CHAPTER 12 Actuarial Funding of Dismissal and Resignation Risks Werner Hürlimann CONTENTS 12.1 I ntroduction 12.2 Dismissal and Resignation Causes of Decrement 12.2.1 Dismissal by the Employer 12.2.2 Resignation by the Employee 12.2.3 Death of the Employee 12.2.4 Survival to the Retirement Age 12.3 Asset and Liability Model for Dismissal Funding 12.4 Dynamic Stochastic Evolution of the Dismissal Fund Random Wealth 12.5 The Probability of Insolvency: A Numerical Example Acknowledgments 28 References 286 268 269 271 272 273 274 276 279 280 B esides t he u sua l p ension benefits, the pension plan of a firm may be forced by law in some countries to offer wage-based lump sum payments due to death, retirement, or dismissal by the employer, but no payment is made by the employer when the employee resigns An actuarial risk model for funding severance payment liabilities is formulated and studied The yearly aggregate lump sum payments are supposed to follow a classical collective model of risk theory with compound distributions The final wealth at an arbitrary time is described explicitly including formulas 267 © 2010 by Taylor and Francis Group, LLC 268 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling for the mean and the variance Annual initial-level premiums required for “dismissal f unding” a re de termined a nd u seful g amma approximations for c onfidence i ntervals o f t he w ealth a re p roposed A spec ific numerical example illustrates the nonnegligible probability of default in case the employee structure of a “dismissal plan” is not well balanced Keywords: Asset and liability management (ALM), solvency, actuarial funding, dismissal risk, resignation risk, compound distributions 12.1 INTRODUCTION In some countries, for example, Austria, modern social legislation stipulates besides usual p ension benefits, fixed wage-based lump sum pa yments by death and retirement as well as through dismissal by the employer of a firm, so-called severance payments (see, e.g., “Abfertigung neu” 2002, Holzmann et al 2003, “Abfertigung neu und alt” 2005, Koman et al 2005, Grund 2006, “Abfindung im Arb eitsrecht” 2007) H owever, if t he co ntract t erminates due to resignation by the employee, no lump sum payment is made by the employer In this situation, there are four causes of decrement, which have a random effect on the actuarial funding of the additional liabilities in the pension plan, referred to in this chapter as the “dismissal plan.” We are interested in actuarial risk models that are able to describe all random lump sum payments until retirement for the dismissal plan of a firm The aggregate lump sum payments in each year are supposed to follow a classical collective model of the risk theory with compound distributions The evaluation of the mean and standard deviation of these yearly payments requires a separate analysis of the four causes of decrement See Section 12.2 for further details Actuarial funding with dismissal payments is based on the dynamic stochastic evolution of the random wealth of the dismissal fund at a spec ific time The final wealth at the end of a time horizon can be described explicitly, and formulas for the mean and the variance are obtained In particular, given the initial capital of the dismissal fund as well as the funding capital, which should be available at the end of a time horizon to cover all expected future random lump sum payments until the retirement of all employees, it is possible to determine the required annual initial level premium necessary for dismissal funding This is described in Section 12.3 Section 12.4 considers the dynamic stochastic evolution of the random wealth at an arbitrary time and proposes a useful gamma approximation for confidence intervals of the wealth © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 269 Section 12.5 is devoted to the analysis of a specific numerical example, which illustrates the nonnegligible probability of insolvency of a dismissal fund in case the employee structure is not well balanced 12.2 DISMISSAL AND RESIGNATION CAUSES OF DECREMENT Consider the “dismissal plan” of a firm, which offers wage-based lump sum payments by death and retirement as well as through dismissal by the em ployer H owever, i f t he co ntract ter minates d ue t o r esignation by t he em ployee, n o l