Description
The report explaining about Risk capital calculation for an idealized bank.
Risk capital calculation for an idealized bank and the new capital adequacy rules
H? akan Andersson and Andreas Lindell 29 March 2006
Agenda 1. Long-term ?nancial models 2. Description of our model bank 3. Purpose of our risk capital calculation model 4. Set of master scenarios 5. Simulation of the pro?t and loss 6. Simulation of the equity 7. The new capital adequacy rules 8. Simulation of Risk Weighted Assets 9. Simulation of the capital ratio 10. Possible extensions
1
1
Long-term ?nancial models
Calculating Value at Risk Simulate movements of relevant risk factors—Calculate the change in portfolio value in each scenario—Extract desired left quantile from the simulated distribution of changes in portfolio value One-day Value at Risk for a speci?c portfolio Risk factors are interest rates, currency rates, stock quotes etc. Due to the short time horizon, it is feasible to specify a model of the time dynamics of the risk factors. Parameters are reasonably stable over time. Five-year Value at Risk for the entire bank Risk factors are interest rates, currency rates, stock quotes, GDP, in?ation, unemployment ?gures, customer volumes, margins and fees, etc etc. NOT feasible to specify a model of the time dynamics on a stand-alone basis. A huge number of parameters would be required. Also, such a model would only be regarded as a ‘black box’.
2
Alternative approach: Long-term what-if scenarios • Specify a small number of ‘master scenarios’ for the long-term movement of the most important risk factors. • Given such a scenario, model the joint movement of the risk factors between today and the speci?ed horizon. Simulate the changes in portfolio value over time. • Report portfolio statistics for each scenario.
Advantages with the approach: Easy to communicate. Not a ‘black box’. Possible to relate to the calculated results. Drawback with the approach: No probability measure on the what-if scenarios. Hence classical Value at Risk cannot be calculated.
3
2
Description of our model bank
Imagine a typical retail bank with only domestic business activities: • Deposit services (Savings and transaction accounts) • Lending services (Loans to households and small corporates) • Brokerage (Equity funds) • Payment services (VISA, Internet etc)
Further assumptions: Treasury function hedges the residual market risk perfectly. Also, no proprietary trading.
4
3
Purpose of our risk capital calculation model
Questions to be answered (1) Given today’s capital and given a master scenario, what is the probability that the capital will stay above zero between today and the speci?ed risk horizon? (2) Given a small number ? > 0, how large capital is needed today to ensure that the bank survives in each of the speci?ed scenarios with probability at least 1 ? ??
Possible applications of the model • Tool for management and board to determine the required capital level • Tool in the communication with rating agencies and the public • Input to the supervisory process (explained later!)
5
4
Set of master scenarios
Generate a set of J scenarios, each scenario specifying risk factor values at the time points T0 = 0, T1, T2, . . . , TI = T . (For instance, equidistant time points with one year in between.) Here: I = 1 and J = 1. Environmental variables Variable Name Today Change Type of change Short interest rate r(t) r0 ?r Absolute Broad equity index S(t) S0 ?S Relative
Bank speci?c variables Variable Deposit volume Lending volume hh* Lending volume corp Equity position Payment services Costs *hh=Households Name VD (t) VLH (t) VLC (t) ?EQ(t) VP S (t) VC (t) Today Change Type of change VD,0 ?D Relative VLH,0 ?LH Relative VLC,0 ?LC Relative ?EQ,0 ?EQ Relative VP S,0 ?P S Relative VC,0 ?C Relative
6
Brownian bridge We need a continuous-time model of the dynamics of the risk factors between t = 0 and t = T . Bank speci?c variables: Linear (deterministic) trend assumed. Short interest rate and Log(Equity index): Brownian bridge assumed. Digression. A Brownian bridge B from a to b on the time interval [0, T ] is Brownian motion started at a and conditioned to arrive at b at time T . The following representation is suitable for simulation purposes: t t B(t) = a + (b ? a) + ?W (t) ? W (T )? ; T T where W is Brownian motion. Variance function: V (t) = t (1 ? t/T ).
? ?
