Lecture 2 of 4-part series. Spring School on Risk Management, Insurance and Finance European University at St. Petersburg, Russia.

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

2 Outline 1 Capital Fair Wang s 2 Capital 3 Fair 4 Wang s 5 page 2

3 Capital Fair Wang s Alternative definitions of economic The Specialty Guide on Economic Capital, page 7, prepared by the SOA Risk Task Force (RMTF) headed by Hubert Mueller in 2003, offers the following alternative definitions of economic : Definition #1 Economic is defined as sufficient surplus to meet potential negative cash flows and reductions in value of assets or increases in value of liabilities at a given level of risk tolerance, over a specified time horizon. Definition #2 Economic is defined as the excess of the market value of the assets over the fair value of liabilities required to ensure that obligations can be satisfied at a given level of risk tolerance, over a specified time horizon. Definition #3 Economic is defined as sufficient surplus to maintain solvency at a given level of risk tolerance, over a specified time horizon. page 3

4 Capital Fair Wang s The purpose of economic Calculating the economic for a firm has its many purposes (this list not all encompassing): understand the company s overall risk profile budgeting pricing purposes company asset and liability management understanding company s risk tolerances and possible constraints performance measurement and incentive compensation regulatory and other external parties (e.g. investors, ratings agencies) Many of these can be found in the Specialty Guide; a lot of development since its publication. page 4

5 Capital and product pricing For simplicity, consider an insurer faced with pricing a risk Z, only for a given short horizon, and assume the company has risk tolerance described by an exponential utility function: Capital Fair Wang s u(x) = 1 a( 1 e ax ), for a > 0. It is rather straightforward to show that the smallest amount of premium this company is willing to accept in exchange of facing the risk can be expressed as: P = 1 a log E[ e az ] Now, suppose the company is faced with the task of pricing a portfolio of n of these identical risks, say X 1, X 2,..., X n so that the aggregate risk is S = X 1 + X X n page 5

6 - continued The total (minimum) premium the company needs to collect is P = 1 a log E[ e a n j=1 X j ], for which we can express as, assuming independent risks: Capital Fair Wang s P = 1 a log n j=1 E [ e ax ] n 1 j = a log E[ e ax ] j. j=1 Each risk then can be assessed a premium of P j = 1 a log E[ e ax j ], for j = 1, 2,..., n However, if we relax the assumption of independence: P = 1 a log n j=1 E [ e ax ] n 1 j a log E[ e ax ] j = P1 + P P n j=1 page 6 Some effect of diversification!!!

7 The problem Capital Fair Wang s Consider a portfolio of n individual losses X 1,..., X n during some well-defined reference period. Assume these random losses have a dependency structure characterized by the joint distribution of the random vector (X 1,..., X n ). The aggregate loss is the sum S = n i=1 X i, or in some instances, it could be a weighted sum S = n i=1 w ix i. Assume company holds aggregate level of K which may be determined from a risk measure ρ such that K = ρ(s) R. Here the (economic) is the smallest amount the company must set aside to withstand aggregated losses at an acceptable level, and tolerance. page 7

8 - continued Capital Fair Wang s The company now wishes to allocate K across its various business units. determine non-negative real numbers K 1,..., K n satisfying: n K i = K. This requirement is referred to as the full requirement. We will see that this requirement is a constraint in our optimization problem (in the optimal paper). i=1 page 8

9 Capital Fair Wang s The literature There are countless number of ways to allocate aggregate. Good overview of methods: Cummins (2000); Venter (2004) Some methods based on decision making tools: Cummins (2000)- RAROC, EVA Lemaire (1984); Denault (2001) - game theory Tasche (2004) - marginal costs Kim and Hardy (2008) - solvency exchange option with limited liability This list needs some updating. page 9

10 - continued Some methods based on risk measures/distributions: Panjer (2001) - TVaR, multivariate normal Landsman and Valdez (2003) - TVaR, multivariate elliptical Capital Fair Wang s Dhaene, et al. (2008) - TVaR, lognormal Valdez and Chernih (2003) - covariance-based, multivariate elliptical Tsanakas (2004, 2008) - distortion risk measures, convex risk measures Furman and Zitikis (2008) - weighted risk s Methods also based on optimization principle: Dhaene, Goovaerts and Kaas (2003); Laeven and Goovaerts (2004); Zaks, Frostig and Levikson (2006) page 10

11 Some notation Denote by X T = (X 1, X 2,..., X n ) the vector of losses, with each entry denoting the loss for the applicable line of business. Define an A to be a mapping A : X T R n such that Capital Fair Wang s A ( X T ) = ( K 1, K 2,..., K n ), where the full is satisfied: K = ρ[z ] = Each component K i of the is viewed as the i-th line of business contribution to the total company. Because must also reflect the fact that each line operates in the presence of the other lines, the notation K i = A ( X i X 1,..., X n ) n i=1 K i may be well suited for this purpose. page 11

