Stock and Flow Diagram
Competitors’ Market Share Ratio =
Competitors’ Market Sharedesired
Competitors’ Market Shareactual.
If the company’s market share is too small compared with
its target, then rates must decrease to reduce the competitors’
market share, thus closing the gap between actual and desired
market share as the competitors’ market share ratio (CMSR)
tends to 1. If the company’s market share is still below target, the cycle repeats itself until the company’s market share
exceeds the target. At this point rates must be increased to
control growth to avoid, for example, resource strain—
forcing the CMSR to move toward 1. As with profitability, the lag
between rate changes and growth can be long, resulting in
overshooting and undershooting the goal. In addition, as with
profitability, it is practically impossible to fine-tune rates to
produce an exact match between actual and expected market
shares. But even if it were possible, few companies would have
the discipline to act based on long-term goals that balance
profit and growth.
The growth (market share) model can be enhanced by
including additional levers, such as re-sloping rates (by age,
deductible), changing plan design, etc., as well as incorporating the feedback effects of corrective actions like antiselection.
The underwriting cycle results from the interplay of the
two loops just described—managing profitability and managing growth. Although their interaction can produce complex
patterns of behavior, Figure 2 gives an overview of the process
by depicting the system structure, identifying its components,
and specifying the rules of interaction.
The mathematics of causal diagrams is summarized below:
> 0 or
> 0 C E (causal diagram with positive polarity) E E C C
< 0 or
< 0 C E (causal diagram with negative polarity) E E C C
where C is the cause and E the effect.
An increase in rates from 100 to 105 (cause), for instance,
could boost profitability from 82 percent to 85 percent (
effect), keeping other variables unchanged. The polarity would
= 0.01 > 0 0.85 – 0.82 105 – 100 .
It’s typical to spend a fair amount of effort quantifying causal
relationships, such as when elasticities of demand are studied
or best response functions are determined.
Stocks and Flows Diagrams
Stocks and flows, depicted in Figure 3, are easier to understand
using a bathtub metaphor.
In a bathtub, the level of water is the accumulation of the
water that flows in less the water that flows out, just as is the
case for any stock like the inventory of claims, which increases
with the number of submitted claims and decreases with the
number of paid claims. The symbols of Figure 3 have the fol-
The source and sink are stocks outside the boundar-
ies of the model. Sources create flows that travel to
the model while sinks absorb flows that leave the model. Sources
and sinks limit model complexity as interactions with, and feed-
backs from, stocks outside the boundary are ignored. When
modeling claims settlement, for example, the population of poli-
cyholders is the source while claimants that receive payment
are one of several possible sinks.
The flow can be an inflow or an outflow. The rate of
claims submission, for instance, is the inflow, while the rate of
claims payment is the outflow. Note that there might be more
than one outflow (the rate of denied claims in addition to the rate
of paid claims, for instance) and, of course, more than one inflow.
The valve regulates the flow. In the individual health insurance business, for example, the rate of application acceptance
is regulated by the intensity of underwriting.
The rectangle represents the stock within the
model boundaries. Examples of stocks include
claims inventory, the number of members of the American Academy of Actuaries, the amount of information in a library, etc.
The mathematics of stocks and flows relies on the generic
stock(t) = ∫ [inflow(x) – outflow(x)]dx + stock(t0)
or equivalently, on the generic differential equation:
= inflow(t) – outflow(t) d[stock(t)] dt .
The integral representation shows that the level of a stock
changes only gradually because it depends on inflow and
outflow rates. This trivial observation, it turns out, is key to
understanding the dynamics of any system containing stock.
It explains, for instance, why even dramatic reductions in the
harvest could be of no use in regenerating the population when
the stock of fish is low.
To illustrate a possible interpretation of Figure 2, consider
a population with 100,000 individuals at t = 0 whose flows of
births and deaths are as shown in Table 1 (See Page 40). Graph
1 (See Page 41) depicts the evolution of the population according to the relationship
population(t) = population(t – 1) + births(t) – deaths(t – 1).
Causal Loop Diagrams
Combined with Stocks and Flows Diagrams
Causal loop diagrams, along with stocks and flows diagrams, are
the basic building blocks of any dynamic model. Consider, for
instance, the dynamics of an investment account as depicted in
Figure 4. The account accumulates as interest is credited, but