ump su m pa yment i s made b y t he em ployer I n this situation, there are four causes of decrement, which have a random effect on the dismissal funding They are described as follows: • Dismissal by the employer with a probability PDx at age x • Resignation by the employee with a probability PR x at age x • Death of the employee with a probability PTx at age x • Survival to the deterministic retirement age s with a probability PSsx at age x Survival to retirement age s of an employee aged x happens if the employee does not die and there is neither dismissal by the employer nor resignation by the employee The probability of this event depends on the probabilities that an employee aged x survives to age x + k, namely, k −1 k PS x = ∏ (1 − PD x + j − PR x + j − PTx + j ), j=0 PS x = 1, (12.1) and equals PSsx = s −x PS x Note t hat i f a n employee attains t he common retirement age s, then retirement payment due to survival takes place and neither d ismissal, resignation, nor death is possible Therefore, it can be assumed that PDx = PR x = PTx = for all x ≥ s We consider an actuarial risk model, which describes all random lump sum pa yments u ntil r etirement f or t he d ismissal p lan o f a firm with M employees at the initial time of valuation t = F or a l onger time horizon H, say 25 or 30 years, and for an initial capital K0, let P be the annual initial level premium of the dismissal fund required to reach at fixed interest rate i the funding capital KH at time H The latter quantity is supposed to cover © 2010 by Taylor and Francis Group, LLC 270 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling at time H all expected future random payments until the retirement of all employees [see formula (12.37)] The considered (overall) funding premium should not be confused with the individual contributions of the employees for t heir ben efits, wh ich ma y va ry be tween em ployees S ince l ump su m payments a re p roportional t o t he wa ges o f t he em ployees, i t i s a ssumed that the annual premium increases proportionally to the wages With an annual wage increase of 100 · g%, the annual premium at time t reads Pt = P ⋅ (1 + g )t −1 , t = 1,…, H (12.2) Let Xt be a t ime-dependent random variable, which represents the aggregate lump sum payments in year t due to the above four causes of decrement We assume that this random variable can be described by a random sum of the type Nt Xt = ∑ Yt , j , t = 1,…, H , j =1 (12.3) where Nt co unts t he number o f em ployee w ithdrawals d ue t o a ny ca use o f decrement Yt,j is t he individual r andom lump s um pa yment g iven t he jth wi thdrawal occurs Under the assumption of a collective model of risk theory, the Yt,j are independent a nd i dentically d istributed l ike a r andom va riable Yt, a nd t hey are i ndependent f rom Nt As shown i n Hürlimann (2007) it is a lso possible to model in a simple way a continuous range of positive dependence between independence (the present model) and comonotone dependence Assuming that Nt has a mean λt = E[Nt] and a standard deviation σ Nt , the mean µ Xt and the standard deviation σ Xt of Xt are given by (e.g., Beard et al 1984, Chapter 3, Bowers et al 1986, Chapter 11, Panjer and Willmot 1992, Chapter 6, Kaas et al 2001, Chapter 3) µ Xt = λ t ⋅µYt , σ Xt = λ t ⋅σY2t + σ N2 t ⋅ µYt , (12.4) where µYt and σYt denote the mean and the standard deviation of Yt The evaluation of these quantities requires a separate analysis for each of the four causes of decrement © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 271 12.2.1 Dismissal by the Employer Let N tD be the random number of dismissals in year t, and let YtD, j ~ YtD be the independent and identically distributed individual random lump sum payments in year t given the jth dismissal by the employer occurs If Ak is the age of the employee number k at the initial time of valuation, then the expected number of dismissals in year t equals M λ tD = E ⎡⎣ N tD ⎤⎦ = ∑ t −1 PS Ak ⋅ PD Ak +t −1 k =1 (12.5) Consider the probability of dismissal of an employer in year t given a population of M employees at initial time defined by the ratio ptD = λ tD M (12.6) Since decrement by the cause of dismissal follows a binomial distribution with parameter ptD , the variance of the number of dismissals is given by σ2NtD = M ⋅ ptD ⋅ (1 − ptD ) = λ tD ⋅ ( M − λ tD ) M (12.7) Furthermore, suppose that at the initial time of valuation, it is known that by d ismissal the kth em ployee w ill r eceive t he l ump su m pa yment B0,k Since the lump sum payment is wage based and the wages increase at the rate of 100 · g%, the effective lump sum payment in year t equals B0,k (1 + g)t−1 U nder t he a ssumption o f a co mpound d istributed m odel f or the aggregate lump sum payments due to dismissal by the employer, that ND t is XtD = ∑ j =1 YtD,j , t = 1,…, H , i t f ollows t hat t he m ean a nd t he va riance of YtD are given by µtD = E[YtD ] = ⋅ E[ XtD ], λ tD [σtD ]2 = Var[YtD ] = D ⋅(Var[XtD ] − σ2NtD ⋅[µtD ]2 ) , λt (12.8) where the mean and the variance of the aggregate lump sum payments are obtained from © 2010 by Taylor and Francis Group, LLC 272 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling M E[ XtD ] = ∑ t −1 PS Ak ⋅ PD Ak + t −1 ⋅ B0,k ⋅ (1 + g )t −1 , k =1 (12.9) M Var[XtD ] = ∑ t −1 PS Ak ⋅ PD Ak +t −1 ⋅[ B0,k ⋅ (1 + g )t −1 ]2 − E[ XtD ]2 k =1 12.2.2 Resignation by the Employee The e valuation i s s imilar t o t he s ituation o f d ismissal b y t he em ployer, with th e d ifference t hat t he f oreseen l ump su m pa yment i s r eleased to the remaining beneficiaries of the dismissal fund Let N tR be the random number of resignations in year t, and let YtR, j ~ YtR be the independent and identically d istributed i ndividual r andom l ump su m pa yments i n y ear t g iven t he kth resignation by t he employee occ urs Again we a ssume a compound distributed model for the aggregate lump sum payments due NR to r esignation b y t he em ployee, t hat i s, XtR = ∑ j =t1 YtR,j , t = 1,…, H The expected number of resignations in year t equals M λ tR = ∑ t −1 PS Ak ⋅ PR Ak +t −1 k =1 (12.10) The probability of resignation of an employer in year t given a population of M employees at the initial time is defined by the ratio ptR = λ tR M (12.11) Since decrement by the cause of resignation follows a binomial distribution with parameter ptR , the variance of the number of resignations is given by σ2NtR = M ⋅ ptR ⋅ (1 − ptR ) = λ tR ⋅ ( M − λ tR ) M (12.12) The mean and the variance of YtR are given by ⋅ E[ XtR ], λ tR [σtR ]2 = Var[YtR ] = R ⋅ (Var[ XtR ] − σ2NtR ⋅[µ tR ]2 ), λt µ tR = E[YtR ] = © 2010 by Taylor and Francis Group, LLC (12.13) Actuarial Funding of Dismissal and Resignation Risks ◾ 273 where the mean and the variance of the aggregate lump sum payments are obtained from M E [ XtR ] = ∑ t −1 PS Ak ⋅ PR Ak + t −1 ⋅ B0,k ⋅ (1 + g )t −1 , k =1 M Var[ XtR ] = ∑ t −1 PS Ak ⋅ PR Ak +t −1 ⋅[B0,k ⋅ (1 + g )t −1 ]2 − E[ XtR ]2 (12.14) k =1 12.2.3 Death of the Employee Suppose that by death of an employee the portion θ of the dismissal payment is due to its legal survivor ( θ = 1/2 in our numerical example) Let N tT be the random number of deaths in year t, a nd let Yt T, j ~ Yt T be t he independent a nd i dentically d istributed i ndividual r andom l ump su m payments i n year t g iven t he jth de ath occ urs We a ssume a co mpound distributed model for the aggregate lump sum payments due to the death NT of an employee, t hat is, XtT = ∑ j =t1YtT,j , t = 1,…, H The expected number of deaths in year t equals M λ tT = ∑ t −1 PS Ak ⋅ PTAk + t −1 k =1 (12.15) The probability of the death of an employer in year t given a population of M employees at initial time is defined by the ratio ptT = λ tT M (12.16) Since decrement by the cause of death follows a binomial distribution with parameter ptT , the variance of the number of deaths is given by σ2NtT = M ⋅ ptT ⋅ (1 − ptT ) = λ tT ⋅ ( M − λ tT ) M (12.17) The mean and the variance of YtT are given by ⋅ E[ XtT ], T λt [σtT ]2 = Var[YtT ] = T ⋅ (Var[ XtT ] − σ2NtT ⋅[µtT ]2 ) , λt µtT = E[YtT ] = © 2010 by Taylor and Francis Group, LLC (12.18) 274 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling where the mean and the variance of the aggregate lump sum payments are obtained from M E[ XtT ] = ∑ t −1 PS Ak ⋅ PTAk +t −1 ⋅θ⋅ B0,k ⋅ (1 + g )t −1 , k =1 M Var[ XtT ] = ∑ t −1 PS Ak ⋅ PTAk + t −1 ⋅[θ⋅ B0, k ⋅ (1 + g )t −1 ] − E[ XtT ]2 (12.19) k =1 12.2.4 Survival to the Retirement Age Let NtS be the random number of retirements in year t, and let YtS, j ~ YtS be the independent a nd identically d istributed i ndividual random lump su m payments generated