UNIVERSITÀ DEGLI STUDI DI TRIESTE

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DIPARTIMENTO DI SCIENZE ECONOMICHE, AZIENDALI, MATEMATICHE E STATISTICHE

“BRUNO DE FINETTI”

Corso di Laurea in Economia, Commercio Internazionale e Mercati Finanziari

Tesi di Laurea in

INSURANCE TECHNIQUE

Structure, Development and Actuarial Valuation of Unit-Linked Policies

in Life Insurance

Laureando: Relatore:

Riccardo Esposito Chiar.mo Prof. Ermanno Pitacco

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Anno Accademico 2013-2014

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TABLE OF CONTENTS

Preface ………………………………………………………………………1

CHAPTER ONE

MAIN CHARACTERISTICS AND DEVELOPMENT OF UNIT-LINKED

POLICIES

1. Preliminary Concepts ………………………………………………….3

2. History and Development in Key Markets …………………………….5

3. Comparison with Other Insurance Policies and Unit Trusts …………..7

4. Advantages and Attractions of Unit-Linked Policies ………………….9

CHAPTER TWO

THE STRUCTURE OF UNIT-LINKED POLICIES

1. Varieties of Benefits ………………………………………………….12

2. Actuarial Pricing ……………………………………………………...13

3. The Investment in the Fund …………………………………………..17

4. The Two Major Configurations of Unit-Linked Policies …………….19

5. Guarantees ……………………………………………………………22

6. Reserving for Unit-Linked Guarantees ………………………………24

CHAPTER THREE

RISK DECOMPOSITION AND MITIGATION OF GMxB RIDERS

1. First Order Market Risks ……………………………………………..27

2. Second Order Market Risks …………………………………………..29

3. Policyholder Behavior Risks …………………………………………31

4. Demographic Risks …………………………………………………...32

5. Other Risks …………………………………………………………...33

CHAPTER FOUR

RISK TRANSFER AND HEDGING TECHNIQUES

1. Dynamic Hedging …………………………………………………….35

2. Static Risk Transfer Solutions ………………………………………..39

Conclusions ………………………………………………………………….43

Bibliography ………………………………………………………………...47

1

Preface

Unit-linked policies have become an important class within life insurance products and a

key driver in premium income. Their development derives from the insurance companies’

will to provide consumers with access to capital markets. Historically, the linking process

was driven by soaring equity markets. Nowadays, given the financial crises and asset

bubbles, consumers do not exhibit blind confidence in stocks and other securities.

Consequently, as consumers’ perspectives shifted towards a more cautious evaluation of

investments, the features of unit-linked policies had to mutate in order to respond to market

demand. Therefore, if the main feature of unit-linked policies was once the linking process

and the participation in equity markets, this shift in perspective led to the creation of a wide

array of unit-linked products and the emergence of guarantees.

Given the vast combinations of different unit-linked products and of their embedded

guarantees, this study does not aim at a simple enumeration of various types of policies.

Conversely, it aims to provide a general framework in which to place an analysis of the

nature, structure, development and valuation of such insurance policies.

The first chapter begins with a brief review of key insurance concepts and a preliminary

introduction of unit-linked life insurance. It proceeds at describing the historical

development and the diffusion of unit-linked policies in important markets over the years.

Following these introductory concepts the dissertation progresses in outlining the main

differences of unit-linked policies with respect to traditional life insurance policies,

participating policies and unit trusts. Lastly, it discusses the main reasons for which unit-

linked policies might prove to be beneficial both for the customer and for the insurance

company.

The second chapter is concerned with the overall structure of unit-linked policies. It is

focused on the elaboration of a simplified but coherent framework in which to analyze the

numerous aspects of these insurance products. It begins by discussing the main classes of

benefits that can be provided depending on the underlying insured event. Then it focuses on

the actuarial pricing, providing a general framework both for single premium and for

periodic premium arrangements. Thirdly, it discusses the process of investing in the unit-

linked fund, followed by a preliminary discussion on the mathematical and economic

difference between the two major forms of unit-linked policies. The study then continues

by examining some of the most common forms of guarantees that can be included in the

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policy and concludes by citing some examples of the process of creating additional reserves

for the guarantees of unit-linked products.

The third chapter is mainly concerned with the enumeration and explanation of the most

prominent risk factors that must be considered when attaching a guarantee to a unit-linked

product. It also hints at possible risk mitigation techniques that can be applied to such risks.

The chapter begins with a broad discussion of market risks, but also focuses on other

classes of risk that pertain less to the financial dimension and more to the insurance

dimension. These classes regard policyholder behavior and demographic risks. The chapter

then concludes with a brief discussion of risks that cannot be included in the previous

categories but that nonetheless have a great impact on insurance companies’ performance.

The last chapter aims at providing an explanation of the risk mitigation and hedging

processes for unit-linked guarantees. It is divided in two main areas: dynamic hedging and

static risk transfer solutions. The dynamic hedging process is explained by referencing the

most common and liquid derivative instruments available on the market and their

utilization in the creation of an effective hedging strategy. The chapter then proceeds in

describing the most common challenges and risks of creating such a hedging strategy. The

last part of the chapter is instead concerned with reinsurance, quasi-reinsurance solutions

and captive reinsurance companies. It concludes by stating the possible advantages,

disadvantages and reasons for adopting these solutions.

The dissertation is then concluded by considering the alternative strategies that an

insurance company can implement in order to market effectively and efficiently a unit-

linked insurance product. The final considerations are made in light of the multitude of

unit-linked products, premium arrangements, benefit structures, embedded guarantees and

hedging solutions.

3

CHAPTER ONE

MAIN CHARACTERISTICS AND DEVELOPMENT OF UNIT-LINKED

POLICIES

1. Preliminary Concepts

In traditional life insurance contracts the insurer binds itself, upon the payment of a

premium, to compensate the insured party upon the happening of an event contingent to

human life. The parties to the insurance contract are the insurer, the insured, the

policyholder and the beneficiary. The insured event is assessed with actuarial methods and

a premium is then charged, reflecting the probability and duration of the risk. Insurance

products aim to provide a monetary amount for a wide spectrum of insured events. This

monetary amount is either fixed or variable according to a predetermined rule. As regards

benefits, the death benefit is a lump sum or an annuity, typical of term or whole life

insurance policies, paid by the insurer to the beneficiary in case of death of the insured. The

survival benefit provides the beneficiary with a lump sum or an annuity in case of survival

and is typical of pure endowments and life annuities. Other insurance products such as the

endowment insurance combine death and survival benefits to provide a certain benefit paid

at a random time. As regards premium arrangements, they may consist of a single premium

paid at policy inception or a sequence of periodic premiums, paid at policy issue and in

subsequent periods according to the insurance contract. Whatever the premium

arrangement might be, the policyholder must always be in a credit position; the financing

condition must hold at all times in order to disincentive policyholders to lapse the contract

without effectively contributing to mutuality1.

The characterizing aspect of traditional life insurance policies is the causality of benefits

and premiums; first the benefits are stated and the expenses are assessed, and only

subsequently are the premiums determined. This process is driven by a premium

calculation principle, commonly the equivalence principle, which states that the expected

present value of premiums and benefits should be equal2. However, a safety loading may

be implicitly or explicitly added to the premium in order to provide the insurer with a

positive expected result and to avoid adverse outcomes deriving from an insufficient yield

1 OLIVIERI A., PITACCO E., “Introduction to Insurance Mathematics”, pp. 226-227. 2NORBERG R., “Basic Life Insurance Mathematics”, p.44.

4

from investments or a wrong estimate of mortality. In this respect, the choice of the

technical basis is of paramount importance.

A technical tool used to define the insurer’s debt position at any time during the policy

duration is the policy reserve. It is defined as the difference between residual benefits and

residual premiums as assessed at time t3. In traditional life insurance policies, the

establishment of benefits drives premium calculation and investment, thus defining the

method for calculating the reserve.

When dealing with a general insurance policy4, one must pay particular attention to the sum

at risk. It is defined as the difference between the death benefit C and the reserve 𝑉!!!,

assessed one year in the future. This amount is not yet available and is financed year by

year via mutuality. With these considerations in mind, the premium can be split in two

components: the risk and the savings premium. The former can be viewed as the premium

for a one-year term insurance covering the sum at risk, while the latter can be viewed as the

amount that maintains the reserving process; the future reserve is thus the pure financial

accumulation of the savings premium5.

Conversely, unit-linked policies present many key differences with respect to traditional

insurance policies. The main difference is that the premium is invested into a reference

fund and, if no explicit guarantee is provided by the contract, the financial risk is borne

entirely by the policyholder. A unit-linked insurance contract may take the form of any

preexisting insurance contract, although the most common form is the endowment

insurance6. In general the reference fund is split into a notional number of units, and the

premium is used to purchase a certain number of such units based on their current market

value. Contrary to traditional insurance policies, the survival benefit is defined as the

current market value of assets and the death benefit is defined as the current market value

of assets plus a sum at risk7. Again, the net premium is split in the risk and savings

component, although the former is referred to as the invested premium and the latter as a

fee for supplementary benefits8. Even though in this kind of policy the policyholder retains

the financial risk, what is possibly the key defining element is the way the reserve is

calculated. As was seen before, traditional policies start from the definition of the benefit,

or the insurer’s contingent liability, and then move to the premiums’ accumulation process, 3 OLIVIERI A., PITACCO E., “Introduction to Insurance Mathematics”, pp.247-248. 4 The general insurance policy refers to a product with a certain term m, an age x at policy issue, a death benefit C, a survival benefit S and annual level premiums P payable for the whole policy duration. 5 OLIVIERI A., PITACCO E., “Introduction to Insurance Mathematics”, p.270. 6 OLIVIERI A., PITACCO E., ibidem, p.346. 7 Defined so as to be greater than zero. 8 It is also common to refer to the savings premium as the fee for rider benefits.

