on the propensity to surrender a variable annuity contract
TRANSCRIPT
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On the Propensity to Surrender a Variable Annuity Contract
An Empirical Analysis of Dynamic Policyholder Behaviour
Working Paper
27th July 2011
Christian Knoller
Institute for Risk Management and Insurance Munich Risk and Insurance Center (MRIC)
Ludwig-Maximilians-Universität München (LMU) Phone: +49 (0)89 2180 3764 Fax: +49 (0)89 2180 99 3764 Email: [email protected]
Gunther Kraut
Institute for Risk Management and Insurance Munich Risk and Insurance Center (MRIC)
Ludwig-Maximilians-Universität München (LMU) Phone: +49 (0)89 2180 2229 Fax: +49 (0)89 2180 99 2229
Email: [email protected] A travel grant by Versicherungsforen Leipzig is gratefully acknowledged. Besides his scientific work, the author
works for Munich Re in the divisional unit Financial Solutions Life.
Pascal Schoenmaekers
Munich Reinsurance Company Divisional unit: Financial Solutions Life
Königinstraße 107 80802 Munich
Germany Phone: +49 (0)89 3891 9857 Fax: +49 (0)89 3891 7 9857
Email: [email protected]
The conclusions are those of the authors and do not necessarily reflect the opinions of any of the institutions mentioned above.
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Abstract
Option theory predicts that a variable annuity contract will be surrendered if and only if the sur-render value exceeds the fair value of the contract that heavily depends on the value of embed-ded options and guarantees. On the other hand, policyholders might surrender their contract due to liquidity constraints. Findings from the field of Behavioural Economics suggest that individuals’ investment behaviour typically is not fully rational. The statistical analysis of individual policy data from a Japanese variable annuity product confirms the conjecture of option theory: moneyness, which is an indicator for the economic value of a guarantee, has the largest explanatory power for the rate at which policyholders surrender their policies. We find evidence that the effect to which option theory explains the surrender rate depends on the size of the policy which we con-sider as a proxy for wealth and financial literacy of the policyholder. Moreover, our results sup-port the emergency fund hypothesis.
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1 Introduction
Unit-linked life insurance products provide an interesting investment opportunity to individuals be-
cause they offer a transparent possibility to participate in the development of the financial markets
and thus to obtain higher average returns compared to traditional participating life insurance prod-
ucts. The disadvantage of these products for the policyholders is that they, in turn, have to bear in-
vestment risk. This is why a number of unit-linked life insurance products have been developed that
contain some form of investment guarantee. Variable annuities are essentially unit-linked products
with such guarantees. In contrast to other products, these guarantees are given by the insurer that
manages them separately from the underlying fund and not within it. Among the most popular guar-
antees offered within these products are guaranteed minimum accumulation benefits (GMAB) that
provide a guaranteed minimum payout at the maturity of the contract regardless of the account
value (often the amount that had been invested). Contracts containing guaranteed minimum death
benefits (GMDB) promise to pay a certain minimum amount in case of the policyholder’s death.
Guaranteed minimum withdrawal benefits (GMWB) allow the policyholder to withdraw a certain
percentage of the invested amount each year, irrespective of the current value of the underlying
fund. Guaranteed minimum income benefits (GMIB) promise the policyholder a minimum annuity
payment during the payout phase, irrespective of the account value at the annuitization date. This
minimum annuity payment is predefined at the conclusion of the contract.1
Typically, the guarantees offered within variable annuities are charged for with a fee that is not paid
in advance but during the lifetime of the policy through a regular deduction of units from the account
value. The contracts usually offer a number of additional options to the policyholder. For GMAB
products, the policyholder typically has the right to surrender the policy in return for the current
value of the underlying fund, possibly subject to a predefined surrender penalty. In rising markets,
the value of the underlying fund will increase and thus the economic value of the guarantee will fall.
The value of the guarantee less the value of the future fees for these guarantees may become nega-
tive. At this stage, the policyholder has a strong incentive to exercise the surrender option. In reality,
insurance companies providing variable annuities usually do not assume this behaviour for all policy-
holders because the economic value of the guarantee might not be fully evident to the policyholders
and there probably are a number of factors independent of the value of the guarantee that will drive
policyholder actions (e.g. some policyholders will surrender because they need liquidity, even if their
guarantees are valuable).
1 For a more detailed description of variable annuities and the guarantees therein see Brown and Poterba (2006) and Condron (2008).
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The demise of the Equitable Life Assurance Society in the United Kingdom was mainly caused by the
fact that the guaranteed annuity options (GAOs) the company provided to its policyholders were
neither appraised correctly nor were these embedded options hedged (please refer to Pelsser 2003
and O'Brien 2006 respectively). White Mountain Life Re, for instance, suffered a loss in 2010 resulting
from a reduction in the surrender assumptions used to calculate their variable annuity guarantee
liability. The change in the surrender assumptions increased the liability by USD 48 million (see White
Mountain Insurance Group 2010).2
It is possible for the insurer to use margins for prudent pricing assumptions, but the size of these
margins may be limited by competitive pressure. Thus, it is usually necessary to apply some form of
lapse supported pricing. In particular, it is usually assumed that fees are collected from some policy-
holders who in fact surrender their policies prior to maturity and by doing so not take advantage of
their guarantees. Milevsky and Salisbury (2006) find that most products only charged 30 to 45 basis
points of the account value as a fee for a GMWB where the no arbitrage hedging cost in the market
ranged from 73 to 160 basis points of assets. Bauer et al. (2008) develop a general pricing framework
for guaranteed minimum benefits in variable annuities. They find evidence that insurers typically
base their calculations on the assumption of irrational surrender behaviour. Swiss Re (2003) suggests
that compared to traditional participating products lapse ratios will be higher and more volatile for
unit-linked life insurance contracts because policyholders are more sensitive to changes in market
conditions and the account value of the underlying fund. Several authors have used dynamic surren-
der assumptions for simulation studies on the impact of policyholder behaviour, e.g. on solvency
capital requirements (Kochanski 2010) and the effectiveness of hedging (Kling et al. 2010). But so far,
there has been no empirical investigation of this question. In the following, we will fill this gap by
analysing if the surrender rates for variable annuities depend on the moneyness of the guarantees
embedded in the contract.