0 ? t ? T,
7
Trajectories of the Brownian bridge
1.5
1
0.5
0
?0.5
0
10
20
30
40
50
60
70
80
90
100
8
5
Simulation of the pro?t and loss
Schematic picture of the pro?t and loss account (everything in MSEK):
Item 2005 2004 Deposit margins 540 590 Lending margins 780 700 Fund commissions 340 280 Revenues from payment services 110 110 Total revenues 1770 1680 Loan losses -120 -100 Costs -1300 -1200 Total pro?t 350 380 Tax -98 -106 Dividends -126 -137 Pro?t after tax and dividends 126 137
9
Deposit margins Assumptions: • Volume according to master scenario: VD (t) = VD,0 (1 + ?D t/T ). • Margin capped by the interest rate: mD (r) = min (mmax, r). Revenue is equal to volume times margin: PD (t) =
t 0
VD (t )mD (r(t )) dt ,
Lending margins Consider household lending. Assumptions: • Volume according to master scenario: VLH (t) = VLH,0 (1 + ?LH t/T ). • Margin mLH is assumed to be constant (too restrictive!). Revenue is equal to volume times margin: PLH (t) = Corporate lending similar.
t 0
VLH (t )mLH dt .
10
Fund commissions Assumptions: • Fund assets follow the equity index level: VEQ(t) = ?EQ(t)S(t). • Commission mEQ is assumed to be constant. Revenue is equal to fund assets times commission: PEQ(t) =
t 0
?EQ(t )S(t )mEQ dt .
Payment services and costs Revenue from payment services according to the master scenario: PP S (t) = The same is true for the costs: PC (t) = ?
t 0 t 0
VP S (t ) dt .
VC (t ) dt .
11
Loan losses Consider household lending. Defaults in the lending portfolio start to build up as the interest rate increases, so we model the process of default events as a Cox process NLH (t) with stochastic intensity ?(r, t) = f (r)VLH (t), where f is increasing with f (0) = 0. Here: f (r) = Cr. The losses given default (i.e. exposures net recoveries) are modelled as iid variables XLH,i, i = 1, 2, . . .. Thus the total credit loss up to time t is given by
NLH (t) i=1
XLH,i.
Corporate lending similar. Note: Typically, no problems with heavy tails! Digression. A Cox process N (t) is de?ned as follows. A nonnegative random process ?(t) is speci?ed. Conditioned upon its realization, N (t) is a non-homogeneous Poisson process with that realization as its intensity.
12
6
Simulation of the equity
De?nition of equity Equity is the di?erence between assets and liabilities. Denote by C(t) the equity at time t, C(0) = C0. Here we simply assume that C(t) = C0 + PD (t) + PLH (t) + PLC (t) + PEQ(t) + · · · Tax and dividends are not taken into account. Trajectory of the capital process
1000
900
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700
600
500
400
300
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100
0
0
10
20
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60
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80
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More trajectories of the capital process
1500
1000
500
0
0
10
20
30
40
50
60
70
80
90
100
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7
The new capital adequacy rules
Basle II background The main purpose of the capital adequacy rules for banks is to ensure a stable global ?nancial system. Banks are required to hold a capital bu?er to be able to cover potential future large losses. Historical background • The ?rst Basle accord (1988) contained very crude calculation rules designed to capture the credit risk that banks are exposed to. • In the late nineties, the rules were complemented with a framework for capturing market risk. • The new Basle accord, Basle II, contains a much more risk sensitive framework designed to capture all risks that the banks are facing. In Sweden, the rules will be in force from 2007 and on.
15
The three “Pillars” of Basle II P1 Minimum capital requirements for credit risk, market risk and operational risk. Several models to choose from. P2 Supervisory review process with the purpose of addressing all major risks that banks are exposed to. P3 Market discipline. Risk strategies etc should be disclosed to the public.
Schematic picture of the minimum capital requirements The Tier 1 capital is given by the equity C. The Risk Weighted Assets are given through supervisory formulas. • RWACR = Risk Weighted Assets for credit risk • RWAM R = Risk Weighted Assets for market risk • RWAOR = Risk Weighted Assets for operational risk Their sum is simply denoted by RWA. Finally, the Tier 1 capital ratio is given by the quotient C/RWA. It is required that the capital ratio stays above 4%.
16
8
Simulation of Risk Weighted Assets
RWA for credit risk (“Standardised approach”) Under this approach, each loan is assigned a risk weight based on the type of counterparty: Counterparty Risk weight, RW Sovereign 0% Bank Often 20% Corporate According to rating Unrated corporate 100% Mortgage 35% . . . . . . At time t we have RWACR (t) = RWiVL,i(t)
i
where VL,i(t) is the total exposure at time t to counterparties of type i, and where we sum over counterparty types. Here we assume that there are only loans with real estate property as collateral and loans to small corporates in the portfolio. It follows that RWACR (t) = 0.35VLH (t) + 1.00VLC (t).
