A Time Series Framework for Pricing Guaranteed Lifelong Withdrawal Benefit Nitu Sharma 1 • S. Dharmaraja 1 • Viswanathan Arunachalam 2 Accepted: 16 May 2020 Ó Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract In this work, the pricing problem of a variable annuity (VA) contract embedded with a guaranteed lifelong withdrawal benefit (GLWB) rider has been considered. VAs are annuities whose value is linked with a sub-account fund consisting of bonds and equities. The GLWB rider provides a series of regular payments to the policyholder during the policy duration when he is alive irrespective of the portfolio performance. Also, the remaining fund value is given to his nominee, at the time of death of the policyholder. The appropriate modelling of fund plays a crucial role in the pricing of VA products. In the literature, several authors model the fund value in a VA contract using a geometric Brownian motion (GBM) model with a constant variance. However, in real life, the financial assets returns are not Normal dis- tributed. The returns have non-zero skewness, high kurtosis, and leverage effect. This paper proposes a discrete-time model for annuity pricing using generalized autoregressive conditional heteroscedastic (GARCH) models, which overcome the limitations of the GBM model. The proposed model is analyzed with numerical illustration along with sensitivity analysis. Keywords Annuities Lifetime income Lifelong guarantee Variable annuity GARCH modelling GLWB pricing & S. Dharmaraja [email protected]Nitu Sharma [email protected]Viswanathan Arunachalam [email protected]1 Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India 2 Departamento de Estadistica, Universidad Nacional de Colombia, Bogota ´, Colombia 123 Computational Economics https://doi.org/10.1007/s10614-020-09999-9
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A Time Series Framework for Pricing Guaranteed LifelongWithdrawal Benefit
Nitu Sharma1 • S. Dharmaraja1 • Viswanathan Arunachalam2
Accepted: 16 May 2020� Springer Science+Business Media, LLC, part of Springer Nature 2020
AbstractIn this work, the pricing problem of a variable annuity (VA) contract embedded
with a guaranteed lifelong withdrawal benefit (GLWB) rider has been considered.
VAs are annuities whose value is linked with a sub-account fund consisting of
bonds and equities. The GLWB rider provides a series of regular payments to the
policyholder during the policy duration when he is alive irrespective of the portfolio
performance. Also, the remaining fund value is given to his nominee, at the time of
death of the policyholder. The appropriate modelling of fund plays a crucial role in
the pricing of VA products. In the literature, several authors model the fund value in
a VA contract using a geometric Brownian motion (GBM) model with a constant
variance. However, in real life, the financial assets returns are not Normal dis-
tributed. The returns have non-zero skewness, high kurtosis, and leverage effect.
This paper proposes a discrete-time model for annuity pricing using generalized
autoregressive conditional heteroscedastic (GARCH) models, which overcome the
limitations of the GBM model. The proposed model is analyzed with numerical
but its conditional variance equals r2t , which may change over time. The quantities
lt and r2t are interpreted as the conditional mean and variance process of the log-
return series Yt. The conditional mean is considered as a constant, as suggested by
the results in Fig. 3 (Sect. 2), where there is no correlation between the daily log-
return series of the S&P 500 index and the Nikkei 225 index. For the MSCI index,
ACF plots in Fig. 3c shows slight correlation up to 1 lag, but this correlation can be
considered while modelling volatility. The model for the varying volatility is dis-
cussed in the following section.
3.1 Volatility Modelling
The time-series modelling literature starts with ARMA models described by Whitle
(1951) in his thesis. These models do not consider varying volatility and thus cannot
account for heteroskedastic effects of the time-series process. Engle (1982)
introduced the well-known ARCH model, which was later generalized as GARCH
model by Bollerslev (1986). The following equation gives the varying volatility rtin case of a GARCH(p, q) model:
r2t ¼ xþXq
j¼1
aje2t�j þ
Xp
i¼1
bir2t�i ð4Þ
Though ARCH and GARCH models both can account for the volatility clustering
and leptokurtosis, but since they are symmetric, they fail to model the leverage
effect. To model this, a family of asymmetric GARCH models exists, some of
which are GJR-GARCH Models, E-GARCH Models and T-GARCH Models. These
models are discussed in details below.
