Univerzita Karlova v Praze Matematicko-fyzikální fakulta. Katedra pravděpodobnosti a matematické statistiky

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1 Univerzita Karlova v Praze Matematicko-fyzikální fakulta DIPLOMOVÁ PRÁCE Tomáš Coufal Výpočet rizikového kapitálu pro investiční životní pojištění Katedra pravděpodobnosti a matematické statistiky Vedoucí diplomové práce: Studijní program: Studijní obor: Mgr. Josef Lukášek matematika Finanční a pojistná matematika Praha 2011

2 Děkuji vedoucímu diplomové práce Mgr. Josefu Lukáškovi za vstřícný přístup a podnětné připomínky. 2

3 Prohlašuji, že jsem tuto diplomovou práci vypracoval samostatně a výhradně s použitím citovaných pramenů, literatury a dalších odborných zdrojů. Beru na vědomí, že se na moji práci vztahují práva a povinnosti vyplývající ze zákona č. 121/2000 Sb., autorského zákona v platném znění, zejména skutečnost, že Univerzita Karlova v Praze má právo na uzavření licenční smlouvy o užití této práce jako školního díla podle 60 odst. 1 autorského zákona. V Praze dne Bc. Tomáš Coufal 3

4 Název práce: Výpočet rizikového kapitálu pro investiční životní pojištění Autor: Bc. Tomáš Coufal Katedra/Ústav: Katedra pravděpodobnosti a matematické statistiky Vedoucí diplomové práce: Mgr. Josef Lukášek vedoucího: Josef.Lukasek@allianz.cz Abstrakt: Investiční životní pojištění je moderním flexibilním druhem životního pojištění. V posledním desetiletí zaznamenáváme výrazný nárůst popularity investičního životního pojištění. Diskuze o dopadu nové direktivy Solventnost II na životní pojišťovny se však zaměřují zejména na tradiční životní pojištění. Tato práce se zabývá problémem výpočtu rizikového kapitálu pro investiční životní pojištění. Součástí práce je analýza dopadu různých druhů garancí v případě smrti, lhůt placení pojistného, doby do splatnosti a tržně dynamického chování pojistníků na výpočet rizikového kapitálu. Klíčová slova: Investiční životní pojištění, Solventnost II, Rizikový kapitál, kapitálový požadavek Title: Risk capital calculation in invesment life insurance Author: Bc. Tomáš Coufal Department/Institute: Department of Probability and Mathematical Statistics Supervisor of the master thesis: Mgr. Josef Lukášek Supervisor s address: Josef.Lukasek@allianz.cz Abstract: Unit linked insurance is a modern and flexible life insurance product. The last decade was marked by the raising popularity of unit linked insurance. The discussions conserning the impact of the new directive Solvency II on the life insurance business focus mainly on the traditional life insurance. This paper examines the issue of the calculation of the risk capital for unit linked insurance. Analysis of the impact of different death guarantees, forms of premium payment, time to maturity and dynamic policyholder bahaviour on the risk capital is presented. Keywords: Unit linked insurance, Solvency II, Risk capital, Solvency capital requirement 4

5 Contents 1 Introduction 8 2 Unit linked insurance Product design Investing the premium Surrender value Maturity benefit Death benefit Revenues, costs and risks of the insurer Risk capital Definition and valuation of risk and risk capital under Solvency II Modular approach Model design Interest rate Risky asset, bid price Mortality Lapses, dynamic policyholder behaviour Expenses Model results Standalone risks Overall results Conclusion 47 Bibliography 48 5

6 A Mortality tables 49 A.1 Males A.2 Females B Deterministic lapse rates 53 C Profits 54 C.1 One year outcome C.2 Remaining profit

7 List of Tables 5.1 Model parameters Modelpoint parameters Standalone results - Market risks Standalone results - Life and Business risks Correlation matrix Overall results - using the dynamic lapse multiplier Overall results - not using the dynamic lapse multiplier Profits A.1 Mortality tables - males A.2 Mortality tables - females B.1 Deterministic lapse rates List of Figures 5.1 Standalone risks - Policy type A Standalone risks - Policy type B Standalone risks - Policy type C Standalone risks - Policy type D C.1 Profits - One year outcome C.2 Profits - Remaining profit

8 Chapter 1 Introduction Unit linked insurance and other flexible products are constantly gaining popularity in the market of life insurance. In the Czech Republic a modern insurance market began to develop along with changes in society after the fall of communism. The stock market growth in the nineties led the insurance companies to focus on unit linked insurance. The beginning of the millennium was marked by the growing popularity of this type of life insurance. The recent financial crisis has led some clients to return to traditional life insurance with guaranteed return. Unit linked insurance, however, still remains the fastest growing type of life insurance in the Czech Republic (see [6, pages 56, 59]). This paper discusses the risks associated with unit linked insurance products. The attention is paid to establishing a model applicable for the calculation of risk capital for unit linked insurance. The aim is to identify the main risks related to this type of life insurance. The impact of different death guarantees and types of premium payments to the structure and size of the risk is discussed as well as the relation between the time to maturity and the risk structure. Furthermore, the issue of dynamic policyholder behavior and its impact on the risk capital is examined. The paper is organized as follows. In chapter 2 we describe the technical desing of a product that is later used as a reference for discussions concerning the risks of the underwriter. In the same chapter we indentify the revenues, costs and risks related to the business. General definition of the risk capital, 8

9 the modular approach and the risk categories are presented in chapter 3. In chapter 4 we describe the model used to calculate the capital requirement for each risk category. The results of the model are presented in chapter 5. The model is performed on both single and regular premium policies. Results obtained for policies of two types of death guarantees, constant sum assured and a guarantee of premium paid, are presented. 9