These examples show the relevance of policyholder behaviour
with respect to these options and guarantees for insurers’ risk management and financial stability. As
insurers might bear significant losses from policyholder behaviour risks, regulators need to know if
the insurers’ assumptions on lapse behaviour are correct. This is especially true as these risks are not
hedgeable. The value of the policyholders’ options heavily depends on the development of the un-
derlying fund and thus on the development of financial markets. This is why the technical specifica-
tions of the fifth quantitative impact study (QIS 5) of Solvency II require that “[i]n general, policy-
holders’ behaviour should not be assumed to be independent of financial markets“ (EIOPA 2010).
2 The CFO, David Foy, stated: "Our previous assumptions reflected our expectation that surrenders would rise as the surrender charges in the underlying annuities continue to decline. However, we have not seen such an uptick, as persistent instability in financial and foreign exchange markets has kept surrenders low. Given this experience, we felt it was prudent to lower our surrender assumptions in the third quarter." (White Mountain Insurance Group 2010)
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The surrender option for a policyholder of a variable annuity contract containing a GMAB resembles
the prepayment option for fixed interest rate mortgage contracts. Deng et al. (2000) find that option
pricing theory can explain default and refinancing behaviour for these mortgage contracts quite well.
However, they also find a significant heterogeneity among borrowers (see also Campbell 2006). We
explore two questions that contribute to the more general literature on how individuals perceive
options embedded in financial contracts:
(i) Does policyholder surrender behaviour for variable annuity products depend on the
value of the embedded option and hence indirectly on the development of the financial
markets?
(ii) Can heterogeneity among policyholders be observed?
Based on surrender data relating to a variable annuity product sold in the Japanese market, we em-
pirically analyse if the policyholders’ lapsing behaviour depends on the development of the underly-
ing fund and thus on the value of the embedded guarantee. Besides that we try to identify additional
factors that might be able to explain a potential heterogeneity in the policyholders’ behaviour. We
find that the moneyness of the guarantee heavily drives surrender rates. The surrender rate in-
creases with the attained age of the insured life and in the first year of the contract also with the
time the policy has been in-force. In addition, the influence of moneyness on the surrender rate de-
pends on the size of the policy. We find that owners of large policies tend to behave more sensitive
towards moneyness: They have a higher probability to surrender their contract if the guarantee is
out of the money. Policy size might to some extent serve as a proxy for wealth and thus also for fi-
nancial literacy. The positive interaction term for policy size and moneyness thus indicates that finan-
cial literary increases the importance of option pricing theory for surrender behaviour.
The remainder of the paper is organised as follows. Section 2 first gives an overview of the variable
annuity market in Japan. The literature on reasons for lapsing life insurance contracts in general is
introduced in section 3. Thereby, the “emergency fund hypothesis” and the “interest rate hypothe-
sis” are introduced. We then explain in more detail how dynamic policyholder behaviour, especially
dynamic surrender, is defined, before we describe the dataset in section 4. In section 5 the hypothe-
ses we want to test are derived from previous empirical and theoretical results. After a short intro-
duction of the empirical model we use in section 6, the results are presented and discussed in section
7. Section 8 concludes.
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2 Market Overview3
There was almost no market for variable annuities in Japan before 2002, when a regulatory change
allowed banks to sell these products. After that, the sales of variable annuity contracts heavily in-
creased. After 2006, new business volumes declined for several reasons. In 2007, stronger regulatory
requirements for the advisory service were introduced that made the sales process more compli-
cated. After the financial crisis, several foreign insurance companies ceased their variable annuity
business in Japan.
Figure 1 shows the development of the variable annuity new business volumes in
Japan from 2001 to 2010:
Figure 1: Development of variable annuity new business volumes in Japan4
The products are almost exclusively sold via bancassurance.
. Source: The Life Insurance Association of Japan (2011)
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3 The following chapter is based on Winkler (2009).
Most insured lives are rather mature
and looking for an investment possibility for their savings in order to bequeath them to their heirs. So
the products predominantly are single premium products and have a strong saving character. Due to
the extremely low interest rates in Japan, savers started to invest in equity. Since the Japanese tend
to be rather risk averse, they highly value investment guarantees in their variable annuities; usually a
GMAB (in some cases a GMIB respectively) is combined with a GMDB. Many products contain a “tar-
get setting”: the policyholder determines a target level for the investment (usually between 110 and
150% of the premium). If the fund reaches this level during the policy term, it is automatically sur-
4 2010 numbers preliminary due to financial year running from 1 April 2010 to 31 March 2011. 5 Only about 1% of the contracts are sold via insurance agents or brokers.
0 5
10 15 20 25 30 35 40 45 50
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
VA Premium Income (in ¥ bn.)
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rendered (“knockout”). 6
Most of the products are invested in a mutual fund consisting of domestic
and foreign bonds as well as domestic and foreign equity and do not allow for fund switches.