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or assets. Unit-linked policies, instead, are asset-driven insofar as they start with the

definition of the policy fund and only subsequently move to the definition of the reserve.

This has key implications for how assets and risks are managed by the insurance company.

2. History and Development in Key Markets

The need to link benefits to investment performance, market interest rates, stock market

indices, mutual funds or other financial indices stems from the will to provide

policyholders with a return on investment that is higher than the technical interest rate. In

traditional insurance policies, the financial accumulation of premiums is achieved by

crediting a guaranteed interest rate that is usually set at a low level in order to avoid an

excessive risk for the insurer. Thus, policyholders investing in such products as endowment

insurance, where there is a large savings component9, have an interest in linking their

product to a financial index. The idea behind this policy design is to share investment

profits, and potential losses, between the insurer and the policyholder.

The origins of unit-linked products date back to the mid-twentieth century. In North

America, they have mainly consisted of the variable annuity variety, commencing with

CREF10 around 195211. Variable annuities are a modification of standard life annuities, in

which the accumulated funds are invested in a portfolio that is managed by a financial

institution. Such products may embed a minimum payment guarantee that reduces the risk

associated with the investment and offers the possibility to participate in financial markets.

In the UK, they have been mainly of the endowment insurance variety, commencing

around 1957 but gaining momentum only in 1963 when the first unit trust groups entered

the market12.

Unit-linked policies then spread and became popular in Europe and the rest of the world,

becoming an important part of modern life insurance. However, it is logical to presume that

the popularity of unit-linked policies depends on the performance of financial markets, so

their role has been increasingly predominant during economic booms and less relevant

during market crashes or financial crises, especially if the policies did not embed any

9 Reference is made to those products that have a large reserve with respect to the insured amount, and whose purpose is not solely insurance protection but also savings. 10 College Retirement Equity Fund. 11 G.L. MELVILLE et al., “The Unit-Linked Approach to Life Insurance”, p.311. 12 Ibidem.

6

explicit guarantee. Thus, though they have had a capillary expansion, the degree of

popularity varied according to many parameters such as, among others, the location, the

development of financial markets and their performance.

As previously stated, after the creation of unit-linked insurance policies there was a

pervasive diffusion. In the USA, market share reached 40% for the variable annuity variety

in 1999, whereas in the UK, unit-linked pension plans accounted for 53% of new issued

premiums in the same year. Italy has seen a rapid growth in unit-linked policies since their

introduction in the 1980s, with new unit-linked premiums gaining a 58% market share in

199913. In general, many other European countries such as Belgium, Spain, Sweden and the

Netherlands have experienced a similar expansion in linked premiums. Excluding the need

to realize a higher return on investment for those policies that include a large savings

element, there are many other factors that contributed to the vast propagation of unit-linked

policies. Among others, it is useful to enumerate such factors as the strong boom of

European stock markets in those years, falling interest rates, increased popularity of shares

as an investment medium, reduced bonuses on conventional products and increased

publicity of stock market indices. However, this rapid growth came to an abrupt end during

the market crash of 2001, resulting in a huge decrease in demand14. This did not lead to a

collapse of the system, but brought forth an increased popularity in unit-linked policies

with guaranteed returns. This proved that customers have had a continuing confidence in

life insurance markets, although they are becoming increasingly more aware of the intrinsic

risks.

13 MUNICH RE GROUP, “Unit Linked Insurance: A General Report”, p.8. 14 SWISS RE, “Unit-Linked Life Insurance in Western Europe: Regaining Momentum?”, p.11.

7

3. Comparison with Other Insurance Policies and Unit Trusts

Unit-linked policies differ from traditional policies in many key aspects. Firstly, if the

policy is sold with no embedded guarantees, the financial risk is transferred to the

policyholder, whereas it is retained by the insurer in traditional life insurance policies.

Secondly, in traditional policies liability calculation drives premium calculation, resulting

in a liability-driven activity. The opposite holds true for unit-linked policies: invested

premiums and the number of fund units credited to the policy generate the insurer’s

liability and drive reserve calculation. It is important to note that when dealing with unit-

linked policies with guarantees, part of the financial risk is transferred to the insurer. This

should drive the creation of an additional reserve according to prudential and regulatory

considerations, thus placing these products between asset-driven and liability-driven.

Thirdly, traditional policies tend to have a fixed benefit structure, or at least a benefit

whose variation is determined in advance. Unit-linked policies, instead, are characterized

by a random return on investment, with a benefit that varies according to investment

performance. Furthermore, unit-linked policies have a predominance of the savings

element over the insurance element, thus new premium income comes largely in the form

of single premiums. This occurs because investors might want to participate immediately in

financial markets’ performance. Finally, an important difference between the two types of

policies arises in flexibility and transparency. Policyholders are usually bound to accept the

insurer’s investment decision and the technical interest rate attributed to the policy. Holders

of unit-linked policies, instead, have the choice of where to invest their money, among a

limited range of investment options proposed by the insurer. The greater degree of

flexibility can best be seen in policies that have a switching option: the option to change

investment target and risk-return profile at a certain time, possibly at policy anniversaries.

As regards transparency, in traditional life insurance policies the policyholder is not aware

of policy charges, the risk premium or the expense loading. Unit-linked policies are said to

be unbundled in that each of the constituent parts of the policy can be identified

separately15. This includes the investment element, expenses and administrative charges,

benefit charges, mortality charges as well as the benefit itself.

15 MUNICH RE GROUP, “Unit-linked Insurance: A General Report”, p.5.

8

Unit-linked policies also differ from participating policies. First, the investment risk in a

participating policy is typically transferred to the insurer to the extent of what was

guaranteed to the policyholder. A substantial part is however retained by the insurer via

bonus fluctuations. Investment value in unit-linked policies depends on the value of the

reference fund and can hence be evaluated almost at any point in time, whereas in

participating policies it is unknown until it is cashed because these policies heavily rely on

a terminal bonus, the size of which has been decreasing in recent years16. This downsizing

of terminal bonuses is one additional factor that contributed to the popularity of unit-linked

policies. Again, participating policies offer a lower degree of transparency with respect to

unit-linked policies because charges are typically hidden. Also, given that asset allocation

is determined solely by the insurer, they offer a lower degree of flexibility. Finally, even

though both policies may embed some form of guarantee, their structure is somewhat

different. Participating policy guarantees may come in the form of guaranteed annual

returns or guaranteed average returns, whereas unit-linked policy guarantees typically come

in the form of survival or death benefits.

Finally, there are important differences to note between unit-linked life insurance policies

and unit trusts. The most obvious difference is that a unit trust is not an insurance product.

Even though unit-linked policies can be considered as being very similar to pure financial

products, they include an insurance element, and may provide guaranteed benefits in case

of death or survival. Purchasing units of a unit trust is equivalent to purchasing a pure

financial product, thus forgoing the insurance element. Another important difference is that

in the event of insolvency, the units in a unit trust will be still available to clients, whereas

the assets of a unit-linked fund are usually available equally with other assets of the insurer

to meet liabilities and there is thus no guarantee that the full value of assets will be

available to policyholders17.

16http://www.pswlaw.co.uk/site/services/pswprivateclient/pswsrvwealthmanagement/unit_linkedorwith_profitswhatsthedifference.html 17 MUNICH RE GROUP, “Unit-Linked Insurance: A General Report”, p.7

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4. Advantages and Attractions of Unit-Linked Policies

In order to efficiently market a new kind of insurance policy, it is necessary for it to

provide advantages not only to the insurance company, but also to the customer. In fact,

there are many reasons why customers could prefer unit-linked products to traditional

policies. During the years of booming equity markets, the advantage was obvious: these

policies offered the possibility of direct participation in soaring markets. As a matter of

fact, the economic prosperity of past years was one of the driving forces behind unit-linked

policies’ growth and market penetration. Nowadays this advantage may seem less relevant

but the protection element embedded in most of these policies enables customers to have

assurances with respect to market crashes or poor performance while retaining the upside

risk; the possibility that the reference fund will perform better than expected. Furthermore,

growing competition between life insurers has led to aggressive marketing techniques such

as offering many different kinds of guarantees at little extra cost for the customer, who can

benefit from this situation. In contrast to pure financial investments in mutual funds,

investments in unit-linked policies may offer tax advantages in certain European

countries18, provided they fulfill specific requirements such as a minimum duration and the

presence of a death benefit. Capital gains and investment income that accrue during the life

of the policy are tax-free, however the paid-out benefits are still taxed as income. Another

advantage is the switching option; if provided for by the contract, the customer has the

choice of changing the investment target and switching between funds. Holders of

traditional or participating policies instead have little or no say in how their premiums are

invested. Moreover, unit-linked policies offer great advantages to clients who want to

monitor the progress of their investment. The disadvantage of traditional policies lies in

their bundled nature; the cash value of the policy at any particular time is not clear to the

client. Unit-linked policies prove superior in this respect because of increased control over

the investment strategy, absolute transparency in charging structure and great flexibility.

