Cs Portfolio Management Basel Accord Var Cvar Avg Shell

8/3/2019 Cs Portfolio Management Basel Accord Var Cvar Avg Shell

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CASE STUDY: Portfolio Management with Basel Accord (var_dev, cvar_dev)

Background

This case study demonstrates an optimization setup for credit portfolio management. It is basedon papers by Theiler, et al. (2003) and Theiler (2004). Similar optimization models for credit risk

were considered in Andersson, et al. (2001). This model maximizes the expected returns of the

credit portfolio under internal and regulatory loss risk limits. From the banks internal

perspective, credit risks are limited by the economic capital, i.e., the capital resources available to

the bank to cover credit losses. The economic capital usually is defined as a subset of the banksequity. At the same time, the bank needs to limit its credit risk from a regulatory perspective. We

consider the loss risk limitation rules set by the Basel Committee on Banking Supervision. We are

considering the prevailing rules of Basel I, Basel (1988, 1996). However, credit risk weights of

the Basel II rules, Basel (2001), can be easily incorporated in similar way. Banks are charged

capital to cover credit risks of their bank book which are limited by the maximum amount of

regulatory capital applicable to cover these risks. We concentrate on a credit portfolio of thebank book. The credit risk of the bank book is limited by the tier_1, i.e. the core capital, and the

tier_2, i.e. the supplementary capital. The tier_1 capital mainly consists of the core capital of

the bank, plus some other components. The tier_2 capital includes supplementary capital

elements, such as the allowance for loan loss reserves and various long-term debt instruments,

such as subordinated debt, see, Basel (1988), and also United (1998), p. 119. This model

integrates assets involving both market and credit risk under internal and regulatory loss risk

limitations. The capital constraints limit the expected profits of the bank in the planning period.

The less economic and regulatory capital are available, the less risk a bank is able to take, and the

more limited the achievable expected profits are in a business period. We assume a planning

horizon of one year for expected returns, one year for credit risk, and one day for market risk. We

combine different horizons for credit and market risks under the assumption that portfolio

positions are constant for the year and the market risk is the same (is constant) for every day of

this year.

To provide background on risk-based regulations we extracted from United (1998) several

relevant citations:

Credit risk

Banks are required to meet a total risk-based capital requirement equal to 8 percent of risk-

weighted assets. At a minimum, a banks capital must consist of core capital, also called tier 1

capital, of at least 4 percent of risk-weighted assets. Core capital includes common stockholders

equity, noncumulative perpetual preferred stock, and minority equity investments in consolidated

subsidiaries. The remainder of a banks total capital can also consist of supplementary capital,

known as tier 2 capital. This can include items such as general loan and lease loss allowances,

cumulative preferred stock, certain hybrid (debt/equity) instruments, and subordinated debt with a

maturity of 5 years or more. The regulation limits the amount of various items included in tier 1

and tier 2 capital. For example, the amount of supplementary (tier 2) capital that is recognized

for purposes of the risk-based capital calculation cannot exceed 100 percent of tier 1 capital.

...

Under the credit risk rules, the adjustments of asset values to account for the relative riskiness ofa counterparty involve multiplying the asset values by certain risk weights, which are percentages

ranging from 0 to 100 percent. A zero risk-weight reflects little or no credit risk. For example, if a


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bank holds a claim on the U.S. Treasury, a Federal Reserve Bank, or the central government or

central bank of another qualifying Organization for Economic Cooperation and Development

(OECD) country, this asset is multiplied by a factor of 0 percent, which results in no capital being

required against the credit risk from this transaction....For an obligation owed by another commercial bank in an OECD country, a bank must multiply

the amount of this obligation by 20 percent, which has the effect of requiring the bank to hold

capital equal to 1.6 percent of the value of the claim on the other bank. Loans fully secured by a

mortgage on a 1-4 family residential property carry a risk weight of 50 percent, thus requiring the

bank to hold capital equal to 4 percent of the value of the mortgage. For an unsecured obligationowed by a private corporation or individual, such as a loan without collateral, a bank must

multiply the amount of the unsecured obligation by 100 percent, which requires the bank to hold

capital equal to a full 8 percent of the value of the unsecured obligation.

...

To adjust for credit risks created by financial positions not reported on the balance sheet, the

regulations provide conversion factors to express off-balance sheet items as an equivalent on-balance sheet item, as well as rules for incorporating the credit risk of interest-rate, exchange-rate,

and other off-balance sheet derivatives. These positions are converted into a credit equivalent

amount, and then the standard loan risk-weight for the type of customer is applied. The risk-

weight is applied according to the type of obligor, except that in the case of derivatives the

maximum risk-weight is 50 percent.