3 Dynamic Policyholder Surrender
Not only variable annuities but also traditional participating life insurance contracts contain a num-
ber of options and guarantees, in particular the option to lapse or surrender the contract before ma-
turity.7
Outreville (1990) analyses the lapse rates for ordinary life insurance contracts in the United States
and in Canada over the period 1955 to 1979. For both countries he consistently finds that an increase
in unemployment drives lapse rates and interprets these results as strong evidence for the emer-
gency fund hypothesis. On the other hand, he finds only weak support for the interest rate hypothe-
sis as a significant influence of the interest rate on lapses is only observed for the United States and
also only for some specific measures of the interest rate. Kuo et al. (2003) look at the unemployment
rate, the interest rate, and lapse rates for all ordinary life insurance policies in force in the United
States in the period 1951 to 1998. Using a co-integration approach these authors find that the un-
employment rate affects the lapse rates in the long and in the short run. The interest rate in contrast
Of course, there may be a lot of reasons for individuals to lapse their contract and the pro-
pensity to do so may be affected by many factors like their family situation, financial goals and
health, the insurer’s financial situation, and the distribution channel. In the academic literature
mainly two hypotheses have been used to explain lapsing behaviour (the interested reader is, for
example, referred to Outreville 1990 and Kuo et al. 2003): the “emergency fund hypothesis” and the
“interest rate hypothesis”. The emergency fund hypothesis states that policyholders need the sur-
render values as an emergency fund if they are in financial need due to e.g. unemployment. There-
fore, the lapse rate would rise in times of recession. Since the market interest rate can be seen as an
opportunity cost of owning a life insurance policy, according to the interest rate hypothesis, lapse
rates should rise when the market interest rate rises. Moreover, if the interest rate goes up, premi-
ums for new contracts usually fall. Therefore, policyholders might lapse their contract to purchase a
new one offering the same coverage for a lower premium or alternatively a higher coverage for the
same premium.
6 This feature appeared to be a valuable marketing instrument and it helps to control for policyholder behav-iour risk: if the guarantees are far out of the money, the contract is automatically surrendered. 7 In contrast to lapsing, early surrender usually results in a cash payment to the policyholder. Nevertheless, the traditional measurement of lapse rates contains lapse and surrender. Hence, following Kuo et al. (2003) we will use the terms lapse and surrender synonymously. In the context of variable annuities surrender is the correct term.
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only drives lapses in the long run. However, the effect of the interest rate on the lapse rate is more
pronounced. Kim (2005) also finds evidence for the emergency fund hypothesis using data relating to
a Korean life insurance company from 1997 to 2000. This author finds that lapses increase when
economic growth decreases and unemployment rises. Differences between the interest rate credited
to the policy and the reference market interest rate also drive lapses. In accordance with the results
of Kuo et al. (2003), the interest rate difference has a stronger effect on lapse rates than the unem-
ployment rate and economic growth.
Summing up, all three analyses propose that both the emergency fund hypothesis as well as the in-
terest rate hypothesis are models that explain surrender rates for traditional participating life insur-
ance contracts. These hypotheses might play a role for the surrender rates of variable annuities as
well. But in the following analysis, we will put our focus on the special characteristics of dynamic
policyholder behaviour concerning variable annuities. According to Kent et al. (2009) dynamic policy-
holder behaviour “reflects the fact that a policyholder’s propensity to exercise options available in a
life insurance policy can be influenced by external factors“, e.g. by economic conditions like the re-
turn on capital markets or the development of the underlying fund. Hence, when assuming dynamic
policyholder behaviour, the value of the contract’s guarantees plays an important role for surrender
rates.
The surrender value is defined as the account value less surrender charge, if any. In general, for each
variable annuity policy, the following inequality holds during the complete policy term:
𝑠𝑢𝑟𝑟𝑒𝑛𝑑𝑒𝑟 𝑣𝑎𝑙𝑢𝑒 ≤ 𝑎𝑐𝑐𝑜𝑢𝑛𝑡 𝑣𝑎𝑙𝑢𝑒
with equality as soon as the surrender charge, which is a function of curtate duration8
The surrender option gives the policyholder the right to sell back the contract to the insurance com-
pany at the surrender value. According to Bacinello (2003), it is an American put option. Since it can
only be used at discrete exercise dates, Gatzert (2009) classifies it as a Bermudan put option on the
cash flow stream of future expected insurance benefits with the surrender value as the strike price. A
, is equal to
0%. The fair value of a variable annuity contract containing a GMAB and a GMDB rider as well as a
surrender option, at least as seen from the perspective of a policyholder, equals:
𝑎𝑐𝑐𝑜𝑢𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 + 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑠
− 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓𝑢𝑡𝑢𝑟𝑒 𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒 𝑓𝑒𝑒𝑠
+ 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑟𝑟𝑒𝑛𝑑𝑒𝑟 𝑜𝑝𝑡𝑖𝑜𝑛9.
8 A typical surrender charge, expressed as a percentage of account value, is: year 1: 6%, year 2: 5%, …, year 6: 1%, year 7 onwards: 0%. 9 To be precise, future surrender fees have to be considered in the calculation of the present value of the sur-render option.
9
GMAB provides a guaranteed minimum payout at the maturity of the contract regardless of the ac-
count value. The payoff of the GMAB contract at the end of the policy term is defined as
max{𝑎𝑐𝑐𝑜𝑢𝑛𝑡 𝑣𝑎𝑙𝑢𝑒;𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑑 𝑣𝑎𝑙𝑢𝑒}.
Hence, the GMAB can either be seen as the guaranteed value plus a European call option on the ac-
count value (with the guaranteed value as the strike price) or as the account value plus a European
put option on the account value (with the guaranteed value as the strike price). The second perspec-
tive appears to be more appealing as here the value of the put option equals the value of the guaran-
tee (see Mahayni and Schneider 2010). A guarantee is defined as “in the money” if the the time value
of the guarantee exceeds the expected present value of the future guarantee fees, “at the money” if
they are equal, and “out of the money” otherwise. Like the GMAB, the GMDB can be seen as a put
option on the account value with the guaranteed value as the strike price. However, since the payoff
structure is between an American and a European option and it is triggered by death, Milevsky and
Posner (2001) name it a “Titanic option”. The time value of the guarantees cannot become negative.
However, the difference between the time value of the guarantees and the expected present value
of the future premiums related to the guarantee might become negative as the guarantee is financed
by recurring premiums (during the lifetime of the policy through a regular deduction of units from
the account value).