10 Chapter 2 Unit linked insurance Unit linked insurance is a type of life insurance that started to emerge in the second half of the twentieth century. It s defined as a type of life insurance, where the investment risk is bared by the policyholder. The premium is invested in mutual funds in form of so-called units. The insurance benefit does not depend on the technical interest rate, but the market value of the mutual funds. The policyholder decides in which fields (mutual funds) the premium is invested. The insurance companies offer a wide variety of funds to choose from. The most general ones are bond and equity funds. By choosing the allocation ratio (ratio defining how the premium will be distributed to various funds) the client defines his investment strategy. The more the allocation ratio favours the equity funds, the more risky is the strategy and the higher is the future expected value of the client s unit fund. 2.1 Product design In following subsections we will describe a model that represents a typical approach to unit linked insurance and can be used as a reference for future discussions concerning the risks of the underwriter. 10

11 2.1.1 Investing the premium Just as in case of traditional life insurance there are three types of premium. Depending on the form of the contract there is single (invested once at the beginning of the contract) or regular (invested every certain period of time) premium. The client usually has an option to invest more money in addition to the regular or single premium payment. In that case we talk about top-up premium. The price used to determine the amount of units to be bought is called offer price. Bid price is used when buying the units from the policyholder as well as for all the other operations concerning units. Offer price is generally slightly higher than the bid price. The difference between the two is called bid/offer spread and represents an income for the insurance company. The amount invested in each fund is then given by the following equation: NU i = AR i IP OP i = AR i IP BP i (1 BOS i ) (2.1) where the following symbols represent NU i number of new units invested in fund i, AR i allocation ratio of fund i, IP invested premium, OP i offer price of fund i, BP i bid price of fund i, BOS i bid/offer spread valid for unit fund i. At the beginning of the contract (for example the first two years) the premium is invested in the initial units, later in accumulation units. This division is convenient, because a significant amount of the initial units is (later) used to cover the initial costs. There are several approaches to the initial costs treatment. The one used in our model is known as Actuarial financing (see [9, pages 22-32]). 11

12 The main idea is that the insurer s view of the amortization of initial costs is different than the policyholder s. While the second one counts on the amount of units bought according to the current offer price and expects certain percentage of the total initial units amount to be cut of annually, the insurer considers all the future fees to be paid at the time of investing the premium and all future costs emerging from this difference are matter of additional fees (in case of lapse of policy) or risk borne by the underwriter (death benefits). Furthermore we will talk about initial units and reduced initial units to make sure which amount we mean. Considering premium invested at the beginning of the policy period we can write the equation 2.1 in this form: NRIU i = AR i IP OP i (1 ICF rate ) n = AR i IP BP i (1 BOS i ) (1 ICF rate ) n where NRIU i number of new reduced initial units invested in fund i, AR i allocation ratio of fund i, IP invested premium, OP i offer price of fund i, ICF rate the rate of Initial costs fee, n number of years (policy s anniversaries) till run-of, BP i bid price of fund i, BOS i bid/offer spread valid for unit fund i. (2.2) Bonus units In our model, the insurer agrees to buy additional amount of units every month. Those are called bonus units. NBU = ABU nom + IP ABU % BP i, (2.3) 12

13 where NBU number of new units invested in bonus fund, IP invested premium, ABU nom nominal amount invested in bonus units every month (compensation for monthly administrative fee), ABU % % of IP invested in bonus units every time a premium is invested (compensation for BOS), bid price of bonus fund. BP bf Surrender value In case of lapse of policy the policyholder has a right to obtain a surrender. By law, [12, Part One., Title I, 23, paragraph 2], the insured has a right to terminate the contract within the first two months of the policy period, hence the calculation of surrender value differs from the one used for policies being in force longer. In our case the surrender value in first two months of the contract equals the value of policyholder s unit fund lowered by certain nominal lapse fee and the debt attributable to the policyholder arising from the agreed policy conditions. That is: SV = where { max[0; P NI AD NLF ] no premium yet invested, max[0; V UF NLF AD + P NI] otherwise, SV surrender value (before taxation), AD accumulation debt (unpaid risk premiums and monthly administrative fee), P NI paid, but not yet invested premium, V UF value of policyholder s unit fund (with no deduction of the ICF ), N LF nominal lapse fee. (2.4) 13

14 After the two month period the value of surrender is determined by the value of Unit fund and the conditions agreed in the insurance contract. In that case it is of a form: SV = max[0; i where BP i RIU i (1 lf iu ) + BP i AU i (1 lf au ) NLF AD] (2.5) SV surrender value (before taxation), BP i bid price of fund i, RIU i reduced initial units of fund i, AU i accumulation units of fund i, lf iu lapse fee valid for initial units, lf au lapse fee valid for accumulation units, N LF nominal lapse fee, AD accumulation debt (unpaid risk premiums and monthly administrative fees). There is a 100% lapse fee for the bonus units, therefore they are not mentioned in the formula Maturity benefit The value of insurance benefit at maturity equals the value of the policyholder s funds net of tax. The tax is calculated from a positive difference in the value of the funds and the premiums used to calculate the tax base. V UF sum = max[0; i V UF i = i BP i (IU i + AU i ) + BP bf BU AD], T = max[0; V UF sum P ] taxr, IB = V UF sum T, (2.6) 14

15 where V UF sum value of policyholder s unit fund, V UF i value of unit fund i, BP i bid price of fund i, BP bf bid price of bonus fund, IU i number of initial units of fund i, AU i number of accumulation units of fund i, BU number of bonus units, AD accumulation debt (unpaid risk premiums and monthly administrative fees), T tax, P premium used to calculate the tax base, taxr tax ratio, IB insurance benefit paid to the policyholder. As the amount of initial units and the reduced initial units equal at maturity, we can replace IU by RIU. The premium used to calculate the tax base, P, equals to the sum of premiums paid to the insurer lowered by the amount used in calculation of the tax base in case of the previous partial surrenders Death benefit The amount of insurance cover in case of death is also derived from the value of policyholder s unit funds, but in this case the insurance company typically guarantees a minimum level of a Death benefit. This minimum level can be for example the value of premium used to calculate the tax base, P, or generally any sum assured writen in the contract. In our case we will study two types of death guarantees. The first will equal to P, the second to a certain constant sum assured. Later it will be interesting to see which impact the form of death guarantee has on the capital requirement. 15

