## Experience Study Method Overview

Experience studies are used by actuaries to determine appropriate assumptions for valuing the liabilities arising from existing contracts, and for pricing new contracts. An actuarial experience study is the analysis of a particular type of behaviour within a population of lives, where the behaviour represents a decrement or exit from the population under study, for example by death i.e. mortality.

In a mortality study, the objective is to calculate the mortality rate which is the probability of death over a period, usually one year. The mortality rate is calculated as the number of deaths from the population over a period divided by the amount of time the population is exposed to the risk of death within the period. As mortality varies by a number of factors such as gender and age, the study divides the population into sub-populations for each set of factors for which mortality rates are calculated. A mortality table is the set of rates calculated for each set of factors, such as gender and age. The study then compares the actual rates against an existing standard mortality table.

There are a variety of methods that can be used to calculate the mortality study. A study method will incorporate a number of components including the exposure method, the age definition, study period and other factors under consideration. The key determinants in the design of a study are the nature of the source data available and the nature of the systems available to process the study, the objectives of the study and the flexibility and interactivity required by the study analysts. The validity of any experience study requires that the same population of lives contributes to both the number of deaths and the exposure. The lives contributing to the number of deaths should also contribute to the exposure, and the exposure should only include lives that would be counted in the deaths if they died.

Experience study systems usually have limitations around the flexibility of the study. Results are normally available at some pre-determined level of aggregation due to the sheer volume of data generated in processing the study. The study parameters are embedded in the design of the study: the system must be reprocessed to change the study period or the standard tables, and the system itself must be modified to add additional mortality factors.

Discussion around the main study factors is set out including formula for rates, exposures, ages, study period. Different jurisdictions can have different terminology and emphases for the same methods, which can lead to some confusion. This is true for experience studies between North America and in the United Kingdom: divided by a common language as ever. A blend of terminology is used here, with reference made to other possible terms.

### Rates

Expected rates are usually based on a theoretical population of lives which can only exit the population by the single decrement under study. Mortality will be used to illustrate the theory and method here, but any type of decrement would be equally valid.

Consider a population of lives starting at age 0, where the number of lives at each exact age is . The number of decrements, e.g. deaths, over the year of age , i.e. from exact age to exact age is . The proportion of lives that exit the population from exact age to exact age is

This proportion is the probability, i.e. the expected rate, of a decrement occurring from a population at exact age over the following year of age. This is referred to as the initial rate of decrement, as it is based on the population at the start of the year of age .

An alternative way is to present the rates is the average rate over the year of age. This is referred to as the central rate as it roughly represents the rate at the middle of the year of age. Instead of using the population at exact age , the average population over year of age , , is used, which is given by the following.

The central rate, , is the ratio of decrements over the year of age to the average population over the year of age. That is,

The relationships between the initial rate and central rate are

and

The force of decrement, , is the instantaneous rate of change of the decrement rate. It can be shown that

The initial and central rates presented above are single decrement rates as there is only one type of decrement by which a life may exit the population. In reality, lives may exit a population by other independent reasons, such as lapses.

Multiple decrement rates can be calculated where more than one type of decrement is under study. Multiple decrement notation uses an “” prefix, parentheses and a superscript to indicate the decrement type. The number of lives at exact age is while the average population over age is . For two decrements, and , the number of decrements over year of age would be and and the total number of decrements is .

For generic decrement , the initial rate and central rate are given by

The total initial rate is , and the total central rate is .

The relationships between the initial rate and central rate are given by

and

Expected rates are usually applied in multiple decrement scenarios, even if presented as single decrement rates, so there is a need to move between single decrement and multiple decrement rates. The single decrement notation needs to be generalized by introducing a superscript to identify the decrement type, e.g. for generic decrement , and qualify the single decrement terms , and . Alternative terminology refers to and as the independent rates for decrement , and and as the dependent rates for decrement .

For generic decrement , the central independent and dependent rates are roughly equal, that is

The approximate relationships between independent and dependent rates involve all of the decrement rates. In this two decrement example,

,

,

These formulae assume decrements occur continuously over the year, but can be adjusted where one decrement occurs at the start or end of the year.

The force of decrement definitions and relationships also hold true for multiple decrements. That is, is the independent force for generic decrement , is the dependent force for generic decrement For two decrements, and , the total force is , the independent and dependent forces are roughly equal, i.e. , and the force is roughly equal to the central rate, .

The initial rate is the standard presentation of rates used by actuaries. This is due to its historical role in life contingencies and commutation functions, in an era of simpler insurance products. Today, actuarial calculations are applied from first principles for more complex products using computer programs, while commutation functions are less frequently used.