The policyholder has a strong incentive to exercise the surrender option if the surrender value ex-
ceeds the fair value of the contract. This is the case the more the account value exceeds the guaran-
teed value and thus the guarantees are out of the money. As stated in Kling et al. (2010), policy-
holder behaviour can be considered as optimal “if the policyholder decides to surrender the contract
whenever the benefit from discounting the contract (…) exceeds the surrender fees.” 10
The emergency fund hypothesis that has been explained above might be relevant for variable annui-
ties in a similar way: there may be reasons for policyholders to surrender their contract independ-
ently of moneyness, i.e. in particular also if the guarantee is in the money. On the other hand, there
might be several reasons for policyholders not to surrender their contract although the guarantee is
out of the money: (i) They possibly do not have a full appreciation of the economic value of the op-
If he or she
actually wants to maintain the contract, the policyholder can nevertheless surrender it. Repurchasing
the same contract afterwards generates a higher guarantee that is at the money. However, transac-
tion costs like expense loadings and surrender charges might make this strategy unattractive.
Milevsky and Salisbury (2002) consider this decision an “optimal stopping problem” that can be
solved by applying American option pricing techniques. They develop a model to value the surrender
option for a GMAB / GMDB variable annuity contract.
10 Assuming a risk neutral investor.
10
tions and guarantees they own, (ii) there might be transaction costs, tax considerations or other
regulatory reasons, and (iii) heuristics and biases that have been shown to determine decision mak-
ing under risk might also influence individuals’ propensity to exercise these options and guarantees.
(i) The guarantee is paid for by regular deductions from the account value. Thus, future
guarantee fees depend on the development of the underlying fund. Additionally, the val-
ue of the contract is affected by mortality risk. Due to this path dependence of cash flow
streams, it is not a trivial task to calculate the present value of the guarantees and the
surrender option and thus the fair value of the contract (see Milevsky and Salisbury
2002). Hence, policyholders having understood that they have a surrender option and
that the value of their contract depends on the development of the account value
probably will not be able to do this calculation and rather use a heuristic approach. To
model dynamic policyholder behaviour, insurers typically use a heuristic approach as
well. Therefore the moneyness of the contract’s guarantees has to be measured. Usually
the moneyness is determined by comparing the surrender value and the guaranteed
value: 11
𝑚𝑜𝑛𝑒𝑦𝑛𝑒𝑠𝑠 = 𝑠𝑢𝑟𝑟𝑒𝑛𝑑𝑒𝑟 𝑣𝑎𝑙𝑢𝑒 (𝑎𝑐𝑐𝑜𝑢𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑙𝑒𝑠𝑠 𝑠𝑢𝑟𝑟𝑒𝑛𝑑𝑒𝑟 𝑓𝑒𝑒𝑠)
𝑔𝑢𝑎𝑟𝑎𝑛𝑡𝑒𝑒𝑑 𝑣𝑎𝑙𝑢𝑒
As our primary goal is to analyse if policyholders surrender dynamically and to gain in-
sights for the insurers’ surrender rate modelling, we will base our analysis on this “quasi-
rational” heuristic, although it does not completely reflect option pricing theory. For the
interpretation of our results, this limitation has to be considered.
(ii) There are potentially a number of factors other than the value of the guarantee that
have influence on policyholder behaviour. There might be transaction costs, tax consid-
erations or other regulatory reasons that make individuals not surrender a contract al-
though the guarantee is not valuable. For example, favourable taxation rules may be
linked to a minimum policy term. This might prevent some policyholders from surrender-
ing their contracts (see Kent and Morgan 2008). Also, without these transaction costs, it
would be possible for policyholders who otherwise do not want to change their invest-
ment strategy to increase the guaranteed value while maintaining the account value by
selling and instantly repurchasing the contract. In reality, this strategy usually involves
transactions costs like e.g. surrender fees or initial expense loadings. Thus, the guaran-
11 The American Academy of Actuaries uses a “dynamic lapse multiplier” calculated at the valuation date (as the ratio of account value and guaranteed value). To model dynamic lapsing behaviour, this factor is multiplied with the deterministic base lapse rate (see American Academy of Actuaries 2005).
11
teed value increases and the account value is reduced. One would expect to observe
heterogeneous surrender behaviour for out of the money policies. Depending on indi-
vidual risk preferences, individuals have different moneyness threshold levels for surren-
dering their policy (and instantly repurchasing it12
(iii) However, the persistency behaviour of policyholders might also be driven by irrational
factors. Inspired by the seminal papers of Tversky and Kahneman (1974) on heuristics
and biases in risk perception and Kahneman and Tversky (1979) on prospect theory,
many experimental and empirical studies have shown that individuals do not always be-
have fully rational concerning decisions under risk and uncertainty. Behavioural Finance
has gained new insights in describing and explaining investors’ behaviour, in particular
what kind of portfolios they choose and how they trade over time. These studies find
that many investors do not sufficiently diversify their portfolio and that diversification
(e.g. within 401(k) plans) depends on the number and composition of funds that have
been offered to the individuals.
). In particular, for individuals that want
to keep this product, it might be rational not to surrender their contract although the
guarantee is out of the money. Hence, the out-of-the-moneyness of the guarantee is a
necessary, but not a sufficient condition for “rational” surrender behaviour.
13
Life insurers usually do not assume fully rational policyholder behaviour: 34 companies
from 6 countries took part in a survey by Kent et al. (2009) about dynamic policyholder
behaviour with respect to traditional participating life insurance products containing dif-
ferent options and guarantees. When they were asked to rate their policyholders’ ra-
tionality for early guaranteed lapse / surrender on a scale from 0 to 10, the average an-
swer was 6.0. Concerning variable annuities, irrationality arises through surrendering
contracts containing a valuable guarantee and, to some extent, through not surrendering
contracts containing non-valuable guarantees. In many variable annuity product designs,
the value of the financial guarantee is less obvious than that of a simple return of pre-
mium guarantee. Variable annuity products are a complex combination of financial op-
tions. The value of these options is not made explicit in many product designs as the sur-
render value is defined as the account value less the surrender charge, if any, and thus
Heath et al. (1999) find that psychological factors influ-
ence stock option exercise. Poteshman and Serbin (2003) analyse the early exercise of
exchange traded options. They find that a large number of option exercises are clearly ir-
rational.
