The effects of projected mortality quotients on accounting Defined Benefit plans under International Accounting Standard 19

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The effects of projected mortality quotients

on accounting Defined Benefit plans under

International Accounting Standard 19

November 2008

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Contents i

International Accounting Standard (IAS) 19 prescribes the accounting for employee benefits. For pension schemes this includes certain calculations which involve mortal-ity assumptions. The past decades Dutch life expectancy showed an upward trend, caused by decreased mortality rates. It can be expected that this trend will continue, causing future life expectancy to rise even further. In 2007 the Dutch association of actuaries published a projection table, which takes future mortality rate development into account. In this thesis the method of construction of the projection table will be looked into, as well as determining the effect of the projection table on the calcula-tions prescribed in IAS 19.

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Introduction 1

1 Introduction

Nearly all Dutch employers offer one or more pension schemes to their employees. To guarantee that sufficient assets are available for payments to pensioners, the required assets have to be placed outside the company such that these assets do not get lost in case the company goes bankrupt. Although pension schemes are carried out by an insurer or pension fund, the company does have to meet certain accounting regula-tions. For companies which are quoted on the Dutch stock exchange these accounting principles are described in International Accounting Standard (IAS) 19.

The requirements of IAS 19 include complex calculations, which are based on a set of actuarial assumptions. One of these assumptions concerns the mortality rates of current and former employees. Based on these mortality rates it can be determined how many and when payments can be expected. It is common practice that the calculations prescribed in IAS 19 are based on (recently) observed mortality rates. However historical mortality rates show that these mortality rates have decreased over the years. If this trend will continue the used mortality rates will turn out to be an underestimation of the actual future mortality rates.

To deal with the anticipated decrease of prospective mortality rates, a projection table can be used, which takes future mortality rate developments into account. In 2007 the Dutch asssociation of actuaries (AG) published such a projection table.

Two main research questions will occupy the centre stage in this thesis. The first question concerns the manner in which the projection table of the AG is constructed and how this projection table affects the life expectancy. The second question con-cerns the effect of the projection table on the calculations for accounting prescribed by IAS 19. These calculations involve mortality assumptions and will therefore be influenced by using the projection table instead of observed mortality rates. This thesis will discuss how these mortality rates are used in the prescribed calculations and how this affects the accounting of the employer.

In order to answer the main questions stated above, Section 2 will discuss IAS 19 and describe the required calculations and the manner in which these calculations should be performed.

In the following section the method of collecting mortality rates resulting in mortality tables is discussed. Also the necessity and used methods of adjusting the collected mortality rates for actuarial purposes is treated.

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2 Introduction

mortality rates.

The effects of the projection table on the calculations prescribed by IAS 19 will be discussed in Section 5. The calculations require the use of so-called actuarial factors and a set of actuarial assumptions. The results of the calculations based on the actuarial assumptions are presented and extensively discussed.

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IAS 19 - Employee Benefits 3

2 IAS 19 - Employee Benefits

The objective of International Accounting Standard (IAS) 19 is to prescribe the ac-counting and disclosure by employers for employee benefits [1]. A consequence of this Standard is that an entity has to recognize:

• a liability when an employee has provided service in exchange for employee benefits to be paid in the future, and

• an expense when the entity consumes the economic benefit arising from service provided by an employee in exchange for employee benefits.

The IASs are published in a series of pronouncements called International Financial Reporting Standards (IFRSs).

2.1 International Financial Reporting Standard

In 2000 the European Commission decided that as from January 1st 2005, all enter-prises quoted on the stock exchange of one of the member countries of the European Union (EU) have to comply with the International Financial Reporting Standards (IFRSs) [2]. The IFRSs are developed by the International Accounting Standards Board (IASB), which forms a part of the International Accounting Standards Com-mittee (IASC) Foundation [3].

The IASB is supported by the Standards Advisory Council (SAC), from which it receives advice. The IASB is also supported by the International Financial Report-ing Interpretations Committee (IFRIC), which makes statements on specific subjects when there is uncertainty in the market.

The aim that the IASB pursues is to develop a single set of high quality, under-standable and enforceable global accounting standards that require transparent and comparable information in general purpose financial statements.

An overview of the organisational structure of the IASC Foundation can be found in Figure 1.

2.2 Defined Benefit and Defined Contribution

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4 IAS 19 - Employee Benefits

Figure 1. An overview of the organizational structure of the IASC Foundation

2. Post-employment benefits are employee benefits (other than termination bene-fits) which are payable after the completion of employment.

3. Other long-term employee benefits are employee benefits (other than post-em-ployment benefits and termination benefits) which do not fall due wholly within twelve months after the end of the period in which the employees render the related service.

4. Termination benefits are employee benefits payable as a result of either: (a) an entity’s decision to terminate an employee’s employment before the

normal retirement date; or

(b) an employee’s decision to accept voluntary redundancy in exchange for those benefits.

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Accounting for short-term employee benefits is, according to paragraph 9 of the Stan-dard, generally straightforward. This is due to the fact that no actuarial assumptions are necessary to measure the cost or obligation.

The Standard prescribes a simpler method of accounting for other long-term em-ployee benefits than for post-employment benefits, because of two reasons mentioned in paragraph 127. First of all the measurement of other long-term employee bene-fits is not usually subject to the same degree of uncertainty as the measurement of post-employment benefits. And secondly, the introduction of, or changes to, other long-term employee benefits rarely causes a material amount of Past Service Cost. The reason that the Standard deals with termination benefits separately from other employee benefits is, according to paragraph 132, that the event which gives rise to an obligation concerns the termination rather than employee service. The method of accounting for termination benefits is more straightforward than it is for post-employment benefits.

While the post-employment benefits are the most comprising, the Standard makes an important distinction in post-employment benefit plans. Namely, it distinguishes be-tween defined contribution (DC) plans and defined benefit (DB) plans. The Standard defines DC plans as:

post-employment benefit plans under which an entity pays fixed contribu-tions into a separate entity (a fund) and will have no legal or constructive obligation to pay further contributions if the fund does not hold sufficient assets to pay all employee benefits relating to employee service in the current and prior periods.

All post-employment benefit plans other than DC plans are defined as DB plans. Examples of DC plans are available premium schemes and savings systems, while final pay schemes, average pay schemes and fixed amounts schemes are examples of DB plans [4]. Accounting for DC plans is, according to paragraph 43, straightforward because the reporting entity’s obligation for each period is determined by the amounts to be contributed for that period. So, similar as for short-term employee benefits, no actuarial assumptions have to be made. How DB plans should be accounted for, will be discussed in the next section.

2.3 Accounting for DB plans

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Defined Benefit Obligation (...) Fair Value of Plan Assets ...

—–

Funded status ...

Unrecognized actuarial (gains)/losses (...) —– Net (liability)/asset ...

Table 1. The construction of the balance sheet liability/asset for Defined Benefit plans

The net liability/asset is included in the balance sheet. The meaning of the compo-nents from Table 1 are discussed next.

• Defined Benefit Obligation (DBO) is the actuarial net present value of all as-signed DB post-employment benefits. The present value of the DBO should be determined with the Projected Unit Credit Method (PUCM). This method will be explained in a later section.

• Fair Value of Plan Assets (Plan Assets) is the ’fair value’ of assets available, for the use of payments to employees. When the market price of the Plan Assets is available then it should be chosen as such. When this is not the case the Plan Assets should be estimated, which for example can be done by discounting expected future cash flows.