upon retirement of the jth employee in year t Taking into account that an employee numbered k and aged Ak attains retirement in year t such that Ak + t − = s and using the definition of the retirement probability PSsx = s −x PS x , one obtains for the expected number of retirements in year t M λ tS = ∑ t −1 PS Ak ⋅ I ( Ak + t − = s), k =1 (12.20) where I(·) is an indicator function such that I(W) = if the statement W is true and I(W) = else The probability of survival to the retirement age of an employer in year t given a population of M employees at initial time is defined by the ratio ptS = λ tS M (12.21) Since decrement by the cause of survival to retirement follows a binomial distribution with parameter p tS , the variance of the number of retirements is given by σ 2NtS = M ⋅ ptS ⋅ (1 − ptS ) = λ tS ⋅ ( M − λ tS ) M (12.22) Furthermore, suppose that at the initial time of valuation, it is known that at re tirement t he kth em ployee w ill r eceive t he l ump su m pa yment Ck Due to wages increase, the effective sum in year t equals Ck(1 + g)t−1 Again, assume a co mpound distributed model for the aggregate lump sum payNS ments due to retirement, that is, XtS = ∑ j =t1YtS,j , t = 1,…, H The mean and the variance of YtS are given by © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 275 ⋅ E[ XtS ], S λt S [σt ] = Var[YtS ] = S ⋅ (Var[XtS ] − σ 2NtS ⋅[µ tS ]2 ), λt µ tS = E[YtS ] = (12.23) where the mean and the variance of the aggregate lump sum payments are obtained from M E[ XtS ] = ∑ t −1 PS Ak ⋅ I ( Ak + t − = s) ⋅ Ck ⋅ (1 + g )t −1 , k =1 M Var[X ] = ∑ t −1 PS Ak ⋅ I ( Ak + t − = s) ⋅[Ck ⋅ (1 + g ) ] − E[ X ] S t t −1 k =1 (12.24) S t The abo ve p reliminaries a re u sed t o o btain t he cha racteristics (12.4) a s follows The expected number of employee withdrawals in year t due to all four causes of decrement equals λ t = λ tD + λ tR + λ tT + λ tS (12.25) Denote by Mt the number of remaining employees in year t Starting with an initial number M of employees, one has M0 = M and for year t > one has Mt = Mt−1 – λt, wh ich shows t hat t he ex pected number of remaining employees dec reases o ver t ime, a s sh ould be The i ndividual l ump su m payment in year t satisfies the following equation: λ t ⋅Yt = λ tD ⋅YtD − λ tR ⋅YtR + λ tT ⋅YtT + λ tS ⋅YtS (1 2.26) Indeed, t he a ggregate lump su m payments i n y ear t a re t he su m of t he payments due to dismissal by the employee, death, and retirement less the payments due to t he resignation of employees Under t he assumption of independence of the different random variables, one obtains for the mean and the variance of Yt the formulas D D (λ t ⋅µt − λ tR ⋅µtR + λ tT ⋅µtT + λ tS ⋅µtS ), λt σYt = Var[Yt ] = (σtD )2 + (σtR )2 + (σtT )2 + (σtS )2 µYt = E[Yt ] = (12.27) Moreover, the variance of the number of withdrawals in year t due to all four causes of decrement is given by © 2010 by Taylor and Francis Group, LLC 276 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling σ2Nt = Var[N t ] = σ2NtD + σ2NtR + σ2NtT + σ2NtS (1 2.28) The cha racteristics (12.4) f ollow i mmediately b y i nserting t he f ormulas (12.25), (12.27), and (12.28) 12.3 ASSET AND LIABILITY MODEL FOR DISMISSAL FUNDING Let Wt be the random wealth of the dismissal fund at time t, where t = is the initial time of valuation The random rate of return on investment in year t is denoted It The wealth at time t satisfies the following recursive equation: Wt = (Wt −1 + Pt − Xt ) ⋅ (1 + It ) (12.29) Taking into account (12.1), the final wealth at the time horizon H is given by H H H t =1 t =1 j =t WH = W0 ⋅ ∏ (1 + It ) + ∑ {P (1 + g )t −1 − Xt } ⋅ ∏ (1 + I j ) (12.30) It is clear that the initial wealth coincides with the initial capital, that is, W0 = K0 For simplicity, assume that accumulated rates of return in year t are independent and identically log-normally distributed such that + It = exp(Zt ), (12.31) where Zt is normally distributed with mean µ and standard deviation σ Consider the products H ∏ (1 + I j ) = exp(Zt, H ), j=t ≤ t ≤ H, (12.32) which r epresent t he acc umulated r ates o f r eturn o ver t he t ime per iod H [t−1,H], wh ere t he su ms Zt, H = ∑ j = t Z j a re n ormally d istributed w ith mean and standard deviation µ t , H = E[Zt , H ] = (H − t + 1) ⋅µ, σt , H = Var[Zt, H ] = H − t + ⋅σ (12.33) The mean and the variance