probable maximum loss

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PROBABLE MAXIMUM LOSS: KEEPING ANEYE ON EXTREMES.

November 30, 2012 bydevvaibhav in Actuarial

Risk selection and decision making lies at the core

of insurance and reinsurance operation. Knowing the worst case scenario forms basis of risk selection

process in insurance. There are multiple reasons to consider worst cases: checking for risk

absorbency, risk comparison and price the product appropriately. In the context of insurance

contracts this consideration becomes important as many of the decision made are prospective in

nature and done in the wake of vast uncertainties.

Underwriters and risk managers while making their decision rely heavily on understanding and

quantifying the worst cases scenarios. Putting it more formally one of the metrics to measure worstscenarios is Probable Maximum Loss (PML). Probable maximum loss is useful both as a metric and

a theoretical concept.

Though estimating PMLs has become a popular phenomenon in many lines of business in P&C

industry, PML find its origin in fire insurance. Underwriters wanted to see what can be the highest

loss they could incur with some amount of confidence.

Probable maximum loss is an obvious term with no obvious meaning. . Despite being around for many

years this term is not yet precisely defined. That being said it is widely used and everybody concerned

with it uses it with complete certainty.

Defining PML

Actuarial Literature defines PMLs as:

Probable maximum loss under a given insurance contract is that proportion of the limit of liability

which will equal or exceed, in a stated proportion of all cases, the amount of any loss covered by thecontract.
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Loosely speaking, if you have all possible losses to a property (Limit of liability) then you are looking

for a reasonably high loss satisfying following condition:

It is not the maximum of all losses

Majority of the losses are below and closer to this.Chances of realizing total loss is very low so, we are looking at the worst loss which has reasonable

chance of occurring.

One of the easiest ways to define a PML is through percentile.

For an insurance contract, generate a loss distribution ( say a random variable X) then m-th

percentile of the this distribution (Lets say Q) will provide the PML. PML can be presented as a

percentage of total liability under contract as well, in that case find what proportion of limit Q

constitutes.

Mean expected loss is an important piece of information which will help in interpreting Q. We can

represent Q as (E(X) +R) then R can be seen as what lies beyond mean expected loss.

Looking into the definition of PML clearly says it cannot be derived in an uni-dimensional fashion.

Key ingredients of defining a PML are: Insured value, Loss distribution, Value of m and Peril/ Line

of business. Out of four quantities two are quite easy to know Insured value and peril/LOB, knowing

loss distribution and value of m presents challenges.

Determination of these two quantities involves much of the work. Loss distribution and value of m

can be derived/calculated from following information:

Historical losses Statistical Theory Judgment Line of business Peril

Applying PMLs

Risk managers, underwriters, actuaries and catastrophe modelers are few of the professionals who

start and end their day with PML, not to forget this is one of the metrics and there is a lot that goes


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into assessing and pricing a risk. PMLs serve to risk assessment, pricing, capacity determination,

capital modeling and reserving with application wide array of lines of businesses. Our discussion has

more of property focus, but the intuition of PML is exploited in non-property domain as well.

Understanding the assumed risk (property risks) brings up many questions to be answered; how is

risk distributed ( by geography, businesses), what is economic value of risk, components of risk

(symmetric and asymmetric), what is worst case scenario in terms of economic loss etc. When risks

are assumed with so much of uncertainties knowing its extreme economic loss becomes one of the

major drivers in either managing/assessing the risk. PMLs provide exactly these answers.

Pricing in insurance is generally based on Average Annual Loss (AALs) and by loading it for profit and

other expenses; this loading also carries information from PML.

Capacity determination easily translates to alternate risk transfer decisions of primary insurers. In the

absence of PMLs, reinsurance program design/decisions and participation in capital markets would

see many limitations. Decision of primary insurers to transfer risk to reinsurance at times can solely

be based on PML. The plain logic would be:

For a primary insurers portfolio of risk, PML is amount Y and loss bearing capacity is L (


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For the most part it is an estimation (done prospectively) therefore inaccuracy in the calculated PML

is inevitable. Probabilistic methods can provide better estimation; deterministic estimation provides

more controlled calculation or underwriters who are doing it for 20 yrs knows it best true are common

arguments in industry. All three arguments hold but best strategy to calculate PML would always be a

mix of all three.

Closing Remark

There many facets attached understanding it, calculating it and applying it. While dealing with PMLs

following guidelines should be followed:

Understand itthoroughly.calculate it cautiously.and.apply it smartly!

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