This version: December 3, 2009

Transcription

1 Rethinking risk capital allocation in a RORAC framework Arne Buch a, Gregor Dorfleitner b,*, Maximilian Wimmer b a d-fine GmbH, Opernplatz 2, Frankfurt, Germany b Department of Finance, University of Regensburg, Regensburg, Germany This version: December 3, 2009 Abstract This paper considers the economic optimization problem of a firm with several subbusinesses striving for its optimal RORAC. An insightful example shows that the implementation of a classical gradient capital allocation can be suboptimal if division managers are allowed to venture into all business whose marginal RORAC exceeds the firm s RORAC. The marginal RORAC requirements are then refined by adding a risk correction term that takes into account the interdependencies of the risks of different lines of business. It is shown that this approach can guarantee that the optimal RORAC will be achieved eventually. JEL classification: C61; D81; D82; G21; G22 Keywords: Risk capital allocation; Gradient allocation principle; Coherent risk measures; Performance measurement; RORAC * Corresponding author. Tel.: ; fax: addresses: gregor.dorfleitner@wiwi.uni-regensburg.de (G. Dorfleitner), maximilian.wimmer@wiwi.uni-regensburg.de (M. Wimmer). 1

2 1. Introduction The allocation of risk capital in financial firms for the purpose of performance measurement and risk-return optimization is well established in theory as well as in practice. While the use of economic capital 1 and its decomposition into a sum of single contributions of sub-businesses has become a standard approach in many banks (see Rosen and Saunders, 2010) and insurance companies (see Myers and Read, 2001), the academic world is still discussing methodological aspects and to an extent, even the significance of this concept. There are several strands of literature which deal with risk capital allocation from various points of view. Most articles can be attributed to the mathematical finance context in which rigorous arguments and axiomatics are the main concern (e.g. Denault, 2001; Kalkbrener, 2005; Tasche, 2004; Buch and Dorfleitner, 2008). Another strand of literature has a definite insurance-linked perspective (e.g. Dhaene et al., 2003; Furman and Zitikis, 2008; Gatzert and Schmeiser, 2008) and seeks to explore the advantages of risk capital allocation for insurance companies. A third strand looks at risk capital allocation from a more financial economics point of view (e.g. Merton and Perold, 1993; Stoughton and Zechner, 2007) and is therefore more closely related to the question concerning why capital allocation is a sensible procedure from an economic perspective. In any case, a sound risk capital allocation framework requires at least two theoretical fundaments, namely a proper definition of a risk measure and an allocation principle. The combination of these two items yields a concrete allocation rule. In addition, several ad hoc allocation rules, like e.g. the covariance allocation rule 2, exist without explicit reference to 1 Throughout this paper we will use the terms risk capital and economic capital synonymously since the risk capital to be allocated by an internal procedure is actually economic capital. This economic capital is calculated by using a risk measure. 2 See e.g. Kalkbrener (2005), who also points out the shortcomings of this allocation rule. Urban et al. (2004) use the covariance principle for calculating relative weights of each segment independently of the overall portfolio risk measure. 2

3 the combination of a risk measure and an allocation principle. Much attention has recently been drawn to coherent risk measures (Artzner et al., 1999), which have several economically favourable properties, and to the Euler allocation principle (Tasche, 2008; Rosen and Saunders, 2010), sometimes also called the gradient allocation principle. The Euler allocation principle is well-suited for firms with homogeneous sub-businesses consisting of a continuum of single contracts whereas in the case of few large single contracts an incremental allocation (Merton and Perold, 1993) seems to be more appropriate, where the risk capital allocated to subbusinesses is derived from looking at the firm with and without the sub-business under consideration and allocating economic capital proportional to the difference in overall risk capital. While many contributions examine technical aspects of risk capital allocation in deep detail and very rigorously, the actual economic justification is mostly verbal. Typically it is stated that the allocation is necessary to control risks ex ante by assigning limits to individual business units and its necessity for performance measurement is emphasized. On the other hand, capital allocation is also subject to criticism. In fact, Gründl and Schmeiser (2007) argue that capital allocation is senseless at all and that firms should rather refrain from using it. Even if one does not want to follow this argument the question emerges concerning why the optimum amounts of every line of business are not more adequately directly optimized by the headquarters. This paper focuses on financial firms with different lines of business for which the managerial decision concerns whether to expand or reduce rather than to create newly or abandon completely. Therefore we base our considerations on the gradient allocation principle. We do not restrict ourselves, however, to certain specific risk measures or distributional assumptions. Our approach comprises banks and insurance companies, both of which are subject to risk capital allocation. In banks the economic capital to be allocated could cover credit risk and 3

