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Lecture notes on Social Security

Massimiliano Menzietti

a.a. 2017/2018

Week 6

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Mortality evolution and projection models – part 1

Mortality experience over the last decades

The mortality experience over the last decades shows some aspects affecting the

shape of curves representing the mortality as a function of the attained age, such as

the curve of deaths (i.e. the graph of the probability density function of the random

lifetime, in the age-continuous setting) and the survival function.

Reduction of the 𝑞𝑥;

Mortality revolution since ‘80ties of the XX century:

o an increasing concentration of deaths around the mode (at old ages) of the

curve of deaths is evident; so the graph of the survival function moves

towards a rectangular shape, whence the term rectangularization to denote

this aspect;

o the mode of the curve of deaths (which, owing to the rectangularization,

tends to coincide with the maximum age ω) moves towards very old ages; this

aspect is usually called the expansion of the survival function;

o higher levels and a larger dispersion of accidental deaths at young ages (the

so-called young mortality hump) have been more recently observed;

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Representing mortality dynamics

When working in a dynamic context (in particular when projecting mortality), the

basic idea is to express mortality as a function of the (future) calendar year 𝑡, where

𝑡 denotes the calendar year in which the population is considered.

In actuarial calculations, however, age-specific measures of mortality are usually

needed. Then in a dynamic context, mortality is assumed to be a function of both

the age 𝑥 and the calendar year 𝑡. In a rather general setting, a dynamic mortality

model is a real-valued or a vector-valued function Γ(𝑥, 𝑡). In concrete terms, a real-

valued function may represent one-year probabilities of death, mortality odds, the

force of mortality, the survival function, some transform of the survival function,

etc.

The projected mortality model is given by the restriction Γ(𝑥, 𝑡)|𝑡 > 𝑡′, where 𝑡′

denotes the current calendar year, or possibly the year for which the most recent

(reliable) period life table is available. The calendar year 𝑡′ is usually called the base

year. The projected mortality model (and, in particular, the underlying parameters)

is constructed by applying appropriate statistical procedures to past mortality

experience.

We assume that both age and calendar year are integers.

Hence, Γ(𝑥, 𝑡) can be represented by a matrix whose rows correspond to ages and

columns to calendar years. In particular, let Γ(𝑥, 𝑡) = 𝑞𝑥(𝑡), where 𝑞𝑥(𝑡) denotes

the probability of an individual aged 𝑥 in the calendar year 𝑡 dying within one year

(namely, the one-year probability of death in a dynamic context).

𝑞𝑥(𝑡) can be represented ina matrix:

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The elements of the matrix can be read according to three arrangements:

a vertical arrangement (i.e. by columns), 𝑞0(𝑡), 𝑞1(𝑡), … , 𝑞𝑥(𝑡), … corresponding

to a sequence of period life tables, with each table referring to a given calendar

year 𝑡;

a diagonal arrangement, 𝑞0(𝑡), 𝑞1(𝑡 + 1), … , 𝑞𝑥(𝑡 + 𝑥), … corresponding to a

sequence of cohort life tables, with each table referring to the cohort born in

year 𝑡;

a horizontal arrangement (i.e. by rows), … , 𝑞𝑥(𝑡 − 1), 𝑞𝑥(𝑡), 𝑞𝑥(𝑡 + 1), … yielding

the mortality profiles, with each profile referring to a given age 𝑥.

Some preliminary ideas on projection by extrapolation

An extrapolation procedure for mortality simply aims at deriving future mortality

patterns (e.g. future probabilities of death) from a database that expresses past

mortality experience. The database typically consists in cross-sectional observations

and, possibly, (partial) cohort observations.

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However, a number of points should be addressed. In particular, consider the

following:

1. How are the items in the database interpreted? Are they correctly interpreted as

observed outcomes of random variables (e.g. frequencies of death), or,

conversely, are they simply taken as ‘numbers’?

2. The projected table, resulting from the extrapolation procedure, is a two-

dimensional array of numbers, providing point estimates of future mortality.

How do we get further information, namely, interval estimates?

If the answer to question (1) is ‘data are simply numbers’, then the extrapolation

procedure does not allow for any statistical feature of the information available, as,

for example, the reliability of the data. Conversely, when the data are interpreted as

the outcomes of random variables, the extrapolation procedure must rely on sound

statistical assumptions and, as a consequence, future mortality can be represented

in terms of both point and interval estimates (whilst only point estimates can be

provided by extrapolation procedures only based on ‘numbers’).

