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Optimal contributions in a defined benefit pension scheme with stochastic new entrants
Insurance: Mathematics and Economics 37 (2005) 335–354
Optimal contributions in a defined benefit pension scheme with stochastic new entrants Luigi Colombo∗ , Steven Haberman Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK Received October 2004; received in revised form February 2005; accepted 11 March 2005
Abstract This paper focuses on the impact of the stochastic evolution of the active membership population on the mismatch between assets and liabilities of a defined benefit pension scheme. Classical results in the actuarial literature on pension plan population theory have been extended to the stochastic case. The paper formulates the trade-off between risk and cost of contribution strategies. Then, using a constrained nonlinear programming approach, optimal contributions strategies have been derived and the trade-off solved by means of identifying an efficient frontier. Finally, a numerical application has been carried out, showing the inefficiency of certain classical normal cost methods. © 2005 Elsevier B.V. All rights reserved. Keywords: Defined benefit; Stochastic new entrants; Optimal normal cost; Nonlinear programming
1. Introduction Defined benefit (DB) pensions schemes are structures which aim to provide workers at retirement with a benefit, which is normally linked to the final pre-retirement salary (or an average of salaries in a short period before retirement). Even though many countries have recently promoted a shift to defined contribution schemes, DB plans are still very common because of the guarantees included in this kind of plan. Pension mathematics provides a scientific approach to the funding of pension schemes, leading to several tools for calculating contributions and valuing assets and liabilities. Specifically, many methods have been developed to calculate the normal costs, i.e. the set of contributions to be paid, which leads to the actuarial equilibrium between assets and liabilities under deterministic conditions. See Winklevoss (1993) for a full description of this approach. However, demographic and financial realisations are very likely to differ from expectations, and these are the two main sources of uncertainty, with which a DB pension scheme must cope. ∗
Corresponding author. E-mail address: [email protected] (L. Colombo).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.02.011
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L. Colombo, S. Haberman / Insurance: Mathematics and Economics 37 (2005) 335–354
Essentially, the financial risk tends to have a greater impact on the wealth of a pension scheme, than the demographic risk. With this justification, scientific works have principally tended to regard the latter as irrelevant. However, a sound management of the risk deriving from the evolution of the membership population can reduce the magnitude of the overall risk faced by a pension scheme. The demographic evolution of a pension plan’s population is a complex phenomenon, in which several factors interact in different directions. On one hand, the active membership population evolves according to the processes of new entrants and withdrawals (with a small effect for mortality and disablement). On the other hand, the retired membership evolves as a result of mortality. There is a considerable actuarial literature on the demographic evolution of populations. However, it mainly focuses on studying the evolution of mortality with the aim of (a) developing projection models and (b) investigating the impact on life insurance and annuity products. An exhaustive review of survival models is presented in Pitacco (2004). The pension plan population theory in the actuarial literature is mainly based on a standard deterministic model, where the resulting population is stable, as employed by Bowers et al. (1976) and Winklevoss (1993). Pension plan theory applies mathematical demography to the specific case of a pension plan population. In this way, it achieves powerful results, showing how equilibria in the cash flows arise under specific conditions on the evolution of the population. A special case of this model (stationary population and fixed entry age) has been further developed in Owadally (1998), where a sequence of iid random variables describes the process of annual new entrants. In this framework, the author isolates the impact of the demographic evolution on the variability of the fund value and of the contributions at any time t. With regard specifically to the accumulation phase in a pension scheme, a model for the active population has been proposed by Janssen and Manca (1997). Specifically, a discrete time non-homogeneous semi-Markov model is used to describe the changes to the internal composition of the active members. However, the total number of members (in each state, i.e. job position) is exogenously given. Mandl and Mazurova (1996) model the new entrants (at a fixed age) in a DB scheme by means of independent random sequences, which are stationary in a wide sense. By using the spectral decomposition of stationary processes and the introduction of frequency transfer functions, the authors obtain closed form expressions of the variance of the fund level, the contributions and the discounted cash flows. A stochastic model for the demographic evolution has been considered in Chang et al. (2002), where optimal decisions for pension plans are derived as a solution of a stochastic control problem, obtained using dynamic programming. However, the demographic random evolution is only accounted for implicitly, as the normal cost process is driven by a Brownian motion. In a similar fashion, Josa-Fombellida and Rinc´on-Zapatero (2004) consider the benefit flow as a stochastic process (geometric Brownian motion), while investigating how to minimise both the contribution rate risk and the solvency risk. Specifically, the random nature of the benefit is introduced in order to describe the disturbances in all of the processes that affect the benefit: such as the evolution of the plan population and of the inflation and salaries. In this work, we focus attention on the dynamics of the active membership, by means of analysing the variability of the mismatch between assets and liability, as well as proposing an optimal contribution strategy which minimises this variability. The work is organised in the following way: in Section 2, the model describing the dynamics of a DB pension scheme is introduced. Well-known results in the actuarial literature are then extended to the case of the stochastic evolution of a population. Section 3 includes the analysis of the role that different variables play in the determination of the asset-liability mismatch variability. We discuss the necessity of including the cost of implementation as a fundamental variable in the analysis of optimal contribution strategies. Finally, optimal normal cost sequences are derived as the solution of a constrained nonlinear program. A numerical application of the case of iid new entrants at a fixed age is implemented: details and results are displayed in Section 4. The work ends with conclusions and suggestions for further research.
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2. The model 2.1. DB pension scheme The pension scheme is modeled using a discrete-time stochastic model, first described and investigated in Dufresne (1988). In this section, the fundamental equations are presented, together with some of the main results derived in the literature. 2.1.1. Fund value The fund value, or fund level, is the value of the assets belonging to the scheme at a specified time t. The literature presents a number of different methodologies to evaluate the assets, among which the market value, the smoothed market value and the discounted cash flow methods are the most commonly used. It is not in the aim of this work to analyse the features of these methodologies, a comprehensive review of which can be found in Exley et al. (1997) and Owadally and Haberman (2004b). Whatever the method of valuation, the dynamic of the fund value may be described by the following recursive equation: f (t + 1) = [f (t) + c(t) − B(t)](1 + r(t))
(1)
where r(t) is the rate of return of the investment portfolio gained during the year(t, t + 1); c(t) = NC(t) + adj(t) is the annual contribution income at time t and it is computed by adding an adjustment, adj(t), to the normal cost, NC(t) and B(t) the benefit paid to the pensioners at the beginning of year t. All of the cash flows are assumed to take place at the beginning of each scheme year and are also evaluated net of price inflation.1 2.1.2. Actuarial liability The actuarial liability (AL), also known as the reserve, is the difference between the actuarial discounted values of liabilities and future contributions for current members (active and retired) of the scheme. Similarly to the fund level, the AL may also be calculated in different ways. An important concern lies in the choice of the valuation discount rate and in the methodology used to determine it. Yet, whatever the valuation discount rate, the following equation describes the expected dynamics of AL: E[AL(t + 1)|Ft ] = [AL(t) + NC(t) − B(t)](1 + i)
(2)
where i is the valuation discount rate and Ft is the filtration summarising the information available at time t. Eq. (2), which we will refer to as the general liability growth (GLG) equation, states that if the assumptions on the demographic evolution are actually borne out by experience, than the AL dynamics can be described by a recursive equation. 2.1.3. Unfunded liability From the previous quantities an amount is computed, which provides valuable information about the scheme’s wealth: the unfunded liability (ul). At any time, ul is defined by the difference between the AL and the fund value; in formula: ul(t) = AL(t) − f (t)
(3)
1 The assumption on inflation might actually overestimate the risk due to the benefit variations. This happens because benefits are usually adjusted according to price inflation; while all the other amounts vary according to salary inflation, which is usually greater than price inflation.
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It is straightforward to note that a positive value happens only when the fund at time t, f(t), is not sufficient to cover the actuarial liability, i.e. the current and future liability of the existing scheme members. Conversely, a negative ul indicates that the value of the assets is higher than the actuarial liability. Thus, it is common practice to refer to ul in order to adjust annual contributions, since it expresses the size of the current surplus/deficit, which needs to be amortised. Specifically, in this work surpluses and deficits are spread over a moving term and the resulting annual adjustment is therefore a fraction of the current ul. Features and properties of spread methods are investigated in Dufresne (1988), Owadally and Haberman (1999), Owadally (2003), Owadally and Haberman (2004a), and Josa-Fombellida and Rinc´on-Zapatero (2004). 2.1.4. Individual funding methods The quantities introduced above, AL, NC and B, are aggregated amounts; i.e. they refer to the whole scheme. However, for the case of individual funding methods, the previous quantities can be disaggregated by age and written as the corresponding individual amount multiplied by the number of existing members. Further decompositions are also possible. So, for example, NC(t) can be broken down by age and expressed in a formula as: NC(t) =
τ
NCα+x n(α + x, t)
(4)
x=0
where n(α + x, t) is the membership function, which represents the number of members aged α + x at time t; α is the minimum age and τ is the length of the working life time (τ = R − α − 1, with R fixed retirement age). NCα+x is the contribution paid by all the members aged α + x. The sequence {NCk } indicates the contribution strategy adopted in the pension plan to finance the final benefit. Several methods – known as funding methods or normal cost methods – have been formulated for the purpose of computing contribution strategies. Theoretically, an infinite number of contribution strategies exists, however, an acceptable funding method must satisfy some requirements, whose rationale lies in the fairness of the actuarial equilibrium, as well as any strict bounds and characteristics which may be imposed by an appropriate supervisory or regulatory authority.2 Similarly to NC(t), also the annual total benefit and the actuarial liability can be disaggregated by age. However, for these quantities, it is necessary to take into account the number of years of past service; i.e. the number of years during which each active member has contributed into the fund. For this purpose, the membership function is further decomposed, as in the following formula: n(z, t) =
M
n(z, t, κ),
z = α, . . . , ω
κ=0
where n(z, t, κ) is the number of members aged z, at time t and with κ years of past service. The sum of n(z, t, κ) over κ is the aggregated membership function as previously defined, i.e. regardless of the years of past service. The upper extreme M varies according to the age z: for post-retirement ages (z ≥ R) M is the longest possible past service, M = R − α; while for pre-retirement ages (z < R) the past service cannot be longer than age z less the minimum age α, so M = z − α. In order to ease the notation, instead of using a varying upper extreme, we assume that the number of active members with a past service longer than their age (less minimum age α) is zero. Thus, 2 See Lee (1986), Aitken (1996) and Sharp (1996) for a fuller discussion. We consider only individual funding methods in this paper. We note that aggregate funding methods do not involve a normal cost or the explicit derivation of the actuarial liability. The contribution rate calculated automatically leads to a nil value for the unfunded liability, because it is designed to eliminate any difference between the values of asset and liabilities.
n(z, t, κ) = 0
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∀κ > z − α
(5)
Henceforth, the actuarial liability and the annual total benefit at time t have the following expressions: AL(t) =
ω−α R−α
ALα+x,κ n(α + x, t, κ)
(6)
x=0 κ=0
B(t) =
ω−R R−α
BR+y,κ n(R + y, t, κ)
(7)
y=0 κ=0
where ω is the extreme age; ALα+x,κ is the actuarial liability corresponding to a member aged α + x with κ years of past service; and, finally, Br+y,κ is the annual benefit received by a pensioner, aged R + y, who contributed for κ years into the fund. For the purpose of this work, there is no need to go in to further detail in analysing the composition of AL and B by past service. It is sufficient to note that this model implicitly assumes that all the members having the same age and past service have accrued the same actuarial liability, and are thus entitled to an equal benefit. Disaggregating the membership function by years of past service allows the model to account for liabilities and benefits that vary according to how long each member has actually contributed into the fund. Moreover, the membership function does not need to be necessarily decreasing, thus allowing for new entrants at any age before retirement. Specifically, the following relation exists between the new entrants function g and the membership function n: n(z, t, κ) = g(z − κ, t − κ) κ pz−κ
(8)
where g(x, t) is the number of new entrants aged x, at time t; and κ pz is the probability that an individual aged z remains in the scheme for κ years. In the following section, we analyse potential models for the membership function, including the case of a randomly evolving population. 2.2. Demographic evolution The membership function n(α + x, t) describes the number of members at different ages and at any time.3 Different types of population correspond to different membership functions. Moreover, for some specific membership functions, several results have been derived in the classical actuarial literature corresponding to specific equilibria. 2.2.1. Mature population A stability condition arises in the population if the age distribution remains constant year after year. This will eventually happen, if the population decrement rates remain constant and if the increment rate grows at a constant rate.4 Part of the actuarial literature (see Winklevoss, 1993, for instance) refers to this population as mature. As a special case of stability, we take into account the case that the increment rate remains constant: the resulting population is then said to be stationary. Moreover, as stated in Winklevoss (1993), if a constant flow of new entrants annually joins the scheme, and decrement rates do not change over time, “a stationary condition will exist after n years, where n equals the oldest age in the population less the youngest.” 3 In order to distinguish whether the value of a variable (say AL) is deterministic or stochastic, subscripts (AL ) are used only to indicate t deterministic variables, while the notation AL(t) indicates a random variable. 4 Refer to Keyfitz (1985), Chapter 4, for a full analysis of the properties of stable populations.
