A Guide to IMF Stress Testing

Chapter 12. Conducting Stress Tests of Defined Benefit Pension Plans

Li Ong
Published Date:
December 2014
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Gregorio ImpavidoThis chapter is an abridged version of IMF Working Paper 11/29 (Impavido, 2011). The author is grateful for the comments received from Karl Habermeier, Deborah Marzouk, Richard Munclinger, Craig Thorburn, Ian Tower, Hasko van Dalen, Manuel Peraita, Asta Zviniene, and participants at IMF seminars at various stages of the project. The author is particularly indebted to Michael Hafeman, an actuary and pension specialist, for all the detailed comments received as well as key suggestions aimed at improving the accompanying spreadsheet tool.

This chapter describes the basic mechanics of defined benefit (DB) plan liability valuation and how to conduct simple stress tests of the solvency ratio. We first review the essential building blocks of liability valuation of DB plans and then, using an Excel-based template with institution-specific data, we take readers through the basics of liability valuation and stress testing of assets and liabilities of a typical DB plan. The template is by no means a substitute for a proper actuarial evaluation, but the simplifications introduced do not affect its usefulness to evaluate sensitivity of the funding ratio.

Method Summary

Method Summary
OverviewThe method implements various single-factor stress tests of the solvency ratio of a defined benefit plan. They include (1) changes of asset values for different asset classes; and (2) changes in actuarial assumptions for valuing liabilities.
ApplicationThe method is appropriate for situations where data are limited and are of poor quality.
Nature of approach
  • Full valuation approach (balance sheet).

  • Assets are valued at market prices.

  • Plan continuation pension liabilities are valued using a closed-group, aggregate, projected benefit obligation method with constant dollar prorating.

Data requirementsPortfolio distribution of assets under management, plan benefit formula, plan demographics, and actuarial assumptions.
  • The method focuses on the liability side of the balance sheet and is independent of solvency regulation.

  • It is very effective at assessing changes in solvency levels as a result of changes in asset values and actuarial assumptions.

  • The data requirement on assets is kept at a minimum, but more granular information on portfolio structure may be used if desired and available.

WeaknessesThe method focuses only on old age pension benefits, and it uses one actuarial valuation method to estimate old age pension liability levels. Hence, it is not a substitute for a full actuarial evaluation of the plan, which could cover (depending on the plan bylaws and relevant regulations) other benefit lines and valuation methodologies that can vary for funding or disclosure purposes.
ToolThe Excel spreadsheet macro is available in the toolkit, which is on the companion CD and at www.elibrary.imf.org/stress-test-toolkit.

This chapter develops a simple methodology for stress testing the funding ratio of defined benefit (DB) pension plans. Using an Excel-based template with institution-specific data, we take readers through the basics of liability valuation and stress testing of assets and liabilities of a typical DB plan. The stress testing framework presented here uses a full asset and liability valuation approach, independent of the solvency regulation specific to any given country. Assets are valued at market prices. Plan continuation pension liabilities (i.e., excluding ancillary benefits) are valued using a closed-group, aggregate, projected benefit obligation method with constant dollar prorating.1 The method developed here is independent of the solvency regulation of any given country as we compare the market value of assets with the value of actuarial liabilities and define the plan funded if the resulting funding ratio is above 100 percent.

The framework presented is no substitute for a full actuarial evaluation of the plan. The Excel-based template is ideally used in an operational context (such as the IMF’s Financial Sector Assessment Program [FSAP] or for technical assistance) to inform: (1) the assessor on the nature of the plan actuarial assumptions; (2) the assessor on the sensitivity of the funding ratio to changes in actuarial assumptions and asset values; and (3) policy dialogue on the desirable actions needed to strengthen the solvency regulatory and supervisory framework, as well as the financial viability of the plan.

The remainder of this chapter is structured as follows. Section 1 introduces basic actuarial factors and functions involved in the estimation of actuarial liabilities of DB plans. Section 2 discusses the valuation method used in the accompanying template for the liabilities of retired and active members. Section 3 takes a real case of a DB pension plan and builds a model plan to conduct simple single-factor stress tests of assets and liabilities. Finally, conclusions and a brief discussion of possible extensions are contained in Section 4.

1. Basic Actuarial Cost Factors and Functions

The calculation of DB pension liabilities relies on assumptions for several key actuarial costs considered in the calculation of liabilities. These are decrement assumptions (like mortality, termination, retirement, and disability) needed for the computation of survival probabilities, salary assumptions, inflation, and discount rate assumptions. We discuss these concepts in turn in the remainder of this section.

A. Decrement assumptions

Decrement assumptions describe how plan members’ liabilities are affected by various risks/contingencies. For instance, retired members are exposed only to mortality risk, whereas active members are exposed to mortality, termination, disability, and retirement risk. In order to describe how members transit from one status to another, rates of decrement (given by available tables) are assumed for each contingency or risk. In other words, the rate of decrement refers to the proportion of the plan population in one status transiting to another status. In this chapter, we indicate withqx(k) the rate at which members of age x transit from one status to another within the period between x and x + 1. The prime indicates we are working with rates (as opposed to probabilities), and k indicates the type of contingency; in this study, we use only mortality (m) and termination (t).

A DB plan would typically have other decrement assumptions. For instance, when disability benefits are provided for, the rate at which members become disabled and—depending on the nature of the disability benefits—the rates on death and continuance of disability could be relevant and would also need to be factored into the funding and costing calculations. Also, retirement decrement rates become relevant when the plan provides for early or delayed retirement. They affect only active members between the earliest and the latest permissible retirement ages, but they affect the level of benefits paid during retirement. In this study, we will not consider other decrement assumptions because doing so can greatly complicate the calculations but is unlikely to materially affect the outcome of stress tests performed in the context of an FSAP.