of the final wealth are given by the following result © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 277 THEOREM 12.1 Under the simplifying assumption that the random rates of return I1,…, IH are independent from the aggregate lump sum payments Xt, the mean of the final wealth is given by the expression r H − (1 + g )H H − ∑ µ Xt ⋅ r H −t +1 r − (1 + g ) t =1 E[WH ] = K ⋅ r H + P ⋅ r ⋅ (12.34) and the variance of the final wealth by the formula H Var ⎡⎣WH ⎤⎦ = K 02 ⋅ r H ⋅ (e H σ − 1) + K ⋅ ∑ r H −t +1 ⋅ (P(1 + g )t −1 − µ Xt ) ⋅ (e (H − t +1)σ − 1) 2 t =1 H H − t +1) + ∑r ( ⋅ t =1 +2⋅ ∑ {(P(1 + g ) t −1 − µ Xt )2 ⋅ (e( H − t +1)σ2 − 1) + σ2Xt ⋅ e( H − t +1)σ r H − t − s + ⋅ (P (1 + g )s −1 − µ Xs ) ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ (e( } H − t +1)σ 1≤ s < t ≤ H − 1), (12.35) where r = exp(µ + 12 σ2 ) is the one-year risk-free accumulated rate of return over the time horizon [0,H] Proof Using the notation (12.32) the expression (12.30) can be rewritten as H WH = W0 ⋅ exp(Z1, H ) + ∑ {P (1 + g )t −1 − Xt } ⋅ exp(Zt ,H ), (12.36) t =1 from which one gets without difficulty (12.34) To get t he expression for the variance, several terms must be calculated One has 2 Var[W0 ⋅ exp(Z1, H )] = K 02 ⋅ (e H (µ+σ ) − e H (2µ+σ ) ) = K 02 ⋅ r H ⋅ (e H σ − 1) © 2010 by Taylor and Francis Group, LLC 278 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling For ≤ t ≤ H one has Var ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )⎤⎦ = Var ⎡ E ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H ) Zt, H ⎤⎦ ⎤ ⎣ ⎦ + E ⎡ Var ⎡⎣(P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H ) Zt, H ⎤⎦ ⎤ ⎣ ⎦ = (P (1 + g )t −1 − µ Xt )2 ⋅ Var [exp(Zt , H )] + σ2Xt ⋅ E [exp(2 ⋅ Zt , H )] = (P (1 + g )t −1 − µ Xt )2 ⋅ (e 2( H −t +1)(µ+σ ) − e ( H −t +1)(2µ+σ 2 ) ) + σ2X t ⋅ e 2(H −t +1)(µ+σ ) t −1 = r 2(H −t +1) ⋅{(P (1 + g ) − µ Xt )2 ⋅ (e( H −t +1)σ − 1) + σ2Xt ⋅ e( H −t +1)σ } 2 For ≤ s < t ≤ H one has Cov ⎡⎣(P (1 + g )s −1 − X s ) ⋅ exp(Z s , H ), (P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )⎤⎦ = (P (1 + g ) s −1 − µ X s ) ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ Cov [exp(Zs ,H ),exp(Zt, H )] , where for the covariance term one gets Cov[ exp(Z s , H ),exp(Zt , H )] = Cov ⎡⎣ E ⎡⎣exp(Z s , H ) Z s , H − Zt, H ⎤⎦ , E ⎡⎣exp(Zt, H ) Z s , H − Zt, H ⎤⎦ ⎤⎦ + E ⎡⎣Cov ⎡⎣exp(Z s , H ),exp(Zt, H ) Z s , H − Zt, H ⎤⎦ ⎤⎦ = E [exp(Z s , H − Zt, H )]⋅ Var [exp(Zt , H )] =e (t − s )(µ+ σ ) 2 ⋅ (e 2( H −t +1)(µ+σ ) − e ( H −t +1)(2µ+σ ) ) = r H −t − s + ⋅ (e( H −t +1)σ − 1) In a similar way, for ≤ t ≤ H one has Cov [W0 ⋅ exp(Z1, H ), (P (1 + g )t −1 − Xt ) ⋅ exp(Zt, H )] = K ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ Cov [exp(Z1, H ),exp(Zt , H )] = K ⋅ (P (1 + g )t −1 − µ Xt ) ⋅ r H −t +1 ⋅ (e ( H −t +1)σ − 1) © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 279 Gathering a ll ter ms t ogether a nd su mming a ppropriately o ne o btains finally (12.35) Note t hat t he r isk-free r ate of re turn r must be realized in order to guarantee w ith c ertainty t he ex pected fi nal wealth We a re n ow i nterested in the determination of the required premium for dismissal funding with dismissal payments Let K0 be the initial capital of the dismissal fund Suppose that at time H the funding capital KH should be available in order to cover all expected future random lump sum payments until the retirement of all employees If Hmax denotes the maximum time horizon at which all employees from the initial population of M employees have be en r etired w ith c ertainty, t hen t he r equired f unding c apital i s given by KH = H max ∑ rH t =H max −t ⋅µ Xt , (12.37) where r is a fi xed one-year guaranteed accumulated rate of return Setting this quantity equal to the expected final wealth, that is, E[WH] = KH, one sees that by fixed r and with the formula (12.34) this equation can be solved for the required annual initial level premium P 12.4 DYNAMIC STOCHASTIC EVOLUTION OF THE DISMISSAL FUND RANDOM WEALTH The dynamic stochastic evolution of the random wealth at time t is determined by the recursive equation (12.29) Similar to (12.30), one obtains the explicit expression t t t j =1 j =1 k= j Wt = W0 ⋅ ∏ (1 + I j ) + ∑ {P (1 + g ) j −1 − Xt } ⋅ ∏ (1 + I k ) (12.38) Applying the same approach as in the proof of Theorem 12.1, one sees that th e m ean