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In terms of the membership function, stationarity is equivalent to assuming that the number of members aged x at time t is described by nα+x,κ . This assumption implies that stationarity holds for the aggregated membership function as well: nα+x,t =
R−α
nα+x,t,κ =
κ=0
R−α
nα+x,κ = nα+x
(9)
κ=0
So, for instance, the actuarial liability broken down by generations, as expressed in Eq. (6) would become constant: ALt =
ω−α R−α
ALα+x,κ nα+x,κ = AL
(10)
x=0 κ=0
Stationarity in the population evolution leads to a remarkable equilibrium between the total contributions, the returns from investments and the payment of benefits. Specifically, the GLG Eq. (2) can be rewritten in the following way: AL = (AL + NC − B)(1 + i)
(11)
As stated in Trowbridge (1952), we can refer to (11) as the equation of maturity. When the plan membership eventually becomes mature, the pension fund reaches a state of financial equilibrium, where the contribution and the expected returns on the reserve match the flow of benefits annually paid out: B = NC + d AL where d = i/(1 + i) is the discount factor associated with i. 2.2.2. Stochastic new entrants In this paper, we are interested in investigating the effects that random perturbations in the membership may have on the process of funding a DB pension. In order to do so, we shall consider the membership function n(α + x, t, κ) as a stochastic process. In detail, we assume that the random nature of this process is due to stochastic new entrants, while the decrements from the scheme population are known. In other words, we assume that the probabilities in Eq. (8) describe the proportion of members not eliminated, and that, these probabilities are deterministic and are known precisely. Clearly, the equilibrium displayed in Eq. (11) no longer holds. However, it can be shown that under equivalent assumptions regarding the stochastic membership process, an equivalent, and more general, form of equilibrium does exist. It has been proved that, assuming deterministic decrements, the GLG equation can be written in the following way5 : AL(t + 1) = [AL(t) + NC(t) − B(t)](1 + i)
(12)
to which we can refer as the liability growth (LG) process, since AL(t) is a stochastic process due to random new entrants. This is an extension of the LG equation, obtained for a deterministic stable population, presented in Bowers et al. (1976). In analogy with the simplification achieved on passing from stability to stationarity, we focus attention on the case that the number of members is random, but where the evolution is not driven by any drift. In another words, 5
Refer to Appendix A for a mathematical proof.
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n(α + x, t, κ) is assumed to be a stationary stochastic process, whose mean, by the definition of stationarity, is independent of time, i.e. E[n(x, t, κ)] = nx,κ . Under this assumption the actuarial liability at time t is a sum of random variables with constant expectation, and therefore its expectation is constant as well: ω−α R−α ω−α R−α E[AL(t)] = E ALα+x,κ n(α + x, t, κ) = ALα+x,κ nα+x,κ = AL, say (13) x=0 κ=0
x=0 κ=0
In the light of this, Trowbridge’s equation of equilibrium holds on average: AL = E[AL(t)] = E[AL(t − 1) + NC(t − 1) − B(t − 1)](1 + i) = [AL + NC − B](1 + i)
(14)
Eq. (14) is the equivalent in a stochastic framework of the equation of maturity (11) which is based on a deterministic model. In fact, Eq. (11) represents a special case of the more general result in (14).
3. Optimal normal cost methods In this section, we aim to obtain a normal cost method which is optimal in the sense that it minimises a given measure of risk, when the membership population randomly varies as described above. Therefore, firstly, a measure of risk is introduced and analysed. Subsequently, we use such a measure in order to identify the optimal contribution strategy. 3.1. Unfunded liability As already mentioned, the unfunded liability, defined in Eq. (3), is an amount which is informative of the scheme’s wealth; and, as a result, the spread method computes the annual contribution adjustment (or supplemental cost) as a proportion of this quantity. In the previous section, we have introduced a potential model to describe random perturbations to the demographic evolution of a DB pension scheme. In fact, as long as the rates of return from investments exactly match the liability valuation rate, such perturbations do not affect the variability of ul(t). This is easily proved: from Eq. (3), if 1 + r(t) = 1 + i then the resulting ul(t) is independent of the demographic variations, and its dynamics are described by the following recursive relationship6 : ul(t) = AL(t) − f (t) = ul(t − 1)(1 − k)(1 + i) where the adjustment at time t − 1 is a proportion of the current unfunded liability: adj(t − 1) = k ul(t − 1). Moreover, if at time 0 the scheme is fully funded, ul(0) = 0, then ul(t) = 0
∀t ≥ 0
However, it is well known that the financial realisations are very likely to differ from the expectations and, hence, an unfunded liability may arise, being either positive or negative. We are interested in how the ul would develop in time and the degree to which it might be sensitive to a stochastic demographic evolution when rates of return from investments differ from the expectations. In fact, we focus on the case in which the investment rate is deterministic and constant, but different from the valuation rate. 6
Refer to Appendix B for mathematical proof.
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Let r denote the deterministic annual rate of return achieved from the investments. Let us also introduce the following parameter ρ = (r − i)/(1 + i), which summarises the degree of mismatching between the rate of return and the valuation discount rate. Thus, the following holds, from (1), (3) and (12): ul(t + 1) = ul(t)(1 − k)(1 + r) − ρ AL(t + 1)
(15)
Furthermore, if the scheme is initially fully funded (i.e.: ul(0) = 0), we obtain: ul(t) = −ρ
t
AL(h)wt−h ,
w = [(1 − k)(1 + r)]
(16)
h=1
It is straightforward to note that ul(t) is null ∀t if • i = r; i.e. if the return rate from investment exactly matches the valuation rate. • k = 1; i.e. if the whole unfunded liability is instantly amortised. It is also of interest to highlight that, when w = 1 and r > i, the annual decrease in the unfunded liability is exactly given by the extra return r − i gained from investing the reserve accrued at the beginning of the year. Conversely, the annual variation is an increment if r < i. Now, w is equal to 1 when the amortised proportion of ul is equal to the discount factor corresponding to the rate r: k=
r = dr 1+r
It is worth noting that, in this case, the ul arithmetically increases/decreases year after year and the adjustment is not sufficient to lead eventually to ul being null.
3.1.1. First moment As soon as the demographic evolution is randomly perturbed, AL(t) becomes a stochastic process and so does the process ul(t). We have seen that, under the assumption of a stationary demographic evolution, the expected actuarial liability is constant at any time; i.e. E[AL(t)] = AL ∀t, as shown in Eq. (13). Furthermore, the expected unfunded liability has the following form: E[ul(t)] = −ρ AL
[(1 + r)(1 − k)]t − 1 (1 + r)(1 − k) − 1
(17)
The condition that (1 − k)(1 + r) < 1 is sufficient for the expectation (17) to converge in the limit. Trivially, ≤ 0 if r ≥ i ρ AL lim E[ul(t)] = = = 0 if r = i t→∞ (1 + r)(1 − k) − 1 ≥ 0 if r ≤ i
(18)
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3.1.2. Second moment From Eq. (16) it is also possible to derive a convenient expression for the variance of the unfunded liability process7 : t t t 2 t−h Var[ul(t)] = ρ Var = ρ2 AL(h)w w2t−h−k Cov[AL(h), AL(k)] h=1
= ρ2
τ τ
h=1 k=1
ˆ NCα+x NCα+y Cov{G(x, t), G(y, t)} + ρ2 Var[B(t)]
x=0 y=0
− 2ρ2
τ
ˆ NCα+x Cov[G(x, t), B(t)]
(19)
x=0
where the function G(x, t) summarises the weights of present and past generations of active members; as given by the following formula: t
G(x, t) =
wt−h
h−1 j=x
h=x+1
uh−j
R−α
n(α + x, j, κ)
κ=0
ˆ is the weighted history of benefits paid to the pensioners. B(t) ˆ is given by the following Further, function B(t) formula: ˆ = B(t)
t
wt−h
h=1
h−1 j=0
uh−j
ω−R R−α
BR+y,κ n(R + y, j, κ)
y=0 κ=0
From Eq. (19), it can be seen that the contribution strategy (identified by the sequence {NCk }) directly affects the variance of the unfunded liability. In fact, a normal cost method, which provides an early accumulation of fund, will lead to a different variability than another method which involves a later degree of accrual. In the development of this work, we will use the variance (19) as a measure of risk, while allowing the demographic evolution to be random. 3.2. Cost of strategies Although this work aims to find a contribution strategy which minimises the variability of the unfunded liability, in this section we discuss the necessity that a comprehensive analysis has to take into account not only measures of risk but also the actual cost, as part of a sound risk management policy. In DB pension schemes, the capital accrued at retirement age should be sufficient to finance the payment of an annual pension to each retiree. PVFB indicates the expected present value of such a benefit at retirement age. In order to accrue this amount, contribution strategies rely on the annual contributions paid by the plan’s sponsor and on the returns from investing the accrued fund value in a portfolio of assets. In this work, the plan’s sponsor is the entity liable for providing this benefit and it often coincides with the employer, who offers this retirement plan to the employees in addition to the salary. However, it is often the case that the employees participate in the pension funding by paying annual contributions linked to their current salary. Usually, in a DB scheme, these contributions are not related to the wealth of the scheme, and hence, the employee 7
Refer to Appendix C for a mathematical proof.
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does not bear any financial risk related to the favourable or unfavourable deviations of the experience from the actuarial assumptions. In such a situation, according to our definition, the employer would still be the plan’s sponsor, as he/she is liable for providing the retirement benefit. Nevertheless, the cost of pension provision would be reduced by an amount given by the expected present value of all the contributions paid by the employee.8 We assume for convenience in the rest of the discussion that the employer meets the full cost of the pension plan Considering that the financial markets provide the returns from investments, it is clear that the plan’s sponsor is expected to pay the normal cost for each member of the scheme. Thus, the higher is the proportion of the benefit funded by these normal cost, the higher is the face value of the total amount that is to be paid by the sponsor. In other words, the cost of a contribution strategy depends on how much of the benefit is funded by the contribution income, when the remainder is funded by the investment income. Bearing this in mind, we can state the following intuitive rule: the more expensive is the strategy, the lower is the risk. This is because of the timing of a contribution strategy: according to the concept of cost introduced above, the earlier is the funding of the pension (which means that the pension is being mainly funded with contributions paid at young ages), the less expensive is the strategy. This happens because an earlier contribution strategy relies more on investment returns, than a later one would do. On the other hand, an earlier contribution strategy would expose the accrued fund to the volatility of all the existing risks for a longer period. So, even though a strategy may optimally reduce the risk, its cost might be excessively high for the plan sponsor. Hence, the plan sponsor might not want to achieve the safest position. Nevertheless, it might prefer to implement a more convenient strategy, for which the resulting level of risk is still acceptable. Clearly, in the decision-making process of choosing a suitable contribution strategy, a trade off between cost and risk arises. With our aim of exploring tools in order to reduce the level of DB pension scheme risk, it appears crucial to include in the model a variable which takes into account the cost of contribution strategies. In the environment described by the current model, inflation is fixed and all of the amounts are being expressed net of price inflation. Hence, a suitable measure of the cost of a contribution strategy may be given by the sum of the normal costs. However, since not all the members are expected to survive up to the retirement age, the expected cost is instead taken into account: E[Cost(t)] =
τ
NCα+x (t) x pα
(20)
x=0
where x pα is the probability that an individual aged α survives for x years. Clearly, by artificially setting the survival probability equal to 1 at all ages, the measure given by the expected cost would collapse to the simple total cost. 3.3. Optimisation problem The aim is to identify an optimal normal costs method, in the sense that it minimises the variance of the unfunded liability, also taking into account the cost. Specifically, the measure of cost can be included in this analysis in several ways. One of these would require us to include the cost as a strict constraint to the minimisation problem. Alternatively, it could be possible to define and subsequently minimise a penalty function, which gives increasing importance to the cost measure according to a scaling (or penalty) factor. 8 The case of sharing the pension funding between the employer and the employee can be simply considered, by adequately modifying the current mathematical structure according to the sharing formula. For instance, in the case that the employee contributes an amount proportional to his/her salary, the sponsor’s annual contribution would be the balance of the normal cost not paid by the employee.