...

In both the banking and securities/futures sectors, capital regulations contain formulas that apply

single risk-weightings to a broad range of riskiness within a single category. For example, in

banking, the same 8 percent capital requirement is imposed on all unsecured loans to private

commercial borrowers regardless of individual creditworthiness, with the result that a high-

risk/high-return loan carries no more regulatory capital than a low-risk/low-return loan. As a

result, the regulation might give firms an incentive to seek the highest returns within a broad class

regardless of underlying risk; or to adjust activities (e.g, develop new products and/or change

operations or corporate structures) in a way that reduces or escapes capital requirements. In other

words, firms may adjust business to achieve the lowest regulatory capital cost rather than an

optimal balance of risk and capital. Also, the securities net capital rule requires registered

broker-dealers to apply a 100-percent haircut to any portion of the trading profits, to the extent

the profits are unsecured, reflecting SECs emphasis on liquidity in its net capital rule.

...

All banks are required to calculate their credit risk for assets, such as loans and securities; and

off-balance sheet items, such as derivatives or letters of credit. The credit risk calculation assigns

all assets and off-balance sheet items to one of four broad categories of relative riskiness

(0, 20, 50, or 100 percent) according to type of borrower/obligor and, where relevant, the nature

of any qualifying collateral or guarantee. Off-balance sheet items are converted into credit

equivalent amounts. The assets and credit equivalent amount of off-balance sheet items in each

category are multiplied by their appropriate risk-weight and then summed to obtain the total risk-

weighted assets for the denominator of the credit risk-based capital ratio. Capital, the numerator

of the capital ratio, is long-term funding sources for the bank that are specified in the regulations.

A bank is to maintain a total risk-based capital ratio (total capital/risk-weighted assets) of at least

8 percent....


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The credit risk regulation requires the use of two sets of multipliers. One set of multipliers

places each off-balance sheet item into one of four categories and converts items in each category

into asset equivalents. These conversion factors are multiplied by the face or notional amount of

the off-balance sheet items to determine the credit equivalent amounts. In addition, forderivatives, these credit equivalent amounts are the value of the banks claims on thecounterparties plus add-on factors to cover the potential future value of the derivative contracts.

Then the other set of multipliers applies the risk-weights to assets and off-balance sheet credit

equivalent amounts according to the type of borrower/obligor (and, where relevant, the nature of

any qualifying collateral or guarantee). The sum of the risk-weighted assets in all categories is the

credit risk-weighted assets for the bank....

Market risk.

Market risk consists of general market and specific risk components. To determine the market

risk-equivalent assets, the risk or capital charges must be calculated for both components.

Market risk capital charges are based on general market and specific risks. Examples of general

market risk factors are interest rate movements and other general price movements. Capitalcharges for general market risks are to be based on internal models developed by each bank to

calculate a VAR estimate, i.e., potential loss that capital will need to absorb. The internal VAR

estimate for general market risks is to be based on statistical analyses that determine the

probability of a given loss, based on at least 1 year of historical data. This VAR estimate is to be

calculated daily using a 99 percent one-tailed confidence interval with a price shock equivalent to

a 10-business day movement in rates and prices; i.e., 99 percent of the time the calculated VAR

would not be exceeded in a 10-day period.

...

Specific risk arises from factors relating to the characteristics of specific issuers of instruments.

Specific risk factors reflect both idiosyncratic price movements of individual securities and

event risk from incidents, such as defaults or credit downgrades, which are unique to the issuer

and not related to market factors. If a banks internal model does not capture all aspects of

specific risk, an add-on to the capital charge is required for specific risk. Specific risk estimates

based on internal models are subject to adjustments based on the precision of the model.

The total market risk capital charge is the sum of the capital charges for general market and

specific risk. The total market risk capital charge is based on the larger of the previous days

VAR estimate and the average of the daily VAR estimates for the past 60 days times the

multiplier. The multiplier ranges from 3 up to a maximum of 4 depending on the results of

backtesting.17 Market risk-equivalent assets are the total market risk capital charges multiplied

by 12.5.

...

