# 20.3 The Top-down Approach to Individual Risks Pricing

As it has been pointed out in the introduction, some premium calculation formulas are additive for independent risks, and then the decomposition of the whole-portfolio premium into individual risks premiums is straightforward. However, sometimes a non-additive formula for pricing the whole portfolio is well justified, and then the decomposition is no more trivial. This is exactly the case of the RBC formula (and also other quantile-based formulas) derived in the previous section. This section is devoted to showing the range, interpretation and applications of some solutions to this problem.

## 20.3.1 Approximations of Quantiles

In the case of the RBC formula decomposition means answering the question what is the share of a particular risk in the demand for capital backing the portfolio risk that in turn entails the premium. In order to solve the problem one can make use of approximations of the quantile by the normal power expansions. The most general version used in practice of the normal power formula for the quantile of order of the variable reads:

where denotes expectation, standard deviation, skewness, and kurtosis of the variable and is the quantile of order of a  variable. Now the premium can be expressed by:

 (20.5)

where coefficients are simple functions of parameters , , , and the quantile of the standard normal variable. The above formula was proposed by Fisher and Cornish, see Hill and Davis (1968), so it will be referred to as FC20.5. The formula reduced by neglecting the last two components (by taking will be referred to as FC20.6:

 (20.6)

and the formula neglecting also the skewness component as normal approximation:

 (20.7)

More details on normal power approximation can be found in Kendall and Stuart (1977).

## 20.3.2 Marginal Cost Basis for Individual Risk Pricing

Premium for the individual risk could be set on the basis of marginal cost. This means that we look for such a price at which the insurer is indifferent whether to accept the risk or not. Calculation of the marginal cost can be based on standards of differential calculus. In order to do that, we should first write the formula explicitly in terms of a function of cumulants of first four orders:

This allows expressing the increment due to extend of the basic portfolio by additional risk in terms of linear approximation:

where , , , denote partial derivatives of the function calculated at the point . By virtue of additivity of cumulants for independent random variables we replace increments by cumulants of the additional risk . As a result the following formula is obtained:

First two components coincide with the result obtained when the whole premium is based on the normal approximation. Setting additionally we obtain the premium for the case when skewness of the portfolio in non-neglectible (making use of FC20.6 approximation), including last two components means we regard also portfolio kurtosis (approximation based on formula FC20.5).

## 20.3.3 Balancing Problem

For each component the problem of balancing the premium on the whole portfolio level arises. Should all risks composing the portfolio be charged their marginal premiums, the portfolio premium amounts to:

that is evidently underestimated by:

The last figure represents a diversification effect obtained by composing the portfolio of a large number of individual risks, which could be also treated as an example of positive returns to scale''.

Balancing correction made so as to preserve sensitivity of premium on cumulants of order 1, 3, and 4 leads to the formula for the basic premium:

Obviously, several alternative correction rules exist. For example, in the case of the kurtosis component any expression of the form:

satisfies the requirement of balancing the whole portfolio premium for arbitrary number . In fact, any particular choice is more or less arbitrary. Some common sense can be expressed by the requirement that a basic premium formula should not produce smaller figures than marginal formula for any risk in the portfolio. Of course this requirement is insufficient to point out a unique solution. Here, the balancing problem results from the lack of additivity of the formula, as it is a nonlinear function of cumulants.

## 20.3.4 A Solution for the Balancing Problem

It seems that only in the case of the variance component some more or less heuristic argument for the correction can be found. The essence of the basic premium for individual risks is that it is a basis of an open market offer. Once the cover is offered to the public, clients decide whether to buy the cover or not. Thus the price should not depend on how many risks out of the portfolio  have been insured before, and how many after the risk in question. Let us imagine a particular ordering of the basic set of risks amended by the additional risk in a form of a sequence . Given this ordering, the respective component of the marginal cost of risk takes the form:

We can now consider the expected value of this component, provided that each of orderings is equally probable (as it was proposed Shapley (1953)). However, calculations are much simpler if we assume that the share of the aggregated variance of all risks preceeding the risk in the total aggregate variance is a random variable uniformly distributed over the interval . The error of the simplification is neglectible as the share of each individual risk in the total variance is small. The result:

is exactly what we need to balance the premium on the portfolio level. The reader easily verifies that the analogous argumentation does not work any more in the case of components of higher orders of the premium formula.

## 20.3.5 Applications

Results presented in this section have three possible fields of application. The first is just passive premium calculation for the whole portfolio. In this respect several more accurate formulas exist, especially when our information on the distribution of the variable extends its first four cumulants.

The second application concerns pricing individual risks. In this respect it is hard to find a better approach (apart from that based on long-run solvability criteria, which is a matter of consideration in next sections), which consistently links the risk relevant to the company (on the whole portfolio level) with risk borne by an individual policy. Of course open market offer should be based on basic valuation , whereas the marginal cost valuation could serve as a lower bound for contracts negotiated individually.

The third field of applications opens when a portfolio, characterized by substantial skewness and kurtosis, is inspected in order to localize these risks (or groups of risks), that distort the distribution of the whole portfolio. Too high (noncompetitive) general premium level could be caused though by few influential risks. Such localization could help in decisions concerning underwriting limits and reinsurance program. Applying these measures could help normalize'' the distribution of the variable . Thus in the preliminary stage, when the basis for underwriting policy and reinsurance is considered, extended pricing formulas (involving higher order cumulants) should be used. Paradoxically, once the prudent underwriting and ceding policy has been elaborated, simple normal approximation suffices to price as well the portfolio as individual risks. Clearly, such prices concern only retained portions of risk, and should be complemented by reinsurance costs.