GJR-GARCH Model Glosten et al. (1993) proposed the GJR-GARCH model for
the time-dependent variability of the log-return series. This model is more flexible
as it models the conditional variance such that it responds differently to past positive
and negative innovations of the same magnitude. The varying volatility ðr2t Þ in case
of a GJR-GARCH(p, q) model is given by the following equation:
r2t ¼ xþXq
j¼1
r2t�jz2t�j aj þ cjIt�j
� �þXp
i¼1
bir2t�i ð5Þ
where x[ 0, aj � 0, aj þ cj � 0 for j ¼ 1; . . .q, and bi � 0, for i ¼ 1; . . .p. If:gdenotes the indicator function which returns one if the innovations are negative and
zero otherwise, i.e., It�j ¼ 0 if zt�j � 0, It�j ¼ 1 if zt�j\0. This function will ensure
that bad news and good news will have different impact on the volatility, where the
magnitude of the difference will be given by the value of the parameter
cj; j ¼ 1; . . .; q. Therefore, if such an effect exists in the return series then the value
of cj; j ¼ 1; . . .; q is expected to be positive making the impact of negative socks on
volatility more compared to the positive ones.
E-GARCH Model Nelson (1991) proposed the E-GARCH model for the time-
dependent variability of the log-return series. This model also models the
conditional variance such that it responds differently to past positive and negative
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A Time Series Framework for Pricing Guaranteed Lifelong...
innovations. This model differs from the previous model as it considers log of the
variance. The log of the varying volatility ðlogðr2t ÞÞ in case of a E-GARCH(p, q)
model is given by the following equation:
logðr2t Þ ¼ xþXq
j¼1
ajzt�j þ cj jzt�jj � Eðjzt�jjÞ� �� �
þXp
i¼1
bi log r2t�i
� �ð6Þ
where x, aj, cj, for j ¼ 1; 2; . . .q and bi, for i ¼ 1; 2; . . .p are parameters not
restricted to positive values. Due to the presence of the term with cj, the volatility
can react asymmetrically to the good and bad news. The coefficient aj captures thesign effect and the coefficient cj captures the size effect of the news on the volatility.
T-GARCH Model Another variant of GARCH models that is capable of
modelling leverage effect is T-GARCH model proposed by Zakoian (1994). Under
this, the conditional standard deviation is modelled as a linear combination of past
innovations ðztÞ and standard deviation ðrtÞ variables. The conditional standard
deviation ðrtÞ in case of a T-GARCH(p, q) model is given by the following
equation:
rt ¼ xþXq
j¼1
ajrt�j jzt�jj � cjzt�j
� �þXp
i¼1
birt�i ð7Þ
where x[ 0, aj � 0, jcjj � 1 for j ¼ 1; . . .q and bi � 0 for i ¼ 1; . . .p. cj is the
parameter which helps volatility to react asymmetrically to positive and negative
innovations.
The asymmetric GARCH models provide a much better fit than a GBM model as
they overcome many deficiencies of a GBM model. However, the discrete-time and
continuous-state nature of the GARCH models makes the market incomplete,
resulting in some difficulties in pricing of investment guarantees (Siu et al. 2004).
Market incompleteness implies the existence of no unique risk-neutral measure
(Tardelli 2011). Hence, an equivalent risk-neutral measure for valuing a guarantee
has to be justified from a different angle. The pricing of guarantees in this situation
is considered in detail in the next section.
3.2 Risk-Neutral Pricing
Risk-neutral pricing is a method to determine the no-arbitrage price of an
investment. For obtaining a risk-neutral price, we need a risk-neutral measure under
which the asset price or index price is a martingale. Equivalently, the risk-neutral
measure is a probability measure under which the expected return on the asset is the
same as the risk-free return. In an incomplete market, there does not exist a portfolio
of risk-free bonds and assets which can replicate the guarantee perfectly. As a result,
no unique risk-neutral measure exists in incomplete markets. Hence, no unique risk-
neutral price for the GLWB guarantee exists in such a scenario.