Application of the market risk capital ratio requires the use of a two-part test. The sum of tiers 1,

2, and 3 capital must equal at least 8 percent of total adjusted risk-weighted assets. The tier 3

capital in this sum is only to be allocated to cover market risk. In addition, the sum of tier 2 and

tier 3 capital for market risk may not exceed 250 percent of tier 1 capital allocated for market risk.

The regulation includes other restrictions on the use of tier 2 and 3 capital.


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Figure 1. Regulatory Capital.

References

Andersson, F., TMausserT, H., Rosen, D., and S. Uryasev (2001): Credit Risk Optimizationwith Conditional Value-At-Risk Criterion. Mathematical Programming, Series B 89, 273-291.

Basel committee on Banking Supervision (1988): International convergence of capitalmeasurement and capital standards, Basel, July 1988.

Basel committee on Banking Supervision committee (1996): Amendment to the capi-tal accord to incorporate market risks, Basel, January 1996.

Basel Committee on Banking Supervision (2001): Consultative Document. The New BaselCapital Accord, January 2001, Basel, January 2001.

Theiler, U., Bugera, V., Revenko, A., and S. Uryasev (2003): Regulatory Impacts on CreditPortfolio Management. Leopold-Wildburger, U. et al. (Eds.), Operations Research

Proceedings 2002, Springer, Berlin, 335-340.

Theiler, U. (2004): Risk Return Management Approach for the Bank Portfolio in: Szego, G.(Ed.), Risk Measures for 21st Century, John Wiley & Sons, Chichester, 403-430.

United States Accounting Office (1998): Risk-Based Capital - Regulatory and Industry Ap-proaches to Capital and Risk, Washington, July 1998.

Several papers in this list can be downloaded from:

http://www.ise.ufl.edu/uryasev/pubs.html#b http://www.ursula-theiler.de/publications.htm TUhttp://www.gloriamundi.org/UT

Credit risk of bank book Market risk of trading book

Regulatory capital

Tier 1 and tier 2 capital

used for credit riskTier 1 and tier 2capital

Tier 3 capital


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Notations

= number of instruments (bonds) in the portfolio;I i={1,,I} index of instruments in the

portfolio;

= number of scenarios;J j={1,,J} index of scenarios;

1x ( ... )

I= x , , x = vector of exposures (in currency) to instruments i=1,I ;

= lower bound on exposure to instrumentil i ;

= upper bound on exposure to instrumenti

u i ;

iq = present value (price) ofi-th instrument;

ir = rate of return (per year) ofi-th instrument in the absence of risk (for instance, yield of the

bond);

bb

ij = future value (in one year) ofi-th instrument in the bank book under the credit risk scenarioj

accounting for credit migration and default;

1

bb

ijbb

ij

i

rq

= = = = = rate of return (per year) ofi-th instrument in the bank book under the credit risk

scenarioj accounting for credit migration and default.

(((( ))))r , ...,bb bb bbj 1j Ijr r==== = vector of rates of return (per year) of instrument i=1,I in the bank bookunder the credit risk scenarioj ;

( )tb

ijr t = rate of return (per 10 trading days) ofi-th instrument in the trading book under the

market risk scenarioj ;

(((( ))))r , ...,tb tb tbj 1j Ijr r==== = vector of rates of return (per 10 trading) of instrument i=1,I in the tradingbook under the credit risk scenarioj ;

0

(x, )I

bb bb

j ij i

i

L r r x====

= = = = = bank book loss under the credit risk scenarioj;

(((( ))))0

x,I

tb tb

ij i

i

L r r x====

= = = = = trading book loss (per 10 trading days) under the market risk scenarioj ;

= confidence level for VaR deviationtb

for trading book;


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= confidence level for CVaR deviationbb

for bank book;

Tier kC = available Tier- kcapital, k=1, 2,3;

a

kx = used for risk management purposes Tier- kcapital, k=1,, 3 (free additional variables);

cr

iw = regulatory credit risk capital weight for security i;

sp

iw = regulatory specific market risk weight for security i;

mrw = regulatory weight for market risk;

econC = maximum amount of economic capital available to cover internal loss risk

(measured by CVaR deviation (((( ))))_ (x,r )bbCVaR DEV L ) .

Simulation of Scenarios

Yearly credit risk scenarios of bond returns,bb

ijr , accounting for credit migration and default can

be simulated using standard methodologies, including CreditMetrics. 10-day market risk

scenarios,tb

ijr , can be calculated with historical Monte Carlo simulations.