• Unrecognized actuarial gains/losses is the part of actuarial gains/losses that has not yet been recognized in the Profit and Loss statement (P&L). Since the valuation of the DBO, and sometimes the Plan Assets, relies on a set of actuarial assumptions, there will occur alterations between the assumptions on forehand and the realization during the year. Besides that, assumptions will change over time. One other effect that has influence is the intake of new employees. When and how actuarial gains/losses have to be recognized will be discussed in the ’corridor’ section.

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Service Cost (...)

Interest Cost (...)

Expected return on Plan Assets ... Amortization of actuarial gains/(losses) (...)

—– Employer Pension Expense ...

Table 2. The construction of the Employer Pension Expense

The meaning of the components from Table 2 will be described next.

• Service Cost (SC) is the cost for purchasing post-employment benefits for one year of employee service. The SC should, just as the DBO, be calculated by means of the PUCM.

• Interest Cost (IC) is the required interest necessary for the growth of the DBO. The IC is calculated by multiplying the discount rate at the beginning of the period and the DBO during that period.

• Expected return on Plan Assets is based on market expectation at the beginning of the period. The difference between the expected return on Plan Assets and the actual return on Plan Assets are actuarial gains/losses.

• Amortization is the amount of the actuarial gains/losses that will be recognized. Unrecognized actuarial gains/losses are amortized according to the corridor ap-proach, which will be explained in the next section.

The balance sheet liability and the Employer Pension Expense can also be influenced by some other effects prescribed in the Standard, such as curtailments, settlements, Past Service Cost or the limit in paragraph 58(b). These effects concern exceptions and will not be discussed in detail.

2.4 Corridor approach

Since some of the calculations mentioned in the previous paragraph require several actuarial assumptions, and are therefore subject to uncertainty, actuarial gains/losses will appear. While the funded status is the ’current best estimate’ of the post-employment benefit obligation, a lot of uncertainty is involved. Therefore actuarial gains/losses may offset one another. That’s why it is sufficient if the obligation lies within a range or ’corridor’ around the current best estimate.

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(a) 10% of the DBO

(b) 10% of the Fair Value of Plan Assets

The portion that has to be recognized, is the excess divided by the expected average remaining working lives of the employees participating in the plan. It is also permitted to recognize actuarial gains/losses faster, as long as this is done consistently.

2.5 Numerical example

For further understanding of the concepts introduced in this section, a numerical ex-ample will be presented. In this exex-ample the fiscal year is assumed to be the same as a calendar year. The Defined Benefit Obligation has been calculated per 1-1-2008 and is equal to 930. The Fair Value of Plan Assets per 1-1-2008 is equal to 800. The total amount of unrecognized actuarial losses per 1-1-2008 is 80. The Service Cost for 2008 has been calculated as 70 and the Interest Cost as 55. In the calculation of the IC only the SC and the DBO per 1-1-2008 have been taken into account, where the IC is 5.5% of the sum of the SC and DBO. The expected return on Plan Assets (RA) is set at 60, which is based on an expected return on Plan Assets of 7.5%. The total contributions (CT) that will be paid to the fund/insurer during 2008 are 50. The expected plan participants contribution (PPC) for 2008 is 10 and the expected benefits that will be paid during 2008 are 40. With these figures the net liability per 1-1-2008, the expected net liability per 31-12-2008 and the Employer Pension Expense (EPE) for 2008 can be calculated. Table 3 presents an overview of all these figures.

Actual Movement Estimated 1-1-2008 —————————— 31-12-2008 P&L Cash DBO (930) SC (70) BP 40 (1025) IC (55) PPC (10) Plan Assets 800 RA 60 CT 50 870 BP (40) Funded Status (130) (65) 40 (155) Unrecognized items 80 0 0 80

Net (liability)/asset (50) EPE (65) 40 (75)

Table 3. Numerical example of the consequences of IAS 19 for a DB plan on the balance sheet and the profit-and-loss account

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Plan Assets minus the unrecognized actuarial losses. Since the unrecognized actuarial losses are within the bounds of the corridor, because 80 < 10% × max(930; 800) = 93, no unrecognized actuarial losses have to be amortized in the P&L during 2008. The Employer Pension Expense for the fiscal year 2008 is 65. This is equal to the SC plus the IC minus the expected return on Plan Assets.

The estimated position of the DBO per 31-12-2008 is 1025, this is equal to the DBO per 1-1-2008 plus the SC plus the IC plus the plan participants contributions minus the expected benefits paid. The estimated position of the Fair Value of Plan Assets per 31-12-2008 is 870, this is equal to the Fair Value of Plan Assets per 1-1-2008 plus the expected return on Plan Assets plus the total contributions minus the expected benefits paid.

The estimated balance sheet provision per 31-12-2008 can be calculated based on the estimated DBO, Fair Value of Plan Assets and unrecognized items, but can also be estimated as the balance sheet provision per 1-1-2008 plus the estimated Employer Pension Expense minus the estimated amount of cash.

2.6 Actuarial assumptions

In calculating the DBO and the Employer Pension Expense, a number of actuarial assumptions have to be made. These assumptions can be grouped in demographic assumptions, which are about the future characteristics of current and former em-ployees eligible for benefits, and financial assumptions. The most important actuarial assumptions are mentioned in the following list.

1. Financial assumptions (a) Discount rate (b) Future salary levels

(c) Benefit levels

(d) Expected rate of return on Plan Assets (e) Indexation

2. Demographic assumptions (a) Mortality rates

(b) Employee turnover rates (c) Disability rates

(d) Early retirement rates

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IAS 19 prescribes in paragraph 72 that actuarial assumptions shall be unbiased and mutually compatible. According to the Standard an assumption is unbiased when it is neither imprudent nor excessively conservative. The Standard defines assumptions to be mutually compatible if they reflect the economic relationships between factors such as inflation, rates of salary increase, the return on Plan Assets and discount rates.

The financial assumptions have to satisfy three conditions, namely they have to be based on market expectations, at the point in time of the balance sheet date and for the period over which the obligations are to be settled. Furthermore, the discount rate has to be determined by reference to market yields on high quality corporate bonds. When no deep market in such bonds is available, the market yields on government bonds should be used. The estimates of future salary levels have to account for inflation, seniority, promotion and other relevant factors.

For demographic assumptions no further guidelines are given, except that an entity has to use the best estimates available.

2.7 Projected Unit Credit Method

In calculating the present value of the DBO and current SC an entity has to use the Projected Unit Credit Method. This method sees each period of service as giving rise to an additional unit of benefit entitlement. The method measures each unit separately to build up the final obligation. This method is illustrated in Figure 2.

Figure 2. The Projected Unit Credit Method measures each unit separately to build up the obligation

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then equal to the sum of the obligation times the corresponding chance of all possible outcomes.

The Coming Service (CS) is the amount of pension entitlements promised to an em-ployee for one year of service. The corresponding cash of the CS for the employer increases yearly, assuming a constant pensionable base, because of two reasons. First of all the employee survived another year and it is therefore more likely that the em-ployee will reach the pensionable age. And secondly, the emem-ployee is one year closer to retirement and therefore the available money has one year less to yield a profit. The corresponding calculations will be discussed further in Section 5.1. When an employee receives a yearly salary increase, and assuming that the pensionable base increases also, the CS will increase yearly as well. In Figure 3 the effect of the CS, Back Ser-vice (BS) and indexation for an average pay and final pay pension scheme can be seen.

