Short and Long-term Risk Control and Mid-Term Recovery Plan in ALM for Dutch Pension Funds

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Plan in ALM for Dutch Pension Funds

Lei Xu

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1 Introduction 3

2 Problem, Approach, and Literature 5

2.1 Problem Description . . . 5 2.2 Approach . . . 6 2.3 Literature Reviews . . . 8 3 Mathematical Models 10 3.1 Modeling . . . 10 3.1.1 Decision Variables . . . 10 3.1.2 State Variables . . . 11 3.1.3 Stochastic Parameters . . . 11

3.1.4 Constraints of Actuarial Principle . . . 12

3.1.5 Constraints of Pension Policies . . . 13

3.1.6 The Mid-Term Recovery Plan . . . 14

3.1.7 The Short-Term Risk Criterion . . . 14

3.1.8 The Long-Term Risk Criterion . . . 17

3.1.9 Objective Function . . . 18 3.2 Mathematical Formulation. . . 20 3.2.1 Constraints . . . 21 3.2.2 Objective Function . . . 22 3.3 Models. . . 22 3.3.1 Basic Model. . . 23 3.3.2 MTRP Model . . . 24 3.3.3 STRCcc+MTRP Model . . . 25 3.3.4 STRCicc+MTRP Model . . . 26 3.3.5 STRCcc+MTRP+LTRCcc Model . . . 27 3.3.6 STRCicc+MTRP+LTRCicc Model . . . 28 4 Numerical Analysis 29 4.1 Data . . . 30

4.2 Scenario Tree Selection. . . 31

4.3 Numerical Results . . . 33

4.3.1 Basic VS MTRP . . . 33

4.3.2 MTRP VS STRCcc+MTRP . . . 37

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4.3.4 STRCcc+MTRP vs STRCcc+MTRP+LTRCcc . . . 47

4.3.5 STRCicc+MTRP vs STRCicc+MTRP+LTRCicc . . . 51

5 Conclusions 57

A Sensitivity Analysis 59

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Introduction

This thesis will address Asset and Liability Management (ALM) for Dutch pension funds. In particular, several modeling issues in FTK [8] and the effects of these issues will be explained and explored later on.

The purpose of a pension fund is to fulfill obligations made to its pensioners. It does so by providing these people with an income when they retire. Active participants, inactive partic-ipants, and a pension sponsor (company) are involved in a pension fund. Active participants pay contributions and inactive participants receive benefit payments. Pension sleepers who switched to other jobs did not contribute to this fund any more, and they might also receive benefit payment of this fund. The pension sponsor uses the contributions from active partic-ipants to invest in different asset classes. Moreover, the pension sponsor may pay remedial contributions for cure of a funding shortfall, which is defined by the value of assets less than a 105 percent of the value of liabilities.

According to types of pensions [10], there are three pensions which are employment-based pensions, state pensions, and disability pensions. Employment-based pensions are arrange-ments to provide employees with financial supplement during their retirement. State pensions are that countries build funds for their citizens in provision of income when they retire. When people suffer from a disability and cannot work, disability pensions will offer them incomes. In the Netherlands, employment-based pensions occupy the most value of assets [10]. This research focuses on employment-based pensions.

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stochastic parameters as explained later on. In addition, pension funds decide which asset classes invest and how much to invest. For example, 50% investments are in stock, 30% in bond, and 20% in cash. An important issue during decision making is that pension funds should keep a certain funding ratio (i.e.105%), which is defined by total assets divided by total liabilities. In case of a funding shortfall, pension funds may decide possible remedial contributions paid by the pension sponsor.

In pension funds, the value of future liability, future total pensionable wages, future benefit payments, and future rates of returns on investments are important for goal achievement and decision-making. For example, the high value of future liability may reap a low funding ratio. High future total pensionable wages possibly increase contributions from active participants. Large future benefit payments mean a great deal of money to inactive participants or pension sleepers. High future rates of returns on investments in assets may increase investments in these assets, and of course it also depends on that whether investors are risk averse, risk neutral, or risk seeking.

In this thesis, the value of future liability, future total pensionable wages, future benefit payments, and future rates of returns on investments will be taken into account as random variables having probabilistic information. For instance, we know the value of liability now, the value of liability next year is random. We assume there are many values of liability next year, and each of them can be reached by a specified probability based on the present state (now).

Pension funds in the Netherlands should comply with rules of supervision on pension funds issued by De Nederlandse Bank (DNB). In this thesis, we will model several supervision rules and investigate effects of these rules on decisions on investments and financing strategies. To explore the effects of supervision rules, we will build several models and compare different outputs of models. The models are built by using stochastic programming, and providing dis-tributions of random parameters under the solvency requirements of DNB with an objective of total funding costs minimized.

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Problem, Approach, and Literature

In this chapter, we introduce the main issues which should be modeled. In Section 2.1,we describe the main problem dealt with and state the research questions for this thesis. In Section 2.2, we argue the ways to cope with the problem. In Section 2.3, we provide a literature review of how other researchers dealt with these issues of research questions.

2.1 Problem Description

The Asset Liability Management problem (ALM) for pension funds is a dynamic decision problem under uncertainty. Asset management needs to make decisions on the asset portfolio, especially for investments in different asset classes and contributions from active participants. Liability management includes pension payment to inactive participants in the future. Due to a long time horizon, typically up to 30 years, the pension problem is certainly dynamic. Uncertain returns on asset investments, the valuation of future liabilities, future total pen-sionable wages, and benefit payments at market values are sources which cause uncertainty. Market values are the true values of future liabilities, future wages, and future payments. In this thesis, we take indexed future liabilities, indexed future wages, and indexed future payments. One of the essential purposes of ALM is to attempt to reach sufficient solvency of the fund during the long time horizon. The solvency is characterized by the funding ratio and defined by the value of assets over the value of the liabilities.

In the Netherlands, solvency for Dutch pension funds is supervised under FTK (Financieel Toetsingskader) issued by De Nederlandse Bank (DNB) [8]. In this thesis we investigate the short-term risk criterion, the mid-term mid-term recovery plan, and the long-term risk criterion of FTK for Dutch pension funds. We will first describe them and then explain how to model them. Descriptions of the short-term risk criterion, the mid-term mid-term recovery plan, and the long-term risk criterion are stated as follows:

• The Short-Term Risk Criterion

In order to be able to meet benefit payment obligations and achieve sufficient solvency, FTK requires that based on a year t, pension funds should at least achieve a funding ratio (105%) in year t + 1 with high reliability (0.975).

• The Mid-Term Recovery Plan

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mandatory to draw up a recovery plan in the event of a funding shortfall. FTK postu-lates this recovery plan that contains a remedy to enable the fund to comply once again with a 105% funding ratio within three years.

• The Long-Term Risk Criterion

Besides the short-term risk criterion and the mid-term recovery plan, FTK also requires pension funds to meet their pension liabilities in the long-term. Hence a 105% funding ratio is demanded at the end of the time horizon.

We are going to explore implications if pension funds implement the short-risk risk criterion, the mid-term recovery plan, and the long-term risk criterion. To do so, we need to adapt models to account for the short-risk risk criterion, the mid-term recovery plan, and the long-term risk criterion. Then research questions in this thesis are presented as follows:

1. How to implement the short-term risk criterion, the mid-term recovery plan, and the long-term risk criterion into models to support decisions making in pension funds? 2. What are effects of the short-term risk criterion, the mid-term recovery plan and, the

long-term risk criterion on funding ratios and decisions on investment and financing policies of pension funds such that enough benefits made to inactive participants, small contributions paid by active participants, and possibly small remedial contributions from the pension sponsor?

2.2 Approach

Recall that the problem is dynamic with uncertainty. Then we should think of an approach that can deal with dynamics and uncertainty. Indeed, this is related to the selection of the mathematical model for the problem. Before answering our research questions, we should answer the following questions:

• Which mathematical model is appropriate for modeling the problem discussed in this thesis?

• How should the interplay between decisions and stochastic parameters be modeled? • How should uncertainty be modeled?

Mathematical Model Selection

As motivated below, we adopt a multistage recourse model in this thesis.

