TRADITIONALLY, ACTUARIES HAVE used stochastic,

scenario, and simulation techniques for a

variety of tasks, and interest in asset liability

management (ALM) has increased significantly

among actuaries in recent years. Looking

ahead, market-consistent valuation techniques may

extend ALM techniques further, to aid fair value measurement

in international accounting standards.

But rather than limiting stochastic modelling to

solving ALM issues, this article explores how simulation

modelling can be applied to multistate problems. Our main example is

healthcare, but we then

take a look at some other

applications, including

customer lifetime value

modelling.

Deterministic

models

We are all familiar with

the life insurance model

where policy status is categorised

as shown in figure

1 (left).This sort of interaction of states is fairly straightforward

for a number of reasons:

Deaths, paid-ups, and lapses/surrenders are

assumed to be independent decrements.

The process of stopping premiums is modelled as

one way; restarting premiums is not generally modelled

deterministically.

Non-contractual options to increase or reduce contributions,

or to switch funds, for example, are

rarely modelled deterministically.

In these models the benefits received and decrement assumptions used depend

only upon the state in

which you are and not on

the path followed to

reach the state. They are

thus not ‘path dependent’

and the assumptions are

not select other than

when leaving the starting

state.

Actuarial systems can adequately cope with these types of models with

single dimension vectors of decrements and survival

factors over time.

Multistate models

While figure 1 is technically a model with multiple

states, the term ‘multistate modelling’ tends to be used when another layer or two of complexity is

added. A couple of example models for income protection

plans (IPP) and long-term care (LTC) are provided

below to illustrate the ideas.

In the IPP model (see figure 2, bottom left), complications

arise because we have introduced a two-way

relationship between ‘well’ and ‘sick’. Recovery rates

from sickness vary enormously between age at the

inception of a claim, the duration of sickness, and the

deferred period. To build an effective model of this

recovery transition, we must track every person that

becomes sick, from the start of their claim. This is

normally done by following each month of claimants

through as a separate tranche, with their own specific

recovery (and death) rates. In effect we have a series

of sub-states: sick with duration zero months, sick

with duration one month, sick with duration two

months, etc.

Historically, actuarial systems designed around the

deterministic models above have ignored recoveries to

the well state and used an inception-annuity

approach in which the full cost of future benefits

while in the sick state is recognised immediately. This

limits the ability to dynamically adjust the termination

probabilities without regenerating new claim

annuity tables. In addition, the models do not provide

information on the timing of claims cashflows, as

these are capitalised at the date of claim.

To overcome these shortcomings, it is necessary to

extend the single dimension vectors (inceptionannuity

value) into two-dimensional vectors in

which each cohort of claim inceptions (the rows) is

run off over time (the columns) using select duration

recovery probabilities based on age at incidence and

duration of claim. It then becomes a simple exercise

to model income payments that vary according to

duration sick, for example. The added benefit is that

‘summing across the diagonal’ provides the cashflow

in a given time period, while discounting along a row

gives the inception annuity (see figure 3 across, top

right).

The approach can then be extended to models with

additional states of sickness. For LTC, two disability

states are often modelled, representing a person who

has failed (say) two of six defined activities of daily living

(ADLs) and a person who has failed three of six

ADLs. Recoveries may or may not be modelled, as the

onset of ADL failures is not as reversible as sickness as

defined under IPP contracts.

So far, so good. We can follow each state through by

duration and therefore allow for select transition

probabilities between various states. Modern actuarial

systems should cope with the above multistate deterministic

models. However, what if: The benefit levels paid are capped at an overall

maximum value across the contract duration (and

therefore multiple sickness claims).

Upon recovery from the first sickness, the inception

likelihood is increased and the recovery rates are

lower.

Upon recovery, the deferred period is waived if the

subsequent claim starts within three months.

Our deterministic model is now looking pretty sick

itself!

The simulation approach

The multistate models described above are effectively

deterministic in nature all contracts are assumed to

follow average decrement rates and filter into the various

states as time goes by. You start with one insured

life and may end up after a few years with say 60- of

a person in-force and healthy, 25- of a person in the

sick state (at various durations) and 15- have lapsed

or died. A student encountering this deterministic

approach for the first time often finds it confusing. In

reality a person can only be in one state at a time, so

why not model it this way?

