2. THE CHARGE FOR RISK: (pages 41 through 43)

Miccolis states that "a major problem with expected value pricing is that it fails to appropriately charge for the risk of being in the insurance business."

For most lines, the profit and contingencies loading compensates for risk.

For a high risk line of coverage, the loading is usually low, and can be seriously deficient. In this paper, risk is defined as the degree of uncertainty in the pure premium.

Two Main Sources of Risk in Ratemaking:

• process risk - variation between actual and expected losses due to the random nature of the frequency and severity of insurance losses.
• parameter risk - the inability to estimate expected losses accurately.

Although parameter risk can be substantial (estimating catastrophes, inflationary trends, changes in business mix, sampling errors, claims practices, etc.), the determination of a risk charge is beyond the paper's scope. This paper only studies the effects of process risk and determines its risk charge.

Variance as a Measure of Risk

The source of risk used in this paper is the random variation in the pure premium (process risk).

This source produces a substantial, measurable difference in risk charge by limit of liability.

Although Lange suggests the standard deviation of the pure premium as an appropriate measure of risk, the variance of the pure premium was selected as more appropriate:

• It satisfied Freifelder's 3 basic axioms.
• It has important theoretical advantages, as discussed by Bühlmann.
• It permits the development of risk adjusted ILFs from the severity distribution alone.

The formula for premium (excluding expenses) with a safety/contingency loading proportional to risk is:

(a) Risk Adjusted Pure Premium = E[y] + l*Var[y], where E[y] and Var[y] are the pure premium and variance of the pure premium, respectively. l is selected judgmentally.

(b) Var[y] = E[n] * Var[g(x)] + Var[n] * (E[g(x)])2 = E[n] *E[g(x)2] + (Var(n) - E(n)) * E[g(x)]2. (assuming independence)

(c) Although Var(n) is usually > E(n), a minimum variance can be set:

Var[y] = E[n]*E[g(x)²] = E[n]* {Var(X) + (E[X])²} (eq. 29).

(d) The second moment, for the cost function for policies limited by k, is:

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