Rethinking risk capital allocation in a RORAC framework

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Rethinking risk capital allocation in a RORAC framework

DGF Annual Meeting 2010·Hamburg

Arne Buch^∗·Gregor Dorﬂeitner^†·Maximilian Wimmer^†

∗d-ﬁne GmbH, Frankfurt

†Department of Finance, University of Regensburg

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Maximilian Wimmer Department of Finance University of Regensburg

Motivation

.Bank .

.Retail lending .Structured ﬁnance .Two lines of business

.Total EC .Diversiﬁcation

.Capital allocation .EC Retail lending .EC Structured ﬁnance .

∆RORACRL= ∆ProﬁtRL

∆ECRL

.

∆RORACSF=∆ProﬁtSF

∆ECSF

.Total RORAC=Total Proﬁt Total EC

.∆RORACRL>Total RORAC .

∆RORACSF<Total RORAC .Expand business .Reduce business .

Total RORAC=Total Proﬁt Total EC

.

Expand business .Reduce business .max?

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Motivation|2 / 17

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Motivation

.Bank .

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Individual EC Retail lending .

Individual EC Structured ﬁnance

∆ECRL

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∆ECSF

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Motivation

.Bank .

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∆ECRL

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∆ECSF

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Motivation

.Bank .

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.Capital allocation .EC Retail lending .EC Structured ﬁnance

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∆ECRL

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∆ECSF

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Motivation

.Bank .

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∆ECRL

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∆ECSF

∆RORACRL>Total RORAC ∆RORACSF<Total RORAC .Expand business .Reduce business .

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Motivation

.Bank .

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∆ECRL

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∆ECSF

∆RORACSF<Total RORAC .Expand business .Reduce business

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Motivation

.Bank .

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∆ECRL

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∆ECSF

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Outline

0 Motivation

1 Literature review

2 Notation and preliminaries

3 Example for failing capital allocation

4 Second-order approach

5 Conclusion

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Outline|3 / 17

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Outline

0 Motivation

5 Conclusion

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Literature review|4 / 17

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Literature review

I Mathematical ﬁnance context

I Denault (2001): Axiomatic approach in a game theory setting

I Kalkbrener (2005): Axiomatic system

I Tasche (2004): Suitability for performance measurement, gradient allocation principle

I Buch and Dorﬂeitner (2008): Coherence of gradient allocation principle

I Insurance-linked perspective

I Dhaene et al. (2003): Coherent risk measures not optimal

I Furman and Zitikis (2008): Weighted allocation

I Financial economics viewpoint

I Merton and Perold (1993): Incremental allocation

I Stoughton and Zechner (2007): Economic optimization problem

. Does gradient allocation lead a ﬁrm to its optimal RORAC?

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Literature review|5 / 17

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Outline

0 Motivation

5 Conclusion

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Notation and preliminaries|6 / 17

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Model and notation

F Firm consisting ofnindividual segments

uk Number of contracts written by segmentk,u= (u1, . . . ,un) Yk(uk) Proﬁt of segmentk,Yk(uk) =Mk(uk) +Xk(uk)

Mk(uk) Expected profit of segmentk,M(u) =∑n k=1Mk(uk) Xk(uk) Profit fluctuation of segmentk,X(u) =∑n

k=1Xk(uk),X(u)is linear WRTu ρ Convex risk measure

ρX Risk functionρX:Rⁿ→R,^ρX:u7→ρ(X(u)) Information asymmetry

I Headquarters can evaluate the riskρ(·), but has only knowledge of the current expected proﬁtM(u)

I Divisions can estimate the expected proﬁtMk(uk+ϵk), but not the overall riskρ(·)

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Preliminaries—RORAC and marginal RORAC

Deﬁnition

The functionrM,ρX:U→Rdeﬁned as

rM,ρX:u7→ M(u) ρX(u)−M(u) is calledreturn function associated withm,X, andρ Deﬁnition

Given per-unit risk contributionak, such that∑

kak(u)uk=ρX(u), one can deﬁne the marginal RORACby

M^′_k(uk) ak(u)−M^′_k(uk)

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Preliminaries—Suitability for performance measurement

Deﬁnition

An allocationa1, . . . ,anis calledsuitable for performance measurementif there holds:

1 For all portfoliosu_∈Uand for all differentiable proﬁt functionsM:U_→RwithρX(u)̸=0 and k∈Nthe inequality

M^′_k(uk)

ak(u)−M^′_k(uk)>rM,ρ_X(u) implies that there is anϵ >0 such that for all τ∈(0, ϵ)we have

rM,ρX(u)<rM,ρX(u+τek) 2 “The other way around”

Theorem

The gradient or Euler allocation, i.e.,

ak(u) =^∂ρ_∂u^X^(u)

k , is the only allocation that is suitable for performance measurement

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Outline

0 Motivation

5 Conclusion

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Example for failing capital allocation

