1. Introduction
Variable annuities (VA) with guarantees of living and death benefits are offered by wealth management and insurance companies worldwide to assist individuals in managing their pre-retirement and post-retirement financial plans. These products take advantage of market growth while providing a protection of the savings against market downturns. Similar guarantees are also available for life insurance policies (
Bacinello and Ortu 1996). The VA contract cash flows received by the policyholder are linked to the choice of investment portfolio (e.g., the choice of mutual fund and its strategy) and its performance while traditional annuities provide a pre-defined income stream in exchange for a lump sum payment. Holders of VA policies are required to pay management fees regularly during the term of the contract for the management of their investment portfolios (wealth accounts).
A variety of VA guarantees, also known as VA riders, can be added by policyholders at the cost of additional insurance fees. Common examples of VA guarantees include guaranteed minimum accumulation benefit (GMAB), guaranteed minimum withdrawal benefit (GMWB), guaranteed minimum income benefit (GMIB) and guaranteed minimum death benefit (GMDB), as well as a combination of them, e.g., guaranteed minimum withdrawal and death benefit (GMWDB), among others. These guarantees, generically denoted as GMxB, provide different types of protection against market downturns, shortfall of savings due to longevity risk or assurance of stability of income streams. Precise specifications of these products can vary across categories and issuers. See (
Bauer et al. 2008;
Kalberer and Ravindran 2009;
Ledlie et al. 2008) for an overview of these products.
The Global Financial Crisis during 2007–2008 led to lasting adverse market conditions such as low interest rates and asset returns as well as high volatilities for VA providers. Under these conditions, the VA guarantees become more valuable, and the fulfillment of the corresponding liabilities become more demanding. The post-crisis market conditions have called for effective hedging of risks associated with the VA guarantees (
Sun et al. 2016). As a consequence, the need for accurate estimation of hedging costs of VA guarantees has become increasingly important. Such estimations consist of risk-neutral pricing of future cash flows that must be paid by the insurer to the policyholder in order to fulfill the liabilities of the VA guarantees.
In this article, we focus on GMWDB, which provides a guaranteed withdrawal amount per year until the maturity of the contract regardless of the investment performance, as well as a lump-sum of death benefit in case the policyholder dies over the contract period. The guaranteed withdrawal amount is determined such that the initial investment is returned over the life of the contract. The death benefit may assume different forms depending on the details of the contract. When pricing GMWDB, one typically assumes either a pre-determined (static) policyholder behavior in withdrawal and surrender, or an active (dynamic) strategy where the policyholder “optimally” decides the amount of withdrawal at each withdrawal date depending on the information available at that date.
One of the most debated aspects in the pricing of GMWDB with dynamic withdrawal strategies is the policyholders’ withdrawal behaviors (
Chen and Forsyth 2008;
Cramer et al. 2007;
Moenig and Bauer 2015;
Forsyth and Vetzal 2014). It is often customary to refer to the withdrawal strategy that maximizes the hedging cost of the VA guarantee, that is, the risk-neutral value of the guarantee alone, as the “optimal” strategy. Even though such a strategy underlies the worst case scenario for the VA provider with the highest hedging cost, it may not coincide with the real-world behavior of the policyholder. Nevertheless, the value of the guarantee under this strategy provides an upper bound of hedging cost from the insurer’s perspective. The real-world behaviors of policyholders often deviate from this “optimal” strategy, as is noted in
Moenig and Bauer (
2015). Different models have been proposed to account for the real-world behaviors of policyholders, including the reduced-form exercise rules of
Ho et al. (
2005), and the subjective risk neutral valuation approach taken by
Moenig and Bauer (
2015). In particular, it is concluded by
Moenig and Bauer (
2015) that a subjective risk-neutral valuation methodology that takes different tax structures into consideration is in line with the corresponding findings from empirical observations.
