Capital allocation: a guided tour

Transcription

1 Capital allocation: a guided tour Andreas Tsanakas Cass Business School, City University London K. U. Leuven, 21 November 2013

2 2 Motivation What does it mean to allocate capital? A notional exercise Is anything physically allocated? Why allocate capital? Performance measurement Sensitivity analysis How to allocate capital? Marginal v stand-alone capital Capital endogenous v exogenous

3 3 Table of Contents Stand-alone v marginal allocations Performance measurement Sensitivity analysis Optimal capital allocations

4 4 Table of Contents Stand-alone v marginal allocations Performance measurement Sensitivity analysis Optimal capital allocations

5 5 The basic set-up Additive portfolio of risks (losses) S = X X d Total available capital K Capital allocation problem: How to calculate K 1,..., K d such that K = K K d and K i is some way reflects the risk of X i. No full references given here See Dhaene et al. (2012) for literature review

6 6 Risk measures A risk measure ρ maps a loss X to the real line ρ : X ρ(x) Examples Expected value, VaR, TVaR, standard deviation... Uses capital setting, premium calculation, valuation... For details see Denuit et al. (2005)

7 Weighted risk measures Representation of risk measures considered in practice (Furman and Zitikis, 2008, Dhaene et al., 2012): ρ(x) = E ( Xh(X) ) where h is a weight function, such that E(h(X)) = 1 Risk is weighed expected loss, where more emphasis is placed on scenarios of interest h is usually (but not always) increasing

8 Risk measures and their weight functions Standard 1 + a X E( X ) deviation TVaR h(x) ρ(x) = E ( Xh(X) ) σ(x) E ( X ) + aσ ( X ) 1 1 p I {X>F 1 X (p)} E ( X X > F 1 X (p)) Distortion φ (F X (X)) E ( Xφ (F X (X)) ) Exponential 1 0 e γax E ( e γax ] dγ 1 a ln E ( e ax)

9 Capital given by a risk measure Let capital for the portfolio be given endogenously by K = ρ(s) = E ( Sh(S) ) d d ρ(x j ) = E(X j h(x j )) K j=1 j=1 Stand-alone capital allocation ρ(x i ) K i = ρ(s) d j=1 ρ(x j) Marginal capital allocation K i = ρ(x i S) = E ( X i h(s) )

10 10 Stand-alone v marginal allocation Stand-alone Reflects risk of each individual X i Diversification credit allocated pro-rata [ ] ρ(s) ρ(x i ) K i = ρ(x i ) 1 d j=1 ρ(x j) Marginal Measures contribution under scenarios that drive aggregate risk Diversification credit reflects dependence of (X i, S) ρ(x i ) K i = E ( X i h(x i ) ) E ( X i h(s) )

11 11 Risk measures and their marginal allocations ρ(s) = E ( Sh(S) ) ρ(x i S) = E ( X i h(s) ) E(S) + aσ(s) E(X i ) + a Cov(X i,s) σ(s) E ( S S > F 1 S (p)) E ( X i S > F 1 S (p)) E (Sφ (F S (S))) 1 a ln E ( e as) E (X i φ (F S (S))) ( ) 1 e γax i E X i E ( )dγ e γax i 0

12 12 Comonotonicity The random variables X 1,, X d are comonotonic if (X 1,..., X d ) d = (h 1 (Z),..., h d (Z)), for non-decreasing functions h 1 (Z),..., h d (Z), see Dhaene et al. (2002) Value-at-Risk is additive for comonotonic risks F 1 S 1 (p) = FX 1 (p) + + F 1 X d (p) No diversification!

13 More on distortion risk measures Defined as (Wang, 1996; Acerbi, 2002) ρ(x) = 1 0 F 1 X (u)φ(u)du = E( Xφ(U X ) ), φ is a non-negative function on [0, 1] with 1 φ(u)du = 1 0 U X U[0, 1] such that F 1 X (U X) = Y Class includes VaR, TVaR, Proportional Hazards transform, CoC valuation principles...

