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Alexander S. Cherny. COHERENT RISKS. AND THEIR APPLICATIONS. PLAN. Why are coherent risks needed? How are coherent risks used?. COHERENT RISKS. Artzner, Delbaen, Eber, Heath (1997) Definition. A coherent risk is a map r ( X ): (i) r ( X + Y ) b r ( X ) +r ( Y );

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Coherent risks

Alexander S. Cherny

COHERENT RISKS

AND THEIR APPLICATIONS


Coherent risks

PLAN

  • Why are coherent risks needed?

  • How are coherent risks used?


Coherent risks

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio


Examples

EXAMPLES

Scenario-based risk:r(X) = -min{X(w1),…,X(wN)},

where N and w1,…,wN are possible scenarios.

[email protected]:r(X) = -E(X|Xbql),

where l(0,1) andql is the l-quantile of X.

[email protected]:(27)r(X) = -Emin{X1,…,XN},

where N and X1,…,XNare independent copies of X.

Numbers in green are the numbers of papers on my website:

http://mech.math.msu.su/~cherny


Operations

OPERATIONS

Maximum:r1,…,rN are coherent risks 

r(X)= max{r1(X),…,rN(X)}

is a coherent risk with D = D1…DN.

Conv. combination:r1,…,rN are coherent risks 

  • r(X)=l1r1(X)+…+lNrN(X)

    is a coherent risk with D = l1D1+…+lNDN.

    Convolution:r1,…,rN are coherent risks 

    r(X) = min{r1(X1)+…+rN(XN):X1+…+XN=X}

    is a coherent risk with D = D1… DN.

(28)


Factor risks i

FACTOR RISKS-I

X-P&L of a portfolio over the unit time period

F - increment of a market factor over this period

Problem:Risk of X driven by F = ?

Definition.(27)Factor risk of X driven by F:

rf(X;F) =r(j(F)), where j(z) =E(X | F=z).

This is a coherent risk with Df = {E(Z|F):ZD}.


Coherent risks

FACTOR RISKS-II

X = X1+…+Xd


Coherent risks

FACTOR RISKS-II

X = X1+…+Xd  rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

[email protected]:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).


Coherent risks

FACTOR RISKS-II

X = X1+…+Xd  rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

[email protected]:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).


Coherent risks

FACTOR RISKS-II

X = X1+…+Xd  rf(X;F) = r(j(F)), where

j(z) = j1(z)+…+jd(z), ji(z) = E(Xi|F=z).

[email protected]:rf(X;F) = -E(j(F)bql),

where qlis the l-quantile of j(F).

[email protected]:rf(X;F) = -Emin{j(F1),…,j(FN)},

where F1,…,FNare independent copies of F.


Coherent risks

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio


Coherent risks

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio


Coherent risks

[email protected]

X=+1 with P=0.96

X=-100 with P=0.04

l=0.05

[email protected](X)=-1

-76 !


Coherent risks

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio


Coherent risks

COHERENT RISKS

Artzner, Delbaen, Eber, Heath (1997)

Definition.A coherent risk is a map r(X):

(i)r(X+Y) b r(X)+r(Y);

(ii) If X b Y, then r(X) r r(Y);

(iii)r(lX) = lr(X) for lr0;

(iv)r(X+m) = r(X)-m for m.

Theorem.ris a coherent risk r(X) = -minQDEQX.

Probabilistic scenarios

Terminal wealth of a portfolio


Quadratic risk i

QUADRATIC RISK-I


Quadratic risk i1

QUADRATIC RISK-I

Do you agree that these two positions have the same risk?

Do you agree that the risk of any position coincides with the risk of the opposite position?


Tail v@r risk of these two positions

Tail [email protected] risk of these two positions:


Quadratic risk ii

QUADRATIC RISK-II

X=-1 with P=0.5 Y=-1 with P=0.5

X=+1 with P=0.5 Y=+0.5 with P=0.48

Y=+13 with P=0.02

EX=0 EY=0


Coherent risks

QUADRATIC RISK-II

X=-1 with P=0.5 Y=-1 with P=0.5

X=+1 with P=0.5 Y=+0.5 with P=0.48

Y=+13 with P=0.02

EX=0, VarX=1EY=0, VarY=7.75

Do you agree that Y is 7 times riskier than X?


Quadratic risk iii

QUADRATIC RISK-III


Coherent risks

QUADRATIC RISK-III

r(X) = -EX+SvarX

is a coherent risk

But there exist better

coherent risks!


Coherent risks

APPLICATIONS

Coherent risks provide a uniform basis for:

  • risk measurement,

  • capital allocation,

  • risk management,

  • pricing and hedging,

  • assessing trades.


Capital allocation

CAPITAL ALLOCATION

X – P&L earned by a company


Capital allocation1

CAPITAL ALLOCATION

X = (X1+…+Xd) – P&L earned by a company

Problem: How is the risk r(X) allocated

between the desks?

r(X1)+…+r(Xd)>r(X) – diversification!