an d th e v ariance o f th e r andom w ealth a t t ime t ar e given by E [W ] = W0 ⋅ r t + P ⋅ r ⋅ © 2010 by Taylor and Francis Group, LLC r t − (1 + g )t H − ∑ µ X ⋅ r H − t +1 , r − (1 + g ) t =1 t (12.39) 280 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling t Var[Wt ] = K 02 ⋅ r 2t ⋅ (et σ − 1) + K ⋅ ∑ r 2t − j +1 ⋅ (P (1 + g ) j −1 − µ X j ) ⋅ (e(t − j +1)σ − 1) 2 j =1 t + ∑ r 2(t − j +1) ⋅ {(P (1 + g ) j −1 − µ X j )2 ⋅ (e(t − j +1)σ − 1) + σ2X j ⋅ e(t − j +1)σ j =1 + 2⋅ ∑ 1≤i < j ≤ t } r 2t −i − j + ⋅ (P(1 + g )i −1 − µ Xi ) ⋅ (P(1 + g ) j −1 − µ X j ) ⋅ (e(t − j +1)σ − 1) (12.40) Let k[Wt] be t he coefficient of va riation of t he wealth at t ime t To e stimate a quantile of the random wealth at time t, we suppose that the wealth is a pproximately g amma d istributed This i s a p ractical a pproximation under the reasonable assumptions of gamma-distributed aggregate lump sum pa yments a nd i ndependent i dentically l og-normally d istributed accumulated rates of return Then, the α -quantile of the wealth at time t, QW−1t (α) = inf x : P (Wt ≥ x ) ≥ α , is { } ⎛ ⎞ ⋅ k[Wt ]2 ⋅ E[Wt ] , QW−1t (α) = Γ α−1 ⎜ 2⎟ ⎝ k[Wt ] ⎠ (12.41) where Γ α−1 (β) is the α -quantile of a standard gamma distribution Γ(β,1) The α-confidence interval of the wealth at time t contains all possible realizations of the wealth in the interval ⎡⎣QW−1t (1 − α), QW−1t (α)⎤⎦ 12.5 THE PROBABILITY OF INSOLVENCY: A NUMERICAL EXAMPLE The features of the present approach will be illustrated at a concrete example, which is not based on a real-life firm The considered situation is chosen to exemplify what could happen in case a d ismissal fund is not well balanced in its employee structure Suppose that the employee structure of the dismissal fund by age, term of ser vice, a nd wage is g iven as in Table 12.1 Each age class is assumed to be represented by 50 employees, of which half is male and half female Therefore the dismissal fund has a total of M = 1000 employees The total wages is equal to 61,500,000 The wage-based lump sum payment is evaluated u sing Table 2.2 u nder t he a ssumption t hat t he wa ges i ncrease b y © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 281 TABLE 12.1 Employee Structure by Age, Term of Service, and Wage Age Term of Service 20 30 30 30 35 35 35 35 40 40 40 40 50 50 50 50 60 60 65 65 Wage 10 15 10 15 10 20 15 10 25 20 30 25 30,000 50,000 40,000 30,000 60,000 50,000 40,000 30,000 70,000 60,000 50,000 40,000 90,000 80,000 70,000 60,000 100,000 90,000 100,000 90,000 TABLE 12.2 Term of Service and Lump Sum Payment Term of Service (in Years) 10 15 20 25 © 2010 by Taylor and Francis Group, LLC Number of Monthly Wage Payments 12 282 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 12.3 TABLE 12.4 Probabilities of Decrement in % Age x PDx PRx PTxm PTxf 20 35 50 65 0.5 2.5 0.2 0.15 0.5 0.05 0.1 0.2 Development of Lump Sum Payments Year Expected Lump Sum Payments Expected Future Lump Sum Payments 10 11 12 13 14 15 16 17 18 19 20 21 22 23 9,429,819 −52,479 −33,653 −31,354 −9,584 10,889,573 −105,101 −85,453 −64,532 −42,279 −41,507 −8,093 27,364 64,955 104,777 22,131,057 −42,264 −12,823 18,422 51,553 90,193 132,906 178,102 50,964,483 43,196,050 44,978,470 46,812,608 48,717,721 50,676,397 41,378,297 43,142,734 44,957,314 46,822,721 48,739,600 50,732,351 52,770,062 54,852,406 56,978,949 59,149,139 38,498,805 40,082,712 41,699,357 43,348,172 45,028,484 46,735,823 48,467,033 Year Expected Lump Sum Payments Expected Future Lump Sum Payments 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 225,889 276,384 23,142,266 143,521 178,203 214,836 253,508 21,921,214 93,753 112,660 132,617 153,670 16,890,209 10,869 14,926 19,217 23,753 28,543 33,599 38,933 44,556 50,482 50,220,489 51,994,383 53,786,720 31,870,232 32,995,779 34,130,279 35,272,061 36,419,295 15,078,004 15,583,621 16,089,800 16,595,471 17,099,473 217,634 215,036 208,114 196,453 179,608 157,108 128,449 93,097 50,482 Note: The required premium to fund the future payments is e valuated for a time ho rizon between H = 25 and 30 years The initial capital is set at K0 = 10 million to avoid insolvency in t he first year because the expected lump sum pa yments for this period are 9.43 million according to Table 12.4 The funding capital at time H is set at KH = 35.272 million, which corresponds to the expected future