14 Distortion risk measures and comonotonic risks If X 1,..., X d are comonotonic, ρ(s) = ρ(x 1 ) + + ρ(x d ) It is then also true that ρ(x i S) = ρ(x i ) Same scenarios driving the risks of X i and S

15 15 Table of Contents Stand-alone v marginal allocations Performance measurement Sensitivity analysis Optimal capital allocations

16 16 Motivation One of the main uses of capital allocation is in measuring performance Given: portfolio Return-on-Capital (RoC) target Needed: profit target for individual lines What should be the target RoC for individual lines? What should be the allocated capital used for working out the RoC of a line? Euler capital allocations (Tasche, 2004) provide a neat solution.

17 7 Return on capital Simple RoC definition RoC(S) = π(s) E(S) ρ(s) E(S) π is a linear pricing functional Question: should we invest (a little bit) in some risk Z? Answer: Yes, if, for small ɛ > 0 RoC(S + ɛz) > RoC(S)

18 8 Signals for investment Define RoC(Z S) = π(z) E(Z) ρ(z S) E(Z) Then for standard deviation, VaR, TVaR, it is RoC(Z S) > RoC(S) = RoC(S + ɛz) > RoC(S) for some small ɛ Marginal allocations give the right signals for investment

19 Performance measurement For a given portfolio S = X X d : (i) Allocate capital K i = ρ(x i S) to X i (ii) Calculate return RoC(X i S) (iii) If RoC(X i S) > RoC(S), can increase (marginally) exposure to X i If RoC(X i S) < RoC(S), reduce (marginally) the exposure to X i or increase profits Profit target π(x i ) E(X i ) RoC(S) (ρ(x i S) E(X i ))

20 Caution! Large risks in the portfolio will be penalised elephant in the boat Small independent risks in the portfolio will get off lightly one mouse in the boat Small highly dependent risks in the portfolio will be penalised many mice in the same sack Potential for perverse incentives diversification masking bad performance

21 Example: allocation of 99%TVaR mean Stand-alone Marginal X1 X2 X3 X1 X2 X3

22 Loss Ratio Target Example: Loss Ratio targets for RoC=20% 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% X1 X2 X3 Stand-alone Marginal

23 21 Table of Contents Stand-alone v marginal allocations Performance measurement Sensitivity analysis Optimal capital allocations

24 22 Motivation Internal risk models are increasingly complex Large number of risk factors and parameters Non-linear effects, interactions... Different modelling teams Sensitivities to risk factors and parameters important but not always well understood What drives aggregate risk? What part of the model should we invest in? Management engagement and regulatory review Practical issues Sensitivity to risk factors and parameters not measured in consistent way Measuring sensitivity to parameters requires repeated model runs

25 Notation X is a random vector of risk factors, taking values in X R d. g : X R is the business structure Y = g(x) is the aggregate loss g(x 1,..., X i + ɛz,..., X d ) is the aggregate loss, with shock ɛz on X i.

26 4 Sensitivity measure Millossovich and Tsanakas (2013) define the sensitivity of ρ ( g(x) ) to the shock Z applied on X i as S i (Z; X, g) = ɛ ρ( g(x 1,..., X i + ɛz,..., X d ) ) ɛ=0. For Z = X i we refer to the sensitivity of ρ ( g(x) ) to X i and write S i (X, g) S i (X i ; X, g).

27 Calculating sensitivities g is differentiable with g = (g 1,..., g d ) Then, subject to technical conditions, it is S i (X, g) = E ( X i g i (X)φ(U Y ) ) If g is linear in X i with g i (X) = β i, we have S i (X, g) = β i E ( X i φ(u Y ) ) = β i ρ(x i Y ) Generalising capital allocation

28 Sensitivity of VaR measures Define as s i (p; X, g) the sensitivity to X i for VaR p : s i (p; X, g) E ( X i g i (X) U Y = p ). Sensitivities for more general distortions recovered by S i (X, g) = 1 0 s i (u; X, g)φ(u)du.