### Exposure

Expected rates cannot be calculated in practice using a single cohort of lives from birth through to death of over a century. Instead a population of lives of different ages is studied over a much short shorter period between a start date and an end date. The population may have new lives entering the population as well as lives exiting the population for reasons other than the decrement or decrements under study. The study period and the migrating lives will result in lives not being in the population under study for a full year of age.

The population used to calculate the rates is generalized to the concept of exposure: the amount of time a life is exposed to the risk of decrement. This can also be referred to as “exposed to risk” or risk time. The exposure units are life-years, or policy-years or more generally contract-years. Where a study uses the amount of insurance or benefit amount as a weight for each life in the study, the exposure units may be expressed as amount-years or benefit-years.

Initial exposure, , is equivalent to the number of lives at age , . Lives that exited by the decrement or decrements under study are assigned a full year of exposure, while lives exiting for other reasons are assigned exposure up to the date of exit. The definition of initial exposure is dependent on the type or types of decrement under study.

Central exposure, , is equivalent to the average number of lives over year of age , . All lives that exited, including for any decrements under study, are assigned exposure up to the date of exit. The definition of central exposure is independent of the type or types of decrement under study in the population. The central exposure is sometimes referred to as the exposure of the force of decrement.

The relationship between initial and central exposure, assuming decrements occur mid-year, is

The initial and central rates for a single decrement study are

Central exposure is a more natural definition of exposure than initial exposure as:

- It is inherent in the population and not dependent of the type or number of decrements under study.
- It represents the amount of time all lives actually spend within the population under study.
- It is calculated consistently for all lives that have exited irrespective of the type and number of decrements.

Initial rates can be derived from central exposure using the following formula.

There are two fundamental methods to calculate exposure: the aggregate census method and the seriatim date method.

The aggregate census method estimates exposure from periodic, usually year-end, snapshots of number of lives active in each subpopulation. The census calculation can be applied on a seriatim, individual life basis. The central exposure is directly calculated for the census method, with the initial exposure calculated using half the decrements as in the formula above. This method may be referred to as the valuation or in-force method. A separate file of decrements is required. The age definition is provided with each snapshot file.

The seriatim date method calculates the exact exposure by age for each life active in the population between the date of entry into, and the date of exit from, the population under study. Central or initial exposures may be directly calculated depending on the purpose of the study. The age is calculated for each life and varies over the study period. The file includes the date of exit and exit reason used to identify decrements under study.

### Ages

Ages within the study are defined by the year over which each age lasts (the rate interval), and by how the age at the start of each year is calculated (i.e. age last). Common definitions of the rate interval are the “life-year” based on the insured birthday, and the “policy-year” based on the issue date of the policy. At the start of each year, the age may be calculated as age last or age nearest.

For the census study, the key determinant is the age definition used for the deaths, with different formula used to estimate exposure for the different types of rate interval definition.

For the seriatim study, the policy-year rate interval is usually used giving age as the age at entry plus the integral number of years since entry.

The age definition for deaths and exposure should be consistent. For the census method, this usually means that the definition of the age used for deaths determines the exposure calculation being applied. For the seriatim method, there is no issue as the exposure and deaths are calculated from the same record and dates.

### Study Period

The study period is the period over which deaths are included in the study, and for which exposures are calculated. This period is usually based on either calendar-years or policy-years. For the census method, the period would be based on calendar-years determined by the years between the first and last year-end snapshot file. For the seriatim method, the period may be based on either calendar-years or policy-years. A calendar year study would only include exposure and deaths between the start and end dates of the study. A policy year study would include exposures and deaths occurring in the full policy years between the start and end of the study, excluding the partial policy years in the first and last calendar years. Calendar year study periods may be referred to as date-to-date, while policy year study periods as anniversary-to-anniversary.

### Conclusion

Understanding how rates are calculated is critical to ensure both meaningful interpretation and applications of the results. For the US, experience studies have been the speciality of pricing actuaries. With the move to Principles Based Reserving (PBR), the use of experience studies will become much more important for valuation actuaries.

### Further Reading

- The Analysis of Mortality, B. Benjamin and J. H. Pollard, Heinemann, 1989 (reprint).
- Life Contingencies, A. Neill, Heinemann, 1989 (reprint).
- Special Note: Exposed-To-Risk, A.S. Puzey, Institute of Actuaries Education Service, 1982.
- Mortality Table Construction, Robert W. Batten, Prentice-Hall, 1978.