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Finally, a multiple objective function minimisation could be carried out, with the effect of simultaneously minimise risk and cost. The work developed here is mainly based on the first approach. However, the first and second approaches require the same techniques and, under particular assumptions, they are equivalent and lead to the same results. The third approach would indeed require different techniques but it could be an interesting extension for further research.9 From Eq. (19) (variance of the unfunded liability), it is possible to state the minimisation problem, for convenience expressed in matrix notation: min Var[ul(t)] = min NC ΣG(t) NC − 2 NC C(t) NC
NCk
(21)
subject to τ
NCα+x x Eα = PVFB ↔ NC E − PVFB = 0
(22)
x=0
E[Cost(NC)] = K ↔ NC p − K = 0,
NCα+k ≥ 0, k = 0, . . . , τ
(23)
where the general (i, j)th element of the square matrix ΣG(t) is given by the covariance Cov{G(i, t), G(j, t)}. Matrix ΣG(t) is then a variance/covariances matrix and therefore it is symmetric and positive defined. Furthermore, its dimension is (τ + 1) × (τ + 1). Similarly, the vector C(t) contains the covariances between the inflow and the outflow processes, i.e. ˆ Cov{G(i, t), B(t)} is the ith element of the vector. Vector p indicates the probabilities of surviving up to the retirement age: so that [p1 = τ pα , p2 = τ−1 pα+1 , . . . , pτ = 1 pα+τ−1 ]. If all the components of the vector p are set equal to 1, then the resulting measure is the simple total cost of the contribution strategy. K is the cost of the strategy that the plan’s sponsor is willing to pay to fund the retirement benefit of each employee. 3.3.1. Kuhn–Tucker conditions Associated with any optimal problem including constraints is the Lagrangian function. For the above problem, the Lagrangian function is: L(NC, λ, ν, µ) = NC ΣG(t) NC − λ(NC E − PVFB) − ν(NC p − K) − µ NC
(24)
The Kuhn–Tucker (K|T) theorem provides necessary and sufficient conditions for a point to be optimal while satisfying the imposed constraints.10 In this specific case, if an optimal solution NC∗ exists, then there must exist multipliers λ, ν and µ0 , . . . , µτ satisfying the following conditions: ∂L(NC∗ ) =0 ∂NC
(25)
NC∗ E − PVFB = 0
(26)
NC∗ p − K = 0
(27)
µk NC∗α+k = 0, λ, ν ≥ 0
k = 0, . . . , τ
(28) (29)
9 A comprehensive review of some of the potential algorithms is illustrated in Rustem (1998). Particularly interesting is Chapter IV, which deals with optimisation problems in uncertainty. An interesting application has been developed in Guo and Huang (1996) in the field of asset allocation within a fuzzy set theory framework. 10 See the original paper by Kuhn and Tucker (1951), or Chiang (1984) for a full introduction to the theory and some applications.
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µk ≥ 0,
k = 0, . . . , τ
(30)
In addition, since the variance of ul is a convex function of the normal costs, and so are the constraints (actually linear), the K|T conditions state that any point NC satisfying the above conditions is an optimal solution of problem (21). Unfortunately, including the non-negativity constraints increases significantly the computational complexity of the system to be solved, leading the constrained quadratic problem to be, in the first place, a combinatorial problem. Indeed, we must consider all the possible 2τ+1 − 1 cases, in which alternatively some or all the multipliers µk (k = 0, . . . , τ) might be null. A possible way to deal with the problem consists of numerically finding the minimum by choosing an algorithm which systematically excludes negative points. This approach has been implemented for a specific case, and details and results are shown in the subsequent Section 4. Remark. One further consideration concerns the sign of the elements involved in the minimisation. The components of both the matrix ΣG(t) and vector C in Eq. (21) are covariances. Since such covariances are ultimately given by the recursive structure of the model, it appears reasonable that such elements are all positive. In fact, the larger is the value of AL at time t − 1 the higher it is expected to be at time t. Thus, in the process of minimisation, the quadratic term will push the variables to be zero, while the term of order 1 should push them to be positive. Since the minimum value of the variance of ul must be positive, it seems reasonable to assume that the optimal solution will lie in the non-negative subspace. In the light of this discussion, the following alternative approach might be found to be worthwhile. 3.3.2. Lagrangian conditions and solution If the non-negativity constraints can be ignored, the Lagrangian function has the following expression: L(NC, λ, ν, µ) = NC ΣG(t) NC − λ(NC E − PVFB) − ν(NC p − K)
(31)
Such a minimum problem may be solved by applying the Lagrange method. Specifically, amongst K|T conditions only (25)–(27) have to hold. Hence, the optimal normal costs sequence can be found as the solution of the following system: ∂L(NC) = 2 NC ΣG(t) − 2C(t) − λE − νp = 0, ∂NC ∂L(NC) ∂L(NC) = NC E − PVFB = 0, = NC p − K = 0 ∂λ ∂ν Such a system consists of (τ + 3) equations in (τ + 3) variables and has the following unique solution: λ∗ ν∗ −1 NC∗ = ΣG(t) C+ E+ p 2 2
(32)
with multipliers λ∗ =
−1 −1 −1 −1 E ΣG(t) p(C ΣG(t) p − K) − p ΣG(t) p(C ΣG(t) E − PVFB)
ν ∗ = K − λ∗
−1 −1 −1 E ΣG(t) Ep ΣG(t) p − (E ΣG(t) p)2 −1 E ΣG(t) p −1 p ΣG(t) p
−1 p − C ΣG(t)
(33)
(34)
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Fig. 1. Variance of ul for several contribution strategies.
4. Numerical application In order to display the effect of using optimal contributions strategy, numerical calculations have been carried out. A suitable expression for the membership stochastic process is therefore needed. Specifically, the membership process has to lead the population to stationarity on average. The assumption of annual new entrants joining the scheme at a fixed aged independently between years satisfies the required condition. The new entrants are thus modeled by means of a stochastic process {gt } of iid random variables with the same mean g and variance σg2 . Hence, the membership function has the following form: n(α + x, t) = gt−x x pα Under this assumption, it is possible to obtain closed expressions for the covariances introduced in the previous section.11 Fig. 1 displays the exact value of the variance of ul during the years 0–80 for different contribution strategies. In detail, setting the cost to its maximum leads us to find the safest strategy.12 The resulting contribution path is known in the actuarial literature as terminal funding (TF), which consists of financing a pension with the payment of a lump sum at the time of retirement. By implementing a TF strategy, the variance of ul is the lowest possible, thus generating in the graph the lowest line. Conversely, by setting the cost equal to its minimum, the minimisation leads to the cheapest and riskiest contribution strategy. In this case, the pension would be financed by the payment of a lump sum at the beginning of the working life-time and by investing this capital during that period. This strategy is known as initial funding (IF). Clearly, the funding capital would be exposed for a long period to the financial risk and thus the resulting variance of ul is the largest possible (the top line in the graph). For intermediate levels of the cost, the minimisation problem leads to contribution strategies that are riskier as the cost increases. 11 Refer to Appendix D for closed formulas and to Appendix E for a detailed summary of the assumptions made and the values of parameters used. 12 It is easy to show that, once the survival probabilities p and the valuation rate i have been set, the cost measure has an unique minimum and an unique maximum.
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Fig. 2. Efficient cost/risk frontier.
It is worthwhile noting that, the contribution strategy does reduce (or increase) the variance of ul, but does not change its fundamental trend over time. All of the lines in Fig. 1 show a sigmoidal shape, which is due to the characteristics of the demographic risk. As argued in Section 3.2, the riskiness of a strategy is directly related to its cost. In fact, a trade off between risk and cost of a strategy exist. Fig. 2 displays such a trade off by means of identifying the efficient frontier. Specifically, the variance of ul at time t = 80 is computed according to Eq. (19), and subsequently plotted on the y-axis against the cost of the corresponding strategy on the x-axis.13 As predicted by the intuitive rule, as the cost increases the variance decreases and vice versa. A detailed examination reveals that, the strategy TF is the safest and the most expensive strategy. Therefore, the resulting point for this method is displayed at the right hand end of the graph. Conversely, the strategy IF is the cheapest and the riskiest possible strategy, and so the corresponding point is displayed at the left hand end of the graph. The black line identifies the efficient frontier, which represents the trade off between cost and risk of a contribution strategy. For the sake of comparison, a set of classical normal costs methods has also been selected. It is interesting to note that all of them have turn out to be inefficient. In fact, the unit credit (UC), fixed entry age (EA) and constant premiums (CP) methods lie above the efficient line. Accepting the level of risk that each of the classical methods implies, it is always possible to find a less expensive contribution strategy. Conversely, given the cost of each of the classical methods, it is possible to find a strategy which implies lower riskiness. As previously mentioned, it is of interest to investigate how the efficient line is sensitive to the spread period. The longer this period is, the more variable is the resulting unfunded liability. Bearing in mind that the covariances between AL at different points in time are positive, it can be seen from Eq. (19) that the variance of ul is an increasing −m )(1 + i), the variance of ul increases function of w. Since w = (1 − k)(1 + r) and k = 1/¨am| ¯ = i/(1 − (1 + i) with the spread period (m).
13 Time and spread period seem to affect the magnitude of the results, but not its meaning. Since the spread period is one of the tools commonly used to calibrate convenient strategies with respect to the solvency and stability of the scheme (see Haberman and Sung, 1994), a sensitivity analysis on the parameter m has been carried out. The results are illustrated at the end of this section.
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Fig. 3. Efficient cost/risk frontier and spread period.
In order to illustrate the effect of this parameter on the efficient frontier, we have computed the variance of ul for a set of spread periods and for different acceptable levels of cost. The resulting variances have been displayed in Fig. 3.14 The displayed lines are consistent with the expected behaviour that the larger is m, the higher is the variance of ul. Specifically, the increments in the spread period have a significant effect on the risk. In fact, it can be seen that for a high value of m (say m = 8 or 9) the most expensive contribution strategy (cost = 1) yields a level of risk which is higher than the variance achievable from combining a cheaper strategy with a shorter spread period (say m ≤ 5 and cost = 0.2). However, the risk can be maintained on the same level by choosing appropriate combinations of cost and m. Table 1 displays the cost necessary to achieve a given level of risk, when combined with a set of possible spread periods. These figures highlight the strong impact that the spread period has on the variability of ul. In order to have a variance equal to 20,000, a very cheap contribution strategy (12% of TF cost) is needed when the surpluses and deficits are spread over 4 years. In order to achieve the same position when m = 5 or 6, the cost of the optimal strategy is respectively 34% and 60% the cost of the TF strategy. Increasing the spread period by one further year, m = 7, a strategy almost as expensive as TF is needed in order to maintain the variance at the chosen level. Finally, it is worthwhile comparing again those figures with those determined by the classical methods considered. In Table 2 the relative cost and the variance of ul, corresponding to each classical method, when m = 5 and t = 80 are displayed. The inefficiency of those methods is still evident. For instance, the entry age method requires a cost slightly higher than 21% and implies a variance higher than the level of 26,000, which is achievable by implementing the optimal strategy for a comparable cost. Similar conclusions apply for the other two methods. 14 In Fig. 3 and in the following tables the actual cost of each contribution strategy has been rescaled in the interval (0, 1), by dividing by the cost of the TF strategy, which is the maximum cost.
