

T ). We then proceed recursively by solving the policyholder’s optimization problem (for each

point in the discretized state space) at time T − 1, T − 2, T − 3, . . . , 0, and determine the corre-

sponding subjective contract values. Since the value function Vt+1(.) is only given on a discrete

grid and since the expected values in (7) do not have an analytic solution, we derive approxima-

tions by interpolating between grid points and by discretizing the return space (see Appendix B for

details). We then rely on those approximations to solve the non-linear equation (7) numerically.

Subsequently, taking into account the optimal withdrawal strategy for all time/state combina-

tions, we run Monte Carlo simulations to obtain relevant valuation measures (by simulating under

Q) and withdrawal statistics (by simulating under P).

All parameters for our base case calibration are summarized in Table I. We consider a male

policyholder that purchased a 15-year VA with a return-of-premium GMWB at age 55, with an

initial premium of $100,000 (and no further premium payments). In our sensitivity analysis, we

also consider a purchase age of 60, as well as 20-year policies sold to investors aged 50 and 55.

We assume mortality follows the 2007 Period Life Table for the Social Security Area Population

for the United States. The considered fee and guarantee structures are typical for contracts offered

in practice (Chen et al., 2008, e.g.). For our base case, we assume a short rate of 5 percent with


Table I: Parameter choices for the base case.

Description Parameter Value

Policyholder & contract specificationAge at inception x 55VA principal P 100,000Years to maturity T 15

Annual guaranteed amount gWt 7,000Excess withdrawal fee st 8%, 7%, . . . , 1%, 0%, 0%, . . .

Financial market parametersInterest rate r 0.05Volatility σ 0.19

Tax ratesIncome tax rate τ 30%Capital gains tax rate κ 23%Early withdrawal penalty sg 10%

sensitivity tests for 3 percent and 7 percent, and a volatility of 19 percent – which roughly equals

the historical volatility of the S&P 500 index between 1989 and 2008 – with sensitivity tests at 16

percent and 22 percent.

The policyholder’s effective tax rates are more difficult to calibrate, since there is considerable

heterogeneity in marginal federal income tax and state tax rates, and since there are substantial

differences in the taxation of dividends, interest, and short-term capital gains (taxed as ordinary

income) versus long-term capital gains (taxed at the lower capital gains tax rate). We assume that in

our base case, earnings from the VA are taxed (upon withdrawal) at an income tax rate of 30 percent

(alternatively, we consider 25 percent and 35 percent in the sensitivity analysis), while earnings in

the replicating portfolio are taxed annually at a capital gains tax rate of 23 percent (alternatively,

20 percent and 25 percent), which is in line with relevant U.S. tax rates. Moreover, consistent with

current U.S. tax policy, we impose a 10 percent early withdrawal penalty for withdrawals made

prior to age 59.5 (IRS, 2003).

We present withdrawal statistics (computed under the physical probability measure P) assum-

ing Sharpe Ratios of 45 percent and 25 percent. The first value reflects the average Sharpe Ratio

of the S&P 500 between 1926 and 2000 (Goetzmann et al., 2002), while the latter value may be


more consistent with post-crisis investment performance.

3 Results for a Basic GMWB

3.1 Optimal Withdrawals in the Base Case

For the base case contract, we obtain a fair GMWB fee that equates fee income and option costs

– and, therefore, yields zero profit for the insurer – of approximately 18.55 bps for a (subjective)

value-maximizing policyholder when accounting for taxes. This figure is in line with fees typically

charged in the VA market. In contrast, when not considering taxation, we calculate a fair fee of

about 52 bps, which accords with corresponding findings in the literature but significantly exceeds

market fees for plain GMWB riders (Milevsky and Salisbury, 2006; Chen et al., 2008).

To obtain an intuition for this discrepancy, Figure 1 plots the policyholder’s optimal withdrawal

strategy with and without taxes as a function of the account value at a fixed value of the benefits

and tax bases for different points in time and two different financial market parametrizations. The

results in the absence of taxes (blue dashed curves) illustrate that the key drivers for the withdrawal

decision are the option value of the guarantee on the one side and the guarantee fees on the other

side. Early in the contract period (panel (a), t = 4), the policyholder always withdraws at least

the guaranteed amount. According to the adjustment in the benefits base (2), it is possible to

increase the option value by withdrawing excessively at low account values. In this case, the

withdrawal level is chosen in order to reduce the benefits base to a level that leaves merely the

guaranteed amount to be withdrawn in each of the remaining contract years. However, for higher

account values, excess withdrawals are not optimal in the considered range due to protection of

the VA account up to the benefits base as well as surrender fees on excess withdrawals. We find

a similar optimal withdrawal pattern towards the middle of the contract period (panel (b), t = 8),

although here in the high-volatility case (left-hand side) there is an interval where it is optimal

not to withdraw from the account in order to preserve option value. The situation changes later

in the contract period when excess withdrawals are no longer subject to surrender fees (panel (c),


t = 12). While withdrawals for relatively low account values are still motivated by maximizing

the value of the protection option, the policyholder now surrenders the VA for large account values

in order to avoid paying for an out-of-the-money guarantee.

In contrast, with taxes (red solid curves in Figure 1), the policyholder’s decision problem is

supplemented by two additional considerations. First, withdrawals are subject to income taxes

as well as the early withdrawal penalty and, second, funds invested inside the VA are subject

to deferred taxation. Both aspects incentivize the policyholder to withdraw less frequently and

smaller amounts than in the absence of taxes. More precisely, we observe that withdrawing is

usually suboptimal whenever the account value exceeds the tax base Ht, since taxes would be due.

In the context of Figure 1, these are exactly the situations where the guarantee is out-of-the-money,

i.e. where the account value exceeds the benefits base, since the benefits and tax bases are chosen

at the same values. This is not atypical, since the two bases frequently evolve in lockstep. Indeed,

they will coincide as long as there are no excess withdrawals and no withdrawals when X−s > Gs,

s ≤ t. It is still optimal to withdraw the guaranteed amount – or even to withdraw excessively

– when the account value has fallen (significantly) below the tax base in order to maximize the

option value of the guarantee. However, also in this range, withdrawals are less frequent due to 1)

the 10 percent early withdrawal penalty before age 59.5 (panel (a), t = 4), and 2) tax deference

that favors inside the VA relative to outside investment (panel (b), t = 8). The interaction of the

various effects leads the optimal withdrawal profile for the high-volatility case late in the contract

period to be characterized by four disjoint intervals (left-hand side of panel (c), t = 12): For low

account values, the policyholder withdraws (excessively) to maximize the option value; at some

point, tax-deference plus the protectional aspect of the guarantee yield the policyholder to defer

withdrawals; for even greater account values, the reduction in option value, fees inside the VA, and

the lower capital gains tax rate for outside investments again trigger the policyholder to withdraw

excessively; and for values beyond the tax base, withdrawing is suboptimal since the withdrawals

are subject to income taxes, whereas growth is tax-deferred inside the VA.

