Pricing riskswhen standard deviation principle

is applied for the portfolio

Wojciech OttoDepartment of Economics

University of Warsaw00-241 Warszawa, Długa Str. 44/50, Poland

wotto@wne.uw.edu.pl

Top-down approach to pricing risks

• At first the premium formula for 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.

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, simultaneous decisions on premium and capital required to back the risk (Bühlmann 1985)

Marginal premium: concept

• independent individual risks:

• aggregate amount of claims for the whole portfolio

• whole portfolio premium

• the price at which the insurer is indifferent whether to accept a risk X or not (marginal premium)

• in our case:

nXXX ,...,, 21

nXXXW ...21

WWW )(

)()()( WXWXm

)()( WXWXm X

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:

W

X

WXW

WXWWXW

2

)(

22WX

2

2

2)(

W

XWXm X

n

jWjm XW

1 21

)()(

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

W

XXb X

2

)(

kW

kX

W

XWXb c

cX

,

,

2

2

22)(

kXc , kWc ,

Borch proposal: Shapley valueUnder the particular orderingof 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)

njj XXXXX ,...,,,,..., 11

221

2221 ......)( jXjXX

ApproximationAssuming that the share of all risks preceding the risk X in the

randomly drawn ordering in the variance of the portfolio is 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:

duuucduuu WWWX

1

0

1

0

222

uu

n

jj

PREjj

u

1

22

1,0Pr

duuu WWX 1

0

222

PREjj

PREjjX

222E

22 / WX

W

XWW cdu

uu

c

21

0

0c

The convergence theorem: assumptions

A. is a basic set of elements,B. is a function that assigns the real

nonnegative number to each element of the basic set , such that

C. M denotes the maximum out of these numbersD. is a basic set E supplemented by the

special elementE. 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

F. The probability function defined on the set of all (n+1)! orderings of elements of the set assigns to each of them the same probability 1/(n+1)!

neeeE ,...,, 21 1,0: Ey

)( jj eyy 1...1 nyy

** eEE *e

jy*e

*E

*E

The convergence theorem: formulationUnder 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

UF

1,0u

uFuFuF U

M

uuF

1

MMu

uF

11u

nyyy ,...,, 21

The case when bounds are binding

The worse is the case of n risks of equal size, i.e. when:

so that:

nM

yy n

1

...1

u

upper bound

FUlower bound

1

1

M

M

M

1

nyy ...1

nM

1

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:

FUFF

222: WXW uuuh

1,01,0

)(E uFduhUhuFduh

Mc

cccM

cUhE

W

1

1132

11)(2323

Mc

cccM

cUhE

W

1

1132

)(2323

cW

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 (M fixed and )

Then the lower bound is still acceptable, but the upper bound tends to infinity, that is no more acceptable

0M

23231 11 ccc

Mconstc 0M

0c

Mixed games: few atoms and the oceanNotations: 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

element in 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

j

nyyyS ...21

*e

n ...1 nyyy ...1 jyj

y

*el

nB 1,0:

Mixed games: distribution of r.v. UThe 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 precedes with 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:

*e

*e

lnl vvvVvSyU 11Pr B

****** yy nyyy ,...,, 21

dvvvvSygUg lnl

B

1)1()(E1

0

The general loading formulaThe 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.

1

)1(2

1 ?1

0

B

lnl

dvvSy

vv

UcUσW E

UcσW 2

1E

cσW 12E

U

The case of one atom and the ocean

In this case we have S=M, and the general loading formula reads:

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.

1

0

1

0 )1(2

)1(

)1(22

1E

vM

dvv

vMM

vdv

U

2

233223

32

31

)1(

)1(

2

1E

M

MMM

U

%10M

n atoms of the same size and the ocean

In this case we have S=Mn, and the general loading formula reads:

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

n

k nk

knk

dvvSS

vvk

n

U 0

1

0 )1(2

1

2

1E

n

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