Shapley Allocation, Diversification and Services in Operational Risk Frankfurt, March, 2015 Dr. Peter Mitic. Santander Bank (Head of Operational Risk Quantitative and Advanced Analytics UK) Disclaimer: The opinions, ideas and approaches expressed or presented are those of the author and do not necessarily reflect Santander s position. As a result, Santander cannot be held responsible for them. The values presented are just illustrations and do not represent Santander losses. Copyright: ALL RIGHTS RESERVED. This presentation contains material protected under International Copyright Laws and Treaties. Any unauthorized reprint or use of this material is prohibited. No part of this presentation may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author.
2 The Allocation problem 1. How should we allocate capital: what is fair? 2. The Shapley method, and its problems 3. How we solve the problems 4. Results
3 Pro-Rata (PR) Allocation 1 Suppose we have 3 business units, and we want to allocate capital of 1m to them. Each has a riskiness measures. Business Unit Investment Bank (IB) Commercial (COM) Retail (R) Riskiness 60 35 5 PR allocation ( 000) 600 350 50
4 Pro-Rata (PR) Allocation 2 This works, but does not account for cooperation between business units. Could they lower their capital charges by forming coalitions? In practice, risky business units complain that their allocation is too high. Suppose that the IB risk manager says that the IB allocation should be 500. We allocate the balance, 100 to the others. Business Unit Investment Bank (I) Commercial (C) Retail (R) Riskiness 60 35 5 PR re-allocation ( 000) 500 350 + 35*100/(35+5) = 437.5 % change -16.7% 25.0% 25.0% 50 + 5*100/(35+5) = 62.5 Sentiment Happy! We are subsidising a risky business unit by paying a high proportion of our capital extra
5 Shapley allocation All capital is shared (exhaustion) Those who do not use a cost element should not be charged for it (dummy player) Everyone who uses a given cost element should be charged equally for it (symmetry) The results of different cost allocations can be added (additive) Theorem (Shapley 1953: A value for n-person Games. Rand Corporation): There exists a unique allocation value satisfying these axioms, given by: The fairness axioms Average over all permutations of coalition members s in C s 1 n s! i v s v s i n! sc Coalition value with the Service Coalition value without the Service
6 Game Theory 1. Game = a mathematical optimality model of interactive decision making 2. Player = participant, member 3. Coalition = group 4. Cost function = a mapping from the set of all players to the real numbers, that describes how much a subset of all players can gain by forming a coalition Formally, with respect to a finite set of players C and a cost function v, C a game is a pair C, v : 2 There are 2 C subsets of C
7 Shapley: example 1 The riskiness measures are: I 60 C 35 R 5 The coalition riskiness measures are: I and C 83 I and R 61 C and R 37 All 90 What can be done if there are many more? How do we get these? The cost function is actually a saving, s, when a new member P enters a coalition C: Saving = Value of the coalition with the new member Value of the coalition without the new member v(c U P) = v(c) + v(p) - s
8 Shapley: example 1 We will fill in all the slots in this table, and use line 3 as an example Coalition Value I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Permutation I allocation C allocation R allocation I R C I C R R I C R C I C R I C I R Sum Shapley Value
9 R enters a coalition first. The marginal allocation to R is 5, entered in the R allocation column. Shapley: example 1 Coalition Value I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Permutation I allocation C allocation R allocation I R C I C R R I C 5 R C I C R I C I R Sum Shapley Value
10 Shapley: example 1 I enters a coalition next. The marginal allocation to I is v(ri) v(r), entered in the I allocation column. Coalition Value I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Permutation I allocation C allocation R allocation I R C I C R R I C 61-5 = 56 5 R C I C R I C I R Sum Shapley Value
11 Shapley: example 1 C enters a coalition last. The marginal allocation to C is v(all) v(ri), entered in the C allocation column. Coalition Value I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Permutation I allocation C allocation R allocation I R C I C R R I C 61-5 = 56 90-61 = 29 5 R C I C R I C I R Sum Shapley Value