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Table 1 Relative cost for different levels of variance of ul (×1000) and spread periods m
2 3 4 5 6 7 8 9 10
Variance (%) 5
11
16
20
26
31
38
11 62 – – – – – – –
– 12 44 94 – – – – –
– – 22 49 86 – – – –
– – 12 34 60 98 – – –
– – – 21 41 64 93 – –
– – – 13 31 50 71 97 –
– – – – 21 37 54 72 93
Table 2 Relative cost and variance for classical methods
CP EA UC
Cost (%)
Risk
15.5 21.6 31.5
31700 28800 24700
5. Conclusions In the development of this work, we have extended existing results in pension plan population theory to the more general case of a stochastic active membership. Specifically, we have shown that a stochastic stationary evolution leads to a condition of maturity on average. Focusing on the unfunded liability, we have used an expression for its variance in order to highlight the role of the contribution strategy in managing the risk of mismatching between assets and liabilities. Specifically, we have focused on the way in which stochastic new entrants may amplify the effect of the mismatch between the expected and actual returns from investments. Subsequently, we have shown how constrained nonlinear programming can be applied to derive optimal contribution strategies. Under specific conditions, it has also been possible to derive an analytical solution. When this has not been possible, a numerical algorithm has been shown to be a suitable alternative for solving the optimisation problem. With regard to the specific case of iid new entrants at a fixed age, optimal strategies have been found numerically. Furthermore, the cost of a strategy has been taken into account, since the cost of a resulting optimal strategy might be too high for adoption by a plan sponsor. Thus, the trade-off between cost and risk of a contribution strategy has been solved, by means of finding an efficient frontier. In addition, classical normal cost methods have been compared to the optimal strategies, showing the inefficiency of these methods in terms of cost and risk. The analysis would be extended by including explicitly for the possibility of sharing of the contribution between the sponsor and the scheme member, following the discussion of Section 3.2. Further research will also focus on natural extensions of the existing model. Financial risk could be included by means of stochastic random returns. However, the resulting expression for the variance of ul might be too cumbersome to allow for analytical solutions, but numerical algorithms could be implemented and will be investigated.
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The possibility that the plan’s sponsor defaults in its payments is a feature that could be included in the actual model. Such a source of uncertainty would affect the risk ordering of contribution strategies, and thus would modify the efficient frontier. Including different risk measures constitutes another area of interest for further extensions. Specifically, downside risk measures, and especially those satisfying Artzner’s axioms of coherence, could be included.15 Rockafellar and Uryasev (2000), and subsequent papers, explored this direction developing algorithms to optimise the conditional value at risk (CVaR) and thus identifying efficient frontier – similar to a Markovitz return/risk plane – in an asset allocation problem. Finally, different techniques could be used to take simultaneously into account the minimisation of cost and risk. A possible avenue identified in the literature comes from the successful attempts in applying algorithms to solving multiple objective functions optimisation problems in finance; see Guo and Huang (1996) for an application to an asset allocation problem in a fuzzy set theory framework.
Appendix A. Proof of liability growth process Under the assumption of deterministic decrements and independently of the fact that the number of members aged x at time t is deterministic or not, the following relation holds: n(x, t) y−x px = n(y, t + y − x) Hence, the GLG equation has the following expression: AL(t) =
∞
ALx n(x, t) =
∞
x=0
=
∞
x=0
(By − NCy )n(y, t) +
y=0
+
n(x, t)
∞
∞
(By − NCy )vy−x y−x px =
y=x ∞ ∞
∞ ∞
(By − NCy )vy−x n(y, t + y − x)
x=0 y=x
(By − NCy )vk n(y, t + k) = B(t) − NC(t)
k=1 y=k ∞
(By − NCy )vh+1 n(y, t + h + 1) = B(t) − NC(t) + v AL(t + 1)
h=0 y=h+1
− vn(0, t + 1)
∞
(Bh − NCh )vh n(0, t + 1)
h=0
= B(t) − NC(t) + v AL(t + 1) − vn(0, t + 1)AL0 = B(t) − NC(t) + v AL(t + 1) where AL0 = 0 for the equivalence principle.
Appendix B. Proof of ul dynamics In Section 3.1 is stated that as long the rates of return exactly match the liability valuation rate and that the annual supplemental cost is proportional to the current unfunded liability, the resulting dynamics of ul are independent of the variations in the number of new entrants in the scheme. As a matter of fact if 1 + r(t) = 1 + i, then the dynamic of the fund value is given by the following equation:
15
Refer to the seminal work of Artzner et al. (1999) for an introduction to coherent risk measures.
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f (t) = (f (t − 1) + NC(t − 1) + k ul(t − 1) − B(t − 1))(1 + i) Hence the following relation holds: ul(t) = AL(t) − f (t) = (AL(t − 1) + NC(t − 1) − B(t − 1))(1 + i) − (f (t − 1) + NC(t − 1) + k ul(t − 1) − B(t − 1))(1 + i) = (AL(t − 1) − f (t − 1) − k ul(t − 1))(1 + i) = ul(t − 1)(1 − k)(1 + i)
Appendix C. Proof of ul variance
Var[ul(t)] = ρ2 Var
t
AL(h)wt−h
= ρ2 Var
t
h=1
= ρ2 Var
= ρ Var 2
t
h=1
τ
wt−h
h=1
wt−h
h−1 j=0
uh−j
τ
NCα+x
x=0
r−α
h−1
[NC(k) − B(k)]uh−k
k=0
ˆ n(α + x, j, κ) − B(t)
κ=0
ˆ NCα+x G(x, t) − B(t)
x=0
= ρ2
τ τ
ˆ NCα+x NCα+y Cov{G(x, t), G(y, t)} + ρ2 Var{B(t)}
x=0 y=0
−2ρ2
τ
ˆ NCα+x Cov{G(x, t), B(t)}
x=0
Appendix D. Closed covariances for iid new entrants Under the assumption of iid new entrants at a fixed age α, it is possible to close the expression for the elements in the matrix ΣG(t) and in the vector C(t):
[x, y] = x pα y pα
G(t)
(ut−x − wt−x )(ut−y − wt−y ) u|x−y|+1 (uw)t−max(x,y) − 1 − (u − w)2 u−w uw − 1
uwt−min(x,y) ut−max(x,y) − wt−max(x,y) uw − 1 u−w 2(t−max(x,y)) 1 u w −1 + wt−min(x,y) + uw − 1 u − w w2 − 1
−
(35)
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Cx (t) = x pα w
t−x
B
353
M 2 (t−1) t−τ−y+1 u y=0
t−x − wt−τ−y+1 u − wt−x uw − u−w (u − w)wt−τ−y uw − 1
2(t−τ−y+1) 1 − 1] −1 u w − + + t−x−1 2 t−τ−y w (u − w)(uw − 1) uw − 1 u − w (w − 1)w t−τ−y+1 t−τ−y+1 −w u + τ+y pα u−w uτ+y−x+1 [(uw)t−τ−y+1
(36)
Appendix E. Assumptions and parameters for calculations and simulations E.1. Model assumptions and parameters • Valuation rate: i = 3%; investment returns rate: R = 3.5%. • Contribution adjustments method is the surplus/deficit spreading over a fixed term: adj(t) = k ul(t) =
ul(t)i (1 − (1 + i)−m )(1 + i)
(37)
where the amortisation period m has been initially set equal to 4. • Fixed entry age α = 20, retirement age r = 65, extreme age ω = 105. • Data from Watson and Wyatt, have been used to derive the service table (probabilities of remaining in the scheme) and salary scale. E.2. Assumptions and parameters for simulations • New entrants process{gt } is normally distributed with mean g = 1000 and standard deviation σg = 250. 10,000 simulations have been carried out, where negative values below 0 have been artificially discarded and set equal to 0.
References Aitken, W.H., 1996. A Problem Solving Approach to Pension Funding and Valuation, 2nd ed. ACTEX Publications, Winsted, CT. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Math. Finance 9 (3), 203–228. Bowers, N.L., Hickman, J.C., Nesbitt, C.J., 1976. Introduction to the dynamics of pension funding. Trans. Soc. Actuaries 28, 177–203. Chang, S.C., Tsai, C.H., Tien, C.J., Tu, C.Y., 2002. Dynamic funding and investment strategy for defined benefit pension schemes: a model incorporating asset-liability matching criteria. J. Actuarial Pract. 10 (1). Chiang, A.C., 1984. Fundamental Methods in Economics, 3rd ed. McGraw-Hill. Dufresne, D., 1988. Moments of pension contributions and fund levels when rates of return are random. J. Inst. Actuaries 115, 535–544. Exley, C.J., Mehta, S.J.B., Smith, A.D., 1997. The financial theory of defined benefit pension schemes. Br. Actuarial J. 3, 835–966. Guo, L., Huang, Z., 1996. A possibilistic linear programming method for asset allocation. J. Actuarial Pract. 4 (1), 67. Haberman, S., Sung, J.H., 1994. Dynamic approaches to pension funding. Insur.: Math. Econ. 15, 151–162. Janssen, J., Manca, R., 1997. A realistic non-homogeneous stochastic pension funds model on scenario basis. Scand. Actuarial J. Josa-Fombellida, R., Rinc´on-Zapatero, J.P., 2004. Optimal risk management in defined benefit stochastic pension funds. Insur.: Math. Econ. 34, 489–503. Keyfitz, N., 1985. Applied Mathematical Demography, 2nd ed. Springer-Verlag.
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Kuhn, H.W., Tucker, A.W., 1951. Nonlinear programming. In: Proceedings of the Second Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, pp. 481–492. Lee, E.M., 1986. Introduction to pension schemes. Inst. Actuaries. Mandl, P., Mazurova, L., 1996. Harmonic analysis of pension funding methods. Insur.: Math. Econ. 17, 203–214. Owadally, M.I., 1998. The dynamics and control of pension funding. PhD Thesis. School of Mathematics, The City University, London, UK. Owadally, M.I., 2003. Pension funding and the actuarial assumption concerning investment returns. ASTIN Bull. 33 (2), 289–312. Owadally, M.I., Haberman, S., 1999. Pension fund dynamics and gains/losses due to random rates of investment return. North Am. Actuarial J. 3 (3), 105–105. Owadally, M.I., Haberman, S., 2004. Efficient gain and loss amortization and optimal funding in pension plans. N. Am. Actuarial J. 8 (1), 21–36. Owadally, M.I., Haberman, S., 2004. The treatment of assets in pension funding. ASTIN Bull. 34 (2), 425–433. Pitacco, E., 2004. Survival models in a dynamic context: a survey. Insur.: Math. Econ. 35 (2), 279–298. Rockafellar, R.T., Uryasev, S., 2000. Optimization of conditional value-at-risk. J. Risk 2 (3), 21–41. Rustem, B., 1998. Algorithms for Nonlinear Programming and Multiple Objective Decisions. John Wiley and Sons Ltd. Sharp, K.P., 1996. Pension funding by normal costs or amortization of unfunded liabilities. J. Actuarial Pract. 4 (1), 257–274. Trowbridge, C.L., 1952. Fundamentals of pension funding. Trans. Soc. Actuaries 4, 17–43 657–683. Winklevoss, H.E., 1993. Pension Mathematics with Numerical Illustrations, 2nd ed. University of Pennsylvania Press.