Optimization Problem

maximizing estimated return (without risk)

=

I

i

iixr1

max (CS.1)

subject to

internal constraint on credit risk

(((( ))))_ (x,r )bb econCVaR DEV L C (CS.2)

regulatory constraint on capital covering credit risk

1 2

1

I

cr a a

i i

i

w x x x====

= += += += + (CS.3)

regulatory constraint on capital covering market risk


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(((( )))) 3 1 1 2 20

_ (x,r ) ( ) ( )I

p mr tb a a a

i i Tier Tier

i

w x w VaR DEV L x C x C x

====

+ + + + + + + + + + + +

(CS.4)constraint limiting unused Tier-2 + used Tier-3 capital

vs. unused Tier-1 capital

3 2 2 1 1( ) 2.5 ( )

a a a

Tier Tierx C x C x

+ + + + (CS.5)

constraint limiting Tier-2 vs. Tier-1 capital

2 1

a ax x (CS.6)

upper/lower bounds on exposures

, 1, ;i i i

l x u i I = = = = KKKK (CS.7)

bounds on usedTier- k capital

0a

k Tier kx C , k=1,,3 . (CS.8)

Comment

According to the Basel accord, see, United (1998), The total market risk capital charge is basedon the larger of the previous days VaR estimate and the average of the daily VaR estimates for

the past 60 days of the minimal return over 10 trading days. As a proxy for this VaR estimate,we considered in the model the VaR estimate of 10 trading days returns. This is an optimistic

estimate of the value which should be included in the model. After solving the optimization

problem the actual risk constraints can be verified for the optimal portfolio. If the actual VaR

constraint included in regulations is not satisfied, then the coefficientmrw can be increased and

the optimization problem can be solved with a higher weight for the market risk.

Implementation within Portfolio Safeguard

Initial DataNumber of instruments in the portfolio, I = 6.

Number of scenarios for internal risk = 10000.Number of scenarios for regulatory risk = 2500.Expected_returnsare in the matrix matrix_returns.


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Scenarios for internal risk are in the matrix matrix_bank_book_scenarios.


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Scenarios forregulatory risk are in the matrix matrix_trading_book_scenarios.


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Regulatory credit risk capital weights,cr

iw , are in the matrix

matrix_credit_risk_capital_weights.

Regulatory specific market risk weight, pi

w , are in the matrix

matrix_specific_market_risk_weights.

Regulatory weight for market risk, mrw = 3.

Lower bounds on exposures in the constraint (CS.7) are in the point point_basle_accord_lb.


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Upper bounds on exposures in the constraint (CS.7) are in the point point_basle_accord_ub.


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Lower and Upper bounds on additional variables,a

kx , k=1,,3, in the constraint (CS.8) as

well as Lower and Upper bounds on exposures in the constraint (CS.7) are in the Box of

Variables variables_1 in the columns LB and UB respectively.

You can view initial data in the File and Data modes. Youcan find the Box of

Variables in the Problems mode.

Functions

The matrix matrix_returns is used for building the Linear functionlinear_return_without_risk incorporated in objective function (CS.1).


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The matrix matrix_bank_book_scenarios is used for building TCVaR DeviationT for Loss

function cvar_dev_internal_constraint_credit_risk incorporated in constraint (CS.2).

The matrix matrix_trading_book_scenarios is used for building TVaR DeviationT for Loss

function var_dev_trading_book incorporated in constraint (CS.4).


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The matrix matrix_credit_risk_capital_weights is used for building the Linear function

linear_capital_covering_credit_risk incorporated in constraint (CS.3).


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The matrix matrix_specific_market_risk_weights is used for building the Linear function

linear_specific_market_risk incorporated in constraint (CS.4).


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Variable functions variable_x1a and variable_x2a are used for modeling Tier-1 and Tier-2

capitals respectively incorporated in constraints (CS.3)-(CS.6), and (CS.8).

Variable function variable_x3a is used for modeling Tier-3 capital incorporated in constraints(CS.4) - (CS.5), and (CS.8).


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You can view information about functions in the File, Functions, and Problems modes.

Problem and Elements of Problem

You can see the structure of the problem (CS.1) - (CS.8) in the Problems mode.


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Name of the problem begins with the string problem_Basle_Accord_C_econ and terminates by

the value of theecon

C in the right-hand side of the constraint (CS.2) (in the pictureecon

C = 65).