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stochastic parameters. The recourse model is suitable to model this so-called the here-and-now decision problem, which is a decision-making problem such that optimal decisions to be made before the realizations of some stochastic parameters.

Pension funds span a long time horizon. Rates of returns on investments, total pensionable wages, benefit payments, and liabilities alter over time. Decisions on investment and financing can and should be adjusted over time. A multi-stage recourse model can capture these features of the pension problem.

A recourse model usually contains at least one recourse variable, which is used to model deviations from targets, and is usually penalized. In the pension problem, the remedial contribution could be modeled as a recourse variable. We explain this as follows:

One target of pension funds is to achieve certain funding ratios. If there are funding shortfalls, remedial contributions may be paid for recovery of the funding ratios. Remedial contributions which can be seen as deviations from the target are undesirable for the pension sponsor. Recourse variables are appropriate modeling tools for this purpose. Considering a pension fund, we specify the targets in term of the funding ratios. A funding shortfall considered as the deviation of targets could be cured by a positive remedial contribution, which is a penalized recourse variable.

Interplays Between Decisions and Stochastic Parameters

As mentioned before, a multistage recourse model is a mathematical model that models fu-ture decisions dependent on fufu-ture observations. For example, following Klein Haneveld and Van der Vlerk [13], we discretize the time horizon of a pension fund into finite stages. Future decisions correspond to certain points along the time horizon, see the future decisions xt,

t ∈ {2, ..., T } in Figure 2.1. Points along the time horizon are stages t as t = 1, 2, ..., T . The first stage t = 1 means ’now’. In Figure2.1, between two consecutive decisions xt and xt−1,

t ∈ {2, ..., T }, we may observe realizations of a set of random parameters ωt, t ∈ {2, ..., T }.

Decisions xtat stage t are dependent on decisions xt−1 at stage t − 1 and on the observations

of ωt at stage t, t ∈ {2, ..., T }. This implies that every future decision xt made depends on

probabilistic information available up to time t along the time horizon. First-stage decisions x1 are made having only probabilistic information on future observations ωt, t ∈ {2, ..., T }.

Decisions xt−1 at stage t − 1 and the future observations of ωt at stage t are assumed to be

independent.

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Modeling Future Uncertainty

Assume there is a random process {ω2, ..., ωT}. The random variables ωt, t ∈ {2, ..., T }

are discretized with finitely many realizations. There are many possibilities of sequences of realizations. Such a sequence of realizations of these random variables is called a scenario. Let values of these random variables ωt, t ∈ {2, ..., T } be revealed at stage t. All scenarios is

the set of all realizations (ω₂s, ..., ωs_T) with s ∈ S := {1, ..., S}, and S presents a finite number of scenarios. Each scenario s has a probability ps with ps > 0 and PS

s=1ps = 1. These

scenarios can be combined and depicted a scenario tree, see Figure 2.2. Some scenarios may have the common history. For example, the first two scenarios at the final stage t = 4 share the same history up to the stage t = 3.

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Figure 2.2: A scenario tree of 4 stages with 18 nodes and 9 scenarios.

In Figure 2.2, the node at the first stage is called the root node. A node at the final stage is called a end node. A path from the source node to the end node is a scenario. Every node in this tree represents a possible state (t, s) at stage t with a scenario s. A path from one node to its successor is a so-called branch, which represents a possible realization of random variables {ω2, ..., ωt} from a given state to the future. Every xst respect to decisions at stage

t in scenario s is made and is relied on the realizations of random variables {ω2, ..., ωt} at the

node (t, s). Obviously, all scenarios passing the node (t, s) have the same history in the stages 1, ..., t. The Figure2.2is a 4-stages’ scenario tree containing 18 nodes and 9 scenarios.

2.3 Literature Reviews

In our review of the literature, we focus on answering the first research question. We would like to see the methods of modeling the short-term risk criterion, the mid-term recovery plan, and the long-term risk criterion by other researchers.

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process to all VaR calculations. The process is composed of the mapping of all financial positions present in the investment portfolio to risk factors, the characters of the probability distribution of risk factor variations, and the computation of the VaR for investment portfolio. They only open the discussion on VaR adaption in pension funds but without any numerical illustrations. We think of that VaR may be an option for modeling the short-term or the long-term risk criterion.

Dert [5] adopt so-called the chance constraint to model the risk of underfunding in scenario-based pension funds problem. Underfunding is defined as the value of assets lower than the specified value of liabilities. The chance constraint measures the probabilistic risk of the underfunding. In his paper the chance constraint is applied on a given node at a given stage t to its successor at the following stage t + 1 in a scenario tree. This chance constraint could be an option for modeling our short-term risk criterion. The chance constraint can be used to modeled the short-term or the long-term risk criterion.

Basak and Shapiro [1] adopt VaR to model the wealth loss at some horizon and analyze optimal and dynamic portfolio in order to maximize utility of investors. Alternatively, the Limited-Expected-Losses (LEL) is introduced for controlling the expectation of the wealth loss. The definition of LEL is that the expected loss of a portfolio over any periods is lower than a threshold. The LEL is actually restricting the size of the expected loss instead of the probability of the expected loss. The approach of LEL could be suitable for modeling our short-term and long-term risk criteria.

Besides VaR and LEL, Bogentoft, Romeijn, and Uryasev [2] employ Conditional Value-at-Risk (CVaR) to measure the risk of underfunding in pension funds. CVaR is similar with VaR, but more sensitive to the shape of the loss distribution in the tail of the distribution. In their paper, CVaR at an α% level is the undefunding in the worst α% of the cases. The CVaR increases as α% increases. Compared with VaR, CVaR takes into account losses exceeding VaR. CVaR may also be adapted to model the short-term or the long-term risk criterion. Streutker, Klein Haneveld, and Van der Vlerk [12] use so-called the integrated chance con-straint to model the one-year risk of funding shortfall for pension funds. In their paper an expected funding shortfall at a stage t + 1 is restricted by a prescribed maximal acceptable expected funding shortfall at the stage t. The objective is also to minimize the total funding costs including present and future costs. To choose this appropriately prescribed maximal acceptable expected funding shortfall, sensitivity analysis is conducted for the selection. The integrated chance constraint is used to control the size of the risk such as funding shortfalls. The idea behind the integrated chance constraint is analogous to LEL. Considering our case, the integrated chance constraint is suitable to model the the short-term or the long-term risk criterion.

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Mathematical Models

This chapter gives a full outline of mathematical models in this paper. In Section3.1variables, constraints and the objective function of models are presented. In Section3.2 mathematical formulations are provided. Finally, in Section3.3, six models are classified.

3.1 Modeling

Note that our model is built based on future information of random variables that are pre-sented by many scenarios s over time. To simplify the notation, we omit the scenario index s and only present time index t.

3.1.1 Decision Variables

In this thesis contribution rates from a time to a time can be flexible, so to decide what value of a appropriate contribution rate is quite important for pension funds. In addition, investments in different assets and remedial contributions paid by pension sponsors are taken into account as decisions.

Contribution Rate

Contribution paid by active participants is a part of assets of a pension fund. The amount of this contribution paid by active participants is quite difficult to decide. The contribution is presented by a proportion of pensionable wages of active participants times the total pension-able wages. This proportion is the contribution rate which should be decided in our model is denoted by ct at time t.

Remedial Contribution

If the value of assets lower than the 105% value of liabilities, then the pension fund confronts a funding shortfall. The pension sponsor should remedy this funding shortfall by paying remedial contributions. To be precise, if a 105% funding ratio is unachieved at time t, such a remedy is required to restore the 105% funding ratio. Remedial contributions in this paper are denoted by Ztat time t.

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Investment decisions are also important for pension funds. If the predicted returns of assets are positively high, then more investments could imply more payoffs. Our assets allocated to an asset class i at the beginning of time t are denoted by Xit.

3.1.2 State Variables

A variable which can be expressed by decision variables and parameters is a so-called state variable. To realize the value of assets before remedial contributions, we introduce a state variable A∗_t that represents the value of the assets just before possible remedial contributions Zt at time t. A∗t is equal to total payoffs of investments plus contributions from active

participants minus benefit payments at time t. The value of assets At at time t is stated by

At, which can be calculated as A∗t plus Zt. Other state variables are A+it and A −

it that stand

for the asset class i bought and sold at time t.