There are various approaches to compute a risk-neutral measure. Some of them
include the utility maximization approach (Rubinstein 2005; Tardelli 2015) and the
conditional Esscher transform method (Siu et al. 2004; Ng et al. 2011). The
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N. Sharma et al.
condition for using these methods is that the returns have independent and stationary
increments, and their underlying conditional distribution is infinitely divisible. The
stock innovations in the proposed model are Normal distributed and hence are
infinitely divisible with a finite moment generating function. Also, the returns
modelled such that they have independent and stationary increments.
Following Siu-Hang Li et al. (2010) and Ng et al. (2011), we have used the
conditional Esscher transform method to obtain the risk-neutral price. The
advantage of using the conditional Esscher Transform method is that it is capable
of incorporating different infinitely divisible distributions for the GARCH
innovations in a unified and convenient framework. Let ðX;F ;PÞ be a complete
probability space, where P is the data generating probability measure. Under the
measure P, returns fYtg are characterized by Eq. (2), with independent and
identically distributed Gaussian innovations. Consider the time index T to be
f1; 2; 3; . . .; Tg and assume that all financial activities take place at t 2 T . Also
assume U ¼ fUtgt2T to be the natural filtration such that, for each t 2 T , Ut
contains all market information up to and including time t, and that UT ¼ F . Under
P, YtjUt�1 �Nðc; r2t Þ, where c is a constant and rt is as described in Sect. 3.1. For
pricing, we construct a martingale pricing probability measure Q equivalent to the
statistical probability measure P on the sample space ðX;FÞ by adopting the
concept of conditional Esscher transforms. Following Buhlmann et al. (1996),
define a sequence fZtgt2T with initial value Z0 ¼ 1 and for t 2 T
Zt ¼Yt
k¼1
ekkYk
E½ekkYk jUk�1�
for some constants fk1; k2; . . .kkg, where E is the expectation under the real world
measure. Since, EðZtjUt�1Þ ¼ Zt�1, therefore, fZtgt2T is a martingale. Take Pt to
be, P restricted on Ut i.e., given information up-to time t. Using the martingale
property of fZtgt2T construct a new family of measures Pt as dPt ¼ ZtdPt and
Pt ¼ Ptþ1jUt, and a probability measure P ¼ PT . To find the conditional distri-
bution of Yt, let A be a Borel measurable set, then
PtðYt 2 AjUt�1Þ ¼EPtIYt2AZt½ � ð8Þ
¼EPtIðYt2AÞ
ektYt
EPt½ekt YtjUt�1�
� �: ð9Þ
Substitute A ¼ ð�1; y�, where y is a real number, to obtain the following distri-
bution function of Yt given Ut�1 under Pt:
FPtðyÞ ¼
R y�1 ktxdFPtðxjUt�1ÞEPt
½ektYt jUt�1�: ð10Þ
where EPtis the expectation under the measure Pt. Then, the moment generating
function of Yt given Ut�1 under the measure Pt is
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A Time Series Framework for Pricing Guaranteed Lifelong...
EPt½eYtz; ktjUt�1� ¼ EPt
eYtðzþktÞjUt�1
EPt½ektYt jUt�1�
� �ð11Þ
Using YtjUt�1 �Nðc; r2t Þ, and Eq. (11), we have
EPt½eYtz; ktjUt�1� ¼ eðcþr2t ktÞzþ1
2r2t z
2
: ð12Þ
To construct the risk-neutral measure Q which is equivalent to P, choose some
Esscher parameters fk0tg such that the expected total return from any asset is the
same as the risk-free interest rate, i.e.,
EPt½eYt ; k0tjUt�1� ¼ er ð13Þ
where r is the continuously compounded risk-free rate. Substituting z ¼ 1 in
Eq. (12) and comparing with Eq. (13) we obtain
EQt½ezYt jUt�1� ¼ eðr�
12r2t Þþ1
2r2t z
2 ð14Þ
where Qt is Q given Ut�1, QT ¼ Q and EQtis the expectation under the measure
Qt. Hence [from Eq. (14)] under Q, YtjUt�1 �Nðr � r2t =2; r2t Þ.
