Christopher C. Finger Austrian Workshop on Credit Risk Management

ViennaFebruary 2, 2001

Enhancing Monte Carlo Techniquesfor Economic Capital Estimation

Outline

• Introduction– Quantities of interest -- portfolio capital, marginal capital

– Troubles with direct Monte Carlo

• Dealing with size -- portfolio compression

• Dealing with the model – importance sampling

• Dealing with the model – analytic marginal capital

• Conclusions

Model-based risk capital

• A natural way to define risk capital is as a level required to guarantee solvency with some (high) degree of confidence.

• Once capital is established for the portfolio, examine the contribution to capital of new positions or increased lines.

• Applications of capital contributions– Internal credit charges or capital allocation, giving hurdle rates

of return or capital budgets– Pricing -- charge for addition to capital, not just expected loss

VaR as risk capital

5500 5600 5700 5800 5900 6000

Expected horizon value or total notional value

Portfolio value at horizon

Worst case horizon value

at level p

Risk capital at level p

Holding risk capital in this way assures that the likelihood of bankruptcy-causing losses is p.

Contribution of a new exposure to portfolio risk capital

Portfolio value at horizon

Increase in capital comes from an increase in the total portfolio value and a decrease in the worst case level.

Risk capital for base portfolio Distribution of base portfolio plus

new exposure

Risk capital for new portfolio

5500 5600 5700 5800 5900 6000

VaR as capital

• Analytic shortcuts are available for marginal standard deviation, but marginal VaR is more difficult.

• A common practice is to define capital as a multiple of standard deviation and use the previous results.

• An established but little used result:– The derivative of VaR with respect to a single exposure weight

is the conditional expectation, given that the realized loss is VaR, of the loss on the exposure in question.

• This leads to the base capital term, but the size penalty is more difficult to obtain.

The trouble with standard Monte Carlo

• The model presents a tough problem:– Small default probabilities– Discrete exposure distributions– Portfolio distribution smoothes out very slowly

• Typical applications make things harder– Large portfolios– Economic capital = extreme portfolio percentiles– Focus on capital contributions; values can be comparable to

portfolio MC error

• “Going faster when you’re lost don’t help a bit.”

Outline

• Introduction– Quantities of interest -- portfolio capital, marginal capital

– Troubles with direct Monte Carlo

• Dealing with size -- portfolio compression

• Dealing with the model – importance sampling

• Dealing with the model – analytic marginal capital

• Conclusions

Naïve compression on ISDA benchmark portfolio

• Initial portfolio– $30 billion total value, 1680 exposures– Investment grade, average rating of A– Diversified across nine industries, average correlation of 43%

• Compressed portfolio– Maintain largest 2.3% of exposures– Bucket remaining exposures homogeneously by industry/rating– Resulting portfolio has 478 (28% of 1680) exposures

Size distribution of original and compressed portfolios

0 20% 40% 60% 80% 100%0

20%

40%

60%

80%

100%

Exposures

Cum

ulat

ive

size

CompressedOriginal

20% of exposures account for

70% of portfolio size

Compressed portfoliois almost homogeneous

Comparison of portfolio results

32.3 32.5 32.7 32.90

20%

40%

60%

80%

100%

Portfolio value ($B)

Cum

pro

b

Original Compressed

27 28 29 30 31 320

0.2%

0.4%

0.6%

0.8%

1.0%

Cum

pro

b

Near perfect fit in middle of distribution

Very good fit in tails as well

Portfolio value ($B)

Comparison of portfolio results

• Mean ($B)

• St. Dev. (bp)

• 5% loss (bp)

• 1% loss (bp)

• 0.1% loss (bp)

Original

32.7

45.9

11.2

150

633

Compressed

32.7

46.2

11.4

150

679

% Diff

0

0.6

1.8

-0.2

7.6

All simulation estimates are within one standard error.

