State Transition Systems

1. Introduction
Insurance executives are naturally interested in how claims operations have performed and affected the firm’s financial profile over time. They would need to ask: what happened between the start and end of a given period? A complete view of the historical development of claims operations gives an insurance executive the ability to determine the financial relevance and financial impact of the activities between two points in time.


Figure 1: What happened between the start and end of a given period?

Among the most important statistics to monitor for any claims inventory are the closed and open claim counts. Although properly regarded as key indicators of the condition of a claim department, such statistics do not highlight the activity between the two points and fail to provide the information needed to spot operational inefficiencies. They are snapshots in time and as such, do not contain enough information to model significant dynamics of the operational flows encountered throughout the claims handling process.

The proposed approach addresses the financial questions by introducing a state transition model. The model provides a methodology for effective monitoring of the operational activities in the claim handling process; knowing the historical development allows a better understanding of the operations and provides insight to assess the financial relevance of changing or addressing key activities.

To the aim of illustrating the state transition modelling concept, let us consider the following situations:

  • Jim, a claim professional, receives the notification of a new loss and opens the file referred to as C1.02_red_dot_c1_revised
  • During the day, working on another loss, Jim made a payment and closed the file called C2. This is a transition from open to close.
  • Susan closed the file C3. Later on, she received an additional bill which resulted in the file reopening. The file transitioned twice. She called the claimant and determined not only that the claimant received an additional bill but he was waiting for the final bills from other doctors. Rather than closing the file, she issued the payment and diaried the file to receive the remaining bills.
  • Jim closed the file C4. When he received the additional bills, reopened the file, paid and closed it again. In speaking with Susan, she asked him: Are you sure there are no outstanding damages?

In considering the transitional picture below, the reader should keep in mind the activities that originated the corresponding state transitions.

Due to activities three files have been closed, performing at least one transition, i.e. C2, C3, C4 and two of them have been reopened (performed at least two transitions), i.e. C3 and C4. One file is re-closed (performed an odd number of transitions) i.e. C4.06_red_dot_c_all_revised

The following figure shows the close conceptual ties among the main steps of the operational flows:


The transitions can result from numerous reasons or activities, such as

  • unanticipated additional damages, delay billing,
  • unforeseen complications, coverage decision, legal decision,
  • recoveries, lack of response from third parties,
  • differring application of technical best practices,
  • regulatory reopenings.

Various activities and behaviours, positive or negative, influence claims transitions. This can result in acceptable practices or premature closings which impact a company’s financials.

A state transition view provides a detailed look at key claims activities, such as file opening, closing, reopening and re-closing over time. Taking a transitional view provides us the opportunity to review our decisions and it will challenge insurance executives to ask: Are we as effective as we can be?

To illustrate, consider the following three scenarios where, from a snapshot standpoint, the state closed and open claim counts are the same, but the state transition view paints substantially different operational pictures.


Figure 2: Susan has ten claims of which eight closed. She reopened five and re-closed additional three.


Figure 3: Jim has ten claims of which eight closed. He reopened two and re-closed none.


Figure 4: Tom has ten claims of which eight closed. He reopened seven and re-closed additional five.

2. Claims As State Transition Systems
Claims are notified to an insurer on dates di (i ≥ 1) satisfying di ≤ dj whenever i < j. We call di notification date. Henceforth, di denotes the date on which the i-th claim is notified. If two (or more) claims, say i and j, are notified on the same date, we would then obviously have di = dj. For any point in time t ≥ 0, we let N(t) denote the number of claims which have been notified by t, that is

N(t) := #{ i ≥ 1 | di ≤  t}, t ≥ 0.

Figure 5 illustrates an example of the notified claims count. The notification dates are highlighted in red. Note that the “claims count” jumps at the notification dates and the size of the jump is given by the number of claims notified on that date.


Figure 5: Example of the claim notification path

The following observations are worth making:

  • For any t ≥ 0, N(t) is a non-negative integer;
  • N(t) is non-decreasing in t;
  • N(0)=0;

The collection of all claims notified in the interval [s, t] is denoted by

N[s,t] = #{ i ≥1 | s ≤ di ≤ t}, 0 ≤ s < t.

We call the interval [s,t] notification period. We can now introduce our next definition:

Definition 2.1 A claim is new with respect to the notification period [s,t] if it belongs to the the set N[s,t].

Having defined the set of new claims with respect to a given time interval, we would like to introduce our main “categorization” of such a set. Note that at each point in time, a claim can only be in one of these two states: open, closed. And therefore only four transitions are possible between two points in times t0 and t1: from open to close, from close to open, from close to close and from open to open (see Figure 6).