4 Pricing Model
In the pricing model, the following notations are considered
x Insured age at time point 0
t The time in years
r Risk-free rate
Wt� Fund value in the beginning of tth year after fee deduction
Wt Fund value at the end of tth year before guarantee deduction
Wtþ Fund value at the end of tth year after guarantee deduction
W0 Initial fund value
St Stock price value at end of tth year
S0 Initial stock price
d Fee charged by the insurance company computed as a fraction of fund value
G Yearly withdrawals which are fixed g% of W0
For simplicity, we consider investing in a single index and assume the premium to
be a one-time lump sum investment of amount W0 done by the insured at the
beginning of the contract. Then, the number of initial stocks will be W0
S0. The fee ðdÞ
charged by the insurance company is deducted from the fund value by the
cancellation of fund units. We assume that the fee is charged at the beginning of the
year and guarantee amount (G) is deducted at the end of the year. Further, the
annual withdrawals by the policyholder are withdrawn at the end of the year.
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N. Sharma et al.
Additionally, assume that the death benefits are paid at the end of the year of death
of insured. Now, Wt� is the fund value at the beginning of the tth year after
deducting the fee. The Wt� amount remains invested in the market for the year t and
grows to Wt by the end of tth year. The insured withdraws G from Wt leaving Wtþ
value remaining in the fund. Again in beginning of year t þ 1, the insured deduct fee
ðdÞ from Wtþ reducing fund balance to Wðtþ1Þ� which grows to Wtþ1 by the end of
year t þ 1. This process of fee and guarantee deduction continues until there is
positive fund value. The time t denotes the number of years after policy inception
ranging for t ¼ 1; 2; 3; . . .;T , where T is the maximum number of years lived by an
individual. The maximum age lived by an individual is considered to be x years.
Therefore, the parameter T equals x� x. Assuming W0þ to be W0, the dynamics of
the fund value is given by:
Wt� ¼ Wðt�1Þþð1� dÞ ð15Þ
Wt ¼ Wt�St
St�1
ð16Þ
Wtþ ¼Wt � G; if Wt [G
0; otherwise
�ð17Þ
for t 2 f1; 2; 3; . . .; Tg. Equation (15) consists of the yearly deduction of fee ðdÞ,Eq. (16) shows changes in fund value corresponding to changes in stock price and
Eq. (17) shows the deduction of guarantee (G) annually. If for any
t 2 f1; 2; 3; . . .; Tg, Wt becomes less than G, then, the fund value after providing
guarantee value becomes 0 and remains 0 after that.
There are two scenarios possible. The first one is that the fund value is always
positive, i.e., until the insured is alive. In this case, the insured will get the
guaranteed amount from the account until death and the remaining balance as a
death benefit to his nominee. Therefore, the insurer is not liable to pay anything. The
second case is when the fund has become 0, and the insured is still alive. In this
case, no death benefit will be paid, but the insurer is liable to pay living guarantees
for the remaining lifetime. Note that, the fund value can become 0 only at the year-
end when the fund is not sufficient to pay the guarantee.
Considering the second case, if the fund value becomes 0 in the end of nth year
for the first time, i.e., Wn\G and Wnþ ¼ 0, then the insurer is liable to pay a
lifetime annuity to the annuitant of amount G. Additionally, the remaining amount
G�Wn is also paid by insurer. Thus, the cost or liability of the company at the end
of year n will be:
ðG�WnÞnpx þXx�x�n
k¼1
e�rkkþnpxG ð18Þ
where e�rk is the discounting factor for k years and kþnpx denotes the probability of a
life aged x surviving till age xþ k þ n. The first term in Eq. (18) is the excess
amount paid by the insurer to fulfil guarantee for nth year and the second term
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A Time Series Framework for Pricing Guaranteed Lifelong...
denotes the annuity for the rest of the life of the annuitant. At time point 0, the
liability cost of insured will be
ðG�WnÞnpx þXx�x�n
k¼1
e�rkkþnpxG
( )e�rn ð19Þ
The above expression can be rewritten as:
npx G 1þXx�x�n
k¼1
e�rkkpxþn
!�Wn npxe
�rn: ð20Þ
To solve for the break-even fee ðdÞ, equate the expected value of the above men-
tioned cost to the expected value of income to the insurer from this fund. The
insurer’s income is the yearly fee charged as a percentage of fund value till the fund
have a positive balance. These are the inflows and outflows from the insurer
perspective.
