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
illustration along with sensitivity analysis.
Keywords Annuities � Lifetime income � Lifelong guarantee � Variableannuity � GARCH modelling � GLWB pricing
& S. Dharmaraja
Nitu Sharma
Viswanathan Arunachalam
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
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Computational Economicshttps://doi.org/10.1007/s10614-020-09999-9(0123456789().,-volV)(0123456789().,-volV)
1 Introduction
Variable annuities (VAs) are life insurance products that combine features of
insurance and securities investments. In a VA, usually, a policyholder pays a single
premium at the beginning. Then, this premium is invested in one or several mutual
funds chosen by the policyholder himself from a variety of different mutual funds.
The embedded options provided by an insurer can be categorized as the guaranteed
minimum living benefit (GMLB), and the guaranteed minimum death benefit
(GMDB) (Hardy 2003; Ledlie et al. 2008). Four main options that offer some
guaranteed minimum living benefit are guaranteed minimum income benefit
(GMIB), guaranteed minimum accumulation benefit (GMAB), guaranteed mini-
mum withdrawal benefit (GMWB), and guaranteed lifelong withdrawal benefit
(GLWB) (Piscopo and Haberman 2011). The first two options, GMIB and GMAB,
offer a guaranteed minimum amount, irrespective of the account value, at the
maturity of the contract. With GMIB, this guarantee is applicable only if the insured
annuitize the account value, at the time of maturity. In GMWB, the insured can
withdraw a pre-specified amount at fixed, regular intervals until the maturity of the
contract. These withdrawals are independent of the sub-account fund value.
Therefore, in case the account value diminishes to zero before the maturity, the
insured can continue to withdraw the guarantees. GLWB is a lifelong version of the
GMWB option, in which the policyholder can withdraw a pre-specified guaranteed
amount at fixed, regular intervals till the time he is alive. If the account value
becomes zero during the lifetime of the insured, he can still continue to withdraw
the guaranteed amount until his death. The guaranteed amount under GMWB and
GLWB options can be either static(constant) or dynamic(varying) depending upon
the withdrawal strategy chosen by the insured. Such riders are congruous for risk-
averse investors. As a result, VAs with a minimum guarantee feature is an alluring
alternative for such investors. Additionally, as the baby boomers approach
retirement, the demand for annuities and savings products will continue to increase
(Condron 2008). Therefore, the fair valuation of VA products is compulsory.
There have been many models introduced to value the VA products with some
embedded options. A modelling framework for valuation of VA was introduced by
Bauer et al. (2008), where they considered pricing of VA with GMDB, GMIB and
GMAB riders. Krayzler et al. (2016) gave closed-form formulas for the pricing of
GMAB and GMDB riders. They have considered a GBM model for the stock price
dynamics with non-constant interest rates and volatility. However, this does not
enable them to capture the leptokurtic behaviour of stock returns. Bacinello et al.
(2011) evaluated different kinds of living and death guarantees (GMDB, GMIB,
GMAB, GMWB) under both the static and mixed approaches.
Since the introduction of GLWB rider in 2004, GLWB riders continue to be the
most popular type of GMLB option in the VA market [according to a research
article by LIMRA (Drinkwater et al. 2014)]. Despite the continued popularity, the
literature for the valuation of GLWBs is very limited. In this direction, Piscopo and
Haberman (2011) has given a theoretical model for the pricing and valuation of
GLWB option embedded in the VA products. They have assumed the sub-account
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fund to be GBM with constant drift and volatility. Similarly, Dai et al. (2008) and
Peng et al. (2012) have assumed GBM for the fund value process of a VA
embedded with a GLWB/GMWB option. Forsyth and Vetzal (2014) developed an
implicit partial differential equation (PDE) method for valuing GLWB option. They
assumed the risky asset follows a Markov regime-switching process. Choi (2017)
computed the indifference price of the VA contract with a GLWB rider using the
concept of equivalent utility. They modelled the risky asset by a GBM model with a
constant rate of return and constant volatility. Assuming fund value to follow GBM
with constant volatility is not always realistic. Also, the returns have a leverage
effect; volatility is not only time-varying, but the future volatility is asymmetrically
related to past innovations. The unexpected negative returns influence future
volatility more than unexpected positive returns (French et al. 1987). The GBM
model, with or without stochastic interest rate, cannot capture the leverage effect
and the volatility clustering effect in the stock returns.
The volatility clustering effect in returns can be captured by the autoregressive
conditional heteroscedastic (ARCH) and the generalized ARCH (GARCH) models
formulated by Engle (1982) and Bollerslev (1986) respectively. However, ARCH
and GARCH approaches have failed to capture asymmetric features of the stock
returns (Ericsson et al. 2016). The family of asymmetric GARCH models can
capture this stylized feature. We used some of the popular asymmetric GARCH
models in our analysis. These include the exponential GARCH (E-GARCH) model
by Nelson (1991), Glosten–Jagannathan–Runkle GARCH (GJR-GARCH) model by
Glosten et al. (1993) and Threshold GARCH model by Zakoian (1994). Apart from
taking into account the volatility clustering effect of stock returns and leverage
effect, these time-series models are also discrete. Since discrete cash flows involved
in the VA contract are incorporated in these models, they may be considered better
models. In the direction of valuation of VA products using asymmetric GARCH
models, Ng et al. (2011) used these models to develop a valuation model for the
investment guarantees: GMDB and GMAB. They have shown that it is not possible
to capture the stylized facts present in the equity index (Nikkei 225) by a GBM
model and hence concluded that an E-GARCH model could provide more realistic
modelling. The prices obtained by them are higher than those obtained by using
GBM method for GMDB and GMAB guarantees. The work by Ng et al. (2011)
incentivizes us to use GARCH type models for the valuation of the recently most
popular living guarantee which enjoys a significant stake in the VA sales, i.e.,
GLWB (Drinkwater et al. 2014). Hence, this paper presents a fair valuation model
for the GLWB guarantee.
