Pricing risks when standard deviation principle is applied f or the portfolio

1. Pricing riskswhen standard deviation principleis applied for the portfolio Wojciech Otto Department of Economics University of Warsaw 00-241 Warszawa, Długa Str. 44/50, Poland wotto@wne.uw.edu.pl

2. Top-down approach to pricing risks • At first the premium formulafor the whole portfolio is set on the basis of risk and return considerations on the level of the whole company • Next the premium formula for individual risks has to be derived by considering the contribution of an individual risk to the aggregate risk of the whole portfolio The problem arises when pricing criteria applied on the company level lead to non-additive premium formula.

3. Standard deviation principleapplied for the whole portfolio Necessary assumption: • aggregate amount of claims over a year is approximately normal • Alternative additional assumptions: • one-year possible loss criterion • predetermined level of the probability of ruin in the long run, simultaneousdecisions on premium and capital required to back the risk (Bühlmann 1985)

4. Marginal premium: concept • independentindividual risks: • aggregate amount of claims for the whole portfolio • whole portfolio premium • the price at which the insurer is indifferent whether to accepta risk X or not (marginal premium) • in our case:

5. Marginal premium: result After transformation: we obtain the approximation, that for reads: However: the sum of marginal premiums suffices to cover a half of the required safety loading only, leaving the remaining half uncovered:

6. Balancing problem: ad hoc solutions • Doubling the marginal contribution (seems reasonable) • Alternatives (seem much more arbitrary) where and denote cumulants of order k of the additional risk X and the basic portfolio W The choice requires justification

7. Borch proposal: Shapley value Under the particular ordering of risks the additional risk X is priced as if the first j risks were already insured. The corresponding marginal premium formula reads: Borch/Shapley solution is the expectation of the above price when each of (n+1)! orderings is equiprobable Problem: • Borch solution is suited for the case when n is small (few companies negotiate pooling their portfolios) • Solution not feasible when n is large (number of individual risks in the portfolio to be priced)

8. Approximation Assuming that the share of allrisks preceding the riskX in the randomly drawn ordering in the variance of the portfoliois uniformly distributed over the unit interval: we come to the elegant and simple result: Denoting by c, and assuming that we obtain the result that justifies the choice of the basic premium formula:

9. The convergence theorem: assumptions • is a basic set of elements, • is a function that assigns the real nonnegative number to each element of the basic set , such that • Mdenotes the maximum out of these numbers • is a basic set E supplemented by the special element • variable U is defined as a sum of assigned to these elements that precede the special element for a given ordering of elements of the set • The probability function defined on the set of all (n+1)! orderings of elements of the set assigns to each of them the same probability1/(n+1)!

10. The convergence theorem: formulation Under the assumptions A.-F. the cdf of U can be bounded from both sides: • Where on the interval • Whereas for negative u and both bounds coincide, and are equal to zero and one, respectively The bounds stated by the theorem cannot be tightened unless we impose additional restrictions on the sequence

11. upper bound FU lower bound 1 u M 1 The case when bounds are binding The worse is the case of n risks of equal size, i.e. when: so that:

12. Bounding the risk premium As cdf’s , , and are stochastically ordered, and the function: is decreasing on the unit interval, we can bound the premium loading for the risk X that is added to the basic portfolio W as well: Bounds for the loading divided by the desired value are:

13. When bounds work, and when they don’t • Fixed c and reflect the scenario when we price a large risk on the background of the portfolio of numerous small risks.Both bounds tend to the same function that for c reasonably small is close to one • and reflects the scenario when the priced risk X is comparable to the largest risk. Also in this case both bounds tend to one • The problem arises when we allow for some large risks in the portfolio and try to price risks that are incomparably smaller (Mfixed and ) Then the lower bound is still acceptable, but the upper bound tends to infinity, that is no more acceptable

14. Mixed games:few atoms and the ocean Notations: • S - the share of n atoms (limited number of large risks) in the variance of the whole portfolio, so that • (1-S) - the share of “ocean” (very large number of very small risks) in the variance of the whole portfolio • - the r.v. that equals 1 when the atom number j precedes the elementin the randomly drawn ordering, and zero otherwise • , - column vectors of and • l – the n-element column vector of ones • - the support for the random vector Resulting expressions: • - the number of atoms preceding the element • - the share of these atoms in the variance of the portfolio

15. Mixed games:distribution of r.v. U The process of drawing randomly an ordering of risks can be reconstructed as the two-stage experiment: • at the first stage the element is randomly located among small risks (ocean), with the resulting share of preceding elements equal V (uniformly distributed over the unit interval) • at the second stage “atoms” are independently located in the same manner, so that each atom precedeswith probability V Assuming (for simplicity) that numbers are such that: We can express the conditional probability function of U given V as: And the expectation of a function g of U can be obtained as:

16. The general loading formula The loading formula is given by: that for small c is approximated by: As the postulated formula for a loading is , the question is whether is close enough to one: The general formula is not practical for the case when the number of atoms is more than a few. Thus only some special cases are analysed in more details.

17. The case of one atom and the ocean In this case we have S=M, and the general loading formulareads: Simple calculations lead to the result: The result for equals 1.017, and even for M as large as 25% is still moderate and equals 1.066.

18. n atoms of the same size and the ocean In this case we have S=Mn, and the general loading formulareads: Despite the simplification, calculations (presented in the paper) are still quite complex. However, general conclusions drawn are as follows: • For fixed S (the overall size of atoms in the game), as n increases, the ratio of the Shapley value to the loading proportional to the variance for the “ocean” decreases, • And converges to one as

19. General conclusions • When the standard deviation principle is used to set the portfolio premium • Then the variance principle (obtained by doubling the marginal contribution of the individual risk in the portfolio loading) can be justified as an approximation to the Shapley value • The approximation is accurate provided the portfolio is in a way balanced – largest risks cannot be too large • However, the same conditions are required to ensure that the distribution of the aggregate amount of claims of the whole portfolio is approximately normal • Accuracy of the normal approximation is needed in turn to justify using the standard deviation principle for the portfolio