15 June 2011
Fixed Income Research
http://www.credit-suisse.com/researchandanalytics
A Nash Equilibrium for Greece
Quantitative Credit Strategy
Research Analysts
Christian Schwarz
+44 20 7888 3161
[email protected]
Helen Haworth
+44 20 7888 0757
[email protected]
Chiraag Somaia
+44 20 7888 2776
[email protected]
Joachim Edery
+44 20 7888 7382
[email protected]
William Porter
+44 20 7888 1207
[email protected]
Applying Game Theory to the Greek debt crisis
The Greek situation is highly complex and very crowded – there are a multitude
of parties involved, each with intertwined and highly intra-dependent strategies,
and different vested interests. In other words, it is a situation that is highly
suitable for analysis using Game Theory.
One of the most famous equilibria is the Nash equilibrium. This is defined as a
combination of strategies whose payoff no player can improve by altering
his/her strategy unilaterally.
In the context of the Greek debt crisis, we analyse two games:
The private sector loss participation game
We find that for a private sector institution the decision to “participate” or not is a
classic “prisoner’s dilemma”. In a single-step game the Nash equilibrium is for
all players to “free ride”, i.e., not to “participate”, irrespective of whether they are
banks or hedge funds. In a multi-step game this strategy ceases to dominate
and is dominated by a mixed strategy between “participate” and “free ride” as
the players have to balance fear of punishment in the next step (e.g., regulation)
with the benefits of not cooperating this time.
The “whole” Greek multiplayer situation
We consider a multiplayer game with the following “players”: the ECB, the private
sector, the IMF, Germany and other northern European countries, France, Greece
and the rating agencies. Each player has the choice of several strategies.
The result is a Nash equilibrium that does not involve private sector loss sharing if
it results in Greek government bonds being downgraded to default and hence not
being eligible as collateral at the ECB. In other words, under these constraints,
private sector loss sharing cannot be part of a Nash equilibrium.
We also show how the result can be stressed by altering our inputs. Greece’s
incentive to implement further austerity measures plays a crucial role here.
In a multi-step game the private sector’s fear of “punishment” for not
participating in the loss-sharing results in a different Nash equilibrium. The
result here is a Nash equilibrium, in which the ECB would accept Greek
government debt as collateral despite it being downgraded to default by
the rating agencies. However, there are likely to be workarounds, for example
a technicality about the timing and duration of a default rating.
We conclude that the ECB's objections will eventually be overcome.
Subsequently, compulsion becomes less of a challenge. Because free-loading
is to be rewarded, we would expect rotation among holders to reduce this fear
of punishment, making compulsion eventually necessary.
ANALYST CERTIFICATIONS AND IMPORTANT DISCLOSURES ARE IN THE DISCLOSURE APPENDIX. FOR OTHER
IMPORTANT DISCLOSURES, PLEASE REFER TO https://firesearchdisclosure.credit-suisse.com.
15 June 2011
Motivation
The Greek situation is highly complex and very crowded – there are a multitude of parties
involved, each with intertwined and highly intra-dependent strategies, and different vested
interests. In other words, it is a situation that is highly suitable to be analysed using Game
Theory.
Game Theory enables us to understand the problem at hand in its full complexity by
breaking it down into its key determining components. Furthermore, it allows us to find and
identify equilibria and assess the measures that are needed to alter those equilibria.
Basics of Game Theory
In mathematics, Game Theory models strategic situations, or games, in which an
individual’s success in making choices depends on the choices of others 1 . In the
mathematical sense, a game consists of a set of players, possible choices (or strategies)
available to each player and the specifications of payoffs (e.g., utility functions) to each
player for each strategy.
Equilibria
One of the aims of game theory is to find equilibria in the respective games. One of the
most famous equilibria is the Nash equilibrium, named after the famous mathematician
John Forbes Nash, who, together with John Harsanyi and Reinhard Selten, won the Nobel
Memorial Price in Economic Sciences in 1994. A Nash equilibrium is defined as a
combination of strategies whose payoff no player can unilaterally improve by altering
his/her strategy. An example is most easily explained by showing when a strategy is not a
Nash equilibrium: if a player can improve his/her payoff from a given combination of
strategies (i.e., combination of each player’s single strategy) by changing his/her choice,
that combination of strategies cannot be a Nash equilibrium.