In this article, we have considered GJR-GARCH, E-GARCH, and T-GARCH
models for modelling stock volatility. The models mentioned above capture all the
‘‘stylized’’ facts present in stock returns. The appropriate model for the considered
data is chosen based on several standard criterion’s values. Following Siu-Hang Li
et al. (2010) and Ng et al. (2011), we obtained a risk-neutral measure for the
proposed model of risky asset. To simplify the model, we considered a static
withdrawal strategy with a constant guarantee withdrawal amount over time. Then
using the risk-neutral measure, we obtained an implicit equation in the fee. We
solved some numerical examples to obtain the break-even fee using the implicit
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equation. For analysis purposes, we consider three different markets: the Japanese
market, the US market and the World market. The first two markets are chosen
based on the selling history of VA products. Moreover, to see the pricing for a
global index, we considered the third market. The Japanese market has a
circumscribed history, where the sale of VA products begins in 1999, whereas
the US market is the oldest one in selling VA products. We obtained fee value for
the three datasets under different scenarios with three different models, namely
asymmetric GARCH model, standardized GARCH model and GBM model. As
observed, the GBM model underestimates the prices for the guarantee. Whereas, the
standardized GARCH model overestimates the prices. We also performed
sensitivity analysis concerning different model parameters. Our results were
consistent with the results conveyed by Quittard-Pinon and Randrianarivony (2011)
for a GMDB guarantee.
The rest of the paper is organized as follows: Sect. 2 comprises finding the
stylized facts present in the stock returns. Section 3 gives a description of the
asymmetric models and designates the corresponding risk-neutral measure.
Section 4 contains the pricing model for the valuation of GLWB. Section 5
consists of selecting the asymmetric GARCH models, obtaining fee using Monte
Carlo simulations and sensitivity analysis of the fund value concerning various
parameters. Section 6 consists of the concluding remarks suggesting some possible
future work.
2 Equity Index: Returns Behavior and Properties
2.1 Historical Data
The datasets include the US market’s S&P 500 Composite Index, the Japanese
market’s Nikkei 225 Average Price Index and MSCI World Index from the Global
market. The indices are chosen based on the availability of 50 years long data and
popularity among the respective country population. We have considered data from
January 1970 to October 2019. We obtained the data for all the three indices from
Thomson and Reuters Datastream.
2.2 Stylized Features in Data
In this section, we will discuss the stylized features present in a log-return series.
Some of them include stationarity, volatility clustering, leverage effect and
conditional heteroscedasticity. In the following paragraphs, we will discuss the
reason behind using log-returns for modelling and some of the features of log-
returns in details.
Non-stationarity Generally, the financial price series data is not stationary. Most
commonly used techniques to make time-series data stationary include differencing,
taking logarithms and log-differencing. In this article, we have considered log-
returns which is the log differencing of the stock price series St
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rt ¼ logðStÞ � logðSt�1Þ: ð1Þ
This process converts non-stationary data to stationary data. Figure 1 shows the
time series plots of the historical returns of the three datasets. From Fig. 1, it is
observed that the returns appear in clusters, positive returns followed by higher
positive returns and the same for negative returns. This phenomenon is called
volatility clustering. To strengthen the observations, we test the stationarity of the
series so obtained using Augmented-Dicky–Fuller (ADF) test. The alternate
hypothesis of the ADF test is ‘‘series is stationary’’. Table 1 shows that the p value
for the ADF test is less than 0.01 which is less than a for a ¼ 1%; 5%; and10%,
resulting in rejection of null hypothesis at 99% level of significance. Hence, giving
evidence in support of stationarity of the three datasets.
Non-normality The GBM assumption with constant volatility considers the daily
log-returns to be independent of each other and identically distributed. The sample
moments given in Table 1 indicate that the empirical distributions have heavy tails
and sharp peaks at the centre compared to the Normal distribution. Further, the
Normal Quantile–Quantile (Q–Q) plot of the returns shown in Fig. 2 supports the
claim of non-normality of the returns data. Normal Q–Q plot is a graphical method
of comparing the returns distribution with a Normal distribution. In this, we plot the
quantiles of the data against the quantiles of the Normal distribution. If the Q–Q
plots of data form a straight line, then it is considered that the quantiles come from
the Normal distribution. The Normal Q–Q plots of the three datasets, shown in
Fig. 2, do not form a straight line, supporting the claim of the returns datasets not
Normal distributed. To formally test the hypothesis of the returns being Normal
Fig. 1 Returns plot of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
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distributed, we applied the Jarque-Bera (JB) test. JB test is a goodness-of-fit test
with the null hypothesis that the skewness and kurtosis of the datasets are the same
as that of the Normal distribution. From Table 1, the resultant p values of JB test are
all 0, rejecting the null hypothesis strongly. Therefore, the datasets are not Normal
distributed.
Autocorrelation Autocorrelation is the correlation between a time-series and a
lagged version of itself. Having a non-zero autocorrelation in the series implies that
the data values are not independent of past information. The equity returns also
exhibit some autocorrelation and are not independent of past innovations. For
instance, for the considered datasets, the dependence between the return series
Table 1 Data statisticsS&P Nikkei MSCI
Mean 0.000271 0.000175 0.000240
Minimum - 0.228997 - 0.161354 - 0.103633
Maximum 0.109572 0.132346 0.090967
Standard deviation 0.010385 0.012573 0.008291
Skewness - 1.013450 - 0.427521 - 0.490187
Kurtosis 26.116752 10.109685 11.510267
ADF test p value \ 0.01 \ 0.01 \ 0.01
JB test p value 0 0 0
Fig. 2 Normal Q–Q plots of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
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values can be seen from the autocorrelation function (ACF) plot of the returns,
partial autocorrelation function (PACF) plot of the returns, ACF plot of squared
returns and ACF plot of the absolute value of the returns shown in Figs. 3, 4, 5 and
6. The plots of ACF and PACF of the returns in Figs. 3 and 4 show no significant
linear dependence between the returns and the lagged values except for the ACF and
PACF of MSCI returns in Figs. 3c and 4c. However, significant dependence with
the previous values is observed from the plot of ACF of squared returns and
absolute returns series, presented in Figs. 5 and 6. The dependency between squared
returns implies that there is a nonlinear dependency in the returns datasets. Hence
conditional heteroscedasticity in the returns datasets is present, indicating the
presence of ARCH effect. The presence of ARCH effect is further verified using the
Engle’s ARCH test whose null hypothesis is ‘‘there is no ARCH effect’’. Table 2
shows the results of Engle’s ARCH test applied to the three datasets. The obtained
p values for the Engle’s ARCH test are 0 at all considered lags. Therefore, we reject
the null hypothesis of no ARCH effect. Hence, ARCH effect is present in the three
datasets.