4 market or interest rate risk (Alessandri and Drehmann, 2009; Breuer et al., 2009) or classically credit risk in a portfolio context (Rosen and Saunders, 2010) while in insurance companies risk capital could be allocated in different lines of insurance contracts (Urban et al., 2004). Here, we aim at economic justification of risk capital allocation with a rather mathematical finance argumentation which is well suited to the many rather axiomatic contributions on risk measures and economic capital in the literature. The contribution of Stoughton and Zechner (2007) is the first to actually consider an economic optimization problem. The authors show that if the firm as a whole pursues maximization of the economic value added it is consistent with allocating capital to the sub-businesses, that are characterized by private information of managers, and letting them maximize the economic value added, based on the allocated capital. However, due to the restriction to normally distributed risks and a very specific incremental value-at-risk allocation rule, that is basically identical with the covariance allocation, their results are only of limited usefulness for practical applications. To our knowledge there is no contribution which shows the necessity of capital allocation in a general setting without restricting the probability distribution of losses and the risk measure chosen, or which argues in such a setting that capital allocation could be sensible when pursuing a maximization problem. In this paper we fill the gap by developing a procedure concerning capital allocation that is designed to maximize the RORAC of a company. Our analysis is based on the work of Tasche (2004) who, however, is not able to achieve the goal of solving a maximization problem due to too simplistic assumptions. We assume the segment managers to have superior knowledge concerning the possible profits induced by segment reductions or expansion while the risk of the portfolio is calculated centrally by the headquarters. Based on this we question RORAC maximization utilizing naive risk capital allocation and develop a more sophisticated rule for RORAC maximization. 4

5 The remainder of this paper is structured as follows: Firstly, we introduce and clarify some notations in section 2. In section 3 we continue with an example showing how a classical risk allocation to sub-businesses can impede a company to attain its optimal RORAC. A refined approach is introduced in section 4 to guarantee that a segment s expansion does not lead to a reduction of a company s overall RORAC, and in section 5 the paper is concluded with an economic discussion of our findings. 2. Notation and preliminaries The risk of a position or a firm is related to the variability of its net worth over the next period. Let the fluctuations of the future be represented by the set X of integrable random variables on the probability space [Ω, A, P]. The risk associated with X X is then quantified by a risk measure ρ; Formally, this measure is a mapping of X into the real numbers R. Artzner et al. (1997) have introduced and justified four desirable axioms defining the coherence of a risk measure. In this spirit, Acerbi and Tasche (2002) show that a certain definition of the Expected Shortfall fulfills the coherence axioms even for discontinuous profit functions. Although coherent risk measures have attracted a lot of attention in the last decade, other classes of risk measures such as spectral risk measures are also discussed in the literature (cf. Acerbi, 2002; Adam et al., 2008). Each class comprises its own set of axioms. However, we do not restrict the risk measure ρ to a certain class; instead we will highlight in the remainder of this article whatever axioms are needed in the places in which they are actually required. More concretely, we consider a firm with n lines of business (subsequently called segments or sub-businesses). Let the vector u = (u 1,..., u n ) U R n symbolize the units of each line. We call u the portfolio. Let us represent the expected profit generated by a portfolio u by the profit function m(u) (with m : U R). The risk in the firm is represented by the pair of variables (X, u), where X = (X 1,..., X n ) 5

6 X n is referred to as the portfolio base. In contrast to Fischer (2003), who defines X k as the actual future period profit of asset k N := {1,..., n}, we interpret the random variable X k as the deviation of the actual future period profit of asset k from its expected future period profit (so E(X k ) = 0) and u k as the number of units held for that type of asset. The future period profit is equal to its expected profit plus its fluctuation, i.e. m(u) + X(u), where X(u) := k N u k X k. The risk capital of portfolio u is assumed to be ρ(m(u) + X(u)). If the risk measure is translation invariant then we have ρ(m(u) + X(u)) = ρ(x(u)) m(u). The term ρ(m(u) + X(u)) can be interpreted as economic capital, which according to Hull (2007) is defined as the amount of capital required to cover the risk of unexpected losses. This implies that the expected losses are already accounted for, here as a part of m(u). Further, we define the function ρ X : R n R by ρ X : u ρ(x(u)) for ease of notation. The variable ρ X (u) therefore represents the risk associated with portfolio u given portfolio base X. Additionally, for a given risk measure ρ and portfolio base X, we let P ρ X = {(X, u) u U} be the set of risky portfolios where U satisfies the following two conditions. Firstly, ρ X is differentiable at every u U. Secondly, we have ρ X (u) 0 for all u U. Next we define a return function linking the yield and the risk dimension of the firm. Definition 2.1. Let ρ be a risk measure, X be a portfolio base, and m be a profit function. Then the function r ρ X m : U R defined as r ρ X m : u m(u) ρ X (u) m(u) is called return function associated with m, X, and ρ. The return function r ρ X m can be interpreted as the return on risk adjusted capital (RORAC). It represents the ratio of the expected profit margin due to business to the extent given by u and the economic capital due to the corresponding risk of the portfolio u. 6