3.1.3 Stochastic Parameters

In the thesis, future rates of returns on investments, total pensionable wages, benefit pay-ments, and liabilities which are stochastic parameters will be modeled as discrete random variables with finitely many mass points.

Rates of Returns on Investments

Decision-making on a portfolio is quite dependent on future rates of returns on investments. The standard deviations of rates of return on investments are usually used to determine the risks of investments. A high standard deviation of a rate of return on the investment commonly indicates a risky investment. The rate of return on investment in asset class i at time t is denoted by rit.

Total Wages

Pensionable wages which is also called the total wages level of active participants. Future amounts of pensionable wages are certainly not deterministic. The pensionable wages are considered as stochastic parameter since the value of them is influenced by many factors such as economic growth, production, and indexation (correction for inflation). At time t, total pensionable wages are denoted by Wt.

Benefit Payments

Benefit payments are money paid for active participants after they retire in the future. Since benefit payments to negative participants are predetermined in Dutch pension funds but the value of benefit payments is fluctuated due to inflations, we model benefit payments as a stochastic parameters. The value of benefit payments at time t is presented by Pt.

Liabilities

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3.1.4 Constraints of Actuarial Principle

ALM for Dutch pension funds cannot accomplished without restrictions. Conversely, re-strictions would strongly affect pension funds. We therefore need to understand what are the restrictions and how can them influence pension funds. Hereby we translate restrictions into mathematical constraints so as to see how these restrictions affect pension funds on a quantitative level.

Assets before Remedial Contributions

The first issue of ALM for pension funds is that we should understand what are assets and what are liabilities in pension funds. Contributions paid by active participants which equals to the contribution rate times total pensionable wages are the money as cash on the asset side of pension funds. Investments and their returns are perceived as the assets of pension funds. Benefit payments to inactive participants are recognized on the side of liabilities. Then we can have the total value of assets before remedial contributions at time t as follow:

A∗_t =

N

X

i=1

(1 + rit) Xit+ ctWt− Pt, t ∈ {2, ..., T } (3.1)

where i is the index of a asset class, i = 1, .., N , and N is total number of asset classes. Assets after Remedial Contributions

Recall that possible remedial contributions used to regain certain funding ratios are paid by pension sponsors. Indeed, it is recognized as a part of cash outflow for pension firms. Remedial contributions are parts of assets. Hence the total value of assets At is equal to A∗t

plus remedial contributions Zt at time t. We model the asset after remedial contributions at

time t as:

At= A∗t + Zt, t ∈ {2, ..., T } (3.2)

Value of Investments

During the pension periods, assets are allocated on different classes. Meanwhile, assets are allowed to be bought or sold, and transaction costs of buy-and-sell are considered. At time t, the total value of asset class i is expressed by (1 + rit) Xit, the net purchase of the value of the

same asset class i is denoted by A+_it− A−_it ,and the transaction costs of buy-and-sell of this asset class i are ki A−it + A

+

it, where ki is the proportional transaction cost for asset class i.

Then the value of the investments in asset class i at time t + 1 can be shown as follows:

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Asset Allocation

The whole assets should be a allocated at each time. This includes all investments and transaction costs of buy-and-sell during the times t − 1 and t which equals to the assets’ value at time t. We model this as follows:

N

X

i=1

Xi,t+1+ ki A−_it+ A+_it = At, t ∈ {1, ..., T − 1} (3.4)

3.1.5 Constraints of Pension Policies

Bounds on Values of Investments

Investments on different asset classes for pension funds are indeed decision-making for a portfolio. To allocate asses in a portfolio pension sponsors may have its own considerations, for example, providing bounds of weights of the portfolio due to investment risks and budgets. In this thesis we set the lower and upper bounds on the value of asset class i such as fraction of the total portfolio. w_il and wu_i are considered as the lower and upper bounds on the value of asset class i, which are pre-determined as deterministic parameters.

wl_i N X j=1 Xjt ≤ Xit≤ wiu N X j=1 Xjt, i = 1, ..., N, t ∈ {2, ..., T } (3.5)

Bounds on Contribution Rates

A high contribution rate each year are unacceptable for active participants. Therefore, we provide the lower and upper bounds on the contribution rate for each year. These lower bound cl and upper bound cu of a contribution rate at time t are specified numerically.

cl≤ ct≤ cu, t ∈ {2, ..., T } (3.6)

Bounds on Changes of Contribution Rates

Besides bounds on contribution rates, we also put bounds on the variation of contribution rates between the time t − 1 and the time t. Too large variations will cause social upheaval. ∆cl and ∆cu are deterministic lower and deterministic upper bounds for these variations at time t.

∆cl ≤ ct− ct−1 ≤ ∆cu, t ∈ {2, ..., T } (3.7)

Funding Ratio at Time Horizon

The value of assets should at least reach the proportion ρ of the value of liabilities at time horizon T even if there is a underfunding UT.

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3.1.6 The Mid-Term Recovery Plan

Essentially, our approach of modeling the mid-term recovery plan is based on following the track of stages in which a funding shortfall occurs. This needs the binary variables in the model. In turn, these binary variables are used to determine when a remedy is necessary in order to restore the α funding ratio. In our model, remedial contributions by the pension sponsor are used for a remedy.

For the viewpoint of modeling, a state variable A∗_t is introduced and stands for the asset’s value just before possible remedial contributions at time t. With Lt denoting liabilities at

time t, the constraint

A∗_t− αLt≥ −M ut, (3.9)

where M is a sufficient large number, forces the binary variable ut to take the value 1 in case

of a funding shortfall. If this occurs at time t, then the constraint

ut−2+ ut−1+ ut≤ 2, (3.10)

ensures that funding shortfalls did not occur in past two consecutive stages and hence restore 105% funding ratios within three stages. With Zt denoting remedial contributions at time t,

the constraint

Zt≤ M zt, (3.11)

force the binary variable zt to equal to the value 1 if remedial contributions Zt are paid at

time t. In case of one or two funding shortfall(s) in two consecutive stages, it should ensure that a remedial contribution can be paid in the later stages after these two consecutive stages. This can be modeled by the constraint

zt≤ ut−1+ ut, (3.12)

If the binary variable zt is 1, then it forces at least one of the two binary variables ut−1 and

ut equal to 1. In other words, there is at least a funding shortfall at time t − 1 or at time t.

3.1.7 The Short-Term Risk Criterion

Before modeling the short-term risk criterion, we would like to introduce how it works on a scenario tree. The short-term risk criterion providing information at a given stage attempts to control the risk of insufficient assets next stage. Figure 3.1depicts that based on a given predecessor (t,s), the short-term risk criterion controls the risk of funding shortfalls on its successor (t+1,s). For instance, at the state (root node) at t = 1, we would like to control the funding shortfalls of three states at t = 2. Then based on the first state of three states at t = 2, we want to control funding shortfalls of the first two states at t = 3 as depicted inside the circles in Figure 3.1. This is the way of how the short-term risk criterion effects.

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STRC

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Figure 3.1: Short-term risk criterion on a scenario tree.

The Short-Term Risk Criterion by Chance Constraint

Dert [4] uses Value-at-Risk (VaR) - based risk management’s constraints to model a certain funding ratio which is guaranteed with a specific probability. These VaR constraints are indeed the chance constraints. We provide a direct translation of the short-term criterion by the chance constraint:

PA∗_t+1− αLt+1≥ 0| (t, s) ≥ θt (3.13)

It measures the probability of a funding shortfall at time t + 1 conditional on the state (t,s). This risk measure implies such risk control is qualitative and not quantitative. Next, we need to transform such a chance constraint into a linear system since an optimization model cannot directly deal with a probabilistic constraint.

