measuring and managing systemic risk
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Measuring and managing systemic risk
Joseph H T Kim
Dept. of Statistics and Actuarial ScienceU of Waterloo
(Joint work with P. Boyle)
September 2010, Presented at U of Toronto
1/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Recent crisisNeed for new regulation
Recent crisis (2007 – Present?)
Triggered by a liquidity shortfall caused by housing bubble inthe United States
Resulted in the collapse of large financial institutions or the”bail out” of banks by national governments
Over 100 mortgage lenders went bankrupt during 2007 and2008Bear Stearns, Lehman Brothers, Merrill Lynch, Fannie Mae,Freddie Mac, Washington Mutual, Wachovia, and AIG.
Stock market plunge around the world
Considered to be the worst crisis since the Great Depression ofthe 1930s
2/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Recent crisisNeed for new regulation
Some thoughts
Current solvency regulation framework looks at individualfinancial institutions in isolation: e.g., VaR or CTE
However this is inadequate, especially in financially bad times
The dependence among companies are weak when economy isgood, but it becomes strong when economy is bad
Need to measure the risk of the whole financial system as wellas individual company
3/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Recent crisisNeed for new regulation
Possible new regulations in the future
Various risk charge for financial institutions are proposed
System insurance scheme can be financed by financial firms’contribtuions
Each company should pay the premium
But how to set the premium that is clear and objective?
Other questions remain (how to distribute the benefit, whokeeps the premium, possible arbitrage, ...)
We introduce the generalized CoCTE and show how to set thepremium (or risk charge)
4/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Recent crisisNeed for new regulation
Current regulations are pro-cyclical
Current regulations (Basel II, Solvency II), based on standardrisk measures, are recently criticized for its procyclicality
A firm’s required capital is smaller in good times and is largerin bad times, which makes the financial situations of individualfirms worse as business goes down during system crisis
We propose a regime-based counter-cyclical premium whichincreases when the economy is good, drastically reducedduring downturn
5/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
CoVaR by Adrian and Brunnermeier (2009)
Denote the net loss of company i as Xi .
If a system contains d institutions, the system loss is theaggregate loss S = X1 + ...+ Xd
CoVaR of Xi given Xj is defined by
VaRp(Xi )|[Xj = VaRp(Xj )]
If Xi ’s are independent, this is just VaR. In reality they arehardly so
CoVaR measures how bad the loss of Xi when Xj is bad
They measure the risk contribution of i th firm towards systemusing
VaRp(S)|[Xj = VaRp(Xj )]− VaRp(S)
6/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
CoCTE defined
Conditional tail expectation (CTE) is defined as
CTEp(X ) = E (X |X > VaRp(X ))
It is known to be a preferred risk measure due to its coherency
Widely used as risk capital in insurance (e.g., Solvency II)
We propose CoCTE:
CTEp(Xi )|[Xj = CTEp(Xj )]
Measures how bad the loss of the i-th firm when the j-th firmis in trouble
Reveals tail dependence between two entities
7/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Using CoCTE
For our purpose we use the CoCTE as follows
Instead of measuring the risk contribution of each firm towardsystem (this was the CoVaR approach) we measure the capitalneeded for each firm when system is in crisis
CTEp(Xi )|[S = CTEp(S)]− CTEp(Xi )
It is implied that the systemic crisis is identified by eventS > CTEp(S)
This makes more sense because the benefit is explicit, unlikethe CoVaR case
Also CoVaR was used in detecting systemic risk; we useCoCTE to set the premium
8/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Generalized CoCTE
CoCTE is good, but has limitations
It assumes the identical confidence level p for system as wellas each firm
This is not realistic, since systemic crisis tends to come afterfirm’s risk capital is wiped out
Need to set a different level for system crisis, say at q(> p)
Thus we define the generalized CoCTE:
CTEp(Xi )|[S = CTEq(S)], 0 < p < q < 1
9/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Risk charge based on CoCTE
Assume all firms set their risk capital at CTEp (This can berelaxed in practice)
The risk charge per dollar is based on RCR (Risk ContributionRate):
di =CoCTEp,q(Xi , S)− CTEp(Xi )
CTEp(Xi )(1)
Represents relative increase in risk capital due to systemiccrisis
Should be the basis of the actual risk charge
The actual charge (or pure premium) for firm i is
Pi = Pr [S > CTEq(S)]︸ ︷︷ ︸Pr( System Insolvency )
· di · CTEp(Xi )︸ ︷︷ ︸Benefit Amount for for firm i
(2)
10/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Generalized CoCTE under MVN
Generalized CoCTE under MVN is analytically available
So is the RCR:
di =CoCTEp,q(Xi , S)− CTEp(Xi )
CTEp(Xi )
=σi ρi ,S h(zq) + σi
(√1− ρ2
i ,S − 1)h(zp)
µi + σih(zp).