Tracing these relatively complicated optimal withdrawal profiles demands a high (and poten-


r = 5%, σ = 19% r = 3%, σ = 16%

0 50 100 150 200 2500

50

100

150

200

250

X4−

w4

0 50 100 150 200 2500

50

100

150

200

250

X4−

w4

(a) t = 4

0 50 100 150 200 2500

50

100

150

200

250

X8−

w8

0 50 100 150 200 2500

50

100

150

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250

X8−

w8

(b) t = 8

0 50 100 150 200 2500

50

100

150

200

250

X12−

w12

0 50 100 150 200 2500

50

100

150

200

250

X12−

w12

(c) t = 12

Figure 1: Optimal withdrawal strategies for simple GMWB.The graphs depict the optimal withdrawal amount wt – with (red solid curves) and without (bluedashed curves) taxes – at different times during the contract period as functions of the VA accountvalue X−t . All units are in $1,000s. For illustrative purposes, the benefits base and tax base areheld constant at the amount of the initial investment, Gt = Ht = 100. The GMWB fee is set as thebreak-even fee with taxes, that is φ = 18.55 bps on the left and φ = 34.25 bps on the right-handside.


tially unrealistic) level of consumer sophistication. However, we note that much of the complexity,

particularly in view of excess withdrawals, is due to the adjustment of the guarantee account (2).

While this specification is common in the actuarial literature, most recent contract designs – and

especially all empirical contracts considered in Section 4 – obey the alternative specifications (2’)

or (2”). In the case of (2’), we obtain the same basic optimal patterns with taxation – withdrawals

are optimal only if the account value is less than the tax base, and more likely if the difference is

larger – but we do not observe optimal excess withdrawals.

This basic pattern conforms with available information on dynamic functions describing GMWB

policyholder behavior as a function of in-the-moneyness of the guarantee, which are based on em-

pirical exercise patterns and used by most companies (Society of Actuaries, 2008). More precisely,

the Variable Annuity Reserve Work Group that reports to the U.S. regulator (NAIC) “on the ongo-

ing work surrounding the development of a reserve methodology for Variable Annuity products,”

in their Analysis Report, Attachment 5: Modeling Specifications, collects Experience Assumptions

with regards to GMWB riders as considered here (American Academy of Actuaries, 2005). These

stipulate that there are no modifications to the surrender rate in the presence of a GMWB unless the

guarantee is significantly in-the-money, in which case they are adjusted downwards.16 Hence, un-

like the situation in Figure 1(c) in the case without taxes, their assumptions do not reflect additional

surrenders due to the GMWB option. The utilization of the GMWB is limited to withdrawals of the

guaranteed amount, which again depends on in-the-moneyness for 80 percent of all policyholders.

More precisely, different partitions of policyholders will start withdrawing at different thresholds

of exceedance of the benefits base over the account value, but there are no triggers to start with-

drawing if the guarantee is out-of-the-money. It is important to note that – given that withdrawals

are limited to the guaranteed amount (or full surrenders) and withdrawals are only prevalent when

the guarantee is in-the-money – the benefits base and the tax base coincide so that moneyness with

16For VAs – as for conventional, front-loaded life insurance contracts – it is standard to include a deterministic

(time-dependent) surrender schedule due to exogenous factors such as liquidity constraints. Thus, for practical appli-

cations, it may be suitable to augment our optimal withdrawal profiles by such an exogenous surrender schedule to

capture aspects outside of the model.


regards to both is equivalent. There is no value-based explanation for these Experience Assumption

profiles in the absence of taxes; we would expect to see withdrawals and even a large number of

complete surrenders for high levels of the account value relative to the guarantee.

Table II demonstrates that this fundamental difference in withdrawal behavior materializes in

much lower aggregate withdrawals during all contract periods when taxes are considered (e.g.

$11,310 vs. $191,780 on aggregate for the base case parameters and a Sharpe Ratio of 25 percent),

and it also leads to a much lower surrender rate (1.0 percent vs. 71.2 percent). Fewer withdrawals

across the board mean that the GMWB is utilized less frequently, and it is thus less valuable, while

the insurer collects fees for a longer period of time. This leads the insurer to break even under the

optimal behavior for the (subjective) risk-neutral valuation approach with taxes – as is clear since

this is how the fair fee was determined – as opposed to losing around two percent of the initial

investment for a policyholder withdrawing optimally in the absence of taxes.

A few aspects warrant additional explanations. First, the positive (though small) surrender fee

in the with taxes case relative to no surrender fees at all without taxes may seem to contradict the

optimal withdrawal profiles in Figure 1. However, this apparent contradiction is resolved when

contemplating foregoing periods: Withdrawing the guaranteed amount is optimal without taxes in

all scenarios, so there are no optimal paths leading to the situation depicted in the figure with the

benefits/tax base equaling the initial investment later in the contract period. Nonetheless, one can

interpret the figure as a GMWB with an alternative initial investment, for which the figure may

align with optimal behavior. Second, aggregate withdrawals without taxes for years 1 through 4

are always 27,400 for T = 15, irrespective of the parameters. This is due to the policyholder

always withdrawing the full guaranteed amount (4 × 7,000 = 28,000). The difference of 600 is

explained by the possibility of dying in this period. Third, aggregate withdrawals appear to be

lower in the later contract years in the without taxes case. This of course is a consequence of

very high withdrawals in preceding contract years, so that the remaining balance is low and even

completely depleted in many scenarios. Finally, the surrender rate increases for the higher Sharpe

Ratio simply because high account value realizations, for which surrendering is optimal (Fig. 1,


Table II: Valuation and withdrawal statistics for a simple GMWB.The table presents the insurer’s profit calculations (under the risk-neutral measure Q) and with-drawal statistics (under the physical measure P) for a VA with a simple return-of-premium GMWBunder the base case contract specification and different parameters. All values are based on the pa-rameter assumptions displayed in Table I unless noted otherwise, and the annual guarantee fee iscalibrated to make the insurer break even for a (subjective) value-maximizing policyholder withtaxes.

r/σ/T 5% / 19% / 15 3% / 16% / 15 5% / 19% / 20

φ 18.55 bps 34.25 bps 16.60 bps

with taxes w/out taxes with taxes w/out taxes with taxes w/out taxes

Contract valuation

GMWB fees 2,370 1,270 4,040 2,270 2,740 1,240Surrender fees 30 0 50 0 0 0

Costs of GMWB 2,400 3,230 4,090 4,760 2,740 3,420

Insurer’s profit 0 -1,960 0 -2,490 0 -2,180

Withdrawals

(a) Sharpe Ratio: 25% µ = 9.75% µ = 7.00% µ = 9.75%

- Aggregate 11,310 191,780 20,100 152,530 7,830 199,490

- Years 1 to 4 420 27,400 1,020 27,400 130 21,140- Years 5 to 8 3,340 152,720 8,230 113,510 1,660 157,500- Years 9 to 15 7,550 11,660 10,850 11,620 2,690 19,230- Years 16 to 20 — — — — 3,350 1,620