Comparison of marginal statistics

• Add an additional exposure in the most concentrated industry– for each investment grade rating– “small” exposure

• average size in the base portfolio ($18M, 0.25%)– “large” exposure

• maximum size in the base portfolio ($74M, 0.06%)

• Capital statistics– increase in portfolio standard deviation– increase in 0.1% loss

• Report increase as percentage of the new exposure size

Aaa Aa A Baa0

5%

10%

15%

20%

25%0.1% loss, large exposure

Aaa Aa A Baa0

5%

10%

15%

20%

25%0.1% loss, small exposure

Aaa Aa A Baa0

0.2%

0.4%

0.6%

0.8%

1.0%St. Dev., large exposure

Comparison of marginal statistics

Original Compressed

Overestimation of capital with

compressed portfolio

Underestimation of capital with

compressed portfolio

The real problem is the lack of convergence with any method.

Aaa Aa A Baa0

0.2%

0.4%

0.6%

0.8%

1.0%St. Dev., small exposure

Outline

• Introduction– Quantities of interest -- portfolio capital, marginal capital

– Troubles with direct Monte Carlo

• Dealing with size -- portfolio compression

• Dealing with the model – importance sampling

• Dealing with the model – analytic marginal capital

• Conclusions

BBBCurrent state

AAA AA A BBB BB B CCC DefaultPossible statesat horizon

100.9% 100.8% 100.7% Par 97.5% 95.8% 83.2% Rec.Instrument value

0.00% 0.11% 5.28% 86.71% 6.12% 1.27% 0.23% 0.28%Probabilities

(determined exogenous to model)

Overview of CreditMetricsSingle exposures follow a discrete distribution

Overview of CreditMetricsCorrelations driven by asset value distributions

• Assume a connection between asset value and credit rating.

Asset return over one yearB BB ZA Z Z AA Z ZBBB ZDef

Set first threshold so tail contains

default probability.

ZCCC

Second threshold so next region

contains CCC probability.

• Transition probabilities give us asset return “thresholds”.

Obligor 1

Obligor 2

Similar asset returns produce joint defaults

Overview of CreditMetricsOne correlation parameter gives all joint probabilities

Opposite asset returnsproduce different credit moves.

Equity factor model gives obligor correlationsbased on mappings to industry indices.

… somewhat on specific

movements Idio-

syncratic

BankingIndex

Obligor 1

First obligor depends strongly on its industry...

Obligor 2

BeverageIndex

Second obligor depends weakly on its industry...

Idio-syncratic

… and mostly on specific

movements Industries

are correlated

With few factors, conditioning on market move makes many calculations easier.

-3 -2 -1 0 1 2 3

Unconditional asset distribution

Asset distribution conditional on

down factor move

Area is conditional default probability

Conditional on factor move, all rating changes are independent.

To be more specific, at least in a simple case …

• Default condition -- obligor assets less than default thresholdZ

• Represent obligors through regressions on a common index.

iiwwXZ 21

• Given a value for the index, conditional default condition leads to conditional default probability

21 wwXZ

21

)(w

wXXp

• Given X, defaults are conditionally independent and portfolio follows a binomial distribution.

Example portfolio

• Proxy a large bank lending book by two homogeneous groups (and one common actor):– A-rated -- 1700 exposures representing 75% of holdings,

20% asset correlation within group– BB-rated -- 400 exposures representing 25% of holdings,

50% asset correlation within group– 32% asset correlation between groups

• Total notional of $60B, “unit exposure” of $25M

• Higher concentration in investment grade, but lower grade exposures are larger and more highly correlated.

Biggest issue is high sensitivity to extreme factor moves

-3 -2 -1 0 1 2 34410

4420

4430

4440

4450

Factor return

Por

tfolio

val

ue

Many factor scenarios where value does not change

Value changes the most where we do not simulate much

Importance sampling involves “cheating” and forcing the scenarios where they are most interesting

New scenarios capture sensitive area better

-5 -4 -3 -2 -1 0 1 2 3 -5 -4 -3 -2 -1 0 1 2 34350

4370

4390

4410

4430

4450

Shift factor scenarios into region where portfolio is more sensitive

Importance sampling stated in mathematical terms

• Goal is to estimate the expectation of V=v(X) where X~f

• Trick is to choose g to reduce the variance of the estimate

Straight MC Imp Sampling

Integral expression

Random variates

Estimate V

dxxfxv )()( dxxgxgxf

xv )()()(

)(

fX i ~ gYi ~

i

iXvn

1

ii

i

i YvYgYf

n1

2,000 simulations,optimized for 5% loss

Result is greater precision with fewer scenarios required, particularly at extreme loss levels

