Miccolis

Miccolis does suggest an exception to the consistency test in the form of anti-selection. Rosenberg expands on the definition of anti-selection and says it can take 2 forms, and provides 2 examples of each:

1. Adverse Selection - Higher limits indicate adverse experience. This can occur because: (a) Insureds that expect higher loss potential could be more inclined to purchase higher limits. (b) Liability lawsuits & settlements may be influenced by the size of the limit. 2. Favorable Selection - Higher limits indicate best experience. This can occur because: (a) Financially secure insureds may be better risks, and may purchase high limits since they have more assets to protect. (b) Insurance companies will be more willing to write high limits coverage for better risks.

There are two reasons that the severity trend on excess layers is greater than total or basic limits severity trends when the severity trend is positive:

1. For losses above the basic limits, the trend is entirely in the excess layer. 2. Losses just under the basic limit are pushed into the excess layer by the trend, creating new losses for the excess layer.

There are 2 main sources of pricing risk:

1. Process risk = The difference between actual & expected losses. 2. Parameter risk = The inability to estimate expected losses accurately. Examples could include: - Natural disasters are difficult to predict. - Unknown inflation. - Changes in mix of business. - Sampling error.

Miccolis lists some problems that occur when trying to create a severity distribution in practice:

1. There could be future development on claims used in creating the distribution. 2. Using loss data from policies with different limits would bias the distribution. 3. The credibility might be low at the high end of the distribution because of relatively few large losses. 4. There might be cluster points at round numbers such as $25k, $50k, etc.

For the purposes of pricing, instead of publishing full rates for every limit, insurers usually use relativities called Increased Limit Factors (ILFs) to the rate for a basic limit:

Rate at Limit k = ILF(k) * Rate at Basic Limit b In determining the ILFs appropriate for each limit, the following assumptions are commonly made: - All UW expenses and profit are variable and don't vary by limit. In practice, profit loads might be higher for higher limits since they are more volatile. We'll look at this later. - Frequency and severity are independent. - Frequency is the same for all limits. Note that this may not be true in practice due to adverse or favorable selection. GLMs or loss ratio methods will reflect these differences, but the following ILF approach will not.

Rosenberg's Consistency Test

Rosenberg guides our understanding of this by giving an extreme example: - An insured with no aggregate limit moves from a $25k occurrence limit to a $50k occurrence limit. He gains a full additional $25k in coverage for each claim. - A different insured with an aggregate limit of $25k moves from a $25k occurrence limit to a $50k occurrence limit. Since the aggregate limit is only $25k, this insured actually gains no coverage at all by making the same switch. - Thus, you can see that more coverage is gained by making this switch for higher aggregate limits than for lower aggregate limits, and as such, the premium change for making the switch should vary accordingly.