measuring and managing systemic risk

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

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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

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BackgroundCoCTE


Numerical example


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

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BackgroundCoCTE


Numerical example


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)

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BackgroundCoCTE


Numerical example


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

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BackgroundCoCTE


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)

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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)

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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)

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BackgroundCoCTE


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

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BackgroundCoCTE


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))

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


Numerical example


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]

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BackgroundCoCTE


Numerical example



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

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BackgroundCoCTE


Numerical example



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.

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BackgroundCoCTE


Numerical example


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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


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


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

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BackgroundCoCTE


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

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BackgroundCoCTE


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

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BackgroundCoCTE


Numerical example

Thank you

Questions ?

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measuring and managing systemic risk

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