B. Contingent, composite survival probabilities

From the aforementioned decrement assumptions, it is possible to derive the composite survival probabilities. The idea is simple: All the various contingencies to which individuals are exposed will affect their probability of survival in the plan and, therefore, future liability cash flows. Composite survival probabilities are very important assumptions as they can affect cash flows into the very distant future, potentially more than 70 years away from the valuation date. Here, it is important to notice that we are moving from a decrement rate space to a decrement probability space. Therefore, we need to discuss how probabilities relate to rates in a multidecrement environment and how to construct composite survival probabilities.2

In a single-decrement environment, the rate of decrement is equal to the probability of decrement. In a multiple-decrement environment, the rate of decrement is higher than the probability of decrement. This is the case for active members who are exposed to multiple risks during the year. For them, the probability of dying (say) will be linked to the other decrement rates and will be smaller than the rate at which the population dies. More formally, if we indicate with qx(k) the rate of decrement for cause k at age x and with qx(k), then qx(k)>qx(k). In practice, by assuming that rates are uniformly distributed over the period, decrement probabilities can be approximated with the following:

Composite survival probabilities give us the probability that an individual will survive from one period to the next. In a single-decrement environment, they are simply the complement of the decrement probability. In a multiple-decrement environment, they are “composite” in the sense that they are the complement of the sum of the relevant decrement probabilities. More concretely, for an individual aged x exposed to mortality, termination, and early retirement contingencies, the composite survival probability of surviving until age x + 1 is given by

where qxk is defined in equation (12.1) and (T) indicates that we are in a multiple-decrement world.

Contingent, composite survival probabilities can be used to derive the survival probability over more than one period. In particular, given the time independence of probabilities, for an individual aged x, the probability of surviving for the next n periods is simply given by

C. Salary assumptions

The future level of salaries is also a key assumption affecting the liabilities of DB plans. Given that benefits accrual can depend on future salaries, there needs to be a way to estimate them. In addition, because such estimations can span many years (like for mortality assumptions), their impact on funding and costing can be important.

Future salary assumptions depend on three key factors. These are merit increase assumptions, productivity improvement assumptions, and inflation assumptions. Merit increase assumptions are intended to reflect the increased contributions of members to the employer as they progress in their careers. Productivity assumptions vary across sectors, but it is not uncommon to observe an assumption of 1 percent annual increases in productivity. Both merit and productivity assumptions are difficult to estimate as they can vary significantly from one employer to another and can change in response to changes in economic and labor market conditions. Finally, the most important assumption for future salaries is typically inflation.

Future wage assumptions are summarized by the wage function. Once the aforementioned assumptions are made, it is possible to estimate the future wage levels with the aid of the wage function. For any plan member of age x who joins the plan at age y with a beginning of the period wage wy,y, future salaries can be calculated as

where wy, y is the beginning of the period entry wage, msx,y is the cumulative merit increase at age x, msy,y = 1 is the cumulative merit increase for service before age y (hence, = 1 to indicate no increase), π is the inflation assumption,3 and pr is the productivity improvement assumption. The wage function wfx,y gives the wage wx,y as a multiple of the entry wage wy, y. More formally:

D. Discount rate assumptions

Discount rates, together with survival probabilities, are one of the most critical assumptions in valuing DB liabilities. The reason is that they are used to discount expected future cash flows, which are often very distant in time, to the present time. For instance, the last pension payment for a new entrant could be more than 80 years in the future.

There is a large debate on what the appropriate discount rate should be for pension liabilities. Traditionally, actuaries have argued that pension liabilities should be discounted at rates that reflect the expected long-term returns on a pension fund’s assets, while financial economists (and increasingly some actuaries) have countered that pension liabilities should be discounted at market rates, either the risk-free government bond yields or the high-grade corporate bond yields. This debate is being superseded by a requirement for consistent (market) valuation of assets and liabilities and the introduction of countercyclical/dynamic solvency buffers.4 For our purpose, we allow for a very general formulation5 of the rate νn that discounts the cash flow expected to occur at the beginning of period n to time zero and given by

where fs is the forward interest rate assumed for the sth year. This, in turn, can be extracted from the government debt yield to maturity curve (ytms) from the following relationship:6

2. Estimation of Actuarial Liabilities

Actuarial liabilities are estimated for each category of plan member. The two most important categories are typically retirees and active members. Other categories can include active members with past service benefits, active but disabled members, and terminated members with deferred benefits. For the sake of simplicity, we will focus in this section on only retired and active members.

A. Individual actuarial liabilities for retired members

The actuarial liability for a retired member can be calculated through the use of annuity functions. Basically, this involves projecting the year-by-year contingent cash flows and discounting them through an annuity function that corresponds to the form of pension provided by the DB plan.

DB plans provide a variety of annuities. These aim at insuring retirees against risks like longevity and inflation. They can cover also spouses and other beneficiaries, insurance can be extended for a certain period or for life, and they sometimes provide for participation in investment and mortality experience. In our template, we focus exclusively on straight individual life annuities.

A straight life nominal annuity insures individuals against longevity and investment risks only. It provides periodical benefit payments fixed in nominal terms B starting with the normal retirement7 age r = 55 until death. The present value of these expected cash flows is given by

where är is the present value of a straight life annuity due (i.e., payable at the beginning of each period), sPr(m) is the contingent survival probability for an individual retired at the normal retirement age r defined in equation (12.3), and νs is the discount rate for period s defined in equation (12.6).

A straight life real annuity insures also against inflation risk. In this case, the value of the periodic benefit payment is not constant anymore. After the initial payment of B, it increases with inflation at a rate π.8 Its present value is given by

Initial periodic payments of real annuities are much lower than nominal annuities that have the same present value. Figure 12.1 reports the expected cash flows for a nominal straight life annuity due paying B=1 and a real annuity starting at the normal retirement age r=55. The areas under the curves are nothing more than the sum of these expected cash flows and given by equations (12.8) and (12.9), respectively. Given that the present values of the two annuities must be the same,9 it follows that the initial payments from the real annuity must be lower than nominal payments (Figure 12.2). Of course, many more annuities can be imagined. However, these are not discussed here as this would be beyond the scope of this introductory presentation.

Figure 12.1Straight Life Annuities—Expected Nominal and Real Cash Flows

Source: Author.