The same situation can be analyzed by considering insured’s inflows and
outflows and equating them to obtain the break-even fee. The insured get two kinds
of benefits: living and death. Living benefit involves the lifelong guaranteed annuity
of G amount, and the death benefit is the positive fund value (if any) given to the
beneficiary in case of demise of the insured.
Now, to find out the break-even fee ðdÞ, equate the expected present value
(E(PV)) of the inflows to the E(PV) of outflows. Hence,
W0 ¼ LB0 þ DB0 ð21Þ
where LB0 is the E(PV) of the living benefit and DB0 is the E(PV) of the death
benefit under the risk-neutral probability measure Q. The living benefits are inde-
pendent of the fund dynamics, and hence are given by a life annuity of a constant G
amount whose present value is given as follows:
LB0 ¼ GXx�x
k¼1
e�rkkpx: ð22Þ
The expected death benefit for a person dying in the mth year will be given by:
DBm ¼EQ½Wm�
¼ W0
S0EQ Smð1� dÞm �
Xm�1
i¼1
gS0
100ð1� dÞm�iSm
Si
!þ" #:
ð23Þ
where ðf Þþ ¼ maxff ; 0g and EQ denotes expectation under the measure Q. Thus,
the present value of death benefit is:
DB0 ¼Xx�x
i¼1
e�riDBi i�1px 1qxþi�1 ð24Þ
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N. Sharma et al.
where kqx denotes the probability of a life aged x dying within next k years. Note
that the condition in Eq. (23) has been replaced by the corresponding probability of
dying in between m� 1 and m years.
The combined benefit to the insured is given by:
GXx�x
k¼1
e�rkkpx þ
Xx�x
i¼1
e�riDBi i�1px 1qxþi�1: ð25Þ
Therefore, the following implicit equation in d is solved to obtain the break-even
value for the fee ðdÞ charged:
W0 ¼ GXx�x
k¼1
e�rkkpx þ
Xx�x
i¼1
e�riDBi i�1px 1qxþi�1: ð26Þ
5 Numerical Results and Sensitivity Analysis
This section consists of modelling the three datasets mentioned in Sect. 2,
simulating returns using the fitted model and calculating fee. To find the best-fitted
asymmetric GARCH models to the partitioned datasets, we applied several
standards criteria such as Akaike Information Criterion (AIC), Bayesian Informa-
tion Criterion (BIC) also known as Schwartz Information Criterion (SIC), and
LogLikelihood (LLK) values. Table 4, shows the fitted asymmetric models to the
Table 4 Fitted asymmetric
models to the partitioned
datasets of S&P 500, Nikkei 225
and MSCI world index
Fitted model
S&P 500
Data 1 GJRGARCH(1,1)
Data 2 GJRGARCH(2,1)
Data 3 EGARCH(2,1)
Data 4 EGARCH(2,1)
Data 5 TGARCH(2,2)
Nikkei 225
Data 1 TGARCH(2,1)
Data 2 TGARCH(1,1)
Data 3 GJRGARCH(2,1)
Data 4 TGARCH(2,1)
Data 5 GJRGARCH(1,1)
Data 6 TGARCH(2,1)
MSCI world
Data 1 GJRGARCH(1,2)
Data 2 EGARCH(2,1)
Data 3 GJRGARCH(1,2)
Data 4 EGARCH(2,1)
Data 5 TGARCH(2,1)
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A Time Series Framework for Pricing Guaranteed Lifelong...
partitioned datasets using models mentioned in Sect. 3. Please refer to ‘‘Ap-
pendix 2’’ for further details regarding fitting asymmetric GARCH models to the
partitioned datasets. The complete unpartitioned datasets are also modelled using
standardized GARCH model and GBM model (see ‘‘Appendices 3, 4’’ for details).
