1. Capital allocation, diversification, and
multivariate stress tests
Andreas Tsanakas
Bayes Business School, City, University of London
joint work with
Yuanying Guan, Pietro Millossovich and Ruodu Wang
LSE Risk and Insurance Day 2022
1
2. Pre-prints
Millossovich, T. & Wang (2021), A theory of multivariate
stress testing
Guan, T. and Wang (2021), An impossibility theorem on
capital allocation
3. Table of Contents
The standard view
An impossibility theorem for capital allocation
Stress-based allocations
3
4. Table of Contents
The standard view
An impossibility theorem for capital allocation
Stress-based allocations
4
7. Allocation rules
Notation
• X = (X1, . . . , Xd) ∈ Xd
: random vector of LoB losses.
• Y =
Pd
j=1 Xj: portfolio loss
• For simplicity assume X, Y continuous
Definition
An allocation rule Λ is a mapping Xd
→ Rd
, such that
Λ(X) = (Λ1(X), . . . , Λd(X)), where Λi(X) represents the
amount of capital allocated to line i.
7
8. Some properties
For a risk measure ρ : X → R, an allocation rule Λ satisfies:1
• Top-down consistency, if
Pn
i=1 Λi(X) = ρ
Pn
i=1 Xi
.
• Diversification, if Λi(X) ≤ ρ(Xi) for all i.
A risk measure ρ is subadditive, if for all X1, X2 ∈ X it
satisfies
ρ(X1 + X2) ≤ ρ(X1) + ρ(X2)
1
These properties are also called efficiency and no-undercut
[Denault, 2001, Kalkbrener, 2005]
8
9. Euler capital allocation with Expected Shortfall
Consider the risk measure and allocation
[Kalkbrener, 2005, Tasche, 2007]:
ρ(Y ) = E
Y | Y F−1
Y (p)
Λi(X) = E
Xi | Y F−1
Y (p)
Subbaditivity of ρ, top-down consistency and diversification of
Λ all hold
• End of story?
10. Real-world example
Simulated scenarios (n = 105
) provided by a UK-based
non-life insurer, from their stochastic economic capital model.
X = (X1, . . . , X16) : losses per line of business
Y : portfolio loss
Consider also a modified portfolio, where we buy some
additional non-proportional reinsurance for X16:
X′
16 = X16 − 0.9 X16 − F−1
X16
(0.8)
+
12. The trouble with Euler allocations
The Euler approach is theoretically justified, but can be
problematic for industry applications.
• Large risks dominate portfolio, while small
independent risks get no risk load
• Reducing risk in part of the portfolio can increase
allocated capital in other parts
• Unstable profit targets, perverse incentives
We will next try to formalise some of these issues
• tl;dr: you can’t have your cake and eat it!
13. Table of Contents
The standard view
An impossibility theorem for capital allocation
Stress-based allocations
13
14. More properties
An allocation rule Λ : Xd
→ Rd
satisfies:
• Shrinking independence, if
Λi(X1, . . . , Xj−1, aXj, Xj+1, . . . , Xd) ≤ Λi(X)
for all j ̸= i and a ∈ (0, 1).
⇝ Reducing exposure in one line does not increase
capital in another
• Vanishing continuity, if
Λ(ϵX) → 0 and
Λj(X1, . . . , Xj−1, ϵXj, Xj+1, . . . , Xd) → 0
as ϵ ↓ 0, for each j = 1, . . . , d .
15. An impossibility theorem
Theorem
An allocation rule Λ satisfies shrinking independence,
vanishing continuity and top-down consistency wrt to a
subbaditive risk measure ρ, if and only if Λ(X) = E[X] for all
X ∈ Xd
.
Remarks:
• Properties are inconsistent: requiring them together
reduces allocation to a trivial one, which ignores
dependence
• Same result if we require the diversification property of
Λ instead of subadditivity of ρ
16. Positive dependence
If elements of X = (X1, . . . , Xd) act as hedges for each
other, shrinking independence is not so meaningful.
• Restrict this property to positive dependence!
• Denote by RS the Spearman rank correlation
Definition
For r ∈ [−1, 1], we say that the random vector X is
r-positively dependent (r-PD), if
RS(Xi, Xj) ≥ r for all i ̸= j,
Denote by Xd
r the set of all r-PD random vectors with
continuous marginals.
16
17. Impossibility theorem for positively dependent risks
The allocation rule Λ satisfies shrinking independence
under r-PD if
Λi(X1, . . . , Xj−1, aXj, Xj+1, . . . , Xd) ≤ Λi(X)
for all j ̸= i, a ∈ (0, 1) and X ∈ Xd
r .