Figure 3. Coming Service, Back Service and indexation for an average pay and final pay pension scheme

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CS will be the same. In an average pay pension scheme the accrued pension entitle-ments receive indexation, while in a final pay pension scheme BS is awarded over the past service years. To clarify these concepts somewhat more a numerical example is shown in Table 4. In this table the CS, BS and indexation for an average pay and final pay pension scheme for the first four years of service is shown. Note that the CS, BS and indexation do not represent the actual value of the pension entitlements, but show the size of the accrued pension entitlements. The average pay and final pay pension scheme both have an accrual rate of 2% and the indexation is set at 1%.

t = 0 t = 1 t = 2 t = 3 Total Pensionable base 10,000 10,500 11,000 12,000 CS 200 210 220 240 870 Final pay BS 0 10 20 60 90 Total 200 220 240 300 960 CS 200 210 220 240 870

Average pay Index. 0 2 4 6 12

Total 200 212 224 246 882

Table 4. Numerical example of an average pay and final pay pension scheme, including Coming Service, Back Service and indexation

The BS in a year is equal to the CS in this year minus the CS in the previous year multiplied with the number of past service years. So for t = 2, the accrued pension entitlements are 220, while for t = 1 the accrued pension entitlements where 210. The BS is therefore equal to (220 − 210) × 2 = 20. The indexation is equal to the total amount of accrued pension entitlements times the indexation percentage. For t = 2 the indexation is equal to (200 + 212) × 1% = 4.12. In this example the BS is much more expensive than the indexation. Though a final pay pension scheme is not by definition more expensive than an average pay pension scheme. This depends among other things on the accrual percentage, the indexation percentage and the expected salary increases.

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Figure 4. Difference between the Service Cost and the cash, which exists of the Coming Service and the Back Service

The employer cash cost for an employee increases yearly and are thus relatively low for a young employee and relatively high for an older employee. The SC is, besides alterations in actuarial assumptions, constant over the years of employment, meaning that these costs are equally spread over the employee’s period of service. Since the costs are higher than the cash in the earlier years of employment, a balance sheet liability will arise. In the latter part of employment the cash will be larger than the costs and consequentially the liability will decrease. The SC is a component of the Employer Pension Expense which is included in the P&L of the employer.

The effect of this method is that the Employer Pension Expense can be calculated at the beginning of the year, since the SC, IC, expected return on Plan Assets and amor-tization are determined on forehand. Fluctuations of one or more of these components are treated as actuarial gains/losses at the end of the year and will not influence the Employer Pension Expense in the current period, which is very convenient for bud-geting reasons.

2.8 Other accounting standards

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Two other international accounting standards have been used in the Netherlands for some time, namely the Financial Accounting Standard (FAS) from the USA and the Financial Reporting Standard (FRS) from the UK [6]. These accounting standards have been used by subsidiary companies of American or British companies, or by Dutch companies quoted on the stock exchange in the USA or UK.

The Financial Accounting Standards Board (FASB) publishes the FASs and these standards apply to American companies, their subsidiary companies and non-American companies with a quotation on an American stock exchange [7]. The Standard that deals with the reporting aspects of pensions is FAS 87 Employers’ Accounting for Pensions, while FAS 88, FAS 106, FAS 112 and FAS 146 deal with other (aspects of) Employee Benefits. Recently the FASB has made some alterations in FAS 87, FAS 88 and FAS 106 which are joined in a new Standard FAS 158.

The Financial Reporting Council (FRC) publishes the FRSs, which apply to British companies, their subsidiary companies and non-British companies quoted on a British stock exchange [8]. FRS 17 Retirement Benefits deals with all post-employment ben-efits.

While IAS 19, FAS 87 and FRS 17 have the same essence, there are some differences between the standards and therefore no further attention will be given to FAS 87 or FRS 17.

One more relevant institution that publishes guidelines for employee benefits is the Raad voor de Jaarverslaggeving (RJa) or translated freely ’Council for Annual Re-porting’. The RJa is responsible for drawing up and publishing of guidelines for the annual report [9]. There is a special guideline which deals with employee benefits, namely RJ 271. This guideline is based on IAS 19, but slightly adjusted to fit the Dutch pension situation. Companies in the Netherlands which are not quoted on the stock exchange have to account according to RJ 271.

2.9 Summary

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Service Cost (SC). The Standard prescribes that the DBO and SC are calculated using the Projected Unit Credit Method (PUCM). This method sees each period of service as giving rise to an additional unit of benefit entitlement. A consequence of the PUCM is that the SC is, based on a set of actuarial assumptions, equal over the years of employment.

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Dutch mortality tables 17

3 Dutch mortality tables

In Section 2.6 the most important actuarial assumptions for IAS 19 calculations were listed. One of the assumptions concerns the mortality rates of current and former employees. Naturally it is unknown when an employee will decease, so an estimate is needed. For the use of the PUCM and other actuarial purposes, yearly chances of mortality are necessary. These estimated yearly mortality rates can be based on historical death rates. In the Netherlands these figures are recorded by the Centraal Bureau voor de Statistiek (CBS) or in English ’Statistics Netherlands’.

3.1 The CBS

The CBS was founded at January 9th 1899 by Royal Decree. The CBS started with only five employees, while in 2007 already approximately 2500 people were employed. The CBS is responsible for: ”collecting, processing and publishing statistics to be used in practice, by policymakers and for scientific research” [10]. The CBS also has a legal basis which is derived from the Statistics Netherlands Act of November 20, 2003.

The CBS has records on mortality rates since 1861, although these are only available for 5-year periods. Yearly mortality rates are available since 1950. Each calendar year the CBS observes how many people of which year of birth and gender decease. Furthermore it is assumed that childbirth is evenly distributed over a calendar year. This means that newly borns are on average 1₂ year old at the end of that calendar year. At the end of the following calendar year they are on average 11₂ years old and so forth. The CBS has records on the number of newly born and migration figures as well. The CBS therefore knows how many people of which age and gender are alive at the end of each calendar year. The mortality rates are taken as the ratio of the number of deaths during a year and the number of people alive at the beginning of that year, distinguished to birth year and gender.

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18 Dutch mortality tables

Figure 5. The life expectancy at birth in the Netherlands per 5-year period since 1861-1865

Based on Figure 5 it is clear that the life expectancy has increased over time for both men and women. There are two clearly visible periods in which the life expectancies declined. The first period is 1916-1920 and the second period is 1941-1945. The rea-son for the first decline is not the first world-war as is mentioned sometimes, but the Spanish Flu, which lasted from 1918 to 1920 [11]. The cause of the second decline is the second world-war. This war caused a decline in life expectancy for both sexes, but especially male life expectancy was heavily influenced.

In the second half of the twentieth century, and in particular the seventies and eight-ies, the difference in life expectancy between men and women became continuously larger. The climax was reached in 1981-1985 when the life expectancies at birth dif-fered over 6.6 years. The cause of this increasing gap, was that more men than women were influenced by unhealthy habits, such as smoking and drinking [12].