Various traditional projection methods consist in extrapolation procedures simply

based on ‘numbers’. First, we will describe these methods which, in spite of several

deficiencies, offer a simple and intuitive introduction to mortality forecasts.

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Let us assume that several period observations (or ‘cross-sectional’ observations)

are available for a given population (e.g. males living in a country, pensioners who

are members of a pension plan, etc.). Each observation consists of the age-pattern

of mortality for a given set 𝒳 of ages, say 𝒳 = {𝑥𝑚𝑖𝑛, 𝑥𝑚𝑖𝑛+1, … , 𝑥𝑚𝑎𝑥}. The

observation referred to calendar year𝑡 is expressed by:

Let us focus on the set of 𝒯 = {𝑡1, 𝑡2, … , 𝑛} observation years then, we assume that

the matrix

constitutes the data base for mortality projections. Note that each sequence on the

right-hand side of 𝑞𝑥(𝑡) represents the observed mortality profile at age 𝑥.

We assume that the trend observed in past years (i.e. in the set of years 𝒯 ) can be

graduated, for example, via an exponential function. Further, we suppose that the

observed trend will continue in future years. Then, future mortality can be

estimated by extrapolating the trend itself (see figure).

Remark: The choice of the set 𝒯 is a crucial step in building up a mortality projection

procedure. Even if a long sequence of cross-sectional observations is available

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(throughout a time interval of, say, more than 50 years), a choice restricted to

recent observations (over, say, 30–50 years) may be more reasonable than the

whole set of data. Actually, a very long statistical sequence can exhibit a mortality

trend in which recent causes of mortality improvement have a relatively small

weight, whereas causes of mortality improvement whose effect should be

considered extinguished are still included in the trend itself (see figure).

Extrapolation of the 𝑞𝑥’s (namely of the mortality profiles) represents a particular

case of the horizontal approach for mortality forecasts (see figure). The horizontal

approach can be applied to quantities other than the annual probabilities of death,

for example, the mortality odds 𝜑𝑥, the central death rates 𝑚𝑥, etc.

Adopting the horizontal approach means that extrapolations are performed

independently for each 𝑞𝑥 (or other age-specific quantity), so that the result is a

function 𝜓𝑥(𝑡) for each age 𝑥. This may lead to inconsistencies with regard to the

projected age-pattern of mortality.

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As far as future mortality is concerned, let us express the relation between

probability of death at age 𝑥, referred to a given year 𝑡′ (e.g. 𝑡′ = 𝑡𝑛) and a generic

year 𝑡 (𝑡 > 𝑡′) respectively, as follows:

The quantity 𝑅𝑥(𝑡 − 𝑡′) is called the variation factor (and usually reduction factor, as

it is expected to be less than 1 because of the prevailing downward trends in

probabilities of death) at age 𝑥 for the interval (𝑡 − 𝑡′).

A simplification can be obtained assuming that the reduction factor does not

depend on the age 𝑥, that is, assuming for all 𝑡 and 𝑥

𝑅𝑥(𝑡 − 𝑡′) = 𝑅(𝑡 − 𝑡′)

Mortality forecasts can then be obtained through an appropriate modelling

procedure applied to the reduction factor. The structure as well as the parameters

of 𝑅𝑥(𝑡 − 𝑡′) should be carefully chosen. Then, projected mortality will be obtained

via 𝑞𝑥(𝑡) = 𝑞𝑥(𝑡′)𝑅𝑥(𝑡 − 𝑡′) (provided that we assume that the observed trend, on

which the reduction factors are based, will continue in the future).

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The exponential model

Let us suppose that the observed mortality profiles are such that the behaviour over

time of the logarithms of the 𝑞𝑥’s is, for each age 𝑥, approximately linear (see

figure).

Then, we can find a value 𝛿𝑥 such that, for ℎ = 1, 2, . . . , 𝑛 − 1, we have

approximately:

Hence:

Or, redefining 𝑟𝑥 = 𝑒−𝛿𝑥:

Assume that, for each age x, the parameter 𝛿𝑥 (or 𝑟𝑥) is estimated, for example via a

least squares procedure. So, the graduated probabilities �̂�𝑥(𝑡) can be calculated.

The constraint �̂�𝑥(𝑡𝑛)= 𝑞𝑥(𝑡𝑛) is usually applied in the estimation procedure.