12 In reality, this effect will often be combined with a change of the investment strategy. 13 See Barberis and Thaler (2003) for a detailed survey on Behavioural Finance.
12
neglecting the time value of the guarantee. According to Kent et al. (2009), guaranteed
annuity options like GMIBs might be more transparent than other types of options since
it is easily possible to compare the GMIB annuity rates to those available in the market.
This is why the average expected level of rationality for these types of guarantees is
higher (7.0) than for early guaranteed lapse / surrender (6.0). On the other hand, Kent
and Morgan (2008) state that GMABs “are clearly defined financial options and could
therefore be assumed to be fully exercised by policyholders when they are in the
money”. In this paper, a GMDB / GMAB single premium product is analysed because of
its simple product design. Hence, the extent to which policyholders exercise their option
to surrender the policy should be a good indicator to test whether option pricing theory
can describe surrender behaviour.
As we have already mentioned, fixed interest rate mortgage contracts contain an option to prepay
the mortgage that is similar to the surrender option in variable annuity contracts and life insurance
contracts respectively (see de Giovanni 2010 for an elaborate discussion): policyholders of a GMAB
variable annuity could surrender their contract and reinvest the money in the same or a similar
product if the guarantee is out of the money. The mortgage holder can prepay the mortgage and
refinance it at a lower interest rate if interest rates decrease (or increase the mortgage for the same
monthly rate). This refinancing strategy incurs substantial transaction costs. Hence, the interest rate
spread must be high enough to cover these transaction costs. There have been several studies look-
ing at the refinancing behaviour of fixed interest rate mortgage holders. Despite the declining inter-
est rates in the U.S. between 1997 and 2003, many households kept their old mortgage contracts and
thus were paying rates clearly above the current rate (see Campbell 2006 for an overview). Deng et
al. (2000) find that a large number of households exercise their options in a fully rational way. But
they also observe a large heterogeneity in the probability to refinance or default a contract. They find
that negative equity (outstanding debt is higher than the value of the house) makes defaults more
probable but refinancing less probable, probably because interest rate conditions would be worse for
them due to the higher loan-to-value ratio. Unemployment and divorce also keep households from
refinancing, presumably also because refinancing is more difficult for them. Hence, not refinancing a
fixed interest rate mortgage contract although market interest rates have decreased is not necessar-
ily irrational: there could be clearly rational reasons that prevent households from refinancing. How-
ever, the lack of refinancing might be an investment mistake indicating that the mortgage holders do
not understand the value of the prepayment option they have. Green and LaCour-Little (1999) find
that many mortgage holders do not prepay their contract when the option is in the money and many
prepay when it is not optimal. They infer from their results that household characteristics like the
loan-to-value ratio can partially explain the observed prepayment behaviour. However, to some ex-
13
tent it appears to be irrational. In the case of variable annuities, the refinancing is not as clearly ob-
servable as the surrender of the contract might often be combined with a change of the investment
strategy. Additionally, most insurance companies will presumably not collect this kind of repurchas-
ing information in their data bases.
4 Data
Product characteristics The data analysed in this paper relates to a sample out of a portfolio of variable annuity policies
originally written in Japan. All policies in the dataset have both a GMDB as well as a GMAB rider. At
inception of the policy, the policyholder fixes a target value, expressed as a percentage of the single
premium. The considered portfolio only contains contracts with a target level of 110%. For each
product a knockout period is defined. Provided the knockout period has expired, the policy termi-
nates automatically as soon as its account value outranges its target value. The company charges an
initial expense loading which effectively leads to an investment of less than 100% of the single pre-
mium. The guarantee fee is periodically charged to the account value of the policy. The product is
exclusively sold via bancassurance. The single premium variable annuity product analysed in this
paper has the following product features:
Policy term ten years
Asset allocation Consists of an investment in domestic and foreign bonds as well as
domestic and foreign equity. The fund is reweighted periodically.
GMAB 100% of single premium
GMDB 100% of single premium
Target level 110% of single premium
Knockout period either one or three years
Surrender charge none
Dataset The dataset presented in this paper consists of 15,180 distinct policies, from which we have observa-
tions from 1 June 2009 up to and inclusive 30 November 2010. For the analysis we have 19,335.7
policy years of central exposure-to-risk at our disposal. The investigated dataset contains policy ex-
posure information, covariates and surrender counts.
14
At the beginning of each calendar month, data relating to the policies are available. The dataset con-
tains the covariates enumerated in Table 1 on a policy level. Moneyness shall mean the ratio of the
surrender value of the policy, measured as at the end of the respective month or at the date at which
the policy went out of force provided this took place in the course of the respective month, over the
guaranteed value. We assume that moneyness remains constant during the respective month.
Covariate Description
Moneyness Rounded to an integer multiple of 0.01
Attained age Attained age of the insured life (in years)
Gender Gender of the insured life
Curtate duration Curtate duration of the policy (in months)
Policy size Single premium (in JPY)
Knockout period Knockout period (in years)
Table 1: Available Covariates
Table 2 summarises the central exposure to risk and surrender count by calendar month. It is inter-
esting to note that the surrender count varies significantly from one month to the other. This is
mainly explained by the evolution of the moneyness on the policy level.14
More details are presented
in chapter 7.
14 Further analysis not presented in this paper did not find evidence for the existence of a strong calendar year effect in this dataset. During the financial crisis, Japanese life insurance companies did not experience a break-down of new business which is a strong indicator for the confidence of the public in the financial strength of life insurance companies offering these products.