Theorem
Suppose that an allocation rule Λ satisfies vanishing continuity
and top-down consistency wrt to a subbaditive risk measure ρ.
For r ∈ (0, 1), Λ satisfies shrinking independence under
r-PD if and only if Λ(X) = E[X] for all X ∈ Xd
r .
17
18. What next?
Top-down consistency diversification are inconsistent with
shrinking independence
• But we like diversification!(?)
Next we adopt a bottom-up view of capital allocation,
dropping the top-down consistency requirement. Precedents:
• Exogenous capital [Dhaene et al., 2012]
• Convex risk measures [Centrone Gianin, 2018]
19. Table of Contents
The standard view
An impossibility theorem for capital allocation
Stress-based allocations
19
20. Stressing mechanisms
Definition
A stressing mechanism is a functional η from Xd
to the set of
Radon-Nikodym densities, satisfying the following properties:
(i) Relevance: η(X) is σ(X)-measurable.
(ii) Law-invariance: (η(X), X)
d
= (η(Z), Z) if X
d
= Z.
In [Millossovich et al., 2021] we study these in the broader
context of stress testing
• Here we focus narrowly on capital allocation applications
21. Univariate stressing
Denote Ui = FXi
(Xi), Ūi = 1 − Ui.
Consider the univariate stressing mechanism:
ηi(X) := (1 − θ)Ū−θ
i , θ ∈ (0, 1).
This stressing mechanism induces a (subadditive) distortion
risk measure – the proportional hazards transform
[Wang, 1996]:
ϕ(Xi) := E[Xiηi(X)]
21
22. Mixture stressing and allocation
Definition
For λ1, . . . , λd ≥ 0,
Pd
j=1 λj = 1, we define the mixture
stressing mechanism and allocation rule by:
ηm
(X) := (1 − θ)
d
X
j=1
λjŪ−θ
j , θ ∈ (0, 1),
Λm
i (X) := E [Xiηm
(X)] .
22
23. Spearman stressing and allocation
Definition
We define the Spearman stressing mechanism and
allocation rule by
ηs
(X) :=
Ū1 · . . . · Ūd
−θ
E
h
Ū1 · . . . · Ūd
−θ
i, θ ∈ (0, 1),
Λs
i(X) := E [Xiηs
(X)]
23
24. Properties
Proposition
The allocation rules Λm
and Λs
satisfy:
a) Copula decomposition: for each i, Λi(X) is
determined by the distribution of Xi and the copula of X.
b) (Strong) shrinking independence: for a ∈ (0, 1),
Λi(X1, . . . , Xj−1, aXj, Xj+1, . . . , Xd) = Λi(X)
c) Vanishing continuity.
25. Back to the example – Euler and Spearman
allocations
25
27. Diversification/aggregation properties
Proposition
i) For mixture allocation rules, it holds that:
a) Λm
i (X) ≤ ϕ(Xi).
b) If X is comonotonic, Λm
i (X) = ϕ(Xi).
⇝ Diversification!
ii) For Spearman allocation rules, it holds that:
a) If X is independent, Λs
i(X) = ϕ(Xi).
b) If X−i is stochastically increasing in Xi, Λs
i(X) ≥ ϕ(Xi).
⇝ Aggregation!
29. Conclusions
There is a fundamental conflict between shrinking
independence and diversification requirements for capital
allocation rules
• Both of practical relevance!
We can reconcile those two properties by dropping the
top-down consistency requirement
• Allocation rules based on multivariate stresses give
plausible alternative approaches
THANK YOU FOR YOUR ATTENTION!
29
30. References I
Centrone, F. Gianin, E. R. (2018).
Capital allocation à la aumann–shapley for
non-differentiable risk measures.
European Journal of Operational Research, 267(2),
667–675.
Denault, M. (2001).
Coherent allocation of risk capital.
Journal of risk, 4, 1–34.
Dhaene, J., Tsanakas, A., Valdez, E. A., Vanduffel, S.
(2012).
Optimal capital allocation principles.
Journal of Risk and Insurance, 79(1), 1–28.
30
31. References II
Kalkbrener, M. (2005).
An axiomatic approach to capital allocation.
Mathematical Finance: An International Journal of
Mathematics, Statistics and Financial Economics, 15(3),
425–437.
Millossovich, P., Tsanakas, A., Wang, R. (2021).
A theory of multivariate stress testing.
Available at SSRN 3966204.
Tasche, D. (2007).
Capital allocation to business units and sub-portfolios: the
euler principle.
arXiv preprint arXiv:0708.2542.
31
32. References III
Wang, S. (1996).
Premium calculation by transforming the layer premium
density.
ASTIN Bulletin, 26(1), 71–92.
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