Leverage Effect Leverage effect is the tendency of future volatility to rise more,
followed by a loss as compared to a gain of the same magnitude. From the returns
plots of Fig. 1, it is observed that the stock volatility tends to be higher
corresponding to a negative shock in the returns as compared to a positive shock of
the same magnitude. Hence, there is a negative correlation between asset returns
and its changing volatility, resulting in the presence of the leverage effect in the
Fig. 3 ACF plot of the returns of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
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Fig. 4 PACF plot of the returns of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
Fig. 5 ACF plot of the squared returns of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
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datasets. For this reason, there is a need to study conditional heteroscedasticity
models which take into consideration the leverage effect present in the datasets.
Structural Breaks Since the daily data of the returns span over a long-duration,
therefore, there is a high probability of having breakpoints. Breakpoints refer to a
significant change in the trend of a data leading to a change in parameters of the
fitted model. For finding breakpoints in the three datasets, we used breakpoints
function in the Strucchange package (Zeileis et al. 2002) assuming a linear trend in
the mean or a linear trend in the variance. Then, we have applied the Chow test
(Chow 1960) to check whether the potential breakpoint obtained from the
breakpoints function is an actual break in the linear trend of mean and variance.
It is pertinent to mention that the Chow test tests the presence of a structural break
assuming a linear trend for mean or variance corresponding to a breakpoint known a
priori. It tests whether the coefficient of linear regression on the datasets before and
after the breakpoints is same or not with the null hypothesis as ‘‘coefficient is
same’’. We performed the test at a 99% level of significance. Therefore, points with
Fig. 6 ACF plot of the absolute returns of S&P 500 (a), Nikkei 225 (b) and MSCI world index (c)
Table 2 Engle’s ARCH test
p valuesLag S&P Nikkei MSCI
1 0 0 0
5 0 0 0
12 0 0 0
50 0 0 0
120 0 0 0
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a p value less than 0.01 are considered to be significant breakpoints. Also, if two
breakpoints have a difference of less than 1000 time points, then the one with the
least p value (mean) and p value (variance) has been considered in our analysis.
Table 3 shows the potential breakpoints and corresponding Chow test statistics
p values. In this table, the p value (mean) term corresponds to p value obtained from
testing for a break in the mean linear trend, and p value (variance) term corresponds
to a p value obtained from testing for break assuming linear trend is in the variance.
From Table 3, it is observed that except for one breakpoint in the mean trend, which
is the point 5216 in the Nikkei 225 index, all other breakpoints are breaks in linear
trend to variance. The final breakpoints considered throughout the analysis are
highlighted in bold in Table 3. Tables 6, 7 and 8 in ‘‘Appendix 1’’ show that the
partitioned series so formed also consists of all the stylized features present in the
original datasets discussed in the above paragraphs.
3 Equity Index Modelling and Risk-Neutral Measure
In Sect. 2, it is shown that the returns series of all the three datasets (S&P 500,
Nikkei 225 and MSCI world index) exhibits leverage effect and strong
heteroscedasticity. In this section, to incorporate the leptokurtic distribution,
volatility clustering and the leverage effect present in the financial datasets, the log-
return series are modelled using an asymmetric GARCH model.
Similar to other GARCH type models the return series ðYtÞ is modelled using the
following time-series equation
Yt ¼ lt þ et ð2Þ
et ¼ ztrt ð3Þ
where zt are independent and identically distributed (i.i.d.) random variables with
zero mean and unit variance. The error term et is serially uncorrelated by definition,
Table 3 Breakpoints in the the return series
S&P 500 composite Nikkei 225 MSCI world
Break
point
p value
(mean)
p value
(variance)
Break
point
p value
(mean)
p value
(variance)
Break
point
p value
(mean)
p value
(variance)
2643 0.2973 0.0002 1950 0.4298 0.0000 1963 0.2839 0.0126
3290 0.1277 0.0001 3265 0.4078 0.0000 2653 0.2801 0.0002
4829 0.6951 0.0860 5216 0.0070 0.0000 4222 0.6617 0.0000
6035 0.8808 0.0016 7019 0.1715 0.0000 7155 0.6379 0.0000
7449 0.3750 0.0000 8129 0.6892 0.0000 7887 0.2466 0.0000
9008 0.9201 0.0069 10223 0.2392 0.2445 9108 0.8843 0.0035
10,222 0.1115 0.1819 11,051 0.2423 0.9155 10,222 0.1115 0.1819
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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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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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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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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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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
Table 5 Break-even fee values (in basis points)
Age g Asymmetric GARCH GARCH GBM
S&P Nikkei MSCI S&P Nikkei MSCI S&P Nikkei MSCI
60 5.4 14.37 97.14 20.67
5.5 11.32 161.74 30.86 15.51 237.89 54.02 96.35
5.6 56.85 461.51 77.37 62.62 525.44 99.54 402.19
5.7 119.80 141.45 127.23 162.52
5.8 210.07 233.63 218.90 252.80 40.52 40.38
5.9 345.96 372.03 356.85 388.67 183.02 183.03
65 6.0 35.92 17.17 2.28 114.73 39.42
6.1 25.92 143.13 47.29 32.78 215.88 69.10 78.81
6.2 63.19 312.28 85.43 70.77 377.56 106.42 251.94
6.3 110.64 602.38 134.14 119.15 658.16 153.77 547.41
6.4 172.49 197.50 182.34 215.75 3.64 3.10
6.5 254.15 280.68 265.13 297.52 88.84 88.63
6.6 363.45 391.60 375.72 406.62 203.67 203.14
70 6.7 16.40 1.51 70.71 36.67
6.8 14.72 51.87 38.56 23.52 124.80 58.60
6.9 40.72 126.44 65.43 50.19 195.54 84.59 64.32
7.0 70.74 226.63 96.66 81.16 290.44 114.86 166.44
7.1 106.54 364.32 133.12 117.39 422.96 150.53 306.26
7.2 150.34 560.14 177.88 161.99 613.18 194.53 505.48
7.3 202.71 856.08 231.48 215.29 900.08 246.83 38.58 806.30 38.59
7.4 265.34 295.50 279.02 309.36 104.80 105.18
7.5 341.46 372.98 356.06 385.31 186.28 186.09
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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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N. Sharma et al.