Set 𝑡′ = 𝑡𝑛, and assume 𝑡 > 𝑡′ the future probabilities are given by:

from which we can express the reduction factor as follows:

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The extrapolation formula originates from the analysis of the mortality profiles, and

hence constitutes an example of the horizontal approach.

Used by CMIB (Continuous Mortality Investigation Bureau) for UK pensioners in

1978 with 𝑟 equal for all the ages (acceptable over 60 years).

Used by SOA (Society Of Actuaries) in USA in 1995.

The exponential model with an assigned asymptotic mortality

In the exponential model usually 𝛿𝑥 > 0 (so 𝑟𝑥 < 1), consequently 𝑞𝑥(∞) = 0.

Although the validity of mortality forecasts should be restricted to a limited time

interval, it may be more realistic to assign a positive limit to the mortality at any age

𝑥. To this purpose, the following formula with an assigned asymptotic mortality can

be adopted:

where 1 ≥ 𝛼𝑥 > 0 ∀𝑥 represents the share of mortality that cannot be reduced.

This formulation represents a generalization of the exponential model.

Under this assumption the reduction factor will be:

And:

The exponential formula can be simplified by assuming that 𝑟𝑥 = 𝑟 ∀𝑥, from which

we obtain:

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Although the mortality decline is not necessarily uniform across a given (wide) age

range, this assumption can be reasonable when a limited set of ages is involved in

the mortality forecast. This would be the case for mortality projections concerning

annuitants or pensioners. In any case, some flexibility is provided by the parameters

𝛼𝑥.

An implementation of the exponential model with an assigned asymptotic mortality

An alternative version of the exponential formula can help in directly assigning

estimates to the parameters 𝑟𝑥. Without loss of generality, we address the simplified

structure represented by equation

so that 𝑟 is independent of the age 𝑥.

The total (asymptotic) mortality decline, from time 𝑡’ on, is given by 𝑞𝑥(𝑡′) − 𝑞𝑥(∞),

whereas the decline in the first 𝑚 years is given by 𝑞𝑥(𝑡′) − 𝑞𝑥(𝑡′ + 𝑚). Let us

define the ratio 𝑓𝑥(𝑚) as follows:

So 𝑓𝑥(𝑚) is the proportion of the total mortality decline assumed to occur by time

𝑚.

Dividing both numerator and denominator by 𝑞𝑥(𝑡′) we obtain:

Since we have assumed 𝑟𝑥 = 𝑟 for all 𝑥, we have 𝑓𝑥(𝑚) = 𝑓(𝑚). So:

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Therefore the choice of the couple (𝑚, 𝑓(𝑚)) unambiguously determines the

parameter 𝑟. The reduction factor will be equal to:

A similar assumption on mortality evolution has been adopted by CMIB in 1990 for

pensioners with 𝑡’ = 1980 as the base year, m = 20, 𝑓(20) = 0.6 and

A recent implementation of formula

by the Continuous Mortality Investigation Bureau is as follows (see CMIB (1999)):

where 𝑐 = 0.13, ℎ = 0.55 and 𝑘 = 0.29.

Parameters have been adjusted so that 𝑡’ = 1992 is the base year.

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A projection model based on odds

Assumption: the logit transform of probabilities 𝑞𝑥(𝑡) can be represented through a

polynomial in 𝑡.

𝑙𝑜𝑔𝑞𝑥(𝑡)

𝑝𝑥(𝑡)= 𝐺𝑥(𝑡)

From which:

𝑞𝑥(𝑡)

𝑝𝑥(𝑡)= 𝑒𝐺𝑥(𝑡) → 𝑞𝑥(𝑡) =

𝑒𝐺𝑥(𝑡)

1 + 𝑒𝐺𝑥(𝑡)→ 𝑝𝑥(𝑡) =

1

1 + 𝑒𝐺𝑥(𝑡)

Specifically, if we assume that 𝐺𝑥(𝑡) = 𝑎𝑥 + 𝑏𝑥𝑡, we obtain:

𝑞𝑥(𝑡) =𝑒𝑎𝑥+𝑏𝑥𝑡

1 + 𝑒𝑎𝑥+𝑏𝑥𝑡

Projecting mortality in a parametric context

When a mortality law is used to fit observed data, the age-pattern of mortality is

summarized by some parameters. Then, the projection procedure can be applied to

the set of parameters (instead of the set of age specific probabilities), with a

dramatic reduction in the dimension of the forecasting problem, namely in the

number of the ‘degrees of freedom’.