Pareto efficiency is defined slightly differently. Given a certain combination of players’
strategies, a change to a different combination of strategies that makes at least one player
better off without deteriorating the outcomes for the other players is called a pareto
improvement. A combination of strategies is called pareto efficient or optimal if no further
pareto improvements can be achieved. Nash equilibria therefore do not have to be pareto
optimal.
An additional important concept in Game Theory is Strategic Dominance. A strategy for a
given player dominates other strategies of the same player if the player would be better off
choosing this strategy no matter what his opponents decide to choose.
Games are commonly expressed in either extensive or normal (strategic) form. Extensive
form games (visualized via a decision tree) are usually used when there is an ordered
sequence of players’ decisions. Games whose players make their decisions at the same
time can be presented in normal form and in the case of a two-player game they are
represented by a matrix illustrating the players, strategies and payoffs. In the case of a
multiplayer game with two or more strategies per player the two-dimensionality of a matrix
is not sufficient and so we chose to represent each combination of strategies by an n-tuple
(a 3-tuple is a triple, a 4-tuple is a quadruple, etc.) where n is the number of players and
each coordinate of this n-tuple represents the choice of strategy of the respective player.
1
A Nash Equilibrium for Greece
Myerson, Roger B. (1991), Game theory: analysis of conflict
2
15 June 2011
Prisoner’s dilemma
One of the most famous games is “prisoner’s dilemma”, illustrated in Exhibit 1. It
demonstrates why two players might not cooperate even if it would be better for each of
them. In the classic form of this game, the strategy of cooperating is strictly dominated by the
strategy of not cooperating or defecting – in other words, both players have a preference for
the “no cooperation” strategy independent of their opponent’s choice – resulting in the only
possible equilibrium being both players defecting. Despite being pareto-suboptimal, the
Nash equilibrium is also present in this game. However, it assumes that all players will act
rationally and that the game is being played in a single instance.
If the game is repeated several times, each player has the opportunity to “punish” the
other player for not choosing to cooperate in the previous game by selecting to defect this
time. If the number of games is known by the players in advance, the theoretic result is still
that every player will choose to not cooperate in every instance of the game played.
However, if the number of instances played is either infinite or not known to the players
then the continuously defecting strategy is not an equilibrium. In this case the incentive to
defect can be overruled by the fear of future punishment.
Exhibit 1: The classic prisoner’s dilemma
Cooperating means to cooperate with the other player, i.e., to not confess
Prisoner B
Prisoner A
Stays silent (cooperates)
confesses (defects)
Stays silent (cooperates)
Each serves 1 month
Prisoner A: 1 year, Prisoner B:
Confesses (defects)
Prisoner A: goes free,
goes free
Each serves 3 months
Prisoner B: serves 1 year
Source: Credit Suisse
There are obviously many further concepts and sub-theories within the subject of Game
Theory; however, our intention is just to give the reader a quick introduction to the field as
a basis for our conclusions in the following sections.
Our aim is to identify the potential equilibria for the current situation in Greece and a
couple of “sub-games”. We believe we have found Nash equilibria in most of these. This
does not mean that the identified strategies have to be adopted by the parties involved or
that they are likely to be, as these conclusions are dependent on further conditions. The
detailed conditions required to guarantee that a Nash equilibrium exists are beyond the
scope of this note, but basically require all players to understand the “game” they are
playing, act rationally and maximize their expected payoff.