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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N. Sharma et al.
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
Data 1 Data 2 Data 3 Data 4 Data 5
Mean 0.0000 0.0005 0.0007 0.0000 0.0003
Minimum - 0.0374 - 0.2290 - 0.0711 - 0.0704 - 0.0947
Maximum 0.0490 0.0871 0.0499 0.0557 0.1096
Standard deviation 0.0086 0.0105 0.0075 0.0130 0.0113
Skewness 0.1798 - 3.9269 - 0.5564 0.0337 - 0.3821
Kurtosis 1.9106 87.0024 8.2467 1.9355 12.4621
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) 0.0000 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 0.0033 0.0000 0.0000 0.0000
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Appendix 2: Asymmetric GARCH Model Selection
To obtain the best-fitted model, we compared the AIC, BIC and Log-Likelihood
values. The difference between AIC and BIC is that the later penalizes free
parameters more strongly compared to the former. The best-fitted model can be
chosen by minimizing AIC, BIC, and by maximizing the log-likelihood function
values. If two or more models have the same AIC, BIC, and log-likelihood values,
Table 7 Statistical analysis of partitioned data of Nikkei 225 index
Data 1 Data 2 Data 3 Data 4 Data 5 Data 6
Mean 0.0004 0.0003 0.0009 - 0.0003 - 0.0005 0.0001
Minimum - 0.0909 - 0.0453 - 0.1614 - 0.0683 - 0.0723 - 0.1211
Maximum 0.0518 0.0445 0.0889 0.1243 0.0766 0.1323
Standard deviation 0.0102 0.0061 0.0089 0.0145 0.0150 0.0145
Skewness - 1.2941 - 0.2388 - 2.7836 0.4564 0.0143 - 0.4237
Kurtosis 11.0159 7.1491 63.5686 5.4348 2.1898 7.2337
ADF test p value \ 0.01 \ 0.01 \ 0.01 \ 0.01 \ 0.01 \ 0.01
JB test p value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
ARCH test p value (lag ¼ 1) 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000
ARCH test p value (lag ¼ 5) 0.0000 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 0.0000
ARCH test p value (lag ¼ 50) 0.0000 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 0.0000
Table 8 Statistical analysis of partitioned data of MSCI world index
Data 1 Data 2 Data 3 Data 4 Data 5
Mean 0.0001 0.0005 0.0004 - 0.0002 0.0002
Minimum - 0.0445 - 0.0316 - 0.1036 - 0.0399 - 0.0732
Maximum 0.0519 0.0288 0.0807 0.0460 0.0910
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
Data 1 Data 2 Data 3 Data 4 Data 5 Whole data
Number of points 3290 2745 1414 1559 4004 13,012
SGARCH
(1,1)
AIC - 6.8380 - 6.5763 - 7.1102 - 5.9846 - 6.6739 - 6.6537
BIC - 6.8306 - 6.5677 - 7.0953 - 5.9709 - 6.6676 - 6.6514
LLK 11,252.51 9030.00 5030.90 4669.03 13,365.13 43,293.15
(1,2)
AIC - 6.8378 - 6.5822 - 7.1088 - 5.9830 - 6.6733 - 6.6536
BIC - 6.8286 - 6.5714 - 7.0902 - 5.9659 - 6.6654 - 6.6508
LLK 11,253.24 9039.08 5030.91 4668.76 13,364.96 43,293.58
(2,1)
AIC - 6.8374 - 6.5753 - 7.1088 - 5.9853 - 6.6759 - 6.6536
BIC - 6.8281 - 6.5645 - 7.0903 - 5.9681 - 6.6681 - 6.6507
LLK 11,252.54 9029.58 5030.94 4670.53 13,370.23 43,293.12
(2,2)
AIC - 6.8372 - 6.5815 - 7.1104 - 5.9857 - 6.6754 - 6.6535
BIC - 6.8261 - 6.5685 - 7.0881 - 5.9651 - 6.6660 - 6.6500
LLK 11,253.24 9039.08 5033.03 4671.87 13,370.23 43,293.57
TGARCH
(1,1)
AIC - 6.8523 - 6.5715 - 7.1309 - 6.0309 - 6.7325 - 6.6782
BIC - 6.8430 - 6.5607 - 7.1124 - 6.0137 - 6.7246 - 6.6753
LLK 11,277.01 9024.41 5046.58 4706.10 13,483.40 43,453.27
(1,2)
AIC NC - 6.5712 - 7.1303 - 6.0283 - 6.7320 - 6.6782
BIC NC - 6.5583 - 7.1080 - 6.0077 - 6.7226 - 6.6747
LLK NC 9024.97 5047.10 4705.09 13,483.45 43,454.26
(2,1)
AIC - 6.8511 - 6.5742 - 7.1284 - 6.0283 - 6.7357 - 6.6779
