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A popular product known as insurance-有一个受欢迎的产品名叫保险(5)

In short, we must choose a suitable estimate using prior experience to categorise policyholders. An appropriate credibility premium can then be charged based on number of years for which data are available. (For more comprehensive understanding regarding the predictive distribution and predictive error of the outstanding, please refer to England and Verrall (2002))

Mack's Model
Mack (1993) proposed that estimates of standard errors can be obtained using an approach that is independent of the distribution underlying the claims. The benefit of this model is that it does not make unrealistic assumptions of the underlying distribution of claims and the development factors.

As summarised by England (2009), the specified mean and variance are as follows:

where is defined as the cumulative claim with year of origin, i, and year of development, j

Using the above equations and the development factors, calculated in the same way as previously defined, we could obtain a squared run-off triangle that can be used to estimate future reserves.

To include the variability factor, we let

By doing so, we are incorporating estimation variance and process variance into the future reserves. Therefore, the mean square error of the reserves of any given year of origin i, as stated by Mack (1993) is:

It should be noted that the residual used in estimating the scale parameters in Mack's model is consistent with assumption of a weighted normal regression model. Therefore, it is a reasonable estimation.

General Insurance - Premium Pricing
Overview
As described earlier, premium can be viewed as the market value of insurance that maximises the wealth of the insurer. At the very least, it should make the policy sustainable even in harsh times. Generally, a premium function must be established in such a way that the solvency of an insurer is assured. The main requirement is for function to be swift enough to manage incoming claims.

However, too high of a value in comparison with rival insurance companies will result in an undesirable outcome. Taylor (1986) developed the fundamental concept of how competition in premium rates has considerable impact on the insurer's strategy. (Please see Taylor (1986) for the full literature)

Simple Capital Asset Pricing Model (CAPM)
Pioneers of this simple, yet fundamental approach were Cooper (1974) and Biger et.al (1978). Premiums obtained using this model reflects valuation in perfect capital markets which is by no means realistic. Despite so, this idea remains to be of much theoretical use because it forms a foundation in many insurance pricing models.

Cummins (1990) states that the derivation of this model begins from:

where Y is the net revenue, I is the revenue from investment, μ is the underwriting profit, is the rate of return on assets, is the rate of return on underwriting, and A and P are assets and premiums respectively.



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