Unit-linked policies may thus be tailored to the need and will of the client, increasing their

attractiveness throughout the market.

Unit-linked policies are not only advantageous for customers, but also for insurance

companies. In a period of low performance in financial markets, insurers have found it

increasingly difficult to deliver the returns on traditional or participating policies. Thus, 18 SWISS RE, “Unit-Linked Life Insurance in Western Europe: Regaining Momentum?”, p.7.

10

unit-linked policies, or mostly those with little or no embedded guarantees, allow the

insurer to shift the investment risk to the policyholder. Another key advantage of unit-

linked business is its lower capital requirement. The solvency requirements for traditional

policies require insurers to hold 4% of mathematical reserves and 0.3% of the sum at risk.

These requirements are much lower for unit-linked policies, 1% of fund value plus 0.3% of

the sum at risk. However, in presence of capital guarantees the reserve requirement can

reach levels up to 4%19. The solvency requirement will in most cases be lower, and in any

case, never be higher than that required for a traditional policy. This lower allocation of

capital has become increasingly appealing to life insurers due to its positive impact on the

return on equity. Moreover, a lower reserve requirement means that unit-linked products

are suitable for insurers in start-up situations. Furthermore, asset management fees for

participating policies range between 1-1.5%, whereas in unit-linked business there is a

management fee which amounts approximately to 0.8-1% and a mutual fund charge in the

range between 1-1.5%20. This results in a double charge and a higher profit margin for the

insurance company. As with any product, an insurer should push for unit-linked policies

either in response to a signal from the market, when it is believed that current market

conditions might support them, or when there is the chance to make a sufficient profit.

Among others, market signals can include stagnating sales of participating policies and

increase in sales of mutual funds or pure investment-linked trusts.

19 SWISS RE, ibidem, p.6. 20 SWISS RE, ibidem, p.6.

11

CHAPTER TWO

THE STRUCTURE OF UNIT-LINKED POLICIES

A unit-linked policy can be defined as an insurance contract in which the savings premium

is linked directly to the value of units in a mutual fund or to the insurance company’s

internal funds, where the investment risk is borne by the policyholder21. Unit-linked

policies are said to be unbundled meaning that the constituent parts of the policy can be

identified separately. The separation concerns the investment element, expenses and

administrative charges, benefit charges and the benefits themselves. The process of

unbundling renders the policy transparent because the client can monitor the progress of the

investment.

Policyholders usually have a limited choice of funds and assets in which to invest their

premiums, based on their desired risk-return profile. The assets include but are not limited

to equities, fixed-interest securities, money market instruments, real estate, derivative

instruments, gold and foreign currency. However, there is an Asset-Liability constraint: the

insurer must be able to buy or replicate the reference units in order to meet its liabilities. In

practice, it is not prudent to link the value of a policy to an asset unless that asset actually

forms part of the reference fund.

21 Ibidem.

12

1. Varieties of Benefits

Even though unit-linked policies are mostly known for the financial or investment part of

the policy, they contain an insurance element that depends on the structure of the policy.

Referring to the general insurance policy22, two distinct kinds of benefits can be defined.

The survival benefit is provided at maturity and in unit-linked business is defined as the

current value at maturity of the fund accumulated with premiums, or the policy fund.

From a mathematical perspective it can be defined as:

𝑆! = 𝐹!

Given the high degree of transparency of the policy, the policyholder can assess the amount

of the benefit that is funded by the current value of the fund, namely:

𝑆! = 𝐹!

The death benefit is defined as the current value of the policy fund at the time of death plus

a sum at risk that is defined so that it is positive, or at least non-negative.

From a mathematical perspective this amounts to:

𝐶! = 𝐹! + 𝐾!    ,        𝐾! ≥ 0

K can be described in various ways, and its definition will shed some light on the presence

or absence of a financial guarantee. In particular, the sum at risk can be set as a function of

fund value, with the death benefit becoming:

𝐶! = 1+∝ 𝐹!

This form does not embed a guarantee, because the death benefit will be zero if the fund

value is zero as well.

22 Refer to footnote 4.

13

Conversely, the death benefit can be defined as:

𝐶! = 𝐹! + 𝐺

G is a fixed number and also a guarantee because the policyholder will receive at least G

even if the policy fund has a value of zero.

A third kind of formula is used to determine the surrender value and it is defined as the

current value of the policy fund at the time of surrender, possibly net of a surrender fee.

The surrender fee usually decreases as the policy reaches maturity, in order to give a larger

penalty for early surrenders. The surrender value is defined as:

𝑅! =  𝜑 𝑡 𝐹!

1− 𝜑(𝑡) represents the surrender fee at time t.

2. Actuarial Pricing

From an actuarial point of view, in order to price a policy it is necessary to calculate the

present value of expected future benefits, according to the equivalence principle. It is

assumed that the difference between the technical bases will provide the insurance

company with a sufficient safety margin to avoid the risk of losses and to profit from the

policy. However, it might be necessary to include further assumptions in order to price the

product profitably.

Firstly, it is important to assess the expected size of the policy that the insurance company

expects to write; this includes the average level of premiums and benefits and the timing of

premiums and charges, conditioned by age, sex and duration of the contract. Secondly, the

company should assess the expected costs of acquiring and administering the product. The

amounts might be divided in initial and renewal costs and can be allocated to a portfolio of

policies or split on a per-policy basis. The company should also consider fluctuations in

sales volumes as they might impact expenses non-proportionally on the portfolio level.

Thirdly, economic assumptions are of paramount importance in unit-linked business

because they include future returns on reference funds and future inflation rates. These

14

factors have a great impact on the value of the policyholder’s investment and, in presence

of guarantees, can have a significant impact on profitability. Furthermore, it is important to

assess expected lapse rates because, as will be stated in the following section, some

expenses are recouped during the life of the policy and the lapse of a contract can generate

losses for the insurance company. Lastly, other assumptions that should be made include

mortality and survival rates of the insured people, calculated using the appropriate tables

and conditioned by age and sex.

In general, the premium that is actually charged is referred to as the expense-loaded

premium. This final premium is composed by the net premium and all the relevant charges

for expenses. In order to analyze pricing, reference is made to the general insurance policy,

with a single or annual premium arrangement.

In the case of a single premium arrangement, the general notation for the net premium is:

𝜋 = 𝐶( 𝐸′! +   𝐴!!!   ) = 𝐶𝐴′!,!˥  !

   

𝐴′!,!˥ is the general notation for the endowment insurance policy and is equal to the

general insurance policy when 𝑆 = 𝐶.

It is split into 𝐸′!  !   and 𝐴!!!

  , where the former refers to the value of a pure endowment

that provides a benefit in case of survival at maturity. The mathematical notation is:

𝐸′!  !   = 1+ 𝑖! !! 𝑝′!!

 

This formula describes a benefit consisting of one monetary unit payable at time m if the

insured, that is currently of age x, is alive at that time. The apostrophe represents the

prudential definition of the interest rate and the probability of survival, calculated in order

to provide a safety margin to the insurance company.

The latter is described as a unitary amount payable at the end of the year of death, if the

event occurs within the m years of policy duration. The mathematical notation is:

𝐴!!!   =   1+ 𝑖! ! !!! 𝑞!!!⃓!

 !!!

!!!

15

This basically represents the sum of 1-year actuarial values from year h, or the sum of 1-

year term insurances.

Once the net premium is calculated, the insurer must assess all the relevant expenses and

charge the policy accordingly. There are three broad categories in which expenses can be

grouped: acquisition, collection and general administration expenses. When dealing with a

single premium arrangement collection expenses can be disregarded, so the definition

becomes:

𝜋 ! =  𝜋 +  𝛩 ! +  𝛩 !

Acquisition and general administration expenses can be calculated in various ways, but for

the sake of simplicity both are assumed to be proportional to the sum insured. Of course

general administration expenses must be forecasted and annuitized or split into annual

amounts that are then loaded on the premium.

As regards the annual premium arrangement, the concepts and mathematical formulae are

simply an extension of the single premium arrangement. The net level premium is defined

as the net single premium conditioned by an annuity-due. This ensures a correct

mathematical division of the single premium, accounting for the accumulation factor due to

the interest rate, and the mortality component due to the probability of survival.

The mathematical notation is:

𝑃 =  𝜋𝑎!:!˥!

The annuity-due is defined as follows:

𝑎!:!˥! =   𝐸′!!  

!!!

!!!

This discounting factor is equal to a temporary life annuity paid in advance. The term s

refers to the premium payment profile: when 𝑠 = 𝑚, premiums are payable for the whole

policy duration.

16

Once the level premium has been calculated, the relevant expense loading must be added,

but it must be split into annual amounts to account for the periodicity of the premium.