In this thesis, At+1 is a decision variable of the assets at time t + 1 and Lt+1 is a random

parameter of the liabilities at time t+1. Assume that Lt+1follows a finite discrete distribution

with realizations Ls_t+1 and their corresponding probabilities ps_t+1, where s ∈ S := {1, ..., S}. Then the chance constraint can be represented by a linear system:

As_t+1+ vs_t+1M ≥ αLs_t+1, s ∈ S X

s∈S

ps_t+1vs_t+1≤ 1 − θ_t v_t+1s ∈ {0, 1} , s ∈ S,

where M is a enough large number. The binary variable v_t+1s equal to 1 implies a funding shortfall at time t + 1. The second inequality means the probabilistic weighted average of binary variables vs_t+1 at time t + 1 then equals the risk of funding shortfall, which is at most 1 − θt.

However, in this thesis we only consider the short-term risk criterion at time 1, so it further means that we include the PA∗

t+1− αLt+1≥ 0| (t, s) ≥ θtonly when t equals to 1.

The Short-Term Risk Criterion by Integrated Chance Constraint

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may cause difficulty of computation for models due to binary variables. Hence, according to Streutker, Klein Haneveld, and Van der Vlerk [12], we introduce another approach called integrated chance constraints for the short-term risk criterion which can measure the size of a funding shortfall without any binary variables.

We introduce an integrated chance constraint for the short-term risk criterion: E

h

A∗_t+1− αL_t+1−

| (t, s)i≤ β_t, s ∈ S. (3.14) It shows that expected funding ratio next times t + 1 should be at most βtthat is decided at

the current time t. The prescribed parameter βt provides the maximal acceptable expected

funding shortfall and should be specified numerically. It is difficult to come up with this value than it would be for the reliability parameter in chance constraints. Note that this integrated chance constraint is restricting the expected funding shortfall at states (t+1,s) based on a given state (t,s). The integrated chance constraint above could be reformulated by a system of linear inequalities: E h A∗_t+1− αLt+1 − | (t, s)i=X s∈S ps_t+1 A∗s_t+1− αLs_t+1− ≤ βt,

where Lt+1 the value of liability which follows a finite discrete distribution with realizations

Ls_t+1 and their corresponding probabilities ps_t+1 at time t + 1, s ∈ S := {1, ..., .S}. The probabilistic weighted average of the expression A∗s_t+1− αLs

t+1

−

at time t + 1 is no larger than the expectedly maximally acceptable funding shortfall by βt,

Now we translate A∗s_t+1− αLs t+1

−

into linear formulations. By definition (a)−:= max {−a, 0}, a ∈ R, we rewrite A∗st+1− αLst+1

−

equals to maxαLs_t+1− A∗s_t+1, 0 . We introduce a de-cision variable y_t+1s , namely maxαLs_t+1− A∗s_t+1, 0 is no larger than ys

t+1. Indeed, yst+1 is

denoted as the maximal value of funding shortfall at time t + 1 in a scenario s. We use y_t+1s to provide following expressions:

A∗s_t+1− αLs_t+1− = maxαLs t+1− A ∗s t+1, 0 ≤ yst+1, s ∈ S ⇐⇒ αLs_t+1− A∗s_t+1≤ y_t+1s , s ∈ S y_t+1s ≥ 0, s ∈ S.

Finally, we collect the whole linear formulations of the integrated chance constraint for the short-term risk criterion such as

E h A∗_t+1− αLt+1 − | (t, s)i≤ βt ⇐⇒ X s∈S ps_t+1y_t+1s ≤ βt αLs_t+1− A∗s_t+1≤ ys t+1, s ∈ S y_t+1s ≥ 0, s ∈ S.

To decide the suitable parameter βt, we will take into count different values of βt, and see

how different values of βt effect outputs of our model. Indeed, we will do sensitivity analysis

of βt in our numerical experiments.

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3.1.8 The Long-Term Risk Criterion

The long-term risk criterion is analogous to the short-term risk criterion. The effects of the long-term risk criterion on a scenario tree depicted in Figure3.2. The long-term risk criterion are aiming for restricting the expected maximal funding shortfall by considering all scenarios at time horizon T based on t = 1. In particular, it should help improve funding ratios at time T − 1 or time T . Similar with the modeling the short-term risk criterion, we also model the

3 2

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LTRC

Figure 3.2: Long-term risk criterion on a scenario tree

long-term risk criterion by using the chance constraint and the integrated chance constraint. The Long-Term Risk Criterion by Chance Constraint

We adopt the chance constraint to model the long-term risk criterion at time horizon consid-ering the reliability ηT:

P {A∗_T − αLT ≥ 0} ≥ ηT (3.15)

The method of formulating the linear inequalities of the chance constraint for the long-term risk criterion is analogous to the way used for the short-term risk criterion. The linear inequalities of this chance constraint as follows:

As_T + δ_TsM ≥ αLs_T, s = 1, ..., S S X s=1 ps_Tδ_Ts ≤ 1 − η_T δ_Ts ∈ {0, 1} , s = 1, ..., S,

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The Long-Term Risk Criterion by Integrated Chance Constraint The integrated chance constraint for the long-term risk criterion is modeled as:

E(A∗T − αLT)− ≤ φT (3.16)

where φT is predetermined maximal acceptable funding shortfall at time horizon T . The

linear transformation of this integrated chance constraint, similar with the short-term risk criterion, is presented as:

αLs_T − A∗s_T ≤ s_T, s = 1, ..., S S X s=1 ps_Ts_T ≤ φ_T s_T ≥ 0, s = 1, ..., S.

where s_T is decision variable which is greater or equal to max {αLs_t − A∗s_t , 0}, and s_T means the maximally acceptable funding shortfall in a scenario s at time horizon T . The selection of the parameter φ5 will be done by sensitivity analysis as well. Further information of selecting

φ5 will be presented during numerical experiments.

3.1.9 Objective Function

The mathematical objective function of a pension fund formulated based on consideration of interests of pension sponsors and active participants. In this thesis a pension fund has a objective to minimize its funding costs, which spans over several periods. Other objective functions of pension models could be to maximize the accrued indexed rights of inactive participants based on decisions on liabilities, see Streutker, Klein Haneveld, and Van der Vlerk [11].

A pension confronts the problem to minimize the total funding costs over a finite horizon. The total fundings consist of contributions paid by active participants, remedial contritions paid by pension sponsors and underfunding at time horizon. Since the target of a pension fund is to fulfill its pension claims, the situation of assets being lower than liabilities will ruin its target. Therefore, remedial contributions and underfunding at time horizon will be penalized if they occur.

Pension funds do not know that in advance how much remedial contributions are, what active participants’ contribution rates could be, and how much the underfunding. Hence, it is appropriate to model its objective in terms of an expectation.

Contributions paid by active participant at time t are denoted by ctWt. Remedial

contri-butions from pension sponsors are presented by Zt and its penalty parameter is denoted by

λZ. Finally, the penalty of underfunding at time horizon T denoted by λTUT where λT is

the penalty parameter of UT. Since it counts future funding costs, to see the present value of

the total funding costs is reasonable. We introduce that γt is the discount factor at time t.

Hence, the total funding costs can be expressed as:

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Consider the discretization of the expectation by introducing a probability pt_s of a scenario s in year t, where s ∈:= S := {1, .., S}, and S is total number of scenarios. The objective function which is indeed a piecewise linear function can be expressed as follows:

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3.2 Mathematical Formulation

Indices

t index of years, t = 1, ..., T

i index of asset classes, i = 1, ..., N s index of scenarios, s = 1, ..., S

I index set of asset classes, I := {1, ..., N } T index set of years, T := {1, ..., T } T1 index set of years, T1 := {1, ..., T − 1}

T2 index set of years, T2 := {2, ..., T }

T₃ index set of years, T3 := {3, ..., T }

T4 index set of years, T4 := {4, ..., T }

Variables

Xit value of investments in asset class i at the beginning of year t

ct+1 contribution rate for year t + 1

Zt remedial contribution by sponsor in year t

At total asset value in year t

A∗_t total asset value in year t just before possible remedial contributions Zt

A+_it asset values in class i bought in year t A−_it asset values in class i sold in year t

ut binary variable equal to 1 if funding ratio lower than 105% in year t, otherwise 0

zt binary variable equal to 1 if remedial contribution needed in year t, otherwise 0