Monte Carlo simulation techniques can be used to

model a particular route that a contract, person, or

company may follow. At the heart of simulation or

stochastic modelling is a random variable (or two or

three). Taking claim incidence as an example, in a

multistate deterministic model we would read in an

incidence rate and create a new tranche of claim incidences

each month. In a simulation we would take

the incidence rate and ‘throw a die’ to determine,

according to that rate, whether the person had

become sick or not. If they do become sick, their state

changes to sick, duration zero. Otherwise they continue

in full health. This process can be modelled very

simply by a Bernoulli random variable, with a parameter

of the incidence rate in that period. Figure 4

(right) illustrates the idea.

What may not be immediately obvious from the

above paragraph is that the simulation model no

longer requires the two- or three-dimensional multistate

vectors, as the person can only be in one state at

a time. You can now collapse the many states down

into a single ‘current’ state, thus reducing the model

complexity significantly, and potentially eliminating

errors that this complexity brings.

Once you have a simulation model working, you

will want to run a number of simulations to produce

a spread of results or calculate a representative

average. Some skill is required to estimate how many simulations should be run to provide reliable estimates

of averages and extreme values. Facilities to provide

simulation rankings, statistics, and graphs of the

results are important to help you efficiently process

the multitude of information available. Customer lifetime value modelling

Other applications of simulation modelling include profit

sharing arrangements under reinsurance contracts and, of

course, simulation techniques are used extensively in general

insurance to model claim frequency and severity on the liability

side. Another application of the simulation technique is

calculating a customer’s profitability to your organisation. You

may believe that their future repurchase and lapse behaviour

is dependent on their length of relationship with your organisation

and their product holdings at any given time.

This type of model can be viewed in terms of an increasingly

sophisticated multistate model. In this case the states do not

refer to levels of sickness but rather to items such as number of

product holdings, level of premium paid, recent activity in

each policy. Further, the transition probabilities can be

impacted by the explicit holdings and path to that status at the

particular projection period. For example, you may have different

premium increment and lapse probabilities for an

investment contract which depend not only on the current

premium level for that contract, but also on whether (say)

there have been increments in each of the prior three years

and whether the associated mortgage with the financial institution

has remained intact.

The multistate methods described above could be used. One

approach may be to assume capitalised fixed future profit

value at point of product purchase. This is analogous to the

inception-annuity approach and would ignore any interactions

between policy holdings other than on the impact to

purchase. To allow for a degree of interaction, a deterministic

multistate model using multi-dimensional vectors could be

employed. However, this can soon become unwieldy as the

complexity grows. In comparison, a simulation approach is

particularly useful in these types of models. You can allow for

dynamic interactions among the product holdings as the customer

follows one particular path, while keeping the model

coding relatively simple.

Applying the techniques

Primarily, all that is required in order to migrate your multistate

deterministic models to the simulation approach is to

place a stochastic process around your transition probabilities

and then run the model through a number of iterations. Given

that the insured only follows one path in each simulation, a

wide range of possible outcomes may require numerous iterations.

This has implications on run times, but is less likely to

be a barrier when using these models for pricing work, for

example.

Simulation modelling is widely adopted by professionals in

numerous industries, for problems such as ‘just in time’ delivery

or telephone queuing periods at call centres. Tools have

been developed to help build these models, and data mining

techniques are used to help extract patterns from customer

databases.

In practice we have found that extending standard profit

testing models for use in both modes deterministic and simulation

is a surprisingly simple exercise. Interpreting results

does require care, but arming actuaries with these techniques

may provide opportunities for us to apply skills in areas where

these techniques are commonly used.

Modelling various scenarios provides a

spread of results that indicate likelihoods of

the outcomes modelled.

An outcome under a simulation may never

happen under a deterministic method for

example, a life moving between well and

sick states numerous times within a short

timeframe thus eliminating the deferred

period on benefits.

Models can be simpler in structure than a

deterministic model, as only the current

state need be modelled.

Modelling in a simulation environment

allows you to develop a rules base that is

far more sophisticated (and, hopefully, more

realistic) than under a deterministic

methodology. This may not actually be used

to provide spreads of results, but just to

calculate averages across complex

multistate decision paths.

Why use a simulation approach?

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