Optimal contributions in a defined benefit pension scheme with stochastic new entrants Luigi Colombo∗ , Steven Haberman Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, UK Received October 2004; received in revised form February 2005; accepted 11 March 2005
Abstract This paper focuses on the impact of the stochastic evolution of the active membership population on the mismatch between assets and liabilities of a defined benefit pension scheme. Classical results in the actuarial literature on pension plan population theory have been extended to the stochastic case. The paper formulates the trade-off between risk and cost of contribution strategies. Then, using a constrained nonlinear programming approach, optimal contributions strategies have been derived and the trade-off solved by means of identifying an efficient frontier. Finally, a numerical application has been carried out, showing the inefficiency of certain classical normal cost methods. © 2005 Elsevier B.V. All rights reserved. Keywords: Defined benefit; Stochastic new entrants; Optimal normal cost; Nonlinear programming
1. Introduction Defined benefit (DB) pensions schemes are structures which aim to provide workers at retirement with a benefit, which is normally linked to the final pre-retirement salary (or an average of salaries in a short period before retirement). Even though many countries have recently promoted a shift to defined contribution schemes, DB plans are still very common because of the guarantees included in this kind of plan. Pension mathematics provides a scientific approach to the funding of pension schemes, leading to several tools for calculating contributions and valuing assets and liabilities. Specifically, many methods have been developed to calculate the normal costs, i.e. the set of contributions to be paid, which leads to the actuarial equilibrium between assets and liabilities under deterministic conditions. See Winklevoss (1993) for a full description of this approach. However, demographic and financial realisations are very likely to differ from expectations, and these are the two main sources of uncertainty, with which a DB pension scheme must cope. ∗
Corresponding author. E-mail address: [email protected] (L. Colombo).
0167-6687/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2005.02.011
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Essentially, the financial risk tends to have a greater impact on the wealth of a pension scheme, than the demographic risk. With this justification, scientific works have principally tended to regard the latter as irrelevant. However, a sound management of the risk deriving from the evolution of the membership population can reduce the magnitude of the overall risk faced by a pension scheme. The demographic evolution of a pension plan’s population is a complex phenomenon, in which several factors interact in different directions. On one hand, the active membership population evolves according to the processes of new entrants and withdrawals (with a small effect for mortality and disablement). On the other hand, the retired membership evolves as a result of mortality. There is a considerable actuarial literature on the demographic evolution of populations. However, it mainly focuses on studying the evolution of mortality with the aim of (a) developing projection models and (b) investigating the impact on life insurance and annuity products. An exhaustive review of survival models is presented in Pitacco (2004). The pension plan population theory in the actuarial literature is mainly based on a standard deterministic model, where the resulting population is stable, as employed by Bowers et al. (1976) and Winklevoss (1993). Pension plan theory applies mathematical demography to the specific case of a pension plan population. In this way, it achieves powerful results, showing how equilibria in the cash flows arise under specific conditions on the evolution of the population. A special case of this model (stationary population and fixed entry age) has been further developed in Owadally (1998), where a sequence of iid random variables describes the process of annual new entrants. In this framework, the author isolates the impact of the demographic evolution on the variability of the fund value and of the contributions at any time t. With regard specifically to the accumulation phase in a pension scheme, a model for the active population has been proposed by Janssen and Manca (1997). Specifically, a discrete time non-homogeneous semi-Markov model is used to describe the changes to the internal composition of the active members. However, the total number of members (in each state, i.e. job position) is exogenously given. Mandl and Mazurova (1996) model the new entrants (at a fixed age) in a DB scheme by means of independent random sequences, which are stationary in a wide sense. By using the spectral decomposition of stationary processes and the introduction of frequency transfer functions, the authors obtain closed form expressions of the variance of the fund level, the contributions and the discounted cash flows. A stochastic model for the demographic evolution has been considered in Chang et al. (2002), where optimal decisions for pension plans are derived as a solution of a stochastic control problem, obtained using dynamic programming. However, the demographic random evolution is only accounted for implicitly, as the normal cost process is driven by a Brownian motion. In a similar fashion, Josa-Fombellida and Rinc´on-Zapatero (2004) consider the benefit flow as a stochastic process (geometric Brownian motion), while investigating how to minimise both the contribution rate risk and the solvency risk. Specifically, the random nature of the benefit is introduced in order to describe the disturbances in all of the processes that affect the benefit: such as the evolution of the plan population and of the inflation and salaries. In this work, we focus attention on the dynamics of the active membership, by means of analysing the variability of the mismatch between assets and liability, as well as proposing an optimal contribution strategy which minimises this variability. The work is organised in the following way: in Section 2, the model describing the dynamics of a DB pension scheme is introduced. Well-known results in the actuarial literature are then extended to the case of the stochastic evolution of a population. Section 3 includes the analysis of the role that different variables play in the determination of the asset-liability mismatch variability. We discuss the necessity of including the cost of implementation as a fundamental variable in the analysis of optimal contribution strategies. Finally, optimal normal cost sequences are derived as the solution of a constrained nonlinear program. A numerical application of the case of iid new entrants at a fixed age is implemented: details and results are displayed in Section 4. The work ends with conclusions and suggestions for further research.
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2. The model 2.1. DB pension scheme The pension scheme is modeled using a discrete-time stochastic model, first described and investigated in Dufresne (1988). In this section, the fundamental equations are presented, together with some of the main results derived in the literature. 2.1.1. Fund value The fund value, or fund level, is the value of the assets belonging to the scheme at a specified time t. The literature presents a number of different methodologies to evaluate the assets, among which the market value, the smoothed market value and the discounted cash flow methods are the most commonly used. It is not in the aim of this work to analyse the features of these methodologies, a comprehensive review of which can be found in Exley et al. (1997) and Owadally and Haberman (2004b). Whatever the method of valuation, the dynamic of the fund value may be described by the following recursive equation: f (t + 1) = [f (t) + c(t) − B(t)](1 + r(t))
(1)
where r(t) is the rate of return of the investment portfolio gained during the year(t, t + 1); c(t) = NC(t) + adj(t) is the annual contribution income at time t and it is computed by adding an adjustment, adj(t), to the normal cost, NC(t) and B(t) the benefit paid to the pensioners at the beginning of year t. All of the cash flows are assumed to take place at the beginning of each scheme year and are also evaluated net of price inflation.1 2.1.2. Actuarial liability The actuarial liability (AL), also known as the reserve, is the difference between the actuarial discounted values of liabilities and future contributions for current members (active and retired) of the scheme. Similarly to the fund level, the AL may also be calculated in different ways. An important concern lies in the choice of the valuation discount rate and in the methodology used to determine it. Yet, whatever the valuation discount rate, the following equation describes the expected dynamics of AL: E[AL(t + 1)|Ft ] = [AL(t) + NC(t) − B(t)](1 + i)
(2)
where i is the valuation discount rate and Ft is the filtration summarising the information available at time t. Eq. (2), which we will refer to as the general liability growth (GLG) equation, states that if the assumptions on the demographic evolution are actually borne out by experience, than the AL dynamics can be described by a recursive equation. 2.1.3. Unfunded liability From the previous quantities an amount is computed, which provides valuable information about the scheme’s wealth: the unfunded liability (ul). At any time, ul is defined by the difference between the AL and the fund value; in formula: ul(t) = AL(t) − f (t)
(3)
1 The assumption on inflation might actually overestimate the risk due to the benefit variations. This happens because benefits are usually adjusted according to price inflation; while all the other amounts vary according to salary inflation, which is usually greater than price inflation.
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It is straightforward to note that a positive value happens only when the fund at time t, f(t), is not sufficient to cover the actuarial liability, i.e. the current and future liability of the existing scheme members. Conversely, a negative ul indicates that the value of the assets is higher than the actuarial liability. Thus, it is common practice to refer to ul in order to adjust annual contributions, since it expresses the size of the current surplus/deficit, which needs to be amortised. Specifically, in this work surpluses and deficits are spread over a moving term and the resulting annual adjustment is therefore a fraction of the current ul. Features and properties of spread methods are investigated in Dufresne (1988), Owadally and Haberman (1999), Owadally (2003), Owadally and Haberman (2004a), and Josa-Fombellida and Rinc´on-Zapatero (2004). 2.1.4. Individual funding methods The quantities introduced above, AL, NC and B, are aggregated amounts; i.e. they refer to the whole scheme. However, for the case of individual funding methods, the previous quantities can be disaggregated by age and written as the corresponding individual amount multiplied by the number of existing members. Further decompositions are also possible. So, for example, NC(t) can be broken down by age and expressed in a formula as: NC(t) =
τ
NCα+x n(α + x, t)
(4)
x=0
where n(α + x, t) is the membership function, which represents the number of members aged α + x at time t; α is the minimum age and τ is the length of the working life time (τ = R − α − 1, with R fixed retirement age). NCα+x is the contribution paid by all the members aged α + x. The sequence {NCk } indicates the contribution strategy adopted in the pension plan to finance the final benefit. Several methods – known as funding methods or normal cost methods – have been formulated for the purpose of computing contribution strategies. Theoretically, an infinite number of contribution strategies exists, however, an acceptable funding method must satisfy some requirements, whose rationale lies in the fairness of the actuarial equilibrium, as well as any strict bounds and characteristics which may be imposed by an appropriate supervisory or regulatory authority.2 Similarly to NC(t), also the annual total benefit and the actuarial liability can be disaggregated by age. However, for these quantities, it is necessary to take into account the number of years of past service; i.e. the number of years during which each active member has contributed into the fund. For this purpose, the membership function is further decomposed, as in the following formula: n(z, t) =
M
n(z, t, κ),
z = α, . . . , ω
κ=0
where n(z, t, κ) is the number of members aged z, at time t and with κ years of past service. The sum of n(z, t, κ) over κ is the aggregated membership function as previously defined, i.e. regardless of the years of past service. The upper extreme M varies according to the age z: for post-retirement ages (z ≥ R) M is the longest possible past service, M = R − α; while for pre-retirement ages (z < R) the past service cannot be longer than age z less the minimum age α, so M = z − α. In order to ease the notation, instead of using a varying upper extreme, we assume that the number of active members with a past service longer than their age (less minimum age α) is zero. Thus, 2 See Lee (1986), Aitken (1996) and Sharp (1996) for a fuller discussion. We consider only individual funding methods in this paper. We note that aggregate funding methods do not involve a normal cost or the explicit derivation of the actuarial liability. The contribution rate calculated automatically leads to a nil value for the unfunded liability, because it is designed to eliminate any difference between the values of asset and liabilities.
n(z, t, κ) = 0
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∀κ > z − α
(5)
Henceforth, the actuarial liability and the annual total benefit at time t have the following expressions: AL(t) =
ω−α R−α
ALα+x,κ n(α + x, t, κ)
(6)
x=0 κ=0
B(t) =
ω−R R−α
BR+y,κ n(R + y, t, κ)
(7)
y=0 κ=0
where ω is the extreme age; ALα+x,κ is the actuarial liability corresponding to a member aged α + x with κ years of past service; and, finally, Br+y,κ is the annual benefit received by a pensioner, aged R + y, who contributed for κ years into the fund. For the purpose of this work, there is no need to go in to further detail in analysing the composition of AL and B by past service. It is sufficient to note that this model implicitly assumes that all the members having the same age and past service have accrued the same actuarial liability, and are thus entitled to an equal benefit. Disaggregating the membership function by years of past service allows the model to account for liabilities and benefits that vary according to how long each member has actually contributed into the fund. Moreover, the membership function does not need to be necessarily decreasing, thus allowing for new entrants at any age before retirement. Specifically, the following relation exists between the new entrants function g and the membership function n: n(z, t, κ) = g(z − κ, t − κ) κ pz−κ
(8)
where g(x, t) is the number of new entrants aged x, at time t; and κ pz is the probability that an individual aged z remains in the scheme for κ years. In the following section, we analyse potential models for the membership function, including the case of a randomly evolving population. 2.2. Demographic evolution The membership function n(α + x, t) describes the number of members at different ages and at any time.3 Different types of population correspond to different membership functions. Moreover, for some specific membership functions, several results have been derived in the classical actuarial literature corresponding to specific equilibria. 2.2.1. Mature population A stability condition arises in the population if the age distribution remains constant year after year. This will eventually happen, if the population decrement rates remain constant and if the increment rate grows at a constant rate.4 Part of the actuarial literature (see Winklevoss, 1993, for instance) refers to this population as mature. As a special case of stability, we take into account the case that the increment rate remains constant: the resulting population is then said to be stationary. Moreover, as stated in Winklevoss (1993), if a constant flow of new entrants annually joins the scheme, and decrement rates do not change over time, “a stationary condition will exist after n years, where n equals the oldest age in the population less the youngest.” 3 In order to distinguish whether the value of a variable (say AL) is deterministic or stochastic, subscripts (AL ) are used only to indicate t deterministic variables, while the notation AL(t) indicates a random variable. 4 Refer to Keyfitz (1985), Chapter 4, for a full analysis of the properties of stable populations.