BIC - 6.8381 - 6.5591 - 7.1024 - 6.0043 - 6.7247 - 6.6738
LLK 11,277.09 9030.05 5046.75 4706.05 13,491.87 43,453.20
(2,2)
AIC - 6.8542 - 6.5740 - 7.1274 - 6.0289 - 6.7366 - 6.6808
BIC - 6.8394 - 6.5567 - 7.0977 - 6.0015 - 6.7240 - 6.6762
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Table 9 continued
Data 1 Data 2 Data 3 Data 4 Data 5 Whole data
Number of points 3290 2745 1414 1559 4004 13,012
LLK 11,283.16 9030.78 5047.10 4707.55 13,494.61 43,473.05
GJRGARCH
(1,1)
AIC - 6.8544 - 6.5873 - 7.1214 - 6.0308 - 6.7198 - 6.6755
BIC - 6.8452 - 6.5765 - 7.1028 - 6.0136 - 6.7119 - 6.6727
LLK 11,280.57 9046.01 5039.83 4706.00 13,457.95 43,435.96
(1,2)
AIC NC NC NC - 6.0292 - 6.7192 NC
BIC NC NC NC - 6.0086 - 6.7097 NC
LLK NC NC NC 4705.78 13,457.78 NC
(2,1)
AIC - 6.8535 - 6.6037 - 7.1239 - 6.0279 - 6.7198 - 6.6766
BIC - 6.8405 - 6.5886 - 7.0979 - 6.0039 - 6.7088 - 6.6726
LLK 11,280.99 9070.57 5043.57 4705.78 13,460.11 43,444.91
(2,2)
AIC - 6.8532 - 6.6030 - 7.1224 - 6.0276 - 6.7200 - 6.6764
BIC - 6.8384 - 6.5857 - 7.0927 - 6.0001 - 6.7074 - 6.6718
LLK 11,281.48 9070.57 5043.57 4706.50 13,461.40 43,444.91
EGARCH
(1,1)
AIC - 6.8515 - 6.5822 - 7.1316 - 6.0371 - 6.7212 - 6.6762
BIC - 6.8422 - 6.5714 - 7.1130 - 6.0200 - 6.7133 - 6.6733
LLK 11,275.64 9039.02 5047.05 4710.95 13,460.82 43,440.28
(1,2)
AIC - 6.8513 - 6.5942 - 7.1308 - 6.0358 - 6.7207 - 6.6763
BIC - 6.8402 - 6.5812 - 7.1085 - 6.0152 - 6.7113 - 6.6729
LLK 11,276.40 9056.50 5047.48 4710.88 13,460.85 43,442.20
(2,1)
AIC - 6.8505 - 6.5908 - 7.1496 - 6.0445 - 6.7255 - 6.6798
BIC - 6.8375 - 6.5757 - 7.1235 - 6.0205 - 6.7145 - 6.6758
LLK 11,276.05 9052.88 5061.74 4718.71 13,471.39 43,466.02
(2,2)
AIC - 6.8501 - 6.5936 - 7.1487 - 6.0435 - 6.7253 - 6.6798
BIC - 6.8353 - 6.5763 - 7.1190 - 6.0160 - 6.7127 - 6.6753
LLK 11,276.47 9057.71 5062.13 4718.89 13,472.04 43,467.09
123
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Table 10 AIC, BIC and LLK of fitted Assymetric model to Nikkei 225 index
Data 1 Data 2 Data 3 Data 4 Data 5 Data 6 Whole data
Number of points 1950 1315 1951 1803 1110 4872 13001
SGARCH
(1,1)
AIC - 6.5907 - 7.5579 - 7.0001 - 5.8380 - 5.6387 - 6.6537 - 6.2845
BIC - 6.5793 - 7.5421 - 6.9887 - 5.8258 - 5.6207 - 6.6514 - 6.2822
LLK 6429.92 4973.29 6832.59 5266.95 3133.49 43,293.15 40,856.44
(1,2)
AIC - 6.5908 - 7.5562 - 7.0051 - 5.8369 - 5.6369 - 6.6536 - 6.2848
BIC - 6.5765 - 7.5365 - 6.9908 - 5.8216 - 5.6144 - 6.6508 - 6.2819
LLK 6431.05 4973.23 6838.49 5266.93 3133.50 43,293.58 40,859.27
(2,1)
AIC - 6.5911 - 7.5665 - 6.9996 - 5.8366 - 5.6385 - 6.6536 - 6.2844
BIC - 6.5768 - 7.5468 - 6.9853 - 5.8214 - 5.6159 - 6.6507 - 6.2816
LLK 6431.36 4979.95 6833.12 5266.70 3134.36 43,293.12 40,856.95
(2,2)
AIC - 6.5906 - 7.5683 - 7.0041 - 5.8358 - 5.6417 - 6.6535 - 6.2846
BIC - 6.5735 - 7.5446 - 6.9869 - 5.8175 - 5.6146 - 6.6500 - 6.2812
LLK 6431.85 4982.13 6838.49 5266.93 3137.12 43,293.57 40,859.27
TGARCH
(1,1)
AIC - 6.6085 - 7.6202 - 7.0112 - 5.8872 - 5.6489 - 5.9221 - 6.3078
BIC - 6.5942 - 7.6005 - 6.9969 - 5.8720 - 5.6263 - 5.9154 - 6.3049
LLK 6448.32 5015.28 6844.41 5312.31 3140.12 14,431.25 41,008.95
(1,2)
AIC - 6.6090 - 7.6187 - 7.0158 - 5.8860 - 5.6466 - 5.9205 - 6.3088
BIC - 6.5918 - 7.5950 - 6.9986 - 5.8677 - 5.6195 - 5.9125 - 6.3053
LLK 6449.75 5015.26 6849.89 5312.20 3139.87 14,428.33 41,016.20
(2,1)
AIC - 6.6198 - 7.6202 - 7.0099 - 5.8924 - 5.6549 - 5.9313 - 6.3142
BIC - 6.5998 - 7.5926 - 6.9899 - 5.8710 - 5.6233 - 5.9219 - 6.3102
LLK 6461.33 5017.26 6845.17 5318.98 3145.48 14,455.57 41,052.52