- Regular 5,110 54,320 9,630 57,400 5,050 45,400- Excess 6,200 137,460 10,470 95,130 2,780 154,090

Surrender rate 1.0% 71.2% 1.6% 68.1% 0.4% 70.6%

(b) Sharpe Ratio: 45% µ = 13.55% µ = 10.20% µ = 13.55%

- Aggregate 3,360 260,460 7,390 194,110 7,550 270,140

- Years 1 to 4 180 27,400 490 27,400 50 21,470- Years 5 to 8 1,070 227,580 3,220 160,970 490 238,090- Years 9 to 15 2,110 5,480 3,680 5,740 710 10,090- Years 16 to 20 — — — — 6,300 490

- Regular 1,480 52,490 3,430 54,090 1,150 44,070- Excess 1,880 207,970 3,960 140,020 6,400 226,070

Surrender rate 0.3% 80.5% 0.8% 77.5% 0.1% 80.9%


Table III: Sensitivity of fair GMWB fee to changes in financial market parameters.The table displays the fair fee rates for a return-of-premium GMWB, for different combinationsof the interest rate r and the fund volatility σ, if the policyholder is a (subjective) value maximizerwith (left-hand side) and without (right-hand side) consideration of taxes. Other parameters are setto the values from Table I. Annual fees are quoted in basis points and charged continuously as apercentage of the concurrent VA account value. We define the fair fee as the fee where the insurerbreaks even (zero profit).

Fair GMWB fee (in bps)

with taxes without taxes

σ = 16% σ = 19% σ = 22% σ = 16% σ = 19% σ = 22%

r = 3% 35 55 78 80 120 169r = 5% 10 19 30 32 52 76r = 7% 3 6 12 12 23 37

panel (c)), are more likely under the physical measure – and this is also the reason for higher

aggregate withdrawal statistics in this case.

3.2 Sensitivity Analysis

Sensitivity to Financial Market Parameters

Overall, we find that qualitatively, optimal withdrawal behavior is consistent across different para-

metrizations of our model. In particular, for any combination of parameters, we find that the con-

sideration of taxes significantly affects the withdrawal behavior – and thus the fair fee. However,

the quantitative impact of different financial parameter assumptions on the fair fee level is substan-

tial. To illustrate, Table III shows fair fees with and without taxes for different combinations of the

risk-free rate r and the fund volatility σ.

As might be expected, generally the fair fee rate increases in the fund volatility and decreases

in the interest rate level. These sensitivities are common for return-of-premium guarantees and

resemble those of a put option: When the underlying investment grows faster on average (high

r), the guarantee is less likely to materialize, whereas an increase in volatility yields a higher

likelihood for the guarantee to be (deep) in-the-money at maturity. These results point to the strong


adverse effects for VA providers offering GMWBs in a low interest / high volatility environment as

observed in the recent past, particularly through the crisis. This is congruent with recent efforts by

insurers to either modify the offered withdrawal guarantees (e.g., by increasing the fees); to limit

the risk exposure of potential investments, particularly when guarantees are elected; or to cease

offering such guarantees altogether (Chopra et al., 2009; Geneva Association, 2013).

As already indicated, we find that the guarantee fee is significantly lower in the with taxes case

for all interest rate / volatility combinations. More precisely, the fee without taxes ranges from

a little more than twice the fee with taxes for the low interest / high volatility assumption to four

times the fee for the high interest / low volatility combination. However, with more than 90 bps, the

absolute difference is largest in the former (low r / high σ) case. It is interesting to note that while

the sensitivities qualitatively and the fair fees in the without taxes case quantitatively conform with

corresponding results for the fair fee in the previous literature on GMWB pricing (97 to 160 bps

in Milevsky and Salisbury (2006); 40 to >100 bps in Bauer et al. (2008); 95 to 214 bps in Chen et

al. (2008)), our with taxes results fall in the range of fees charged in the market as reported in this

literature (30 to 45 bps in Milevsky and Salisbury (2006); 40 to 65 bps in Bauer et al. (2008); less

than 50 bps in Chen et al. (2008)).

Sensitivity to Tax Rates

Table IV presents fair fee rates for different combinations of the income tax rate τ and the capital

gains tax rate κ. We find that the guarantee fee increases in the income tax rate τ but decreases

in the capital gains tax rate κ. The intuition is that, ceteris paribus, a higher (income) tax rate on

VA earnings leads the VA to lose some of its tax advantage over outside investments, increasing

the likelihood for the policyholder to withdraw prematurely – which in turn increases the value of

the guarantee and reduces fee payments. Conversely, an increase in the (capital gains) tax rate on

earnings from the replicating portfolio renders the VA more attractive, and the policyholder will

be less inclined to withdraw money early on. Nonetheless, the relatively small magnitude of the

variations in the fair fee across different tax regimes (between 14 bps and 24 bps in contrast to 52


Table IV: Sensitivity of fair GMWB fee to changes in tax parameters.The table displays fair fee rates for a return-of-premium GMWB, for different levels of the incometax rate τ and capital gains tax rate κ, if the policyholder is a subjective value maximizer. Otherparameters are set to the values from Table I. Annual fees are quoted in basis points and chargedcontinuously as a percentage of the concurrent VA account value. We define the fair fee as the feewhere the insurer breaks even (zero profit). The fair fee without taxes (τ = κ = 0) is approximately52 bps.


τ = 25% τ = 30% τ = 35%

κ = 20% 20 22 24κ = 23% 16 19 21κ = 25% 14 17 20

bps in the absence of taxes) illustrates that irrespective of the absolute level of the applicable tax

rates, the opportunity for tax-deferred investment considerably affects optimal withdrawal behavior

and leads the policyholder to remain invested in the VA.

Sensitivity to Contract Specifications

Table V provides fair fee rates for various contract specifications for the with and without taxes

cases. The first four rows collect results for different combinations of the policyholder’s age (x)

and contract maturity (T ). Interestingly, the sensitivities with respect to x and T go in different

directions between the with and without taxes cases although the effects are relatively small. For

age x, mortality probabilities increase as x increases, so that the likelihood of being able to take

advantage of the guarantee decreases in the absence of taxes – and, thus, so does the fair fee.