lump sum payments at time H = 30 Table 12.5 disp lays t he s ensitivity o f t he required p remium dep ending o n t he time horizon and the variation of the expected aggregate lump sum pa yments (mean ± multiple of the standard deviation) In these tables, µt, σt stand for µ Xt , σ Xt © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 283 100 · g = 3% per y ear The u sed probabilities of w ithdrawal for t he four causes of decrement are summarized in Table 12.3, where linear interpolation is applied for ages between two values These probabilities are only rough values, but correspond qualitatively to real-life data A d istinction is made between male and female probabilities of death Following the formulas of Section 12.2, it is now possible to calculate the expected aggregate lump sum payments for an arbitrary year t (formulas (12.3), (12.19), and (12.22)) as well as the expected aggregate future lump sum payments (formula (12.37) for an arbitrary time horizon H ≤ Hmax = 45) The a ssumed g uaranteed r ate o f r eturn i s se t a t % The obtained results are summarized in Table 12.4 One notes that until every employee attains w ith certainty t he retirement age of 65 years, t here is expected a total of 75 dismissals by the employer, 170 resignations by the employee, 121 deaths, and t he remaining 634 employees are expected to attain t he retirement age The dynamic stochastic development of the random wealth is displayed in Table 12.5 The calculation is done with a volatility σ = 2% and a logarithmic r ate o f r eturn µ = ln(1.04) − 12 σ2 = 3.902% The skew em ployee structure implies a % probability of default in t he 6t h year, a nd a s ituation close to i nsolvency i n t he first a nd 16th year w ith a p robability of 1% Also, there is a nonnegligible probability that the overall goal at time H = 30 will not be attained Traditionally, t he l ife i nsurance sec tor s se t a nnual p remiums a t a constant level It i s i nteresting to compare t his situation w ith t he above one To calculations one has to replace the wage increase factor + g by a fac tor of one in the relevant formulas Tables 12.5 and 12.6 are then being replaced by Tables 12.7 and 12.8 TABLE 12.5 Sensitivity of the Required Premium H mt – 2st mt – st mt mt + st mt + 2st 25 26 27 28 29 30 1,129,202 1,374,935 1,309,253 1,249,018 1,193,612 1,142,511 1,326,777 1,600,598 1,528,853 1,463,125 1,402,754 1,347,173 1,524,352 1,826,262 1,748,452 1,677,231 1,611,895 1,551,836 1,721,926 2,051,925 1,968,052 1,891,338 1,821,037 1,756,499 1,919,501 2,277,589 2,187,652 2,105,444 2,030,178 1,961,161 © 2010 by Taylor and Francis Group, LLC 284 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling TABLE 12.6 Mean Coeffici ent of Variation 2,206,897 4,012,079 5,919,757 7,952,719 10,097,264 1,046,963 3,125,238 5,324,024 7,648,551 10,104,250 12,720,547 15,471,813 18,363,276 21,400,334 24,588,562 5,070,224 7,906,838 10,903,994 14,068,569 17,407,696 20,925,103 24,626,232 28,518,472 32,609,476 36,907,161 17,694,659 21,733,726 26,002,703 30,511,889 35,272,000 0.530 0.307 0.219 0.171 0.142 1.941 0.684 0.422 0.309 0.246 0.205 0.177 0.157 0.141 0.129 0.784 0.529 0.404 0.329 0.280 0.245 0.219 0.199 0.183 0.170 0.393 0.337 0.296 0.265 0.241 Time 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Dynamic Stochastic Development of the Random Wealth TABLE 12.7 95% Confidence Interval 692,822 2,223,832 3,961,998 5,851,223 7,859,784 34 602,880 2,239,834 4,220,898 6,397,179 8,755,150 11,255,590 13,896,046 16,677,694 19,603,453 677,515 2,487,624 4,803,000 7,409,025 10,239,056 13,266,041 16,479,967 19,879,381 23,466,319 27,244,777 7,997,966 11,251,124 14,769,885 18,539,418 22,555,892 4,413,146 6,227,286 8,198,931 10,319,397 12,566,069 4,976,769 7,257,567 9,474,570 11,900,489 14,501,908 17,292,620 20,239,259 23,343,486 26,608,985 30,040,568 12,852,005 15,799,055 19,006,586 22,451,506 26,120,528 30,006,842 34,110,296 38,435,301 42,988,043 47,775,880 30,476,084 34,999,255 39,809,538 44,909,763 50,306,448 99% Confidence Interval 399,041 1,710,377 3,327,517 5,128,543 7,063,043 264,374 1,503,764 3,239,545 5,237,385 7,447,177 9,815,828 12,332,805 14,994,805 17,802,014 239,218 1,434,460 3,300,388 5,560,046 8,099,782 10,867,754 13,840,080 17,006,984 20,365,200 23,915,138 5,567,451 8,373,491 11,492,782 14,891,006 18,552,088 5,785,244 