Definition.Risk contribution of Y to X:

rc(Y;X) = -EQ*Y,

where Q*=argminQDEQX.

Capital allocation:rc(X1;X),…, rc(Xd;X).

P&L of a subportfolio

P&L of a portfolio


Examples1

EXAMPLES

Scenario-based risk:r(X) = -min{X(w1),…,X(wN)},

where N and w1,…,wN are possible scenarios.

[email protected]:r(X) = -E(X|Xbql),

where l(0,1) andql is the l-quantile of X.

[email protected]:(27)r(X) = -Emin{X1,…,XN},

where N and X1,…,XNare independent copies of X.

Numbers in green are the numbers of papers on my website:

http://mech.math.msu.su/~cherny


Risk contribution

RISK CONTRIBUTION

Scenario-based risk:rc(Y;X) = -Y(wn*),

where n*=argminn=1,…,NX(wn).

[email protected]:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

[email protected]:rc(Y;X) = -EYn*,

where n*=argminn=1,…,NXn,

(X1,Y1),…,(XN,YN) areindependent copies of (X,Y).

Properties:rc(X1;X)+…+rc(Xd;X) = r(X),

rc(Y;X) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y)  r(X)+ rc(Y;X).


Risk contribution1

RISK CONTRIBUTION

Scenario-based risk:rc(Y;X) = -Y(wn*),

where n*=argminn=1,…,NX(wn).

[email protected]:rc(Y;X) = -E(Y|Xbql),

where qlis the l-quantile of X.

[email protected]:rc(Y;X) = -EYn*,

where n*=argminn=1,…,NXn,

(X1,Y1),…,(XN,YN) areindependent copies of (X,Y).

Properties:rc(X1;X)+…+rc(Xd;X) = r(X),

rc(Y;X) = lime0 e-1[r(X+eY)-r(X)],

YX r(X+Y)  r(X)+ rc(Y;X).


Risk management i

RISK MANAGEMENT-I

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

Theorem. (25) If (X1,…,Xd) is optimal, then

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

whereX = X1+…+Xd.


Risk management i1

RISK MANAGEMENT-I

Problem:E(X1+…+Xd) max,

XiAi – P&Ls available to the i-th desk,

r(X1+…+Xd)bC- firm’s capital.

Theorem. (25) If (X1,…,Xd) is optimal, then

EX1/rc(X1;X) =…= EXd/rc(Xd;X),

 

RAROCc(X1 ; X) RAROCc(Xd ; X),

whereX = X1+…+Xd.


Risk management ii

RISK MANAGEMENT-II

Question: Is it possible to decentralize the

procedure of imposing risk limits?

Yes!

Theorem. (27) If the limits are imposed on the

risk contributions and the desks are allowed

to trade these limits within the firm, then the

equilibrium is an optimal solution, and vice versa.


Pricing and hedging

PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F+x)b0 and x is as small as possible.


Pricing and hedging1

PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F)bx and x is as small as possible.

Price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

Risk-adjusted price:EPF+aVar(X*-F)

Which r to apply?


Pricing and hedging2

PRICING AND HEDGING

F - contingent claim

A – space of P&Ls of possible trading strategies

Problem:Find x and XA such that

r(X-F)bx and x is as small as possible.

Risk-adjusted price: minXAr(X-F)

Hedge: argminXAr(X-F)

Quadratic risk:P – pricing measure

Price:EPF

Hedge: argminXAVar(X-F)

Risk-adjusted price:EPF+aVar(X*-F)

Which r to apply?


Coherent risks

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W) = r(X*+W) = EQ**W

rm(W-F)  -EQ**W+EQ**F if FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q** = argminQDREQW.

Market-modified risk


Coherent risks

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F)  -EQ**W+EQ**F if FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk


Coherent risks

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F)  -EQ**W+EQ**F if FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge


Coherent risks

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F)  -EQ**W+EQ**F if FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure


Coherent risks

Theorem.Ifr(Z)=-minQDEQZ, then

rm(Z) := minXAr(X+Z) = -minQDREQZ,

whereR={Q:EQX=0 XA}.

Risk-adjusted price of F equals

minXAr(X-F) = maxQDREQF = EQ*F

W – P&L of the firm’s overall portfolio

rm(W)= r(X*+W) = EQ**W

rm(W-F)  -EQ**W+EQ**F if FW

 Risk-adjusted price contribution of F to W

equalsEQ**F, where Q**=argminQDREQW.