Acquisition expenses are thus split into a sequence of annual amounts, each loaded on the

related premium. Acquisition expenses are progressively recovered and this generates a

potential risk for the insurer related to the lapse of the contract. Administration expenses

are attributed for the whole policy duration. If the premiums are payable for the same

duration each one will be loaded with the annual share of expenses, whereas if premium

payment is shorter a higher share will be loaded on each premium. Collection expenses,

which were not present in the single premium arrangement, are loaded year by year. As

before, administration and acquisition expenses are assumed to be proportional to the sum

insured, whereas collection expenses are assumed to be proportional to the expense-loaded

premium.

Acquisition expenses are defined as follows:

𝛬 ! =  𝛼𝐶𝑎!:!˥!

General administration expenses are defined as follows:

𝛬 ! =  𝛾𝐶𝑎!:!˥!

𝑎!:!˥!

It is worth noting how in the case of premiums payable for the whole policy duration, the

notation simplifies to a constant share of the sum insured.

Lastly, collection expenses are defined as follows:

𝛬 ! = 𝛽𝑃 !

With some algebraic calculations, the expense-loaded annual premium can be defined as:

𝑃 ! =  𝛼 + 𝛾 𝐶𝑎!:!˥!

(1− 𝛽)𝑎!:!˥!

Of course, results may vary according to the various definitions of expenses, but this

notation provides a simplified framework to calculate the expense-loaded premium with

17

acquisition and administrative expenses proportional to the sum insured and collection

expenses proportional to the expense-loaded premium itself. Furthermore, it allows for the

possibility to have a payment profile in which the number of annual premiums is different

from the policy’s duration.

In the framework of unit-linked policies there are various expenses and charges, some of

which can be included in the general framework that was previously discussed. In general

the types of charges that can be levied are initial charges, surrender charges, renewal

charges, fund management charges and switch or redirection charges.

Initial charges are intended to cover the marketing, distribution and other new business

costs relating to the policy. These charges can be identified as acquisition expenses in the

previous framework. Surrender charges are applied when a policy is surrendered and are

used to recover costs already incurred to the extent that they have not been recovered from

the charges made prior to surrender. Renewal charges are intended to cover the costs of

administering the policy and any renewal commissions payable. Fund management charges

are not present in fixed-income insurance but are a specific feature of linked contracts as

they relate to the ongoing costs of managing the investments of the policy fund. These

charges are usually calculated as a percentage of the funds under management23.

Other charges that are specific to unit-linked business are the switch or redirection charges.

These are intended to cover the additional administration costs associated with switching

investments between funds and redirecting premium flows. Another objective of these

charges is to discourage excessively frequent switches that would entail a change in the

insurer’s position towards market risk and are themselves further risks for the company.

3. The Investment in the Fund

In order to analyze the mechanism of investment in the unit fund, the case of a unit-linked

endowment policy with annual premiums, a survival and a death benefit is taken into

consideration. The insurer usually pays most of the acquisition commission upfront and

pays the rest over the course of the following years. This initial outflow is amortized over

the life of the policy and recouped from the policyholder over the course of many years.

23 MUNICH RE GROUP, “Unit-Linked Insurance: A General Report”, p.20.

18

The policyholder pays an expense-loaded premium that is immediately split in two

components and used to invest in the reference fund of choice. One part of the premium

represents the charges that are placed in a non-unit fund whereas the net premium is

invested in the unit fund according to the formula:

𝑃 ! =  𝑃 +  𝛬 ! +  𝛬 ! +  𝛬 !

The net premium is used to buy units on behalf of the policyholder. The number of units

that can be acquired with the net premium is:

𝑛! =𝑃!𝑤!

𝑤! represents the value of one unit at a specific point in time.

The presence of the sum at risk generates a mutuality cost that must be financed. As in

traditional insurance, the premium is split into risk and savings premium, with the only

difference that in unit-linked business the premium is financed out of the fund by cashing

units. The units would then be split in two components:

𝑛! =  𝑛!  ! +  𝑛!!

The accumulation of the premiums in the fund would then result only by the accumulation

of the savings premium.

𝑁! =   𝑛!!!!!

!!!

With these considerations in mind, the policy fund at any point in time is defined as the

product of the accumulated units and the value of each unit according to the formula:

𝐹! = 𝑁!𝑤!

19

Given that unit-linked policies are asset-driven to the extent that the insurer’s liability is

defined only as a consequence of the definition of the assets, the reserve is simply equal to

the policy fund at any point in time:

𝑉! = 𝐹!

At the end of the year, a return is paid on the accumulated units in the form of unit growth,

to which management fees are charged before being passed over to the life insurer to cover

expenses. The non-unit fund is credited with interest at the end of the period and it is used

to pay administrative expenses.

In case of death the amount invested in the fund is released along with a sum at risk and

used to pay the death benefit, whereas in case of survival at maturity the current value of

the fund is released in the form of units.

4. The Two Major Configurations of Unit-Linked Policies

Once the general structure of unit-linked policies is defined, it is important to make a

characterizing distinction between two major forms of policies regarding the presence or

absence of guarantees.

Unit-linked policies without guarantees transfer the investment risk to the policyholder,

who relies on market conditions to determine the value of the investment. As in the general

framework, the net premium is used to acquire units of the reference fund, some of which

are then transferred to the non-unit fund in order to fund mutuality costs. Given that there

are no guarantees on investment performance or minimum benefits, the death benefit is

simply defined as the current value of the policy fund plus a sum at risk that depends on

such fund. This ensures that in the limit case in which the fund reaches a value of zero,

there will be no guaranteed death benefit to be provided to the beneficiary. An example of

such an arrangement is:

𝐶! = 1+ 𝛼 𝐹!

In this case the sum at risk is defined as a percentage of the policy fund, it is a positive

amount but there is no guarantee whatsoever in case of death.

20

Intuitively, this arrangement is less risky for the insurer with respect to a minimum

guarantee, and this also occurs because all the quantities are deterministic at time t, after

premium payment. In order to analyze this particular aspect, reference is made to the

Kanner equation. The Kanner equation is a recursive equation that splits the death benefit

in two components, the future value of the reserve owned by the insurance company, and

the sum at risk that must be financed via mutuality. It links the current with the future value

of the reserve and is described as follows:

𝑉! + 𝑃 1+ 𝑖! = 𝐶 − 𝑉!!! 𝑞!!!! + 𝑉!!!

In the context of unit-linked policies, this formula can be adapted and applied to a more

familiar framework:

𝐹! + 𝑃𝑤!!!𝑤!

= 𝐶!!! − 𝐹!!! 𝑞!!!! + 𝐹!!!

In this framework the yield of the fund is random, as opposed to a traditional policy in

which a technical interest rate is credited. Whatever happens, the value of the policy fund at

time t + 1 will be available, whereas in case of death, the sum at risk must be added to pay

the death benefit 𝐶!!! to the beneficiary.

Assuming a sum at risk proportional to the policy fund, by using the definitions provided

above the recursive equation becomes:

𝑁! + 𝑛! 𝑤!!! =  𝛼𝑁!!!𝑤!!!𝑞!!!! + 𝑁!!!𝑤!!!

This equation can then be simplified further to yield:

𝑁! + 𝑛! =  𝛼𝑁!!!𝑞!!!! + 𝑁!!!

All the quantities involved are deterministic at time t, and from this equation it is possible

to calculate the accumulated units at time t+1.

The relevant deterministic quantities are:

𝑁!!! =𝑁! + 𝑛!𝛼𝑞!!!! + 1

21

𝑛!! =𝑛! − 𝛼𝑞!!!!

𝛼𝑞!!!! + 1

𝑛!! = (𝑛! + 1)𝛼𝑞!!!!  

𝛼𝑞!!!! + 1  

These conditions imply that after premium payment there is no financial risk for the

insurer. However, a financial risk may arise before time t because it is unknown how many

units will be purchased each year.

In order to see why and how a guarantee entails a risk for the insurer, a minimum death

benefit guarantee G is considered. The death benefit thus becomes:

𝐶! = 𝐹! + 𝐺

Even in the case where the policy fund reaches a value of zero the beneficiary will still

receive the guaranteed amount G in case of death of the insured.

Returning to the recursive equations, the balance condition becomes:

𝑁! + 𝑛! 𝑤!!! =  𝐺𝑞!!!! + 𝑁!!!𝑤!!!

It is clear that it is not possible to cancel out 𝑤!!!, so in order to calculate all the relevant

quantities an estimate is required. This generates a financial risk for the insurer caused by

the presence of the minimum death benefit. The relevant quantities thus become:

𝑁!!! = 𝑁! + 𝑛! −𝐺𝑞!!!!

𝑤!!!

𝑛!! = 𝑛! −𝐺𝑞!!!!

𝑤!!!

𝑛!! =𝐺𝑞!!!!

𝑤!!!

22

The risk element, the savings element and the future accumulation of units all depend on

the performance of the fund and from the insurer’s perspective are random quantities that

must be estimated and that generate a risk. This is one example of how a guarantee can

influence the risk position of an insurance company, but it represents only one of the

myriad possible guarantee combinations, some of which will be further analyzed in the

following section.