UT underfunding at year horizon with respect to the funding ratio ρ

Stochastic Parameters

rit random return on asset class i in year t

Wt random total wages of active participants in year t

Pt random total benefit payments in year t

Lt random value of liabilities after year t

Deterministic parameters

α lower bound on the funding ratio

ρ prescribed funding ratio at year horizon (ρ ≥ α) ki proportional transaction cost for asset class i

wl

i lower bound on the value of asset class i as a fraction of the total value portfolio

w_iu upper bound on the value of asset class i as a fraction of the total value portfolio cl lower bound on the contribution rate

cu _{upper bound on the contribution rate}

∆cl lower bound on the contribution rate change ∆cu upper bound on the contribution rate change θt reliability of no funding shortfall in STRC in year t

βt maximal acceptable expected funding shortfalls in STRC in year t

ηt reliability of no funding shortfall in LTRC in year t

φt maximal acceptable expected funding shortfalls in LTRC in year t

γt discount factor for a cash flow in year t

λZ penalty cost for a remedial contribution

λU penalty cost for underfunding at year horizon

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3.2.1 Constraints

• Constraints based on actuarial principles:

A∗_t = N X i=1 (1 + rit) Xit+ ctWt− Pt, t ∈ T2 (3.18) At= A∗t+ Zt, t ∈ T2 (3.19) Xi,t+1= (1 + rit) Xit− A−_it+ A_it+− ki A−_it+ A+_it , i ∈ I, t ∈ T1 (3.20) N X i=1 Xi,t+1+ ki A−it+ A+it = At, t ∈ T1 (3.21)

• Constraints based on policies of pension funds: w_il N X j=1 Xjt ≤ Xit ≤ wui N X j=1 Xjt, i ∈ I, t ∈ T2 (3.22) cl≤ ct≤ cu, t ∈ T2 (3.23) ∆cl≤ ct− ct−1≤ ∆cu, t ∈ T2 (3.24) AT + UT ≥ ρLT (3.25)

• Constraints of Strict 105% Funding Ratio:

At≥ αLt, t ∈ T2 (3.26)

• Constraints of Short-Term Risk Criterion using chance constraints: PA∗

t+1− αLt+1≥ 0|t ≥ θt, t = 1 (3.27)

• Constraints of Short-Term Risk Criterion using integrated chance constraints: E

h

A∗_t+1− αL_t+1−

|ti≤ β_t, t = 1 (3.28)

• Constraints of Mid-Term Recovery Plan:

A∗_t− αL_t≥ −M u_t, t ∈ T (3.29a) ut−2+ ut−1+ ut≤ 2, t ∈ T3 (3.29b)

• Constraints of Possible Remedy during Mid-Term Recovery Plan:

Zt≤ M zt, t ∈ T (3.29c)

zt≤ ut−1+ ut, t ∈ T2 (3.29d)

• Constraints of Long-Term Risk Criterion Using Chance Constraints: PA∗

t+1− αLt≥ 0 ≥ ηt, t = T (3.30)

• Constraints of Long-Term Risk Criterion Using Integrated Chance Constraints:

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3.2.2 Objective Function min E " _T X t=1 γt(ctWt+ λZZt) + γTλTUT # . (3.32)

3.3 Models

To see impacts of the mid-term recovery plan, the short-term risk criterion and the long-term risk criterion on the decisions, we compare outputs between two models. From each com-parison, we will see the effects of the mid-term recovery plan, the short-term risk criterion and the long-term risk criterion on remedial contributions, contribution rates, portfolios and funding ratios.

We first build the Basic model seen as the benchmark, then replace the hard constraints of At ≥ αLt by constraints of the mid-term recovery plan, and then constitutes the MTRP

model. Next we model the term risk criterion based on the MTRP model. The short-term risk criterion can be modeled by the chance constraint or the integrated chance con-straint. Hence, we add the chance constraint of the short-term risk criterion to the MTRP model recognized as the STRCcc+MTRP model. For its integrated chance constraint ap-proach, we build the STRCicc+MTRP model. After that, the chance constraint of the long-term risk criterion embeds into the STRCcc+MTRP model, and the STRCcc+MTRP+LTRCcc model is created. In addition, we put the integrated chance constraint into the STRCicc+MTRP model, and giving the STRCicc+MTRP+LTRCicc model. The ways of how the models come up are presented in Figure3.3.

Figure 3.3: Model structure and development

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3.3.1 Basic Model

The Basic model which is the benchmark includes constraints of actuarial principles, pension policies, and strict 105% funding ratios. Its objective function is to minimize the total funding costs. The complete Basic model is presented as follows:

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3.3.2 MTRP Model

The MTRP model is aiming to find out the effects of the mid-term recovery plan. The differences between the Basic and the MTRP models are that the MTRP model adopts the constraints of the mid-term recovery plan and excludes the hard 105% funding ratios. We present the whole MTRP model as:

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3.3.3 STRCcc+MTRP Model

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3.3.4 STRCicc+MTRP Model

The STRCicc+MTRP Model is used for digging out influences of the short-term recov-ery plan modeled by the integrated chance constraint. Compare with STRCcc+MTRP Model, we replace the constraint P {A∗₂− αL₂ ≥ 0|t} ≥ θ₂ and cast into the constraint E(A∗2− αL2)−|t ≤ β1. The STRCicc+MTRP model is given as follows:

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3.3.5 STRCcc+MTRP+LTRCcc Model

We add the constraint of P {A∗_T − αLT ≥ 0} ≥ ηT into STRCcc+MTRP Model, and

con-stitute the STRCcc+MTRP+LTRCcc Model, which is used for investigating effects of the long-term risk criterion modeled by the chance constraint. We present it as follows:

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3.3.6 STRCicc+MTRP+LTRCicc Model

Finally, we want to analyze influences of the long-term risk criterion modeled by the integrated chance constraint. We add the constraint E(A∗T − αLT)− ≤ φT into the STRCicc+MTRP

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Numerical Analysis

In this chapter we give numerical experiments to evaluate the models introduced in Chapter3. The essential issue is to investigate the effects of the mid-term recovery plan, the short-term risk criterion, and the long-term risk criterion on funding ratios, contribution rates, remedial contributions, and investments. In Section 4.1, we illustrate the data of our case study. In Section 4.2, we explain the method of scenario tree selection. In Section 4.3, we give the optimal values and optimal solutions of the six models.

Computer Package

To perform the numerical experiments we use computer package Xpress-MP. Its module mmsp can solve models built by stochastic programming. We refer to the reference manual Xpress-SP [3] especial for the modeling part in Xpress-MP. The computer package adopts Dual Simplex to deal with stochastic multistage recourse models.

Round-off Solution

Since plenty of binary variables are involved in models, but not all, it is difficult to find optimal solutions in those models. Instead, the Branch and Bound algorithm is adopted in order to get approximately optimal solutions. All models can be solved within 300 seconds by a desktop with Intel Duo 2, 2.4 GHs and 2 GB memory.

Due to approximation adopted in the model, it is quite necessary to see its approximation ratio, which measures the integrality gap. Next, I will explain the integrality gap.

Consider the objective value corresponding to the current best solution of the mixed-integer programming in a model. Let OV be the optimal value of objective function and BUB be the value of the best upper bound of it found by a global search. We consider the best upper bound here because of a minimal objective considered. The global search for the optimal solution is terminated if:

kBU B − OV k ≤ δ × BU B

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Investment class i wl_i wu_i ki Initial investments

Cash 1 0 0.2 0.005 400

Bond 2 0.2 0.6 0.0015 1500

Stock 3 0.3 0.6 0.00425 1600 Table 4.1: Data on investments.

4.1 Data

These numerical experiments take a 5 − year span. Decisions are made at t = 1 (now) until the time horizon T = 5. For year t depicts a year span [t − 1, t), where t = 2, 3, 4, 5. The data set is referring to stochastic parameters considered as finitely discrete mass points from year 2 to year 5 based on the scenario tree presented in later Section4.2.

First, we introduce deterministic parameters. Deterministic Parameters

The values of the deterministic parameters presented are shown in Table 4.2. We provide initial values of assets, liabilities, benefit payments, investments on different asset classes, and the contribution rate realized as inputs at year 1, see Table 4.1 and Table 4.2. Initial asset A1 of 3652.31 million euro can be computed by data presented in Tables 4.1 and 4.2.