Applying Game Theory to the Greek debt crisis
Private sector participation
One of the most pressing decisions that currently has to be made by various parties
currently is whether the private sector needs to participate in the loss sharing in a potential
Greek restructuring. We start analysing this situation by assuming two players, a bank that
is holding Greek government bonds in its banking book and another player that can be
thought of as all other banks that are in the same situation. The strategies available to
each of these two players are either to voluntarily participate in the loss sharing or to “free
ride”, i.e., not participate. A single instance of this game is illustrated in Exhibit 2 . For both
players the “participate” strategy is strictly dominated by the “free ride” strategy – i.e., this
is always their best strategy regardless of what the other players decide, resulting in the
double “free ride” strategy being a Nash equilibrium. It can easily be shown that the
absolute numbers do not matter in the sense that only their order has an impact on the
A Nash Equilibrium for Greece
3
15 June 2011
results, i.e., the equilibria of this game. This is a classic representation of the prisoner’s
dilemma. If all private sector institutions choose not to participate, no institution will take
losses, therefore likely aggravating the severity of the situation for Greece and hence
increasing the chance of potential future losses exceeding the ones suffered by a
combination of strategies in which every player participated.
Exhibit 2: “Private sector loss sharing game between two banks”
Cells in the bottom right represent the payoff functions of each player in brackets. The left number is the payoff for the row
player, in this case Bank A and the second number is the payoff for the column player in this case Bank B.
Bank A
Participate
Free Ride
Bank B
Participate
(-1,-1)
(0,-10)
Free Ride
(-10,0)
(-5,-5)
Source: Credit Suisse
In a slight alteration of this game we replace one of the banks by a hedge fund. Most likely
a hedge fund has less incentive to voluntarily participate in the private sector loss sharing,
as it can be less coerced by officials to do so. An example of such a game can be seen in
Exhibit 3. We adjust the cells in the matrix according to our idea that hedge funds would
be less incentivized to participate on a voluntary basis, therefore, for example, making the
payoff for the hedge fund even worse than in the previous game if it participates and the
banks do not.
Exhibit 3: “Private sector loss sharing game between a bank and a hedge fund”
Cells in the bottom right represent the payoff functions of each player in brackets. The left number is the payoff for the row
player, in this case the Bank and the second number is the payoff for the column player in this case the Hedge Fund.
Bank
Participate
Free Ride
Hedge Fund
Participate
(-1,-2)
(0,-12)
Free Ride
(-10,1)
(-5,-5)
Source: Credit Suisse
This would suggest that both highly regulated and less constrained institutions would
choose not to participate in the loss sharing as, for example, recently envisaged by the
German finance ministry. We don’t think that a private sector loss sharing, once it is
agreed by the EU, is actually likely to end up having a minimal participation rate among
banks and other institutions, as suggested by this.
And in fact, in the more realistic multi-stage version of this game, the double “free ride”
strategy will not be an equilibrium in every instance of the game, provided the number of
games is not known. The conclusion here is that the players, i.e., the private sector, will
choose their strategy in every instance of the game depending on how likely and painful a
punishment there might be in the next game versus the potential benefit they might gain
from choosing to “free ride” this time. There are two interpretations that apply to the Greek
situation in our opinion:
1. There might quite likely be a second or third round of potential loss sharing
agreements and players not participating in the first one will be punished in the next
one.
2. Regulated entities deal with the public sector in many different ways and occasions.
Not participating in this situation, i.e., the loss sharing of Greek debt, might potentially
result in negative outcomes in other areas, e.g., new capital requirements.
Banks, in particular, therefore will not necessarily choose to “free ride” in this instance if
they feel the potential gain could be offset by detrimental effects elsewhere.
A Nash Equilibrium for Greece
4
15 June 2011
Applying Game Theory to the Greek debt crisis, part two
The players and their strategies we consider in this game are
1. The ECB: Continue to allow Greek government bonds as collateral for its repo
operations: Yes or No
2. The private sector: Voluntarily participate in the loss sharing: Yes or No
3. The IMF: Pay the 5th tranche of the Greek bail-out package: Yes or No
4. Northern Europe, e.g., Germany, the Netherlands, Finland: Create additional bailout package for Greece: Yes or No
5. France2: Create additional bail-out package for Greece: Yes or No
6. Greece: Implement further austerity measures and privatizations. Yes or No
7. The rating agencies: Determine a potential voluntary private sector participation a
(selective) default rating event: Yes or No
As explained above, such a multidimensional game is difficult to represent in matrix form.