Figure 6: Claim as state transition system

But out of these possible transitions, only two concern a change in state between any two points in time t0 and t1: from open to close and from close to open – as highlighted in red in Figure 7. We call them state transitions.


Figure 7: State transitions

We are now in the position to define the attributes of a claim i in terms of number of state transitions performed from the notification date di up to an arbitrary point in time t (di ≤ t). Note that claims are considered to be in an open state when they are notified.

Definition 2.2. A claim i is open at time t if no state transition is performed by time t from its notification date di.

Definition 2.3 A claim i is closed at time t if exactly one state transition is performed by time t from its notification date di.

Definition 2.4 A claim i is reopened at time t, if 2n (n > 0) state transitions are performed by time t from its notification date di.

Definition 2.5 A claim i is re-closed at time t, if 2n+1 (n > 0) state transitions are performed by time t from its notification date di.

Remark 2.6 Note that the four just-defined attributes are mutually exclusive and collectively exhaustive.

Remark 2.7 Note also that for n = 0, the definitions of re-closed and reopened are equivalent to the definitions of closed and open, respectively.

Remark 2.8 One of the benefits of our approach – reducing the attributes of claims in terms of number of state transitions performed over time – is that it makes very easy to define and analyze new metrics like the average number of transitions per reopened and/or re-closed claim, average duration of transitions, etc.

3. Example
The following picture illustrates a fictional set of possible state transitions in a given notification period [s, t]. Note that the transitions displayed have all the same duration, which is clearly not the case in reality: the real duration of a transition might range from a few hours to several years.


We see that claims C1 and C8 performed no transitions from their notification dates and are therefore open at time t. Claims C3, C5, C7, C9 and C10 performed one state transition and are closed at time t. Claim C2 performed 3 state transitions and is in the state closed at time t (re-closed claim). Claims C4 and C6 performed 2 and 4 transitions, respectively, and are both in the state open at time t (reopened claims). In general, we note that all claims which performed an odd number of state transitions by time t (from their notification dates) are re-closed at time t and all claims which performed an even number of state transitions by time t (from their notification dates) are reopened at time t.

Another way of reading-off information from our picture is as follows. We see that 10 new claims have been reported. Out of these 10 claims, 8 have been closed (moved at least once in the state closed, e.g. performed at least one state transition), i.e. C2, C3, C4, C5, C6, C7, C9 and C10. Out of these 8 claims, 3 have been reopened (moved at least once in the state open, e.g. performed at least 2 state transitions), i.e. C2, C4 and C6. Out of these 3 claims, 1 is re-closed at time t (performed an odd (> 1) number of transitions), i.e. C2. We further note that:

  • The number of open claims at time t is just the difference between the new claims and the ones which have been closed, 10-8=2.
  • The number of closed claims at time t is the difference between the claims which have been closed and the ones which have been reopened, 8-3=5.
  • The number of reopened claims at time t is the difference between the claims which have been reopened and the re-closed claims 3-1=2.
  • The number of re-closed claims at time t is 1.


The state transition approach outlined in the paper allows us to combine together “point in time” and “through time” metrics and can therefore be regarded as a very simple evolutionary analytical model. The model allows for early identifications of changes in the historical data patterns which can be indicative of the current workload capacity, the impact of very significant operational changes in the claims department, potential sources of inefficiencies in the claims systems and changes in the claims handling practices.

I would like to express my gratitude to Ron Pacini and Luigi Pistis for their inputs, valuable guidance and continued support.

[1] Hans Bühlmann, Mathematical Methods in Risk Theory. Springer, Berlin, 1970.
[2] Thomas Mikosch, Non-Life Insurance Mathematics. Springer, Berlin, 2004.
[3] Vincenzo Salipante, GI Claims Definitions – Reference Guide. Zurich Insurance Company Ltd, Switzerland, Internal report, 2nd Edition, 2012.

Vincenzo Salipante is Head of Claim Analytics with Zurich Insurance Company Ltd., where he has been providing key analytical services to Actuarial, Reinsurance and Claims functions for eight years. Prior to his current role in Claim Analytics, he was with Group Reinsurance and the Global Corporate Division. He holds a MSc and PhD from the University of Rome and The Institute of Applied Mathematics and Computer Science of the University of Bern, respectively. He is currently studying for his Executive MBA at the International Institute for Management Development (IMD), Lausanne, Switzerland.

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