However, when accounting for taxes – since the 10 percent early withdrawal penalty does not

apply after age 60 – older policyholders have an increased incentive to withdraw money from the

VA, leading to an increase in the fee in this case. For an increased maturity T , policyholders have

more time to take advantage of the guarantee; while this potentially leads to a higher fee income,

it also provides the possibility for the policyholder to optimize usage of the guarantee and enjoy

protection for a longer time period. As demonstrated in Table II by comparing columns 2/3 with


Table V: Sensitivity of fair GMWB fee to changes in contract specifications.The table displays the fair fee rates for a return-of-premium GMWB, under various contract spec-ifications. These include: variations in the policyholder’s age at purchase (x) and maturity (T );whether the VA account is fully or partially annuitized at maturity; a dynamic investment strategy,where the policyholder reduces equity exposure over time (here we assume a linear reduction of 5percent per year, from 100 percent exposure to the S&P 500 index in year 1 to 30 percent in year15); a GMWB rider with fees assessed in proportion to the remaining benefits base Gt rather thanthe VA account value Xt; and two GMWB products with alternative specifications of adjustmentsto the benefits base in case of excess withdrawals (2’) and (2”). Other product specifications andparameters are consistent with our earlier discussion and Table I. We define the fair fee as the feewhere the insurer breaks even (zero profit).


with taxes without taxes

Base case 19 52

x = 50, T = 20 16 57x = 55, T = 20 17 54x = 60, T = 15 21 48

Annuitize 100% 14 52Annuitize 50% 16 52

Reduce Equity 12 39

Fee prop. to benefits base 29 67

Adjustment to Gt as in (2’) 12 53Adjustment to Gt as in (2”) 21 55


columns 6/7, the former aspect dominates in the with taxes case leading to a lower fair fee, whereas

without taxes the second aspect overbears, increasing the guarantee fee level.

Our base case assumption is that upon survival until maturity, the policyholder takes out the

remaining VA account value as a lump sum payout. Alternatively, the policyholder may choose to

annuitize all or part of the account value. Clearly, in the absence of taxation, this possibility will

not affect the fee since annuitization does not entail additional guarantees. However, since annu-

itizing VA accounts enjoys preferred tax treatment (cf. Section 2.2), it improves the policyholder’s

valuation of the terminal benefit and therefore puts a further penalty on withdrawing prematurely

– reducing early withdrawals and thus also the fair fee, as illustrated in rows 5 and 6 of Table V.

As the next sensitivity, we consider an underlying fund that follows a dynamic investment

strategy by annually decreasing the proportion of equity (from 100 percent in year one to 30 percent

in year 15). Such a structure is akin to life-cycle or target funds that entail a reduction of exposure

to risky investments, to e.g. account for reductions in human capital as the policyholder ages. As

is evident from Table V, the fair fee decreases in both cases. This decrease is in line with the

sensitivities to volatility (Table III), as a reduction of equity exposure effectively results in a time-

varying, decreasing volatility parameter.

Over the years, some VA providers have started offering riders against guarantee fees that are

charged in proportion to the remaining benefits base Gt rather than the VA account value Xt. This

modification partially mitigates the misalignment of fees and liabilities when fees are charged as a

proportion of the account value (Bauer et al., 2008) – since here fee income is low when guarantee

liabilities are high and vice versa. From row 8 of Table V, we see that the fair fee rate increases in

both cases. This is intuitive for our simple return-of-premium GMWB as – ignoring withdrawals

– the fund grows in expected value terms whereas the benefits base stays at the same level.

Finally, we consider the impact of the two alternative specifications of the adjustments to the

benefits base Gt in the case of an excess withdrawal described by (2’) and (2”). Table V illustrates

that for specification (2”), the effect is minor and leads to a small increase in the fee levels. This

is not surprising, since the equations both entail pro-rata adjustments when the guarantee is in-


the-money – which is where we observe excess withdrawals that are not full surrenders. However,

the alternative specification is slightly more generous since pro-rata adjustments only affect the

excess withdrawal amount, leading to a small increase in withdrawals and thus the fair fee rate.

For adjustment (2’), on the other hand, the effect is more substantial in the with taxes case. The

key insight is that here we do not obtain excess withdrawals to optimize the option value, since

the benefits base is adjusted (down) to the account value in case of an excess withdrawal. Thus,

(excess) withdrawals are less frequent, leading to a lower fair fee rate. In the without taxes case,

however, this reduction in excess withdrawals is counteracted by an increase in full surrenders,

effectively increasing the guarantee fee.

Therefore, all in all, while modifications can have a significant impact, they do not change the

key observations: Tax considerations substantially affect optimal withdrawal behavior and lead to

a far lower fair fee for a simple withdrawal guarantee.

4 Results for Empirical GMWB Products

VA plus GMWB products offered in the marketplace typically are significantly more complex than

the simple contracts outlined in the previous sections (usually VA contract descriptions are sev-

eral hundred pages in length). They differ in their fee structure, the guaranteed annual withdrawal

amount, the aggregate guaranteed amount, the availability and frequency of step-ups, the adjust-

ment of the benefits base upon excess withdrawals, etc. To analyze the impact of these various

modifications on the contract valuation, and particularly to assess the empirical performance of

the proposed pricing approach, we implement 13 VA plus GMWB products from seven different

providers offered at different points in time. The first subsection provides more details on these

modifications and describes all the considered contract designs as well as their implementation in

an extended version of the model from Section 2. The second part presents our valuation results.

REVISITING THE RISK-NEUTRAL APPROACH TO OPTIMAL POLICYHOLDER BEHAVIOR 29Ta

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4.1 Product Descriptions

The VA market has changed drastically over the last decade, particularly in the wake of the financial

crisis. To account for these changes, we consider two generations of GMWB riders: some offered

on Jan. 1, 2007 and some offered on Jan. 1, 2014. A summary of the key features of these contracts

is presented in Table VI.

Each VA carries a Mortality & Expense Risk Charge and an Administration Charge, which

we combine in the VA fee provided in column 5 of Table VI. This annual fee rate is deducted

continuously in proportion to the VA account value Xt and covers the insurer’s expenses and

administrative costs. These expenses include compensation for a return-of-premium GMDB that

is automatically embedded in all considered VA products. Thus, if the policyholder dies before

the VA matures, policy beneficiaries will receive the greater of the VA account value at the time

and the initial premium (minus adjustments in the case of withdrawals). Another feature that all

considered contracts share is a premature annuitization option. More precisely, the policyholder

can elect to annuitize the VA policy prior to maturity subject to age restrictions (we consider the

possibility to annuitize every year after age 65). In that case, the insurer converts the VA account

value at the time into a whole life annuity with level annual benefit payments. Otherwise, the

policy matures and is annuitized automatically on the policyholder’s 95th birthday.

Changes in the interest rate environment may explain revisions in fees for these automatic

guarantees between 2007 and 2014. Moreover, some of the substantial differences in VA fees

across products/providers can be attributed to discrepancies in surrender fees, including the time

period they are assessed (Column 9 in Table VI), exempt withdrawal levels (Note 2 in Table VI),

and the absolute fee levels (we refer to the product prospectuses for details). However, another

part of the inhomogeneity is due to differences in investment fees and (unobserved) variations in

expense structures and profit margins across companies.