We therefore display it as a list of 7-tuples (since there are 7 players), where the first of the
seven components represents the ECB’s decision (a “1” for Yes and a “0” for No), the
second corresponds to the private sector’s choice (again, a “1” stands for Yes and a “0” for
No) and so forth. The table of 128 (2 to the power of 7) possible combinations of strategies
can be found in Exhibit 5. So far, so good.
In order to fully specify the game we now need to determine the payoffs of each of the
players in each of the strategy combinations (i.e., for each 7-tuple). For this game that
would be 7*128=896 payoff values. Clearly too many. We therefore simplify the payoff
functions by assuming that not every player’s payoff value is a function of every other
player’s strategy choice. In fact, some of the above players might even have payoff values
that are almost entirely determined by their own choice and another single player’s
strategy. For example the rating agency’s payoff values might simply depend on its own
choice of strategy and whether the private sector shares losses in a voluntary debt
restructuring. This small simplification enables us to value the number of payoffs.
Single step game
Nash equilibrium rules out a private sector participation if it causes a
rating default
The payoff functions we have chosen for each player are in Exhibit 4. These are obviously
highly subjective and we would be interested if readers have opposing views. However, it
is important to keep in mind that the absolute values of the payout functions actually are
not important; only their relative rankings impact the results.
Our choice of payoff functions is driven by the following assumptions:
1. The ECB is positioning itself in a way that rules out any kind of restructuring that
causes a rating downgrade. This is reflected in the preferential payoff values under
the “no default” scenario. Furthermore the ECB clearly wants to prevent any concern
regarding the soundness of the financial system and therefore is clearly incentivized
to continue allowing Greek collateral for repo operations.
2
A Nash Equilibrium for Greece
France is a separate player in this setup because of its public stance and therefore its payoff function in this game is different than
Germany's and other northern European countries.
5
15 June 2011
2. According to its statutes, the IMF will not lend to a creditor if it doesn’t deem it to be
on a sustainable debt path, which includes a 12-month window of available
financing. For this to be the case, we think additional bail-out funds from the EU and
further austerity measures will be needed. Furthermore we think that the IMF pulling
out of the existing program will have negative implications, particularly for the
probability of a successful further EU bail-out package and thus for the likelihood of
the IMF getting its previously paid tranches of the original package back.
3. As stated in his letter and thereafter in a speech to the German parliament, Wolfgang
Schaeuble, the German finance minister, expressed the view that Germany is in
favour of voluntary private sector loss participation and in particular a voluntary
seven-year maturity extension. Without this and also without further Greek austerity
measures and privatisations it will be difficult to ‘sell’ an additional bail-out package
to the German electorate.
4. France in contrast is opposed to a private sector loss participation, even on a
voluntary basis, if it has the potential to trigger a rating default and therefore
potentially contagion effects to other sovereigns and the European banking sector.
Similar to Germany, it will find it politically difficult to approve additional bail-out funds
without further Greek austerity measures and privatisations.
5. From Greece’s perspective, there is no need to implement further austerity
measures and privatisations if there are to be no additional bail-out funds. The
crucial variable for the Nash equilibrium we find in the analysis below is whether
there is an incentive for Greece to implement extra measures when additional bailout funds from the IMF and the EU are forthcoming. In this section we assume this to
be the case, i.e., no extra measures causes a payoff of -3 and further measures of 2, respectively, (with both conditional on further bail-out funds). As we explain, the
solution changes if we change this assumption.
6. For Banks, which we use to represent the private sector, as discussed above there is
no incentive to participate in some sort of loss sharing on a voluntary basis if they do
not have to fear repercussions. Furthermore, a failure of the Troika-led bail-out has
the potential to be disastrous for some banks, as some sort of hard restructuring
event might unfold, leading to further contagion effects.
7. The rating agencies have been increasingly verbose in the recent past, more or less
indicating that they will deem pretty much every kind of private sector involvement in
a (voluntary) restructuring a (selective) default.
A summary of the resulting pay-off functions per player and for each set of individual
strategies can be found in Exhibit 6. The last column highlights the results of our algorithm to
identify the Nash equilibrium. The first number in this column shows which player can alter
his strategy to arrive at a better outcome. The second number identifies the strategy that the
player can switch to from the current one to improve his situation. As an example “4
dominated by 13” in the fifth row of the table, means that player 4 can deviate from the
current strategy (number five) to strategy number 13, thereby improving his payoff without
anybody else changing his/her strategy. In this example, Germany improves its payoff from 10 to -5 by choosing to pay additional bail-out moneys rather than not as in strategy five.