The difference in life expectancy at birth of men between periods 2001-2005 and 1861-1865 is 40.0 years. This large difference is largely caused by high infant mortality in the nineteenth century. The life expectancy at for example age 5.5, between men in 2001-2005 and 1861-1865 is already reduced to 22.0 years. The effect of the high infant mortality in 1861-1865 is made visible in Figure 6, where the life expectancy at age x in 1861-1865 and 2001-2005 can be seen. Clearly visible is the difference in shape for the two periods at young ages, caused by the high infant mortality in 1861-1865. In that period it meant for young children that their expected remaining lifetime increased for each year they survived.

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Figure 6. Life expectancy at age x in 1861-1865 and 2001-2005 in the Netherlands for both men and women

used for actuarial purposes, because these mortality rates can show a rather erratic course. Therefore actuaries often work with mortality tables published by the Actu-arieel Genootschap (AG), or ’Actuarial Association’. Section 3.3 will discuss the AG and their mortality tables. The following section will first introduce some definitions and notation on future lifetime.

3.2 Future lifetime

A person aged x years has a future lifetime which will be denoted as T (x). Where T (x) is a random variable and has the following probability distribution function:

G(t) = Pr[T (x) ≤ t], t ≥ 0, (1) where t denotes the time in years. It will be assumed that G(t) is continuous and has probability density g(t) = G0_{(t). While probabilities and expected values can be}

expressed in terms of the distribution and density functions, in actuarial circles it is common to use a different notation.

The symboltqxdenotes the probability that a person of age x will die within t years.

Similarly the symbol tpx denotes the probability that a person of age x will survive

at least t years. The relations with the distribution function are as followed:

tqx = G(t),

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When t = 1, the index t is omitted. When a cohort of lives is imagined, who are all of the same age and are observed over time, lx can be denoted as the number still

alive at age x. The relation of lx and the symbols described in formula (2) are the

following: tpx = lx+t lx , tqx = lx− lx+t lx . (3)

The expected remaining lifetime of a person of age x is E[T (x)] and will be denoted by the symbol ˚ex, which is defined as:

˚ex=

Z ∞

0

tg(t)dt. (4)

The force of mortality of (x) at age x + t is denoted by µx+t and defined as:

µx+t=

g(t) 1 − G(t)= −

d

dtln[1 − G(t)]. (5) The force of mortality can be used to express the expected future lifetime. This follows from rewriting formula (4) as:

˚ex = Z ∞ 0 tg(t)dt = Z ∞ 0 tµx+t[1 − G(t)]dt = Z ∞ 0 ttpxµx+tdt. (6)

Based on formula (2) and the last equation in (5), integration yields the following expression:

tpx= e− Rt

0µx+sds_, ₍₇₎

which shows the relation of the probability of surviving t years and the force of mortality.

3.3 Mortality tables from the AG

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preceding five years. The first table published by the AG was based on the observa-tion years 1961-1965. The tables are compiled by the Mortality Table Committee of the AG. The most recent table is 2000-2005 which was published in 2007 [14]. Just as for table 1995-2000 [15] the AG used a ’smoothing’ algorithm developed by a member of the Committee, named van Broekhoven [16]. The algorithm transforms the ’raw’ mortality quotients of the CBS to ’smoothed’ mortality quotients, such that these are less erratic and better fit for actuarial purposes.

Table 2000-2005 of the AG was published in a publication named Over sterfte en overleven, or in English ’On mortality and life expectancy’. The model description used for the mortality table 2000-2005 in this publication contains several incorrect formulae. The errors are listed in Appendix A. In the following model description the correct formulae are used.

To use the algorithm the ’raw’ mortality quotients determined by the CBS for the stated observation period for age x are described as qrx and the ’smoothed’

mortal-ity quotients are described as qx. To determine qx for a given x, a total of eleven

observations are used, namely the mortality quotients qru for u = x − 5 up to and

including x + 5. The standard for smoothness is based on the sum of squares of the third differences, which is represented by the following formula:

x0−2 X x=2 (∆3qrx)2 or alternatively as x0−2 X x=2 (∆3qx)2, (8)

where x0 is the maximum age for which the algorithm is used. The smoothing that

is applied is a sort of ’progressive’ average. With the eleven observations used for each age x, the smoothed mortality quotients qx are determined with the use of a

smallest-square method. This method is not directly applied to the raw mortality quotients, but to the following transformation:

f r(x) = ln[− ln(1 − qrx)]. (9)

The function that is selected to be rounded is the following:

f (x) = ln[− ln(1 − qx)] = a + bx + cx2. (10)

For each x the variables a, b and c can be solved in the following function:

min

5

X

k=−5

[f (x + k) − f r(x + k)]2. (11)

With a, b and c solved for all x the smoothed mortality quotients qx are estimated

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qx= 1 − e−e

f (x)

. (12)

The f (x) can also be determined with matrix calculations. In that case f (x) can be directly calculated by:

f (x) = [1 x x2](X0X)−1X0Y, (13) with: X =     1 x − 5 (x − 5)2 .. . ... ... 1 x + 5 (x + 5)2     , (14) and: Y =     f r(x − 5) .. . f r(x + 5)     . (15)

When f (x) is calculated with formula (13), the mortality quotients can again be es-timated with formula (12).

The reason that the AG chose formula (10) as the function to be smoothed is the-oretically justified by it being approximately equal to the mortality intensity of the Gompertz formula. In 1824 Gompertz postulated that the force of mortality grows exponentially [17]:

µx+t= βγx+t, t > 0. (16)

The justification follows from formula (10) which with the help of formula (7) can be rewritten as: f (x) = ln[− ln(px)] = ln h − lne−R01µx+sds i , (17)

which leads to:

ef (x)= Z 1

0

µx+sds. (18)

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Dutch mortality tables 23 ef (x) = Z 1 0 µx+sds ≈ µx+ µx+1 2 = βγ x_{+ βγ}x+1 2 = β(1 + γ) 2 γ x_. ₍₁₉₎ And thus: f (x) ≈ ln β(1 + γ) 2 + x ln γ. (20)

So under the assumption of Gompertz, f (x) is approximately a linear function of x. Since the transformed mortality quotients of the CBS f r(x) are approximately linear and for the f (x) a quadratic function is used instead of a function according to the Gompertz formula, the result will be a good approximation. Therefore the AG considers the smoothing algorithm as a ’progressive improved Gompertz-rounding’.

The algorithm requires eleven observations and can therefore not be applied to the lowest ages. The algorithm is used for x ≥ 6, the reason that it is not already applied to x = 5 is that the mortality at birth (x = 0) is much higher than for other young ages. Therefore f r(0) is not used in smoothing other quotients. Because no other information is available, q0 and q1 are set equal to qr0 and qr1. For 2 ≤ x ≤ 5 the qx

are determined by the smoothing algorithm described before, but instead of eleven observations, 2x − 1 observations are used.

For (very) high ages the algorithm cannot be applied directly. The reason for this is that only a relatively small number of people reach high ages and consequentially the mortality quotients of the CBS become very erratic. Therefore the algorithm does not provide the desired result. Another reason is that the algorithm does not work properly anyway for observed mortality quotients equal to 0 or 1, because of the transformation in formula (9).

It is of course not obvious until what age the algorithm is satisfactory. Therefore experimental calculations were made to determine the optimal age x0as from which

an alternative rounding is used. In selecting the x0 which was used, the desire of

realizing a smooth transition between the mortality quotients which were smoothed using the ’standard’ algorithm and the mortality quotients which were smoothed on the basis of the alternative method was taken into account.