(2,2)
AIC - 6.6188 - 7.6182 - 7.0165 - 5.8924 - 5.6531 - 5.9309 - 6.3159
BIC - 6.5959 - 7.5866 - 6.9937 - 5.8680 - 5.6170 - 5.9202 - 6.3113
LLK 6461.33 5016.95 6852.64 5319.96 3145.48 14,455.58 41,064.61
GJRGARCH
(1,1)
AIC - 6.6105 - 7.6124 - 7.0338 - 5.8790 - 5.6500 - 5.9223 - 6.3055
BIC - 6.5962 - 7.5927 - 7.0195 - 5.8637 - 5.6275 - 5.9156 - 6.3026
LLK 6450.21 5010.16 6866.49 5304.91 3140.77 14,431.61 40,993.64
(1,2)
AIC - 6.6110 - 7.6106 - 7.0373 - 5.8778 - 5.6482 - 5.9217 - 6.3058
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compared to a GARCH model. The bold values in Tables 9, 10, and 11 corresponds
to the best-fitted models for the partitioned datasets based on the AIC, BIC and log-
likelihood values. Note that NC denotes not convergent models in Tables 9, 10, and
11. The final fitted models to all the partitioned series based on above-mentioned
criteria are given in Table 4 in Sect. 5. The parameters corresponding to the fitted
models are specified in Tables 12, 13 and 14. For the goodness of fit check of the
fitted models, we applied the Engle’s ARCH test to residuals, and the Ljung–
Box Portmanteau test to the squared residuals. The results are mentioned in
Table 10 continued
Data 1 Data 2 Data 3 Data 4 Data 5 Data 6 Whole data
Number of points 1950 1315 1951 1803 1110 4872 13001
BIC - 6.5939 - 7.5870 - 7.0202 - 5.8595 - 5.6212 - 5.9137 - 6.3024
LLK 6451.77 5009.98 6870.93 5304.82 3140.78 14,431.29 40,997.03
(2,1)
AIC - 6.6106 - 7.6185 - 7.0421 - 5.8784 - 5.6477 - 5.9249 - 6.3092
BIC - 6.5906 - 7.5909 - 7.0221 - 5.8571 - 5.6161 - 5.9156 - 6.3052
LLK 6452.32 5016.16 6876.58 5306.39 3141.49 14,440.08 41,020.13
(2,2)
AIC - 6.6096 - 7.6171 - 7.0411 - 5.8773 - 5.6496 - 5.9245 - 6.3091
BIC - 6.5867 - 7.5856 - 7.0182 - 5.8529 - 5.6135 - 5.9138 - 6.3045
LLK 6452.32 5016.28 6876.58 5306.39 3143.55 14,440.08 41,020.13
EGARCH
(1,1)
AIC - 6.6104 - 7.6186 - 7.0229 - 5.8835 - 5.6472 - 5.9237 - 6.3121
BIC - 6.5961 - 7.5989 - 7.0086 - 5.8683 - 5.6246 - 5.9171 - 6.3092
LLK 6450.11 5014.25 6855.84 5309.00 3139.19 14,435.20 41,036.82
(1,2)
AIC - 6.6112 - 7.6177 - 7.0325 - 5.8821 - 5.6456 - 5.9231 - 6.3133
BIC - 6.5940 - 7.5941 - 7.0153 - 5.8638 - 5.6185 - 5.9151 - 6.3098
LLK 6451.92 5014.64 6866.18 5308.68 3139.29 14,434.62 41,045.53
(2,1)
AIC - 6.6103 - 7.6173 - 7.0363 - 5.8833 - 5.6488 - 5.9296 - 6.3175
BIC - 6.5903 - 7.5897 - 7.0163 - 5.8619 - 5.6172 - 5.9203 - 6.3135
LLK 6452.08 5015.38 6870.89 5310.76 3142.09 14,451.49 41,073.75
(2,2)
AIC - 6.6153 - 7.6170 - 7.0360 - 5.8812 - 5.6503 - 5.9296 - 6.3176
BIC - 6.5924 - 7.5855 - 7.0131 - 5.8568 - 5.6142 - 5.9190 - 6.3130
LLK 6457.90 5016.20 6871.63 5309.88 3143.91 14,452.59 41,075.44
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Table 11 AIC, BIC and LLK of fitted Assymetric model to MSCI world index
Data 1 Data 2 Data 3 Data 4 Data 5 Whole data
Number of points 2653 1569 3665 1221 3904 13,012
SGARCH
(1,1)
AIC - 7.4449 - 7.2243 - 7.1237 - 6.5911 - 6.9251 - 7.0868
BIC - 7.4360 - 7.2107 - 7.1169 - 6.5743 - 6.9187 - 7.0845
LLK 9879.68 5671.50 13,058.11 4027.84 13,525.32 46,114.27
(1,2)
AIC - 7.4559 - 7.2268 - 7.1268 - 6.5897 - 6.9246 - 7.0868
BIC - 7.4448 - 7.2097 - 7.1183 - 6.5688 - 6.9166 - 7.0845
LLK 9895.30 5674.39 13,064.88 4028.04 13,525.35 46,114.27
(2,1)
AIC - 7.4443 - 7.2226 - 7.1230 - 6.5904 - 6.9254 - 7.0866
BIC - 7.4332 - 7.2055 - 7.1145 - 6.5695 - 6.9174 - 7.0838
LLK 9879.81 5671.14 13,057.93 4028.42 13,526.92 46,114.24
(2,2)
AIC - 7.4552 - 7.2255 - 7.1263 - 6.5888 - 6.9250 - 7.0866