29 Calculating sensitivities Law of X and function g known in principle But internal model can in practice be a black box Work with Monte-Carlo sample from (X, Y = g(x)) Approximate g by a smooth function ĝ, based on simulated sample Gradients calculated numerically, by local-linear regression (Li and Racine, 2004) We use the NP package in R (Hayfield and Racine, 2008) Estimate sensitivity from the simulated sample Can use structural information in hierarchical models to simplify calculations

30 Example: Two LoB with RI default and inflation Total loss is C = X 3 (X 1 + X 2 ) Y = C (1 X 4 ) min{(c λ) +, l} where X 1 LogNormal, E(X 1 ) = 153, Var(X 1 ) = 44 2 ; X 2 Gamma, E(X 2 ) = 200, Var(X 1 ) = 10 2 ; X 3 LogNormal, E(X 3 ) = 1.05, Var(X 3 ) = ; X 4 Binomial(1, 0.1); X 1, X 2, X 3 are independent; C, X 4 have Gaussian copula with r = 0.6. Conditional on C, X 4 is independent of X 1, X 2, X 3 ; λ = 380, l = 30.

31 29 Business structure ĝ(x), x 3 = 1.05, x 4 = 0

32 30 ŝ i (u; X, g)

33 31 TVaR risk measure: φ(u) = 1 1 p1{u > p} zeta(u) p=0.99 p= u

34 32 Exponential distortion (ED): φ(u) = λ exp(λu) exp(λ) 1 zeta(u) lambda=20 lambda= u

35 33 Scaled sensitivities S i (g, X)/ρ(Y ) TVaR TVaR ED ED (p=0.99) (p=0.95) (γ=20) (γ=5) X X X X

36 Sensitivity to business structure parameters Business structure parameters (e.g. reinsurance deductible) so far suppressed We view such parameters as degenerate risk factors Can still work out their sensitivity Rate of change in risk measure under proportional stress in the parameter One extra step required for estimating the gradient

37 35 Sensitivity to deductible

38 36 Further extensions Sensitivity to statistical parameters Choose distribution of parameter Simulate with parameter uncertainty Express X i as function of random parameter and process uncertainty Work out sensitivities Sensitivity to drivers of dependence Drivers can be explicit or implicit (e.g. in copula models) Express X i as function of common and idiosyncratic factors Work out sensitivities More examples and implementation details in Millossovich and Tsanakas (2013)

39 37 Table of Contents Stand-alone v marginal allocations Performance measurement Sensitivity analysis Optimal capital allocations

40 Motivation Capital K is often exogenously given Not given by a risk measure Capital allocation is an answer to a question But what was the question? Return to first principles: Allocated capital should match the risk as close as possible, subject to risk preferences

41 9 An optimal capital allocation problem (Dhaene et al., 2012) Given the aggregate capital K > 0, determine the allocated capitals K i, i = 1,..., d, as the solution of the optimisation problem: min K 1,...,K n [ ] d (X j K j ) 2 E ζ j, such that v j j=1 d K j = K. j=1 v i > 0 such that d j=1 v j = 1 ζ i 0 is a random variable such that E(ζ i ) = 1

42 General solution Obtained in Dhaene et al. (2012): ) d K i = E(ζ i X i ) + v i (K E(ζ j X j ) j=1 Risk of X i plus proportional share of the excess capital Special case: v i = E(ζ i X i ) d j=1 E(ζ jx j ) = K i = K E(ζ ix i ) d j=1 E(ζ jx j ) Proportional allocation

43 Stand-alone and marginal allocations Let ζ i depend on X i ζ i = h i (X i ) E(X i ζ i ) = ρ i (X i ) = ) d K i = ρ i (X i ) + v i (K ρ j (X j ) j=1 Let now ζ i depend on S for all i ζ i = h(s) E(X i ζ i ) = ρ(x i S) = K i = ρ(x i S) + v i (K ρ(s))

44 2 Market-driven allocation Pricing functional with state-price deflator ζ M π(s) = E(ζ M S), π(x i ) = E(ζ M X i ) Portfolio solvency ratio (Sherris, 2006) K π(s) π(s) Let ζ i = ζ M and v i = π(x i) π(s) K i π(x i ) π(x i ) = K π(s) π(s)

45 3 Allocation with the default option Scenario weights correspond to portfolio default events I(S > K) ζ i = P[S > K] E(ζ i X i ) = E(X i S > K) Optimal allocation satisfies E[(X i K i ) I(S > K)] = v i E[(S K) + ] Allocation of the default option value (Myers and Read, 2001)