15
Month Central Exposure to Risk (in years)
Surrender Count
Jun 09 641.4 13 Jul 09 619.9 11
Aug 09 617.6 44 Sep 09 594.2 44 Oct 09 1,228.6 47 Nov 09 1,264.1 69 Dec 09 1,215.9 104 Jan 10 1,207.5 75 Feb 10 1,122.6 52 Mar 10 1,316.4 87 Apr 10 1,182.3 105 May 10 1,214.7 48 Jun 10 1,172.7 32 Jul 10 1,170.2 13
Aug 10 1,246.5 37 Sep 10 1,165.9 46 Oct 10 1,122.4 60 Nov 10 1,232.8 83
Total 19,335.7 970 Table 2: Data description
Figure 2 - Figure 4 provide a short descriptive analysis of the dataset:
Figure 2
16
Many policyholders of a single premium product purchase their contract around the retirement age
as can be seen from the mode of the age distribution in Figure 2.
Figure 3
Figure 3 shows that the moneyness distribution in our data set is well centred around the value 1.0.
To be able to test the hypotheses presented in section 5, it is important that both exposure for in the
money policies and for out of the money policies is observed.
17
Figure 4
The observed portfolio is open for new business as can be seen in Figure 4.
5 Hypotheses
Most of the theoretical papers and simulation studies presented above propose to assume some kind
of functional relationship between surrender behaviour and the development of the underlying fund
and the moneyness of the guarantee. The analyses of the refinancing behaviour of fixed interest rate
mortgages suggest that some people appear to behave irrational and there might be rational reasons
that prevent households from exercising their prepayment option. However, a large share of bor-
rowers behaves at least approximately according to option pricing theory. This suggests that policy-
holders of variable annuity contracts might do so as well. Hence, fewer contracts should be surren-
dered if the guarantee is in the money than if it is out of the money, or more general:
Hypothesis 1:
The probability for a contract to be surrendered increases with the moneyness of the guarantee.
18
On the other hand, the emergency fund hypothesis provides a reason to surrender a variable annuity
contract or a life insurance contract respectively although its guarantees are valuable.15 De Giovanni
(2010) integrates these findings in his rational expectation model. He distinguishes between rational
surrender depending on the value of the implicit options and irrational surrender due to e.g. financial
difficulties. Of course, we cannot observe if policyholders are in a situation in which they suddenly
need a considerable amount of money. But there might be some determinants we can observe that
potentially drive the probability of such a situation. A GMAB is valuable if the account value is low. As
the account value depends on the development of stock prices and thus on the economic environ-
ment in general, the guarantee should be in the money in case of a poor economic situation (when
stock prices go down). Since emergency fund situations might come along with such a situation (e.g.
due to unemployment), it seems to be plausible that more policyholders surrender their contracts
when moneyness is low due to emergency fund situations.16
Hypothesis 2.1
When people get older, medical emer-
gencies and severe diseases arise more often. As a considerable amount of money might be needed
to cover health care or long-term care costs, the age of the insured life might be positively correlated
with the probability to surrender the contract.
Age is positively correlated with the probability to surrender.
Smaller policies might be an indication for a policyholder with lower income or wealth. As emergency
fund situations are more likely if wealth and income are low, we suppose that the surrender rate will
be higher for small policies.
Hypothesis 2.2
Policy size is negatively correlated with the probability to surrender.
Even if not all policyholders behave according to option pricing theory, we might be able to identify a
number of factors that drive this behaviour.17
15 This would not necessarily be the case if a well-developed secondary market for variable annuities existed. At the moment, this is not the case for any of the existing variable annuity markets. Nevertheless, as secondary markets might develop in the future, surrender behaviour might be subject to a considerable change (see Kent and Morgan 2008).
Beside institutional factors like regulation and secon-
dary markets, the main driving force presumably is financial literacy. Although we cannot directly
observe the policyholders’ financial literacy, we have a number of variables that are likely to be cor-
related with financial literacy: individuals with a higher financial literacy will better understand the
16 Unfortunately, we cannot test this hypothesis because we cannot observe whether policyholders surrender their contract because of an emergency fund situation when the guarantee is out of the money. 17 See Kent and Morgan (2008) for an elaborate discussion of possible factors.
19
concept of financial options. Behrman et al. (2010) find a positive correlation between wealth accu-
mulation and financial literacy. Agnew (2006) finds that higher salaried employees show fewer be-
havioural biases and make significantly better choices concerning their 401(k) plans. So there is evi-
dence that on average wealthier people have a higher financial literacy. As we have already dis-
cussed, policy size should be a good proxy for the policyholder’s wealth. Hence, we suppose that
surrender behaviour for large policies is more sensitive towards the value of the embedded options
and guarantees.
Hypothesis 3:
For large policies, surrender rates are higher when the guarantee is out of the money.
Assuming a negative interaction between policy size and moneyness if the guarantee is in the money
would also be clear evidence for the dependence of policyholder rationality on policy size. On the
other hand, it seems to be rather easy to understand that surrendering a contract when the account
value is lower than the guaranteed value would be a bad decision. Thus, we do not expect to observe
such an effect.