7 The natural optimization problem for the management of a shareholder-oriented financial firm would be to maximize r ρ X m (u) and therefore employ the limited equity capital in the most beneficial way. Clearly, if both the profit function and the risk function are known to the headquarters, it can directly optimize the firm s RORAC. In this case, both capital allocation, and division managers become superfluous. Hence, throughout this paper it is assumed that the headquarters can evaluate the risk of the whole portfolio. We regard this as realistic since risk modeling and risk calculations indeed take place at the headquarters. However, in general the return function m is unknown on the corporate level except at the present portfolio extent u. Additionally, we model the profit function as the sum of the sub-business profits, i.e. m(u 1,..., u n ) = k N m k (u k ). We can assume that the return m(u + ɛe k ), where e k denotes the kth unit vector, is known to the kth segment for at least some ɛ. This depicts the idea that the segments know much better than the headquarters which profit margins can be generated by signing additional business of the same kind due to their direct negotiations with the their business partners. The risk capital ρ X (u) of a firm is assumed to depend on the total portfolio fluctuations, but has to be fully allocated to the portfolio positions (X 1, u 1 ),..., (X n, u n ) in the current period. The rule for applying risk capital is called an allocation principle, and is defined as follows. Definition 2.2. Let the set of risky portfolios P be defined as a set of pairs (X, u), with X X n, u R n, and n N + = N\{0}. Given a risk measure ρ, an allocation principle on 7

8 P is defined as a mapping A ρ : P R n A ρ : (X, u) A ρ 1(X, u). A ρ n(x, u) such that k N A ρ k (X, u) = ρ X(u). The expression A ρ k (X, u)/u k =: a k (u) is called the per-unit risk contribution of position k. Definition 2.3. For a given risk measure ρ and portfolio base X, the mapping A ρ : P ρ X R n defined as A ρ : (X, u) A ρ (X, u) = u ρ X (u) is called the gradient allocation principle associated with ρ on PX. ρ Here, denotes the Hadamard or component-wise product. Thus, A ρ (X, u) is a vector. Evidently, according to Euler s theorem, if ρ is homogeneous, the gradient allocation principle A ρ is an allocation principle on P ρ X. Given the notion of a per-unit risk contribution one can define the marginal RORAC by m(u) u k. a k (u) m(u) u k With the above-mentioned additive structure of m the marginal RORAC is simplified to m k(u k ) a k (u) m k (u k). This ratio expresses the expected additional profits in relation to the additional risk capital for the additional business. It is natural to state that a business extension is useful for every segment k whenever its marginal RORAC is higher than the present RORAC. Based on this idea we define the notion of suitability for performance measurement of an allocation principle. This term was coined by Tasche (2004) and therefore we adhere to it. However, 8

9 it turns out below that it is not sufficient for an allocation principle simply to fulfill the following definition. Definition 2.4. Let U be a non-empty set in R n and r : U R, be some function on U. An allocation principle A ρ is called suitable for performance measurement with ρ X if there holds: 1. For all portfolios u U and for all differentiable profit functions m : U R with ρ X (u) 0 and k N the inequality m k(u k )ρ X (u) > m(u)a k (u) (1) implies that there is an ɛ > 0 such that for all τ (0, ɛ) we have r(u) < r(u + τe k ). 2. For all portfolios u U and for all differentiable profit functions m : U R with ρ X (u) 0 and k N the inequality m k(u k )ρ X (u) < m(u)a k (u) (2) implies that there is an ɛ > 0 such that for all τ (0, ɛ) we have r(u τe k ) > r(u). Remark 2.5. Note that if a k (u) m k(u k ) > 0 and r ρ X m (u) > 0, it is easy to verify that equation (1) is equivalent to m k(u k ) a k (u) m k (u k) > rρ X m (u) 9

