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A popular product known as insurance-有一个受欢迎的产品名叫保险(7)

Performing the steps described above and simplifying the expression, Cummins (1990) obtained the following expression:

where P is the premium, E( L + e ) is the expected losses, is the risk-free rate, to be adjusted risk factor , is the corporate income tax rate, and to be the surplus over premium.

From the equation above, we are able to deduce that positive risk premium suggests lower premium and vice versa. Although an explicit expression for the premium has been obtained, this is not so useful because it fails to consider the element of market risk.

Myers-Cohn Model Using CAPM
In this section, we fuse the concept of CAPM and Myers-Cohn model in the more general situation whereby investment balance for tax and underwriting profit are included. The resulting expression as pointed out by Mahler (1998) is as follows:

where represents present value discounted at risk-free rate, r, represents present value discounted at adjusted risk factor, represents premium after tax, represents Investment Balance discounted at a risk-free rate that has been taxed, and is the underwriting profit. All other variables are defined as before. [Note that this may be slightly different from what we have defined in section 3.3. Reason being that in this section we include all the real-world elements such as corporate tax and investment balance after tax]

Rearranging the above equation yields two important results:


where is the risk-adjusted rate for L + E, is the risk-free discounted rate for P, is the risk-free discounted for IB, is the risk-free discounted rate for U, is the risk-adjusted rate for U after tax, is the risk-free discounted rate for revenue, is the revenue offset rate for tax, and is the provision to be held.

Hence it is popular among the property and liability insurance companies in USA to set  as the target provision.

(For more information regarding the derivation of this result and a real-life example with numerical solutions, please see Mahler (1998), pg 728-731, Exhibit 5)

Validation Methods
Looking at different classes of pricing approach, we should not neglect the fact that premiums should be fair. Fair in this context refers to premiums being forward looking, as stated by Harrington and Niehaus (2003). (The explanation of forward looking is on pg 149-151, Harrington and Niehaus (2003))

Thus, we should investigate whether the premium obtained is a reasonable figure. A basic method of checking, as proposed by England (2003) is to use the expected claims and add a few risk adjustments, usually standard deviation in most cases. Certainly, premiums charged should not be lower than the risk-adjusted expected loss, but also not on the other extreme as to making it uneconomical and unethical to the insured.

Another method of validation proposed by Wang (1999) has been used extensively. His method uses proportional hazards model to calculate a risk-adjusted price through the control of the parameter, ρ. Essentially the loss distribution is transformed by raising it to the power of 1/ ρ. Wang's method can be easily applied in excel spreadsheets and the parameter, ρ could be manipulated to obtain the true risk-adjusted price. However, choosing a value of ρ could prove to be very subjective.

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