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In terms of the membership function, stationarity is equivalent to assuming that the number of members aged x at time t is described by nα+x,κ . This assumption implies that stationarity holds for the aggregated membership function as well: nα+x,t =
R−α
nα+x,t,κ =
κ=0
R−α
nα+x,κ = nα+x
(9)
κ=0
So, for instance, the actuarial liability broken down by generations, as expressed in Eq. (6) would become constant: ALt =
ω−α R−α
ALα+x,κ nα+x,κ = AL
(10)
x=0 κ=0
Stationarity in the population evolution leads to a remarkable equilibrium between the total contributions, the returns from investments and the payment of benefits. Specifically, the GLG Eq. (2) can be rewritten in the following way: AL = (AL + NC − B)(1 + i)
(11)
As stated in Trowbridge (1952), we can refer to (11) as the equation of maturity. When the plan membership eventually becomes mature, the pension fund reaches a state of financial equilibrium, where the contribution and the expected returns on the reserve match the flow of benefits annually paid out: B = NC + d AL where d = i/(1 + i) is the discount factor associated with i. 2.2.2. Stochastic new entrants In this paper, we are interested in investigating the effects that random perturbations in the membership may have on the process of funding a DB pension. In order to do so, we shall consider the membership function n(α + x, t, κ) as a stochastic process. In detail, we assume that the random nature of this process is due to stochastic new entrants, while the decrements from the scheme population are known. In other words, we assume that the probabilities in Eq. (8) describe the proportion of members not eliminated, and that, these probabilities are deterministic and are known precisely. Clearly, the equilibrium displayed in Eq. (11) no longer holds. However, it can be shown that under equivalent assumptions regarding the stochastic membership process, an equivalent, and more general, form of equilibrium does exist. It has been proved that, assuming deterministic decrements, the GLG equation can be written in the following way5 : AL(t + 1) = [AL(t) + NC(t) − B(t)](1 + i)
(12)
to which we can refer as the liability growth (LG) process, since AL(t) is a stochastic process due to random new entrants. This is an extension of the LG equation, obtained for a deterministic stable population, presented in Bowers et al. (1976). In analogy with the simplification achieved on passing from stability to stationarity, we focus attention on the case that the number of members is random, but where the evolution is not driven by any drift. In another words, 5
Refer to Appendix A for a mathematical proof.
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n(α + x, t, κ) is assumed to be a stationary stochastic process, whose mean, by the definition of stationarity, is independent of time, i.e. E[n(x, t, κ)] = nx,κ . Under this assumption the actuarial liability at time t is a sum of random variables with constant expectation, and therefore its expectation is constant as well: ω−α R−α ω−α R−α E[AL(t)] = E ALα+x,κ n(α + x, t, κ) = ALα+x,κ nα+x,κ = AL, say (13) x=0 κ=0
x=0 κ=0
In the light of this, Trowbridge’s equation of equilibrium holds on average: AL = E[AL(t)] = E[AL(t − 1) + NC(t − 1) − B(t − 1)](1 + i) = [AL + NC − B](1 + i)
(14)
Eq. (14) is the equivalent in a stochastic framework of the equation of maturity (11) which is based on a deterministic model. In fact, Eq. (11) represents a special case of the more general result in (14).
3. Optimal normal cost methods In this section, we aim to obtain a normal cost method which is optimal in the sense that it minimises a given measure of risk, when the membership population randomly varies as described above. Therefore, firstly, a measure of risk is introduced and analysed. Subsequently, we use such a measure in order to identify the optimal contribution strategy. 3.1. Unfunded liability As already mentioned, the unfunded liability, defined in Eq. (3), is an amount which is informative of the scheme’s wealth; and, as a result, the spread method computes the annual contribution adjustment (or supplemental cost) as a proportion of this quantity. In the previous section, we have introduced a potential model to describe random perturbations to the demographic evolution of a DB pension scheme. In fact, as long as the rates of return from investments exactly match the liability valuation rate, such perturbations do not affect the variability of ul(t). This is easily proved: from Eq. (3), if 1 + r(t) = 1 + i then the resulting ul(t) is independent of the demographic variations, and its dynamics are described by the following recursive relationship6 : ul(t) = AL(t) − f (t) = ul(t − 1)(1 − k)(1 + i) where the adjustment at time t − 1 is a proportion of the current unfunded liability: adj(t − 1) = k ul(t − 1). Moreover, if at time 0 the scheme is fully funded, ul(0) = 0, then ul(t) = 0
∀t ≥ 0
However, it is well known that the financial realisations are very likely to differ from the expectations and, hence, an unfunded liability may arise, being either positive or negative. We are interested in how the ul would develop in time and the degree to which it might be sensitive to a stochastic demographic evolution when rates of return from investments differ from the expectations. In fact, we focus on the case in which the investment rate is deterministic and constant, but different from the valuation rate. 6
Refer to Appendix B for mathematical proof.
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Let r denote the deterministic annual rate of return achieved from the investments. Let us also introduce the following parameter ρ = (r − i)/(1 + i), which summarises the degree of mismatching between the rate of return and the valuation discount rate. Thus, the following holds, from (1), (3) and (12): ul(t + 1) = ul(t)(1 − k)(1 + r) − ρ AL(t + 1)
(15)
Furthermore, if the scheme is initially fully funded (i.e.: ul(0) = 0), we obtain: ul(t) = −ρ
t
AL(h)wt−h ,
w = [(1 − k)(1 + r)]
(16)
h=1
It is straightforward to note that ul(t) is null ∀t if • i = r; i.e. if the return rate from investment exactly matches the valuation rate. • k = 1; i.e. if the whole unfunded liability is instantly amortised. It is also of interest to highlight that, when w = 1 and r > i, the annual decrease in the unfunded liability is exactly given by the extra return r − i gained from investing the reserve accrued at the beginning of the year. Conversely, the annual variation is an increment if r < i. Now, w is equal to 1 when the amortised proportion of ul is equal to the discount factor corresponding to the rate r: k=
r = dr 1+r
It is worth noting that, in this case, the ul arithmetically increases/decreases year after year and the adjustment is not sufficient to lead eventually to ul being null.
3.1.1. First moment As soon as the demographic evolution is randomly perturbed, AL(t) becomes a stochastic process and so does the process ul(t). We have seen that, under the assumption of a stationary demographic evolution, the expected actuarial liability is constant at any time; i.e. E[AL(t)] = AL ∀t, as shown in Eq. (13). Furthermore, the expected unfunded liability has the following form: E[ul(t)] = −ρ AL
[(1 + r)(1 − k)]t − 1 (1 + r)(1 − k) − 1
(17)
The condition that (1 − k)(1 + r) < 1 is sufficient for the expectation (17) to converge in the limit. Trivially, ≤ 0 if r ≥ i ρ AL lim E[ul(t)] = = = 0 if r = i t→∞ (1 + r)(1 − k) − 1 ≥ 0 if r ≤ i
(18)
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3.1.2. Second moment From Eq. (16) it is also possible to derive a convenient expression for the variance of the unfunded liability process7 : t t t 2 t−h Var[ul(t)] = ρ Var = ρ2 AL(h)w w2t−h−k Cov[AL(h), AL(k)] h=1
= ρ2
τ τ
h=1 k=1
ˆ NCα+x NCα+y Cov{G(x, t), G(y, t)} + ρ2 Var[B(t)]
x=0 y=0
− 2ρ2
τ
ˆ NCα+x Cov[G(x, t), B(t)]
(19)
x=0
where the function G(x, t) summarises the weights of present and past generations of active members; as given by the following formula: t
G(x, t) =
wt−h
h−1 j=x
h=x+1
uh−j
R−α
n(α + x, j, κ)
κ=0
ˆ is the weighted history of benefits paid to the pensioners. B(t) ˆ is given by the following Further, function B(t) formula: ˆ = B(t)
t
wt−h
h=1
h−1 j=0
uh−j
ω−R R−α
BR+y,κ n(R + y, j, κ)
y=0 κ=0
From Eq. (19), it can be seen that the contribution strategy (identified by the sequence {NCk }) directly affects the variance of the unfunded liability. In fact, a normal cost method, which provides an early accumulation of fund, will lead to a different variability than another method which involves a later degree of accrual. In the development of this work, we will use the variance (19) as a measure of risk, while allowing the demographic evolution to be random. 3.2. Cost of strategies Although this work aims to find a contribution strategy which minimises the variability of the unfunded liability, in this section we discuss the necessity that a comprehensive analysis has to take into account not only measures of risk but also the actual cost, as part of a sound risk management policy. In DB pension schemes, the capital accrued at retirement age should be sufficient to finance the payment of an annual pension to each retiree. PVFB indicates the expected present value of such a benefit at retirement age. In order to accrue this amount, contribution strategies rely on the annual contributions paid by the plan’s sponsor and on the returns from investing the accrued fund value in a portfolio of assets. In this work, the plan’s sponsor is the entity liable for providing this benefit and it often coincides with the employer, who offers this retirement plan to the employees in addition to the salary. However, it is often the case that the employees participate in the pension funding by paying annual contributions linked to their current salary. Usually, in a DB scheme, these contributions are not related to the wealth of the scheme, and hence, the employee 7
Refer to Appendix C for a mathematical proof.
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does not bear any financial risk related to the favourable or unfavourable deviations of the experience from the actuarial assumptions. In such a situation, according to our definition, the employer would still be the plan’s sponsor, as he/she is liable for providing the retirement benefit. Nevertheless, the cost of pension provision would be reduced by an amount given by the expected present value of all the contributions paid by the employee.8 We assume for convenience in the rest of the discussion that the employer meets the full cost of the pension plan Considering that the financial markets provide the returns from investments, it is clear that the plan’s sponsor is expected to pay the normal cost for each member of the scheme. Thus, the higher is the proportion of the benefit funded by these normal cost, the higher is the face value of the total amount that is to be paid by the sponsor. In other words, the cost of a contribution strategy depends on how much of the benefit is funded by the contribution income, when the remainder is funded by the investment income. Bearing this in mind, we can state the following intuitive rule: the more expensive is the strategy, the lower is the risk. This is because of the timing of a contribution strategy: according to the concept of cost introduced above, the earlier is the funding of the pension (which means that the pension is being mainly funded with contributions paid at young ages), the less expensive is the strategy. This happens because an earlier contribution strategy relies more on investment returns, than a later one would do. On the other hand, an earlier contribution strategy would expose the accrued fund to the volatility of all the existing risks for a longer period. So, even though a strategy may optimally reduce the risk, its cost might be excessively high for the plan sponsor. Hence, the plan sponsor might not want to achieve the safest position. Nevertheless, it might prefer to implement a more convenient strategy, for which the resulting level of risk is still acceptable. Clearly, in the decision-making process of choosing a suitable contribution strategy, a trade off between cost and risk arises. With our aim of exploring tools in order to reduce the level of DB pension scheme risk, it appears crucial to include in the model a variable which takes into account the cost of contribution strategies. In the environment described by the current model, inflation is fixed and all of the amounts are being expressed net of price inflation. Hence, a suitable measure of the cost of a contribution strategy may be given by the sum of the normal costs. However, since not all the members are expected to survive up to the retirement age, the expected cost is instead taken into account: E[Cost(t)] =
τ
NCα+x (t) x pα
(20)
x=0
where x pα is the probability that an individual aged α survives for x years. Clearly, by artificially setting the survival probability equal to 1 at all ages, the measure given by the expected cost would collapse to the simple total cost. 3.3. Optimisation problem The aim is to identify an optimal normal costs method, in the sense that it minimises the variance of the unfunded liability, also taking into account the cost. Specifically, the measure of cost can be included in this analysis in several ways. One of these would require us to include the cost as a strict constraint to the minimisation problem. Alternatively, it could be possible to define and subsequently minimise a penalty function, which gives increasing importance to the cost measure according to a scaling (or penalty) factor. 8 The case of sharing the pension funding between the employer and the employee can be simply considered, by adequately modifying the current mathematical structure according to the sharing formula. For instance, in the case that the employee contributes an amount proportional to his/her salary, the sponsor’s annual contribution would be the balance of the normal cost not paid by the employee.