Note: Calculations assume: r = 55, π = 0.035, B = 1 for nominal annuity, B = 0.6246 for real annuity with same present value, mortality rates defined and discount rate defined in Table 12.1.

Figure 12.2Straight Life Annuities—Nominal and Real Cash Flows per Surviving Annuitant

Source: Author.

Note: Calculations assume: r = 55, π = 0.035, B = 1 for nominal annuity, B = 0.6246 for real annuity with same present value, mortality rates defined and discount rate defined in Table 12.1.

B. Individual actuarial liabilities for active members

The calculation of actuarial liabilities for active members is a considerably more complex affair than the calculation of actuarial liabilities for retired members. This depends on the plan benefit formula and on the actuarial cost method used. Unfortunately, many benefit formulae exist and many cost methods can be used. We discuss these two concepts in turn.

Benefit Functions

The accrued benefit function is used to determine the level of benefits paid at retirement. It sums the amount of benefits that individuals have accrued over time before retirement. Hence, if we define by bx,y the amount of benefits accrued between age x and age x+1 by a plan member who entered at age y (sometimes called “accrual factor”), the accrued benefit function is simply given by the sum of past accrued benefits:

Accrued benefit functions can take many forms, and in this study, we consider only the final average salary.10

Final average salary benefits are based on the average compensation over a period defined by the plan rules. In its simplest case, plan members are credited units of benefits during their career through a given accrual factor, and these will be applied to their average pensionable salary during a specified period (say, 1, or 3, or 5, or 10 years immediately prior to retirement or termination).11 In a more complex case, the average pensionable salary is calculated over the years of highest compensation (rather than the last years prior to retirement), which might not need to be consecutive. Another variant is to average the highest compensation over n years in the final m>n years of employment (rather than over the full career). If the average is calculated over the last five years immediately prior to retirement or termination, the accrued benefit function is given by

where wx,y:5 is the final five years’ average salary. When the average is calculated only on the last year of service, the accrued benefit collapses to a final salary function, which we will be using in this study.

Projected Unit Credit Cost Method

The second element needed for the calculation of liabilities for active members is the actuarial cost method. Actuarial cost methods define how expected costs of a pension plan are allocated over the period of active membership. These methods (also known as “funding methods”) generate an annual normal cost (NCx,y) representing the present value of the benefits earned under the plan that have been allocated to that year.12 The NCx,y for a member for a particular year is given by

where bx,y* is the benefit accrual allocated to the period between attained age x and attained age x + 1 for a member who joined the plan at age y and stemming from the use of a cost method yet to be defined (and therefore, marked by the asterisk), rxpx(T) is the composite probability that the individual aged x will survive until the normal retirement age r, νr-x is the discount factor used to calculate the present value of the accrued benefits, and är is the present value at the normal retirement age r of the annuity provided by the plan.

In general, the sum of the normal costs up to the attained age provides the actuarial liability under the given cost method. Hence, the actuarial liability for an individual with attained age x and entry age y can be defined as

where Bx,y* is the value of accrued benefit up to the attained age x for a member who joined the plan at age y and stemming from the use of a cost method yet to be defined (and therefore marked by the asterisk).

Two general categories of cost methods are more common: (1) benefit allocation methods; and (2) cost allocation methods. The two methods are radically different and will produce very different values of actuarial liabilities for the same set of underlying data and assumptions, as shown later. In general:

  • Benefit allocation methods define how to allocate the total benefits that will be earned by a person to the various years of service and then value the annual allocations.

  • Cost allocation methods value the cost of the total benefits that will be earned by a person and then allocate the total cost among the various years of service.

As a result, they produce different values of actuarial liabilities, ALx,y*. In this study, we focus exclusively on benefit allocation methods.

Benefit allocation methods allocate units of benefits to the various plan years. With these methods, the actuary would calculate the present value of accrued benefits that all plan members may become entitled to (if they survive) either from past service (i.e., rendered up to the valuation date) or also adding future service (i.e., projected until retirement). Two types of benefit allocation methods are described here: the unit credit method and the projected unit credit method.13

The unit credit method considers only past service. The accrual function bx,y is determined by the benefit function/formula defined by the plan (typically, flat dollar). The accrued benefit function Bx,y is simply the sum of the accruals between entry age y and attained age x. Finally, the actuarial liability is called accrued benefit obligation (ABO), which is equal to the present value of the benefits accrued up to the attained age x. Hence:

The projected unit credit method considers also future service until the normal retirement age. The accrual function bx,y is determined by applying the benefit function/formula defined by the plan (flat dollar, career average salary, or final average salary). The accrued benefit function Br, y is simply the sum of the accruals between entry age y and the normal retirement age r. The method therefore produces an actuarial liability that is equal to the present value of the benefits that will be accrued at the normal retirement age evaluated with the accrued benefit function/formula defined by the plan. Given the definition, we will call this liability retirement benefit obligation (RBO).14 Hence:

The RBO method is not used in practice as it breaks any relationship between costs and service rendered. The RBO method (also known as initial funding method) forces the sponsor to reserve for the present value of all possible benefits accruable until retirement right at entry age y, independently of whether the plan member will remain on payroll until then. This goes against rational actuarial and accounting practices, which attempt to allocate pension costs in reasonable relationship to the services rendered. Hence, it is not used in practice. Nonetheless, it is a useful concept as all other accepted methods are essentially a weighted function of the RBO method.

For this reason, regulations allow fund managers to prorate the RBO obligation for funding purposes. Given that the RBO liability is often much larger than the ABO liability and because it is not certain that workers will remain on payroll until retirement, regulations allow plan managers to prorate the RBO liability over time. What proration achieves is to allocate projected benefits equally to all years of service. Notice that accrued benefit functions are very flat during the early years of an individual career and very steep toward retirement. Prorating reduces the amount of reserves that the sponsor has to allocate toward retirement by anticipating factors that are expected to increase the ultimate level of benefits and recognizing their cost throughout the working years. Proration of the RBO liability produces an actuarial liability called projected benefit obligation (PBO).