BIC - 7.4419 - 7.2050 - 7.1161 - 6.5637 - 6.9154 - 7.0838
LLK 9895.29 5674.39 13,064.87 4028.45 13,527.13 46,114.24
TGARCH
(1,1)
AIC - 7.3833 - 7.2286 - 7.1333 - 6.6311 - 6.9679 - 7.0877
BIC - 7.3722 - 7.2116 - 7.1249 - 6.6101 - 6.9599 - 7.0848
LLK 9798.8860 5675.8569 13,076.8480 4053.2647 13,609.8022 46,121.2455
(1,2)
AIC - 7.3976 - 7.2270 - 7.1363 - 6.6299 - 6.9674 - 7.0891
BIC - 7.3843 - 7.2065 - 7.1261 - 6.6048 - 6.9577 - 7.0856
LLK 9818.8737 5675.6013 13,083.1881 4053.5375 13,609.7637 46,131.1041
(2,1)
AIC - 7.4492 - 7.2268 - 7.1318 - 6.6289 - 6.9715 - 7.1003
BIC - 7.4336 - 7.2029 - 7.1199 - 6.5996 - 6.9603 - 7.0962
LLK 9888.3135 5676.4037 13,076.0057 4053.9251 13,618.8758 46,204.8740
(2,2)
AIC - 7.4484 - 7.2283 - 7.1352 - 6.6275 - 6.9710 - 7.1018
BIC - 7.4307 - 7.2010 - 7.1216 - 6.5940 - 6.9582 - 7.0972
LLK 9888.31 5678.64 13,083.19 4054.07 13,618.97 46,215.85
GJRGARCH
(1,1)
AIC NC - 7.2324 - 7.1452 - 6.6256 - 6.9559 - 7.1045
BIC NC - 7.2154 - 7.1367 - 6.6047 - 6.9479 - 7.1017
LLK NC 5678.8438 13,098.4911 4049.9076 13,586.4917 46,230.6935
(1,2)
AIC - 7.4608 - 7.2334 - 7.1479 - 6.6245 - 6.9553 - 7.1059
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Tables 15, 16, and 17. Based on the p values of the Engle’s ARCH test mentioned
in Tables 15, 16, and 17, the null hypothesis of having ‘‘no ARCH effect’’ in the
residual series is not rejected in most of the cases, except for a few cases with lag
120. The results for the Ljung–Box Portmanteau test to the squared residuals series
are mentioned in Tables 15, 16, and 17. The null hypothesis of the test is ‘‘the
observations are random and independent’’. Based on p values for the Ljung–
Table 11 continued
Data 1 Data 2 Data 3 Data 4 Data 5 Whole data
Number of points 2653 1569 3665 1221 3904 13,012
BIC - 7.4475 - 7.2129 - 7.1377 - 6.5994 - 6.9456 - 7.1024
LLK 9902.7366 5680.6062 13,104.5110 4050.2317 13,586.1363 46,240.2320
(2,1)
AIC - 7.4504 - 7.2293 NC - 6.6233 - 6.9559 - 7.1070
BIC - 7.4349 - 7.2054 NC - 6.5940 - 6.9446 - 7.1030
LLK 9890.0111 5678.3894 NC 4050.5067 13,588.3563 46,248.5345
(2,2)
AIC - 7.4601 - 7.2316 - 7.1535 - 6.6216 - 6.9554 - 7.1068
BIC - 7.4424 - 7.2043 - 7.1399 - 6.5882 - 6.9425 - 7.1022
LLK 9903.88 5681.21 13,116.75 4050.51 13,588.36 46,248.57
EGARCH
(1,1)
AIC - 7.3881 - 7.2324 - 7.1393 - 6.6297 - 6.9616 - 7.0882
BIC - 7.3771 - 7.2153 - 7.1309 - 6.6087 - 6.9536 - 7.0853
LLK 9805.3658 5678.7923 13,087.8086 4052.4056 13,597.6091 46,124.2856
(1,2)
AIC - 7.3988 - 7.2319 - 7.1416 - 6.6284 - 6.9611 - 7.0895
BIC - 7.3855 - 7.2115 - 7.1315 - 6.6033 - 6.9514 - 7.0861
LLK 9820.4565 5679.4621 13,093.0341 4052.6365 13,597.4957 46,133.9022
(2,1)
AIC - 7.3886 - 7.2339 - 7.1475 - 6.6352 - 6.9641 - 7.0909
BIC - 7.3730 - 7.2100 - 7.1356 - 6.6059 - 6.9529 - 7.0869
LLK 9807.9396 5681.9894 13,104.7093 4057.8037 13,604.3860 46,143.8536
(2,2)
AIC - 7.4366 - 7.2336 - 7.1476 - 6.6336 - 6.9637 - 7.0909
BIC - 7.4188 - 7.2063 - 7.1341 - 6.6002 - 6.9509 - 7.0863
LLK 9872.58 5682.76 13,106.01 4057.84 13,604.66 46,144.67
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N. Sharma et al.
Box Portmanteau test observed from Tables 15, 16, and 17, the null hypothesis is
not rejected in most of the cases, showing that the squared residuals are also
independent. The resulting p values are significantly greater than the level of
significance, a for a ¼ 1%; 5%; 10%, hence, supporting the efficacy of the estimated
models in both the cases. Hence, the residuals do not have any ARCH effect.