10 and that equation (2) is equivalent to m k(u k ) a k (u) m k (u k) < rρ X m (u). The following theorem, which has been proofed for linear profit functions in Tasche (2004), links the concepts of suitability for performance measurement with the gradient allocation principle: Theorem 2.6. Let U be a non-empty set in R n and r ρ X m : U R be a return function on U. Let ρ X : U R be a function that is partially differentiable in U with continuous derivatives. Then A ρ is suitable for performance measurement with ρ X if and only if A ρ = A ρ. Proof. Sufficiency: We have r ρ X m (u) u k = (ρ X (u) m k (u k )) 2 ( ρ X (u)m k(u k ) m(u) ρ X(u) u k ). (3) If a k (u) = ρ X(u) u k then ρ X (u)m k(u k ) > m(u)a k (u) implies that ρ X (u)m k(u k ) m(u)a k (u) > 0. (4) Equation (4) in conjunction with equation (3) implies that r ρ X m (u) u k > 0. Hence, there is an ɛ > 0 such that for all τ (0, ɛ) we have r(u) < r(u + τe k ). 10

11 Analogously (by replacing > through < ), part 2. in Definition 2.4 is proved. Necessity: The proof of the necessity exactly follows the steps of Tasche (2004) with m(u) = m u. Note that Theorem 2.6 can be seen as an additional justification for the gradient allocation principle. Since firms strive to maximize their return as stated above, the next lemma gives sufficient conditions to ensure that the return function r ρ X m has a finite global maximum. Lemma 2.7. Let R R n 0 be a closed convex set. Assume that there holds: (a) The profit function m is differentiable, positive, and concave on R. (b) The risk function ρ X is differentiable and positive on R. Moreover, ρ X is positive homogeneous, and sub-additive, hence convex, on R. (c) For all k N we have lim uk m k(u k ) = 0. Then, the return function r ρ X m either attains its global maximum in the interior of R where r ρ X m = 0, or on some boundary point of R. Proof. Clearly, r ρ X m is pseudoconcave, as its numerator is concave and its denominator is convex. Therefore, each local maximum of r ρ X m is also a global maximum on R (Cambini and Martein, 2009). Since r ρ X m is defined on the convex set R, we have to consider two cases: Case 1: R is compact. In this case, r ρ X m either attains its global maximum in the interior of R where r ρ X m = 0, or on some boundary point of R. Case 2: R is not compact. Note that since r ρ X m (u) u k =m k(u k )(ρ X (u) m k (u k )) 1 ( ) ρx (u) m k (u k ) m u k(u k ) (ρ X (u) m k (u k )) 2, k 11

12 and by assumption (c), m k (u k ) ( ρ X (u) u k m k(u k ) ) (ρ X (u) m k (u k )) 2 m k (u > 1 k)(ρ X (u) m k (u k )) 1 for u k sufficiently large, we have r ρ X m (u) u k < 0 for u k sufficiently large. Therefore, the return r ρ X m is reduced when increasing u k, if u k is sufficiently large. Hence, we can safely ignore all portfolios exceeding a certain u k and thus fall back upon case 1. While the adherence to condition (a) of Lemma 2.7 is straightforward in an economic context, and condition (b) is necessary for ρ X to be a coherent risk measure, it is advisable to briefly discuss condition (c). This assumption corresponds to the decreasing profitability of additional business of the same kind. Moreover, we assume the marginal profits as to eventually converge to zero. We regard this assumption as uncritical from a theoretical viewpoint as we assume an incomplete market, i.e. we implicitly assume that X(u) cannot be traded on the capital market at a unique price. From a practical point of view such behavior also seems realistic as one has to offer better conditions when expanding into a market, which is becoming increasingly satiated. 3. An example for failing capital allocation Consider a firm with two risky segments. Let (X 1, X 2 ) be multivariate normally distributed with zero mean and variance 1. Let the correlation γ between the two assets be 0.5. The firm now strives to improve its RORAC in each period until it obtains its optimal RORAC. Let (u (1) 1, u (1) 2 ) = (1.5, 1.7) in the first period. Let the risk measure be the value-at-risk (VaR) at 12

