Lecture 3 of 4-part series. Spring School on Risk Management, Insurance and Finance European University at St. Petersburg, Russia.

Transcription

1 Principles and Lecture 3 of 4-part series Spring School on Risk, Insurance and Finance European University at St. Petersburg, Russia 2-4 April 2012 University of Connecticut, USA page 1

2 Outline page 2

3 Dhaene, J., Tsanakas, A., Valdez, E.A., and S. Vanduffel (2012). Optimal capital allocation principles. Journal of Risk and Insurance, 79(1), page 3

4 Our contribution to the literature We re-formulate the problem as minimum distance problem in the sense that the weighted sum of measure for the deviations of the business unit s losses from their respective capitals be minimized: essentially distances between K i and X i Takes then into account some important decision making allocation criteria such as: the purpose of the allocation allowing the risk manager to meet specific target objectives the manner in which the various segments interact, e.g. legal and/or organizational structure Solution to minimizing distance formula leads to several existing allocation methods. New allocation also emerge. page 4

5 Some important concepts already discussed Capital allocation why important? Risk measures economic capital calculation VaR and CTE or Tail-VaR Some popular allocation page 5

6 α mixed inverse distribution function For p (0, 1), we denote the Value-at-Risk (VaR) or quantile of X by F 1 (p) defined by: X F 1 X (p) = inf {x R F X (x) p}. We define the inverse distribution function F 1+ X (p) of X as F 1+ X (p) = sup {x R F X (x) p}. The α mixed inverse distribution function F 1(α) X F 1(α) X (p) = αf 1 1+ (p) + (1 α)f (p). X X of X is: It follows for any X and for all x with 0 < F X (x) < 1, there exists an α x [0, 1] such thatf 1(αx ) X (F X (x)) = x. page 6

7 Some familiar allocation methods Allocation method ρ[x i ] K i Haircut allocation (no known reference) F 1 (p) Xi K n j=1 F 1 (p) F 1 (p) Xi Xj Quantile allocation Dhaene et al. (2002) Covariance allocation Overbeck (2000) F 1(α) Xi (F S c (K )) F 1(α) (F S c (K )) Cov[X i, S] Xi K Var [S] Cov [X i, S] CTE allocation Acerbi and Tasche (2002), Dhaene et al. (2006) [ ] S E X i > F 1 S (p) K CTE p [S] E [ X i S > F 1 S (p) ] page 7

8 The optimal capital allocation problem We reformulate the allocation problem in terms of : Given the aggregate capital K > 0, we determine the allocated capitals K i, i = 1,..., n, from the following problem: min K 1,...,K n n j=1 [ v j E ζ j D such that the full allocation is met: n K j = K, j=1 ( )] Xj K j and where the v j s are non-negative real numbers such that n j=1 v j = 1, the ζ j are non-negative random variables such that E[ζ j ] = 1 and D is a non-negative function. v j page 8

9 The components of the Elaborating on the various elements of the problem: Distance measure: the function D( ) gives the deviations of the outcomes of the losses X j from their allocated capitals K j. squared-error or quadratic: D(x) = x 2 absolute deviation: D(x) = x Weights: the random variable ζ j provides a re-weighting of the different possible outcomes of these deviations. Exposure: the non-negative real number v j measures exposure of each business unit according to for example, revenue, premiums, etc. page 9

10 The case of the quadratic Consider the special case of quadratic where D(x) = x 2 so that the is expressed as [ n E min K 1,...,K n j=1 ζ j (X j K j ) 2 v j This optimal allocation problem has the following unique solution: n ) K i = E[ζ i X i ] + v i (K E[ζ j X j ], i = 1,..., n. j=1 ]. page 10

11 allocations Risk measure ζ i = h i (X i ) E[X i h i (X i )] Standard deviation principle Buhlmann (1970) Conditional tail expectation Overbeck (2000) Distortion risk measure Wang (1996), Acerbi (2002) Exponential principle Gerber (1974) Esscher principle Gerber (1981) 1 + a X i E[X i ] σ Xi, a 0 E[X i ] + aσ Xi 1 ( ) 1 p I X i > F 1 (p), p (0, 1) CTE Xi p [X i ] g ( F Xi (X i ) ), g : [0, 1] [0, 1], g > 0, g < E [ X i g ( F Xi (X i ) )] e γaxi E[e γaxi ] dγ, a > 0 1 a ln E [ e axi ] e axi E[e axi ], a > 0 E[X i e axi ] E[e axi ] page 11