12 Shapley: example 1 Fill in all other entries similarly Coalition Value I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Permutation I allocation C allocation R allocation I C R 60 83-60 = 23 90-83 = 7 I R C 60 90-61 = 29 61-60 = 1 R I C 61-5 = 56 90-61 = 29 5 R C I 90-37 = 53 37-5 = 32 5 C R I 90-37=53 35 37-35 = 2 C I R 83-35 = 48 35 90 83 = 7 Sum Shapley Value
13 Shapley: example 1 Sum the allocations in each column. The Shapley value for each player is the mean allocation in each player s column Coalition I 60 C 35 R 5 I and C 83 I and R 61 C and R 37 All 90 Value Check: 55 + 30.5 + 4.5 = 90 Permutation I allocation C allocation R allocation I R C 60 83-60 = 23 90-83 = 7 I C R 60 90-61 = 29 61-60 = 1 R I C 61-5 = 56 90-61 = 29 5 R C I 90-37 = 53 37-5 = 32 5 C R I 90-37 = 53 35 37-35 = 2 C I R 83-35 = 48 35 90 83 = 7 Sum 330 183 27 Shapley Value 330/6 = 55 183/6 = 30.5 27/6 = 4.5
14 Shapley: example 1 Compare with PR allocation: all have lower capital values because they cooperated. Everybody is happy! Method I allocation C allocation R allocation Allocation of 1m 1m x 55/90 = 611111 1m x 30.5/90 = 338889 1m x 5/90 = 50000 Shapley 55 30.5 4.5 Pro Rata 60 35 5 % Capital reduction 8.33% 12.86% 10%
15 Shapley: example 2 There are 3 players: P 1, P 2 and P 3 Their riskiness values are v 1, v 2 and v 3 When a new player P enters a coalition C, we define a cost function for riskiness v: constant factor. (it s really a saving function) v(c U P) = v(c) + v(p) - dv(p) where 0 < d < 1 is a There are 6 permutations of the 3 players: P 1 P 2 P 3 P 1 P 3 P 2 P 2 P 1 P 3 P 2 P 3 P 1 P 3 P 1 P 2 P 3 P 2 P 1
16 Shapley: example 2 We will fill in all the slots in this table, and use line 3 as an example Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 P 1 P 3 P 2 P 2 P 1 P 3 P 2 P 3 P 1 P 3 P 1 P 2 P 3 P 2 P 1 Sum Shapley Value
17 Shapley: example 2 P 2 enters a coalition first. The marginal allocation to P 2 is therefore v 2, entered in the Allocation to P 2 column. Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 P 1 P 3 P 2 P 2 (P 1 P 3 ) v 2 P 2 P 3 P 1 P 3 P 1 P 2 P 3 P 2 P 1 Sum Shapley Value
18 Shapley: example 2 P 1 is the next player to enter the coalition. The marginal allocation to P 1 is the difference of the allocation to the coalition {P 2, P 1 } and the allocation to P 2 alone. v(p 2 P 1 ) - v(p 2 ) = v(p 1 ) - dv(p 1 ) = v 1 dv 1. It s entered in the Allocation to P 1 column. Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 P 1 P 3 P 2 P 2 P 1 (P 3 ) v 1 - dv 1 v 2 P 2 P 3 P 1 P 3 P 1 P 2 P 3 P 2 P 1 Sum Shapley Value
19 Shapley: example 2 P 3 is the last player to enter the coalition. The marginal allocation to P 3 is the difference of the allocation to the coalition {P 2, P 1, P 3 } and the allocation to {P 2, P 1 }. v(p 2 P 1 P 3 ) - v(p 2 P 1 ) = v(p 3 ) - dv(p 3 ) = v 3 dv 3 : entered in the Allocation to P 3 column Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 P 1 P 3 P 2 P 2 P 1 P 3 v 1 - dv 1 v 2 v 3 dv 3 P 2 P 3 P 1 P 3 P 1 P 2 P 3 P 2 P 1 Sum Shapley Value
20 Process the five other cases in the same way. Shapley: example 2 Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 v 1 v 2 - dv 2 v 3 - dv 3 P 1 P 3 P 2 v 1 v 2 - dv 2 v 3 - dv 3 P 2 P 1 P 3 v 1 - dv 1 v 2 v 3 dv 3 P 2 P 3 P 1 v 1 - dv 1 v 2 v 3 dv 3 There is a pattern... P 3 P 1 P 2 v 1 - dv 1 v 2 - dv 2 v 3 P 3 P 2 P 1 v 1 - dv 1 v 2 - dv 2 v 3 Sum Shapley Value
21 Shapley: example 2 The Shapley value for each player is the mean allocation in each column. Permutation P 1 allocation P 2 allocation P 3 allocation P 1 P 2 P 3 v 1 v 2 - dv 2 v 3 - dv 3 P 1 P 3 P 2 v 1 v 2 - dv 2 v 3 - dv 3 P 2 P 1 P 3 v 1 - dv 1 v 2 v 3 dv 3 P 2 P 3 P 1 v 1 - dv 1 v 2 v 3 dv 3 P 3 P 1 P 2 v 1 - dv 1 v 2 - dv 2 v 3 P 3 P 2 P 1 v 1 - dv 1 v 2 - dv 2 v 3 Sum 6v 1-4 dv 1 6v 2-4dv 2 6v 3-4dv 3 Shapley Value v 1-2dv 1 /3 v 2-2dv 2 /3 v 3-2dv 3 /3 There is a pattern here too...
22 n players: constant diversification If you do the same analysis for 4 players, the pattern becomes clearer. For n players P 1, P 2,,P n with values v 1, v 2,,v n, define a cost function by v(c U P) = v(c) + v(p) - dv(p) where 0 < d < 1 is a constant factor. Then the Shapley value of player P r is given by SH n, r vr 1 dvr 1 n Proof by considering how many cases there are, and the value of those cases, where player P r is the first to enter a coalition, and where P r is not the first to enter a coalition.