13 the 99.97% level. 3 Let the profit functions m k, k = 1, 2 be defined by m 1 : u 1 log(u ), and m 2 : u 2 log(u ). Then ρ X (u (1) ) = 3.43 u γu 1 u 2 + u 2 2 = The per-unit risk contribution a k (u (1) ), k = 1, 2 is then a 1 (u (1) ) = ρ X(u) u 1 (u (1) 1, u (1) 2 ) = and a 2 (u) = ρ X(u) u 2 (u (1) 1, u (1) 2 ) = The RORAC for the present portfolio is r ρ X m (u (1) ) = %. Marginal RORAC analysis according to the above ratio leads to m 1(u (1) 1 ) a 1 (u (1) ) m 1(u (1) 1 ) 2(u (1) 2 ) m = % and a 2 (u (1) ) m 2(u (1) 2 ) = %. We assume that the firm follows a marginal RORAC expansion or reduction strategy, respectively, i.e., that segments whose marginal RORAC exceeds the firm s overall RORAC will expand, while segments whose marginal RORAC falls below the firm s RORAC will reduce their business. Therefore segment 1 would expand and segment 2 would reduce. However they do not know to which extent. We assume an extension and reduction, respectively, to the new values (u (2) 1, u (2) 2 ) = 3 Note that the VaR in general is not a coherent risk measure since it fails to meet the sub-additivity property (see e.g. Tasche, 2002). However, when restricted to the multinormal distribution it is in fact sub-additive (Artzner et al., 1999). 13

14 (1.85, 1.55). The new RORAC now is r ρ X m (u (2) ) = % and thus has decreased slightly although the intention was to increase it. In fact the real optimum of r ρ X m (u opt ) = % at (u opt 1, u opt 2 ) = (1.6555, ) has been missed, as Figure 1 displays. Figure 1: Contour plots of the return function r ρ X m as defined in section 3 and the sequence of portfolios calculated in section 3. The boxes indicate the feasible regions where m i (u i +ɛ i ) m i (u i ) ɛ i a i (u) (m i (u i +ɛ i ) m i (u i > r(u). It is possible for a firm to swap between portfolios )) 1 and 2 all the time u opt Now since the functions m k are known to the segments one could consider a more sophisticated allocation: Instead of approximating the additional yield by ɛ k m k(u k ) one can use m k (u k + ɛ k ) m k (u k ) and divide it by the approximated additional risk capital ɛ k a k (u) (m k (u k + ɛ k ) m k (u k )). 14

15 In fact the values above of the new (u (2) 1, u (2) 2 ) were chosen in such a way that the sophisticated marginal RORAC of the extension was still higher than the original RORAC (segment 1: %) and the marginal RORAC of the reduction was still smaller than the original RORAC (segment 2: %). In the next period, the per-unit risk contributions become a 1 (u (2) ) = ρ X(u) u 1 (u (2) 1, u (2) 2 ) = and a 2 (u (2) ) = ρ X(u) u 2 (u (2) 1, u (2) 2 ) = , and the marginal RORACs are m 1(u (2) 1 ) a 1 (u (2) ) m 1(u (2) 1 ) 2(u (2) 2 ) m = % and a 2 (u (2) ) m 2(u (2) 2 ) = %. This time, segment 1 would reduce and segment 2 would expand. In fact, the new portfolio for period 3 could be (u (3) 1, u (3) 2 ) = (1.5, 1.7), where we started in period 1. In this case, the sophisticated marginal RORAC of the reduction was still smaller than the RORAC (segment 1: %) and the marginal RORAC of the extension was still higher than the RORAC (segment 2: %). Therefore, this example proves that proceeding in the way described above does not necessarily lead the firm to the optimum. 4. A refined approach Definition 2.4 merely states that if a section s marginal RORAC exceeds the firm s RORAC, then there is an ɛ k such that the expansion by ɛ k units increases the firm s RORAC, i.e., that r(u + ɛ k e k ) > r(u), but provides no conditions on the size of ɛ k. As the example in the previous section indicates, this can lead to over-expanding a section s business and hence 15