7,414,207 9,341,901 11,462,095 13,729,412 9,859,078 10,073,121 11,883,455 14,181,928 16,753,370 19,560,732 22,551,984 25,719,470 29,062,113 32,582,106 18,457,196 20,705,055 23,652,256 27,016,366 30,698,009 34,654,113 38,866,622 43,330,197 48,045,416 53,016,289 37,753,115 42,261,068 47,148,505 52,389,783 57,975,579 Sensitivity of the Required Premium H mt − 2st mt − st mt mt + s t mt + 2st 25 26 27 28 29 30 1,129,202 1,374,935 1,309,253 1,249,018 1,193,612 1,142,511 1,326,777 1,600,598 1,528,853 1,463,125 1,402,754 1,347,173 1,524,352 1,826,262 1,748,452 1,677,231 1,611,895 1,551,836 1,721,926 2,051,925 1,968,052 1,891,338 1,821,037 1,756,499 1,919,501 2,277,589 2,187,652 2,105,444 2,030,178 1,961,161 © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 285 TABLE 12.8 Dynamic Stochastic Development of the Random Wealth for a Level Premium Time 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Mean Coeffici ent of Variation 2,941,511 5,462,274 8,064,287 10,767,991 13,557,202 5,122,858 7,785,600 10,534,419 13,371,433 16,298,785 19,342,428 22,473,065 25,692,053 29,000,706 32,400,289 13,028,525 15,942,145 18,941,690 22,028,723 25,204,781 28,467,695 31,816,704 35,252,670 38,776,376 42,388,516 22,364,624 25,458,471 28,640,002 31,910,696 35,272,000 0.397 0.225 0.160 0.126 0.105 0.395 0.273 0.212 0.176 0.152 0.134 0.121 0.112 0.104 0.098 0.304 0.261 0.231 0.209 0.192 0.179 0.169 0.160 0.153 0.147 0.310 0.286 0.267 0.252 0.240 95% Confidence Interval 1,318,252 3,612,278 6,065,441 8,637,159 11,297,847 2,303,596 4,644,543 7,143,772 9,753,547 12,459,791 15,280,195 18,183,285 21,168,118 24,234,430 27,382,281 7,270,643 9,762,520 12,352,045 15,029,295 17,789,267 20,625,684 23,534,829 26,515,276 29,565,925 32,685,827 12,311,133 14,772,358 17,311,308 19,923,914 22,607,494 5,086,515 7,624,833 10,297,587 13,093,124 15,987,332 8,844,611 11,585,772 14,463,928 17,458,432 20,562,819 23,800,275 27,138,816 30,578,994 34,121,858 37,768,730 20,145,833 23,354,574 26,681,485 30,123,576 33,680,084 37,347,836 41,125,765 45,014,849 49,016,294 53,131,449 34,845,671 38,503,688 42,287,094 46,196,695 50,233,935 99% Confidence Interval 913,730 3,017,245 5,368,605 7,864,281 10,458,998 1,599,411 3,698,604 6,035,767 8,516,055 11,107,927 13,820,489 16,619,029 19,500,096 22,461,838 25,503,221 5,610,061 7,869,927 10,251,288 12,733,256 15,304,421 17,954,575 20,677,303 23,469,223 26,327,767 29,250,836 9,437,490 11,611,309 13,870,655 16,206,918 18,614,135 6,310,620 8,714,272 11,365,858 14,174,014 17,096,979 10,966,572 13,572,662 16,424,579 19,438,873 22,588,170 25,887,037 29,299,116 32,822,585 36,457,189 40,203,525 23,950,965 27,194,971 30,601,159 34,153,059 37,842,703 41,662,938 45,610,332 49,684,428 53,885,625 58,214,874 41,547,960 45,388,811 49,390,007 53,547,485 57,859,685 ACKNOWLEDGMENTS I a m deeply g rateful to t he referees of t he fi rst International Actuarial Association L ife C olloquium f or t heir de tailed co mments a nd h elpful suggestions on a first version of this contribution Special thanks go to B Sundt for corrections in formula (12.35) © 2010 by Taylor and Francis Group, LLC 286 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling REFERENCES “Abfertigung neu” 2002 On-line a 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Proceedings of the th E USFLAT C onference, O strava, Czec h Rep ublic, pp 205–212 Kaas, R., Goovaerts, M., Dhaene, J., and M Denuit 2001 Modern Actuarial Risk Theor y Kluwer Academic Publishers, Dordrecht, the Netherlands Koman, R , U lrich S chuh, U., and A W eber 2005 The Austrian S everance Pay Reform: Toward a Funded Pension Pillar Empirica 32(3–4), 255–274 Panjer, H.H and G.E Willmot 1992 Insurance Risk Models Society of Actuaries, Schaumburg, IL © 2010 by Taylor and Francis Group, LLC ... by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 271 12. 2.1 Dismissal by the Employer Let N tD be the random number of dismissals in year t, and let YtD,... racteristics (12. 4) f ollow i mmediately b y i nserting t he f ormulas (12. 25), (12. 27), and (12. 28) 12. 3 ASSET AND LIABILITY MODEL FOR DISMISSAL FUNDING Let Wt be the random wealth of the dismissal. .. y © 2010 by Taylor and Francis Group, LLC Actuarial Funding of Dismissal and Resignation Risks ◾ 281 TABLE 12. 1 Employee Structure by Age, Term of Service, and Wage Age Term of Service 20 30 30