Consider a law, for example, describing the force of mortality:

In a dynamic context, the calendar year 𝑡 enters the model via its parameters:

Let 𝒯 = {𝑡1, 𝑡2, … , 𝑡𝑛} denote the set of observation years, Hence, for a given set 𝒳

of ages, the data base is represented by the set of observed values:

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For each calendar year 𝑡ℎ, we estimate the parameters to fit the model:

(e.g. via least squares, or minimum 𝜒2, or maximum likelihood) so that a set of 𝑛

functions of age 𝑥 is obtained

Trends in the parameters are then graduated via some mathematical formula, and

hence a set of functions of time 𝑡 is obtained:

It is worth noting that the above projection procedure follows a vertical approach to

mortality forecast, as the parameters of the chosen law are estimated for each

period table based on the experienced mortality (see figure).

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Conversely, a diagonal approach can be adopted, starting from parameter

estimation via a cohort graduation (see figure). In this case, parameters depend on

the year of birth 𝜏:

For each year of birth 𝜏ℎ, ℎ = 1, 2, . . . , 𝑚, we estimate the parameters to fit the

model

so that a set of 𝑚 functions of age 𝑥 is obtained

Trends in the parameters are then graduated via some mathematical formula, and

hence a set of functions of time 𝜏 is obtained:

Example: A Makeham’s law, representing mortality dynamics according to the

vertical approach, can be defined as follows

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where 𝑡 represents the calendar year. When the diagonal approach is adopted, the

dynamic Makeham law is defined as follows:

where 𝜏 = 𝑡 − 𝑥 denotes the year of birth of the cohort.

From a double-entry to a single-entry projected table

From a strictly practical point of view, the simultaneous use of various cohort tables

may have some disadvantages. Moreover, probabilities concerning people with the

same age 𝑥 at policy issue vary according to the issue year 𝑡. These disadvantages

have often led to the adoption, in actuarial practice, of one single-entry table only,

throughout a period of some (say 5, or 10) years. The single-entry table must be

drawn, in some way, from the projected double-entry table.

Single-entry tables can be derived, in particular, as follows (see figure)

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1. A birth year 𝜏 is chosen and the cohort table pertaining to the generation born in

year.𝜏 is only addressed; so, the probabilities

where 𝑥𝑚𝑖𝑛 denotes the minimum age of interest, are used in actuarial

calculations. Thus, just one diagonal of the matrix {𝑞𝑥(𝑡)} is actually used. The

choice of 𝜏 should reflect the average year of birth of annuitants or pensioners to

whom the table is referred.

2. A (future) calendar year 𝑡 is chosen and the projected period table referring to

year 𝑡 is only addressed; and so the probabilities

are adopted in actuarial calculations. Thus, just one column of the matrix is used.

The choice of 𝑡 should be broadly appropriate to the mix of life annuity business

in force over the medium-term future.

Of course, both approaches lead to biased evaluations. Notwithstanding this

deficiency, approach (1) can be ‘adjusted’ reducing such a bias. A common

adjustment is the so-called “age shifting”.

The age shifting

For people born in year 𝜏 = 𝑡 − 𝑥, the probabilities related to the year of birth 𝜏

should be used, whereas approach (1) leads to the use of probabilities which are

independent of the actual year of birth. To reintroduce a dependence on 𝜏, at least

to some extent, we use the following probabilities:

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Note that all the probabilities involved belong to the same diagonal referred to

within approach (1).This adjustment (often called Rueff’s adjustment) involves an

age-shift of ℎ(𝜏) years. Assuming a mortality decline, the function ℎ(𝜏) must satisfy

the following relations:

As regards the determination of the age-shift function ℎ(𝜏), various criteria can be

adopted. We just mention that most criteria are based on the analysis of the

actuarial values of life annuities calculated using the appropriate probabilities and,

respectively, using the probabilities obtained by age shifting, with the aim of

minimizing the ‘distance’ (conveniently defined) between the sets of actuarial

values. When a criterion of this type is adopted, the function ℎ(𝜏) depends on the

interest rate used in calculating the actuarial values.

Alternatively ℎ(𝜏) can be obtained as an average value of the correction obtained

considering also the age 𝑥:

ℎ(𝜏) =∑ ℎ(𝜏,𝑥)𝑥

𝑛𝑥

Age shifting correction is applied in Italy for RG48, IPS55 and A62 tables and in

France for the TPRV one.