a time series framework for pricing guaranteed lifelong ...web.iitd.ac.in/~dharmar/paper/ce2020.pdfa...

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

[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 Economicshttps://doi.org/10.1007/s10614-020-09999-9(0123456789().,-volV)(0123456789().,-volV)

http://orcid.org/0000-0003-2892-0864

http://orcid.org/0000-0003-0905-3208

http://crossmark.crossref.org/dialog/?doi=10.1007/s10614-020-09999-9&domain=pdf

https://doi.org/10.1007/s10614-020-09999-9

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

123

N. Sharma et al.

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

123

A Time Series Framework for Pricing Guaranteed Lifelong...

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

123

N. Sharma et al.

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)

123


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)

123

N. Sharma et al.

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)

123


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)

123

N. Sharma et al.

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

123


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

123

N. Sharma et al.

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

123


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

123

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

123


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.

123

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

123


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

123

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 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 4 TGARCH(2,1)


Data 6 TGARCH(2,1)

MSCI world


Data 2 EGARCH(2,1)


Data 4 EGARCH(2,1)

Data 5 TGARCH(2,1)

123


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

123

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

123


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

123

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

123


Fig. 8 Relation between the GLWB value and the guarantee amount for S&P 500 (a), Nikkei 225 (b) andMSCI world (c) indices

123

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

123


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

123

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





123


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





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






123

N. Sharma et al.

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

123


Table 9 continued


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

N. Sharma et al.

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

123


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

123

N. Sharma et al.

Table 11 AIC, BIC and LLK of fitted Assymetric model to MSCI world index


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

123


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


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

123

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

123


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

123

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)


Nikkei 225 index

Lag Data 1 Data 2 Data 3 Data 4 Data 5 Data 6


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


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


MSCI world index

Lag Data 1 Data 2 Data 3 Data 4 Data 5


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


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

123


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.

References

Bacinello, A. R., Millossovich, P., Olivieri, A., & Pitacco, E. (2011). Variable annuities: A unifying

valuation approach. Insurance: Mathematics and Economics, 49(3), 285–297.

Bauer, D., Kling, A., & Russ, J. (2008). A universal pricing framework for guaranteed minimum benefits

in variable annuities. ASTIN Bulletin: The Journal of the IAA, 38(2), 621–651.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Economet-

rics, 31(3), 307–327.

Buhlmann, H., Delbaen, F., Embrechts, P., & Shiryaev, A. N. (1996). No-arbitrage, change of measure

and conditional esscher transforms. CWI Quarterly, 9(4), 291–317.

Choi, J. (2017). Indifference pricing of a GLWB option in variable annuities. North American Actuarial

Journal, 21(2), 281–296.

Chow, G. C. (1960). Tests of equality between sets of coefficients in two linear regressions.

Econometrica: Journal of the Econometric Society, 28, 591–605.

Condron, C. M. (2008). Variable annuities and the new retirement realities. The Geneva Papers on Risk

and Insurance-Issues and Practice, 33(1), 12–32.

Dai, M., Kuen Kwok, Y., & Zong, J. (2008). Guaranteed minimum withdrawal benefit in variable

annuities. Mathematical Finance, 18(4), 595–611.

Drinkwater, M., Iqbal, J., & Montiminy, J. (2014). Variable anuity guaranteed living benefits utilization:

2012 experience. A joint study sponsored by the Society of Actuaries and LIMRA.

Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united

kingdom inflation. Econometrica: Journal of the Econometric Society, 50(4), 987–1007.

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

123

N. Sharma et al.

Ericsson, J., Huang, X., & Mazzotta, S. (2016). Leverage and asymmetric volatility: The firm-level

evidence. Journal of Empirical Finance, 38, 1–21.

Forsyth, P., & Vetzal, K. (2014). An optimal stochastic control framework for determining the cost of

hedging of variable annuities. Journal of Economic Dynamics and Control, 44, 29–53.

French, K. R., Schwert, G. W., & Stambaugh, R. F. (1987). Expected stock returns and volatility. Journal

of Financial Economics, 19(1), 3.

Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and

the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779–1801.

Hardy, M. (2003). Investment guarantees: Modeling and risk management for equity-linked life

insurance. Hoboken: Wiley.

Krayzler, M., Zagst, R., & Brunner, B. (2016). Closed-form solutions for guaranteed minimum

accumulation and death benefits. European Actuarial Journal, 6(1), 197–231.

Ledlie, M., Corry, D., Finkelstein, G., Ritchie, A., Su, K., & Wilson, D. (2008). Variable annuities.

British Actuarial Journal, 14(2), 327–389.

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica:

Journal of the Econometric Society, 59(2), 347–370.

Ng, A. C.-Y., Li, J. S.-H., & Chan, W.-S. (2011). Modeling investment guarantees in Japan: A risk-

neutral garch approach. International Review of Financial Analysis, 20(1), 20–26.

Peng, J., Leung, K. S., & Kwok, Y. K. (2012). Pricing guaranteed minimum withdrawal benefits under

stochastic interest rates. Quantitative Finance, 12(6), 933–941.

Piscopo, G., & Haberman, S. (2011). The valuation of guaranteed lifelong withdrawal benefit options in

variable annuity contracts and the impact of mortality risk. North American Actuarial Journal,

15(1), 59–76.

Quittard-Pinon, F., & Randrianarivony, R. (2011). Impacts of jumps and stochastic interest rates on the

fair costs of guaranteed minimum death benefit contracts. The Geneva Risk and Insurance Review,

36(1), 51–73.

Rubinstein, M. (2005). The valuation of uncertain income streams and the pricing of options. In S.

Bhattacharya & G. M. Constantinides (Eds.), Theory of valuation, 2nd edn (pp. 25–51). World

Scientific.

Siu, T. K., Tong, H., Yang, H., et al. (2004). On pricing derivatives under GARCH models: A dynamic

Gerber–Shiu approach. North American Actuarial Journal, 8, 17–31.

Siu-Hang Li, J., Hardy, M. R., & Tan, K. S. (2010). On pricing and hedging the no-negative-equity

guarantee in equity release mechanisms. Journal of Risk and Insurance, 77(2), 499–522.

Tardelli, P. (2011). Utility maximization in a pure jump model with partial observation. Probability in the

Engineering and Informational Sciences, 25(1), 29–54.

Tardelli, P. (2015). Partially informed investors: hedging in an incomplete market with default. Journal of

Applied Probability, 52(3), 718–735.

Whitle, P. (1951). Hypothesis testing in time series analysis (Vol. 4). Stockholm: Almqvist & Wiksells.

Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and control,

18(5), 931–955.

Zeileis, A., Leisch, F., Hornik, K., & Kleiber, C. (2002). Strucchange. An R package for testing for

structural change in linear regression models. Journal of Statistical Software, 7(2), 1–38.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps

and institutional affiliations.

123


a time series framework for pricing guaranteed lifelong ...web.iitd.ac.in/~dharmar/paper/ce2020.pdfa...

Documents