16 V UF sum = i (IU i + AU i ) BP i, DB = max[v UF sum ; IP sum P R sum ] + P BNIP + BP bf BU, or, = MAX[V UF sum ; SA] + P BNIP + BP bf BU. (2.7) where DB insurance benefit paid to the policyholder, V UF sum value of policyholder s unit fund, IP sum sum of invested premium payments, P R sum sum of previous partial surrenders, SA sum assured, P BNIP paid, but not yet invested premium, IU i number of initial units of fund i, AU i number of accumulation units of fund i, BU number of bonus units, BP i bid price of fund i, BP bf bid price of bonus fund. To avoid poor readability of the text, we will furthermore always write SA when talking about the guaranteed minimum level of Death benefit. Later we will discuss the difference between the two death guarantees when calculating the capital requirements related to such a policy. Unlike the previous case, there is a difference between using the initial units as seen by the client or the reduced ones while calculating the value of policyholder s unit fund. The insurance companies almost never use the reduced initial units in this case and so won t we. Risk premium To finance the guarantee of a minimum level of Death benefit the insurer charges the policyholder a risk premium. This premium is usually calculated only to finance the difference between the sum assured and the value of policyholder s unit fund determined from the amount of accumulation and initial units. 16

17 V UF sum = i BP i (IU i + AU i ), SAR = max[0; SA V UF sum ] + AD, RP = SAR q x /12, (2.8) where V UF sum value of policyholder s unit fund, (calculated using non-reduced initial units), BP i bid price of fund i, IU i number of initial units of fund i, AU i number of accumulation units of fund i, SA sum assured, SAR sum at risk, AD accumulation debt (unpaid risk premiums and monthly administrative fees), RP risk premium, SA sum assured, q x probability of a x-year-old to die in the ongoing year, (for women we write q y ), We assume the probability of death of a person at the age of x to be uniformly distributed over the year, therefore q x /12 is an approximation of mortality of such a person within one month. The risk premium is paid by the sale of client s units. AFI reserve While the risk of paying max[sa i (IU i + AU i ) BP i ; 0] is covered by the risk premiums, the risk of paying the difference between the value of initial and reduced units i (IU i RIU i ) BP i represents an uncovered risk for the insurer. To cover the expected future losses the insurer holds an aditional reserve called AFI (actuarial financing) reserve. For each contract we calculate the size of this reserve at the beginning of the policy year t as: 17

18 V IU t = i V RIU t = i BP i (t) IU i (t), BP i (t) RIU i (t), where AF I t = n t 1 j=0 [V IU t (1 ICF rate ) j j p x+t q x+t+j ] +V IU t (ICF rate ) n t n t p x+t V RIU t, (2.9) AF I t AFI reserve hold for the contract at time t, x the client s entry age, n lenght of the period of insurance, t number of anniversaries of the contract already past, ICF rate the rate of Initial costs fee, jp x probability of survival of a man at the age of x over next j years, (for women we write j p y ), q x mortality probability of a man at the age of x, (for women we write q y ), V IU t value of initial units at time t, V RIU t value of reduced initial units at time t, BP i (t) bid price of fund i at time t, IU i (t) number of initial units of fund i at time t, RIU i (t) number of reduced initial units of fund i at time t. Although the formula 2.9 is correct, the actual calculations are done using commutating numbers. Finally, the value of AFI reserve at time t (which is generally any real number lower than the policy period n) equals: AF I t = AF It t2 t + AF I t + t t 1 (2.10) t 2 t 1 t 2 t 1 ( ) AF It = V IU t Mx+t 1 M x+n lx+n + V RIU t 1 D x+t1 l x+t1 AF I + t = V IU t v Mx+t 2 M x+n D x+t2 + V RIU t 18 ( lx+n l x+t2 1 )

19 where AF I t AFI reserve hold for a contract at time t, t time (in years) already past since the beginning of the policy, t 1 time at the previous anniversaries of the contract, t 2 time at the following anniversary, n lenght of the period of insurance, l x number of surviving at age x, d x number of death cases at age x, v = (1 ICF rate ), C x = d x v x, D x = l x v x, M x = ω j=x C j. 2.2 Revenues, costs and risks of the insurer There are risks the underwriter takes on himself and there are expenses related to the policy. On the other hand there are allowances coming mostly from the fees charged. Revenues bid/offer spread Initial costs fee The Initial costs fee is proportional to the length of the insurance period. The longer the insurance period, the higher the commissions to brokers and therefore the higher fee. Monthly administrative fee The monthly administrative fee is paid by selling the client s units just like the risk premium. 19

20 Lapse fees In case of lapse of the policy the surrender value is calculated using the amount of accumulation and reduced initial units. The difference in value of initial units and the reduced ones represents a fee charged to the insured and covers the unpaid initial costs. The additional lapse fees represent income of the insurer to compensate the loss of business opportunities. Note that in case of lapse in the first two month of the insurance period there is no compensation of the initial costs. risk premium The risk premium covers only the risk of paying the difference between the sum assured and the value of unit fund. The risk releted to the unpaid initial costs is not covered. Management fees Costs and risks initial costs The investment fund management pais part of its profit back to the insurer as a benefit for large business valumes. For more details about management fees see section 4.2. Commissions to brokers and other initial administrative expenses. Administrative/other costs Claims handling expenses bonus units Costs of claims handling in case of death, maturity or surrender (to avoid poor readibility of the text we keep the costs of processing the surrender under this category). There is no premium to cover the investment in bonus units. Death guarantee 20