46 44 Extensions Many more examples in Dhaene et al. (2012) Percentile-based allocations for absolute deviations Generalisation and explicit solution of the optimisation problem in Cheung et al. (2013) Capital invested in risky assets (Zaks, 2013) Hierarchical setting (Zaks and Tsanakas, 2013) Insurance group consisting of d legal entities, each writing d i lines of business Insurance company in d lines of business, with d i policies in each Discrepant risk preferences within the organisation

47 5 Hierarchical optimal allocations setup Portfolio and sub-portfolios S = X X d X i = X i1 + + X idi Top- and bottom-level capital K = K K d K i = k i1 + + k idi Preferences represented by weighting variables ζ 1,..., ζ d ζ i1,..., ζ idi

48 Hierarchical allocations optimisation problem min K,k { (1 λ) d 1 E [ ζ i (K i X i ) 2] v i i=1 d d i 1 +λ E [ } ζ ij (k ij X ij ) 2] v ij i=1 j=1 s.t. d K i = K, i=1 d i j=1 v ij = v i d i j=1 k ij = K i

49 Hierarchical allocations solution Full solution stated in Zaks and Tsanakas (2013) Simple interpretation as compromise solution Two stage process: Bottom-up Calculate optimal K 1,..., K d using averaged preferences Top-down Given K i, work out optimal k i1,..., k idi as before

50 Example Company with d lines, each with d i policies Board preferences ζ i = h(s) K = E(ζ i S) = ρ(s) E(ζ i X i ) = ρ(x i S) Line manager preferences ζ ij = h i (X i ) E(ζ ij X i ) = ρ i (X i ) E(ζ ij X ij ) = ρ i (X ij X i )

51 49 Example Optimal solution K i = (1 λ)ρ(x i S) + λρ i (X i ) ( ) + λ v d i ρ(s) ρ r (X r ) v k ij = ρ(x ij X i ) + v ij v i r=1 ( ) d i K i ρ s (X is X i ) s=1

52 9 C. Acerbi. Spectral measures of risk: a coherent representation of subjective risk aversion. Journal of Banking and Finance, 26(7): , K. C. Cheung, J. Dhaene, Y. Rong, and P. S. C. Yam. Explicit solutions for a general class of optimal allocation problems, URL http: // M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial Theory for Depedent Risks. Wiley, J. Dhaene, M. Denuit, M.J. Goovaerts, R. Kaas, and D. Vyncke. The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics, 31:3 33, J. Dhaene, S. Vanduffel, Q. Tang, M.J. Goovaerts, R. Kaas, and D. Vyncke. Risk measures and comonotonicity: a review. Stochastic Models, 22(4): , 2006.

53 49 J. Dhaene, A. Tsanakas, E. A. Valdez, and S. Vanduffel. Optimal capital allocation principles. Journal of Risk and Insurance, 79(1):1 28, E. Furman and R. Zitikis. Weighted premium calculation principles. Insurance: Mathematics and Economics, 42(1): , T. Hayfield and J. S. Racine. Nonparametric econometrics: The np package. Journal of Statistical Software, 27(5), Q. Li and J. Racine. Cross-validated local linear nonparametric regression. Statistica Sinica, 14: , P. Millossovich and A. Tsanakas. Sensitivity analysis of internal risk models, URL com/sol3/papers.cfm?abstract_id= S.C. Myers and J.A. Read. Capital allocation for insurance companies. Journal of Risk and Insurance, 68(4): , 2001.

54 49 M. Sherris. Solvency, capital allocation and fair rate of return in insurance. Journal of Risk and Insurance, 73(1):71 96, D. Tasche. Allocating portfolio economic capital to sub-portfolios. In A. Dev (ed.), Economic Capital: A Practitioner s Guide, Risk Books, pp , S.S. Wang. Premium calculation by transforming the premium layer density. ASTIN Bulletin, 26(1):71 92, Y. Zaks. The optimal asset and liability portfolio for a financial institution with multiple lines of business. European Actuarial Journal, 3(1):69 95, Y. Zaks and A. Tsanakas. Optimal capital allocation in a hierarchical corporate structure, URL abstract_id=