Column 6 in Table VI gives the GMWB rider fee. As is evident, this fee also varies substantially

across products, at levels between 20 bps and 105 bps. In particular, fees have noticeably increased

between 2007 and 2014, which – at least to some extent – can be attributed to the decline in


interest rates (see the sensitivities in Table III). The variation of GMWB fees within each contract

generation is influenced by differences in the underlying VA as well as different compositions

of features in the GMWB rider. These differ in their fee structure, including whether fees are

charged as a percentage of the account value or the benefits base (Column 6 of Table VI) and fee

forgiveness (Note 1 in Table VI); the level of the guarantee (Column 7, Note 4, and Note 5 in

Table VI); and how the benefits base is adjusted in case of an excess withdrawal (MN, RAVA, and

RAVA< adhere to adjustment mechanism (2’); ASL2, AG5, AG6, HF, HFP, SB5, SB6, and SB7

follow (2”); and CIA4S as well as CIAS apply (2”) to the annual withdrawal amount). Moreover,

most contract designs contain step-up or ratchet options, which allow the policyholder to increase

the benefits base to the VA account value at certain policy anniversaries (Column 8 of Table VI).

These step-ups may also increase the guaranteed annual amount.

With the exception of the CIA4S and CIAS, all considered products also contain a spousal

continuation option that provides the possibility for a surviving spouse to continue the GMWB

rider. We model this by assuming that upon the policyholder’s death, the beneficiaries choose

between receiving 1) the current VA account value as a lump sum; 2) the initial investment adjusted

for withdrawals (due to the basic GMDB rider embedded in the VA policy); and 3) the continuation

of the GMWB rider in the form of annual payments of the guaranteed annual amount until the

benefits base depletes. As described in Section 2.2, all past and future earnings are taxed upon

withdrawal, so that this decision is based on which alternative has the highest present value under

subjective risk-neutral valuation with taxes.

We implement the contracts one-by-one in an extended version of the model introduced in

Section 2. To reflect the financial circumstances surrounding VA investments in 2007 and 2014,

we use interest rates implied by the yield curves from Jan. 1, 2007, and Jan. 1, 2014, for the

two generations of contracts, respectively. While VA contracts differ in the available investment

opportunities, they all entail restrictions with regards to the proportion of equity investments –

particularly when including a GMWB. To account for these investment constraints, we rely on

a typical restriction and assume that only three quarters of the investments can be allocated into


the S&P 500 – which effectively results in a fund volatility of 14 percent based on our base case

calibration from Section 2.4. We also report results for an increased volatility of 16 percent. We

take mortality rates from the 2012 Individual Annuity Mortality (IAM) Basic Table – Male for a

55-year old male investor. As for the simple GMWB in Sections 2 and 3, we assume an initial

investment of $100,000, and tax rates of τ = 30 percent for income (including withdrawals from

the VA), and κ = 23 percent for capital gains (earnings in the replicating portfolio).

4.2 Valuation Results

Valuation results for the thirteen VA plus GMWB products from Table VI are presented in Table

VII. For the with and without taxes cases, we report the (subjective) risk-neutral value to the

policyholder (V0); the VA, GMWB, and surrender fee income (risk-neutral present value) to the

insurer; corresponding VA and GMWB costs (risk-neutral present value) to the insurer; and the

surplus from offering the GMWB to the insurer calculated as the associated fees minus the costs.

In the absence of taxes, of course V0 – the value of the contract to the policyholder – can

be calculated as the initial premium minus fees plus VA/GMWB costs, which are benefits to the

policyholder. This is no longer true with taxes. Here, the VA fees (on average $11,740) are far

larger than all cost components combined, leading to a significant positive risk-neutral value of all

cash flows from the perspective of the VA provider. At the same time, the subjective risk-neutral

value to the (value-maximizing) policyholder is slightly positive in all but one case, and also here

the shortfall is a mere $130 or 0.13 percent of the premium. This wedge between the policyholder’s

and the company’s valuation is a consequence of taxation, especially the deferred tax treatment of

the VA investment.

As a consequence, when not considering taxes, a value-maximizing consumer will not pur-

chase any of the VAs. And conditional on being endowed with one of them, policyholders will

surrender their policies in most scenarios, similarly to the analysis in Section 3. In contrast, with

taxation, (subjective) value-maximizing policyholders will choose to enter the policies – or, in

other words, the VA market appears to be incentive compatible. Indeed, the observation that the


Table VII: Valuation results for empirical GMWB contracts from Table VI with σ = 14%.

2007 Contracts: ASL2 HF HFP MN SB5 SB6 SB7

With Taxes

V0 99,870 102,300 103,490 102,300 102,820 101,690 100,740

VA fees 14,640 10,720 11,800 11,840 10,290 10,710 11,010GMWB fees 2,770 4,620 2,040 4,690 4,670 4,860 5,000Surrender fees 0 0 0 0 0 0 0

VA cost 610 550 570 660 490 530 710GMWB cost 4,100 4,250 1,160 4,200 7,120 4,620 2,340

Surplus from GMWB -1,330 370 880 490 -2,450 240 2,660

Without Taxes

V0 98,330 95,400 94,630 95,080 97,920 95,330 94,830

VA fees 2,550 6,240 6,150 6,930 6,620 5,970 5,640GMWB fees 540 2,690 1,060 3,210 3,010 2,710 2,560Surrender fees 0 30 30 10 880 1,240 1,370

VA cost 190 320 410 340 370 360 330GMWB cost 1,230 4,040 1,460 4,890 8,060 4,890 4,070

Surplus from GMWB -690 -1,350 -400 -1,690 -5,050 -2,180 -1,510

2014 Contracts: AG5 AG6 CIA4S CIAS RAVA RAVA< (σ = 10%)

With Taxes

V0 100,530 100,240 101,880 100,140 104,520 102,410

VA fees 11,290 10,790 15,460 16,670 8,550 8,790GMWB fees 8,290 9,620 3,250 6,990 9,360 5,510Surrender fees 0 0 0 0 0 0

VA cost 1,510 1,320 1,810 2,060 3,060 2,630GMWB cost 6,470 8,830 6,050 7,940 9,520 3,350

Surplus from GMWB 1,820 790 -2,800 -950 -160 2,160

Without Taxes

V0 94,130 95,590 95,550 94,100 100,100 97,350

VA fees 6,960 7,600 6,760 6,330 6,330 6,490GMWB fees 5,840 7,420 1,420 2,650 6,920 4,070Surrender fees 1,130 110 0 0 0 0

VA cost 1,140 950 940 900 2,640 2,400GMWB cost 6,920 9,770 2,790 2,180 10,710 5,510

Surplus from GMWB -1,090 -2,350 -1,430 470 -3,790 -1,440


(subjective) risk-neutral value is close to the initial premium indicates that VA pricing is approxi-

mately demand-based.

The situation is different for the GMWB. Here, in the with taxes case, the VA providers roughly

break even when offering the (optional) rider. More precisely, the average surplus from offering

the VA is roughly $132 (0.13 percent of the premium) with an average deviation of $1,280 (1.28

percent of the premium). These figures are similar across the two contract generations. In contrast,

without taxes, the average surplus is -$1,730 (-1.7 percent of the premium) with an average devia-

tion of $990 (0.99 percent of the premium). This is again in line with Section 3 and the previous

literature on GMWB pricing in that the market GMWB fees do not equalize associated costs and

fee income: The fair fee for a value-maximizing policyholder without the consideration of taxes

will be larger for all but one of the GMWB riders (the CIAS).