Then, with these models, the prices of the GLWB guarantee were obtained. Finally,
we did a sensitivity analysis concerning the parameters involved in pricing.
To find the appropriate fee value that should be charged for the GLWB contract,
we simulate the daily log-returns of the index using the fitted asymmetric GARCH
model, standardized GARCH model and GBM model under the risk-neutral
measure. With the help of these returns, the value of St,Wt, LBt and DBt is obtained.
Consider the following assumptions for the pricing model:
1. Independent mortality rates for the Nikkei 225 index follows the standard life
tables of Japan combine mortality rates of the year 2017. And for S&P 500 and
MSCI world index, they follow US combine mortality rates of the year 2017.
(Obtained from the Human Mortality Database)
2. Premium paid is a lump sum amount of 100, i.e., W0 ¼ 100.
3. For numerical analysis, if not mentioned then the risk-free rate (r) considered to
be 3% per annum, fee ðdÞ equal to 200 basis points (bp) and guarantee (g) 6%
are the default values.
4. The range of break-even fee is considered to be 0 to 1000 basis points (bp).
5. Only one state of decrement, i.e., death.
6. Static withdrawal strategy is considered, i.e., constant withdrawals of amount
g% of the initial premium are made every year.
Consider the following algorithm for finding out the fee ðdÞ for the GLWB contract.
1. Simulate 10,000 sample paths of Yt under measure Q, that is, on the basis of
Yt �Nðr � rt=2; rtÞ (see Sect. 3.2). For each path, rt are first generated from
the fitted models (see Table 1 for fitted models).
2. For each sample path, obtain the fund value ðWtÞ as the function of the break-
even fee ðdÞ.3. Average simulated values of Wt to obtain EQ½Wt�, and hence obtain the death
benefit values DBt.
4. Calculate the value of d using Eq. 26.
The comparison between the asymmetric GARCH, standardized GARCH and GBM
model is obtained by observing the break-even fee value obtained from each model.
The break-even fee is the value of d for which the E(PV) of the outflows equals the
premium W0. Table 5 shows the value of the break-even fee for ages 60, 65 and 70
correspondings to different guarantee amount g% of W0. As mentioned earlier, the
fee is charged as a percentage of the fund value till the time the fund value is non-
negative. From Table 5, it is observed that higher guaranteed withdrawal amount
comes with a higher fee. Also, the existence of a break-even fee corresponding to
every guarantee percentage is not necessary. As a very low guarantee (For instance,
1% for a person aged 60) cannot make the present value of the contract equal to the
premium even if the fee charged is 0 bp. Similarly, a very high guarantee (For
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N. Sharma et al.
instance, 25% for a person aged 60) cannot make the present value of the contract
equal to the premium even if the fee charged is 2500 bp. Therefore, many of the
blocks in Table 5 are empty as no break-even fee in the range 0–1000 bp exists for
those cases. Additionally, Table 5 shows that a high guarantee implies a high fee
charged by the insurer.
To perform a comparison between constant and variable volatility of the returns,
we obtained pricing results with stock price modelled as GBM (see Table 5).
Further, to show the impact of consideration of leverage effect, comparison with the
standardized GARCH model is also shown in Table 5. From Table 5, it is observed
that the resulting break-even fee for the proposed model lies between the fee
obtained by GBM and the fee obtained by standardized GARCH. Therefore, the
GBM model may underestimate the contract value. Similarly, the GARCH model
may overestimate the contract value if the underlying index has a leverage effect
feature. Additionally, the fund behaviour is dependent on the guarantee amount, and
the fee charged. In case of a high guarantee, the fund will diminish before the
A Time Series Framework for Pricing Guaranteed Lifelong...
expected time resulting in a liability to the VA provider and hence considerable
losses to them. Further, charging a low fee will hamper the financial stability of the
insurance firm in the long run. Therefore, the insurer should have enough funds
(which they get from the fee charged and pooling similar individuals), so that they
could pay the living benefits in case the fund gets exhausted. These results show the
need for an appropriate fee to be charged, corresponding to a fixed-guarantee by the
insurance company. Now, the question is, how much fee should be charged? The
answer is dependent on the prevailing risk-free rate and the guarantee provided. For
instance, a higher guarantee will imply a higher fee to be charged, keeping the risk-
free rate fixed.