Very interestingly, the result is that there is only one Nash equilibrium, namely strategy 95,
where
1. The ECB continues to take Greek collateral
2. The private sector does not participate in the loss sharing
3. The IMF pays the 5th tranche
4. Germany and France pay additional bail-out funds
5. Greece takes further austerity measures and privatises state assets
6. The rating agencies do not downgrade Greek government debt to (selective) default
A Nash Equilibrium for Greece
6
15 June 2011
The result therefore is a Nash equilibrium that does not involve private sector loss sharing
if it results in Greek government bonds being downgraded to default and hence not being
eligible as collateral at the ECB. Under these constraints, private sector loss sharing
cannot be part of a Nash equilibrium. That said, a Nash equilibrium does not need to be
the actual final outcome of this situation, as further conditions must be met, as we
mentioned earlier.
In addition, we believe this also shows where the parties involved need to make the most
progress in order for the Greek situation to improve. In other words, in our view, they need
to find a way in which the desired level of participation does not trigger a default or,
alternatively, convince the ECB to continue accepting Greek collateral for its Repo
operations even with a default rating.
Another very interesting observation is that if we change Greece’s pay-off function such
that further austerity measures and privatisations are not beneficial to Greece (conditional
on extra bail-out funds) – one could, for example, think about the Greek population and
their incentives – the Nash equilibrium would be a strategy that is the same as the one
discussed above, except for the player Greece, which would deviate from its strategy and
not implement further steps. We find this very interesting and wonder if the recent calls for
EU and/or IMF oversight in the Greek austerity measures and privatisation efforts are
ways the authorities are trying to “change the pay-off function of Greece” to one that
favours the original Nash equilibrium.
A Nash Equilibrium for Greece
7
15 June 2011
Exhibit 4: Conditional payoff functions for each player
Some pay-off functions are only dependent on the choice of strategy of the individual player and one further condition.
Some others are dependent on a multitude of other conditions, connected by “&” signs.
ECB
Allow Greek collateral
yes
no
IMF
Rating default
yes
no
-5
-10
0
-10
Pay 5th Tranche
yes
no
Additional EU bail-out & Greek austerity
yes
no
0
-7
-5
-10
Pay additional bailout
yes
no
Private participation & Greek austerity
yes
no
0
-5
-7
-10
Pay additional bailout
yes
no
No rating default & Greek austerity
yes
no
0
-7
Germany
France
Greece
Further austerity & privatisation
yes
no
Banks
Participate
yes
no
Rating Agencies
Default rating
yes
no
IMF & EU bail-out
yes
-5
-10
no
-2
-3
-10
-5
IMF & EU bailout & austerity
yes
no
-2
0
Private sector losses
yes
-10
-5
no
-1
-3
-2
0
Source: Credit Suisse
A Nash Equilibrium for Greece
8
15 June 2011
Exhibit 5: Possible outcomes of combined strategies in the multiplayer game (1 of 2)
Each row is a 7-tuple, where each component represents the choice a player can make
Player
Strategy
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
A Nash Equilibrium for Greece
ECB
Allow Greek
collateral
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Banks
Participate
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
IMF Pay 5th
Tranche
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Germany Pay
additional
bailout
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
France Pay
additional
bailout
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Greece further
Austerity &
Privatisation
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Rating A
Rate Default
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
9
15 June 2011
Exhibit 5: Possible outcomes of combined strategies in the multiplayer game (2 of 2)
Each row is a 7-tuple, where each component represents the choice a player can make
Player
Strategy
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
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90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
ECB
Allow Greek
collateral
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Banks
Participate
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
IMF Pay 5th
Tranche
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Germany Pay
additional
bailout
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
France Pay
additional
bailout
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
Greece further
Austerity &
Privatisation
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
Rating A
Rate Default
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Source: Credit Suisse
A Nash Equilibrium for Greece
10
15 June 2011
Exhibit 6: Payoff values per combined strategy and player (1 of 2)
Each column holds the payoff of the respective player. Each row corresponds to a combination of strategies. The Nash equilibrium is highlighted in yellow.