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Finally, a multiple objective function minimisation could be carried out, with the effect of simultaneously minimise risk and cost. The work developed here is mainly based on the first approach. However, the first and second approaches require the same techniques and, under particular assumptions, they are equivalent and lead to the same results. The third approach would indeed require different techniques but it could be an interesting extension for further research.9 From Eq. (19) (variance of the unfunded liability), it is possible to state the minimisation problem, for convenience expressed in matrix notation: min Var[ul(t)] = min NC ΣG(t) NC − 2 NC C(t) NC
NCk
(21)
subject to τ
NCα+x x Eα = PVFB ↔ NC E − PVFB = 0
(22)
x=0
E[Cost(NC)] = K ↔ NC p − K = 0,
NCα+k ≥ 0, k = 0, . . . , τ
(23)
where the general (i, j)th element of the square matrix ΣG(t) is given by the covariance Cov{G(i, t), G(j, t)}. Matrix ΣG(t) is then a variance/covariances matrix and therefore it is symmetric and positive defined. Furthermore, its dimension is (τ + 1) × (τ + 1). Similarly, the vector C(t) contains the covariances between the inflow and the outflow processes, i.e. ˆ Cov{G(i, t), B(t)} is the ith element of the vector. Vector p indicates the probabilities of surviving up to the retirement age: so that [p1 = τ pα , p2 = τ−1 pα+1 , . . . , pτ = 1 pα+τ−1 ]. If all the components of the vector p are set equal to 1, then the resulting measure is the simple total cost of the contribution strategy. K is the cost of the strategy that the plan’s sponsor is willing to pay to fund the retirement benefit of each employee. 3.3.1. Kuhn–Tucker conditions Associated with any optimal problem including constraints is the Lagrangian function. For the above problem, the Lagrangian function is: L(NC, λ, ν, µ) = NC ΣG(t) NC − λ(NC E − PVFB) − ν(NC p − K) − µ NC
(24)
The Kuhn–Tucker (K|T) theorem provides necessary and sufficient conditions for a point to be optimal while satisfying the imposed constraints.10 In this specific case, if an optimal solution NC∗ exists, then there must exist multipliers λ, ν and µ0 , . . . , µτ satisfying the following conditions: ∂L(NC∗ ) =0 ∂NC
(25)
NC∗ E − PVFB = 0
(26)
NC∗ p − K = 0
(27)
µk NC∗α+k = 0, λ, ν ≥ 0
k = 0, . . . , τ
(28) (29)
9 A comprehensive review of some of the potential algorithms is illustrated in Rustem (1998). Particularly interesting is Chapter IV, which deals with optimisation problems in uncertainty. An interesting application has been developed in Guo and Huang (1996) in the field of asset allocation within a fuzzy set theory framework. 10 See the original paper by Kuhn and Tucker (1951), or Chiang (1984) for a full introduction to the theory and some applications.
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µk ≥ 0,
k = 0, . . . , τ
(30)
In addition, since the variance of ul is a convex function of the normal costs, and so are the constraints (actually linear), the K|T conditions state that any point NC satisfying the above conditions is an optimal solution of problem (21). Unfortunately, including the non-negativity constraints increases significantly the computational complexity of the system to be solved, leading the constrained quadratic problem to be, in the first place, a combinatorial problem. Indeed, we must consider all the possible 2τ+1 − 1 cases, in which alternatively some or all the multipliers µk (k = 0, . . . , τ) might be null. A possible way to deal with the problem consists of numerically finding the minimum by choosing an algorithm which systematically excludes negative points. This approach has been implemented for a specific case, and details and results are shown in the subsequent Section 4. Remark. One further consideration concerns the sign of the elements involved in the minimisation. The components of both the matrix ΣG(t) and vector C in Eq. (21) are covariances. Since such covariances are ultimately given by the recursive structure of the model, it appears reasonable that such elements are all positive. In fact, the larger is the value of AL at time t − 1 the higher it is expected to be at time t. Thus, in the process of minimisation, the quadratic term will push the variables to be zero, while the term of order 1 should push them to be positive. Since the minimum value of the variance of ul must be positive, it seems reasonable to assume that the optimal solution will lie in the non-negative subspace. In the light of this discussion, the following alternative approach might be found to be worthwhile. 3.3.2. Lagrangian conditions and solution If the non-negativity constraints can be ignored, the Lagrangian function has the following expression: L(NC, λ, ν, µ) = NC ΣG(t) NC − λ(NC E − PVFB) − ν(NC p − K)
(31)
Such a minimum problem may be solved by applying the Lagrange method. Specifically, amongst K|T conditions only (25)–(27) have to hold. Hence, the optimal normal costs sequence can be found as the solution of the following system: ∂L(NC) = 2 NC ΣG(t) − 2C(t) − λE − νp = 0, ∂NC ∂L(NC) ∂L(NC) = NC E − PVFB = 0, = NC p − K = 0 ∂λ ∂ν Such a system consists of (τ + 3) equations in (τ + 3) variables and has the following unique solution: λ∗ ν∗ −1 NC∗ = ΣG(t) C+ E+ p 2 2
(32)
with multipliers λ∗ =
−1 −1 −1 −1 E ΣG(t) p(C ΣG(t) p − K) − p ΣG(t) p(C ΣG(t) E − PVFB)
ν ∗ = K − λ∗
−1 −1 −1 E ΣG(t) Ep ΣG(t) p − (E ΣG(t) p)2 −1 E ΣG(t) p −1 p ΣG(t) p
−1 p − C ΣG(t)
(33)
(34)
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Fig. 1. Variance of ul for several contribution strategies.
4. Numerical application In order to display the effect of using optimal contributions strategy, numerical calculations have been carried out. A suitable expression for the membership stochastic process is therefore needed. Specifically, the membership process has to lead the population to stationarity on average. The assumption of annual new entrants joining the scheme at a fixed aged independently between years satisfies the required condition. The new entrants are thus modeled by means of a stochastic process {gt } of iid random variables with the same mean g and variance σg2 . Hence, the membership function has the following form: n(α + x, t) = gt−x x pα Under this assumption, it is possible to obtain closed expressions for the covariances introduced in the previous section.11 Fig. 1 displays the exact value of the variance of ul during the years 0–80 for different contribution strategies. In detail, setting the cost to its maximum leads us to find the safest strategy.12 The resulting contribution path is known in the actuarial literature as terminal funding (TF), which consists of financing a pension with the payment of a lump sum at the time of retirement. By implementing a TF strategy, the variance of ul is the lowest possible, thus generating in the graph the lowest line. Conversely, by setting the cost equal to its minimum, the minimisation leads to the cheapest and riskiest contribution strategy. In this case, the pension would be financed by the payment of a lump sum at the beginning of the working life-time and by investing this capital during that period. This strategy is known as initial funding (IF). Clearly, the funding capital would be exposed for a long period to the financial risk and thus the resulting variance of ul is the largest possible (the top line in the graph). For intermediate levels of the cost, the minimisation problem leads to contribution strategies that are riskier as the cost increases. 11 Refer to Appendix D for closed formulas and to Appendix E for a detailed summary of the assumptions made and the values of parameters used. 12 It is easy to show that, once the survival probabilities p and the valuation rate i have been set, the cost measure has an unique minimum and an unique maximum.
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Fig. 2. Efficient cost/risk frontier.
It is worthwhile noting that, the contribution strategy does reduce (or increase) the variance of ul, but does not change its fundamental trend over time. All of the lines in Fig. 1 show a sigmoidal shape, which is due to the characteristics of the demographic risk. As argued in Section 3.2, the riskiness of a strategy is directly related to its cost. In fact, a trade off between risk and cost of a strategy exist. Fig. 2 displays such a trade off by means of identifying the efficient frontier. Specifically, the variance of ul at time t = 80 is computed according to Eq. (19), and subsequently plotted on the y-axis against the cost of the corresponding strategy on the x-axis.13 As predicted by the intuitive rule, as the cost increases the variance decreases and vice versa. A detailed examination reveals that, the strategy TF is the safest and the most expensive strategy. Therefore, the resulting point for this method is displayed at the right hand end of the graph. Conversely, the strategy IF is the cheapest and the riskiest possible strategy, and so the corresponding point is displayed at the left hand end of the graph. The black line identifies the efficient frontier, which represents the trade off between cost and risk of a contribution strategy. For the sake of comparison, a set of classical normal costs methods has also been selected. It is interesting to note that all of them have turn out to be inefficient. In fact, the unit credit (UC), fixed entry age (EA) and constant premiums (CP) methods lie above the efficient line. Accepting the level of risk that each of the classical methods implies, it is always possible to find a less expensive contribution strategy. Conversely, given the cost of each of the classical methods, it is possible to find a strategy which implies lower riskiness. As previously mentioned, it is of interest to investigate how the efficient line is sensitive to the spread period. The longer this period is, the more variable is the resulting unfunded liability. Bearing in mind that the covariances between AL at different points in time are positive, it can be seen from Eq. (19) that the variance of ul is an increasing −m )(1 + i), the variance of ul increases function of w. Since w = (1 − k)(1 + r) and k = 1/¨am| ¯ = i/(1 − (1 + i) with the spread period (m).
13 Time and spread period seem to affect the magnitude of the results, but not its meaning. Since the spread period is one of the tools commonly used to calibrate convenient strategies with respect to the solvency and stability of the scheme (see Haberman and Sung, 1994), a sensitivity analysis on the parameter m has been carried out. The results are illustrated at the end of this section.
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Fig. 3. Efficient cost/risk frontier and spread period.
In order to illustrate the effect of this parameter on the efficient frontier, we have computed the variance of ul for a set of spread periods and for different acceptable levels of cost. The resulting variances have been displayed in Fig. 3.14 The displayed lines are consistent with the expected behaviour that the larger is m, the higher is the variance of ul. Specifically, the increments in the spread period have a significant effect on the risk. In fact, it can be seen that for a high value of m (say m = 8 or 9) the most expensive contribution strategy (cost = 1) yields a level of risk which is higher than the variance achievable from combining a cheaper strategy with a shorter spread period (say m ≤ 5 and cost = 0.2). However, the risk can be maintained on the same level by choosing appropriate combinations of cost and m. Table 1 displays the cost necessary to achieve a given level of risk, when combined with a set of possible spread periods. These figures highlight the strong impact that the spread period has on the variability of ul. In order to have a variance equal to 20,000, a very cheap contribution strategy (12% of TF cost) is needed when the surpluses and deficits are spread over 4 years. In order to achieve the same position when m = 5 or 6, the cost of the optimal strategy is respectively 34% and 60% the cost of the TF strategy. Increasing the spread period by one further year, m = 7, a strategy almost as expensive as TF is needed in order to maintain the variance at the chosen level. Finally, it is worthwhile comparing again those figures with those determined by the classical methods considered. In Table 2 the relative cost and the variance of ul, corresponding to each classical method, when m = 5 and t = 80 are displayed. The inefficiency of those methods is still evident. For instance, the entry age method requires a cost slightly higher than 21% and implies a variance higher than the level of 26,000, which is achievable by implementing the optimal strategy for a comparable cost. Similar conclusions apply for the other two methods. 14 In Fig. 3 and in the following tables the actual cost of each contribution strategy has been rescaled in the interval (0, 1), by dividing by the cost of the TF strategy, which is the maximum cost.