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’standard’ smoothing algorithm is also the basis for the smoothing of the mortality quotients for the higher ages. This involves only a single parameter value, therefore following on to this the Gompertz formula is used.

From formula (7) it can be derived that: px px0 = e −R1 0 µx+sds e−R01µx0+sds . (21)

When the mortality intensity of the Gompertz formula is assumed, pxcan be rewritten

as: px = e− R1 0µx+sds = e−R01βγ x+s_ds = e− h_{βγx γs} ln(γ) i1 0 = e−βγx γln γ + βγx ln γ = eβγxln γ(1−γ). (22)

From formulas (21) and (22) it follows that: ln px

ln px0

= eα(x−x0)_, ₍₂₃₎

where α is defined as α = ln γ. From this equation the expression for qxfor x ≥ x0is

derived:

qx= 1 − ee

α(x−x0)_{ln p}

x0_{, x ≥ x}₀_. ₍₂₄₎

The value of α is determined by calculating the average remaining lifetime for age x0

based on the observed mortality quotients qrx and to equate it to the corresponding

average lifetime of the Gompertz assumption.

This procedure is carried out for the higher ages x0 between 90 and 105 years. This

results in series of mortality quotients, for both men and women and for each value of x0. With the use of the ’standard’ algorithm the smoothed mortality quotients qxare

determined for each x0. For both men and women the most optimal table is chosen

using the smallest-square method for ages up to and including 105. This means x0is

estimated by minimizing: v u u t 105 X x=0 (qx− qrx)2, (25)

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into account. For table 2000-2005 this has led to the conclusion that for men x0= 100

is optimal and for women x0= 101. While for table 1995-2000 x0= 97 for men and

x0= 101 for women were chosen.

3.4 Smoothing algorithm results

As explained in Section 3.1 the mortality tables published by the CBS show a rather irregular trend, which makes them unfit for actuarial purposes. This is the reason that these mortality tables are smoothed by the AG, such that a more regular trend is realized. The smoothing can be based on different objectives, which also shows for different editions of the AG mortality tables as will be discussed in Section 3.5. The objectives for tables 1995-2000 and 2000-2005 are as follows:

• The rounded mortality quotients show a smoother trend than the raw mortality quotients. Smoothness is a concept which can be interpreted in various ways. In this case the standard for smoothness is based on the sum of squares of the third differences, as is represented in formula (8).

• The quadratic deviation between the raw mortality quotients and the rounded mortality quotients is minimal for the selected degree of smoothness.

For table 2000-2005 a third objective is selected, namely:

• The number of sign changes in the difference between the raw and the rounded mortality quotients must be large.

A notable fact is that while the same smoothing method was selected for table 1995-2000, this third objective is named as a consequence of the selected method of smooth-ing instead as an objective.

To investigate the results of the smoothing procedure, the objectives will be tested. For the first objective the sum of squares of the third differences as in formula (8) can be calculated. The outcomes are given for both table 1995-2000 and table 2000-2005 and for the raw and smoothed mortality quotients. These are calculated over the age interval from 0 to x0− 2, where the x0 are as mentioned in Section 3.3 and depend

on gender and differ per table. The results can be found in Table 5.

One of the striking results is the increase in third differences from table 1995-2000 to 2000-2005 of the raw mortality quotients for both men and women. Especially for women this result is notable since x0remained the same. In conclusion the CBS table

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Raw quotients Smoothed quotients Percentage 1995-2000 Male 0.00068068 0.00004278 6.285%

Female 0.00071436 0.00003529 4.940% 2000-2005 Male 0.00215444 0.00018730 8.694% Female 0.00283478 0.00007275 2.566%

Table 5. Reached smoothness of the mortality tables of the AG compared to the mortality rates of the CBS measured according to formula (8)

need for a smoothed table for actuarial purposes. Measured by the sum of squares of the third differences, the results of the smoothing procedure are more or less compa-rable, since the procedure smoothed between 91.3% and 97.4% of the original erratic table.

The measure of smoothness is closely related with the fit of the smoothed mortality quotients on the raw mortality quotients. Since a good fit of the mortality quotients will in general lead to a reduced smoothing success. The other way around holds true as well, because heavily smoothed quotients will in general differ more from the original quotients. An interesting measure is therefore the expected remaining life-time ˚ex at a given age x of the raw quotients and the smoothed quotients. These life

expectancies should be approximately equal, otherwise the data has been smoothed too much. In Table 6 the life expectancies in years for table 2000-2005 for both sexes at different ages are given.

Male Female

Age Raw Smoothed Difference Raw Smoothed Difference 0 76.243 76.247 -0.004 80.960 80.963 -0.003 20 56.972 56.970 0.002 61.526 61.527 -0.001 40 37.641 37.640 0.001 41.960 41.960 0.000 60 19.723 19.722 0.000 23.709 23.711 -0.002 80 6.716 6.719 -0.003 8.565 8.563 0.002

Table 6. Effect of smoothing on the life expectancy at age x for mortality table 2000-2005

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The second objective named is easily fulfilled. The objective states that the quadratic deviation between the raw mortality quotients and the rounded mortality quotients should be minimal. Therefore the rounding procedure is based on a smallest-square method, such that the quadratic deviation is kept minimal.

For the third objective the number of sign changes in the difference between the raw and the rounded mortality quotients is calculated. This has been done for both mor-tality table 1995-2000 and 2000-2005, over the age interval from 0 to x0. These results

can be found in Table 7.

1995-2000 2000-2005 Male Female Male Female

0-97 0-101 0-100 0-101 Sign changes 52 49 65 66

Table 7. Number of sign changes in the difference between the raw and the rounded mortality quotients for mortality tables 1995-2000 and 2000-2005

For men, but even more for women the number of sign changes for mortality table 2000-2005 has increased significantly compared to mortality table 1995-2000. This is remarkable, because the AG stated in the publication of mortality table 1995-2000 that the number of sign changes that occurred in that period were very close to the maximum number that can be achieved. Though for mortality table 2000-2005 these numbers are much higher.

It can be concluded that these numbers are quite high meaning a lot of sign changes occurred and therefore the objective is fulfilled.

To conclude the results, the effect of the smoothing can be made visible. In Figure 7 the raw mortality quotients and the smoothed mortality quotients for men during period 2000-2005 can be seen. This has been done for two age periods in which the smoothing effect becomes clear.

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Figure 7. Raw and smoothed mortality quotients for men in period 2000-2005 for the age intervals 15-35 and 95-110

3.5 Previous AG methods

For mortality tables 1995-2000 and 2000-2005 the smoothing is mainly based on the ’van Broekhoven’ algorithm, as described in section 3.3. In the decades preceding these most recent two tables, the smoothing was based on the mortality law of Make-ham [18] [19] [20] [21] [22]. In 1860 MakeMake-ham postulated a generalization of the Gompertz formula (16), in which he added a constant, age independent factor α > 0:

µx+t= α + βγx+t, t > 0. (26)

Two important advantages of this assumption are that the mortality tables can be described by a limited number of constants and that the ’Law of Uniform Seniority’ is valid. This means that actuarial quantities depending on n different ages can be reduced to quantities depending on n equal ages. Since the results from Makeham were not always satisfactory, many artifices have been used in the past. Some of the artifices and their necessity will be explained later, but the procedure of smoothing will be discussed first.