5. Guarantees

The central purpose of the guarantees in the form of Guaranteed Minimum Benefits is to

ensure that the client receives benefits that are contingent upon the greater of the future

value of the policy fund or a guaranteed payout function24. There are many different types

and combinations of guarantees in unit-linked insurance policies, but one of the most

common is the Guaranteed Minimum Death Benefit25.

It can be defined in various ways, it may be a fixed amount or it may vary according to

some parameter, but for the sake of simplicity two examples are addressed.

The first is slightly different from the guarantee that was analyzed in the previous section.

It provides the higher of two amounts: the policy fund or the GMDB. The death benefit is

thus defined as:

𝐶!!! =  max{F!!!  ,G}

Then the sum at risk becomes:

𝐾!!! = 𝐶!!! − 𝐹!!! = 𝑚𝑎𝑥{G− F!!!  ,0}

From a financial point of view, this represents the pay-off of a put option and it has

important implications for the insurer. When a financial guarantee is underwritten, a risk

emerges for the insurer. This risk must be hedged properly through a suitable hedging

strategy. Therefore, it is not uncommon for the insurer to investigate the hedging 24 MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, p.16. 25 From this point forward the Guaranteed Minimum Death Benefit shall be referred to as GMDB, for the sake of brevity.

23

opportunities on the market and only subsequently decide which types of guarantees it can

offer to its policyholders. This simple form of guarantee might thus be easier to hedge with

respect to a complexly structured policy guarantee and might provide advantages in terms

of risk mitigation for the insurer.

Another kind of GMDB aims to provide the highest value of the policy fund as evaluated at

the previous policy anniversaries. It is mathematically described with the following

notation:

𝐶!!! =  𝑚𝑎𝑥{𝐹!!!,𝑚𝑎𝑥 𝐹! !!!,!,…,!}

The sum at risk becomes:

𝐾!!! =  𝐶!!! − 𝐹!!! = 𝑚𝑎𝑥{𝑚𝑎𝑥 𝐹! !!!,!,…,! − 𝐹!!!, 0}

This corresponds to the pay-off of a ratchet option.

Another common type of guarantee is the Guaranteed Minimum Accumulation Benefit. An

example of such an arrangement is that at maturity the value of the accumulated funds must

be at least equal to the total premiums26 paid in. The survival benefit with this kind of

guarantee becomes:

𝑆! = 𝑚𝑎𝑥 𝐹!,𝐺!!!

where the guarantee is defined as:

𝐺!!! =   𝑃!! 1+ 𝑖! !!!!!!

!!!

This is an attractive guarantee for clients because it limits their downside risk. Of course, it

will generate a risk for the insurer that must be hedged either with derivatives or with some

other financial instrument. Other common solutions might include a simple guarantee on

total premiums paid, net of withdrawals, or a ratchet guarantee on the policy fund.

26 With “total premiums”, a reference is made to the invested premium 𝑃!! = 𝑛!!𝑤!.

24

Another common guarantee is known as the Guaranteed Minimum Withdrawal Benefit27. It

is usually underwritten in the context of unit-linked whole-life assurances and protects the

policyholder against downside market risk. A GMWB permits the policyholder to

withdraw up to a stated percentage of total premiums paid each year, irrespective of the

value of the underlying funds. In order to better explain the advantage for consumers, an

example is provided.

A certain client underwrites a unit-linked whole-life insurance policy with a single

premium arrangement of €100,000, with a GMWB rider and with a maximum withdrawal

right of 10%. He thus has the right to withdraw €10,000 per year irrespective of the current

value of the policy fund. If, due to some market crash, his investment were to lose 90% of

its value in the first year he would still be able to withdraw €10,000. This would

completely drain the policy fund, but the GMWB would enable the client to continue to

withdraw that same amount each year until the initial investment has been exhausted. Of

course a catastrophic event such as a market crash is a limit case that would put the insurer

in a difficult position in the presence of a GMWB, but under more usual conditions such a

risk can be mitigated or hedged with derivative products.

6. Reserving For Unit-Linked Guarantees:

Constructing the mathematical reserve for unit-linked policies with no guarantees seems to

be a fairly simple process, at least from a mathematical point of view, because the reserve

creation is asset-driven. Of course the matching of assets and liabilities is no easy task

when dealing with unit-linked funds that present a diverse basket of securities. However,

when unit-linked policy guarantees start to come into play, the need for additional reserves

becomes obvious. By underwriting such guarantees, the insurance company exposes itself

to market fluctuations and risks and must construct additional reserves to back these new

liabilities. The need for additional reserves stems from legal requirements as well as simple

protection, as it is in the interest of the company to be solvent and not to be excessively

exposed to the risk of guaranteed products. There is no unique way to establish additional

reserves, as legal requirements vary across the globe and each company has its own internal

mechanism to determine projections of future liabilities, based on different market

27 From this point forward the Guaranteed Minimum Withdrawal Benefit shall be referred to as GMWB, for the sake of brevity.

25

assumptions that lead to different conclusions. However, there are some major classes of

reserves that are common to many insurance companies and that constitute the backbone of

the reserving process for guarantees. As the distinction between insurance products can be

blurry in certain cases, not all the additional reserves will be exclusive to unit-linked

products. These are the reserve for demographic risk, the reserve for mortality risk and the

reserve for unit-linked guarantees.

The demographic risk reserve is implemented in case of unit-linked products that come in

the form of deferred life annuities with guaranteed conversion coefficients. As the name

suggests, this reserve is intended to cover the risk that the longevity or mortality profiles

assumed at contract inception differ from those observed at the time of conversion.

The reserve for mortality risk is created in the case of unit-linked policies that come in the

form of whole life insurance with a GMDB. As stated above28, GMDBs can come in

various forms, but this reserve is intended to cover the guarantee to receive, in case of

death, at least all the premiums paid into the fund. This reserve is calculated by adding to

the mortality component of the premium the amount that each year is expected to exceed

that reserve, where the last amount arises from the presence of the guarantee. Instead, when

the value of the guarantee is simply defined as an excess over the value of the policy fund

at any point in time, no reserve is set up and the relationship between inflows and outflows

is frequently monitored to see if the hypotheses hold true.

Lastly, the reserve for unit-linked guarantees is intended to guarantee the payment of the

invested premiums, at least. Given recent market scenarios and downturns, it has been

increasingly difficult for insurance companies to guarantee invested premiums and this

consequently has created the risk of not having adequate reserves to face the issue. The

reserve construction is established on a per-case basis; for example, when dealing with a

minimum survival benefit, the company evaluates the risk by monitoring the underlying

investment. When this investment proves to be inadequate in presence of the guarantee, an

appropriate reserve is set up. So, in practice, the reserve is established by projecting the

policy fund’s value to maturity and reserving for the difference between the expected

projection and the maturity guarantee.

These examples serve to capture the complex nature of creating additional reserves.

Creating these reserves is a complicated process, with specific formulae that can vary

among insurers, but it is also worth noting that, given the many different combinations and

structures of unit-linked policies and guarantees, the reserves will also come in many

28 Refer to pp.22-23.

26

forms. As such, one cannot rely on a single universal rule to decide how much money to

allocate, but must decide on a per-case or portfolio basis and must also evaluate current

market conditions, legal requirements and risks. Given the wide variety of reserves that can

and must be created to face market fluctuations, it is clear that the insurance company will

face a myriad of market and other risks when underwriting unit-linked guarantees. The

enumeration and explanation of such risks will be the subject of the following chapter.

27

CHAPTER THREE

RISK MITIGATION AND DECOMPOSITION OF GMxB RIDERS

Once a GMxB29 has been sold, it generates a liability on the insurer’s balance sheet. This

liability is typically valued on a per-policy seriatim basis as the present value of future

guarantee claims less the present value of guarantee charges. The value of a certain

guarantee depends on the value of the policy fund and on the proportion of policyholders

that exercise the guarantee. As a consequence, anything that impacts the number of units

held and the price of a unit or of the underlying funds will impact the value of the

guarantee. There is a wide array of risks that the insurance company must consider; some

can be appropriately hedged whereas others are harder to assess. In general, the risks of

guarantees can be grouped into five broad categories: first order market risks, second order

market risks, policyholder behavior risks, demographic risks and other risks30.

1. First Order Market Risks

As regards first order market risks, the primary focus rests on the level of funds, interest

rates and their variability. The risks covered in this section are the most commonly valued

and hedged sensitivities within guaranteed portfolios and are known as delta, rho and vega.

Delta risk is possibly the main factor that impacts the value of a guarantee. It is defined as

the ratio of the change in price of a derivative to the change in the price of the underlying

asset31. It arises from all factors that impact returns on underlying funds. After analyzing

the exposure to this source of risk the insurance company might seek to enter in a trade that

29 As the name suggests, a GMxB refers to a Guaranteed Minimum Benefit of type x, where the x represents death, accumulation or any other kind of guaranteed benefit. 30 “Other risks” are those risks that cannot be identified as being part of the previously cited categories but nonetheless have an impact on the value of a guarantee or the performance of the insurance company. As regards the classification of risks, for further insight refer to MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”. 31 http://www.investopedia.com/terms/d/delta.asp

28

has the opposite effect, creating what is referred to as a “delta neutral” position32. This kind

of position may be achieved instantaneously with a dynamic hedging strategy or

permanently in the case of reinsurance or quasi-reinsurance33.