17
RWA for market risk (“Internal model”) Under this approach, RWAM R is based on the bank’s daily Value at Risk ?gures. Here, we simply keep RWAM R constant in time. RWA for operational risk (“Basic indicator approach”) Under this approach, it is assumed that large revenues indicate vivid business activity, which in turn indicates large operational risk. Therefore, RWAOR is based on the bank’s total revenues, namely the average of the revenues for the three previous years. Here we assume that P +(?3 + t, 0) + P +(0, t) RWAOR (t) = C 3 + where P (t1, t2) is the total revenue between t1 and t2. The ?rst term of the numerator is calculated using historical data, the second term is simulated. Of course, if t > 3 we have RWAOR (t) = CP +(t ? 3, t)/3.
18
9
Simulation of the capital ratio
Management typically speci?es an internal target capital ratio ? well above the required 4%. Hence, it is required that C(t) ?? for all t. RWA(t) This means that we want to study the probability that the process C(t) ? ?RWA(t) reaches zero somewhere within the risk horizon t = T . (Work in progress.) Note that risky loans are punished twice. Compare a mortgage loan and a corporate loan. • The capital C is more a?ected by the corporate loan due to the higher default intensity. • Also, RWA is more a?ected by the corporate loan due to the higher risk weight.
19
10
Possible extensions
More sophisticated bank business • FX trading and foreign business activity—introduces currency risk and complicated dependencies between markets • Proprietary trading—introduces a volatile component that is hard to model • Active management of interest rate positions—introduces strategic interest rate risk • Loans to large corporates—requires di?erent credit risk models
20
Dependency between margins and market shares It is easy to increase the pro?t—just increase margins and fees! In reality, the bank’s market share (i.e., customer volumes) tends to decrease with increasing margins due to competition. Assume that functions VEQ = VEQ (mEQ), etc, can be speci?ed somehow (game theory??). Then it would be interesting to, given a master scenario, study the problem of maximizing E (C(T )) over mD , mLH , mLC , mEQ under the constraint E (C(t) ? ?RWA(t)) ? 0 for all t within the risk horizon.
21
doc_919832069.pdf
The report explaining about Risk capital calculation for an idealized bank.
Risk capital calculation for an idealized bank and the new capital adequacy rules
H? akan Andersson and Andreas Lindell 29 March 2006
Agenda 1. Long-term ?nancial models 2. Description of our model bank 3. Purpose of our risk capital calculation model 4. Set of master scenarios 5. Simulation of the pro?t and loss 6. Simulation of the equity 7. The new capital adequacy rules 8. Simulation of Risk Weighted Assets 9. Simulation of the capital ratio 10. Possible extensions
1
1
Long-term ?nancial models
Calculating Value at Risk Simulate movements of relevant risk factors—Calculate the change in portfolio value in each scenario—Extract desired left quantile from the simulated distribution of changes in portfolio value One-day Value at Risk for a speci?c portfolio Risk factors are interest rates, currency rates, stock quotes etc. Due to the short time horizon, it is feasible to specify a model of the time dynamics of the risk factors. Parameters are reasonably stable over time. Five-year Value at Risk for the entire bank Risk factors are interest rates, currency rates, stock quotes, GDP, in?ation, unemployment ?gures, customer volumes, margins and fees, etc etc. NOT feasible to specify a model of the time dynamics on a stand-alone basis. A huge number of parameters would be required. Also, such a model would only be regarded as a ‘black box’.
2
Alternative approach: Long-term what-if scenarios • Specify a small number of ‘master scenarios’ for the long-term movement of the most important risk factors. • Given such a scenario, model the joint movement of the risk factors between today and the speci?ed horizon. Simulate the changes in portfolio value over time. • Report portfolio statistics for each scenario.
Advantages with the approach: Easy to communicate. Not a ‘black box’. Possible to relate to the calculated results. Drawback with the approach: No probability measure on the what-if scenarios. Hence classical Value at Risk cannot be calculated.
3
2
Description of our model bank
Imagine a typical retail bank with only domestic business activities: • Deposit services (Savings and transaction accounts) • Lending services (Loans to households and small corporates) • Brokerage (Equity funds) • Payment services (VISA, Internet etc)
Further assumptions: Treasury function hedges the residual market risk perfectly. Also, no proprietary trading.