In order to see the impact of change in age at inception x on the GLWB value, we
consider its impact on death benefit and living benefit separately. With the increase
in x, the contract length T decrease, resulting in a reduction in the number of
withdrawals and the living benefit amount. Further, a decrease in T implies a
reduction in the duration of discounting the death benefit value, hence an increase in
death benefit. For the younger policyholders, the magnitude of decrease in living
benefit value is comparatively higher then the increase in the death benefit value,
therefore, an increase in x decreases the overall GLWB value (see Fig. 7). However,
for older policyholders (age 80 and above), the magnitude of increase in death
benefit value is higher than the decrease in the living benefit value. Therefore, the
overall GLWB value increases slightly with an increase in x.
Continuing the analysis, consider the relationship between the GLWB value for
policyholders and the parameters r, g, and d, shown in Figs. 7, 8 and 9 respectively.
Similar to the effect of x, the risk-free rate (r) also affects both living benefit and
death benefit. With an increase in the value of r, the living benefit value reduces
because the present value of each guarantee decreases. Whereas the death benefit
may increase or decrease, as for death benefit, r is used both as a discounting factor
and in the expected rate of return of the risky asset. Overall, an increase in
r decreases the GLWB value as shown in Fig. 7. However, this decrease in the value
diminishes for older policyholders.
The other parameters affecting the GLWB value are the guaranteed value (g) and
the fee ðdÞ. Figure 8 shows that the GLWB value is an increasing function of g. An
increase in g increases each of the guaranteed withdrawals amounts and decreases
the death benefit value. However, the increase in the value of living benefit with an
increase in g is higher compared to the decrease in death benefit. Therefore, the
overall GLWB value increases with an increase in the value of g. However, the
impact is stronger for younger policyholders as the difference in GLWB values
corresponding different g values reduces and becomes almost negligible as age at
inception increases. Therefore, for older policyholders, a high guarantee value can
be offered without increasing the fee significantly.
In comparison to the behaviour of x, g and r on the GLWB value, a change in daffects the death benefit only. Higher the value of d, lesser will be the death benefit
and vice versa. The same conclusion can be drawn from Fig. 9. For a fixed age, the
GLWB value decreases with the increase in the fee charged by the VA provider.
Figure 9 shows that the difference in GLWB value with a varying fee is very less for
the young policyholders. The reason for this behaviour is that the major part of
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N. Sharma et al.
Fig. 7 Relation between the GLWB value and the risk-free interest rate for S&P 500 (a), Nikkei 225(b) and MSCI world (c) indices
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Fig. 8 Relation between the GLWB value and the guarantee amount for S&P 500 (a), Nikkei 225 (b) andMSCI world (c) indices
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Fig. 9 Relation between the GLWB value and the fee charged for S&P 500 (a), Nikkei 225 (b) and MSCIworld (c) indices
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GLWB value for the young policyholders came from the living benefits and
expected death benefit value for them is comparatively insignificant. As the age at
inception increases, this ratio of the proportion of living benefit to death benefit
decreases. For older people (aged 80 and above), it is the death benefit value which
is more important compared to living benefits. Hence, charging a high fee will result
in reduced death benefits for older people and correspondingly reducing the GLWB
value of the contract.
6 Conclusion and Future Work
Lifelong guarantees are very much useful for people approaching retirement as they
protect the insured against outliving their resources. Also, they provide participation
in the equity market as well as protection against the downside movement of the
stock market indices. However, these guarantees should be priced in a way that
neither the insurers incur losses over the long term nor they levy a very high fee as it
will result in reduced demands for the product.