16 to a decline of the firm s RORAC. The following theorem fills this gap by ensuring that an expansion cannot reduce the firm s RORAC. Theorem 4.1. Let H(u) = [ ] 2 ρ X (u) u i u j be the Hessian of ρx (u). Assume that H(u) is bounded on a convex set R R n 0. Let Λ max u R λ max (H(u)) be an upper bound for the largest eigenvalue of H(u), u R. If u R, u + ɛ R, m(u) > 0, r(u) > 0, and if for all k N there holds m k (u k + ɛ k ) m k (u k ) (ɛ k a k (u) ɛ2 k Λ) (m k(u k + ɛ k ) m k (u k )) r(u), (5) with strict inequality given for at least one k N, than there also holds r(u + ɛ) > r(u). (6) Proof. Equation (5) is equivalent 4 to m k (u k + ɛ k )ρ X (u) m k (u k )ρ X (u) m(u)ɛ k ρ X (u) u k 1 2 ɛ2 km(u)λ 0. Summing up over all k N, we derive m(u + ɛ)ρ X (u) m(u)ρ X (u) m(u)ɛ ρ X (u) 1 2 m(u)ɛ ɛλ > 0, (7) since m(u) = n i=1 m i (u i ). Clearly, as ρ X is convex, H is positive definite and therefore Λ > 0. Moreover, since by the Rayleigh-Ritz-Theorem (Horn and Johnson, 1990) Λ max ɛ 0 ɛ H(υ)ɛ ɛ ɛ, the inequality 1 2 m(u)ɛ ɛλ 1 2 m(u)ɛ H(υ)ɛ 4 Notice that the denominator of the LHS of equation (5) must be positive: If it were negative, the numerator also has to be negative, because the whole LHS needs to be positive since r(u) > 0. However, if the numerator of the LHS is negative, the denominator of the LHS is clearly positive, which is a contradiction. 16

17 holds for all υ R. Therefore, equation (7) yields m(u + ɛ)ρ X (u) m(u)ρ X (u) m(u)ɛ ρ X (u) 1 2 m(u)ɛ H(υ)ɛ > 0, which is equivalent 5 to m(u + ɛ) ρ X (u) + ɛ ρ X (u) ɛ H(υ)ɛ m(u + ɛ) > m(u) ρ X (u) m(u). (8) Note that according to the Mean Value Theorem (Nocedal and Wright, 2006) there is some υ [u, u + ɛ] such that ρ X (u + ɛ) = ρ X (u) + ɛ ρ X (u) ɛ H(υ )ɛ. Letting υ = υ in equation (8) yields r(u + ɛ) > r(u). Note that the additional term of 1 2 ɛ2 kλ in equation (5) is quadratic on ɛ k. While it vanishes for small values of ɛ k, it decreases the marginal RORAC with the size of the expansion. Theorem 4.1 gives rise to the following algorithm: Algorithm 4.1. Start with i = 1 and an arbitrary u (1) R R n 0. Let α (0, 0.5]. 1. For each k N check if m k(u (i) k )/(a(i) k m k(u (i) k )) = > r(u (i) ): < a) If = holds, set ɛ (i) k = 0. b) If > holds, calculate ɛ (i) k,max = max ɛ(i) k > 0 such that equation (5) is fulfilled and u (i) k + ɛ(i) k,max R. Choose ɛ (i) k [α ɛ (i) k,max, (1 α) ɛ(i) k,max ]. c) If < holds, calculate ɛ (i) k,min = min ɛ(i) k < 0 such that equation (5) is fulfilled and u (i) k + ɛ(i) k,min R. Choose ɛ (i) k [α ɛ (i) k,min, (1 α) ɛ(i) k,min ]. 5 A same argument as in the footnote above shows that the denominator of the LHS of equation (8) has to be positive. 17

18 2. If ɛ (i) k = 0 for all k N, terminate. Otherwise, set u (i+1) = u (i) + ɛ (i). 3. Set i = i + 1 and continue with step 1. It is noteworthy to show that the algorithm guarantees convergence to the optimal RORAC, which is carried out in the following corollary. Corollary 4.2. Assume that the conditions (a) (c) of Lemma 2.7 apply and max u R r ρ X m (u) lies in the interior of R R 0. Then Algorithm 4.1 converges, i.e., lim i u (i) = u opt, where r ρ X m (u opt ) = max u R r ρ X m (u). Proof. We refer to the Convergence Theorem A from Zangwill (1969) to show the convergence of Algorithm 4.1. Clearly, the algorithm terminates at critical points of r ρ X m (u). Since r ρ X m pseudoconcave, each critical point is a global maximum (Cambini and Martein, 2009). Hence it suffices to show: 1. Compactness: By assumption (c) of Lemma 2.7, ɛ k < 0 for u k sufficiently large, since the sign of ɛ k follows the sign of rρ X m (u) u K (cf. proof of Theorem 2.6). Therefore, to show that the sequence of points generated by Algorithm 4.1 is bounded, we only need to show that ɛ k is bounded if rρ X m (u) u k > 0: Due to the concavity of m, there holds is m k (u k + ɛ k ) m k (u k ) ɛ k m k(u k ). Therefore, equation (5) yields m k(u k ) a k (u) ɛ kλ m k (u k) rρ X m (u). Clearly, the right hand side is bounded from below, since the algorithm creates an increasing sequence of return function values, and r ρ X m (u (1) ) > 0 as m(u (1) ) > 0. On the other hand, m k(u k ) is bounded from above, and therefore ɛ k is bounded from above. 18