12 Capital Fair Wang s for a fair Let N = {1, 2,..., n} be the set of the first n positive integers. No undercut: For any subset M N, we have ] X i i M A( X i X 1,..., X n) ρ[ i M Symmetry: Let N = N {i 1, i 2 }. If M N (strict subset) with M = m, X T m = ( X j1,..., X jm ) and if A ( X i1 X T m, X i1, X i2 ) = A ( Xi2 X T m, X i1, X i2 ) for every M N, then we must have K i1 = K i2. Consistency: For any subset M N with M = m, let X T n m = ( X j1,..., X jn m ) for all jk N M where k = 1, 2,..., n m. Then we have i M A( X i X 1,..., X n) = A( i M X i X i, X T n m i M ). page 12

13 Capital Fair Wang s page 13 Intuitive interpretations The no undercut criterion: recognizes the benefits of diversification risk allocated to a business unit may not exceed the risk allocated if these were offered on a stand-alone basis analogous to sub-additivity for a coherent risk measure The symmetry criterion: states that two lines of business that equally contribute to the risk within the firm must have equal must therefore recognize only the level or degree of contribution to risk within the firm, and nothing else sometimes called equitability property The consistency criterion: ensures that a unit s cannot depend on the level at which the occurs is independent of the hierarchical structure of the firm

14 Relative or proportional method Capital Fair Although quite a popular method, partly because of its simplicity, the relative formula according to A ( X i X 1,..., X n ) = ρ[x i ] ρ[x 1 ] + + ρ[x n ] ρ[s] violates all three possible criteria for a fair. See Valdez and Chernih (2003) for proofs. Wang s page 14

15 Capital Fair Wang s The covariance method The covariance formula is based on A ( X i X 1,..., X n ) = λi ρ[s], λ T = (λ 1,..., λ n ) denotes a vector of weights that adds up to one so that full is satisfied. To determine these weights λ i, one approach is to minimize the following quadratic loss function: [ ((X E µ) λ(s µs ) ) T ( W (X µ) λ(s µs ) )] where the weight-matrix W is assumed to be positive definite. Optimization results to the following unique values of: λ i = E[ (X i µ i )(S µ S ) ] E [ (S µ S ) 2] = Cov[X i, S], Var[S] for i = 1,..., n. page 15

16 - continued Capital Fair Wang s The formula based on the covariance principle satisfies the three criterion or properties of a possible fair : no undercut, symmetry, and consistency. See Valdez and Chernih (2003) for proofs. page 16

17 Wang s Preserving the notation used by Wang (2002), define and denote the expectation of X i,q by Capital Fair Wang s H λ [X i, S] = E[X i,q ] = E[X exp(λs)] E[exp(λS] and the expectation of the aggregate loss Z Q by H λ [S, S] = E[S Q ] = E[S exp(λs)]. E[exp(λS] Note that this last equation exactly gives the Esscher Transform of S. The price of a random payment X i traded in the market is H λ [X i, S] so that one can think of the difference ρ[x i ] = E[X i,q ] E[X i ] = H λ [X i, S] E[X i ] as the risk premium. page 17

18 Capital Fair Wang s - continued For the aggregate loss S, its risk premium is given by [ n ] ρ[s] = ρ X i = H λ [S, S] E[S]. i=1 It is rather straightforward to show that ρ[x i ] = Cov[X i, exp(λs] E[exp(λS)] and ρ[s] = Cov[S, exp(λs]. E[exp(λS)] Wang (2002) proposed computing the of to individual business unit i based on the following formula: K i = H λ [X i, S] E[X i ]. Assuming an aggregate of K, the parameter λ can be computed by solving then K = H λ [S, S] E[S]. page 18 It is not difficult to show that the full requirement is met in this case: K = n i=1 K i.

19 Capital Fair Wang s Special case: multivariate normal In the special case where the vector of losses (X 1,..., X n ) follows a multivariate normal distribution, we have that the Wang s method reduces to the covariance method. Some straightforward calculation yields the results: ( E[S exp(λs)] = exp λµ S + 1 ) 2 λ2 σs 2 (µ + λσs 2 ) and ( E[X i exp(λs)] = exp λµ S + 1 ) 2 λ2 σs 2 (µ i + λσ i,s ) Then it follows that K = λσ 2 S and K i = λσ i,s which clearly is equivalent to the covariance method. page 19

20 Capital Fair Wang s Dhaene, J., Tsanakas, A., Valdez, E.A., and S. Vanduffel (2012). Optimal principles. Journal of Risk and Insurance, 79(1), Gerber, H.U. (1979). An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph, Wharton School, University of Pennsylvania. Valdez, E.A. and A. Chernih (2003). Wang s formula for elliptically contoured distributions. Insurance: Mathematics and Economics, 33, Wang, S. (2002). A set of new methods and tools for enterprise risk management and portfolio optimization. Working paper. SCOR Reinsurance Company. page 20

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