21 The insurer agrees to pay max[sa; i (IU i + AU i ) BP i ] in case of death. The risk of the insurer then equals max[sa i (RIU i + AU i ) BP i ; 0] + i (IU i RIU i ) BP i. The risk of paying max[sa i (IU i + AU i ) BP i ; 0] is covered by the risk premium. The risk of paying (IU i RIU i ) BP i is not. i difference in the fund value between the record date and the date of the actual redemption of units The bid prices change in time. The date determining the bid prices used to calculete the value of unit fund is set by the terms of policy. The difference in the value of the fund due to the change in bid prices is therefore a risk of the insurer. cost of AFI reserve The insurer has to hold a reserve to cover the difference between the value of initial and reduced initial units in case of death. unpaid accumulation debt There is not such a thing as a negative maturity benefit, hence unpaid accumulation debt in case the value of unit fund is bellow the accumulation debt is a risk barred by the policyholder. Although in case of surrender the insurer would probably ask the policyholder for the unpaid accumulation debt, for simplicity reasons we assume there may never be a negative surrender value. The unpaid accumulation debt in case of death is covered by the risk premium. 21

22 Chapter 3 Risk capital Risk capital is a capital required to cover unexpected losses. The traditional definition (see [5, page 241]) says that the Risk capital is a supplement of the net reserve to level the sum assured. However, this definition is rather vague and doesn t even consider all potentional losses related to a business. Solvency I as factor based framework defines Risk capital as a factor of technical reserves. In this framework only the liability side of the balance sheet is taken into account. The risk profile of the asset side is not considered. The actual risks are not properly measured and the calculation is only based on the size of the portfolio. 3.1 Definition and valuation of risk and risk capital under Solvency II Solvency II is reffered to as risk based framework. Under this framework we think of a risk capital as of a value that somehow corresponds to the costs related to all the risks tied with the business. To define such a risk we need to consider all potential future events and assign probabilities to each of them. The theoretical values of unexpected losses may be very high and it s practically impossible to hold all that value as a risk capital. Therefore we usually set up a risk tolerance level. In practice it means to set up risk 22

23 capital to withstand worst case scenarios of (1 α) probability, where α is the risk tolerance level. Under Solvency II one year time horizon is used in accordance with the procedures for determining the credit rating of a company. We have: RC α = argmin x (P [X E(X) > x] 1 α), = argmin x (P [X > x] 1 α) E[X], = V ar α (X) E[X], (3.1) where X stays for all the risk related to a business, α is a risk tolerance level and E[X] denotes expected losses that are covered by technical reserves. Although one year time horizon is used, it would be rather naive to only count on the difference between best estimate and shocked cash-flow values. As stated in [11, page 2]: Re/insurance companies should have enough capital on their balance sheet to cover the risks that can emerge over a 12- month timeframe, and allow for a (theoretical) transfer of all (contractual) liabilities at the end of this balance-sheet period. This means that companies have to be able to calculate the impact of such shocks on their end-of-year balance sheets, and value these in such a way that they can be transferred to a third party. That leads us to a modified definition risk capital: where RC α = V ar 1Y α (X) E 1Y [X] + n T P (j), (3.2) j=2 V arα 1Y (X) one-year horizon Value at Risk as defined in 3.1, E 1Y [X] expected (one-year horizon) losses, T P (j) change in technical provisions in the subsequent year j, conditional on the first year being in distress, n number of years till run-off. 23

24 3.2 Modular approach However the formula 3.2 defines a risk capital perfectly, it is to complicated to describe all risks related to a business of an insurance company in one stochastic model. While calculating the risk capital, we usually calculate all risk drivers separatelly and sum them up using a correlation matrix. There are several categories of risks the life insurance company has to face. The list of risks bellow is an example of general division of risks the insurance company may use. Only risks directly related to an unit linked insurance are listed. Other risks relevant to the insurance business in general such as operational risks are not covered by this paper. Risk categories Market risks Interest rate risk - The shift of yield curve has an effect on both sides of the balance-sheet and reflects in the change of the Asset-Liability match (ALM). Equity risk - caused by the change in stock prices. Life risks Mortality risk - caused by higher or lower mortality than expected. Lapse risk - caused by fluctuations in lapse rates on existing insurance contracts. Business risks Expense risk - caused by fluctuations expenses. The CEIOPS s standard formula (see [3, pages ], [4, pages ]) introduces deterministic scenarios that are supposed to represent worst scenarios of (1 α) probability, where α = Assuming all (n) risks to be multivariete normally distributed the risk capital is calculated as follows SCR i α = V ar α (X i ) E[X i ], SCR α = (SCRα; 1 SCRα; 2 ; SCRα) n T, RC α = SCRαCorrSCR T α, (3.3) 24

25 where SCRα i solvency capital requirement related to a risk factor i, Corr correlation matrix. Our internal model is based on the concepts presented in the standard formula. However, the test scenarios used and the correlation matrix will be based on insurer s own assumptions. The capital requirement is calculated separately for each risk category presented in the list above and aggregated using the correlation matrix. 25

26 Chapter 4 Model design The basic idea of the model is Monte Carlo modeling of stochastic processes describing each risk factor and their impact on the casf flow. For each risk module only the relevant processes are considered to be stochastic. Concepts introduced in the section 2.1 are respected while modeling the fund development. Every income of the insurer is invested on a bank account whose development is described by a short rate r t. Similarly all the insurance benefits and other expenses of the insurer are financed by withdrawing the money from the bank account (we allow the value of bank account to be negative). As stated in [10, page 12]: Cash flow projection should be based on policy-by-policy approach but suitable model points are permitted, hence risk capital related to a single policy will be the object of our studies. Following sequence of simulation steps is used while modeling. All of this take place at the beginning of each month. Values of interest rates, risky assets and bid and offer prices are determined. Premiums are invested (in case of single premium policy the premium equals zero for all other times than t=0). The value of remaining accumulation debt from last month is multiplied by (1 + tr) 1/12, where tr is a technical interest rate. Monthly administrative fees, risk premiums and accumulation debt are paid whenever it is possible (the value of accumulation units is higher 26