From formulas (7) and (26) it can be derived that:

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The AG uses the following variables:

s = e−α

c = γ

g = e−ln(γ)β _, ₍₂₈₎

such that formula (27) can be rewritten as:

tpx= stgc

x₍_ct₋₁

). (29)

To calculate the necessary constants there are multiple possible methods. The one applied by the AG is the method of King-Hardy. For this purpose four sums of m consecutive values of ln(lx) are determined, where lx is as in formula (3). Since

lx+1= pxlx, the following formula for lx is used:

lx= ksxgc

x

. (30)

So there are four unknowns (k, s, g and c), which can be solved by means of successive elimination from the system of four equations derived from the sums described above.

The found constants may not be optimal, therefore another method is used which improves the connection with the observed mortality quotients. The method is based on the logarithm of the one-year survival probabilities, which can be derived from formula (29) as:

ln(px) = ln(s) + cx(c − 1) ln(g). (31)

The method minimizes the sum of squares from the difference of the logarithms of the smoothed and the logarithms of the observed one-year survival probabilities, multi-plied with an age dependent weight.

The variables a and b are introduced and defined as:

a = ln(s),

b = (c − 1) ln(g), (32)

such that ln(px) = a + bcxis linear in a and b. The optimal values for s, g and c are

then determined by applying an iteration process. At first the constants a, b and c are approximated by the values a0, b0 and c0, which are determined by the results

from the method of King-Hardy.

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law of Makeham has been the basis for many mortality tables. The methods applied for parts where the results of the Makeham assumption were not satisfactory will be discussed next.

One of the objectives stated for the tables which used the mortality law of Makeham is that the smoothed mortality quotients should be monotonically non-decreasing. However, this does not match with the observed mortality quotients. In Figure 8 the male mortality quotients in 2000-2005 for ages 0-18 can be seen, which is illustrative for previous tables as well.

Figure 8. Raw mortality quotients for men in the age interval 0-18 in period 2000-2005

The raw mortality quotients are nothing like monotonically non-decreasing. In fact, the high mortality quotient for age 0 in 2000-2005 is not surpassed until age 55. Thus the Makeham assumption is not very realistic for the youngest ages, though the ob-jective of the monotonically non-decreasing mortality quotients remained. Therefore since mortality table 1976-1980 the mortality quotients have been assumed to be con-stant for all ages from 0 up to 11 to 14, varying per table. This resulted in a negative effect on the fit of the smoothed and the raw mortality quotients, which for example shows in an approximate half a year difference in life expectancy at birth between the smoothed and raw data.

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quotients according to Makeham. In periods 1976-1980 and 1980-1985 the mortality quotients were based on fourth-degree polynomial functions. While in periods 1985-1990 and 1985-1990-1995 the mortality quotients were based on third-degree polynomial functions.

Two problematic age categories have already been described by artifices to improve the fit of the raw and the smoothed mortality quotients. Though the course of the remaining mortality quotients, which were smoothed according to Makeham with the use of a single set of parameters, does not match the course of the raw mortality quotients. From mortality table 1980-1985 onwards a division of the remaining ages was made. For both age categories a set of Makeham parameters were determined. Though parameter γ as in formula (26) was held constant, such that the Law of Uni-form Seniority remained valid.

In Section 3.3 the objectives of the AG as from mortality table 1995-2000 were men-tioned. These objectives are radically different from the old objectives, which were abandoned for the following reasons. Because the monotonically non-decreasing objec-tive showed not to correspond with the raw mortality quotients for both the youngest ages and age category 20 to 25, the AG abandoned this objective. The Law of Uni-form Seniority objective was abandoned as well, since this law, because of modern technical aids, no longer provides any advantage. And thirdly the mortality law of Makeham is no longer used, because this did not provide the desired results.

3.6 Other Dutch mortality tables

Several other mortality tables are in use in the Netherlands. The main reason for the existence of these other tables is that the data of the CBS, and therefore the mortality tables of the AG, are based on the entire Dutch population. The deter-mined mortality quotients are solely based on two properties, namely age and gender. Though the mortality quotients required for actuarial purposes should be based on the section of the population for which it is used. Take for example pension provision calculations, where the relevant population group is not the same as the entire Dutch population. Population groups like the disabled and the unemployed have, based on age and gender, lower life expectancies than which is average for the Dutch popula-tion. So when an AG mortality table is used for IAS 19 calculations these calculations will be based on overestimated mortality quotients, causing an underestimation of the required provision to guarantee lifelong payments.

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the experienced mortality quotients and the mortality quotients of the entire Dutch population. First there is the factor for insured mortality which concerns the relation of the average mortality quotients of the insured to the mortality quotients of the entire Dutch population. And secondly there is the factor for sum mortality which concerns the relation of the mortality quotients measured in the sum insured to the mortality quotients measured in numbers. The product of the insured mortality and the sum mortality is called the experienced mortality. The experienced mortality is defined as ex,g, with age x and gender g where g ∈ {1, 2} = {male, female}. The

relation between the one-year mortality quotients for the entire population (qx,g) and

the one-year mortality quotients of the insured (qi x,g) is:

qi_x,g= ex,gqx,g. (33)

The factor ex,g depends not only on age and gender but of course also on the type of

insurance. The following example can clarify the different mortality concepts:

Assume a group of 10.000 men of age 50. All men have accrued pension, but 8.000 men have accrued ¤10.000 while the other 2.000 men have accrued ¤20.000. According to the AG mortality table 2000-2005, q50,1≈ 0.004. So the expected number of men

who will die during the coming year is 10.000 ∗ 0.004 = 40. Suppose that at the end of the year it turned out that only 35 men have died. From this 35 men, 30 men had accrued ¤10.000 and the other 5 men had accrued ¤20.000. This leads to the following mortality factors:

Insured mortality = 35 40 = 0.875 Sum mortality = 3035∗¤10.000+ 5 35∗¤20.000 8.000 10.000∗¤10.000+ 2.000 10.000∗¤20.000 ≈ 0.952 Experienced mortality = 30∗¤10.000+5∗¤20.000_{32∗¤10.000+8∗¤20.000} ≈ 0.833. (34) First there is the insured mortality factor which is the fraction of the actual num-ber of deceased and the expected numnum-ber of deceased. Secondly there is the sum mortality factor which is the fraction of the average sum of accrued pension of the deceased and the expected sum of accrued pension for a deceased. And thirdly there is the experienced mortality factor which is the fraction of the total sum of accrued pension of the deceased and the expected number of deceased times the expected sum of accrued pension. From this it follows that the experienced mortality is indeed the product of insured mortality and sum mortality as described in formula (33).

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factor is that in case of ex,g < 1, instead of age x an alternative xa < x is used. As

a consequence ˚exa > ˚ex causing a similar effect as the experienced mortality factors.

When for example for males an age correction factor of −1 is used, so xa = x − 1, the

mortality quotients qx−1,1, qx,1, qx+1,1, . . . are used instead of qx,1, qx+1,1, qx+2,1, . . ..