Strategy
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
A Nash Equilibrium for Greece
ECB
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
Banks
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-2
-2
IMF
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
Germany
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
France
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
Greece
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
Rating
Agencies
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
Dominated
5 dominated by 5
7 dominated by 1
6 dominated by 1
7 dominated by 3
4 dominated by 13
7 dominated by 5
6 dominated by 5
7 dominated by 7
5 dominated by 13
7 dominated by 9
6 dominated by 9
7 dominated by 11
3 dominated by 29
7 dominated by 13
6 dominated by 13
7 dominated by 15
5 dominated by 21
7 dominated by 17
6 dominated by 17
7 dominated by 19
4 dominated by 29
7 dominated by 21
6 dominated by 21
7 dominated by 23
5 dominated by 29
7 dominated by 25
6 dominated by 25
7 dominated by 27
6 dominated by 31
7 dominated by 29
1 dominated by 95
7 dominated by 31
7 dominated by 34
5 dominated by 38
7 dominated by 36
6 dominated by 34
7 dominated by 38
4 dominated by 46
7 dominated by 40
6 dominated by 38
7 dominated by 42
5 dominated by 46
7 dominated by 44
6 dominated by 42
7 dominated by 46
3 dominated by 62
7 dominated by 48
6 dominated by 46
7 dominated by 50
5 dominated by 54
7 dominated by 52
6 dominated by 50
7 dominated by 54
4 dominated by 62
7 dominated by 56
6 dominated by 54
7 dominated by 58
5 dominated by 62
7 dominated by 60
6 dominated by 58
7 dominated by 62
6 dominated by 64
7 dominated by 64
2 dominated by 32
11
15 June 2011
Exhibit 6: Payoff values per combined strategy and player (2 of 2)
Each column holds the payoff of the respective player. Each row corresponds to a combination of strategies. The Nash equilibrium is highlighted in yellow.
Strategy
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
ECB
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
Banks
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
-5
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-2
-2
IMF
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
Germany
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
France
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
Greece
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
Rating
Agencies
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
5
7
6
7
4
7
6
7
5
7
6
7
3
7
6
7
5
7
6
7
4
7
6
7
5
7
6
7
6
7
Dominated
dominated by 69
dominated by 65
dominated by 65
dominated by 67
dominated by 77
dominated by 69
dominated by 69
dominated by 71
dominated by 77
dominated by 73
dominated by 73
dominated by 75
dominated by 93
dominated by 77
dominated by 77
dominated by 79
dominated by 85
dominated by 81
dominated by 81
dominated by 83
dominated by 93
dominated by 85
dominated by 85
dominated by 87
dominated by 93
dominated by 89
dominated by 89
dominated by 91
dominated by 95
dominated by 93
7 dominated by 95
7 dominated by 98
5 dominated by 102
7 dominated by 100
6 dominated by 98
7 dominated by 102
4 dominated by 110
7 dominated by 104
6 dominated by 102
7 dominated by 106
5 dominated by 110
7 dominated by 108
6 dominated by 106
7 dominated by 110
3 dominated by 126
7 dominated by 112
6 dominated by 110
7 dominated by 114
5 dominated by 118
7 dominated by 116
6 dominated by 114
7 dominated by 118
4 dominated by 126
7 dominated by 120
6 dominated by 118
7 dominated by 122
5 dominated by 126
7 dominated by 124
6 dominated by 122
7 dominated by 126
6 dominated by 128
7 dominated by 128
2 dominated by 96
Source: Credit Suisse
A Nash Equilibrium for Greece
12
15 June 2011
Multi-step game
Nash equilibrium: The ECB continues to accept Greek collateral despite
a rating downgrade to default
In a multi-step game it becomes very interesting what happens after the first step has
been played. As explained for the two-player, multi-step private sector participation game,
the players “losing” in the first step will likely punish the players “free-riding” in the second
step. According to their payoff values in the original Nash equilibrium, the player “losing”
the most is Germany with a payoff of minus 5, lower than everyone else’s. We think this
will likely have consequences for a second-step game, where exactly as described above,
the players have to price in the chance that the player Germany will punish them for not
cooperating in the first step. The player we consider the most sensitive to such a
punishment is the private banking sector; junior to most other players, it is probably the
most vulnerable to action by Germany.