Thus, the initial funding ratio F1 = A1/L1 is 0.95.

We select θ1 = 0.93 of the chance constraint for the short-term risk criterion in spite the

reliability 0.975 is required by FTK. This selection is concerning to the structure of the scenario tree mentioned later. The scenario tree has 16 nodes at t = 2, so the source node visits each node at t = 2 with the probability 0.0625. To allow at least one node that can have a funding shortfall, we choose β1 = 0.93. On the other hand, there many probabilities

of η5 can be chose. Corresponding to 0.975 reliability by FTK, we select η5 = 0.975 since

the probability of visiting each node is round to 0.00098, which is less than 0.025. In other words, there could be 25 nodes that may have funding shortfalls.

To choose the right hand side parameter β1 of the integrated chance constraint for the

short-term risk criterion, we take different β1as different percentages of A1and discover the different

optimal solutions and optimal values of the STRCicc+MTRP model. The value of β1is chosen

when the optimal solutions of the STRCicc+MTRP model are equivalent to the optimal solutions of the STRCcc+MTRP model by θ1 = 0.93. Then we can conclude the value of β1

yields 0.93 reliability. Moreover, it states the money needed for satisfying the requirement of P {A∗₂− L2≥ 0} ≥ 0.93.

For selection of φ5, first its sensitivity analysis is taken, and then choosing φ5 when its

corresponding optimal solutions that are equivalent to the optimal solutions of the STR-Ccc+MTRP+LTRCcc model with the reliability η5 = 0.975 at t = 5.

Stochastic Parameters

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cl= 0 A1 = 3652.31 γ1 = 1 λZ = 100 θ1 = 0.93 cu _{= 0.2} _L 1 = 3847.87 γ2 = 0.987137 λU = 110 η5 = 0.975 ∆cl= −0.05 W1 = 1015.41 γ3 = 0.962859 α = 105% β1= 1.72 ∆cu= 0.05 P1 = 0 γ4 = 0.930959 ρ = 110% φ5 = 14.6 c1 = 0.15 γ5 = 0.89432

Table 4.2: Values of deterministic parameters, observed random parameters, and contribution rate in year 1.

Year(t) Investment Mean Standard Other Random Mean Standard

returns deviation Parameters Deviation

r12 0.013 0 W2 1029.67 0 t = 2 r22 0.017 0.041 P2 170.52 0 r32 0.091 0.104 L2 3764.25 363.29 r13 0.024 0.017 W3 1053.57 10.45 t = 3 r23 0.025 0.054 P3 179.03 1.6 r33 0.115 0.217 L3 3752.74 440.15 r14 0.024 0.013 W4 1059.72 9.87 t = 4 r24 0.029 0.067 P4 190.25 3.44 r34 0.12 0.278 L4 3796.53 481.16 r15 0.033 0.019 W5 1085.72 16.37 t = 5 r25 0.037 0.078 P5 204.49 5072 r35 0.127 0.386 L5 3802.33 509.18

Table 4.3: Summary statistics of random investment returns, random wages, random benefit payments, and random liabilities from year 2 to year 5.

used in our thesis was generated three years ago. Assume that we only invest in cash, bond, and stock. Hence, we are only need to know the rates of returns on cash, bond, and stock, denoted by r1t, r2t, and r3t, respectively, where t is a number of years. The latest moment

that total wage levels, benefit payments, liabilities, and returns on cash, bond, and stock were taken into account are fixed to ultimo 2000. Hence, if t=2006 there are 5 different years. The pension data are provided by ORTEC finance b.v. and generated with the Pension Assets and Liabilities scenario Model (PALM). The wage increase factors for years 2003 to 2006 derive from Statistics Netherlands. For simplicity, we start at the begging of the year 2002 by considering t = 1. The coming year is the year 2003 by setting t = 2. We finish in 2006 and set t = 5. The statistics on the distribution of the investment returns for t = 2 until t = 5 are presented in Table4.3.

4.2 Scenario Tree Selection

Initially, we choose a scenario tree having 10000 scenarios, then for solving the mixed-integer problem, it can not be solved by realistic times. Instead, we shrink the size of the scenario tree, and details of the tree is presented in next paragraph.

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4.3 Numerical Results

This section mainly presents the results of six models based on the given data. In Subsection

4.3.1, we analyze results of the Basic model and the MTRP model. In Subsection 4.3.2, we compare results of the MTRP model and the STRCcc+MTRP model. In Subsection 4.3.3, we first give sensitivity analysis for the STRCicc+MTRP model then compare results of the MTRP model and the STRCicc+MTRP model. In Subsection 4.3.4, we analyze outputs of the STRCcc+MTRP model and the STRCcc+MTRP+LTRCcc model. Finally, in Subsection

4.3.5, we adopt sensitivity analysis for the STRCicc+MTRP+LTRCicc model then analyze outputs of the STRCicc+MTRP model and the STRCicc+MTRP+LTRCicc model.

4.3.1 Basic VS MTRP

The total funding costs of the Basic model are 168% as large as the total funding costs of the MTRP model, see optimal values in Table4.4and Table 4.5. The present costs for both models are the same. The expected future costs of the Basic model are greater than it of the MTRP model. The excessive amount of the expected future costs of the Basic model is 68.6% of the MTRP model’s expected future costs. The optimal solutions of all decisions on expected, minimal and maximal levels at t = 1 in the Basic and the MTRP models are the same, see the third columns in both Tables 4.4 and 4.5. That is why the present costs for both models are the same.

Effects on Funding Ratio by Mid-Term Recovery Plan

The funding ratios for both models are the same at t = 1. The MTRP model yields equal or lower funding ratios rather than the Basic model at the expected and minimal levels at t = 2, 3, 4, 5, see Figure4.1.

In Figure4.1, there are six columns corresponding to each time t. For each time t, the first two columns stand for the expected funding ratios of the Basic model and the MTRP model at time t; the third and fourth columns denote the minimal funding ratios of two models at time t; the last two mean the maximal funding ratios of two models at time t.

The greatest difference of the expected funding ratios between two models is 0.06 at t = 4. The largest gap of the minimal funding ratios is 0.12 at t = 4. The maximal funding ratios of the MTRP model are larger than maximal funding ratios of the Basic model at t = 4, 5. Effects on Financing Policy by Mid-Term Recovery Plan

The remedial contributions in the two models are quite different from each other at t = 2, 3, 4, 5 by comparing remedial contributions in Table 4.4 and Table 4.5. The minimums in both models are all 0s at t = 2, 3, 4, 5, and the maximums for two models are not the same. Compare with the Basic model, the MTRP model yields the relatively smaller maximal and smaller expected remedial contributions at t = 2, 3, 4. For example, in the MTRP model at t = 2, 3, 4, the expected remedial contributions are 0.21 (0% of A1), 0.66 (0% of A1), and 1.4

(0.04% of A1) respectively. The Basic model obtains the relatively larger expected remedial

contributions 94.15 (2.6% of A1), 8.34 (0.2% of A1), and 8.92 (0.2% of A1) at t = 2, 3, 4

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1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 t F t E[F] B. E[F] MT. minF B. minF MT. maxF B. maxF MT.

Figure 4.1: Funding ratios in Basic and MTRP models at t = 1, 2, 3, 4, 5.

To take into account funding ratios, the MTRP model has lower expected, lower minimal, and lower maximal funding ratios and pays less expected, less minimal, and less maximal remedial contributions at t = 2, 3, 4. However, at t = 5, the MTRP model yields higher expected, higher minimal, and higher maximal funding ratios and pays larger expected, larger minimal, and larger maximal remedial contributions. Particularly, the maximal remedial contributions are 2259.23 (61.9% of A1) at t = 5. It shows that the MTRP model can avoid the remedial

contributions Zt in case of a funding shortfall at t = 2, 3, 4. For instance, at t = 2, the

expected funding ratio in the MTRP model is 1.03 in which a funding shortfall occurs, but only expected remedial contributions 0.21 (0% of A1) are paid. Nevertheless, the expected

funding ratio is 1.1 at t = 5, but still expected remedial contributions 132.27 (3.6% of A1)

are demanded.