The problem consists of the following Elements of Problem:

Objective objective_Basle_Accord_return corresponding to (CS.1) Constraint constraint_internal_credit_risk corresponding to (CS.2) Constraint constraint_capital_covering_credit_risk corresponding to (CS.3) Constraint constraint_capital_covering_market_risk corresponding to (CS.4) Constraint constraint_Tier1_Tier2_Tier3 corresponding to (CS.5) Constraint constraint_Tier1_Tier2 corresponding to (CS.6) Box of Variables corresponding to (CS.7) - (CS.8)The icon means that the problem is of maximization type.

The Objective objective_Basle_Accord_return (CS.1) includes the TLinear functionlinear_return_without_risk.


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The Constraint constraint_internal_credit_risk (CS.2) includes CVaR Deviation for Loss(Cvar_Dev) function cvar_dev_internal_constraint_credit_risk.

The lower bound in constraint_internal_credit_risk (CS.2) is -Infinity (see the third row in

the right-hand side of the Problems screen) and the upper bound in this constraint,econ

C , is set

to 65 (see the first row in the right-hand side of the Problems screen). This value is used as thesuffix in the name of the problem.

The Constraint constraint_capital_covering_credit_risk (CS.3) is modeled in thefollowing modification:


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1 2

1

0I

cr a a

i i

i

w x x x====

= = = = . (CS.3)

The left-hand side of(CS.3) is a linear combination of the following three functions:Linear function linear_capital_covering_credit_risk included into

constraint_capital_covering_credit_risk with the coefficient 1

Variable function variable_x1a included into constraint_capital_covering_credit_risk with the coefficient -1

Variable function variable_x2a included into constraint_capital_covering_credit_risk with the coefficient -1

The equality to zero in (CS.3) is set by the zero lower bound (see the fifth row in the right-hand

side of the Problems screen) and the zero upper bound (see the first row in the right-hand side

of the Problems screen).

The Constraint constraint_capital_covering_market_risk (CS.4) is modeled in thefollowing modification

(((( ))))1 2 3 1 20

_ (x,r )I

sp a a a mr tb

i i Tier Tier

i

w x x x x w VaR DEV L C C

====

+ + + ++ + + ++ + + ++ + + + . (CS.4)

The left-hand side of(CS.4) is a linear combination of the following five functions:


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Linear function linear_specific_market_risk included intoconstraint_capital_covering_market_risk with the coefficient 1

Variable function variable_x1a included intoconstraint_capital_covering_market_risk with the coefficient 1

Variable function variable_x2a included intoconstraint_capital_covering_market_risk with the coefficient 1

Variable function variable_x3a included intoconstraint_capital_covering_market_risk with the coefficient -1

VaR Deviation for Loss function var_dev_trading_book included intoconstraint_capital_covering_market_risk with the coefficient mrw = 3

The lower bound in the constraint_capital_covering_market_risk (CS.4) is -Infinity (see

the seventh row in the right-hand side of the Problems screen) and the upper bound in thisconstraint,

1 2Tier TierC C

++++ , is set to 10 ( 1 210, 0)Tier TierC C = == == == = ) (see the first row in the

right-hand side of the Problems screen).

The Constraint constraint_Tier1_Tier2_Tier3 (CS.5) is modeled in the followingmodification

1 2 3 1 22.5 2.5

a a a

Tier Tierx x x C C

+ + + + . (CS.5)


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The left-hand side of(CS.5) is a linear combination of the following three functions:Variable function variable_x1a included into constraint_Tier1_Tier2_Tier3 with the

coefficient 2.5

Variable function variable_x2a included into constraint_Tier1_Tier2_Tier3 with thecoefficient -1

Variable function variable_x3a included into constraint_Tier1_Tier2_Tier3 with thecoefficient 1

The lower bound in constraint_Tier1_Tier2_Tier3 (CS.5) is -Infinity (see the fifth row inthe right-hand side of the Problems screen) and the upper bound in this constraint,

1 22.5

Tier TierC C

, is set to 25 ( 1 210, 0)Tier TierC C = == == == = ) (see the first row in the right-hand

side of the Problems screen).

The Constraint constraint_Tier1_Tier2 (CS.6) is modeled in the following modification

2 1 0a a

x x (CS.6)

The left-hand side of(CS.6) is a linear combination of the following two functions:Variable function variable_x2a included into constraint_Tier1_Tier2with the

coefficient 1

Variable function variable_x1a included into constraint_Tier1_Tier2with thecoefficient -1


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The lower bound in constraint_Tier1_Tier2 (CS.6) is -Infinity (see the fourth row in the

right-hand side of the Problems screen) and the upper bound in this constraint is set to 0 (seethe first row in the right-hand side of the Problems screen).