6 Method
Modelling count data is essential in the analysis of insurance data. Count regression analysis allows
the identification and quantification of risk factors and prediction of the expected frequency of sur-
render given the characteristics of the insured lives and their policies respectively. There is a vast
amount of literature on count regression analysis. The interested reader is referred to Cameron and
Trivedi (1998) or Denuit et al. (2007). To test our hypotheses, we will use a Poisson generalised linear
model with logarithmic link function as follows:
𝑆𝑚,𝑐,𝑠,𝑎 ~ Poisson(𝐸𝑚,𝑐,𝑠,𝑎 ∙ 𝜇𝑚,𝑐,𝑠,𝑎)
whereas ln 𝜇𝑚,𝑐,𝑠,𝑎has the following functional form
𝛽0 + 𝛽1(𝑚−𝑚1)+ + 𝛽2(𝑚−𝑚2)+ + 𝛽3𝑐𝐼(𝑐≤12) + 𝛽4𝑎 + 𝛽5𝐼(𝑐>12) + 𝛽6𝐼�𝑐≥24 𝑚<1� + 𝛽7𝐼�𝑠≥8 𝑚>1�
Here, 𝑆𝑚,𝑐,𝑠,𝑎 denotes the number of surrendered policies by moneyness 𝑚, curtate duration 𝑐, pol-
icy size 𝑠 and attained age 𝑎. The term 𝐸𝑚,𝑐,𝑠,𝑎 is an exposure constant that gives the length of time
(in years) for which a policy has been under observation. 𝜇𝑚,𝑐,𝑠,𝑎 denotes the surrender rate, i.e. the
expected number of surrender per policy year exposure to risk. 𝐼 denotes the indicator function. The
last interaction term allows to differentiate for out of the money policies between surrender behav-
iour of policyholders having a large policy and those who have a small policy. 𝑚1 and 𝑚2 are fringes
20
of hockey stick functions. A hockey stick function is a real-valued function, whose function value
equals zero up to its fringe and whose function value increases monotonically with a constant slope
of 1 from the fringe onwards, i.e. it is formally defined by
(𝑚 −𝑚1)+ = � 0, 𝑚 < 𝑚1𝑚 −𝑚1, 𝑚 ≥ 𝑚1
�
The categorisation of the covariates which was used in the analysis is given in Table 3. As insignificant
variables are taken out of the analysis, the results in the following chapter do not list all of these
variables.
Covariate Categorisation
Moneyness m continuous
Attained age a in years
Gender male
female
Curtate duration c in months
Policy size s < 8m JPY
≥ 8m JPY
Knockout period 1 year
3 years
Table 3: Categorisation of covariates
21
7 Findings
The empirical analysis reported in Table 4 is largely consistent with our hypotheses.18
Regression Coefficient β Covariate Regression Term Estimate Std. Error z value p value
(Intercept) 𝛽0 * * -21.491 0.0000
Moneyness hockey stick 1 𝛽1(𝑚−𝑚1)+ 24.9828 1.6354 15.28 0.0000
Moneyness hockey stick 2 𝛽2(𝑚−𝑚2)+ -20.5529 5.2090 -3.95 0.0000
Curtate duration 𝛽3𝑐𝐼(𝑐≤12) 0.1266 0.0247 5.14 0.0000
Attained age 𝛽4𝑎 0.0093 0.0032 2.86 0.0042
Curtate duration 𝛽5𝐼(𝑐>12) 1.4490 0.2465 5.88 0.0000
Interaction term 1 𝛽6𝐼�𝑐≥24 𝑚<1� -0.4446 0.1976 -2.25 0.0245
Interaction term 2 𝛽7𝐼�𝑠≥8 𝑚>1� 0.7588 0.0858 8.84 0.0000
Table 4: Parameter estimates of regression model
The surrender rate is an increasing function in moneyness 𝑚, as both �̂�1 > 0 and �̂�1 + �̂�2 > 0:
ln �̂�(𝑚) = �̂�0 + �̂�1(𝑚−𝑚1)+ + �̂�2(𝑚−𝑚2)+
wherein 𝑚1 < 𝑚2. This is in line with option pricing theory. Although the economic value of the
guarantee is not made explicit to the policyholders, it appears, however, that they take the value of
the guarantee provided to them into account for their surrender decision. Hence, our results strongly
support Hypothesis 1.
Figure 519
18 The parameter values for the intercept are taken out because of confidentiality reasons. This does neither derogate the qualitative conclusions of the analysis nor the quantitative conclusions about the relative size of observed effects.
shows that the surrender rate only increases slightly while the guarantee is in the money.
Around a moneyness value of 1, the graph shows a sharp twist. Out of the money, the surrender rate
increases exponentially at a higher rate than in the money. At both ends of the interval of the ob-
served moneyness values, the precision of the estimation decreases due to a decreasing number of
observations.
19 In the process of finding a suitable model fit, we used generalised additive models as explorative tools. Figure 5 - Figure 7 are generated with this technique and are included here for illustrative purposes only.
22
Figure 5
All other things being equal, the surrender rate increases linearly with the attained age of the insured
life (�̂�4 > 0). This result is in line with our hypothesis 2.1 and supports the emergency fund hypothe-
sis. This is also illustrated by Figure 6.
Figure 6
23
The inclusion of policy size as a main effect proved to be not significant, i.e. the covariate policy size
does not enter the linear predictor of the final regression model. As we conjectured that policy size
might be a proxy for the policyholder’s wealth or income, and emergency fund situations should be
more likely if wealth and income are low, this result does not confirm hypothesis 2.2 that had been
derived from the emergency fund hypothesis. Thus, we find mixed evidence with regard to that hy-
pothesis.
Curtate duration plays a significant role in explaining the observed policyholder behaviour. For young
policies (curtate duration ≤ 12 months), the surrender rate increases linearly with the time the policy
has been in-force, as �̂�3 > 0. After one year, the surrender rate remains more or less constant, as
can be seen from �̂�5 ≈ 12 · �̂�3 > 0 as well as graphically in Figure 7. We are not aware of a theoreti-
cal foundation for this. Note that the direction of this effect is contrary to what is mostly seen for
traditional participating life insurance contracts. Furthermore, more mature policies (curtate dura-
tion ≥ 24 months) are less likely to be surrendered if out of the money, as �̂�6 < 0. This might, how-
ever, be due to a selection effect as the insured lives that remain in the sample over the first two
years may systematically differ from those insured lives that surrender earlier. Moneyness sensitive
policyholders have a high probability to already reach their surrender threshold level during the first
two years. Hence, older policies still remaining in the sample have a high chance of being less sensi-
tive towards moneyness.20
20 Stanton (1995) and Deng et al. (2000) discuss the question of unobserved heterogeneity among fixed interest rate mortgage borrowers. They argue that surviving borrowers might be less aware of this option or less inter-est rate sensitive than those that prepay their mortgage contract earlier.