Considering all scenarios, optimal remedial contributions at t = 1, 2, 3, 4, 5 are resented in Figure 4.2. For all scenarios, it clearly shows remedial contributions of the Basic model are paid more rather than the MTRP model at t = 2, 3, 4.

Maximal and minimal contribution rates for two models are the same at each time and expected funding ratios of the MTRP model are slightly larger than expected funding ratios of the Basic model at t = 4, 5 see contribution rates in Tables4.4and 4.5.

Effects on Investment Strategy by Mid-Term Recovery Plan

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Basic Model total = 21610.31 present = 152.31 future= 21458 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 Funding ratio E [F ] 0.95 1.05 1.19 1.3 1.38 min F 1.05 1.05 1.05 1.1 max F 1.07 1.39 1.67 1.97 Contribution rate E [c] 0.15 0.2 0.188 0.161 0.123 min c 0.2 0.15 0.1 0.05 max c 0.2 0.2 0.2 0.2 Remedial contribution E [Z] 0 94.15 8.34 8.92 110.16 min Z 0 0 0 0 max Z 232.42 330.14 422.98 2189.58 Investment in cash E [X1] 400 0 664.01 665.92 415.49 min X1 0 0 0 0 max X1 0 884.75 1086.21 1337.5 Investment in bond E [X2] 1500 1459.81 1406.27 2139.97 2595.63 min X2 1459.81 763.77 736.74 684.26 max X2 1459.81 2211.87 3265.71 47304 Investment in stock E [X3] 1600 2189.72 2061.54 1707.37 1751.6 min X3 2189.72 1174.25 1126.89 913.27 max X3 2189.72 2832 3553.17 4261.72 Portfolio E [%cash] 11.4 0 16.1 14.8 8.7 E [%bond] 42.9 40 34 47.4 54.5 E [%stock] 45.7 60 49.9 37.8 36.8

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MTRP Model total = 12877.02 present = 152.31 future = 12724.7 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 Funding ratio E [F ] 0.95 1.03 1.17 1.24 1.35 min F 1 0.98 0.93 1.1 max F 1.07 1.31 1.71 3.01 Contribution rate E [c] 0.15 0.2 0.188 0.162 0.125 min c 0.2 0.15 0.1 0.05 max c 0.2 0.2 0.2 0.2 Remedial contribution E [Z] 0 0.21 0.66 1.4 132.27 min Z 0 0 0 0 max Z 3.29 42.4 359 2259.23 Investment in cash E [X1] 400 0 662.5 753.39 412.4 min X1 0 0 0 0 max X1 0 852.51 1163.11 1299.22 Investment in bond E [X2] 1500 1459.81 1315.36 1989.65 2497.7 min X2 1459.81 761.58 683.96 656.29 max X2 1459.81 2131.26 3165.46 4472.67 Investment in stock E [X3] 1600 2189.72 2060 1671.98 1742.76 min X3 2189.72 1076.08 1051.9 909.71 max X3 2189.72 2692.4 3489.34 4467.27 Portfolio E [%cash] 11.4 0 16.4 17.1 8.9 E [%bond] 42.9 40 32.6 45.1 53.7 E [%stock] 45.7 60 51 37.8 37.4

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1 2 3 4 5 0 100 200 300 400 500 t Z t Basic 1 2 3 4 5 0 100 200 300 400 500 t Z t MTRP

Figure 4.2: Remedial contributions for all scenarios in Basic and MTRP models at t = 1, 2, 3, 4, 5.

4.3.2 MTRP VS STRCcc+MTRP

Now we turn to compare the results of the MTRP and the STRCcc+MTRP models and to see their differences.

The total funding costs of the STRCcc+MTRP model are larger than the MTRP model. This is because the present costs of the STRCcc+MTRP model are significantly greater then the MTRP model. The huge value of the present costs of the STRCcc+MTRP model is caused by the larger remedial contributions paid at present. Due to larger remedial contributions made at present, the STRCcc+MTRP model has relatively low contribution rates and relatively small remedial contributions at later stages. Moreover, the future costs are lower in the STRCcc+MTRP model, see future in Tables4.6 and 4.7.

Effects on Funding Ratio by Short-Term Risk Criterion

The funding ratio increases by 0.05 in the STRCcc+MTRP model compared with the MTRP model at t = 1. The increased funding ratio is triggered by the constraint of 93% reliability of the sufficient fund at t = 2 (P {A∗₂− 1.05L2 ≥ 0} ≥ 0.93) demanding extra remedial

contributions 168.29 (4.6% of A1) at t = 1. Expected, minimal, and maximal funding ratios

in the STRCcc+MTRP model are equal to or larger than the MTRP model at t = 2, 3, 5. However, the maximal funding ratio in the STRCcc+MTRP model at t = 4 is 0.07 less than in the MTRP model, see Figure4.3.

In Figure 4.3, At each time t, there are six columns. The first two columns stand for the expected funding ratios of the MTRP model and the STRCcc+MTRP model at time t; the third and fourth columns denote the minimal funding ratios of two models at time t; the last two mean the maximal funding ratios of two models at time t.

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1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 t F t E[F]_MT. E[F]_SMT. minF_MT. minF_SMT. maxF MT. maxF_SMT.

Figure 4.3: Funding ratios in MTRP and STRCcc+MTRP models at t = 1, 2, 3, 4, 5.

Effects on Financing Policy by Short-Term Risk Criterion

The minimal and maximal contribution rates for both models are the same at t = 2, 3, 4, 5. Whereas the expected contribution rates in the STRCcc+MTRP model are decreasing com-pared with the expected ones in the MTRP model at t = 4, 5, see optimal expected contribu-tion rates in Tables4.6and 4.7. The decreases in contribution rates in the STRCcc+MTRP model at later stages are triggered by payment of relatively high remedial contribution at t = 1.

Remedial contributions for the two models are significantly distinguished at the expected and maximal levels but the same at the minimal levels, See remedial contributions in Table 4.6

and Table4.7. At t = 1, the MTRP model does not require remedial contributions. Again, the chance constraint P {A∗₂− 1.05L2 ≥ 0} ≥ 0.93 results in remedial contributions 168.29

(4.6% of A1) at t = 1 in the STRCcc+MTRP model. These remedial contributions 168.29 at

t = 1 seen as pre-paid remedy make no remedial contributions at t = 2, 3. Furthermore, the high remedial contribution paid at t = 1 also helps pay less remedial contributions at t = 4, 5. Effects on Investment Strategy by Short-Term Risk Criterion

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STRCcc+MTRP Model total = 26395.26 present = 16981.65 future = 9413.61 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 Funding ratio E [F ] 1 1.08 1.22 1.29 1.39 min F 1.03 1.05 0.95 1.1 max F 1.12 1.36 1.64 3.01 Contribution rate E [c] 0.15 0.2 0.187 0.157 0.118 min c 0.2 0.15 0.1 0.05 max c 0.2 0.2 0.2 0.2 Remedial contribution E [Z] 168.29 0 0 0.76 97.08 min Z 0 0 0 0 max Z 0 0 195.41 2039.93 Investment in cash E [X1] 400 0 680.24 779.45 442.43 min X1 0 0 0 0 max X1 0 891.36 1182.48 1422.25 Investment in bond E [X2] 1500 1526.97 1421.97 2155.79 2565.15 min X2 1526.97 796.29 697.45 617.65 max X2 1526.97 2228.49 3448.32 4488.36 Investment in stock E [X3] 1600 2290.45 2116.2 1662.69 1824.43 min X3 2290.45 1151.92 1099.33 956.41 max X3 2290.45 2784.2 3268.27 4266.74 Portfolio E [%cash] 11.4 0 16.1 17 9.2 E [%bond] 42.9 40 33.7 46.9 53.1 E [%stock] 45.7 60 50.2 36.1 37.7

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4.3.3 MTRP VS STRCicc+MTRP

In this part, we model the short-term risk criterion by using integrated chance constraints. The first issue is to choose the appropriate parameter of β1 for the STRCicc+MTRP model.