L. Colombo, S. Haberman / Insurance: Mathematics and Economics 37 (2005) 335–354
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Table 1 Relative cost for different levels of variance of ul (×1000) and spread periods m
2 3 4 5 6 7 8 9 10
Variance (%) 5
11
16
20
26
31
38
11 62 – – – – – – –
– 12 44 94 – – – – –
– – 22 49 86 – – – –
– – 12 34 60 98 – – –
– – – 21 41 64 93 – –
– – – 13 31 50 71 97 –
– – – – 21 37 54 72 93
Table 2 Relative cost and variance for classical methods
CP EA UC
Cost (%)
Risk
15.5 21.6 31.5
31700 28800 24700
5. Conclusions In the development of this work, we have extended existing results in pension plan population theory to the more general case of a stochastic active membership. Specifically, we have shown that a stochastic stationary evolution leads to a condition of maturity on average. Focusing on the unfunded liability, we have used an expression for its variance in order to highlight the role of the contribution strategy in managing the risk of mismatching between assets and liabilities. Specifically, we have focused on the way in which stochastic new entrants may amplify the effect of the mismatch between the expected and actual returns from investments. Subsequently, we have shown how constrained nonlinear programming can be applied to derive optimal contribution strategies. Under specific conditions, it has also been possible to derive an analytical solution. When this has not been possible, a numerical algorithm has been shown to be a suitable alternative for solving the optimisation problem. With regard to the specific case of iid new entrants at a fixed age, optimal strategies have been found numerically. Furthermore, the cost of a strategy has been taken into account, since the cost of a resulting optimal strategy might be too high for adoption by a plan sponsor. Thus, the trade-off between cost and risk of a contribution strategy has been solved, by means of finding an efficient frontier. In addition, classical normal cost methods have been compared to the optimal strategies, showing the inefficiency of these methods in terms of cost and risk. The analysis would be extended by including explicitly for the possibility of sharing of the contribution between the sponsor and the scheme member, following the discussion of Section 3.2. Further research will also focus on natural extensions of the existing model. Financial risk could be included by means of stochastic random returns. However, the resulting expression for the variance of ul might be too cumbersome to allow for analytical solutions, but numerical algorithms could be implemented and will be investigated.
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351
The possibility that the plan’s sponsor defaults in its payments is a feature that could be included in the actual model. Such a source of uncertainty would affect the risk ordering of contribution strategies, and thus would modify the efficient frontier. Including different risk measures constitutes another area of interest for further extensions. Specifically, downside risk measures, and especially those satisfying Artzner’s axioms of coherence, could be included.15 Rockafellar and Uryasev (2000), and subsequent papers, explored this direction developing algorithms to optimise the conditional value at risk (CVaR) and thus identifying efficient frontier – similar to a Markovitz return/risk plane – in an asset allocation problem. Finally, different techniques could be used to take simultaneously into account the minimisation of cost and risk. A possible avenue identified in the literature comes from the successful attempts in applying algorithms to solving multiple objective functions optimisation problems in finance; see Guo and Huang (1996) for an application to an asset allocation problem in a fuzzy set theory framework.
Appendix A. Proof of liability growth process Under the assumption of deterministic decrements and independently of the fact that the number of members aged x at time t is deterministic or not, the following relation holds: n(x, t) y−x px = n(y, t + y − x) Hence, the GLG equation has the following expression: AL(t) =
∞
ALx n(x, t) =
∞
x=0
=
∞
x=0
(By − NCy )n(y, t) +
y=0
+
n(x, t)
∞
∞
(By − NCy )vy−x y−x px =
y=x ∞ ∞
∞ ∞
(By − NCy )vy−x n(y, t + y − x)
x=0 y=x
(By − NCy )vk n(y, t + k) = B(t) − NC(t)
k=1 y=k ∞
(By − NCy )vh+1 n(y, t + h + 1) = B(t) − NC(t) + v AL(t + 1)
h=0 y=h+1
− vn(0, t + 1)
∞
(Bh − NCh )vh n(0, t + 1)
h=0
= B(t) − NC(t) + v AL(t + 1) − vn(0, t + 1)AL0 = B(t) − NC(t) + v AL(t + 1) where AL0 = 0 for the equivalence principle.
Appendix B. Proof of ul dynamics In Section 3.1 is stated that as long the rates of return exactly match the liability valuation rate and that the annual supplemental cost is proportional to the current unfunded liability, the resulting dynamics of ul are independent of the variations in the number of new entrants in the scheme. As a matter of fact if 1 + r(t) = 1 + i, then the dynamic of the fund value is given by the following equation:
15
Refer to the seminal work of Artzner et al. (1999) for an introduction to coherent risk measures.
L. Colombo, S. Haberman / Insurance: Mathematics and Economics 37 (2005) 335–354
352
f (t) = (f (t − 1) + NC(t − 1) + k ul(t − 1) − B(t − 1))(1 + i) Hence the following relation holds: ul(t) = AL(t) − f (t) = (AL(t − 1) + NC(t − 1) − B(t − 1))(1 + i) − (f (t − 1) + NC(t − 1) + k ul(t − 1) − B(t − 1))(1 + i) = (AL(t − 1) − f (t − 1) − k ul(t − 1))(1 + i) = ul(t − 1)(1 − k)(1 + i)
Appendix C. Proof of ul variance
Var[ul(t)] = ρ2 Var
t
AL(h)wt−h
= ρ2 Var
t
h=1
= ρ2 Var
= ρ Var 2
t
h=1
τ
wt−h
h=1
wt−h
h−1 j=0
uh−j
τ
NCα+x
x=0
r−α
h−1
[NC(k) − B(k)]uh−k
k=0
ˆ n(α + x, j, κ) − B(t)
κ=0
ˆ NCα+x G(x, t) − B(t)
x=0
= ρ2
τ τ
ˆ NCα+x NCα+y Cov{G(x, t), G(y, t)} + ρ2 Var{B(t)}
x=0 y=0
−2ρ2
τ
ˆ NCα+x Cov{G(x, t), B(t)}
x=0
Appendix D. Closed covariances for iid new entrants Under the assumption of iid new entrants at a fixed age α, it is possible to close the expression for the elements in the matrix ΣG(t) and in the vector C(t):
[x, y] = x pα y pα
G(t)
(ut−x − wt−x )(ut−y − wt−y ) u|x−y|+1 (uw)t−max(x,y) − 1 − (u − w)2 u−w uw − 1
uwt−min(x,y) ut−max(x,y) − wt−max(x,y) uw − 1 u−w 2(t−max(x,y)) 1 u w −1 + wt−min(x,y) + uw − 1 u − w w2 − 1
−
(35)
L. Colombo, S. Haberman / Insurance: Mathematics and Economics 37 (2005) 335–354
Cx (t) = x pα w
t−x
B
353
M 2 (t−1) t−τ−y+1 u y=0
t−x − wt−τ−y+1 u − wt−x uw − u−w (u − w)wt−τ−y uw − 1
2(t−τ−y+1) 1 − 1] −1 u w − + + t−x−1 2 t−τ−y w (u − w)(uw − 1) uw − 1 u − w (w − 1)w t−τ−y+1 t−τ−y+1 −w u + τ+y pα u−w uτ+y−x+1 [(uw)t−τ−y+1
(36)
Appendix E. Assumptions and parameters for calculations and simulations E.1. Model assumptions and parameters • Valuation rate: i = 3%; investment returns rate: R = 3.5%. • Contribution adjustments method is the surplus/deficit spreading over a fixed term: adj(t) = k ul(t) =
ul(t)i (1 − (1 + i)−m )(1 + i)
(37)
where the amortisation period m has been initially set equal to 4. • Fixed entry age α = 20, retirement age r = 65, extreme age ω = 105. • Data from Watson and Wyatt, have been used to derive the service table (probabilities of remaining in the scheme) and salary scale. E.2. Assumptions and parameters for simulations • New entrants process{gt } is normally distributed with mean g = 1000 and standard deviation σg = 250. 10,000 simulations have been carried out, where negative values below 0 have been artificially discarded and set equal to 0.
References Aitken, W.H., 1996. A Problem Solving Approach to Pension Funding and Valuation, 2nd ed. ACTEX Publications, Winsted, CT. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Math. Finance 9 (3), 203–228. Bowers, N.L., Hickman, J.C., Nesbitt, C.J., 1976. Introduction to the dynamics of pension funding. Trans. Soc. Actuaries 28, 177–203. Chang, S.C., Tsai, C.H., Tien, C.J., Tu, C.Y., 2002. Dynamic funding and investment strategy for defined benefit pension schemes: a model incorporating asset-liability matching criteria. J. Actuarial Pract. 10 (1). Chiang, A.C., 1984. Fundamental Methods in Economics, 3rd ed. McGraw-Hill. Dufresne, D., 1988. Moments of pension contributions and fund levels when rates of return are random. J. Inst. Actuaries 115, 535–544. Exley, C.J., Mehta, S.J.B., Smith, A.D., 1997. The financial theory of defined benefit pension schemes. Br. Actuarial J. 3, 835–966. Guo, L., Huang, Z., 1996. A possibilistic linear programming method for asset allocation. J. Actuarial Pract. 4 (1), 67. Haberman, S., Sung, J.H., 1994. Dynamic approaches to pension funding. Insur.: Math. Econ. 15, 151–162. Janssen, J., Manca, R., 1997. A realistic non-homogeneous stochastic pension funds model on scenario basis. Scand. Actuarial J. Josa-Fombellida, R., Rinc´on-Zapatero, J.P., 2004. Optimal risk management in defined benefit stochastic pension funds. Insur.: Math. Econ. 34, 489–503. Keyfitz, N., 1985. Applied Mathematical Demography, 2nd ed. Springer-Verlag.
354
L. Colombo, S. Haberman / Insurance: Mathematics and Economics 37 (2005) 335–354
Kuhn, H.W., Tucker, A.W., 1951. Nonlinear programming. In: Proceedings of the Second Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, pp. 481–492. Lee, E.M., 1986. Introduction to pension schemes. Inst. Actuaries. Mandl, P., Mazurova, L., 1996. Harmonic analysis of pension funding methods. Insur.: Math. Econ. 17, 203–214. Owadally, M.I., 1998. The dynamics and control of pension funding. PhD Thesis. School of Mathematics, The City University, London, UK. Owadally, M.I., 2003. Pension funding and the actuarial assumption concerning investment returns. ASTIN Bull. 33 (2), 289–312. Owadally, M.I., Haberman, S., 1999. Pension fund dynamics and gains/losses due to random rates of investment return. North Am. Actuarial J. 3 (3), 105–105. Owadally, M.I., Haberman, S., 2004. Efficient gain and loss amortization and optimal funding in pension plans. N. Am. Actuarial J. 8 (1), 21–36. Owadally, M.I., Haberman, S., 2004. The treatment of assets in pension funding. ASTIN Bull. 34 (2), 425–433. Pitacco, E., 2004. Survival models in a dynamic context: a survey. Insur.: Math. Econ. 35 (2), 279–298. Rockafellar, R.T., Uryasev, S., 2000. Optimization of conditional value-at-risk. J. Risk 2 (3), 21–41. Rustem, B., 1998. Algorithms for Nonlinear Programming and Multiple Objective Decisions. John Wiley and Sons Ltd. Sharp, K.P., 1996. Pension funding by normal costs or amortization of unfunded liabilities. J. Actuarial Pract. 4 (1), 257–274. Trowbridge, C.L., 1952. Fundamentals of pension funding. Trans. Soc. Actuaries 4, 17–43 657–683. Winklevoss, H.E., 1993. Pension Mathematics with Numerical Illustrations, 2nd ed. University of Pennsylvania Press.