When the management fee of the policyholder’s wealth account is zero, and deterministic withdrawal behavior is assumed,
Hyndman and Wenger (
2014) and
Fung et al. (
2014) show that risk-neutral pricing of guaranteed withdrawal benefits in both a policyholder’s and an insurer’s perspectives will result in the same fair insurance fee.
Feng and Volkmer (
2016) obtains similar results based on an application of an identity of hitting times. Several studies that take management fees into account in the pricing of VA guarantees include
Bélanger et al. (
2009),
Chen et al. (
2008) and
Kling et al. (
2011). In these studies, fair insurance fees are considered from the insurer’s perspective with the management fees as given.
Feng and Volkmer (
2012),
Feng and Jing (
2016),
Feng and Huang (
2016) show that it is possible to obtain closed-form solutions for the valuation and risk measures of guaranteed benefits under certain assumptions including deterministic withdrawal behaviours.
Feng and Vecer (
2017) studies a PDE approach for the calculation of risk-based capital for GMWB. A comonotonic approximation approach for the calculation of risk metrics for VA is considered in (
Feng et al. 2017) where a dynamic lapse rate is taken into account.
Cui et al. (
2017) studies the pricing of VA with VIX-linked fee structure under a Heston-type stochastic volatility model.
Similar to the tax consideration in (
Moenig and Bauer 2015), the management fee is a form of market friction that would affect policyholders’ rational behaviors. Despite a large range of papers mentioned above on VA guarantee pricing with management fees, the important question of how the management fees as a form of market friction will impact withdrawal behaviors of the policyholder, and hence the hedging cost for the insurer, is yet to be specifically examined in a dynamic withdrawal setting. The main goal of the paper is to address this question.
The paper contributes to the literature in three aspects. First, we consider two pricing approaches based on the policyholder’s and the insurer’s perspective. In the literature, it is most often the case that only an insurer’s perspective is considered, which might result in mis-characterisation of the policyholder’s withdrawal strategies. Second, we characterize the impact of management fees on the pricing of GMWDB, and demonstrate that the two afore-mentioned pricing perspectives lead to different fair insurance fees due to the presence of management fees. In particular, the fair insurance fees from the policyholder’s perspective is lower than those from the insurer’s perspective. This provides a possible justification of lower insurance fees observed in the market. Third, the sensitivity of the fair insurance fees to management fees under different market conditions and contract parameters are investigated and quantified through numerical examples.
The paper is organized as follows. In
Section 2, we present the contract details of the GMWDB guarantee together with its pricing formulation under a stochastic optimal control framework.
Section 3 derives the total value function of the contract under the risk-neutral pricing approach, followed by the net guarantee value function in
Section 4. In
Section 5, we introduce the wealth manager’s value function that relates the total value and guarantee value functions and the two optimal withdrawal strategies corresponding to these value functions.
Section 6 demonstrates our analysis via numerical examples.
Section 7 concludes with remarks and discussion.
2. Formulation of the GMWDB Pricing Problem
We begin with the setup of the framework for the pricing of GMWDB contract and describe the features of this type of guarantees. (In the sequel, we refer to the VA contract with GMWDB rider simply as the GMWDB contract, unless explicitly stated otherwise.) The pricing problem is formulated under a general setting so that the resulting pricing formulation can be applied to different contract specifications. Besides the general setting, we also consider a specific GMWDB contract, which will be subsequently used for illustration purposes in numerical experiments presented in
Section 6.
The VA policyholder’s retirement fund is usually invested in a managed wealth account that is exposed to financial market risks. A management fee is usually charged for this investment service. In addition, if the GMWDB rider is selected, extra insurance fees will be charged for the protection offered by the guarantee provider (insurer). We assume the wealth account guaranteed by the GMWDB is subject to continuously charged proportional management fees independent of any fees charged for the guarantee insurance. The purpose of these management fees is to compensate for the wealth management services provided, or perhaps merely for the access to the guarantee insurance on the investment. This fee should not be confused with other forms of market frictions, e.g., transaction costs, if any, that must incur when tracking a given equity index. Given the proliferation of index-tracking exchange-traded funds in recent years, with much desired liquidity at a fraction of the costs of the conventional index mutual funds, see, e.g., (
Agapova 2011;
Kostovetsky 2003;
Poterba and Shoven 2002), regarding these management fees as additional costs to policyholders beyond the normal market frictions seems to be a reasonable assumption.