Sensitivity Analysis of β1

Different values of β1 as percentages of A1 in the STRCicc+MTRP model are taken to explore

effects of the integrated chance constraints on funding ratios, contribution rates, remedial contributions, and investments. The expected optimal solutions and optimal values of the STRCicc+MTRP model by considering different values of β1 are presented in Table A.1 in

Appendices. We only present the expected optimal solutions because optimal expected values reflect the explicit changes when it takes different values of β1 as percentages of A1. Another

reason is the most minimal values of the same decisions are highly similar at each time when β1 alters, so using minimal values of solutions to inspect effects of the integrated chance

con-straint is inexplicit. The maximal values are highly linked with the expected values, so only inspection on the expected values is convenient and effective.

We first solve the MTRP model without the integrated chance constraint at t = 1. It displays expected funding shortfall of the assets with respect to α times liabilities at t = 2 equal to 0.21 million euro (0.006% of A1), which is a quite small value. However, at the horizon t = 5,

expected remedial contribution is 130.89 (3.6% of A1). Then the integrated chance constraint

which restricts the expected funding shortfall at t = 2 by several percentages of A1 are added

to the MTRP model, see the STRCicc+MTRP model. Optimal values, optimal first-stage’s solutions and optimal later-stages’ solutions in terms of their expected, minimal and maximal values with different percentages of A1 are presented in A.1.

Each increase in β1 is 18.26 million euro (0.5% of A1). The total optimal costs decreases with

increasing β1. Present costs are decreasing when β1 is increasing, but expected future costs

are rising as β1 is increasing, see Figure 4.4. For instance, over the range (0%-0.5% of A1),

present costs decrease (42.6%), and expected future costs increase (16.1%).

Contribution rates change slightly, or even keep stable when β1 is increasing as depicted in

Figure4.5. At t = 2, contribution rates fix at 0.2 even if β1 changes. This is caused by high

penalty for remedial contributions in the objective and initial poor funding ratio (F1 = 0.99).

Note that the initial funding ratio increases from 0.95 to 0.99 because extra remedial con-tributions are required at t = 1. The expected contribution rates are non-decreasing with increasing β1 at t = 1, 2, 3, 4, 5. This could be explained by the decreasing remedial

contribu-tion by increasing β1 at t = 1, 2, 3, 4, 5, which will be presented later.

As the percentages keep increasing, the expected, minimal and maximal amounts of invest-ments in cash, bond, and stock insignificantly fluctuate at t = 1, 2, 3, 4. The expected fractions of portfolio keep quite stable when β1 changes.

The remedial contributions are strongly altering at t = 1, 5 if β1 changes, see Figure4.6. For

β1 is lower than 10.96 (3% of A1), remedial contributions Z1 are positive in the optimal

solu-tion at t = 1 in TableA.1in Appendices. At t = 1, by decreasing β1, remedial contributions

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0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3x 10 4 % of A₁ (β₁) Costs Present Future Total

Figure 4.4: Optimal values as function of β1 in STRCicc+MTRP model.

0 0.5 1 1.5 2 2.5 3 0 0.05 0.1 0.15 0.2 0.25 % of A 1 (β1) ct E[c₂] E[c 3] E[c₄] E[c₅]

Figure 4.5: Contributions rates as function of β1 in STRCicc+MTRP model.

decreasing β1. In particular, at time horizon t = 5, the decease in remedial contributions is

obvious when β1 is declining. For instance, to employ a decrease of the percentages of A1

from 3% to 2.5% (18.26), one thinks of that an increase of Z1 with 18.26 is enough. Due to

its penalty parameter λZ = 100, this could increase the present costs by 1826. However, in

the opitmal solution, Z1 is increased by 20.86 with additional present costs of 2086, the extra

values are compensated by the decrease in expected future costs.

Now we integrate the expected optimal solutions between contributions rates, remedial con-tributions, and portfolios. As deceasing β1, expected contribution rates are non-increasing at

t = 1, 2, 3, 4, 5. On the other hand, as decreasing β1, remedial contributions at t = 1 (Z1)

are increasing. This illustrates that higher remedial contributions paid at t = 1, then contri-bution rates will be relatively lower at t = 1, 2, 3, 4, 5. In addition, as deceasing β1, remedial

contributions at t = 2, 3, 4, 5 (Z2,Z3,Z4, and Z5) are non-increasing. It suggests that if a large

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0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 % of A₁ (β₁) Zt Z₁ E[Z₂] E[Z 3] E[Z 4] E[Z₅]

Figure 4.6: Remedial contributions as function of β1 in STRCicc+MTRP model.

be paid at t = 2, 3, 4, 5. Finally, portfolios almost keep fixed at t = 1, 2, 3, 4 when β1 changes.

We conclude that in this numerical experiment, the integrated chance constraint on the expected funding shortfall is suitable for modeling short-term risk criterion. It gives the im-pression of7 lower short-risk risk in exchange for marginally higher expected total funding costs.

Finally, we need to select an appropriate value of β1, but it is difficult since it cannot be

easily specified by the right hand side parameter of the chance constraint. We select β1linked

with the reliable level required by FTK. What we do is to consider the optimal solutions of STRCcc+MTRP model including the reliability constraints P {A∗₂− αL2 ≥ 0} ≥ 0.93. We

solve STRCcc+MTRP model by θ1 = 0.93 required by FTK, then compared with optimal

solutions of STRCicc+MTRP model by the different β1. It appears that β1 = 1.72 million

euro (0.047% of A1) yields the reliability 0.93 required by FTK. The optimal solutions and

optimal values of the STRCicc+MTRP model by taking β1 = 1.72 are presented in Table4.9.

Effects on Funding Ratio by Short-Term Risk Criterion

However, funding ratios of these two models are quite different. The constraint of restricting expected maximal shortfall β1 = 1.72 at t = 2 demands extra 168.36 million euro (4.4% of

A1) at t = 1. This affects the funding ratio (F1) to increase from 0.95 to 0.99 at t = 1. The

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1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 t Ft E[F]_MT. E[F]_SMT. minF_MT. minF_SMT. maxF_MT. maxF_SMT.

Figure 4.7: Funding ratios in MTRP and STRCicc+MTRP models at t = 1, 2, 3, 4, 5.

Effects on Financing Policy by Short-Term Risk Criterion

The remedial contributions for the MTRP model and the STRCicc+MTRP model at t = 1, 2, 3, 4, 5 are presented in Table4.8 and Table 4.9. No remedial contributions are paid at t = 1 in the MTRP model, however, there are remedial contributions 168.36 (4.4% of A1)

paid at the same time in the STRCicc+MTRP model. At t = 2, 3, 4, 5, expected, minimal, and maximal remedial contributions are paid less in the STRCicc+MTRP model than them paid in the MTRP model.

The optimal expected, minimal and maximal investments and contribution rates are quite analogous in the MTRP model and the STRCicc+MTRP model at t = 1, 2, 3, 4, 5.

Effects on Investment Strategy by Short-Term Risk Criterion

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STRCicc+MTRP Model total = 26401.36 present = 16988.59 future = 9617.56 t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 Funding ratio E [F ] 0.99 1.08 1.22 1.29 1.39 min F 1.04 1.05 0.95 1.1 max F 1.12 1.38 1.73 2.72 Contribution rate E [c] 0.15 0.2 0.187 0.157 0.118 min c 0.2 0.15 0.1 0.05 max c 0.2 0.2 0.2 0.2 Remedial contribution E [Z] 168.36 0 0 0.76 96.99 min Z 0 0 0 0 max Z 0 0 195.34 2039.84 Investment in cash E [X1] 400 0 680.25 738.77 439.97 min X1 0 0 0 0 max X1 0 891.41 1105.57 1545.33 Investment in bond E [X2] 1500 1526.99 1423.38 2107.33 2551.66 min X2 1526.99 796.3 697.46 617.67 max X2 1526.99 2228.53 3517.22 4254.15 Investment in stock E [X3] 1600 2290.49 2118.08 1757.78 1856.09 min X3 2290.49 1151.94 1099.35 956.45 max X3 2290.49 2815.2 3316.71 4635.99 Portfolio E [%cash] 11.4 0 16.1 16 9.1 E [%bond] 42.9 40 33.7 45.8 52.6 E [%stock] 45.7 60 50.2 38.2 38.3