Constraints (CS.7) - (CS.8) are modeled in the Box of Variables (see the table in the right-hand side of the Problems screen).


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The table consists of five columns:

Id of variableName of variableLower Bound (LB) on variable (Constrants (CS.7) - (CS.8))Value of variableUpper Bound (UB) on variable (Constrants (CS.7) - (CS.8))If the problem was not optimized, the column Value is empty, otherwise it contains optimal

solution of the problem.

Generation of Optimal Solutions of the Problem

The problem was run with several values ofecon

C = 35, 40, 45, 50, 55, 60, 65 in the

constraint (CS.2).

For every value ofecon

C we set as the upper bound of the

constraint_internal_credit_risk and we rename the last two digits in the name of theproblem in the Problems mode. Then, we run the modified problem in the Optimization mode.

Correspondingly, seven optimal points were generated. Name optimal points are labeled

by the values ofecon

C :

point_problem_Basle_Accord_C_econ_35,

point_problem_Basle_Accord_C_econ_40,point_problem_Basle_Accord_C_econ_45,


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point_problem_Basle_Accord_C_econ_50,point_problem_Basle_Accord_C_econ_55,

point_problem_Basle_Accord_C_econ_60,

point_problem_Basle_Accord_C_econ_65.Components of these points are expressed in currency. You can view these points in the

File or Data modes.

Analyses

Functions and generated optimal solutions are embedded in the analysis under the Analysis mode.


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All functions included in the analysis were evaluated on all included points (see columnValue in the table at the right-hand side of the Analyses screen built for one selected

point point_problem_Basle_Accord_C_econ_35).

Graphs

Graphs mode presents nine charts:

Graph graph_efficient_frontier shows dependence of the portfolio return vs. CVaR Deviationof credit risk (see constraint (CS.2))


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Graph graph_portfolio_RORAC shows dependence of the portfolio Return on RiskAdjusted Capital (RORAC) as defined in Theiler (2004) vs. bound

econC in constraint

(CS.2); bound valuesecon

C are reflected in the names of the points.


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Graph graph_portfolio_ROE shows dependence of the portfolio Return on Equity (ROE)as defined in Theiler (2004) vs. bound

econC in the constraint (CS.2); bound values

econC

are reflected in the names of the points.


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The portfolio Return on Equity (ROE) is calculated as the ratio of the function

linear_return_without_risk to the sum (x1a+x2a+x3a). The first function is available in

the problem formulation, while the sum is not available. To calculate the sum(x1a+x2a+x3a) we defined the constraint constraint_sum_Tier in the Function mode.


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The Constraint constraint_sum_Tier is a linear combination of the following threefunctions:

Variable function variable_x1a included into constraint_sum_Tierwith the coefficient1

Variable function variable_x2a included into constraint_sum_Tierwith the coefficient1

Variablefunction variable_x3a included into constraint_sum_Tierwith the coefficient1

The lower bound in the constraintconstraint_sum_Tier is -Infinity (see the first row in the

right-hand side of the Functions screen), and the upper bound in this constraint is set toInfinity(see the fifth row in the right-hand side of the Functions screen). This constraint is notused in the problem formulation.

Graph graph_marginal_vs_exposure shows marginal and exposures for the four assetswith non-zero exposures in the optimal point point_problem_Basle_Accord_C_econ_40(similar graphs can be found in Andersson, et al. (2001)). To get the exact numericalinformation about the point coordinates in the graph, left-click on a point in the graph.


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Graphgraph_asset_ROE shows the dependence of the Return on Equity (ROE) of a singleasset as defined in Theiler, et al. (2003) and Theiler (2004) vs. components of the pointpoint_problem_Basle_Accord_C_econ_40.


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Graph graph_asset_RORAC shows dependence of the Return on Risk Adjusted Capital(RORAC) of a single asset as defined in Theiler, et al. (2003) and Theiler (2004) vs.components of the point point_problem_Basle_Accord_C_econ_40.


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Graph graph_exposures shows the structure of several of the optimal points (withecon

C =35, 45, 65).


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The histogram histogram_bank_book_loss" shows the histogram of the credit risk forthe selected optimal points with

econC =40.


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The histogram histogram_trading_book_loss" shows the histogram of the market riskfor the selected optimal points with

econC =40.

Documents

Cs Portfolio Management Basel Accord Var Cvar Avg Shell