The most commonly used proration is the constant dollar method.15 With the constant dollar pro rata method, the accrual factor bx,y is constant and defined as a fixed share of retirement benefits—that is, bx,y = Br,y/(r – y). Hence, the accrued benefits Bx,y at age x are simply the product of the constant accrual benefit and the number of years between attained and entry age—that is, Bx,y = (x–y) Br,y/(r–y). Hence:

C. From individual to aggregate values

So far we have focused on the liabilities of a single individual—retired or in active service. Aggregate liabilities for the whole plan can be calculated in two different ways. Under the individual method, the actuary calculates the individual liabilities and other functions (as done in this section) and then sums them across all different individual members. Under the aggregate method, no reference is made to individuals, and values are calculated on an aggregate basis by averaging values across all individuals or specified cohorts. In the next section, we will use an aggregate method by annual cohorts as this allows us to be much more parsimonious in terms of data requirement. The template can also use average information on cohorts of more than one year.16

3. Stress Test Methodology

This section refers to a real-life plan to discuss a possible stress test methodology implemented with the accompanying template Model.xls. The model plan uses the definitions developed in the previous sections, and the underlying data and assumptions are taken from a real DB plan. This section can be read in conjunction with the Appendix, which focuses on the mechanics of the accompanying Excel template.

Even for a simple plan like the one chosen, several simplifications are necessary in order to make the template easy to use in an operational context.

A. The real pension plan

The real plan has the following characteristics:

  • Benefits. The plan provides for retirement, death, and disability benefits with various withdrawal options in the form of single gender-specific and/or joint straight life real inflation-indexed annuities or cash lump sums.

  • Retirement benefits. Retirement benefits represent the largest liability of the plan and are based on a final five-year average salary formula with an accrual rate of 1 percent. Entry age varies from a minimum of y=20 to the normal retirement age, r=55. Early retirement is allowed starting with age x=45, and in a few cases delayed retirement was granted. We know that early retirement reductions are less than actuarially fair, but no information is available on benefit accrual in the case of delayed retirement.

  • Assets and liabilities. The investment portfolio is reported by asset classes and economic sectors, as well as domestic and foreign. Total assets are expressed in local currency units (LCU) and amount to LCU3,773 million. The plan actuary uses a projected benefit obligation constant dollar (PBOcd) actuarial method to calculate active members’ individual actuarial liabilities. These are added up to yield the plan aggregate liability. The latest actuarial report shows liabilities amounting to LCU4,000 million.

  • Solvency regulation. No minimum solvency margin is required in the jurisdiction. The scant rules issued simply require that a fund has assets in excess of liabilities to be considered “funded.” The reported solvency ratio is 94 percent.

  • Actuarial cost factors. The plan actuary uses the male mortality and termination rates defined in Table 12.2. In addition, we could obtain inflation, salary increase, and discount rate assumptions reported in Table 12.1. We could not obtain information on other decrement factors as the actuarial report was not available.

  • Plan membership. The relevant distributions are reported in Figure 12.3. The first panel reports the density distribution of the number of active workers with two peaks around ages 40–44 and 49–53. The second panel reports the density distribution of the wage remuneration of active workers with two peaks around ages 40 and 52. The third and fourth panels report the density distributions of the number of retired workers and their pensions, with a large concentration soon after the normal retirement age of 55. Notice that the active distributions include individuals active beyond the normal retirement age and the retired distributions include individuals who retired before the normal retirement age. Actual data are reported in Table 12.1.

Table 12.1Actuarial Factors Used in the Model Plan (Series)
Source: Actual defined benefit plan.
Table 12.2Actuarial Factors Used in the Model Plan (Cells)
SymbolValueInput CellsAssumption
y20Input!D12Entry age
r55Input!D13Normal retirement age
Xmax115Input!D14Max age
Pr1.0%Input!D17Labor productivity
πw3.5%Input!D18Inflation (wages)
πa3.5%Input!D21Inflation (annuities)
b1.0%Input!D22Accrual rate
Source: Actual defined benefit plan.

Figure 12.3Active and Retired Member Distributions

Source: Actual DB plan.

B. The model pension plan

On the basis of the information collected, we constructed a model plan with the following characteristics:17

  • Benefits. We consider only retirement benefits in the form of a single life inflation-indexed annuity.

  • Retirement benefits. Retirement benefits are calculated on the basis of a final salary formula with a constant accrual rate. Effective entry and retirement ages are assumed the same for all members and set at y=20 and r=55.

  • Assets and liabilities. For assets, we use investment portfolio data reported by the real plan. Regarding liabilities, we use an aggregate (by annual cohorts) PBOcd method to calculate active member liabilities. Notice that we are not interested in the level of the actuarial liabilities per se but in their change for given shocks in the actuarial factors. Once the model plan’s actuarial liabilities are estimated, they are re-scaled to coincide with the liabilities calculated by the plan actuary. Once the shock is applied to the actuarial factors and the change in the model plan’s actuarial liability is calculated, this same change is applied to the real plan’s actuarial liability to calculate the change in the funding ratio. The error made by moving from the change in the model liability to the change in the plan liability is immaterial for our purpose.

  • Actuarial cost factors. Given that we could not obtain additional information on actuarial cost factors, we use only male mortality for all members and termination rates defined in Table 12.1. Wage projections are calculated using actual merit scale, inflation, and productivity assumptions reported in Table 12.2.

  • Plan membership. We use real plan distributions reported in Figure 12.3. Given the cutoff date imposed by our normal retirement assumption at age r=55, early retirees are ignored in our calculations while late retirees are assumed to have retired at the normal retirement age; all members are assumed to have full career service.

C. Impact of simplifications made

As it is, we have had to make several simplifications in this very simple plan, but they do not reduce the usefulness of the accompanying template. The simplifications might result in either an overestimation of real liabilities or an underestimation. Typically, however, they should have an ambiguous impact on liability estimation:

  • Simplifications that imply an underestimation of true liabilities. We consider only pension benefits and disregard ancillary benefits like death and disability. We also ignore lump sum commutation, which can result in large underestimation of retirees’ liabilities.18 We also consider only single life annuities and disregard joint life annuities. This underestimates the annuity factor for active members and therefore produces a smaller valuation of their liabilities. Naturally, retired members’ liabilities also are underestimated, and such underestimation could be significant if many retired members have joint life annuities and we value them as single life annuities.