Analytic form not available beyond normality
11/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Graph of RCR under MVN (µi = 0)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−4
−3
−2
−1
0
1
2
Correlation coefficient
p=0.99, q=0.99p=0.95, q=0.99p=0.8, q=0.99
Figure: Graph of the RCR di against the correlation coefficient undernormality
12/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Comments on the graph
Concave and maximum exists at
ρmax =
(h(zq)
h(zp)
)/√(h(zq)
h(zp)
)2
+ 1 (3)
The RCR is negative when ρi ,S < 0 and positive whenρi ,S > 0.
As p increases the RCR decreases, reflecting that conservativecapital should reduce the risk charge
A firm can reduce the RCR, thus the risk charge, by stayingaway from ρmax .
Shifting to the left side means more diversification
Shifting to the right side perhaps means larger market share(For a dominant player, its own risk is the systemic risk)
13/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
Estimating CoCTE via CTE regression
For non-normal cases we estimate the CoCTE based on data
We can adapt the quantile regression for the CTE regression
Consider a sample (xi , yi ), i = 1, . . . , n for regression. thequantile regression coefficients β is specified by
minβ∈R2
n∑i=1
(yi − xiβ) (p − I (yi < xiβ)) (4)
with the minimizing β,
y = xβ = β0 + xβ1 (5)
estimates the p quantile of y .
This applies to CoVaR as the VaR is the quantile
14/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
CoVaRCoCTE
Generalized CoCTE
CTE regression
To apply this to CTE, we transform the original points to(xiyi − xiβ) and collect only the points lying above zero, thenapply the least squares regression.
If the coefficients from this step are γ = (γ0, γ1), the resultingCTE regression is
y = xβ + xγ = x(β + γ) = (β0 + γ0) + x(β1 + γ1)
The gen. CoCTE is simply obtained by setting x = CTEq(S):
CoCTEp,q(Xi , S) = (β0(p)+γ0(p))+CTEq(S)·(β1(p)+γ1(p))
15/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Current procyclical regulationsNeed of countercyclical premium
Current regulations are procyclical
Current regulations (Basel II, Solvency II), based on standardrisk measures, are recently criticized for its procyclicality
There is a big concern however around all of these:Procyclicality of the capital requirement
It means that the firm’s required capital is smaller in goodtimes and is larger in bad times → worsens the financialsituations of individual firms during system crisis
Same can be said for the system insurance premium
All co-risk measures (CoVaR, CTE allocation, CoCTE,generalized CoCTE) can measure the systemic risk and thusserve as premium basis for the system insurance, but they leadto procyclical premiums
16/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Current procyclical regulationsNeed of countercyclical premium
Countercyclical premium
Ideally the system insurance premium should becountercyclical
Means lower premium when economy is bad, and higherpremium when economy is good
Essentially firms should save more in good days
How can we do this?
We attempt to solve this using regime switching models
17/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Regime switching modelPremium under regime switching model
Regime switching models
Popular in modeling non-linear time series to describe businesscycle
Massive amount of literature in econometrics and finance
Several proposals are available regarding how the underlyingregime switches
Markovian. No feedback from past observationsEndogenous. Past history drives future regime changeExogenous. Regime change depends on variables outside data,e.g., economic cycles
We use discrete Markov Regime switching
For our purposes, any reasonable choice would work, as longas the filtered regime probabilities can be computed
18/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Regime switching modelPremium under regime switching model
Risk charge under regime switching economy
Pi (t) =Pr [S(t + 1) > CTEq(S0)]
× [CTEp(Xi (t + 1))|(S = CTEq(S0))− ECi ]
The risk charge = Prob(system insolvency) × Benefit of thesystem fund for Firm i
S0 is the unconditional distn of system wide net loss. Thesystemic crisis threshold is fixed over time
Risk charge is pro-cyclical. In particular, the probability partThe probability Pr [S(t + 1) > CTEq(S0)] can be seen as
K∑k=1
Pr(rt+1 = k) · Pr [S(t + 1) > CTEq(S0)|rt+1 = k]
19/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Regime switching modelPremium under regime switching model
Risk charge under regime switching economy
Pr [S(t + 1) > CTEq(S0)] =K∑
k=1
Pr(rt+1 = k)
×Pr [S(t + 1) > CTEq(S0)|rt+1 = k]
The current regime is denoted by rt = j (j = 1, 2, . . . ,K )
WLOG, we order the regimes based on CTE of S0
CTEq(S0|rt = 1) < CTEq(S0|rt = 2) < . . . < CTEq(S0|rt = k).