The hedging cost of the guarantee, on the other hand, is paid by proportional insurance fees continuously charged to the wealth account. The fair insurance fee rate, or the fair fee in short, refers to the minimal insurance fee rate required to fund the hedging portfolio, so that the guarantee provider can eliminate the market risk associated with the selling of the guarantees.
We consider the situation where a policyholder purchases the GMWDB rider in order to protect his wealth account that tracks an equity index
at time
, where 0 and
T correspond to the inception and expiry dates. The equity index account is modelled under the risk-neutral probability measure
following the stochastic differential equation (SDE)
where
is the risk-free short interest rate,
is the volatility of the index, which is time-dependent and can be stochastic, and
is a standard
-Brownian motion modelling the uncertainty of the index. Here, we follow standard practices in the literature of VA guarantee pricing by modelling under the risk-neutral probability measure
, which allows the pricing of stochastic cash flows to be given as the risk-neutral expectation of the discounted cash flows. The risk-neutral probability measure
exists if the underlying financial market satisfies certain “no-arbitrage” conditions. Adopting risk-neutral pricing here assumes that stochastic cash flows in the future can be replicated by dynamic hedging without transaction fees. For details on risk-neutral pricing and the underlying assumptions, see, e.g.,
Delbaen and Schachermayer (
2006) for an account under very general settings.
The wealth account
over the lifetime of the GMWDB contract is invested into the index
S, subject to management fees charged by a wealth manager at the rate
. An additional charge of insurance fees at rate
for the GMWDB rider is collected by the insurer to pay for the hedging cost of the guarantee. We assume that both fees are deterministic, time-dependent and continuously charged to the wealth account. (Sometimes, the insurance fees are charged to the guarantee account mentioned shortly.) Discrete fees may be modelled similarly without any difficulty. The wealth account in turn evolves as
for any
at which no withdrawal of wealth is made. Here,
is the total fee rate. The GMWDB contract allows the policyholder to withdraw from a guarantee account
on a sequence of pre-determined contract event dates,
. The initial guarantee
usually matches the initial wealth
. The guarantee account stays constant unless a withdrawal is made on one of the event dates, which changes the guarantee account balance. If the policyholder dies on or before the maturity
T, the death benefit will be paid at the next event date immediately following the death of the policyholder. Additional features such as early surrender can be included straightforwardly but will not be considered in this article to avoid unnecessary complexities.
To simplify notations, we denote by
the vector of state variables at
t, given by
where we assume that all state variables follow Markov processes under the risk-neutral probability measure
, so that
contains all the market and account balances information available at
t. For simplicity, we assume the state variables
,
and
are continuous, and
and
are left continuous with right limit (LCRL). We include the index value
in
, which under the current model may seem redundant, due to the scale-invariance of the geometric Brownian motion type model (
1). In general, however,
may determine the future dynamics of
S in a nonlinear fashion, as is the case under, e.g., the minimal market model described in
Platen and Heath (
2006).
We define as the life indicator function of an individual policyholder as the following: if the policyholder was alive on the last event date on or before t; if the policyholder was alive on the second-to-the-last event date prior to t but died on or before the last event date; if the policyholder died on or before the second-to-the-last event date prior to t. We assume the policyholder is alive at . The life indicator function therefore starts at , is right continuous with left limit (RCLL), and remains constant between two consecutive event dates. Note that mortality information contained in (RCLL) comes before any jumps of the LCRL account balances and on the event dates, reflecting the situation that any jumps in these account balances may depend on the mortality information. We denote the vector of state variables including as , and we denote by the risk-neutral expectation conditional on the state variables at t, i.e., . Note that the risk-neutral measure is assumed to extend to the mortality risk represented by the life indicator .