Remaining deviations for some of the products may in part be attributable to heterogeneity in

investment restrictions. For instance, Table VIII presents results for an increased volatility of 16

percent. While the qualitative differences between the with and without taxes cases are similar

to Table VII, we observe that generally the surplus from the GMWB decreases. In particular, the

three products with the largest with taxes surplus from offering the GMWB for σ = 14 percent,

AG5, RAVA<, and SB7, now are substantially less profitable with regards to the GMWB, with

surplus from the latter two changing signs. Of course, similarly, a negative with taxes GMWB

surplus may turn positive when decreasing the volatility level for some of the products.

Moreover, additional aspects such as exogenous surrenders due to liquidity constraints or hard-

ship may further affect withdrawal (or, at least, surrender) behavior and thus the valuation. How-

ever, the (sole) consideration of taxes produces valuation results that explain consumer participa-

tion and that are, at least to first order, consistent with withdrawal patterns and fair pricing of the

optional GMWB riders in the marketplace. Hence, at the very least, our approach provides a new

benchmark for VA pricing – improving on conventional risk-neutral valuation methods applied in

the literature.


Table VIII: Valuation results for empirical GMWB contracts from Table VI with σ = 16%.

2007 Contracts: ASL2 HF HFP MN SB5 SB6 SB7

With Taxes

V0 102,200 105,870 106,530 106,280 105,400 104,420 104,540

VA fees 15,300 15,690 16,770 17,090 12,090 13,280 14,320GMWB fees 2,990 6,770 2,900 7,750 5,490 6,040 6,510Surrender fees 0 0 0 0 0 0 0

VA cost 780 730 870 740 660 710 740GMWB cost 6,300 8,530 1,830 10,700 9,370 7,380 7,810

Surplus from GMWB -3,310 -1,760 1,070 -2,950 -3,880 -1,340 -1,300

Without Taxes

V0 98,740 96,800 95,390 96,650 99,520 96,770 96,230

VA fees 2,750 6,370 6,230 7,110 6,890 6,340 6,150GMWB fees 580 2,750 1,080 3,360 3,130 2,880 2,800Surrender fees 0 20 30 0 650 950 930

VA cost 260 400 550 420 460 450 410GMWB cost 1,810 5,540 2,180 6,700 9,730 6,490 5,700

Surplus from GMWB -1,230 -2,790 -1,100 -3,340 -6,600 -3,610 -2,900

2014 Contracts: AG5 AG6 CIA4S CIAS RAVA RAVA< (σ = 12%)

With Taxes

V0 102,550 103,200 104,660 103,380 108,050 104,760


VA cost 1,980 1,700 2,120 2,410 1,240 820GMWB cost 8,010 12,210 8,210 10,930 14,830 5,690

Surplus from GMWB 790 -1,620 -4,980 -3,630 -3,270 -140

Without Taxes

V0 95,910 97,590 96,700 94,890 102,580 99,580


VA cost 1,330 1,050 1,460 1,320 2,720 2,490GMWB cost 9,860 12,150 4,360 3,630 13,290 7,710

Surplus from GMWB -2,820 -4,360 -2,780 -660 -6,270 -3,620


5 Conclusion

Various papers in the actuarial and quantitative finance literatures suggest that popular GMWBs

in VA contracts are underpriced in the marketplace – alluding to sub-optimal or irrational policy-

holder behavior. In contrast, we show that when accounting for tax benefits within VA investments,

a value-maximizing approach yields exercise patterns and prices that square well with empirical

evidence.

Our focus is on GMWB pricing. However, of course our valuation approach is not limited

to these guarantees, but may be applied with little modification to other products such as surren-

der guarantees in conventional participating life insurance contracts or other types of GLBs. We

conjecture that for some products, the findings will be similar. However, for other products, we

suspect that the incompleteness of the market with respect to consumption profiles that can be

attained based on existing savings and retirement products will become material – particularly in

higher age ranges. Hence, in these cases, a risk-neutral approach may no longer be appropriate

and it may be necessary to resort to life-cycle modeling. In this context, the question of how to

quantify the level of incompleteness to discern when policyholder behavior is mainly driven by

value maximization (accounting for taxes) and when it is affected by risk-aversion and other be-

havioral factors – possibly following ideas by Koijen et al. (2014) – is an interesting avenue for

future research.

Furthermore, as already indicated in the Introduction, research is necessary to better understand

the drivers of financial innovation in the VA market. In particular, the question of whether new

products are designed with an eye on policyholder needs or to obfuscate is an open and important

problem for policy in view of the size of the VA market.

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Appendix A Valuation of Investments with Differing Taxation

Consider an agent with endowment A and access to regular investments (stocks, bonds, mutual

funds, etc.) that are subject to capital gains taxation. We assume that the pre-tax market for these

investments is complete. Hence, there exists a unique equivalent martingale measure, denoted by

Q, such that the cost of setting up a replicating portfolio for any pre-tax cash flow is given by

its expected discounted value under Q with respect to the numéraire (Bt)t≥0 (savings account).

Furthermore, for simplicity and without much loss of generality, we assume that all cash flows

to the agent are realized at the end of each year only, and that investment gains in the replicating

portfolio are taxed annually at the constant capital gains tax rate κ.

We first focus on a single year (t, t + 1] and consider the (post-tax) cash flow Xt+1 = X at

time t + 1, which originates from some separate investment opportunity and is potentially subject

to different tax rules. We are interested in the time-t value of this investment/cash flow. Denote by

At+1 the investor’s (individual, state-specific, post-tax) endowment at time t+ 1, with given value

At at time t. If an amount Vt is necessary at time t in order to replicate the time t + 1 post-tax

cash flow Y = Xt+1 + At+1 , then we define the (marginal and subjective) value of Xt+1 by

Xt = Vt −At . It is subjective in the sense that it depends on the agent’s endowment and tax rates.

Hence, the valuation problem reduces to determining the (pre-tax) replicating portfolio, and thus

Vt.

Define Z as the corresponding pre-tax cash flow required to attain Y after tax payments, i.e.:

Y = Z − κ× (Z − Vt)+ . (8)

Inverting the function on the right-hand side, Equation (8) may be restated as:

Z = Y +κ

1− κ× (Y − Vt)+ . (9)

On the other hand, since Vt is the cost of setting up the pre-tax cash flow Z, and since the pre-tax


market is complete, we have:

Vt = EQt

[Bt

Bt+1

× Z]. (10)

Thus, combining Equations (9) and (10), we obtain:

Vt = EQt

[Bt

Bt+1

× Y]

+κ

1− κ× EQ

t

[Bt

Bt+1

× (Y − Vt)+], (11)

with the unknown Vt depending on the state of the world at time t. Hence, Equation (11) presents

a (non-linear) valuation rule for Y – and, thus, Xt+1 – which gives a unique value Vt – and, thus,

Xt – as shown by the following result.