  • Simplifications that imply an overestimation of true liabilities. We assume that all plan members join the plan at entry age y with immediate vesting. However, in reality, some individuals would enter later and not necessarily with immediate vesting, therefore accruing smaller vested benefits by the time they terminate employment or retire.19 In addition, we consider only final salary pensions that imply an overestimation of true liabilities relative to the more common case of career-average salary pensions. Finally, we consider full indexation of pension rights to inflation and full longevity insurance, but in many jurisdictions, these indexations are more and more often conditional on the performance of the plan so that inflation, longevity, and investment risk are shared between providers and retirees. Where these forms of risk sharing are present, our model provides an over-estimation of liabilities.

  • Simplifications that have an ambiguous impact. We assume that active members all retire at the normal retirement age of r. Depending on the degree of actuarial fairness of early retirement provisions, the model estimation may be higher or lower than the plan liabilities. We also assume that the pensionable salary (the base of the accrual rate) is equal to total remuneration. This overestimates (underestimates) liabilities when pensionable salary is in fact lower (higher) than total remuneration.

  • Plan termination or continuation valuation. The choice of method (PBOcd) implies that we are conducting a plan continuation (ongoing concern) valuation as opposed to a plan termination valuation. This may differ from the regulatory requirements in the local jurisdictions where the template is used.

The simplifications used in our template do not affect the validity of the model for stress testing purposes. The simplifications made are immaterial for our purpose. Again, we are not interested in the absolute level of liabilities, the estimation of which is best left to the plan actuary. Here, we are merely interested in their rate of change for given shocks. Hence, we can rescale our model liabilities to coincide with the true liabilities and apply the model rate of change stemming from the shock to the true liabilities to obtain the change in the funding ratio.

D. Stress testing the funding position—asset shocks

This section uses the enclosed template Model.xls to conduct single-factor stress test of the plan funding ratio. The plan is reasonably healthy, with a support ratio of 413 percent and a funding ratio of 94 percent. However, we observe that the plan is exposed to concentration risk in key asset classes and suspect that the assumptions used by the actuary to calculate liabilities are overly optimistic. For instance, the level interest rate of 9 percent used for discounting cash flows appears excessively high in relation to market interest rates, and the mortality table used is static and not dynamic. In addition, discussions with the actuary have revealed that the termination rates used have not been updated for many years: the economy is booming and the sponsor is retaining far more staff than what it used to when the termination rates were estimated. Finally, we have initial signs that the economy is overheating and inflation expectations are increasing. We conduct both asset and actuarial factor shocks to see how sensitive the funding ratio is to market risk (on the asset side) and to the actuarial assumptions used for liability valuation purposes (on the liability side).

Asset stress tests focus on concentration risk. We collected information on the investment portfolio of the model plan in sheet BAL. The data are disaggregated by type of issuer (government, financial sector, and real sector; both domestic and foreign) and by type of investment vehicle. The analysis can design any type of scenario (macro, credit, or other) that has an impact on asset values that will translate into a change in the funding ratio. For the purposes of our analysis, we assume up to a 25 percent symmetrical shock in the value of assets, issued by the government, financial sector, and real sector, in the value of bonds, stocks, and foreign assets (amounting to a change in the exchange rate).20 Results are reported in Table 12.3. Plan assets are fairly diversified, with marginally higher concentration in real sector investments but almost equally divided among bonds and stocks. Foreign exchange (FX) risk is unhedged, and a 25 percent local currency appreciation (or a decrease in FX denominated assets by the same amount) would reduce the funding ratio from 94 percent to 88 percent.

Table 12.3Model Plan—Stress Tests (Asset Concentration Risk)
Market shockGovernmentFinancialRealBondsStocksFXTOT
Source: Author.Note: FX = foreign exchange; TOT=total assets.

E. Stress testing the funding position—liability shocks

Liability stress tests are more varied, and the results can differ significantly depending on the specific circumstances of the pension plan. The crucial issue about liabilities is that their valuation relies critically on assumptions that may not be validated ex post by experience. The mismatch between assumptions and reality creates unfunded liabilities that might develop very slowly over time and become evident only when it is too late for the sponsor to easily—if at all—remedy the situation. Hence, adequate liability measurement becomes critical even in the short term.21 We consider interest rate, inflation, longevity, and termination rate shocks.

Interest Rate Shock

We assume two types of shocks: (1) we substitute the level 9 percent interest rate used for discounting future cash flows with more reasonable level rates; and (2) we derive market discount rates from a AAA government debt yield curve, which we then shock. In general, changes in the discount rate will affect the present value of retirees’ liabilities by changing νs and äxπ, in equation (12.18)22 and the present value of active members’ liabilities by changing νr−x and ärπ in equation (12.22).23 We expect liabilities for active members to be more sensitive to interest rate changes because of their higher duration: total liability duration is 15 years, versus 17 years for active members and 10 years for retired members.

Interest rate shocks have a large impact on liabilities. If the interest rate assumption decreases from 9 percent to 4 percent, the funding ratio decreases from 94 to 38 percent, or an average 11 percent for every percentage point change in the interest rate (Table 12.4).

Table 12.4Model Plan—Stress Tests (Level Interest Rates)
Funding ratio94.3380.7668.3357.1147.0838.25
Source: Author.

With AAA market rates, the funding ratio is much smaller than reported. When we derive a market discount rate from a AAA government debt yield curve (in our case, the end of 2009 U.S. domestic debt curve24), the funding ratio is around 33 percent (Table 12.5). This dramatic change in the solvency position underlines the importance of having a reasonable discount factor to value liabilities. The plan actuary has assumed that a 9 percent average rate of return on assets yields a 94 percent funding ratio. However, the risk-free rate at 30 years is around 4 percent, and even if the equity risk premium is realized over the full duration of liabilities, the plan will never be able to meet its obligation. Finally, the plan termination funding ratio varies between 26 and 39 percent if we assume parallel shifts of the yield curve of up to 150 basis points25 and derive corresponding discount rate assumptions (Table 12.5).