On event dates
, a nominal withdrawal
from the guarantee account is made. The policyholder, if alive, may choose
on
. Otherwise, a liquidation withdrawal of
is made. That is,
where
denotes the indicator function of an event, and
is referred to as the
withdrawal strategy of the policyholder. The real cash flow received by the policyholder, which may differ from the nominal amount, is denoted by
. This is given by
where
is the payment received if the policyholder is alive, and
denotes the death benefit if the policyholder died during the last period. As a specific example,
may be given by
Here, the contracted withdrawal
is a pre-determined withdrawal amount specified in the GMWDB contract, and
is the penalty rate applied to the part of the withdrawal from the guarantee account exceeding the contracted withdrawal
. Note that the
term in (
6) accommodates the situation that, at expiration of the contract, both accounts are liquidated, but only the
guarantee account withdrawals exceeding the contracted rate
will be penalized. Excess balance on the wealth account after the guaranteed withdrawal is not subject to this penalty. An example of the death benefit may simply be taken as the total withdrawal without penalty, i.e.,
Upon withdrawal by the policyholder, the guarantee account is changed by the amount
, that is,
where
denotes the guarantee account balance “immediately after” the withdrawal. For example,
may be given by
i.e., the guarantee account balance is reduced by the withdrawal amount if the policyholder is alive and the policy has not expired. Otherwise, the account is liquidated. The guarantee account stays nonnegative, that is,
if chosen by the policyholder must be such that
. The wealth account is reduced by the amount
upon withdrawal and remains nonnegative. That is,
where
is the wealth account balance immediately after the withdrawal. It is assumed that
, i.e., no withdrawals at the start of the contract. Both the wealth and the guarantee account balance are 0 after contract expiration. That is,
The total value function at time
t is denoted by
, which corresponds to the risk-neutral expected value of all cash flows to the policyholder on or after time
t. The remaining policy value after the final cash flow is thus 0, i.e.,
3. Calculating the Total Value Function
We now calculate the policyholder’s total value function
as the risk-neutral expected value of policyholder’s future cash flows at time
. The risk-neutral valuation of the policyholder’s future cash flows can be regarded as the value of the remaining term of the VA contract from the policyholder’s perspective. As mentioned in the beginning of
Section 2, valuation under the risk-neutral pricing approach assumes that the cash flows may be replicated by hedging portfolios without market frictions. This may be carried out by a third-party independent agent, if not directly by the individual policyholder.
Following
Section 2, the total value function on an event date
can be written as
which by (
5) can be further written as
since, if the policyholder died during last period, the death benefit is the only cash flow to receive. Taking the risk-neutral expectation
, we obtain the jump condition
where
is the risk-neutral probability that the policyholder died over
, given that he is alive on the last withdrawal date
. That is,
Here, we assume that the mortality risk is independent of the market risk under the risk-neutral probability measure. Under the assumption that the mortality risk is completely diversifiable, the risk-neutral mortality rate may be identified with that under the real-world probability measure and inferred from a historical life table. (Since an individual policyholder cannot hedge the mortality risk through diversification, risk-neutral pricing of the total value function essentially assumes that the policyholder is risk-neutral toward the mortality risk.) Here, the mortality information over is revealed at , thus at such information is not yet available. This assumption is not a model constraint since all decisions are made only on event dates.
The total value at
is given by the expected discounted future total value under the risk-neutral probability measure, given by
where
is the discount factor. The initial policy value, given by
, can be calculated backward in time starting from the terminal condtion (
12), using (
15) and (
17), as described in Algorithm 1.
