Transiciones agroecológicas hacia la sustentabilidad

To finance the pension fund, employers and employees make on a regular basis payments to the fund. These payments are a percentage of a part of the total wages of the participants. This part is called the pensionable salary, and is denoted by

Ws

t for year t. The board of the pension fund has to determine the level of the

percentage of these pensionable salaries. For time t and scenario s, the level of this

percentage is denoted by cs

t. This level is determined at the beginning of every year.

As a result, for year t + 1, this level is specified at time t.

only has to take into account bounds on this level, it also considers the wish of a relatively stable contribution rate through time. Moreover, from the perspective of the sponsor it is reasonable to assume that the level of the contribution rate should be sufficiently high if the sponsor of the fund makes a remedial contribution, so that also active participants contribute to a better financial position of the fund. Details with respect to these policy rules and the mathematical formulations are discussed in this section.

Although the board can determine the level of the contribution rate, this level is bounded. Even if the funding ratio is so low, that not all future benefit payments

can be guaranteed, cs

t cannot increase indefinitely. This upper bound on cst is de-

noted by c, and is assumed to be time independent. Also a time independent lower

bound on cs

t, which is denoted by c, exists. The numerical specification of c and c

may be fund dependent, and may be stated in the fund’s pension regulation. We assume that the following conditions hold: 0 ≤ c < c < 1.

The lower and upper bounds on the contribution rate cs

tgive rise to the follow-

ing constraint in our ALM model:

c≤ cs

t ≤ c. (2.12)

Not only the level of the contribution rate is important, but also its stability, since too much variability is undesirable.

We can model this refinement as

−η ≤ cst− cst−1≤ ρ, (2.13)

where cs

t − cst−1 represents the change in the contribution rate in two consecutive

years (t and t − 1) and η and ρ are fixed bounds for decreases and increases of the contribution rates. In the ALM model presented in Kouwenberg [55], these ‘hard’ constraints are used.

The numerical values for η and ρ also fund dependent. In addition, these values are expected to be not too low. On the other hand, if the values of η and ρ are large, (2.13) loses its meaning in modeling the undesirability of a contribution rate which changes rapidly.

However, if the funding ratio is relatively low for a number of years, it may be preferable to increase the contribution rate rather than to ask for a remedial contribution. This may lead to an increase in the contribution rate which exceeds ρ. Hence, it is better to specify (2.13) as a goal constraint: increases (decreases) greater than ρ (η) are allowed (although constraint (2.12) still has to be satisfied), but they are penalized in the objective function. Now, η and ρ denote the maximum decrease and increase in the contribution rate in two consecutive years, such that no penalties are incurred due to a rapidly changing contribution rate.

We model this in a linear programming formulation by the introduction of addi-

tional decision variables cis

t, representing the amount by which the increase in the

contribution rate exceeds ρ at time t compared with the contribution rate at time

t− 1. The second inequality of (2.13) is replaced by

cis_t ≥ cs

t− cst−1− ρ

2.6 CONTRIBUTION RATE 47

In the objective function, we penalize cis

t = (cst − cst−1− ρ)+, which is positive if

cs

t> cst−1+ ρ.

We can use an analogous reasoning when the funding ratio is relatively high for a number of successive years. In this situation it may be preferable to lower the contribution rate in two successive years by more than η. This may be more desir- able than, for example, making a restitution to the sponsor, because the board of the fund would like that the active participants profit from the financial prosperity of the fund.

We can model this by the introduction of additional decision variables cds

t, rep-

resenting the amount by which the decrease of the contribution rate in year t ex- ceeds η, as compared with the contribution rate at time t − 1:

cds_t≥ cs

t−1− cst− η

cds_t≥ 0.

In the objective function, we penalize cdst = (cst−1− cst− η)+, which is positive if

cs_t< cst−1− η.

We penalize cis

t and cdst by positive parameters ζciand ζcd(usually ζcd ≤ ζci),

whereas no penalty is imposed if (2.13) is satisfied. Figure 2.5 shows an example of such a penalty function. In stochastic programming, this structure, that models

Penalty

c_t−c_t−1

−η 0 ρ

(−ζ_cd)

(ζ_ci)

Figure 2.5: Penalty costs associated with a change in the contribution rate at time t, as compared with the contribution rate at time t − 1.

piecewise linear increasing costs for shortages and surpluses, is known as multiple simple recourse. For details on multiple simple recourse, see Klein Haneveld [53]

and Van der Vlerk [91]. We have never seen this structure in ALM models in the literature before.

The formulation described above makes sense in practise: relatively large in- creases in contribution rates are faced with a lot of resistance by active participants. As a result, if large deviations of the current level of the contribution rate could be avoided, this would be the best alternative for the parties involved in the de- cision making process. On the other hand, if large increases are needed, because otherwise the financial position would become too weak, such an increase can be considered.

Note that the above introduced formulation is also very flexible: if managers do not want to penalize large deviations in the contribution rate, they can choose to

either specify ζcd= ζci= 0, or to set η = ρ ≥ c − c.

Contribution rate in case of a remedial contribution

As we have seen before, we included in our model the policy rule, that the sponsor of the fund has to restore the funding ratio if there is a shortage with respect to the level α in a number of consecutive years. It seems reasonable that if such a remedial contribution is made, the sponsor requires that the active participants also provide a sufficient contribution to the financing of the fund in the form of a sufficiently high contribution rate. Therefore, we will include this condition in a policy rule of our model, as follows. We denote the minimum required contribution rate in case

of a remedial contribution by c∗_{. The numerical value of c}∗_{may for example be the}

fund’s actuarial premium. The constraint, which serves the rule mentioned above, is given by:

cs_t− c∗_{≥ M (z}s

t− 1). (2.14)

In this case, big M may be taken equal to c−c. If the sponsor of the fund has to make

a remedial contribution, and, as a result, zs

t = 1, (2.14) leads to the requirement

cst ≥ c∗. On the other hand, if zts = 0, no additional requirement with respect to

the level of the contribution rate is made. Note that if one does not want to model

this relationship between cs

tand Zts, one can simply eliminate this rule by choosing

c∗_{= c, since c}s

t≥ c always has to be satisfied.

Of course, there are a variety of alternative formulations for requirements with respect to the level of the contribution rate if the sponsor of the fund makes a reme- dial contribution. A few alternatives are presented below.

• Require that the contribution rate does not decrease if zs

t = 1. This can be

accomplished by the constraint cs_t− cst−1≥ M (zts− 1).

The disadvantage of this modeling is, however, that even if the sponsor of the fund makes a remedial contribution, it is allowed that the contribution rate may still be very low.

2.7 INDEXATION 49

• Penalize the deviation of the contribution rate from its upper bound c. This can be accomplished by the constraint

c− cs

t− cdust≤ M (1 − zst), (2.15)

and penalize the nonnegative decision variable cdus

tin the objective function.

If the sponsor has to make a remedial contribution, (2.15) results in cdus_t≥ c − cs

t,

and, as a result, the deviation of the contribution rate from its upper bound

can be penalized. On the other hand, if zs

t = 0, constraint (2.15) is non-

binding.

A disadvantage of this modeling is that it may be difficult to find an appro-

priate value for the penalty parameter associated with cdus

t.

Which constraint(s) are added to the ALM model may depend on the contri- bution policy of the fund. Because we think formulation (2.14) is the most impor- tant for real world practice, we have chosen to include this constraint in our ALM model.