24
Figure 7
Regarding the interaction between policy size and moneyness, it turns out that in as much as a policy
is in the money, no differences can be found between small and large policies. On the other hand, for
policies having a moneyness in excess of 1, we observe that owners of large policies are more prone
to surrender their contracts, as �̂�7 > 0. Hence, hypothesis 3 is confirmed: Owners of larger policies
seem to react more sensitive to the moneyness of their contract. This is in line with the expected
higher financial literacy of these policyholders. We do not observe a negative interaction between
policy size and moneyness if the guarantee is in the money: not only should only a small number of
policyholders surrender their in the money policies. The in the money surrenders should also be
dominated by surrenders due to emergency fund situations. Figure 821
21 The GLM underlying the graphs in
visualises the interaction ef-
fect between policy size and moneyness: the slope of the graph is steeper for large policies when
moneyness is in excess of 1.
Figure 8 differs from the one presented in Table 4 due to illustrative rea-sons. Covariates are moneyness, policy size and their interaction. The included raw observations are to be in-terpreted as realised surrender probability on the aggregated exposure for all observations with the corre-sponding values for moneyness and policy size. For illustrative purposes, a few raw observations are cut off due to the scaling of the graph.
25
Figure 8
Several studies show that brokers and distribution channels in general potentially influence investor
behaviour. Poteshman and Serbin (2003) analyse early exercising of exchange traded stock options.
They find that customers of discount brokers and full-service brokers both exhibit a significant num-
ber of irrational exercises while traders at large investment houses do not conduct irrational early
exercises. Shapira and Venezia (2001) compare clients making individual decisions to investors whose
accounts were managed by brokerage professionals. They find that professionally managed accounts
were better diversified and that the disposition effect was stronger for individual investors.22
As we have already mentioned, the out-of-the-moneyness of the guarantee is only a necessary, but
not a sufficient condition for rational surrender behaviour: Transaction costs, tax considerations, and
risk preferences might prevent policyholders from surrendering their contract. If policyholders own-
ing large contracts differ from those owning smaller contracts with respect to these characteristics,
Owners
of large policies are presumably advised more intensely and by better agents or brokers and thus
might better understand the value of the options in their contracts; this could give another reason
for a positive interaction between policy size and moneyness if the guarantee is out of the money.
22 Weber and Camerer (1998) define the disposition effect as “the tendency to sell assets that have gained value (‘winners’) and keep assets that have lost value (‘losers’).” Usually, this effect is explained by the proper-ties of prospect theory’s value function: individuals behave risk seeking for losses and risk averse for gains rela-tive to a reference point.
26
this might explain differences in surrender behaviour, in particular a positive interaction term be-
tween policy size and moneyness if the guarantee is out of the money. However, as for the analysed
product all direct costs (especially the initial expense loading) are charged as a certain share of the
amount invested or the account value, differences in transaction costs cannot explain the observed
surrender behaviour. We are also not aware of any differences in taxation between small and large
policies. As the considered variable annuity product contains no surrender fees, risk preferences
should not influence the surrender behaviour of policyholders that want to reinvest the surrender
value in a different product. However, the propensity of policyholders to surrender their policy and
instantly repurchasing it might depend on individual risk preferences: Due to the up-front expense
loading, the account value would decrease. At the same time, the value of the guarantee less the
expected present value of future guarantee fees would increase. Depending on risk preferences, this
might be more or less favourable. So if many policyholders wanted to maintain their contract, also
those that surrendered their contract and thus wanted to instantly repurchase it, and if the owners
of big and small policies differ in their risk preferences in a way that surrendering and repurchasing
the contract is more attractive for the owners of large policies, this might to some extent explain the
observed surrender behaviour as well.
27
8 Conclusion Dynamic policyholder behaviour forms a significant risk for life insurance companies offering variable
annuity products because it has influence on the pricing of the options and guarantees within the
contracts, on solvency capital requirements, and hedging effectiveness. Hence, it is important for
insurers to know what drives their policyholders’ behaviour, in particular if surrender rates depend
on the development of the underlying fund and thus on the moneyness of the guarantees. We have
analysed this question empirically using data on a GMAB/GMDB product offered in Japan, the second
biggest market for variable annuities in the world.
We find that the majority of the surrender behaviour is explained by the moneyness of the guaran-
tee. We identified a number of additional factors that explain the propensity to surrender. Regarding
the more general formulation of our research question, we find that:
(i) Policyholder surrender behaviour for variable annuity products depends on the value of
the embedded option and hence indirectly on the development of the financial markets.
(ii) Heterogeneity among policyholders can be observed.
Surrender rates increase with attained age of the insured life. However, it is not correlated with the
policy size. Hence, we find mixed evidence for the emergency fund hypothesis. It might be reason-
able to include unemployment and / or economic growth into a future regression. In addition, the
influence of moneyness on the surrender rate depends on the size of the policy. We find that owners
of large policies tend to behave more sensitive towards the moneyness of the guarantee. According
to existing literature, financial literacy is positively correlated with income and wealth. In our study,
policy size serves as a proxy for wealth and thus to some extent also for financial literacy. The posi-
tive interaction term for policy size and moneyness thus indicates that financial literary increases the
influence of the value of the embedded options and guarantees for surrender behaviour.
The data we analyse covers a rather limited range of moneyness values. It remains open for future
investigation to see if policyholders behave differently should we have access to a broader range of
moneyness values. This is particularly important for regulatory considerations because the question
how much regulatory capital is needed to cover risks originating in dynamic policyholder behaviour is
especially relevant for tail events. The definition of moneyness used in this paper is only one of many
meaningful possibilities to measure the economic value of guarantees contained in variable annuity
products. Nevertheless, it gives a reasonable proxy. Further research will be needed to evaluate if
results change if the economic value is more precisely taken into account. This research project fo-
cuses on the economic analysis of the propensity to surrender in variable annuity products. The sta-
tistical analysis uses generalised linear models and their extensions. A more elaborate discussion of
this methodology for this purpose may be worthwhile.
28
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