# 20.2 Single-period Criterion and the Rate of Return on Capital

In this section the problem of joint decisions on premium and required capital is considered in terms of shareholder's choice of the level of expected rate of return and risk. It is assumed that typically the single-year loss (when it happens) is covered by the insurance company through reduction of its own assets. This assumption can be justified by the fact that in most developed countries state supervision agencies efficiently prevent companies to undertake too risky insurance business without own assets being large enough. As shareholders are unable to externalize the loss, they are enforced to balance the required expected rate of return with the possible size of the loss. The risk based capital concept (RBC) formalizes the assumption that premium loading results from the required expected rate of return on capital invested by shareholders and the admitted level of risk.

## 20.2.1 Risk Based Capital Concept

Let us denote by the amount of capital backing risk borne by the insurance portfolio. It is assumed that the capital has a form of assets invested in securities. Shareholders will accept risk borne by the insurance portfolio provided it yields expected rate of return larger than the rate of return on riskless investments offered by the financial market. Let us denote by the required expected rate of return, and by the riskless rate. The following equality holds:

 (20.1)

For simplicity it is assumed that all assets are invested in riskless securities. This means that we neglect shareholder's capital locked-up in fixed assets necessary to run the insurance operations of the company, and we also assume prudent investment policy, at least with respect to those assets, which are devoted for backing the insurance risk. It is also assumed that all amounts are expressed in terms of their value at the end of year (accumulated when spent or received earlier, discounted when spent or received after the year end).

Let us also assume that company management is convinced that the rate of return is large enough to admit the risk of technical loss in amount, let us say, , with a presumed small probability . The total loss of capital amounts then to . The assumption could be expressed in the following form:

 (20.2)

where denotes the cdf of random variable .

Combining equations (20.1) and (20.2), one obtains the desired amount of capital backing risk of the insurance portfolio:

 (20.3)

 (20.4)

In both formulas, only the difference is relevant. We denote it by . The obtained premium formula is just a simple generalization of the well-known quantile formula based on the one-year loss criterion. This standard formula is obtained by replacing the coefficient by one. Now it is clear that the standard formula could be interpreted as a result of the assumption , so that shareholders are not ready to suffer a technical loss at all (at least with probability higher than ).

## 20.2.2 How to Choose Parameter Values?

Parameters , , and of the formula are subject to managerial decision. However, an actuary could help reducing the number of redundant decision parameters. This is because parameters reflect not only subjective factors (shareholder's attitude to risk), but also objective factors (rate of substitution between expected return and risk offered by the capital market). The latter could be deduced from capital market quotations. In terms of the Capital Asset Pricing Model (CAPM), the relationship between expectation and standard deviation of the excess of the rate of return over the riskless rate is reflected by the so-called capital market line (CML). The slope coefficient of the CML represents just a risk premium (in terms of an increase in expectation) per unit increase of standard deviation. Let us denote the reciprocal of the slope coefficient by . We will now consider shareholder's choice between two alternatives: investment of the amount RBC in a well diversified portfolio of equities and bonds versus investment in the insurance company's capital. In the second case the total loss exceeds the amount with probability  . The equally probable loss in the first case equals:

where denotes the quantile of order of the standard normal variable. This is justified by the fact that the CAPM is based on assumption of normality of fluctuations of rates of return. The shareholder is indifferent when the following equation holds:

provided that expected rates of return in both cases are the same: . Making use of our knowledge of the substitution rate and putting the above results together we obtain: .

In the real world the required rate of return could depart (ceteris paribus) from the above equation. On the one hand, required expected rate of return could be larger, because direct investments in strategic portions of the insurance company capital are not as liquid as investments in securities traded on the stock exchange. On the other hand, there is empirical evidence that fluctuations in profits in the insurance industry are uncorrelated with the business cycle. This means that having a portion of insurance company shares in the portfolio improves diversification of risk to which a portfolio investor is exposed. Hence, there are reasons to require smaller risk premium.

The reasonable range of the parameter is from to . The rate of return depends on shareholder's attitude to risk and market conditions, but it is customary to assume that the range of the risk premium is from to . A reference point for setting the parameter could also be deduced from regulatory requirements, as the situation when the capital falls below the solvency margin needs undertaking troublesome actions enforced by supervision authority that could be harmful for company managers. A good summary of the CAPM and related models is given in Panjer et al. (1998), Chapters 4 and 8.