Market-modified risk

Risk

Hedge

Extreme measure


Coherent risks

STATIC MODEL

Sn – price of the underlying at time n =0,1

W – P&L of a portfolio

r(h(S1-S0)+W)  min, h

Pflug-Rockafellar-Uryasev method: r – [email protected]

l-1E(q-h(S1-S0)-W)+- q min, h,q

m*, h*, q*

Risk:m*

Hedge:h*

Extreme measure:P(  |h*(S1-S0)+W <q*)


Example

EXAMPLE

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0


Example1

EXAMPLE

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))


Example2

EXAMPLE

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))

Risk:-E(f(S1) | S1(a,b))


Example3

EXAMPLE

W = f(S1), f is concave

Find a<b: P(S1(a,b)) = l,

E(S1 | S1(a,b)) =S0

Extreme measure:P( | S1(a,b))

Risk:-E(f(S1) | S1(a,b))

Hedge:-(f(b)-f(a))/(b-a)


Example4

EXAMPLE

W=+1 with P=0.95

W=-19 with P=0.05

Quadratic hedging:

Coherent hedging:


Dynamic model

DYNAMIC MODEL

Sn – price of the underlying at time n =0,…,N

sn – volatility at time n = 0,…,N

(Sn,sn) is a Markov process

Examples: GARCH, SV.

W– P&L produced by a portfolio

X– P&L produced by trading

r(X+W)  min, XA


Dynamic model1

DYNAMIC MODEL

Sn – price of the underlying at time n =0,…,N

sn – volatility at time n = 0,…,N

(Sn,sn) is a Markov process

Examples: GARCH, SV.

W =(W1,..,WN)– stream of payments of a portfolio

X =(X1,..,XN)– stream of payments produced by trading

r(X+W)  min, XA

Theorem.(32) If W corresponds to a portfolio of

European options, then the risk-adjusted price and

the hedge of W at time n are functions of n,Sn,Qn.


Example5

EXAMPLE

Sn = S0 exp{X1+…+Xn}, where Xnare i.i.d.

Wn=jn(Sn), where jnare concave

  • – dynamic Tail [email protected] of order l

    Find 0<a<b: P(exp(X)(a,b))=l,

    E(exp(X)| exp(X)(a,b))=1.

    Q** = P( |exp(X1)(a,b),…, exp(XN)(a,b))


Assessing trades

ASSESSING TRADES

X – P&L of a trade

Quality of X= Reward/Risk

Sharpe ratio:EX/VarX

RAROC: EX/[email protected](X)

Gain-Loss ratio:EX+/EX-,

X+=max(X,0), X-=max(-X,0)

Coherent RAROC:EX/r(X),

 r – coherent risk


Coherent risks

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.


Coherent risks

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.


Coherent risks

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.


Acceptability indices

ACCEPTABILITY INDICES

[email protected] acceptability index:

a(X) = inf{l:E(X|Xbql)>0}-1.

Definition.An acceptability indexis a map

a(X) = max{z[0,) :rz(X)<0},

where (rz)z[0,)is a family of coherent risks

increasing in z.


Comparison of performance measures

COMPARISON OF PERFORMANCE MEASURES

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY  A(X)b A(Y)

Arbitrage consistency:A(X)=+  X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + +


Comparison of performance measures1

COMPARISON OF PERFORMANCE MEASURES

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY  A(X)b A(Y)

Arbitrage consistency:A(X)=+  X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + + (SP1)


Comparison of performance measures2

COMPARISON OF PERFORMANCE MEASURES

A(X) – performance measure

Convexity:A(X)rz, A(Y)rz A(X+Y)rz

Monotonicity:XbY  A(X)b A(Y)

Arbitrage consistency:A(X)=+  X is an arbitrage

Measure Conv. Mon. Arb.

Sharpe ratio +

RAROC +

Gain-Loss ratio + + +

Coherent RAROC + +

Acceptability index + + +(SP1)


Application to pricing and hedging

APPLICATION TO PRICING AND HEDGING

(rz)z[0,) – family of coherent risks increasing in z

A – space of P&Ls of possible trading strategies

W – P&L of a portfolio

For any z[0,), find Xz = argminXArz(W+X)

Find z*such that rz*(W+Xz*)=0

 a(W+X)  max, XA

Hedge of W:Xz*

Risk measure:rz*

Pricing measure: Q**=argminQD*REQW

rz (W+Xz)


Coherent risks

F – P&L of an additional trade

For any z[0,), find X’z =argminXArz(W+F+X)

Find z*’such that rz*’ (W+F+Xz*’)=0

Hedge of F:X’z*’- Xz*

F is profitable  z*’ > z*

Theorem:(SP2)If FW, then F is profitable 

EQ**F > 0.

Fair price of F:EQ**F


Coherent risks

EQ**W= 0

= EQ**F


Summary

SUMMARY

  • Risk measurement Scenarios, [email protected], [email protected]

    Conv. comb., max, convolution

    Factor risks

  • Capital allocation Risk contribution

    Extreme measure

  • Risk management Trading risk limits

    imposed on risk contributions

  • Pricing and hedging Market-modified risk, Q*, Q**

    Pflug-Rockafellar-Uryasev

  • Assessing trades Acceptability indices

    Appl. to pricing and hedging


Coherent risks

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