The design of the multi-step game is exactly the same as the first game with one small but
important alteration: the way we incorporate the banking sector’s fear of punishment is by
subtracting 3 from its payout function in the case where it chooses not to participate. The
resulting payoff diagram can be seen in Exhibit 7.
Exhibit 7: Payoff values for banks in a multi-step game
We subtracted 3 from the bottom two numbers to reflect the fear of punishment in a future step
Banks
Participate
yes
no
IMF & EU bailout & austerity
yes
no
-2
-3
-10
-8
Source: Credit Suisse
A summary of the resulting pay-off functions per player and for each set of individual
strategies can be found in Exhibit 8. The last column shows that the original Nash
equilibrium strategy is now dominated by strategy 127 because of the 2nd player, i.e., the
banks, changing their choice from “no participation” to “participate”. This strategy in turn is
dominated by the new Nash equilibrium strategy 128 because of the 7th player, i.e., the
rating agencies, in turn changing their decision and downgrading Greek debt to default.
So, for the multi-step game the result is that there is again only one Nash equilibrium,
namely strategy 128, where
1. The ECB continues to take Greek collateral
2. The private Sector does participate in the loss sharing
3. The IMF pays the 5th tranche
4. Germany and France pay additional bail-out funds
5. Greece takes further austerity measures and privatises state assets
6. The rating agencies do downgrade Greek government debt to (selective) default
The result therefore is a Nash equilibrium, in which the ECB would accept Greek
government debt as collateral despite it being downgraded to default by the rating agencies3.
3
A Nash Equilibrium for Greece
As before this comes with the warning that a Nash equilibrium does not need to be the actual final outcome of this situation, as
further conditions must be met, as we mentioned earlier.
13
15 June 2011
There are likely to be workarounds to this dilemma for the ECB, potentially of a rather
technical nature. As some of the rating agencies have already suggested there is the
possibility that Greek government bonds are actually not downgraded to default on the day
the voluntary private sector loss participation is announced but rather on the day of the
actual implementation. This potentially leaves the door open for the rating agencies
downgrading Greece to default on a Friday evening and by the time for example a debt
exchange or maturity extension has been executed over the weekend the rating agencies
could upgrade the rating back to non-default. As a result, the ECB would actually not need
to take defaulted government debt as collateral and therefore associated negative
implications, for example a threat to the banking system, could be averted.
Finally, the threat of punishment probably will work less and less over time, as we think
that the banking sector will gradually exit its Greek government bond positions by selling
these (at a manageable trade-off between market prices and par) to less regulated entities,
which will be less sensitive to coercion. Should private sector participation be necessary in
the future it will therefore become more likely to be forced rather than voluntary.
A Nash Equilibrium for Greece
14
15 June 2011
Exhibit 8: Payoff values per combined strategy in the multi-step game (1 of 2)
Each column holds the payoff of the respective player. Each row corresponds to a combination of strategies. The new Nash equilibrium is highlighted in
turquoise.