4
3
Purpose of our risk capital calculation model
Questions to be answered (1) Given today’s capital and given a master scenario, what is the probability that the capital will stay above zero between today and the speci?ed risk horizon? (2) Given a small number ? > 0, how large capital is needed today to ensure that the bank survives in each of the speci?ed scenarios with probability at least 1 ? ??
Possible applications of the model • Tool for management and board to determine the required capital level • Tool in the communication with rating agencies and the public • Input to the supervisory process (explained later!)
5
4
Set of master scenarios
Generate a set of J scenarios, each scenario specifying risk factor values at the time points T0 = 0, T1, T2, . . . , TI = T . (For instance, equidistant time points with one year in between.) Here: I = 1 and J = 1. Environmental variables Variable Name Today Change Type of change Short interest rate r(t) r0 ?r Absolute Broad equity index S(t) S0 ?S Relative
Bank speci?c variables Variable Deposit volume Lending volume hh* Lending volume corp Equity position Payment services Costs *hh=Households Name VD (t) VLH (t) VLC (t) ?EQ(t) VP S (t) VC (t) Today Change Type of change VD,0 ?D Relative VLH,0 ?LH Relative VLC,0 ?LC Relative ?EQ,0 ?EQ Relative VP S,0 ?P S Relative VC,0 ?C Relative
6
Brownian bridge We need a continuous-time model of the dynamics of the risk factors between t = 0 and t = T . Bank speci?c variables: Linear (deterministic) trend assumed. Short interest rate and Log(Equity index): Brownian bridge assumed. Digression. A Brownian bridge B from a to b on the time interval [0, T ] is Brownian motion started at a and conditioned to arrive at b at time T . The following representation is suitable for simulation purposes: t t B(t) = a + (b ? a) + ?W (t) ? W (T )? ; T T where W is Brownian motion. Variance function: V (t) = t (1 ? t/T ).
? ?
0 ? t ? T,
7
Trajectories of the Brownian bridge
1.5
1
0.5
0
?0.5
0
10
20
30
40
50
60
70
80
90
100
8
5
Simulation of the pro?t and loss
Schematic picture of the pro?t and loss account (everything in MSEK):
Item 2005 2004 Deposit margins 540 590 Lending margins 780 700 Fund commissions 340 280 Revenues from payment services 110 110 Total revenues 1770 1680 Loan losses -120 -100 Costs -1300 -1200 Total pro?t 350 380 Tax -98 -106 Dividends -126 -137 Pro?t after tax and dividends 126 137
9
Deposit margins Assumptions: • Volume according to master scenario: VD (t) = VD,0 (1 + ?D t/T ). • Margin capped by the interest rate: mD (r) = min (mmax, r). Revenue is equal to volume times margin: PD (t) =
t 0
VD (t )mD (r(t )) dt ,
Lending margins Consider household lending. Assumptions: • Volume according to master scenario: VLH (t) = VLH,0 (1 + ?LH t/T ). • Margin mLH is assumed to be constant (too restrictive!). Revenue is equal to volume times margin: PLH (t) = Corporate lending similar.
t 0
VLH (t )mLH dt .
10
Fund commissions Assumptions: • Fund assets follow the equity index level: VEQ(t) = ?EQ(t)S(t). • Commission mEQ is assumed to be constant. Revenue is equal to fund assets times commission: PEQ(t) =
t 0
?EQ(t )S(t )mEQ dt .
Payment services and costs Revenue from payment services according to the master scenario: PP S (t) = The same is true for the costs: PC (t) = ?
t 0 t 0
VP S (t ) dt .
VC (t ) dt .
11
Loan losses Consider household lending. Defaults in the lending portfolio start to build up as the interest rate increases, so we model the process of default events as a Cox process NLH (t) with stochastic intensity ?(r, t) = f (r)VLH (t), where f is increasing with f (0) = 0. Here: f (r) = Cr. The losses given default (i.e. exposures net recoveries) are modelled as iid variables XLH,i, i = 1, 2, . . .. Thus the total credit loss up to time t is given by
NLH (t) i=1
XLH,i.
Corporate lending similar. Note: Typically, no problems with heavy tails! Digression. A Cox process N (t) is de?ned as follows. A nonnegative random process ?(t) is speci?ed. Conditioned upon its realization, N (t) is a non-homogeneous Poisson process with that realization as its intensity.