Proposition 1. Any time t + 1 post-tax cash flow Xt+1 can be valued uniquely by the investor at

time t, and its time-t value is given by Vt − At , where Vt is the unique solution to Equation (11).

Proof. All that is left to show is the existence and uniqueness of the solution to Equation (11),

which can be represented as:

Vt − EQt

[Bt

Bt+1

× Y]− κ

1− κ

∫Bt

Bt+1

× (Y − Vt)× 1Y >Vt dFQ = 0 ,

where 0 ≤ κ ≤ 1 and 0 ≤ Bt ≤ Bt+1 . Denote the left-hand side by f(Vt). Existence follows

immediately from f(−∞) = −∞, f(∞) = ∞, and continuity of f by the Intermediate Value

Theorem. For uniqueness, it suffices to show that f(.) is strictly increasing. Consider V 2 > V 1.

Then:

f(V 2)− f(V 1) = V 2 − V 1 − κ

1− κ

[∫Bt

Bt+1

(Y − V 2)1Y >V 2dFQ−∫

Bt

Bt+1

(Y − V 1)1Y >V 1dFQ]

= V 2 − V 1 +κ

1− κ

[∫Bt

Bt+1

×

(Y − V 1)1V 1<Y≤V 2 +

(V 2 − V 1)1Y >V 2dFQ]

> 0,


which completes the proof.

To generalize this method for payoffs multiple years ahead, consider again the cash flow

Xt+1 = X at time t + 1. As we have argued, its time-t value is given by Xt = Vt − At ,

which again can be interpreted as a post-tax cash flow. Hence, its value at time t − 1 is given by

Xt−1 = Vt−1 − At−1 , where Vt−1 is the setup cost of a replicating portfolio for the post-tax cash

flow Xt +At = Vt as above.17 Therefore, the time t− 2 value – and similarly the values at times

t − 3, t − 4, . . ., and, eventually, the time-zero value – can be determined recursively by serially

solving the corresponding Equation (11).

This procedure allows us to evaluate every combination of post-tax cash flows uniquely as

the marginal increase required in today’s outside portfolio in order to replicate the aggregate cash

flow. As indicated, this value is subjective in that it depends on the agent’s endowment – and thus

on investment and consumption decisions. In the main text, we consider the basic case with no

offsetting investments. Numerical experiments in the context of a life-cycle model with investment

opportunities outside the VA and where the VA only makes up a fraction of the agent’s wealth show

that the effects of outside investments on the valuation are small (Moenig, 2012).

17Note that for simplicity of exposition, we disregard consumption and income that would change At. Generaliza-

tions are straightforward.


Appendix B Numerical Evaluation of Equation (7)

A key component of the policyholder’s optimization problem is the solution of integral Equation

(7). However, since the values for V −t+1(.) are given only on a discrete grid, this equation cannot be

solved analytically. We solve the integral numerically by discretizing the underlying return space.

Since the integral entails the standard normal distribution function, one conventional approach

for its numerical solution is to rely on a Gauss-Hermite quadrature. To ascertain the accuracy of

our approximation, we additionally consider a second approach building on ideas by Tanskanen

and Lukkarinen (2003). Note that the integral entails terms of the form:

K =

∞∫−∞

φ(u)F (γ(u)) du , (12)

where φ(u) =1√2π

exp

(1

2u2)

is the standard normal density function,

γ(u) = exp

(σu+ r − 1

2σ2

)

corresponds to the annual (risk-neutral) stock return St+1/St, and we have:

F (γ) = qx+t × bt+1(γ) + px+t × V −t+1 (. |γ )

and:

F (γ) =(qx+t × bt+1(γ) + px+t × V −t+1 (. |γ )− V +

t (.))+

for EQ[Y ] and EQ[(Y − V +

t

)+

], respectively. Here, “.” stands for the state variables at the time

the value function is assessed, chosen on the respective grid.

Lemma 1. Dividing the return space (−∞,∞) into M > 0 subintervals [uk, uk+1), for k =

0, 1, . . . ,M − 1, where we set −∞ = u0 < u1 < . . . < uM−1 < uM = ∞, a consistent


approximation of the integral (12) is given by:

K ≈M−1∑k=0

Φ(uk+1)× [ak − ak+1] + exp(r)× Φ(uk+1 − σ)× [bk − bk+1]. (13)

Here, Φ(.) is the standard normal cdf; ak =xk+1 × ψk − xk × ψk+1

xk+1 − xk; and bk =

ψk+1 − ψkxk+1 − xk

for

k = 0, . . . ,M − 1 ; aM = bM = 0 ; xk = γ(uk) represent the gross returns; and ψk = F (xk)

are the function values evaluated at returns xk.

Proof. For arbitrary 0 < x <∞ , we can approximate the function value linearly as:

F (x) ≈M−1∑k=0

(ψk +

x− xkxk+1 − xk

× (ψk+1 − ψk))× 1[xk,xk+1)(x)

=M−1∑k=0

(ak + bk × x)× 1[xk,xk+1)(x) .

Plugging this into Equation (12), we have:

K =

∞∫−∞

φ(u)F (γ(u)) du ≈∞∫

−∞

φ(u)×M−1∑k=0

(ak + bk × γ(u))× 1[xk,xk+1)(γ(u)) du

=M−1∑k=0

uk+1∫uk

φ(u)× (ak + bk × γ(u)) du

=M−1∑k=0

ak ×uk+1∫uk

φ(u) du+ bk ×uk+1∫uk

φ(u)× γ(u) du ,

and since:

φ(u)× γ(u) =1√2π× exp

(−1

2u2)× exp

(σu+ r − 1

2σ2

)= er

1√2π

exp

(−1

2(u− σ)2

)= erφ(u− σ) ,

we obtain:

K ≈M−1∑k=0

ak × [Φ(uk+1)− Φ(uk)] + exp(r)× bk × [Φ(uk+1 − σ)− Φ(uk − σ)].


Reordering of the summation terms yields Equation (13).

With this approach, we have the discretion to choose the number (M − 1) and location (xk) of

all nodes, providing more flexibility than the Gauss-Hermite quadrature method. It is important

to note, however, that the values ψk cannot be calculated directly, but need to be derived from the

value function grid at time t + 1. Here, we rely on multilinear interpolation when necessary. We

find very similar results for both approaches, and therefore limit our presentation in the main text

to results based on the approximation via Equation (13).


100

100u

100 d

Figure 2: Annual evolution of the risky asset (pre-tax).

Table IX: Parameter choices for the 2-Period model.The financial parameters u, d, and r describe the binomial tree and the risk-free investment. Thefee is calculated so as to make the insurer break even if the policyholder withdraws under allcircumstances.