12 driven allocations Reference ζ i = h(s) E[X i h(s)] Overbeck (2000) Overbeck (2000) Tsanakas (2004) Tsanakas (2008) Wang (2007) 1 + a S E[S] σ S, a 0 E[X i ] + a Cov[X i, S] σ S 1 ( ) 1 p I S > F 1 S (p), p (0, 1) E[X i S > F 1 S (p)] g (F S (S)), g : [0, 1] [0, 1], g > 0, g < e γas E[e γas ] dγ, a > 0 E [ E [ X i g (F S (S)) ] X i 1 e as E[e as ], a > 0 E[X i e as ] E[e as ] 0 ] e γas E[e γas ] dγ page 12

13 allocations Let ζ M be such that market-consistent values of the aggregate portfolio loss S and the business unit losses X i are given by π[s] = E[ζ M S] and π[x i ] = E[ζ M X i ]. To determine an optimal allocation over the different business units, we let ζ i = ζ M, i = 1,..., n, allowing the market to determine which states-of-the-world are to be regarded adverse. This yields: K i = π[x i ] + v i (K π[s]). Using market-consistent prices as volume measures v i = π[x i ]/π[s], we find K i = K π[s] π[x i], i = 1,..., n. Rearranging these expressions leads to K i π[x i ] π[x i ] = K π[s], i = 1,..., n. π[s] page 13

14 Allocation with respect to the default option An alternative choice for the weighting random variable is ζ i = I(S > K ), i = 1,..., n, P[S > K ] such that only those states-of-the-world that correspond to insolvency are considered. The allocation rule then becomes K i = E [X i S > K ] + v i (K E [S S > K ]). which can be rearranged as follows: E[(X i K i ) I(S > K )] = v i E[(S K ) + ], i = 1,..., n. Quantity E[(S K ) + ] represents the expected policyholder deficit. Marginal contribution of each business unit to the EV of the policyholder deficit is the same per unit of volume, and hence consistent with Myers and Read Jr. (2001). page 14

15 Additional items considered in the We also considered other criterion: absolute value deviation: D(x) = x combined quadratic and shortfall: D(x) = ((x) +)) 2 shortfall: D(x) = (x) + Shortfall is applicable in cases where insurance market guarantees payments out of a pooled fund contributed by all companies, e.g. Lloyd s. Such allocation can be posed as an problem leading to that have been considered by Lloyd s. [Note: views here are the authors own and do not necessarily reflect those of Lloyd s.] page 15

16 Illustrative case study - same as previous For purposes of showing illustrations, we consider an insurance company with five lines of business: auto insurance - property damage auto insurance - liability household or homeowners insurance professional liability other lines of business We measure loss on a per premium basis and denote the random variable by S for the entire company and X i for the i-th line of business, i = 1, 2, 3, 4, 5. Same model assumption as previously explored. For convenience and for illustrative purpose only, we set ζ i = 1 so that risk aversion is ignored. page 16

17 Results of allocation for different distance measure allocation based on the distance measure Line of business squared x 2 absolute x positive (x) + Auto (PD) Auto (liab) Household Prof liab Other Total page 17

18 Allocation by for different distance measures other prof liab household auto (liab) auto (PD) squared absolute positive page 18

19 Graph of densities - by lines of business density fxi(x) auto (PD) auto (liab) household prof liab other x page 19 * reproduced here again for convenience

20 Concluding remarks We re-examine existing allocation that are in use in practice and existing in the literature. We re-express the allocation issue as an problem. No single allocation formula may serve multiple purposes, but by expressing the problem as an problem it can serve us more insights. Each of the components in the can serve various purposes. This allocation methodology can lead to a wide variety of other allocation. page 20