27 than the sum of accumulation debt and fees). In other cases the unpaid fees are added to the client s accumulation debt. AFI reserve is calculated. The cost of holding AFI reserve is withdrawn from the bank account. Number of deaths is determined. Death benefits are paid. Units related to those policies are sold. Number of lapses is determined. Surrender values are paid. Units related to those policies are sold. The previous two steps determine the number of in-force policies at the beginning of the following month. 4.1 Interest rate The short rate model used is known as Two-factor Black-Karasinski Model. The short rate and the mean reversion level are assumed to develop according to the dynamics given by following stochastic differential equations: dln(r) = a 1 (ln(m) ln(r))dt + σ 1 dwt r, dln(m) = a 2 (ln(µ) ln(m))dt + σ 2 dwt m, (4.1) where r short rate, m mean reversion level, a 1 mean reversion speed, a 2 reversion speed, µ long run short rate, σ 1 volatility per annum - short rate, σ 2 volatility per annum - mean reversion level, Wt r, Wt m independent Brownian motions. The distribution of r t is lognormal, hence only positive interest rates. As there is no closed-form solution for this model, the Euler method of 27

28 discretization presented in [1] is used. The unit linked insurance is usually modeled on monthly bases, hence time step of one month is used. Applying the discretization on 4.1 we obtain the following expression for the short rate r t : r tj r tj 1 exp{ t (j) a 1 (ln(m tj ) ln(r tj 1 )) + σ 1 t (j) Zj r }, m tj m tj 1 exp{ t (j) a 2 (ln(µ) ln(m tj 1 )) + σ 2 t (j) Zj m }, t (j) = t j t j 1, (4.2) where Z r j, Z m j, j = 1, 2,..., are independent standart normal random variables and t (j) stays for the length of a time step j. The forward and swap rates can be derived from r t equations hold: as the following tj fr tj 1,t j = exp{ r u du} 1 exp{ t (j) r tj 1 } 1, t j 1 sr tj = ( (1 + sr tj 1 ) t j 1 (1 + fr tj 1,t j ) t(j)) 1/t j 1, (4.3) where fr tj 1,t j forward rate for time [t j 1, t j ], sr tj annual zero coupon swap rate at maturity t j. 4.2 Risky asset, bid price We assume the investment fund to consist only of risky assets. Investment fund fees are taken into account while modeling the price of an unit. Standart Black-Scholes setting is used to model the risky asset: ds t = r t S t dt + σ ra S t dw ra t, (4.4) 28

29 where S t r t σ ra Wt ra value of one share of the risky asset, short rate, volatility per annum - risky asset, Brownian motion (independent on Wt r, Wt m ). The equation 4.4 has the following solution (see [2, pages 66-69]): S tj = S 0 exp = S tj 1 exp S tj 1 exp { tj 0 { tj { r u du σ2 ra 2 t + σ ra Wt ra j r u du σ2 ra t j 1 2 t(j) + σ ra (Wt ra j W ra t (j) r tj 1 σ2 ra 2 t(j) + σ ra }, t j 1 ) t (j) Z ra j } },, (4.5) where Zj ra, j = 1, 2,... are independent standart normal random variables. Note that while the volatility σ ra is annual, the lenght of a time step [t j 1, t j ] is one month. Price of an unit of investment fund is derived from the price of the risky assets (see [8, pages 24-25]): BP tj = BP tj 1 S tj S tj 1 (1 iff), OP tj = BP tj 1 BOS, iff = 1 (1 aiff) 1/12, mf per unit = BP tj mf, mf = amr/12, (4.6) 29

30 where S t BP t OP t aif f if f amf mf value of one share of the risky asset, bid price, offer price, annual investment fund fee, monthly equivalent of the annual investment fund fee, annual management fee, monthly equivalent of the annual management fee. Management fees are paid by the investment fund management (third party) to the insurer as a benefit for large business valumes, hence the parameter amf is always smaller than aiff. 4.3 Mortality 2009 Czech mortality tables [7] are used (see A.1, A.2). The probabily of x-years old to die is assumed to be uniformly distributed throughout an underwriting year. The insured is assumed to die in the underwriting year [t, t + 1] with probability q x+t, where x is his/her entry age. The mortality tables [7] are considered to be a best estimate. This assumption is rather prudent as the life expectancy is commonly believed to protract and probabilities q x to decrease. The actual mortality for simulation path s is then assumed to develop according to the following equation: q s (t j ) = [ q x+tj t (j + 1) ] = q x+tj tj+1 t j 1 t (j + 1) tj+1 t j Xu mort du, X mort u du, (4.7) where q s (t j ) is the actual mortality for time interval [t j, t j+1 ] (one month) and simulation path s. The process describing the fluctuation of mortality 30

31 Xt mort is assumed to develop according to the dynamics given by the following stochastic differential equation: where X mort a mort σ mort Wt mort dx mort = a mort (1 X mort )dt + σ mort dw mort t, (4.8) mortality fluctuation describing process, mean reversion speed - mortality, volatility per annum - mortality, Brownian motion. The mean value of Xt mort calculated at time 0 equals 1 for all t, hence E [ q s (t j )] = q x+tj t (j + 1). Using the discretization again we get the following: [ ] Xt mort j+1 Xt mort j + t (j + 1) a mort (1 Xt mort j ) + σ mort t (j + 1) Zj+1 mort, q s (t j ) q x+tj t (j + 1) X mort t j, (4.9) where Zj mort, j = 1, 2,... are independent standart normal random variables. 4.4 Lapses, dynamic policyholder behaviour There are many reasons that may cause the policyholder to cancel the policy. Generally, if the decision was based on the observation of his/her insurance contract we consider that act as an example of dynamic policyholder behaviour and we talk about rational lapse. As unit linked insurance is a variable product and there is a lot of uncertainty bared by the policyholder, dynamic policyholder behaviour should be taken into account while designing models of processes influenceable by the policyholder s decision. In our case it s mainly the issue of lapses. There are of course other reasons causing the policyholder to cancel the policy (personal and others). Lapses based on those decisions are called irational lapses and cannot be modeled with regard to the policy development. 31