Rho risk is referred to as the variation in the price of a derivative relative to a change in the

risk-free rate34. This risk is important due to the fact that GMxB liabilities are calculated by

discounting a future set of obligations to the present date. Thus the valuation will be subject

to movements in this market curve. Given that guarantees might be valued in different

points in time and that changes in the interest rate can be contrasting for distinct terms,

guarantees and liabilities should be considered independently as each one is exposed to rho

risk in a unique way. A further consideration that might be overlooked is that the variation

in the “risk-free rate” or the discount rate will have implications for the bonds held within

certain unit funds, so it may be appropriate to consider rho risk together with the delta risk

for bonds35. In this framework risk mitigation proves to be difficult, and constructing

appropriate hedges is becoming increasingly important, because yield curves might not

move in parallel ways. Nonetheless, the main instruments used to hedge rho risk are

interest rate swaps, as they are highly liquid and easily tradable.

Vega risk is defined as the measurement of an option’s sensitivity to changes in the

volatility of the underlying asset36. In parameterizing a model to evaluate future movements

of underlying funds the scholastic approach is to assume that volatility is stable throughout

time. In reality, volatility has very diverse values for different assets and in general is not

stable. Therefore, the impact of changes in volatility must be taken into consideration when

evaluating guaranteed products. As a result, all valuations that are subject to changes in

volatility parameters are exposed to vega risk. Among the many existing sources of vega

risk perhaps the most important ones are: the underlying equity index or equity vega, the

volatility of interest rates or rate vega and the volatility of bond funds or bond vega37. The

risk mitigation process for vega risk might include the use of volatility or variance swaps,

32MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, p.18. 33 These hedging strategies will be better analyzed in Chapter Four. 34 http://www.investopedia.com/terms/r/rho.asp 35 MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, p.19. 36 http://www.investopedia.com/terms/v/vega.asp 37 MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., ibidem, p.20.

29

the purchase of options or swaptions and the purchase of reinsurance or quasi-reinsurance

solutions.

2. Second Order Market Risks

First order market risks, as explained above, are identifiable and in general can be

appropriately hedged. On the other hand, other risks can be interrelated and cannot be

easily hedged. These risks are known as second order market risks and their main

categories are higher sensitivities, correlation, basis risk, asset allocation risk, credit and

counterparty default risks, inflation and new business pricing risk. The pricing and the

allocation of capital must thus be assessed in the light of the following considerations.

In the discussion of first order market risks, the focus was on the first and second moments

of a given model distribution. Higher sensitivities refer to higher order risks that contribute

to the skewedness and long-tailed form of some distributions when compared to the

simplified Gaussian model38. Among these higher sensitivities the most prominent risks are

gamma risk and cross-greek risk. Gamma is defined as the rate of change for delta with

respect to the underlying asset's price39. It is important to consider gamma risk because the

adoption of a delta neutral strategy might lead to losses if the movement of the parameter is

more volatile than expected. Cross-greek risk arises from the nature of the decomposition

of each risk factor. Delta, rho and vega are considered independently and then integrated in

a consistent valuation, but it is necessary to consider the interaction between these risk

factors or “greeks”. In fact, it might not be clear whether the simultaneous impact of two

adverse risk factors will be greater, lower or equal to the sum of those risk factors

considered independently. This is why it is important to understand higher sensitivities, as

an effective risk mitigation strategy that only considers delta, rho and vega risk

management might not lead to a risk-free result for the insurer. This result stems from the

presence of higher sensitivities, and in general does not depend on the failure of market-

related assumptions.

38 Also known as the “bell curve” or the normal distribution, this statistical model can be completely described by its first and second moments. http://math.about.com/od/glossaryofterms/g/Bell-Curve-Normal-Distribution-Defined.htm 39 http://www.investopedia.com/terms/g/gamma.asp

30

As anticipated above, the correlation between market risk factors is an important aspect to

consider when dealing with GMxBs. However, what might be troublesome for the

insurance company is the fact that the realized correlation is non-stationary and correlation

itself can be correlated to market conditions. There are various methods to deal with this

kind of issue, some of which include the use of exotic options of varying liquidity, but in

general such a risk is either retained or transferred by using reinsurance or quasi-

reinsurance solutions.

Basis risk is defined as the risk that the offsetting strategy used in a hedging solution will

not experience exactly opposite price changes with respect to the underlying40. This can

create the potential for excess gains or losses but will in general add risk to the insurer’s

position. From a financial point of view, if a delta hedge relies on market-based

instruments then it is described as beta hedging, and the basis risk will refer to the presence

of an alpha component in the underlying fund. There are three main broad sources of basis

risk: tracking error, mapping error and proxy risk. The former refers to the risk controlled

by the asset manager and it relates to the difference between fund return and benchmark

return. Other than the mere presence of a deviation with respect to the benchmark, it is also

important to note that the tracking error itself might not be stationary. Secondly, the

mapping error is the risk of an incorrect valuation of price changes due to mistakes in the

mapping process. Lastly, the proxy risk refers to the risk that the predictive model does not

accurately represent reality. The risk mitigation process for basis risk heavily depends on

the understanding of the problem and the nature of the specific risk. As such, mitigation

options include the use of passive index tracking funds, a sufficient diversification across

funds, real time analysis of information related to risk and the ability to replace an

underperforming fund.

Asset allocation risk arises when the actual asset allocation varies from what was originally

assumed. For example, during a market crash equity investments tend to fall causing a

change in portfolio volatility. This risk cannot be mitigated with the use of derivatives but

must be handled at the product design stage, by setting and implementing adequate asset

allocation rules.

40 http://www.investopedia.com/terms/b/basisrisk.asp

31

Credit and counterparty default risk manifest themselves in the presence of a risk

mitigation strategy. As some or all the risk can be transferred to financial markets, the

insurance company transfers market risks and assumes credit or counterparty risks. The use

of derivatives exposes the company to this risk that is inherent to the marketplace, the

impact of which will depend on the degree of liquidity and collateralization of the financial

instrument.

Inflation risk is not a common risk factor for unit-linked insurance policies, as inflation

linked benefits might be more popular in participating policies. Nonetheless, if an exotic

form of guarantee were to be described as a function of inflation or if the underlying fund

was exposed to inflation-linked securities, the insurer would be exposed to inflation risk.

This type of risk can be hedged by using inflation swaps.

The last second order market risk, new business pricing risk, is not purely financial but

depends on price stickiness and on the company’s ability to predict changing market

conditions. The price of a guarantee will depend on market conditions and hedging

opportunities and, while market conditions can change on a daily basis, guarantee prices

can hardly behave in the same fashion.

3. Policyholder Behavior Risks

Policyholder behavior risks are those risks that arise from certain sets of actions taken by

clients. They can be divided in lapse risk, fund switching risk and business mix risk. As this

category of risks mainly depends on policyholders’ decisions and behavior, no hedging

instruments are available to mitigate them. However, risk mitigation solutions might come

in the form of product design, pricing assumptions and the accurate selection of choices

that are given to policyholders with respect to fund switching and guarantee structure.

Lapse risk is important for the insurer to consider, also because the recovery of initial

acquisition costs occurs during the whole policy duration41. The degree of lapses will in

general depend on such factors as the level of aggressive competition, the value of the

guarantees and the ability in asset management. However, when dealing with policyholder

41 In general, the recovery of acquisition costs occurs throughout the stream of premium payments, but the underlying assumption is that premiums are payable for the whole policy duration.

32

or general human behavior, one must always consider that there might be some degree of

irrationality in decision-making, even though in most market models some form of

rationality is a common assumption.

Fund switching risk arises when the policyholder decides to shift the investment. Therefore

the value of the insurer’s liability will change as the other fund experiences a different level

of volatility. A way to control this risk is through product design, by applying fund-

switching limits.

Lastly, business mix risk relates to the risk that the pricing structure is not accurate. Since it

is not practical to apply a pricing structure that is a function of every single risk factor, risk

factor buckets are commonly used. This might create a potential for adverse selection as

there could be some cross-subsidization occurring among policyholders.

4. Demographic Risks

Even though the main focus of the evaluation of risk factors has been on financial risks,

one must not forget that unit-linked policies are insurance contracts. This means that

demographic risks such as mortality and longevity must be accounted for. Mortality risk

relates to the risk of policyholders dying earlier than expected and is especially important

for GMDB products. Longevity risk, instead, relates to the risk of policyholders dying later

than assumed and is important for GMWB products. In general these risks are either

hedged via reinsurance or retained by applying the conservative technical bases in order to

allow for a safety margin.

33

5. Other Risks

These risks do not fall in any of the previous categories but must be evaluated to have a

complete picture of the risk scenario. They are expense risk and operational risk.

Expense risk relates to the cost structure of the insurance company. Overhead costs are

incurred while applying a risk mitigation solution and it is important to achieve economies

of scale to reduce them. Asset management is also a source of expenses because of

brokerage fees, taxes and transaction costs. It is important to consider these costs while

pricing the product in order not to incur in losses.