. ^.1.4 ^.1.5 ^.1.6 ^.1.7 ^.1.8 ^.1.9

.1.4.1.5.1.6.1.7.1.8.1.9

. u^opt

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General setup:

Two divisions M1(u1) =log(u1+¹₂) M2(u2) =log(u2+¹₂)

X1,2∼N(0,1) corr(X1,X2) =0.5

ρ=99.97%-VaR

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Example for failing capital allocation

. ^.1.4 ^.1.5 ^.1.6 ^.1.7 ^.1.8 ^.1.9

.1.4.1.5.1.6.1.7.1.8.1.9

. u^opt .1

Let

u⁽¹⁾₁ =1.5,u⁽¹⁾₂ =1.7 Then,

rM,ρX(u⁽¹⁾) =18.451% Marginal RORAC analysis leads to

M^′₁(u⁽¹⁾₁ ) a1(u⁽¹⁾)−M^′₁(u⁽¹⁾₁ )

=20.775% and

M^′₂(u⁽¹⁾₂ ) a2(u⁽¹⁾)−M^′₂(u⁽¹⁾₂ )

=17.647%

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Example for failing capital allocation

. ^.1.4 ^.1.5 ^.1.6 ^.1.7 ^.1.8 ^.1.9

.1.4.1.5.1.6.1.7.1.8.1.9

. u^opt .1

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How far to go?

Approximate additional proﬁt by Mk(uk+ϵk)−Mk(uk) instead ofϵ_kM^′_k(uk)and replace

ϵkM^′_k(uk)

ϵkak(u)−ϵkM^′_k(uk) >rM,ρ_X(u) by

Mk(uk+ϵk)−Mk(uk)

ϵ_kak(u)−(Mk(uk+ϵ_k)−Mk(uk)) >rM,ρX(u)

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Example for failing capital allocation

. ^.1.4 ^.1.5 ^.1.6 ^.1.7 ^.1.8 ^.1.9

.1.4.1.5.1.6.1.7.1.8.1.9

. u^opt .1

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Let

u⁽²⁾₁ =1.85,u⁽²⁾₂ =1.55 Then,

M^′₁(u⁽²⁾₁ ) a1(u⁽²⁾)−M^′₁(u⁽²⁾₁ )

=16.190% and

M^′₂(u⁽²⁾₂ ) a2(u⁽²⁾)−M^′₂(u⁽²⁾₂ )

=20.397%

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Example for failing capital allocation

. ^.1.4 ^.1.5 ^.1.6 ^.1.7 ^.1.8 ^.1.9

.1.4.1.5.1.6.1.7.1.8.1.9

. u^opt .1

.2

Let

u⁽²⁾₁ =1.85,u⁽²⁾₂ =1.55 Then,

M^′₁(u⁽²⁾₁ ) a1(u⁽²⁾)−M^′₁(u⁽²⁾₁ )

=16.190% and

M^′₂(u⁽²⁾₂ ) a2(u⁽²⁾)−M^′₂(u⁽²⁾₂ )

=20.397%

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Outline

0 Motivation

5 Conclusion

Rethinking risk capital allocation in a RORAC framework|October 8, 2010 Second-order approach|12 / 17

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A second-order approach

Theorem Assume that

I H(u) =[_∂2ρ_X(u)

∂ui∂uj

]

is the Hessian ofρX(u)

I ∥H(u)∥is bounded on a convex set U⊆Rⁿ_≥0

I Λ≥max

u∈U λmax(H(u))is an upper bound for the largest eigenvalue of H(u)

I u_∈U, u+ϵ∈U, M(u)>0,rM,ρ_X(u)>0

I For all k∈N there holds

Mk(uk+ϵk)−Mk(uk)

(ϵkak(u) +¹₂ϵ²_kΛ)−(Mk(uk+ϵk)−Mk(uk)) ≥rM,ρX(u), with strict inequality given for at least one k∈N

Then there also holds

rM,ρX(u+ϵ)>rM,ρX(u)

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0 Motivation

5 Conclusion

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Conclusion

I (Slightly) extended setting as inTasche (2004): Concave expected proﬁt function

I Here: Only strictly stationary proﬁt process considered

I The implementation of a naïve gradient capital allocation in ﬁrms can be suboptimal if division managers are allowed to venture into all business whose marginal RORAC exceeds the ﬁrm’s RORAC

I If the marginal RORAC requirements are reﬁned by adding a risk correction term that takes into account the interdependencies of the risks of different lines of business, it can be guaranteed that the optimal RORAC will be achieved eventually (under the assumption of a strictly stationary proﬁt process)

Financial crisis check

I Higher requirements on the yields of signed contracts

I Less piling of tons of CDO tranches

X

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Dr. Maximilian Wimmer Department of Finance 93040 Regensburg Germany

ph: + 49 (941) 943 - 2672 fax: + 49 (941) 943 - 81 2672

maximilian.wimmer@wiwi.uni-regensburg.de http://www-finance.uni-regensburg.de

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