32 The deterministic lapses rates (see B.1) s t where t is the year of policy are estetimated based on historical observations. As it is practically impossible to differentiate the rational and irational lapses in historical data, the rates s t therefore represent a sum of both. As there is a problem collecting data showing the proportions between rational and irational lapses, the lapse function (dynamic lapse multiplier) has to be designed based on expert judgement (e.g. insurance assessor experience), hence a simplicity is an appropriate property of such a function. The dynamic lapse multiplier used in our model is simple. Assuming a bad performance of the clients unit fund leads to higher lapse rates and good fund performance reduces lapses, the lapse function can be described as follows: where dlm s (t j ) = V UF t s j 1, (4.10) V UFt BE j 1 lr s (t j ) = dlm s (t j ) [s tj t (j + 1) ] = dlm s (t j ) s tj tj+1 t j 1 t (j + 1) tj+1 t j X lp u du, X lp u du, (4.11) dlm s (t j ) dynamic lapse multiplier at time (month) t j, V UFt s j 1 value of unit fund at time t j 1, simulation path s, V UFt BE j 1 value of unit fund at time t j 1, best estimate, s t deterministic lapse rates, lapse fluctuation process, X lp t Similarly as in 4.8 we assume the fluctuation factor X lp to develop according to the following dynamics: dx lp = a lp (1 X lp )dt + σ lp dw lp t, (4.12) 32

33 where X lp a lp σ lp W lp t lapse fluctuation process, mean reversion speed - lapses, volatility per annum - lapses, Brownian motion. Finally using the discretization we obtain: [ ] X lp t j+1 X lp t j + t (j + 1) a lp (1 X lp t j ) + σ lp t (j + 1) Z lp j+1, lr s (t j ) dlm s (t j ) s tj t (j + 1) X lp t j, (4.13) where Z lp j, j = 1, 2,... are independent standart normal random variables. 4.5 Expenses The expenses related to an insurance contract can be devided into three groups: Initial expenses Those directly related with the cost of starting a new business. The major portion of these costs is the commission to the broker. Claims handling expenses In case of death, maturity or surrender. Administrative/other expenses Includes expenses related to the insurance business itself. In our model we have to add the cost of buying bonus units. The three types of expenses are modeled differently. While the initial expenses represent certain fraction of the premium agreed in the contract to be paid by the policyholder (the higher is the premium and the longer is the insurance 33

34 period, the higher is the commission to the broker), the administrative expenses are considered to be the same for every policy in-force (regardless the premium or sum assured). Finally the claims handling expenses are modeled as a combination of an fixed rate and an additional rate if the sum assured exceeds a certain limit. Initial expenses { (commreg + oie ie = reg ) n Y P regular premium policy, (comm sin + oie sin ) SP single premium policy, (4.14) where ie comm oie Y P SP n initial expenses, quotient - commission paid to the brokers, quotient - other initial expenses, one year premium, single premium, insurance period (in years). We assume the broker has to pay the full commission back to the insurer in case of lapse of the policy or death of the policyholder both if happens in the first year of the policy. Furthermore, we assume half of the commission is returned if lapse or death of the policyholder happens in the second year of the policy. Administrative/other expenses where ae s (t j ) = iae s eif s (t j ), (4.15) iae s = iae BE σ ae Y s, ( eif s (t j ) = eif s (t j 1 ) 1 + if 1 t (j + 1) tj+1 t j s Xu eif du ) t(j+1) ae s (t j ) administrative/other expenses at time t j, simulation path s, iae s initial value of administrative expenses, simulation path s, iae BE initial value of administrative expenses, best estimate, administrative expenses -initial volatility, σ ae 34,

35 Y s independent standart normal random variables, eif s (t j ) inflation factor at time t j, simulation path s, if deterministic inflation - expenses, fluctuation process - expenses inflation. X eif t Let a eif be the mean reversion speed, σ eif the annual volatility and W eif t a Brownian motion, then the fluctuation process X eif t is assumed to have the following dynamics: dx eif = a eif (1 X eif )dt + σ eif dw eif t (4.16) Using the discretization we obtain: [ X eif t j X eif t j 1 + t (j) a eif (1 X eif t j 1 ) + σ eif t (j) Z eif j eif s (t j ) eif s (t j 1 ) ( 1 + if s X eif t j ) t(j), (4.17) where Z eif j, j = 1, 2,... are independent standart normal random variables. ], Claims handling expenses (iche mort + Id SA SAlimit iche hsa mort) eif s (t j ) che s (t j ) = (iche lp ) eif s (t j ) (iche mat ) eif s (n) where death, lapse, maturity, (4.18) che s (t j ) claims handling expenses at time t j, simulation path s, iche mort initial value of claims handling expenses - mortality, iche hsa mort initial value of claims handling expenses - high sum assured, iche lp initial value of claims handling expenses - lapses, iche mat initial value of claims handling expenses - maturity, eif s (t j ) inflation factor at time t j, simulation path s. 35

36 Chapter 5 Model results MATLAB R2009a software was used to implement the model described in chapter 4. All risks factors were calculated separatelly and aggregated using the correlation matrix. The setting of parameters is presented in table 5.1: Category Parameter Description Value ns number of simulations 5000 General t length of a time step 1 month ( 1 12 Y ) α risk tolerance level 0.5% tr technical interest rate 2.4% r 0 starting value 3% a 1 mean reversion speed 0.65 Short rate a 2 reversion speed 0.1 µ long run short rate 5% σ 1 volatility p. a. - short rate 0.75 σ 2 volatility p. a. - mean reversion 0.12 Risky asset S 0 starting value 100 CZK σ ra volatility p. a. - risky asset 0.2 Bid price BP 0 starting value 100 CZK aif f annual investment fund fee 1.2% amf annual management fee 0.6% Fees, Rates BOS bid/offer spread 5% ICF rate the rate of Initial costs fee 9.5% amf Monthly administrative fee 60 CZK Bonus units ABU nom ABU % nominal monthly bonus % of IP invested in bonus fund 30 CZK 0.3% Mortality a mort mean reversion speed - mortality 0.3 σ mort volatility p. a. - mortality