Table 12.5Model Plan—Stress Tests (Shifts in the Yield Curve)
Shock−150 bps−100 bps−50 bps0+50 bps+100bps+150 bps
Funding ratio26.5228.5130.5932.7034.8637.0439.24
Source: Author.Note: bps = basis points.

Inflation Shock

Inflation shocks affect the values of annuities and wage projections. Inflation affects (1 + πa) in equation (12.18)26 and therefore the annuitization factor ärπ in equation (12.22).27 In addition, it affects (1 + πw) in equation (12.20)28 and therefore the value of accrued benefits Br,x in equation (12.22).29Table 12.6 reports the sensitivity of the funding ratios to changes in inflation assumptions used to project wages and to calculate the annuity factor. A 100 basis point (bps) increase in the inflation assumption for annuity valuation πa reduces the funding ratio from 94 percent to 85 percent. A 100 bps increase in the inflation assumption for wage projections πw reduces the funding ratio from 94 to 89 percent. A 100 bps increase in both inflation assumptions reduces the funding ratio from 94 to 80 percent.30

Table 12.6Model Plan—Stress Tests (Inflation Shocks)
−150 bps−100 bps−50 bpsπa+50 bps+100bps+150 bpsεFR,πw
−150 bps118.44112.85107.36101.9796.7091.5386.49−10.24
−100 bps115.45110.01104.6699.4094.2689.2284.31−10.24
−50 bps112.49107.19101.9796.8691.8486.9382.14−10.24
+50 bps106.66101.6396.6891.8387.0782.4277.87−10.25
+100 bps103.7898.8894.0789.3584.7280.1975.77−10.25
+150 bps100.9496.1791.4986.9082.3977.9973.69−10.25
Source: Author.Note: bps = basis points.

The funding ratio is more sensitive to inflation assumptions used for the projection of increases in pension payments than for wage projections. Notice that the elasticity of the funding ratio, with respect to the inflation assumption for annuity valuation εFR,πa is much larger than the elasticity with respect to the inflation assumption used for wage projections εFR,πw. This is due to the fact that πa increases both ärπ and Br,x in equation (12.22).31

Longevity Shock

We model longevity shocks by projecting mortality improvements over a number of years. In order to capture longevity improvements not captured by the static table used by the plan actuary, we project over a number of improvement years, t, the period mortality rates.

The projected mortality rate in calendar year t, tPRJqx(m), is the rate from the static table multiplied by (1 – rx)t. Thus:

where rx represents annual rates of mortality the multiplicative improvement in longevity for each cohort is x. As an example, Figure 12.4 reports the change in the conditional survival probabilities for an individual aged 55 by applying the U.S. mortality improvement rates used for the 1994 Group Annuity Reserving Table and projecting the static rates for t=30 years.32 The area between the two curves is the change in life expectancy at age 55 (Δe55= 2.70)

Figure 12.4Impact of Longevity Improvements on Survival Probabilities

Source: Author.

Note: PER-sp55(m) = conditional survival probabilities (using the period table) for a male at age 55. PRJ-sp55(m) = projected (at 30 years) conditional survival probabilities for a male at age 55.

Longevity improvements translate into higher values of life expectancy at different ages. For a given rx, an increasing number of improvement years translates into higher values of life expectancy. Table 12.7 reports the increase in life expectancy at the plan normal retirement age of 55 (e55) from 28.39 to 33.96 years and the related decrease in the funding ratio between 94.33 to 85.26 for increasing number of improvement years between zero and 70.

Table 12.7Model Plan—Stress Tests (Longevity Shocks)
Funding ratio94.3389.4688.2187.1086.1385.26
Source: Author.

Termination Rate Shock

Changes in termination rate assumptions will affect liabilities only of active members. A recent report of the personnel department concludes that retention rates have increased by at least 10 percent across the board. On the basis of this information, we investigate the impact on the funding ratio of decreases in termination rates of between 10 and 30 percent. Results are reported in Table 12.8.

Table 12.8Model Plan—Stress Tests (Termination Shocks)
Funding ratio94.3392.1891.1190.0488.9787.91
Source: Author.

4. Conclusion

This chapter describes the basic mechanics of DB plan liability valuation and how to conduct simple stress tests of the solvency ratio, using the accompanying Excel template. We review the essential building blocks of liability valuation of DB plans. We then construct an Excel template to analyze changes in funding ratio of DB plans for alternative values of actuarial factors and asset values. The accompanying template, Model.xls, uses a last salary DB formula and a PBOcd actuarial method to value liabilities. The template is by no means a substitute for a proper actuarial evaluation, but the simplifications introduced do not affect its usefulness to evaluate sensitivity of the funding ratio.

Section 3 provides a few examples of the stress tests that can be conducted with the template, and extensions are, of course, possible. The stress test methodology used in this note is parsimonious and aimed at quantifying key risk exposures by nonactuarial analysts. As already mentioned, it is not a substitute for a full actuarial evaluation. However, it can be extended in several ways:

  • Refinements. On the asset side, it would be worthwhile to extend the analysis to identify sources of risk stemming from interest rate shocks at various maturities and credit risk shocks of large exposure, or of the sponsor (ability to pay contributions), so as to improve the connection of the stress tests with macro scenarios. On the liability side, the methodology could be refined by (1) improving the granularity of the age, wage, and pension distributions; (2) considering additional “decrement factors” beyond the mortality tables such as the distribution for entry into and exit from the labor force (e.g., retirement, disability, voluntary unemployment); (3) reflecting gender and types of pensions in the calculations; and (4) also considering the possibility of decreases in longevity that are due to health, famine, or natural catastrophe events. Finally, other tests could be conducted to assess the impact of plan changes. Indeed, the template lends itself to studying various parametric reforms such as changes in accrual rate, retirement ages, any actuarial assumption, and so forth.