Operational risk covers the risk of various kinds of failure during the pricing, selling or

hedging processes. This risk includes misselling, technology failures, hedge management

failures, third party risks and fraud. Also, an important operational risk stems from the

volatility of financial markets. A market crash will not only affect premium income, but

also asset management fees42. Thus financial market volatility might translate into profit

volatility for the insurance company.

42 SWISS RE, “Unit-Linked Life Insurance in Western Europe: Regaining Momentum”, p.23.

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35

CHAPTER FOUR

RISK TRANSFER AND HEDGING TECHNIQUES

The process of underwriting guarantees in unit-linked business is extremely risky and must

therefore be implemented in light of adequate risk mitigation or hedging strategies. Risk

management is fundamental because it enables the insurer not to be exposed to varying

market conditions and all the other risk factors enumerated above. In fact, it is possible that

the insurance company decides first to investigate the available hedging solutions on the

market and only subsequently which kinds of guarantees to offer to its customers. There are

many different combinations of risk management strategies, with some being static and

others dynamic. In general, these strategies can be grouped in dynamic hedging, internal

reinsurance, external reinsurance and quasi-reinsurance solutions. Lastly, there is the option

to keep certain risks unhedged, even if it is not a very common solution. This last option

could be implemented if the required hedging strategy either does not currently exist on the

market or is unreasonably expensive to actuate. When leaving risks unhedged the company

must have enough economic capital43 to ensure a low probability of default.

As regards the other hedging instruments and opportunities, the following sections will

provide an overview of the manufacturing and risk mitigation processes.

1. Dynamic Hedging

Dynamic hedging is described as the ultimate hedging solution44. It is ultimate in the sense

that all other hedging solutions will ultimately depend on dynamic hedging to transfer the

risks to capital markets. This solution can be viewed as the last link of the risk transfer

chain. Whether the risks are hedged by the insurance company or transferred via

reinsurance, dynamic hedging provides the tools to reach instantaneous risk neutrality. As

explained in the previous chapter, market risks are defined in terms of their sensitivities to

market conditions. These risk factors are referred to as greeks. An effective risk mitigation

strategy aims at purchasing financial instruments with equivalent but opposite signed 43 The amount of capital that a firm needs to ensure that the company stays solvent. Source: http://www.investopedia.com/terms/e/economic-capital.asp 44 MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, p.28.

36

greeks. Given the volatility of financial markets, the effectiveness of a dynamic hedging

strategy will diminish over time and must be therefore updated frequently and monitored

constantly. Many challenges are faced in the valuation and creation of a dynamic hedging

strategy and some of the most important ones are the choice of appropriate hedging

instruments, collateral costs, volatility hedging, basis risk and taxation.

There is a wide array of financial instruments available to hedge market risks such as delta,

rho and vega and must be identified in terms of the sensitivity to a particular risk factor.

Derivative instruments are commonly used in this respect and can be exchange traded or

sold over the counter. Such instruments may also be used for speculative purposes and to

increase the exposure to a particular kind of asset, but this is out of the scope of the

dissertation and in general not a prudential investment for an insurance company. Liquid

derivative instruments are particularly attractive because they are not subject to credit risk

as they are widely available and traded on a daily basis. The major types of derivative

instruments include futures, forwards, options and swaps. The next section briefly lists

some of the most common derivative instruments along with the kinds of market risks they

are used to hedge.

Equity index futures are liquid exchange traded tools that protect against the risk of falling

equity markets. For this reason, they can be effectively used to hedge delta risk. A similar

consideration can be made for currency futures: they can be used to hedge delta risk to the

extent that the unit-linked fund is exposed to foreign exchange risk. If the insurance

company needs to find a currency-risk exposed derivative tailored to their specific needs

they might resort to currency forwards. These derivatives protect against delta risk but

sacrifices credit risk to promote personalization. In fact, the main difference between a

future and a forward is that the latter is a private transaction45.

Interest rate swaps are over the counter agreements in which two parties exchange a stream

of fixed interest payments for a stream of floating interest payments, possibly linked to

another reference rate such as LIBOR or EURIBOR46. As such, interest rate swaps can be

used to hedge rho risk and, in certain cases, delta risk.

Bond futures are exchange traded instruments that can be used to hedge both rho and delta

risk. The measure in which bond futures are effective in hedging delta risk will depend on

45 http://www.investopedia.com/exam-guide/cfa-level-1/derivatives/futures-versus-forwards.asp 46 http://www.investopedia.com/terms/i/interestrateswap.asp

37

the nature of the unit-linked fund, namely the presence of bonds or similar instruments in

its composition47.

The derivative instruments used to hedge vega risk are called volatility swaps. They are

expressed as the difference between realized volatility and fixed volatility as established at

the time of trading48, multiplied by a notional volatility that represents the notional amount

paid per volatility point.

Last, but not least, there are vanilla equity options. These derivative instruments give the

right, but not the obligation, to buy or sell an asset at a predetermined price49. They are

extremely versatile and provide hedging opportunities for delta, rho and vega as well as

gamma risk.

The use and combination of the aforementioned derivative instruments can result in an

effective hedging strategy that has as an objective the creation of a market position that has

an equal and opposite greek with respect to the unit-linked guarantee.

Collateral costs are the costs associated with the provision or receipt of collateral to back

hedging instruments. If an option is fully collateralized then the contract will be credit risk

free to the buyer and seller. However, the presence of a full collateralization will come at a

price that is either implicit or explicit. An explicit price is simple to picture because it arises

through the supply of two distinct instruments, one with and one without collateral. An

implicit price would instead arise through a difference in the reference rate at which the

obligation is deemed to accrue. The cost of collateral is sensitive to market conditions as it

can widely vary in times of financial distress and must be therefore taken in to

consideration when constructing an appropriate hedging strategy.

Another key consideration in dynamic hedging relates to the appropriate understanding of

volatility. In general, there are two terms that refer to volatility and they are statistical and

implied volatility. Statistical or historical volatility refers to the realized volatility over a

given time period50. It describes the price process of a given asset and is usually

represented by the annualized standard deviation. In contrast, implied volatility does not

necessarily depend on the historical pricing of a certain stock. It is what the market implies

the volatility of the stock will be in the future, based on option price fluctuations. Therefore

47 Refer to p.28. 48 BENNETT C., GIL. M., “Volatility Trading”, p.51. 49 http://www.investopedia.com/terms/v/vanillaoption.asp 50 http://www.investopedia.com/terms/h/historicalvolatility.asp

38

a deep understanding of the difference between these terms is required in order to construct

an effective hedge, because an option’s implied volatility might result in a different pricing

pattern with respect to the actual stock, generating the potential for excess gains or losses.

A great challenge in constructing a successful replication of a market risk is the evaluation

of the basis risk. As described in the previous chapter51, basis risk represents the risk that

the replicating portfolio will not experience an exactly offsetting gain or loss during the

hedging process. In evaluating or forecasting basis risk, it is necessary to understand the

underlying factors such as tracking error, mapping error and proxy risk that can generate

this risk.

Lastly, a complete analysis of the challenges in dynamic hedging cannot disregard the

problem of taxation. Even though theoretical models often sacrifice the inclusion of

taxation regimes in favor of simplicity, real world considerations must include the effects

of such rules. Of course, the nature and impact of any taxation regime will be country

specific and as such must be evaluated on a per-case basis. The most common differences

in taxation rules include but are not limited to the tax rates that apply to capital gains and

income and the differing tax treatment for different assets. These factors will not only

impact the valuation of the liability but also the effectiveness of the dynamic hedging

strategy.

Once the challenges and opportunities of dynamic hedging have been adequately evaluated

and understood, it is necessary to implement an efficient method or model for the

manufacturing process. As this regards internal procedure, it is company specific and thus

there is no single way of manufacturing a dynamic hedging strategy. However, there are

major areas or activities that all companies must consider in one form or another:

administration, liability management, risk management and hedging and operational

governance52.

51 Refer to page 30. 52 For an example of the operational manufacturing activities refer to MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, pp.36-42.

39

2. Static Risk Transfer Solutions

An insurance company might not wish to implement a dynamic hedging solution for a

variety of reasons. First, it might lack the necessary skills and expertise to engage in such

activities. Secondly, it might lack the economies of scale that are necessary to have an

efficient allocation of transaction costs. Lastly, it might not find the necessary hedging

strategies on the market and must therefore rely on a well-diversified third party.

Therefore, for these and other reasons a static risk transfer solution might be in order. This

solution includes reinsurance agreements and investment bank quasi-reinsurance

agreements.

Reinsurance is defined as the practice whereby one party called the reinsurer, in

consideration of a premium paid to him, agrees to indemnify another party, called the

reinsured, for part or all of the liability assumed by the latter party under a policy or

policies of insurance which it has issued53. The main principle in this agreement is that of

risk diversification; it mitigates or reduces the insurer’s exposure to risk. Reinsurance

contracts may come in various forms, such as quota-share or surplus reinsurance, but

further considerations must be made in presence of unit-linked policies and its guarantees.

These considerations will be in the nature of variations in the agreement and limitations or

restrictions applied to the cedant.