37 Category Parameter Description Value Lapses Expenses a lp mean reversion speed - lapses 0.55 σ lp volatility p. a. - lapses 0.4 N LF nominal lapse fee { 400 CZK 100%, t < 2 lf iu lapse fee - initial units 4%, t [2, 3) { 2%, t 3 6%, t < 1 lf au lapse fee - accumulation units 4%, t [1, 2) 2%, t 2 a eif mean reversion speed - expenses inflation 0.10 σ eif volatility p. a. - expenses inflation 0.25 σ ae initial volatility - administrative e comm reg commission - regular premium policy 2.5% comm sin commission - single premium policy 4% oie reg other initial e. - regular premium policy 0.5% oie sin other initial e. - single premium policy 0.8% iae BE administrative e. - initial value, best est. 30 CZK if deterministic inflation - expenses 3% SA limit limit of high sum assured CZK iche mort iche hsa mort iche lp iche mat initial values of claims handling e. Table 5.1: Model parameters 3000 CZK 5000 CZK 1000 CZK 1000 CZK 5.1 Standalone risks As a reference model point a twenty years lasting policy where the policyholder is a fourty years old (entry age) male was chosen. The calculations were done for both regular and single premium policies. Policies of different death guarantees were also a case of the study. Two types of death guarantees were considered. For the first one we considered the sum assured being a constant agreed in the terms of policy. For the second one the death guarantee was simply the premium paid (untill the death occured). As a result of the combinations we have four types of insurance contracts: regular premium policy with a constant death guarantee (policy type A), regular premium policy where the death guarantee equals the premium paid (policy type B), single premium policy with a constant death guarantee (policy type C) and single premium policy where the death guarantee equals 37

38 the premium paid (policy type D). In the first two years of the contract the regular premium (policies A and B) is allocated in initial units, later in accumulation units. In case of single premium policy (policies C and D) the premium is fully allocated in accumulation units and invested at the beginning of the contract. Following table shows parameters used for policies A-D: Parameter Description Value P reg regular (monthly) premium 1500 CZK P sin single premium CZK SA sum assured (policies A and C) CZK Table 5.2: Modelpoint parameters For each type of policy we considered four different times to maturity (ttm) and calculated each standalone capital requirement for all four values of ttm. Values of ttm = 20, 18, 10, 1 were chosen. The calculations were done both - with and without the use of dynamic lapse multiplier. Tables 5.3 and 5.4 present standalone capital requirements related to a single policy. All the values are in Czech crowns (CZK). Policy A Policy B Policy C Policy D ttm : dl no dl dl no dl dl no dl dl no dl Interest rate risk Equity risk , Table 5.3: Standalone results - Market risks 38

39 ttm : Policy A Policy B Policy C Policy D Lapse risk Mortality risk Expense risk Table 5.4: Standalone results - Life and Business risks The usage of dynamic lapse multiplier only matters when talking about the market risks. In other cases there is zero shock on the bid price and the value of unit fund, hence dlm equals always one. The type of death benefit has a significant impact on the composition of risk capital. While in case of policies B and D the relative size of SCRα mortality is almost negligible, for policies with constant death guarantee (A and C) the mortality risk seems to be one of the key risk drivers especialy when talking about policies with short time to maturity. Higher value of death guarantee means higher risk premium (see section 2.1.4) and lower storage component of the premium. That is the reason why the equity risk is higher for policies B and D than for A and C, respectively. Figures 5.1, 5.2, 5.3 and 5.4 show the proportions of the standalone results SCR i α. 39

40 Figure 5.1: Standalone risks - Policy type A Figure 5.2: Standalone risks - Policy type B Despite the definition of unit linked insurance as a type of life insurance where the investment risk is bared by the policyholder, market risk is the major risk driver in most cases. Looking closer to the market risks, interest rate risk dominates over the equity risk for most parts. The shock on inter- 40

41 Figure 5.3: Standalone risks - Policy type C Figure 5.4: Standalone risks - Policy type D est rate not only influences the unit fund development through the change of the drift, but also the bank account dynamics and discounting of future cash flows. The impact of equity risk depends on the amount of premium already invested. The risk is higher for single premium policies and policies 41

42 with short time to maturity. The usage of dynamic lapse multiplier (dlm) has an impact mainly on the size of equity risk. In case of policies with long time to maturity the usage of dlm increases the equity risk as bad behaviour of the risky assets (and the bid price) lowers the value of unit fund and hence increases the lapse rates. For policies with short time to maturity (ttm = 1) higher lapse rates mean higher profit of the insurer as the lapse fees are applied while calculating the surrender value. Therefore, usage of dlm for such policies lowers the equity risk as this risk is mainly driven by the loses from death benefits and the values of management fees. While bad behaviour of the risky asset has a negative effect on the results of both death benefits and management fees, the effect of higher lapses goes the opposite way. 5.2 Overall results Correlation matrix 5.5 is used while aggregating the standalone results presented in section 5.1. Risk factor Interest rate Equity Lapses mortality Expenses Interest rate 100% 25% 50% 0% 50% Equity 25% 100% 50% 0% 50% Lapses 50% 50% 100% 0% 50% Mortality 0% 0% 0% 100% 0% Expenses 50% 50% 50% 0% 100% Table 5.5: Correlation matrix Tables 5.6 and 5.7 present the overall results. The risks is highest when the policy is in the middle of its term. Single premium policies proved to be generally riskier than regular premium policies. As a result of higher mortality risk, policies with constant death guarantee (A and C) are generally riskier than those with death guarantee of the premium paid (B and D, respectivelly). However, as far as the overall results, higher value of death guarantee lowers the equity risk in most cases and therefore lowers the effect of higher mortality risk. 42