Regime 1 represents the best economic state; Regime Krepresents the worst, from aggregate perspective
We manipulate regime probabilities to achievecounter-cyclicality
20/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Regime switching modelPremium under regime switching model
Risk charge under regime switching economy
Define the economic cycle score, a weighted average:∑Kk=1 Pr(rt+1 = k) · B(k)
B(k) is a suitable discrete function that represents economicstate in regime k , e.g., B(k) = k
In 2 regime case, for example, the score lies in [0,1]. Economicboom will give score close to 0; recession gives score close to1 ⇒ Score shows how good economy is at given time
Record the all past scores to create histogram and identify thescore of the current time period as the α quantile.
The counter-cyclical counterpart of the current economy isthe (1− α) quantile in the histogram
Denote the past period that corresponds to the (1− α)quantile by tcc < t + 1.
21/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Regime switching modelPremium under regime switching model
Then the counter-cyclical counterpart of the current regimeprobability vector is given by Pr(rtcc = k), k = 1, . . . ,K .
Replace the pro-cyclical regime probs
Pr(rt+1 = k)
with the counter-cyclical regime probs
Pr(rtcc = k), k = 1, . . . ,K
This converts PC risk charge to CC risk charge
That is, PPCi (t) = Prob(sys insolv) × Benefit for Firm i
becomes PCCi (t) = ProbCC (sys insolv) × Benefit for Firm i
No change to the benefit part to avoid moral hazard
22/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Numerical example
Hypothetical but realistically constructed system
System has three financial assets: Small cap, large cap, andCorporate Bond
Model this economy using Markov-switching vector AR with 3regimes and lag 1 based on monthly returns during 1958-2008
Assume three firms
Firm A: Invest 100% in Small capFirm B: Invest 30% in Small cap and 70% in Large capFirm C: Invest 100% in Large cap
Liability of all firms follows the same bond dynamic
EC is set at CTE 90% of net loss with max 20% of asset toreflect limited ability to raise capital during recession
Systemic crisis is set at CTE 95% of system-wide net loss
23/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
1975 1980 1985 1990 1995 2000 20050
0.2
0.4
0.6
0.8
1Regime probability over time
1975 1980 1985 1990 1995 2000 20050
0.5
1
1.5
2
2.5
3
3.5
4Economic cycle based on regime probability above
Figure: Filtered regime probabilities and economic score24/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
1975 1980 1985 1990 1995 2000 20050
5
10
15
20
25
30
35
40
45
50Stand alone Economic Capital: Required (Black) VS. Actual (Red):
1975 1980 1985 1990 1995 2000 20050
0.5
1
1.5
2
2.5
3
3.5
4Risk charge comparison: Pro−cyclical (Red) VS. counter−cyclical (Blue)
Figure: Systemic risk charge of Firm B (Blue for CC and Red for PC)25/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Comparison of risk charges: PC vs. CC
Correl. between Correl. between Correl. betweenEcon. cycle and Econ. cycle and PC risk charge and
Firm PC risk charge CC risk charge CC risk charge
A 0.617 -0.363 0.102
B 0.524 -0.490 0.097
C 0.628 -0.410 0.050
Table: Correlations between the pro-cyclical risk charge, thecounter-cyclical risk charge and the economic cycle score
26/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Accumulated systemic fund amount
Under Pro-cyclical Under Counter-cyclicalFirm risk charge risk charge
A 1,025.2 1,360.9B 377.0 832.8C 755.2 1175.3
System-wide Amount 2,157.4 3,369.0
Table: Accumulated system risk charge as of Dec. 2008
27/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Concluding remarks
The current economic crisis has prompted an active discussionon creating systemic risk fund financed by participatingfinancial institutions
Systemic risk charge in its natural form is pro-cyclical,worsening the already weak financial position of participants
In this paper a sensible systemic risk charge has been definedbased on the generalized CoCTE
A new counter-cyclical risk charge is proposed via regimeprobability modification
A numerical example shows the proposed method works welland is strongly counter cyclical
28/ 29
BackgroundCoCTE
Beyond procyclicalityRegime switching economy
Numerical example
Thank you
Questions ?
29/ 29
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