19 2. Adaption: Follows directly from the proposition of Theorem Closedness: Holds since ɛ k is chosen from a closed set. Example (continued) Let us return to the example from section 3. Let R = {(u 1, u 2 ) : u 1 1, u 2 1}. Since the eigenvalues of the Hessian H(u) of ρ X (u) are 0 and (u u 2 2)/(u u u 1 u 2 ) 3/2, we can use the upper bound Λ = H(1, 1) = Figure 2: Contour plots of the return function r ρ X m as defined in section 3 and a sequence of portfolios generated by Algorithm 4.1. The boxes indicate the feasible regions according to equation (5) with R = {(u 1, u 2 ) : u 1 1, u 2 1}. The sequence of portfolios converges to u opt u opt Starting again at (u (1) 1, u (1) 2 ) = (1.5, 1.7), we choose ɛ (i) k = 0.5 ɛ (i) k,max or ɛ(i) k respectively, for each k = 1, 2 in each step. In the first period, equation (5) yields = 0.5 ɛ (i) k,min, ɛ (1) 1,max = and ɛ (1) 2,min =

20 Thus, we can calculate u (2) 1 = u (1) ɛ(1) 1,max = and u (2) 2 = u (1) ɛ(1) 2,min = The new RORAC becomes r ρ X m (u (2) ) = %. For the next period, the maximum step sizes become ɛ (2) 1,max = and ɛ (2) 2,min = , which leads to new values of u (3) 1 = u (2) ɛ(2) 1,max = and u (3) 2 = u (2) ɛ(2) 2,min = , and to a RORAC of r ρ X m (u (3) ) = %. Figure 2 shows the progress of the portfolios u (i) for i = 1, 2, Concluding discussion The main advantage of a classical RORAC capital allocation is its straightforward implementation in firms. The headquarters calculates the firm s overall risk according to its risk model once every period and therewith the overall RORAC. The headquarters also supplies to each segment its individual per-unit risk contribution. With these two parameters, the overall RORAC and the individual per-unit risk contribution, each segment can decide individually whether an additional business is profitable by calculating the marginal RORAC the business yields. Yet, traditional capital allocation with the gradient approach in the marginal RORAC framework linearizes the risk function embedded and can lead to over-expansions and over-reductions of businesses and may even lead to a reduction of the overall RORAC, as highlighted in the example in section 3. 20

21 The approach suggested by Theorem 4.1 is similar. However, instead of using the linear approximation ɛ k a k (u) of the risk function for the calculation of the marginal RORAC, the risk is adjusted by the additional risk correction term (RCT) 1 2 ɛ2 kλ in the denominator. Now, instead of venturing all business whose marginal RORAC exceeds the overall RORAC, only those businesses are undertaken whose marginal RORAC with RTC exceeds the overall RORAC to prevent the overall RORAC from decreasing. Compared with the classical RORAC allocation, the only new parameter in this model is Λ in the RCT. For simple risk structures like those shown in the example above, Λ can be calculated analytically. In most cases however, especially if the overall risk is computed with Monte Carlo simulations, Λ needs to be estimated numerically. Note that Λ does not need to be fixed for all periods. Hence, a viable approach could be to let R = {u U : u u m} be some neighborhood of u allowing each segment a change by at most m units per period. On the one hand, this ensures that the computation of Λ is technically feasible; on the other hand, it also warrants that Λ does not become so large as to restrict the segments in their individual business too much. At a first glance, Algorithm 4.1 looks technical. However, it can also be interpreted from an economical standpoint. Each step in the algorithm corresponds to one period (e.g. one month, one quarter, or one year) in a firm s lifecycle. At the beginning of each period, the headquarters publishes the firm s current RORAC and the per-unit risk contributions for each segment. Those segments, whose marginal RORACs equal the firm s RORAC, should neither expand nor reduce their business; if the marginal RORACs of all segments equal the firm s RORAC, the firm has obtained its optimal RORAC. All the other segments should either expand or reduce their business, depending on whether their marginal RORAC exceeds or falls below the firm s RORAC. However, their allowance in venturing into new business or giving up existing business is limited to the cases where the marginal RORAC with RTC 21