For financial institutions, pricing and hedging of such guarantees are of utmost
importance. In this regard, obtaining a fair fee for the GLWB contract is a crucial
problem to be addressed. If the fee charged by them is very high, then the product
may not be attractive to the investors, and if it is very low, then the insurance
company may run out of funds to pay the lifelong guarantees. For a fair fee
computation, the fund value has to be modelled in such a way that all the stylized
features present in the underlying assets are appropriately captured. The prominent
GBM model fails to capture the common stylized features present in the underlying
assets. Therefore, we employed the asymmetric GARCH models to account for
these features. The proposed fee pricing model is based on the asymmetric GARCH
model. To show the significance of the leverage effect, we considered modelling
with a standardized GARCH model. The fee obtained by using the asymmetric
GARCH model is higher than that obtained by the GBM model and is lower than the
one calculated by the standardized GARCH model. Hence, it can be concluded that
the GBM model may not be reliable for pricing for an investment guarantee. Also, if
the considered dataset contains the leverage effect, then the standardized GARCH
model also provides false results compared to the asymmetric GARCH model.
The future work in this direction comprises of considering a more realistic
scenario by removing assumptions such as constant risk-free rate, static withdrawal
strategy, etc.. The sensitivity analysis of GLWB fund value corresponding to
varying the risk-free rate (see Fig. 7) shows that, small variations in the risk-free
rate, causes significant changes in the GLWB value. Therefore, the assumptions of a
constant risk-free rate can be replaced by an appropriate model. Also, the model can
be further extended by incorporating a dynamic withdrawal strategy and taking
surrender into account. This will imply a more realistic insured behaviour and
hence, will result in a better realistic model.
Acknowledgements The excellent comments of the anonymous reviewers are greatly acknowledged and
have helped a lot in improving the quality of the paper. This research work is supported by the
Department of Science and Technology, India. One of the authors (NS) would like to thank UGC, India
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for providing her financial support through Senior Research Fellowship. The third author (VA) was
supported by the project SaSMoTiDep from MinCiencias (ColCiencias) MATH-AMSUD program.
Appendix 1: Statistical Analysis of Partitioned Datasets
Tables 6, 7 and 8 displays the value of some statistics such as mean, variance and
kurtosis for the partitioned datasets. These tables also show the results of some
important tests for all the partitions of the three datasets. The high kurtosis values of
the partitioned series shown in the tables signifies that the distribution of all
partitioned datasets is leptokurtic. Further, from these tables, it is evident that the
null hypothesis of ADF Test is rejected at 99% level of significance for all the
partitions of the three datasets. The normality hypothesis got rejected by the results
of the JB test. The ARCH test p values are less than 0.01 for all considered lags
rejecting the null hypothesis of ‘‘no ARCH effect’’ at 99% level of significance for
all the partitions of the three datasets. Hence, ARCH effect is present in the
partitioned data series.
Table 6 Statistical analysis of partitioned datasets of S&P 500 index
Standard deviation 0.0063 0.0068 0.0078 0.0098 0.0098
Skewness 0.0099 0.1542 - 0.9986 0.0346 - 0.5264
Kurtosis 8.3496 1.5154 18.6461 1.8099 10.3513
ADF test p value \ 0.01 \ 0.01 \ 0.01 \ 0.01 \ 0.01
JB test p value 0.0000 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 1) 0.1443 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 5) 0.0000 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 12) 0.0000 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 50) 0.0000 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 120) 0.0000 0.0000 0.0000 0.0000 0.0000
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then the model having the least number of parameters is chosen. Also, if there exists
no single model with highest AIC, BIC, and log-likelihood values, then a
comparison between the models is based only on BIC and log-likelihood values.
We have considered the possible values for parameters p and q to be f1; 2g. FromTables 9, 10 and 11, the AIC, BIC, and log-likelihood values shows that the GJR-
GARCH, E-GARCH and T-GARCH models provide better fit to all the series
Table 9 AIC, BIC and LLK of fitted Assymetric model to S&P 500 index