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29

Insurance Structure Optimisation

In Chapter 4, we looked at heuristic rules on how to choose the deductibles and limits of poli-

cies. These rules may suggest, for example, that the annual aggregate deductible (AAD) for

a commercial liability policy should be hit with a probability of 5% to 20%, or that an each-

and-every-loss deductible should lter out at least 90% of the claims. However, these heuristic

rules are exactly what the name says – heuristics – and there is no rigorous justication for

them, only common sense reasoning. It is no surprise that these rules are often and cheerfully

outed.

Is it possible to improve on this and have a more scientic criterion for an insurance

buyer to choose their insurance structure in an optimal way, or at least to prune out the

‘inefcient’ insurance structures and leave us with the ‘efcient’ ones? This chapter will

illustrate some of the approaches that have been devised to identify efcient (Section 29.1)

and optimal (Section 29.2) insurance structures.

29.1 Efficient Frontier Approach

The idea of the efcient frontier approach is quite simple: use a cost–benet analysis to

compare different insurance structures.

29.1.1 Cost of Insurance

There is no doubt as to what the main cost of insurance is: the premium paid to the insurer

(plus possibly some management costs for keeping track of the insurance purchase and

monitoring its performance).

29.1.2 Benefits of Insurance

The calculation of the benets of insurance is more complicated. General insurance (and

reinsurance) policies are bought for many purposes:

• Reducing nancial volatility

• Giving peace of mind

• Abiding by the law and regulatory constraints

• Satisfying shareholders

• Obtaining related risk management services from underwriters and brokers, and

much more

520 Pricing in General Insurance

For example, if you buy car insurance, you will see that it obviously satises the rst

three purposes: it avoids sudden nancial outgoings to pay chunky claims, it gives you

peace of mind and even if you didn’t care about nancial outgoings and your peace of

mind is unaffected by the thought of the consequences of a car accident, you will have

to buy it anyway because it is compulsory. If you own a pharmaceutical company, you

may have other reasons to buy product liability insurance, for example, demonstrating to

shareholders that you are taking care of that risk properly, and gain access (through the

a broker/consultant or directly through the insurer) to market information on product

liability risk and different ways of managing it.

In practice, most of the benets above are either not amenable to quantitative analysis

(such as peace of mind) or do not help in differentiating between different insurance pro-

grammes (e.g. not abiding by laws and regulations is not really an option), and we can

focus on volatility reduction.

29.1.3 Efficient Structures

Given a metric for nancial volatility, some insurance programmes are obviously better

than others:

• If A and B achieve the same nancial volatility reduction, but A is cheaper than B,

then A is better than B (A ≻ B)

• If A and B have the same premium, but A achieves more volatility reduction than

B, then A ≻ B

Two programmes may, however, be ‘not comparable’:

• If A is cheaper than B but B achieves more volatility reduction than A, then A and

B are not comparable.

In mathematical terms, one can say therefore that volatility reduction and pre-

mium impose a partial ordering on the set of insurance structures. This in turn is

simply a consequence of the fact that we have two different criteria according to

which we judge insurance structures, premium and volatility reduction, and no

means is provided to weigh one against the other.

Having decided to focus on volatility reduction as the criterion for differentiat-

ing between different insurance programmes, we need a metric to measure vola-

tility reduction (or, more simply, the volatility itself). Three measures of volatility

reduction are commonly used:

• Reduction of the standard deviation (or variance) of the retained losses

• Reduction of the value at risk at the pth percentile, that is, of the maximum

losses expected to be retained with probability p

• Reduction of the tail value at risk at the pth percentile, that is, of the mean losses

expected to be retained above the pth percentile

Each of these measures has its own advantages and disadvantages. All of them, how-

ever, have one general disadvantage: they need a reference point – ‘reduction’ with respect

to what? The obvious choice would be ‘reduction with respect to the case where no insur-

ance is purchased at all’. If we want to avoid this complication altogether, we can simply