Parameter Value Parameter Value Parameter Value Parameter Value

P 100 gWt 50 q0 0 q1 0u 1.5 d 0.5 r 0.25 φ 6.15%

Appendix C Illustration in a Two-Period Model

To illustrate the impact of deferred taxation on optimal GMWB withdrawal behavior, we consider

the following 2-period binomial model. Each year, the underlying investment fund evolves – on

a pre-tax basis – according to Figure 2. In contrast to Section 2.1, r denotes the effective annual

interest rate, and the GMWB fee φ is charged entirely at the beginning of each period. The initial

investment is P = 100, with guaranteed annual withdrawals of gW1 = gW2 = 50 at times 1

and 2. To simplify the exposition, we fix the financial parameters (see Table IX) and restrict the

policyholder’s withdrawal options to wt ∈ 0, 50 for t = 1, 2.

Valuation of the Withdrawal Guarantee

Since the decision at maturity is obvious, the sole choice variable for the policyholder is whether or

not to withdraw at time 1. This choice is based on the time-1 investment value, which takes on one


of two outcomes, depending on the movement of the underlying asset (“up” or “down”). Figure

3 depicts the evolution of the VA account in the case where the policyholder chooses to withdraw

either in both states or in the “down” state only; the investment tree for other exercise strategies

can be constructed analogously.

Starting with an initial investment of 100, the insurer first deducts 6.15% for the guarantee fee

at the beginning of the first year. The remaining amount is invested in the risky asset for one year,

and – following the assumptions in Table IX – moves to either 140.78 or 46.93. If the policyholder

chooses to withdraw, the VA account value will be reduced by 50 (down to a minimum of 0). The

insurer once more deducts the guarantee fee at t = 1 and invests the remaining amount for another

year. At maturity, if the VA account has fallen below gW2 = 50, the insurer makes up the difference.

With the information provided in Figure 3(a), it can be easily verified that the assumed guaran-

tee fee of 6.15% makes the insurer break even under the assumption of withdrawals in both time-1

states. The time-0 risk-neutral present value of the collected fees and guarantee payouts both equal

9.50. We now determine under what conditions this is the policyholder’s optimal strategy.

Optimal Withdrawal Strategy, Without Taxes

First we consider the policyholder’s optimal withdrawal strategy in the absence of taxes. Here,

the valuation is straight-forward and based on the risk-neutral value of current and future payouts.

More precisely, in the “up” state (with an account value of 140.78), Figure 3(a) shows the account

evolution and payouts if the policyholder withdraws 50, while Figure 3(b) shows the corresponding

results if there is no withdrawal. The value when withdrawing is:

V 50u = 50 +

p∗ × 127.79 + (1− p∗)× 50

1 + r= 136.67 ,


100 93.85

140.78

46.93

90.78

0

85.19

0

127.79

42.60

0

0

127.79

50

50

50

0.00

7.40

50.00

50.00

Fees

w = 50

w = 50

Fees

Fees

Taxes

Guar.

Guar.

Guar.

Issuer:

Issuer:

Issuer:

Issuer:

(a) Policyholder withdraws 50 in both states.

100 93.85

140.78

46.93

140.78

0

132.12

0

198.18

66.06

0

0

198.18

66.06

50

50

0.00

0.00

50.00

50.00

Fees

w = 0

w = 50

Fees

Fees

Taxes

Guar.

Guar.

Issuer:

Issuer:

Issuer:

Issuer:

(b) Policyholder withdraws only in the “down” state.

Figure 3: Evolution of investment account under different withdrawal strategies.


where p∗ = (r − d)/(u − d) = 0.75 denotes the risk-neutral probability of an upwards tick. In

contrast, the value in the “up” state in the absence of a withdrawal is:

V 0u = 0 +

p∗ × 198.18 + (1− p∗)× 66.06

1 + r= 132.12 .

Therefore, the policyholder optimally exercises the withdrawal option at time 1 in the “up” state.

Analogously one can show that it is also optimal to withdraw in the “down” state.

Optimal Withdrawal Strategy, Under Deferred Taxation

Consider now the case when the policyholder bases withdrawal decisions on expected after-tax

payouts. Following the valuation methodology described in Appendix A, we again contemplate

the withdrawal decision in the “up” state.

If the policyholder withdraws in the “up” state (Figure 3(a)), the withdrawal amount will be

subject to taxation. Since the initial investment of 100 can be taken out tax-free, and since earnings

must be withdrawn before the principal, out of the 50 that are withdrawn at time 1, 40.78 are

earnings and 9.22 are tax-free principal. This yields a time-1 net payout of:

Y2 = 9.22 + (1− τ) 40.78 = 50− 40.78 τ .

The withdrawal reduces the tax base for year 2 to H2 = 100 − 9.22 = 90.78 . Hence, if the VA

investment further appreciates to 127.79, the policyholder receives:

Y2 = 90.78 + (1− τ) (127.79− 90.78) = 127.79− 37.01 τ ,

net of taxes. On the other hand, if the account value decreases in period 2, the terminal payout will

be 50 (tax-free).

According to the valuation Equation (11), at time 1 the policyholder values this random time-2


payout Y2 as V1, given implicitly via:

(1 + r)× V1 = E[Y2] +κ

1− κ× E

[(Y2 − V1)+

].

Under our parametrization, we obtain:

E[Y2] = p∗ × (127.79− 37.01 τ) + (1− p∗)× (50) = 108.34− 27.76 τ

and hence:

V1 =108.34− 27.76 τ − 12.50κ

1.25− 0.5κ.

Combined with the net payout from the withdrawal, we thus obtain that the policyholder’s value

is:

V 50u = (50− 40.78 τ) + V1 =

170.84− 78.74 τ − 37.50κ+ 20.39 τ κ

1.25− 0.5κ(14)

when withdrawing 50 in the “up” state at time 1.

Similarly, when the policyholder does not withdraw in the “up” state, we find (following Figure

3(b)) a time-1 value of:

V 0u =

165.15− 73.64 τ − 16.51κ

1.25− 0.5κ. (15)

Combining Equations (14) and (15) leads to the following result:18

Lemma 2. Under taxation, a value-maximizing policyholder will not withdraw in the “up” state

if and only if:

5.1τ + 20.99κ− 20.39 τ κ > 5.69 .

Figure 4 displays the joint region of κ and τ where this inequality is satisfied. We observe

that the withdrawal decision is mainly driven by the capital gains tax rate κ, with the intuition

that a larger value of κ makes outside investments less attractive, thus reducing the benefit from

18Following the same procedure, we can show that withdrawing is optimal in the “down” state under more general

conditions.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

κ

τ

Figure 4: Tax parameters (κ,τ ) for which a subjective value-maximizing policyholder should notwithdraw in the time-1 “up” state.

withdrawing early. This implies that – while withdrawing is always optimal under standard risk-

neutral valuation – for certain tax parameters the policyholder is better off when foregoing the right

to withdraw in the time-1 “up” state. This reduces the insurer’s obligations due to the GMWB and

also increases the fees collected in the second year. As a result, this change in optimal withdrawal

behavior (purely due to the consideration of appropriate tax treatments of the various investments)

allows the insurer to break even at a much lower fee rate of 4.55%.

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