There are five other mortality tables which are (still) used in the Netherlands. These tables are all published by the Verbond van Verzekeraars (VvV) or in English ’Dutch Association of Insurers’ [23]. In the early nineties the VvV researched the net foun-dations of pension insurances. This led to a mortality table named Collectief 1993 [24]. Ten years later a new version of the mortality table was published under the name Collectief 2003 [25]. The most recent table for pension insurances of the VvV was published in 2005 and is called Pensioentafel 2006 or ’Pension table 2006’ [26]. For annuity insurances two other tables have been published by the VvV. In the late nineties the VvV published a mortality table for annuity insurances named DIL98, but the most recent table for annuity insurances is the Lijfrentetafel 2006 or ’Annuity table 2006’ [27].

The majority of the market supply their mortality observations to the Centrum voor Verzekeringsstatistiek (CVS) or in English ’Centre of Insurance Statistics’, which is part of the VvV. Analysis of the CVS showed that mortality of pension insured and annuity insured is significantly lower than for the entire Dutch population. This analysis has also shown that insured with a large sum insured generally have lower mortality quotients than comparable insured with a lower sum insured. Based on the collected mortality observations, the CVS has determined experienced mortality fac-tors for both pension insurances and annuity insurances. The aforementioned tables of the VvV are adjusted for experienced mortality and are therefore useful.

3.7 Foreign life expectancies

Mortality tables are usually published for each country separately. The records for European mortality quotients are collected by Eurostat [28]. There are also worldwide records available which are recorded by the World Healthcare Organization (WHO) [29]. To get an idea of the life expectancies at birth of nearby countries, Figure 9 shows the life expectancies of European citizens in 1995 and 2005.

From a few countries there were no available records in 1995. It is very interesting to see how the life expectancies in other European countries are proportioned. While many European countries have quite similar life expectancies, especially the East-European countries have much lower life expectancies.

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Figure 9. Life expectancies in 1995 and 2005 of both men and women for some European countries

life expectancy of 79.6. This is 14.3 years higher than the male life expectancy in Lithuania, which had the lowest male life expectancy in 2005. In 1995 Sweden had the highest male life expectancy with 76.2 years, which was 14.7 years higher than in Estonia. On average the European male life expectancies have increased with 3.0 years in a period of only 10.0 years.

For women the differences in life expectancies are much smaller than for men. In 2005 it was Liechtenstein with the highest female life expectancy which was 84.1 years. The lowest female life expectancy was recorded in Romania, where it was 75.7 years. This is a difference of 8.4 years, which is much smaller than the male difference of 14.3 years. In 1995 the lowest female life expectancy was recorded also in Romania which was 73.5 years, while the highest female life expectancy was recorded in Switzerland with 81.9 years. The female life expectancies have on average increased with 2.2 years in the 10-year period.

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3.8 Summary

In the Netherlands, Statistics Netherlands (CBS) collects yearly observed mortality rates. These mortality rates are distinguished to age and gender. Based on the ob-served mortality rates over a certain period the corresponding life expectancy can be calculated. The observed mortality rates of the CBS can show a rather erratic course and are therefore unfit for actuarial purposes such as the PUCM. Therefore the Dutch asssociation of actuaries (AG) uses a smoothing algorithm which transforms the ob-served mortality rates to less erratic mortality rates. An important objective for the smoothing procedure is that the life expectancy based on an AG mortality table does not significantly differ from the life expectancy based on the observed mortality rates. Studying historical mortality rates leads to the important conclusion that these rates have dropped considerably, leading to a nearly continuous increase of life expectancy at birth during the past 140 years.

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The projection table of the AG 37

4 The projection table of the AG

The AG mortality table 2000-2005 as described in Section 3.3 is, just as the previous AG tables, based on actual observed mortality quotients. These kind of tables are called period tables, because they refer to observations from a certain period. There-fore period tables do not take future mortality quotient development into account. From Figure 5 it already showed that the life expectancy at birth has been subject to change, caused by decreasing mortality quotients. In general the mortality quotients have decreased for nearly all ages. For calculating premiums and provisions it is nec-essary to use a best estimate as is also prescribed by the supervisory organ called De Nederlandsche Bank (DNB) or in English ’Dutch Central Bank’ [30]. This is in accordance with the Financieel Toetsingskader or in English ’Financial Assessment Framework’ of the Ministry of Social Affairs and Employment [31].

Determining proper provisions requires the development of mortality quotients by the year of birth. These tables are called generation tables and are two-dimensional, in contrary to the period tables which are only one-dimensional. When a generation ta-ble uses not only past observations, but also takes future developments into account it is called a projection table.

4.1 Sorts of projection models

There are different models that can describe future developments in life expectancy. Roughly speaking four types of general models can be distinguished:

1. Extrapolation from causes of death. In these models the projection is based on studying the causes of death over time. An important challenge is how to deal with causes of death that disappear over time and how to model new causes of death that arise.

2. Extrapolation from the structure of observed mortality. These models are based on recognizing a trend in the observed mortality quotients. Difficulty here is that usually multiple trends can be recognized, based on the point of view. 3. Extrapolation of the life expectancy at birth. These models predict the life

ex-pectancy at birth, instead of the mortality quotients. Therefore future mortality quotients have to be derived of the estimated life expectancy at birth, based on some selected assumptions.

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38 The projection table of the AG

There are three Dutch organizations that have published projection tables: the CBS, the VvV and the AG. The CBS used to extrapolate the life expectancy at birth in drawing up the population projection. For that cause no yearly mortality quotients are required. For the many purposes for which yearly mortality quotients are re-quired the CBS regularly publishes a population projection based on extrapolation from causes of death. For this model the CBS defines a number of age categories and for each age category a number of causes of death are specified. Then these causes of death are extrapolated over time.

In the next two sections the projection models of the VvV and the AG will be dis-cussed. After that some results of the projection tables will be shown.

4.2 The VvV projection model

The projection model of the VvV is made by a committee named the Commissie Re ferentietarief Collectief (CRC) or in English the ’Group Reference Rate Commis-sion’. Therefore the model is known as the CRC-model. First some new notation is introduced. Symbol qt,x,g is defined similarly as qx,g in Section 3.6 but with an

additional index t representing the year. So qt,x,g is the one-year mortality quotient

for an x-year old with gender g in year t, where t = T + 1, T + 2, . . . , T + 50 and T is chosen as the most recent mortality observation year. Symbol ax,g is defined as

the yearly reduction factor for the mortality quotient belonging to age x and gender g.

The CRC-model is based on a fixed annual percentual decline of the one-year mortality quotients qt,x,g, which leads to the following recursive relationship:

qt,x,g = ax,gqt−1,x,g. (35)

This means that only the ax,g have to be determined. This is done based on the

observed mortality quotients between the years from 1986 till T for ages 141₂ up to and including 951₂. The reason that 1986 has been chosen as starting year is because an apparent structural trend shift occurred in period 1986-1990 for both men and women. The model exists of the following seven steps:

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2. For each age x ∈ {20, 21, . . . , 90} a five-year progressive average is taken leading to smoothed mortality quotients over the years from 1988 till T −2. The resulting mortality quotients for year 1988 are defined as Q1988,x,g and the resulting

mortality quotients for year T − 2 are defined as QT −2,x,g.

3. The reduction factors ax,g are estimated by:

ax,g =

QT −2,x,g

Q1988,x,g

T −2−19881

. (36)

4. The resulting reduction factors ax,g are then smoothed over the ages by taking

a progressive average.

5. For the ages 20 up to and including 90 the projection for year t becomes:

qt,x,g = (ax,g)t−T +2QT −2,x,g. (37)

6. For x < 20 and x > 90 the most recent AG mortality table is used.

7. When for any x, qt,x,1 < qt,x,2 for one or more years, the female mortality

quotients are adjusted in such a way that qT +50,x,1= qT +50,x,2.