As an illustrative example, we assume
,
and
as constants. Since
and
are constants between the withdrawals, the only effective state variable between the withdrawal dates are
. Thus, we can simply write
for
without confusion. We now derive the partial differential equation (PDE) satisfied by the value function
V through a hedging argument. We consider a delta hedging portfolio that, at time
, takes a long position of the value function
V and a short position of
shares of the index
S. Here,
is the partial derivative of
with respect to the second argument, evaluated at
. Denoting the total value of this portfolio at
as
, the value of the delta hedging portfolio is given by
By Ito’s formula and (
1), the SDE for
can be obtained as
for
. Since the hedging portfolio
is locally riskless, it must grow at the risk-free rate
r, that is
. This along with (
18) implies that the PDE satisfied by the value function
is given by
for
and
. The boundary conditions at
are specified by (
12) and (
15). The valuation formula (
17) or the PDE (
20) may be solved recursively by following Algorithm 1 to compute the initial policy value
. It should be noted that (
17) is general, and does not depend on the simplifying assumptions made in the PDE derivation.
Algorithm 1 Recursive computation of |
1: choose a withdrawal strategy |
2: initialize |
3: set |
4: |
5: compute the withdrawal amount by by (4) |
6: compute by applying jump condition (15) with appropriate cash flows |
7: compute by solving (17) or (20) with terminal condition |
8: |
9: end while |
10: return |
4. Calculating the Net Guarantee Value Function
The GMWDB contract may be considered from the insurer’s perspective by examining the insurer’s liabilities to the guarantee, given by the risk-neutral value of the cash flows that must be paid by the insurer in order to fulfill the guarantee contract. We define the net guarantee value function as the time-t risk-neutral value of all future payments on or after t made to the policyholder by the insurer, less that of all insurance fees received over the same period.
The insurance fees, charged at the rate
, is called fair if the total fees exactly compensate for the insurer’s total liability, such that the net guarantee value is zero at time
. That is,
and if
is a constant, its value can be found by solving (
21). The fair insurance fees represent the hedging cost for the insurer to deliver the GMWDB rider to the policyholder. We emphasize here that the net guarantee value may not be equal to the added value of the GMWDB rider to the policyholder’s wealth account, as we will show in
Section 5.
On an event date
, the actual cash flow received by the policyholder is given by (
5). This cash flow is first paid out of the policyholder’s real withdrawal from the wealth account, which is equal to
, the smaller of the nominal withdrawal and the available wealth. If the wealth account has an insufficient balance, the rest must be paid by the insurer. If the real withdrawal exceeds the cash flow entitled to the policyholder, the insurer keeps the surplus. The payment made by the insurer at
is thus given by
To compute
for all
we first note that, at maturity
T, the terminal condition on
G is given by
i.e., no further liability or insurance fee income after maturity. Analogous to (
13) and (
15), the jump condition of
G on event date
is given by
and
where the insurance payments under
and
are given by
and
, respectively. See (
4), (
5) and (
22).
At
, the net guarantee value function
is given by the risk-neutral value of the net guarantee value just before the next withdrawal date
less any insurance fee incomes over the period
, discounted at the risk-free rate. Specifically, we have
Note that the net guarantee value at t is reduced by expecting to receive insurance fees over time. Since this reduction decreases with time, the net guarantee value increases with time over .
To give an example, we again assume constant
,
,
,
. Under these simplifying assumptions, we have
for
. To derive the PDE satisfied by
, consider a delta hedging portfolio that, at time
, consists of a long position in the net guarantee value function
G and a short position of
shares of the index
S. The value of the delta hedging portfolio, denoted as
, is given by
By Ito’s formula and (
1), we obtain the SDE for
as
where
. Since the hedging portfolio
is locally riskless and must grow at the risk-free rate
r, as well as increase with the insurance fee income at rate
(see remarks after (
26)), we must also have
. This along with (
27) implies that the PDE satisfied by
is given by
for
. The initial net guarantee value can thus be computed by recursively solving (
26) or (
29) from terminal and jump conditions (
23) and (
25), as described in Algorithm 2.
Algorithm 2 Recursive computation of |
1: choose a withdrawal strategy |
2: initialize |
3: set |
4: do |
5: compute the withdrawal amount by (4) |
6: compute by applying jump condition (25) with appropriate cash flows |
7: compute by solving (26) or (29) with terminal condition |
8: |
9: end while |
10: return |