22 exceeds or falls below the firm s RORAC. The minimal bound for ɛ k in Algorithm 4.1 is needed to ensure that the segments actually move in the right direction and to exclude the possibility that the segments simply maintain their current business as long as they have not reached their individual optimal RORAC. The main advantage of our approach is stated in Corollary 4.2. While the example in section 3 showed that a classical RORAC allocation may prevent a firm from attaining its optimal RORAC, our approach ensures that the optimal RORAC if existent will eventually be obtained. To achieve this goal, we have used most of the axioms for a coherent risk measure according to Artzner et al. (1997): We needed translation invariance to split up the risk in the denominator of the return function in Definition 2.1, homogeneity to ensure that the gradient allocation principle is well-defined, and homogeneity as well as sub-additivity in Lemma 2.7 to ensure the existence of an optimal RORAC. References Acerbi, C., Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance 26, Acerbi, C. and Tasche, D., On the coherence of expected shortfall. Journal of Banking & Finance 26, Adam, A., Houkari, M., and Laurent, J.-P., Spectral risk measures and portfolio selection. Journal of Banking & Finance 32, Alessandri, P. and Drehmann, M., An economic capital model integrating credit and interest rate risk in the banking book. Journal of Banking & Finance, forthcoming. 22

23 Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Thinking coherently. Risk 10, Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D., Coherent measures of risk. Mathematical Finance 9, Breuer, T., Jandačka, M., Rheinberger, K., and Summer, M., Does adding up of economic capital for market- and credit risk amount to conservative risk assessment? Journal of Banking & Finance, forthcoming. Buch, A. and Dorfleitner, G., Coherent risk measures, coherent capital allocations and the gradient allocation principle. Insurance: Mathematics and Economics 42, Cambini, A. and Martein, L., Generalized Convexity and Optimization. Springer: New York. Denault, M., Coherent allocation of risk capital. Journal of Risk 4, Dhaene, J., Goovaerts, M., and Kaas, R., Economic capital allocation derived from risk measures. North American Actuarial Journal 7, Fischer, T., Risk capital allocation by coherent risk measures based on one-sided moments. Insurance: Mathematics and Economics 32, Furman, E. and Zitikis, R., Weighted risk capital allocations. Insurance: Mathematics and Economics 43, Gatzert, N. and Schmeiser, H., Combining fair pricing and capital requirements for non-life insurance companies. Journal of Banking & Finance 32, Gründl, H. and Schmeiser, H., Capital allocation for insurance companies What good is it? Journal of Risk and Insurance 74,

24 Horn, R. A. and Johnson, C. R., Matrix Analysis. Cambridge University Press: Cambridge. Hull, J. C., Risk Management and Financial Institutions. Prentice Hall International: Upper Saddle River. Kalkbrener, M., An axiomatic approach to capital allocation. Mathematical Finance 15, Merton, R. and Perold, A., Theory of risk capital in financial firms. Journal of Applied Corporate Finance 6, Myers, S. C. and Read, J. A., Capital allocation for insurance companies. Journal of Risk and Insurance 68, Nocedal, J. and Wright, S. J., Numerical Optimization. Springer series in operations research and financial engineering, Springer: New York. Rosen, D. and Saunders, D., Risk factor contributions in portfolio credit risk models. Journal of Banking & Finance 34, forthcoming. Stoughton, N. and Zechner, J., Optimal capital allocation using RAROC and EVA. Journal of Financial Intermediation 16, Tasche, D., Expected shortfall and beyond. Journal of Banking & Finance 26, Tasche, D., Allocating portfolio economic capital to sub-portfolios. in Economic Capital: A Practitioner Guide, ed. by A. Dev, Risk Books: London, Tasche, D., Capital allocation to business units and sub-portfolios: the Euler principle. in Pillar II in the new basel accord: The challenge of economic capital, ed. by A. Resti, Risk Books: London,

25 Urban, M., Dittrich, J., Klüppelberg, C., and Stölting, R., Allocation of risk capital to insurance portfolios. Blätter DGVFM 26, Zangwill, W. I., Nonlinear Programming. Prentice-Hall International Series in Management, Prentice-Hall: Englewood Cliffs. 25