Variations can come in a myriad of forms, such as the obligation to aid the insurer in

persistency management. A common problem in life insurance, especially in unit-linked

business, is the high lapse rate of contracts. Such an obligation would ensure good faith

from the insurer and possibly help in maintaining a high level of persistence. The contract

might also include special provisions stating that only significantly adverse market

outcomes can be transferred, while “normal” losses are to be retained by the insurer. While

these are only two examples of contract modifications, it must be kept in mind that the

reinsurance contract will be tailored to the specific need of the company, and as such can

contain many modifications and many limitations.

As regards limitations, the reinsurer could limit the portion of behavioral risks that it

wishes to assume. Since policyholders’ fund switching activities can generate substantial

risks for the insurer and, in this case, for the reinsurer, it is also possible for the reinsurer to

53 http://www.captive.com/service/signetstar/GlosRein.html

40

limit the amount of freedom the insurance company gives to its policyholders in switching

or choosing funds.

Investment bank quasi-reinsurance solutions provide a similar protection with respect to

traditional reinsurance but differ in the structure of such protection. In this type of

reinsurance, protection comes in the form of derivative instruments. A common derivative

used in these complicated risk transfers is the total return swap. The total return swap is a

swap agreement in which one party makes payments based on either a fixed or variable rate

while the other party makes payments based on the return of an underlying asset, which

will contain both capital gains and income54.

When considering the implementation of static risk transfer solutions, the company must be

aware of its cost. Given that the reinsurer must actually diversify the undertaken risks and

charge a price for the service, the total price will obviously be higher than the cost of risk

mitigation. It is also important to consider that a comprehensive reinsurance solution might

also include coverage not only against first order market risks, but also against higher

sensitivities, cross-greeks, correlation and long-term volatility. Thus the true price of the

solution might be perceived as high given a naïve interpretation of the process, and this is

why a deep understanding of all the risk factors is required when evaluating not only

reinsurance or quasi-reinsurance solutions, but also any kind of hedging strategy.

It is worth noting that there is not a large availability of comprehensive risk transfer

solutions in the market55. It is thus possible that the insurance company will not find a

solution tailored to its needs, or it might not want to rely completely on a third party for

hedging; this gives rise to internal group structures and captive arrangements.

An integrated solution to the risk transfer problem might be given by a captive reinsurance

company. It provides the ability to centralize risks into a single group balance sheet as well

as the possibility to tailor the risk transfer to an exact need. Of course there might be other

reasons for which an insurance company might want to set up a captive reinsurer for these

purposes. Some key reasons include the retention of margins that would be otherwise

54 http://www.investopedia.com/terms/t/totalreturnswap.asp 55 MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, p.45.

41

handed over to the reinsurer, the centralization of know-how and expertise and the

protection of sensitive information on the manufacturing process of unit-linked guarantees.

Once the primary objectives of the captive reinsurer have been established, its optimal

location can be determined. Taxation and other industry specific regulations have a great

impact on company performance; it is therefore necessary to establish the captive in an

appropriate location for these purposes.

There are many options available to the insurance company to transfer risks, but there is no

perfect or optimal solution applicable to every portfolio. Each portfolio will have to be

evaluated with respect to the hedging or the reinsurance opportunities available in the

market in order to establish the optimal solution for every case on a seriatim basis. While

dynamic hedging is a clean solution, the insurer exposes itself to capital market failures.

Market fluctuations do not necessarily affect the effectiveness of the hedge, if it is

constructed properly, but market crashes can obliterate the market for certain derivatives,

hindering their liquidity and exposing the company to a further risk along with all the

dangers of a market crash. Reinsurance might then seem a good solution, but its limited

availability and high price do not make it an optimal solution in terms of actual cost. This

solution also depends on the company’s reliance and trust of third party reinsurance

companies. Finally, the company might decide to establish its own captive reinsurer, but it

must make sure to have the right expertise and group structure to be up to the task. So,

every solution has its own benefits and drawbacks, and in general the insurance company

will have a certain degree of freedom to decide which one suits its needs best.

42

43

Conclusions

Unit-linked products were launched in the mid-twentieth century with the objective to

allow customers a direct participation in equity markets, while retaining in many aspects

the defining elements of insurance contracts. This study shows how these policies can be

extremely diverse, creating the opportunity to tailor products and investments to specific

consumer needs. The mere nature of the policy fund, that can include a variety of

securities, allows for the first and foremost element of diversification. Then, the structure

of premium arrangements and underlying insurance covers can create a spectrum of

different products. Finally, the manufacturing of guarantees can lead the policy to divergent

results as each guarantee can manifest in different forms and offer protection against a

variety of both market and demographic risks. In light of these considerations, perhaps

there is no unique way of determining an optimal unit-linked insurance product, but the

combination of contrasting features can lead to the creation of products that are more

favorable to the insurance company or to the customer.

Unit-linked products are structured and placed on existing insurance solutions such as term,

endowment, pure endowment and whole life insurance. Insurance companies might offer

many different products or focus on a specific kind of policy, but must always consider

market signals and the opportunity to push it to the public. For example, a decline in the

use and efficiency of a public pension system might be a signal for the introduction of

private pension products, among which a company can include a unit-linked accumulation

solution with a guaranteed conversion option in a life annuity.

This dissertation also shows that even though there are many premium arrangements, the

single premium arrangement proves to be advantageous both for the insurance company

and for the customer. The insurance company has the advantage of receiving a lump sum

that can be used to cover administrative expenses, thus forgoing the risk of a loss due to a

potential lapse of a contract. The customer might not have the funds to choose a single

premium arrangement and might be irrationally deterred by the explicit presence of the

policy’s charges, but the opportunity to invest and take advantage of market conditions on a

full scale can lead to the immediate realization of profits.

Of course, considerations of the full-scale impact of a single premium solution are a

double-edged sword; market conditions have not proven to be particularly stable and

market crashes have created some skepticism that can result in decreased premium income

and losses for the consumer. However, history has shown that a market crash such as the

44

burst of the Internet bubble in 2002 has led to a less than proportional depauperation in

unit-linked policies. In fact, this crash forced the evolution of guarantees and enabled a

revival in consumer confidence. It thus appears that in volatile market conditions the

effective manufacturing and marketing of unit-linked guarantees is of paramount

importance.

Given that guarantees have now become a common feature in unit-linked products, most

consumers will expect such arrangements and insurance companies must be able to

manufacture and deliver these riders, even in unstable market conditions. Fierce

competition among insurers could lead to excessive risk taking and inadequate risk

management solutions. It is thus necessary for insurers to place the solvency and stability

of the company in high regard and not to focus solely on profit-seeking activities because

they could prove to be counterproductive. A hypothetical future market crash, in presence

of unstable risk mitigation, could bring forth a collapse of the whole system with

consequences on consumer income, insurers’ defaults and the level of confidence in unit-

linked policies. However, even if the insurance company is effective in determining its

exposure to all the types of risks and is able to offer a suitable guarantee on a unit-linked

product, it must also be able to manufacture and implement an effective hedging solution.

As such, this study suggests that it might be proper first to seek the availability of hedging

instruments on the market and only subsequently decide which kinds of guarantees to offer.

Furthermore, if the company has the possibility and availability of funds to expand, it can

attempt to adopt the captive reinsurance solution to the risk management problem.

Alternatively, it can develop the expertise and governance structure suited to implement a

dynamic hedging solution without the need to rely on third parties. This does not mean that

reinsurance or quasi-reinsurance solutions are less effective in risk management, but means

that the internalization of these activities can prove to be efficient in cost reduction and has

the advantage of retaining sensitive information about the processes used to manufacture

and market unit-linked guarantees.

In conclusion, notwithstanding the myriad of combinations that unit-linked products offer

to the market, they have proven to be a successful product. Their pervasive diffusion was

hindered but not eliminated by market crashes that instead brought forth their evolution.

They are advantageous both to customers and insurance companies, subject to adequate

risk decomposition and hedging strategies. Thus, unit-linked products can be placed among

45

the most prominent products that have attempted and succeeded in constructing a bridge

between the domain of finance and that of traditional insurance.

46

47

BIBLIOGRAPHY

Authors

BENNETT C., GIL. M., “Volatility Trading”, 2012

G.L. MELVILLE et al., “The Unit-Linked Approach to Life Insurance”, 1969

MAHER J., CORRIGAN J., BENTLEY A., DIFFEY W., “An Executive’s Handbook for

Understanding and Risk Managing Unit-Linked Guarantees: A Discussion Paper”, 2010

MUNICH RE GROUP, “Unit Linked Insurance: A General Report”, 2000

NORBERG R., “Basic Life Insurance Mathematics”, 2002

OLIVIERI A., PITACCO E., “Introduction to Insurance Mathematics”, 2011

SWISS RE, “Unit-Linked Life Insurance in Western Europe: Regaining Momentum?”,

2003

Sitography

math.about.com

www.captive.com

www.investopedia.com

www.pswlaw.co.uk

48

49

Acknowledgements

It is now time to acknowledge people that I believe to have been fundamental for their

support in the development of this dissertation.

A first and most special thanks goes to Professor Pitacco, for his valuable support and

expertise.

I would also like to thank my family, in particular my mother, father and sister for their

ongoing and unconditional support not only throughout my years at the university, but also

during my life in general.

Last, but not least, I would like to thank my friends for helping me in times of need and for

creating a pleasant environment that enabled me to thrive as a student and as a person.