  • Liquidity shocks. Data on asset liquidity are not used in the template. Asset shocks should also include tests on the portion of assets that might need to be used to cover short-term liabilities. These types of shocks are potentially more severe than the ones considered, as they affect the ability of the plan to meet short-term liabilities (rather than long-term) and could force plan managers to sell assets at (potentially) distressed prices, further undermining the ability to meet long-term liabilities. These shocks are very important for closed plans, with no active members accruing benefits or plans with very low support ratios.

  • Multi-asset shocks. Multi-asset (factor) shocks have not been considered. This would require the estimation of asset classes return correlations.

  • Asset-liability correlations. Asset shocks have been considered in isolation of liability shocks. When liabilities are discounted using a market yield curve, this becomes an unreasonable simplification. In such a case, the analyst should attempt to offset changes in liabilities, with changes in the value of the portion of assets that are interest-rate sensitive. This would, of course, require knowledge of the durations of these assets.

  • Expected cash flow analysis. The analysis conducted in Section 3 is merely focused on changes in the funding ratio. An alternative, potentially appealing way to present the same results is to analyze the shocks in terms of impact on future cash flows to determine when assets (always on a termination basis) are enough to meet liabilities. The template already produces expected liability cash flows, and it would be possible to project asset cash flows, by assuming future rates of returns on assets, dispositions of assets, and the allocation of future cash flows to different types of assets.

Appendix. The Accompanying Template (Model.xls) and Assumptions

The accompanying Model.xls template contains the following assumptions:

  • Basic Assumptions (Table 12.2). Entry age is set at age y =20 (Input!D12) and normal retirement age at r=55 (Input!D13).

  • Decrement assumptions (Table 12.1). Mortality rates qx(m) (Input!I11:I136) are derived from the 1996 U.S. male annuitant table (t887). Termination rates qx(t) (Input!J11:J136) are assumed between age 20 and 54. Mortality and termination rates are used to calculate the mortality probabilities qx(m) the termination probabilities qx(t) and the composite survival probabilities qx(T) following the methodology discussed in Section 1.33

  • Salary assumptions (Table 12.1 and Table 12.2). Wage growth is set at about 7 percent and composed of 3 percent average merit increase (Input!T11:T136), 3.5 percent inflation (πw=0.035 in Input!D18), and 1 percent productivity improvements (pr=0.01 in Input!D17).

  • Discount rate assumptions (Table 12.1). A level 9 percent interest rate is assumed for the purpose of discounting liabilities. Nonetheless, the template allows for almost any type of discount curve derived from seven easily obtainable yield-to-maturity rates (Input!011:0136) following the methodology discussed in Section 1, equations (12.6) and (12.7).

  • Retirement benefit assumptions (Table 12.2). Retirement benefits are calculated on the basis of a final salary formula with constant accrual rate (b=0.01 in Input!D11). Effective entry and retirement ages are assumed the same for all members and set at y=20 and r=55. We consider only retirement benefits in the form of a single life inflation-indexed annuity, with the inflation assumption defined as πa=0.035 in Input!D20.

  • Distributions of active and retired members (Table 12.1). For each age cohort, we collected the number of workers (Input!V11:V136) and retirees (Input!W11:W136), as well as the cohort wage remuneration (Input!X11:X136) and cohort retirement benefit paid (Input!Y11:Y136). These data are used to calculate the total number of workers (Input!V6), retirees (Input!W6), remuneration (Input!X6), retirement benefits (Input!Y6), and their density distributions (Input!Z11:Z136 to Input!ACll:AC136).

Actuarial Liabilities for Retired Cohorts

Actuarial liabilities for retired cohorts are calculated in sheet AL-R. This is implemented in two steps:

In the first step, we calculate for each cohort x∊ [r, ∞) (in row 'AL-R'!F8:CH8) the present value of $1 real life annuity using equation (12.9). This is given by

where the inflation term structure (1 + πa)s rendered in 'AL-R'!B11:B136, the discount factor curve νs is derived using equation (12.6) and rendered in 'AL-R'!C11:C136, the conditional probabilities of survival px(m)s are derived using equation (12.3) and rendered in matrix 'AL-R'!F11:CH136, and the present value äxπ is rendered in row 'AL-R'!F7:CH7.

In the second step, we aggregate the actuarial liabilities for all retired cohorts. This is given by

and rendered in 'AL-R'!D1.

Actuarial Liabilities for Active Cohorts

Actuarial liabilities for active cohorts are calculated in sheets Brx and AL-A. Among the actuarial cost methods described in Section 3, we use the projected unit-credit, constant-dollar method (PBOcd). Again, there are two steps involved:

In the first step, we calculate in the Brx sheet the accrued benefit at retirement for each active cohort. This is done by first projecting wages until retirement for all active cohorts. These are needed because we are performing a projected benefit obligation constant-dollar evaluation as discussed in Section 1, equation (12.5). Hence, in matrix Brx!F11:CH136, we project for eachcohort x∊ [y,r) (in row Brx!F8:CH8) the future wages for each period s∊ [x,r) (in column Brx!A11:A136) as a multiple of today’s wage using the following formula:

We then calculate for each cohort in row Brx!F1:CH1 the total accrued benefits at retirement, using the plan final salary benefit formula and the distributions of active members and salaries:

where wfr,x=max(wfs,x).34

In the second step, we aggregate in the sheet AL-A the actuarial liabilities for all active cohorts. As discussed in Section 3, these are given by the product of the present value of life annuity at retirement (px(T)r-xυr-xärπ), the accrued benefit at retirement (Br,x), and the constant dollar method prorated (x–y)/(r–y). The plan actuarial liabilities for active cohorts are nothing less than the sum of cohort liabilities across all active cohorts rendered in 'AL-A'!D1. Thus:

where px(T)r-x is the conditional composite probability of survival between age x and age r rendered in matrix 'AL-A'!E11:BM136 and row 'AL-A'!E3:BM3, νr–x is the discount rate for (rx) periods rendered in column 'AL-A'!C11:C136 and row 'AL-A'!F4:CH4, and ärπ is the present value of a real life annuity at the normal retirement age rendered in cell 'AL-R'!F7.