Table 12 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the S&P 500 index
Parameter Data 1 Data 2 Data 3 Data 4 Data 5
l 0.000086 0.000511 0.000621 - 0.000359 0.000186
x 0.000001 0.000001 - 0.178861 - 0.149810 0.000474
a1 0.008314 0.001478 - 0.264287 - 0.218060 0.084660
a2 0.018446 0.198109 0.110076 0.119310
b1 0.954071 0.961873 0.981408 0.982822 0.242224
b2 0.501502
c1 0.061821 0.258241 - 0.034574 - 0.103214
c2 - 0.247926 0.163592 0.181625
g1 1.000000
g2 0.071013
Table 13 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the Nikkei 225 index
Parameter Data 1 Data 2 Data 3 Data 4 Data 5 Data 6
l 0.000524 0.000320 0.000876 - 0.000455 - 0.000531 0.000188
x 0.001299 0.000468 0.000003 0.000338 0.000010 0.000462
a1 0.161291 0.105142 0.059280 0.044351 0.018726 0.067913
a2 0.190407 0.000001 0.062634 0.077510
b1 0.683961 0.839054 0.813740 0.890410 0.893421 0.881342
b2c1 0.464501 0.089754
c2 - 0.267226
g1 0.670378 1.000000 1.000000 1.000000
g2 - 0.810832 - 0.338117 - 0.724967
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Table 14 Parameters of fitted Assymetric GARCH model mentioned in Table 4 for the MSCI world
index
Parameter Data 1 Data 2 Data 3 Data 4 Data 5
l 0.000215 0.000388 0.000468 0.000149 0.000265
x 0.000001 - 0.163938 0.000004 - 0.157250 0.000195
a1 0.062920 0.026628 0.084505 - 0.199284 0.075232
a2 - 0.061924 0.122041 0.049759
b1 0.120606 0.983573 0.396040 0.983656 0.899272
b2 0.763513 0.334662
c1 0.050564 0.184575 0.222889 - 0.046053
c2 - 0.103080 0.137180
g1 1.000000
g2 - 0.755871
Table 15 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of
S&P 500 index
Lag Data 1 Data 2 Data 3 Data 4 Data 5
ARCH test on residuals
1 0.426533203 0.72305888 0.7807073 0.951559 0.947043765
5 0.843228897 0.99847108 0.9981231 0.980055 0.998932385
12 0.526579122 0.99999996 0.9999994 0.999823 0.998311979
50 0.011380724 1 1 0.99547 0.999999906
120 0.018142514 1 1 5.81E-06 0.999999642
Ljung–Box test on squared residuals
1 0.210865315 0.66722615 0.6442405 0.770663 0.121708286
5 0.163685546 0.30799076 0.6456445 0.74912 0.645226104
12 0.284061115 0.84254928 0.7258412 0.95291 0.050579576
50 0.106585383 0.98872841 0.60701 0.0437 0.692479151
120 0.01611489 0.99972616 0.7872885 0.000115 0.218651523
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N. Sharma et al.
Appendix 3: Standarrdized GARCH Model
For comparing the effect of not taking leverage effect in the modelling of return
series, we considered modelling with a standardized GARCH model. The
parameters (p, q) are decided based on the AIC, BIC, and Log-Likelihood values
mentioned in the last column of first block in Tables 9, 10 and 11. The AIC, BIC,
and Log-Likelihood values are almost the same for all the fitted GARCH(p, q)
Table 16 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of
Nikkei 225 index
Lag Data 1 Data 2 Data 3 Data 4 Data 5 Data 6
ARCH test on residuals
1 0.954837226 0.44245394 0.3207574 0.885943 0.875251445 0.149618
5 0.999995866 0.71608445 0.959422 0.672467 0.999792675 0.755577
12 1 0.97633856 0.9999746 0.972185 0.999999936 0.864248
50 1 0.99995148 0.9999896 1 1 1
120 1 0.87949889 1 1 1 1
Ljung–Box test on squared residuals
1 0.646970642 0.54290805 0.1039804 0.951583 0.431388315 0.436315
5 0.864785265 0.51120898 0.3684142 0.189817 0.466987717 0.939537
12 0.996453615 0.65232024 0.7877093 0.43448 0.790077387 0.368873
50 0.999859144 0.09093599 0.0387843 0.217479 0.057855242 0.783925
120 0.999925617 0.05829188 0.031555 0.605142 0.217550874 0.450965
Table 17 Results of Engle’s ARCH test and Ljung–Box Portmanteau test applied to squared residuals of
MSCI world index
Lag Data 1 Data 2 Data 3 Data 4 Data 5
ARCH test on residuals
1 0.832684865 0.98735473 0.6791415 0.722706 0.822266237
5 9.85E-06 0.90783394 0.9925967 0.832254 0.999032623
12 0.000845608 0.99909996 0.9999426 0.988149 0.999938146
50 0.383819557 0.9999988 1 0.872812 1
120 0.289683043 0.95423124 1 0.998086 0.001169245
Ljung–Box test on squared residuals
1 0.365037124 0.55897844 0.8660144 0.894178 0.771728701
5 4.07E-06 0.78356147 0.8766658 0.801388 0.940752556
12 3.13E-12 0.95390798 0.8038386 0.977058 0.766670669
50 0 0.79096471 0.291068 0.906944 0.901663003
120 0 0.33749568 0.0042286 0.467533 0.045761651
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A Time Series Framework for Pricing Guaranteed Lifelong...
models for p; q ¼ 1; 2. Therefore, in order to minimize the number of parameters,
GARCH(1, 1) model for all the three datasets is chosen. The parameters of the fitted
model are mentioned in Table 18.
Appendix 4: Geometric Brownian Motion (GBM) Model
To show the significance of varying volatility, we considered returns modelling with
GBM model also. The parsimonious GBM model considers the return volatility to
be a constant over time. The stock price dynamics for a GBM under risk-neutral
measure is given by
St ¼ S0er�1
2r2ð ÞtþrBt ð27Þ
here r is the continuously compounded risk-free rate, r2 is the variance and Bt is a
Wiener process. The estimated value of r for the S&P 500, Nikkei 225 and MSCI
world index is given by 0.010385, 0.012573 and 0.008291 respectively.
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Table 18 Parameters of fitted
GARCH(1, 1) model to the
three datasets
Parameter S&P Nikkei MSCI
l 0.000495 0.000622 0.000480
x 0.000001 0.000002 0.000001
a1 0.078449 0.123909 0.093377
b1 0.909432 0.875091 0.890288
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A Time Series Framework for Pricing Guaranteed Lifelong...