43 Policy ttm Profits RC Profit-Solvency SCR type ratio ratio % 0.39% A % 0.53% % 1.72% % 0.42% % 0.38% B % 0.49% % 1.66% % 0.31% % 0.50% C % 0.72% % 2.43% % 0.49% % 0.50% D % 0.71% % 2.42% % 0.41% Table 5.6: Overall results - using the dynamic lapse multiplier The presented profit values represent expected one year outcome of the cash flow. Discounting is applied. There are two ratios presented in tables 5.6 and 5.7. Profit-Solvency ratio is defined as the expected one year outcome over the risk capital valid for the same policy year. The SCR ratio is defined as the risk capital over the sum of all premium payments agreed in the contract RC P tot. The required capital is highest when the policy is in the middle of its period. The value of unit fund is already relatively high and there is still a long period in which the shock will have an impact. Market shocks cause a significant change of the sum at risk (SAR) over a long period of time. In addition to that, unfavourable development of the market negatively influences the income from management fees. The more is invested in the unit fund the more significant the effect is. Finally, the same relative change in mortality rates has a different impact depending on the value of the deterministic rate. The older the policiholder, the higher actual change, hence the higher impact on the cash flows. 43

44 The usage of dlm generally increases the overall capital requirement. However, at the tail of the policy period the effect of increased lapse rates tends to offset the negative impact of a market shock. In fact, it is questionable whether the usage of dlm is appropriate in these cases. The lapse function defined in 4.10 may not be applicable to policies with short time to maturity. As there are lapse fees applied while calculating the surrender value we can expect the policyholder to rather wait till the maturity despite the unfavourable development of the unit fund. Policy ttm Profits RC Profit-Solvency SCR type ratio ratio % 0.38% A % 0.51% % 1.68% % 0.42% % 0.37% B % 0.49% % 1.62% % 0.33% % 0.48% C % 0.69% % 2.41% % 0.51% % 0.48% D % 0.69% % 2.40% % 0.43% Table 5.7: Overall results - not using the dynamic lapse multiplier Table 5.8 presents one year outcomes and remaining profits for different times to maturity. By remaining profit we mean the result of all discounted expected future cash flows related to the policy at time t = 20 ttm. Note the remaining profit is not a simple sum of the future one year outcome 44

45 values as discounting over different time periods is applied. The one year outcome and remaining profit values for all possible ttm are shown in figures C.1 and C.2, respectivelly. ttm Policy A Policy B Policy C Policy D one year outcome remaining profit Table 5.8: Profits The distribution of profits throughout the policy period depends on the type of the product. In case of the regular premium policies monthly administrative fees and risk premiums are not paid the first two years as there are no accumulation units to be sold, hence all the profit coming from these fees is realized in the third year of the policy (the year when ttm = 18 at its beginning). According to the principles presented in section 4.5 the broker agrees to pay back the full commission in case of lapse of the policy or death of the policyholder within the first year of the policy and half the commission if any of the two events occurs in the second year of the policy. At the same time, lapse fees are applied while calculating the surrender value, hence higher profit. Finally, all the profit coming from the initial costs fee in case of the regular premium policies is realized in the first two years. The insurers invests only the amount that corresponds to the value of reduced initial units. Regular premium policies proved to be more profitable and less risky than single premium ones. The reason for the higher profitability is mainly 45

46 the fact that there is the allocation in initial units the first two years of the contract. Policies with lower death guarantee proved to be more profitable. Higher risk premium lowers the value of the unit fund, hence also the size of the business. The worse risk profile of policies with higher death guarantee also lowers the profit as there is a higher cost of holding the required capital. 46

47 Chapter 6 Conclusion The aim of this paper was to identify the main risks related to the unit linked insurance. Analysis showed the market risk is a major risk factor related to this type of life insurance. Interest rate risk dominates over the equity risk in most cases. As expected, the type of death guarantee has an impact on the size of mortality risk. However, the effect of higher mortality risk of policies with higher death guarantees is partly compensated by lower equity risk. The type of premium payment changes the structure of market risks. While the interest rate risk remains the same-order when we shift from a regular to single premium policy, the equity risk is significantly higher. The time to maturity influences the structure of the risks and the amount of required capital. The risk is highest when the policy is in the middle of its period. The dynamic lapse multiplier used to simulate the dynamic behaviour of the policyholder increases the risk capital in most parts. However, the effect is opposite for policies with short time to maturity. For reasons of prudence we recommend not to use the multiplier when calculating the risk capital for such policies. 47

48 Bibliography [1] Bayer Ch.: Discretization of SDEs: Euler Methods and Beyond, Royal Institute of Technology, Stockholm, [2] Björk T.: Arbitrage Theory in Continuous Time (3rd edn), Oxford University Press Inc., New York, [3] CEIOPS: QIS4 Technical Specifications, [4] CEIOPS: QIS5 Technical Specifications, [5] Cipra T.: Pojistná matematika teorie a praxe, Ekopress, Praha, [6] Czech Insurance Association: Výroční zpráva 2009, Entree, Praha, [7] Czech statistical office: Life Tables for the Czech Republic, Areas and Regions, [8] Kochanski M.: Capital Requirement for German Unit-Linked Insurance Products, Universität Ulm, [9] Křižanová H.: Rozklad technického zisku pro investiční životní pojištění, Seminář z aktuárských věd, Praha, [10] Lozsi I.: Technical provisions, Methods and statistical techniques to calculate the best estimate, Seminář z aktuárských věd, Praha, [11] Munroe D., Odell D.: Solvency II - One-year and ultimate year risk horizons and IFRS 4, Insureware, [12] Zákon č. 37/2004 Sb.: O pojistné smlouvě a o změně souvisejících zákonů,

49 Appendix A Mortality tables A.1 Males males v = (1 9.5%) age q x l x d x C x D x M x N x S x R x

50 males v = (1 9.5%) age q x l x d x C x D x M x N x S x R x Table A.1: Mortality tables - males 50

51 A.2 Females females v = (1 9.5%) age q y l y d y Cx Dx Mx Nx Sx Rx 0 0, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

52 males v = (1 9.5%) age q y l y d y C y D y M y N y S y R y 61 0, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table A.2: Mortality tables - females 52

53 Appendix B Deterministic lapse rates policy year s t % % % % % Table B.1: Deterministic lapse rates 53

54 Appendix C Profits C.1 One year outcome Figure C.1: Profits - One year outcome 54

55 C.2 Remaining profit Figure C.2: Profits - Remaining profit 55