Finally, total model plan liabilities are calculated in sheet AL-TOT. These are nothing less than the sum of the actuarial liabilities for active and retired members—that is, AL(TOT) = AL(R) +AL(A).


Given the summary nature of this chapter, we do not provide definitions here but refer the reader to Impavido (2011) and references contained therein for a more pedagogical text.

The accompanying spreadsheet uses two decrement factors (mortality and termination) to derive composite survival probabilities for active members.

This does not need to be constant. In particular, when current levels of inflation are significantly different than long-term expectations, it is common to grade the assumptions over time.

In addition, there is a debate regarding the appropriate discount rate to be used for financial reporting versus funding standards. Regulations typically would require some form of market rate for financial reporting standards, while they would allow for a more stable rate for funding standards. See Rocha and Vittas (2010) for a discussion.

In other words, we do not enter in the debate of the appropriate discount rate. In our template, we simply allow for any type of rate, whether market determined or fixed in regulation.

The use of the forward rates, rather than the spot rates, in calculating discount rates implies an underestimation of the discount rate curve. This stems from the fact that yield rates typically contain an illiquidity premium to compensate for the higher risk involved in investing at longer maturities. Estimating such premium and correcting for it would be beyond the scope of this introductory text. For our purpose, we ignore the difference that is likely to be trivial in very liquid markets.

Throughout Sections 3 and 4, we calculate individual liabilities assuming that plan members all retire at the normal retirement age. In practice, for valuing liabilities, one needs to begin with the actual retirement age of each retiree. We discuss later the impact of this simplification, which is also made in the accompanying template Model.xls.

For example, the rate of increase might be one-half the inflation rate, or it may be discretionary or ad hoc. For instance, many pension plans that provide indexed pensions in the United States do not provide full indexation or automatic cost-of-living adjustment. As discussed in Section 4, we will be able to account for this possibility by assuming alternative inflation indexation rates for annuities in the accompanying template Model.xls.

Intuitively, if är is the premium that would buy a straight nominal annuity that pays periodically B = 1, then the same premium would buy a real annuity that pays less (initially, although increasing with inflation).

Again, Impavido (2011) and references contained therein include a more general discussion of benefit functions.

For instance, with a 2 percent accrual factor, an average salary during the final five years of US$100,000, and 40 years of service, the retired member would receive an intitial pension worth US$80,000.

Unfortunately, things are not as simple because many other types of costs also should be considered when calculating the actuarial liability. Three are more common: supplemental costs, past service costs, and ancillary costs. The first type of cost is generated when the experience of the plan deviates from the assumed cost factors creating actuarial gains or losses. The second type of cost is generated when the plan recognizes liabilities for past service prior to the introduction of the plan (or an amendment of the plan that applies improvements to past service). The third is associated with ancillary benefits like death and disability (in addition to retirement/pension benefits). For sake of simplicity, we will not discuss these types of costs, and we consider only normal costs associated with retirement benefits based on the normal retirement age r.

Again, Impavido (2011) and references contained therein include a more general discussion of alternative cost methods.

We owe this very intuitive RBO label to Milevsky (2006). This is also known as the present value of future benefits. Notice that the way we use “projected unit credit” in this text to indicate the general class of methods that project benefits until the normal retirement age can be misleading in the United States. In this country, this term is generally used to indicate the constant dollar variation that we will discuss later.

Indeed, in the United States, this is the only method that can be used for preparing financial statements, although many others can be used for funding purposes.

An aggregate approach is acceptable if the data necessary for individual calculations are unavailable or if the purpose of the valuation does not require much precision (such as a stress test done as part of an FSAP). However, individual calculations are greatly preferable.

The Appendix contains a detailed description on how the accompanying template works. Table 12.7 and Table 12.8 summarize the actuarial cost factors used.

In the United States, for instance, the discount rate used to calculate lump sum commutation is neither the one used for financial reporting nor the one used for funding purposes. It is linked to the Fed rate and established year by year by the Internal Revenue Service. Such disconnection can result in very large unfunded liabilities, depending on the monetary policy stance.

Also, the calculations assume that all benefit payments are made annually, at the beginning of the year. On average, they will be made one-half year later.

This is a rough test. Flat percentage changes are more commonly used to test the effects of changes in equities and real estate, but for bonds and mortgages, it would be more common to calculate or estimate the average duration of the portfolio and then test the effects of a given change in interest rates. For this exercise, information on duration of fixed income instruments was not available.

According to a recent survey, MetLife Assurance Limited (2010) finds that the five risk factors ranked highest in importance in the United Kingdom were Measurement of Technical Provisions/Liabilities, Longevity Risk, Employer Covenant, Investment Management Style, and Funding Deficits, whereas in the United States, the “most important” risk factors were Liability Measurement, Underfunding of Liabilities, Plan Governance, Asset Allocation, and Advisor Risk.

Equation reported in the Appendix.

Equation reported in the Appendix.

This is done for illustrative purposes only: the relevant yield curve for the jurisdiction where the template is applied should be used.

Negative shifts are not strictly parallel at shorter maturities as yield rates cannot become negative.

Equation reported in the Appendix.

Equation reported in the Appendix.

Equation reported in the Appendix.

Equation reported in the Appendix.

In principle, there is no reason why we should have two different inflation assumptions for wage projections and annuity. The only reason why this is done here is that, typically, plans do not provide full inflation indexation. By disconnecting inflation assumptions for wages and annuities, we are able to test the impact of alternative inflation annuity guarantees, while maintaining full inflation indexation in wage projections.

Equation reported in the Appendix.

These rates are as good as any, and in any case, they should be discussed with the local actuary and amended as needed.

qx(m) and qx(t) are calculated using equation (12.1) and rendered in Input!K11:K136 and Input!L11:L136, respectively. px(T) is calculated using equation (12.2) and rendered in Input!M11:M136.

The template can be easily changed by substituting max(wfs,x) with Σj=0n-1wfr-j,xn to implement a final n years’ average salary formula.

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