23 n players: constant diversification The impact of this result... The Shapley allocation of a player P r is simply its value reduced by an amount proportional to its value. The Shapley allocation amount is always less than the corresponding Proportional Allocation amount by So Business Managers are happy! 1 1 dv r n
24 A service with constant diversification A service does not generate income in its own right. However, a service does have associated risk (e.g. model risk, technology risk, people risk). We model this by making the service absorb allocation from one or more business functions. We think of The Risk Department as a service. IT is another. The value of a service is set initially to a small number, but not zero. A small portion of the value of each of the other players is transferred to the Service before allocation starts. This is a technicality to ensure that the service is not treated as a dummy player: one that is effectively ignored. The cost function for a service is v(c US) = v(c) + v(s) + f(d, P 1, P 2, ), where f is a function of d and the values of P 1, P 2,
25 A service with constant diversification In a 3-player case, players A and B give the service S some (small) value e (e.g. 1) v(a) = v a - e; v(b) = v b - e; v(s) = 2e. When a non-service player P enters a coalition C, the cost function is v(c U P) = v(c) + v(p) - dv(p) - dm When a service player S enters a coalition C, the cost function is v(c U S) = (n-1)dm (n = 3 in this case) m is the median of the values of the non-service players.
26 A service with constant diversification Results for all 6 permutations: patterns are also apparent Permutation P 1 allocation P 2 allocation P 3 allocation S A B 2e v a - e - dv a - dm v b - e - dv b - dm S B A 2e v a - e - dv a - dm v b - e - dv b - dm A B S 2e + 2dm v a - e v b - e - dv b - dm A S B 2e + 2dm v a - e v b - e - dv b - dm B A S 2e + 2dm v a - e - dv a - dm v b - e B S A 2e + 2dm v a - e - dv a - dm v b - e Sum 12e + 8dm 6v a - 6e - 4dv a - 4dm 6v b - 6e - 4dv b - 4dm Shapley Value 2e + 4dm/3 v a - e - 2dv a /3-2dm/3 v b - e - 2dv b /3-2dm/3
27 A service with constant diversification Generalisation for n players: The Shapley values for the non-service players P r (2 r n) and the Service P 1 are: SH n, r v r e dv r 1 1 dm1 n 1 n 2 r n SH n,1 n 1e n 1dm 1 1 n Proof by considering how many cases there are, and the value of those cases. Treat P 1 (the service) and P r (2 r n) separately.
28 A service with constant diversification What has been achieved so far? By introducing the service we have acknowledged that the service contributes to risk, and should incur a charge for that. The charge is given by the Shapley calculation. This charge covers a risk of failure by the service. The capital allocation is fair We have allowed a migration of allocation amounts from the non-services to the service. Risk managers are even happier. The service manager should pay the service capital charge, so he or she is not happy. There is a closed-form formula for calculating Shapley values. Time-consuming combinatorial calculations are eliminated
29 Shapley Allocation: Diversification and Services A service with diminishing diversification When a new member joins a coalition, the added value decreases with increasing coalition size. The revised cost function is: v(cu P) = v(c) + v(p) - d C v(p) - d C m The Shapley values for the non-service players P r (2 r n) and the Service P 1 are given by Proof as before but more complex. 1 2 1 1,1 2, n n n n r r d d d D D n m n e n n SH n r D n v m e v r n SH
30 Operational risk losses for 11 units of measure, mid-2009 to mid-2014 Plus a service: the Risk Department. In Shapley-world, 11 is large! Results Lognormal distribution fitted to the 11 UoMs, LDA process, extract 99.9% VaR, which are the values v r of each UoM. Calculate diversification, d: Aggregate all losses and calculate total capital value, C all For each UoM, r, do the following sub-steps Remove losses for r from aggregate losses Calculate capital values, C r, for all losses except losses for r using Calculate C r ' = 100*(C r - C all )/ C all (the % deviation from C all ) Calculate the median C r ' (which is the diversification factor d, expressed as a %)
m Shapley Allocation: Diversification and Services 31 Results d ~ 4.174% (used as 0.04174) The graphs show that PR > SH for all business units except the service (#1). The service has absorbed allocation from the other business units. The diminishing diversification factor results are only marginally better than PR results. 4 3 2 1 0-1 -2-3 -4-5 -6 Pro Rata - Shapley allocation 1 2 3 4 5 6 7 8 9 10 11 12 UoM Constant diversification factor Diminishing diversification factor
32 Summary Shapley allocation can be done for a large number of UoMs. The allocation is seen to be fair by risk managers Shapley allocation uses closed-form formulae Combinatorial problems are eliminated Calculations are quick Risk managers are happy Service Manager agrees to share the capital