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