Strategy
ECB
Banks
IMF
Germany
France
Greece
Rating
Agencies
Dominated
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-3
-3
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-2
-2
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
5 dominated by 5
7 dominated by 1
6 dominated by 1
7 dominated by 3
4 dominated by 13
7 dominated by 5
6 dominated by 5
7 dominated by 7
5 dominated by 13
7 dominated by 9
6 dominated by 9
7 dominated by 11
3 dominated by 29
7 dominated by 13
6 dominated by 13
7 dominated by 15
5 dominated by 21
7 dominated by 17
6 dominated by 17
7 dominated by 19
4 dominated by 29
7 dominated by 21
6 dominated by 21
7 dominated by 23
5 dominated by 29
7 dominated by 25
6 dominated by 25
7 dominated by 27
6 dominated by 31
7 dominated by 29
2 dominated by 63
7 dominated by 31
7 dominated by 34
5 dominated by 38
7 dominated by 36
6 dominated by 34
7 dominated by 38
4 dominated by 46
7 dominated by 40
6 dominated by 38
7 dominated by 42
5 dominated by 46
7 dominated by 44
6 dominated by 42
7 dominated by 46
3 dominated by 62
7 dominated by 48
6 dominated by 46
7 dominated by 50
5 dominated by 54
7 dominated by 52
6 dominated by 50
7 dominated by 54
4 dominated by 62
7 dominated by 56
6 dominated by 54
7 dominated by 58
5 dominated by 62
7 dominated by 60
6 dominated by 58
7 dominated by 62
6 dominated by 64
7 dominated by 64
1 dominated by 128
A Nash Equilibrium for Greece
15
15 June 2011
Exhibit 8: Payoff values per combined strategy in the multi-step game (2 of 2)
Each column holds the payoff of the respective player. Each row corresponds to a combination of strategies. The new Nash equilibrium is highlighted in
turquoise.
Strategy
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
ECB
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
0
-5
Banks
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-8
-3
-3
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-2
-2
IMF
Germany
France
Greece
Rating
Agencies
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
-7
0
0
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-10
-10
-10
-10
-10
-10
-5
-5
-5
-5
-5
-5
-5
-5
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-7
-10
-10
-7
-7
-5
-5
0
0
-5
-5
0
0
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-10
-10
-7
-10
-5
-5
0
-5
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-5
-5
-10
-10
-3
-3
-2
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
0
-2
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
-3
-1
Dominated
5 dominated by 69
7 dominated by 65
6 dominated by 65
7 dominated by 67
4 dominated by 77
7 dominated by 69
6 dominated by 69
7 dominated by 71
5 dominated by 77
7 dominated by 73
6 dominated by 73
7 dominated by 75
3 dominated by 93
7 dominated by 77
6 dominated by 77
7 dominated by 79
5 dominated by 85
7 dominated by 81
6 dominated by 81
7 dominated by 83
4 dominated by 93
7 dominated by 85
6 dominated by 85
7 dominated by 87
5 dominated by 93
7 dominated by 89
6 dominated by 89
7 dominated by 91
6 dominated by 95
7 dominated by 93
2 dominated by 127
7 dominated by 95
7 dominated by 98
5 dominated by 102
7 dominated by 100
6 dominated by 98
7 dominated by 102
4 dominated by 110
7 dominated by 104
6 dominated by 102
7 dominated by 106
5 dominated by 110
7 dominated by 108
6 dominated by 106
7 dominated by 110
3 dominated by 126
7 dominated by 112
6 dominated by 110
7 dominated by 114
5 dominated by 118
7 dominated by 116
6 dominated by 114
7 dominated by 118
4 dominated by 126
7 dominated by 120
6 dominated by 118
7 dominated by 122
5 dominated by 126
7 dominated by 124
6 dominated by 122
7 dominated by 126
6 dominated by 128
7 dominated by 128
Source: Credit Suisse
A Nash Equilibrium for Greece
16
Credit Strategy and Quantitative Research
Eric Miller, Managing Director
Global Head of Fixed Income and Economic Research
+1 212 538 6480
William Porter, Managing Director
Group Head
+44 20 7888 1207
[email protected]
Helen Haworth, CFA, Director
Christian Schwarz, Vice President
+44 20 7888 0757
[email protected]
+44 20 7888 3161
[email protected]
Chiraag Somaia, Associate
Joachim Edery, Analyst
+44 20 7888 2776
[email protected]
+44 20 7888 7382
[email protected]
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Corporate Bond Fundamental Recommendation Definitions
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Credit Suisse’s Distribution of Global Credit Research Recommendations* (and Banking Clients)
Global Recommendation Distribution**
Buy
4%
(of which 92% are banking clients)
Outperform
35%
(of which 95% are banking clients)
Market Perform
45%
(of which 95% are banking clients)
Underperform
15%
(of which 89% are banking clients)
Sell
<1%
(of which 67% are banking clients)
*Data are as at the end of the previous calendar quarter.
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