12
6
Simulation of the equity
De?nition of equity Equity is the di?erence between assets and liabilities. Denote by C(t) the equity at time t, C(0) = C0. Here we simply assume that C(t) = C0 + PD (t) + PLH (t) + PLC (t) + PEQ(t) + · · · Tax and dividends are not taken into account. Trajectory of the capital process
1000
900
800
700
600
500
400
300
200
100
0
0
10
20
30
40
50
60
70
80
90
100
13
More trajectories of the capital process
1500
1000
500
0
0
10
20
30
40
50
60
70
80
90
100
14
7
The new capital adequacy rules
Basle II background The main purpose of the capital adequacy rules for banks is to ensure a stable global ?nancial system. Banks are required to hold a capital bu?er to be able to cover potential future large losses. Historical background • The ?rst Basle accord (1988) contained very crude calculation rules designed to capture the credit risk that banks are exposed to. • In the late nineties, the rules were complemented with a framework for capturing market risk. • The new Basle accord, Basle II, contains a much more risk sensitive framework designed to capture all risks that the banks are facing. In Sweden, the rules will be in force from 2007 and on.
15
The three “Pillars” of Basle II P1 Minimum capital requirements for credit risk, market risk and operational risk. Several models to choose from. P2 Supervisory review process with the purpose of addressing all major risks that banks are exposed to. P3 Market discipline. Risk strategies etc should be disclosed to the public.
Schematic picture of the minimum capital requirements The Tier 1 capital is given by the equity C. The Risk Weighted Assets are given through supervisory formulas. • RWACR = Risk Weighted Assets for credit risk • RWAM R = Risk Weighted Assets for market risk • RWAOR = Risk Weighted Assets for operational risk Their sum is simply denoted by RWA. Finally, the Tier 1 capital ratio is given by the quotient C/RWA. It is required that the capital ratio stays above 4%.
16
8
Simulation of Risk Weighted Assets
RWA for credit risk (“Standardised approach”) Under this approach, each loan is assigned a risk weight based on the type of counterparty: Counterparty Risk weight, RW Sovereign 0% Bank Often 20% Corporate According to rating Unrated corporate 100% Mortgage 35% . . . . . . At time t we have RWACR (t) = RWiVL,i(t)
i
where VL,i(t) is the total exposure at time t to counterparties of type i, and where we sum over counterparty types. Here we assume that there are only loans with real estate property as collateral and loans to small corporates in the portfolio. It follows that RWACR (t) = 0.35VLH (t) + 1.00VLC (t).
17
RWA for market risk (“Internal model”) Under this approach, RWAM R is based on the bank’s daily Value at Risk ?gures. Here, we simply keep RWAM R constant in time. RWA for operational risk (“Basic indicator approach”) Under this approach, it is assumed that large revenues indicate vivid business activity, which in turn indicates large operational risk. Therefore, RWAOR is based on the bank’s total revenues, namely the average of the revenues for the three previous years. Here we assume that P +(?3 + t, 0) + P +(0, t) RWAOR (t) = C 3 + where P (t1, t2) is the total revenue between t1 and t2. The ?rst term of the numerator is calculated using historical data, the second term is simulated. Of course, if t > 3 we have RWAOR (t) = CP +(t ? 3, t)/3.
18
9
Simulation of the capital ratio
Management typically speci?es an internal target capital ratio ? well above the required 4%. Hence, it is required that C(t) ?? for all t. RWA(t) This means that we want to study the probability that the process C(t) ? ?RWA(t) reaches zero somewhere within the risk horizon t = T . (Work in progress.) Note that risky loans are punished twice. Compare a mortgage loan and a corporate loan. • The capital C is more a?ected by the corporate loan due to the higher default intensity. • Also, RWA is more a?ected by the corporate loan due to the higher risk weight.
19
10
Possible extensions
More sophisticated bank business • FX trading and foreign business activity—introduces currency risk and complicated dependencies between markets • Proprietary trading—introduces a volatile component that is hard to model • Active management of interest rate positions—introduces strategic interest rate risk • Loans to large corporates—requires di?erent credit risk models
20
Dependency between margins and market shares It is easy to increase the pro?t—just increase margins and fees! In reality, the bank’s market share (i.e., customer volumes) tends to decrease with increasing margins due to competition. Assume that functions VEQ = VEQ (mEQ), etc, can be speci?ed somehow (game theory??). Then it would be interesting to, given a master scenario, study the problem of maximizing E (C(T )) over mD , mLH , mLC , mEQ under the constraint E (C(t) ? ?RWA(t)) ? 0 for all t within the risk horizon.
21
doc_919832069.pdf