In Section 3.6 among other things the most recent mortality tables of the VvV were discussed, namely the ’Pension table 2006’ and the ’Annuity table 2006’. These tables are both corrected for experienced mortality, but they are also corrected for future developments in mortality using the aforementioned CRC-model. The most recent mortality observation year T at the time of publishing was 2003.

As described in the introduction of this chapter, projection tables are two-dimensional. Because insurers are used to one-dimensional mortality tables in their actuarial soft-ware, the VvV decided that the Pension table 2006 and the Annuity table 2006 should be published not only as projection tables, but as one-dimensional tables as well. For the Annuity table the VvV has selected the future mortality quotients belonging to a certain age according to the projection table. They chose the average starting age from annuity insured which appeared to be, based on research of the CVS, 62 for men and 60 for women. For ages beneath the average starting age the one-dimensional mortality quotients have been chosen as the projection table mortality quotients in 2008. As a consequence, ages beneath the starting ages are an underestimation of the projection table and ages above the starting ages are an overestimation of the projection table.

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the one-dimensional table are chosen in such a way that the fit of the calculated premiums according to the projection table and the premiums calculated with the constructed one-dimensional table is as good as possible. Because information is lost by dropping the year as index, changes will occur between the projection table and the one-dimensional table. The sort of effect and the size of the effect depends on more factors than only age.

4.3 The AG projection model

Together with the 2007 publication of the AG mortality table 2000-2005 the AG also published a projection table. This projection table is called the AG projection table 2005-2050 and is the first projection table published by the AG. The model which is described in the publication of the AG has not been used for the actual projection table. This is further explained in Appendix A. In this section will only be referred to the correct model. The model used for table 2005-2050 is derived from the CRC-model from Section 4.2. The AG CRC-model is therefore just as the CRC-CRC-model based on the annual percentage decline of mortality quotients as in formula (35). The ax,g are

based on the observed mortality quotients from 1986 till 2005. A small difference is that the AG receives raw mortality quotients by whole age directly from the CBS. The following procedure, which differs at a few points from the CRC-model, leads to the AG table 2000-2050:

1. The mortality quotients between 1986 and 1990 are smoothed according to the procedure described in Section 3.3 based on the ’van Broekhoven’ algorithm. The resulting mortality quotients (0 ≤ x ≤ 120) are defined as Q1988,x,g. The

mortality quotients from the AG table 2000-2005 are defined as Q2003,x,g.

2. For each age x ∈ {0, 1, . . . , 120} the reduction factors ax,g are estimated by:

ax,g =

Q2003,x,g

Q1988,x,g

151

. (38)

3. The estimated mortality quotients which would follow from the recursive for-mula (35) are defined as qrt,x,g, with t = 2004, 2005, . . . , 2053, and are equal

to:

q_t,x,gr = ax,gqrt−1,x,g, (39)

or similarly as:

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4. When for any x holds true that qr

2053,x,1< qr2053,x,2, then the female mortality

quotients qt,x,2 are adjusted in such a way that q2053,x,1= q2053,x,2. Also if for

some pair {x, g} follows that ax,g > 1, indicating increasing mortality quotients,

all mortality quotients qt,x,g are adjusted such that q2053,x,g = Q2003,x,g. To

adjust when necessary the qt,x,g, an additional term is introduced: rx,g, with x

and g as before. The definition for rx,g differs for men and women and rx,1 and

rx,2 are therefore defined separately:

rx,1=    1 (ax,1)50 ₁₂₇₅1 ax,1> 1 1 ax,1≤ 1, (41) rx,2=          _qr 2053,x,1 qr 2053,x,2 12751 q2053,x,1r < q2053,x,2r 1 (ax,2)50 12751 q_2053,x,1r ≥ qr 2053,x,2, ax,2> 1 1 q_2053,x,1r ≥ qr 2053,x,2, ax,2≤ 1. (42)

The explanation behind these formulae is explained after the following and final step.

5. The future mortality quotients qt,x,g are defined as:

qt,x,g=        ax,g(rx,g)t−2003qt−1,x,g x ≤ 100 1 −x−100 5 qt,100,g qt−1,100,g + x−100 5 qt−1,x,g 101 ≤ x ≤ 104 qt−1,x,g x ≥ 105 . (43)

For ages x ≤ 100 formula (43) can be written as:

qt,x,g= at−2003x,g r Pt−2003

i=1 i

x,g Q2003,x,g. (44)

When according to formulae (41) and (42) rx,g 6= 1, then qt,x,g is affected by an

ad-ditional power term rx,g. If q2053,x,1r < qr2053,x,2, then rx,g is chosen in such a way

that q2053,x,1 = q2053,x,2. The mortality quotients are also adjusted if ax,g > 1. In

that case rx,g is chosen such that q2003,x,g= q2053,x,g. Formula (44) contains the term

r

Pt−2003 i=1 i

x,g , which for t = 2053 becomes r1275x,g . Hence the value 1275 in formulae (41)

and (42). The effect of the power term rx,2 on qt,x,2 and the course of qt,x,2 without

including rx,2 is shown in Figure 10 for x = 50 and x = 100.

In the left part of Figure 10 the qt,50,2 and the qrt,50,2 can be seen. The qrt,50,2 are

increasing over time, because for 44 ≤ x ≤ 51, Q1988,x,2 < Q2003,x,2 and therefore

ax,2> 1. Because qr2053,50,1< qr2053,50,2, the qt,50,2 are adjusted for this. The effect of

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Figure 10. Female mortality quotients for the ages 50 and 100 with and without rx,gterm for the years

2003-2053

than qr 2053,50,2.

In the right part of Figure 10 the qt,100,2 and the qrt,100,2can be seen. Since a100,2> 1,

the qt,100,2 are adjusted such that q2003,100,2= q2053,100,2. The effect is that q2053,100,2

is more than 25% smaller than qr

2053,100,2.

A phenomenon that has been observed for many years is that the mortality quotients for high ages have been increasing over time. Reason for this is that more and more people reach these high ages and consequentially a larger portion of the elderly decease yearly. For some ages the AG considered that the assumption q2003,x,g = q2053,x,g

should be let go. Therefore for 101 ≤ x ≤ 104 the AG chose a different systematic for qt,x,g as already described in formula (43). Though the effect on q2053,x,g is very

small, since:

q2053,x,g

Q2003,x,g

< 1.00012, 101 ≤ x ≤ 104. (45) Also the course over time of qt,x,g for 101 ≤ x ≤ 104 is quite similar to the course of

qt,x,g for x ≤ 100 with ax,g > 1. For x ≥ 105 the AG chose to keep qt,x,g constant

over time, because there is too few data available. Besides Q1988,x,g and Q2003,x,gfor

x ≥ 105 are based on the mortality law of Gompertz instead of observed mortality quotients. Similarly as ˚ex and the aforementioned notation, ˚et,x,g is defined as the

remaining lifetime of a person in year t of age x and gender g. Figure 11 shows the effect of the chosen power functions for qt,x,g, as in formula (43), on the course of

˚et,x,g, where ˚et,x,g is based on the mortality quotients in year t.

The effect of the power functions for qt,x,g can be clearly seen in Figure 11. The ages