Percentile level0.1 %

0

1000

2000

3000

4000

5000

6000

Loss

1 %5 %

Direct Monte Carlo Importance Sampling

10,000 simulations for Direct MC,100 for Importance Sampling

0

1000

2000

3000

4000

5000

6000

Loss

0.1 %

Outline

• Introduction– Quantities of interest -- portfolio capital, marginal capital

– Troubles with direct Monte Carlo

• Dealing with size -- portfolio compression

• Dealing with the model – importance sampling

• Dealing with the model – analytic marginal capital

• Conclusions

Intuition for the conditional loss result comes from considering Monte Carlo estimation of VaR

• Suppose 1000 scenarios to estimate 1% VaR• Losses in each scenarios (in descending order)

1 2 3 … 9 10 11Pos. 1 37 35 29 32 31 28Pos. 2 12 39 27 10 31 23

…Pos. N 60 57 58 … 62 54 53Total 2500 2312 2297 … 1689 1500

1476• A small position change will not change the ordering, so VaR will

change by amount that position loss changes in scenario 10

Examine the conditional portfolio distribution, given the factor return

The factor is the greatest determinant of portfolio value

Factor return

Port

folio

val

ue ($

M)

Cond SD bandsMC scenarios

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.55750

5800

5850

5900

5950

6000

Since exposures are conditionally independent, the conditional portfolio distribution is close to normal

5860 5870 5880 5890 5900 5910Portfolio value ($M)

Practical assumptions on the conditional distribution

• Assumptions1. One factor drives all correlations2. Factor is normally distributed3. Portfolio is conditionally normal4. Exposures are independent given factor returns

• First is not necessary, though results are only practical for a reduced set of factors

• Second is part of CreditMetrics assumptions, but can be relaxed• Third is not essential (results require small modification for

arbitrary standardized distribution)• Fourth is crucial to the analysis

Size penalty analytic results

• Notation:– L - portfolio loss, lq - portfolio VaR at level q– l(Z) - conditional portfolio loss, li(Z) - conditional exposure loss

– (Z) - conditional portfolio SD, i (Z) - conditional exposure SD

• Capital estimates are expectations over the conditional distribution of the factor, given that VaR is realized

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.55750

5800

5850

5900

5950

6000

-2 -1.5 -1

Size penalty analytic results

Variance contribution

of new exposure

Variance ofconditional mean

Conditionalvariance of new

exposure

Positive contribution if factor move is

less than expected, negative otherwise

2

22

21

)())(()())((E

ZBZlZZll ii

qlL q

• Size penalty

)(E ZlB ilL q• Base capital (conditional expected loss on unit exposure)

Why is this useful?Capital at 50bp for additional investment grade exposure

2 4 6 8 10

-20

0

20

40

60

80

100

120

140

160

Unit exposure

Cap

ital (

bp)

Capital and size penalty for additional investment grade exposure

0 20 40 60 80 1000

20

40

60

80

100

120

140

Unit exposure

Cap

ital (

bp)

5% loss

1% loss

50bp loss

10bp loss

Capital and size penalty for additional speculative grade exposure

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50

Unit exposure

Cap

ital (

%)

5% loss

1% loss50bp loss

10bp loss

Express capital charges as a grid

• Base capital (basis points)5% 1% 50bp 10bp

– Inv grade 26.8 47.7 57.3 90.8– Spec grade 695 1070 1220 1650

• Incremental capital (basis points) per unit exposure5% 1% 50bp 10bp

– Inv grade 0.15 0.23 0.27 0.43– Spec grade 22.2 26.2 27.5 30.9

Conclusions

• Inherent features of a direct Monte Carlo approach will cause convergence problems, particularly with capital calculations

• Practical assumptions go a long way, regardless of the model

– Portfolio capital based on compressed portfolio

– Capital contribution based on generic new exposures rather than for each unique exposure in the portfolio

• For CreditMetrics particularly, a reduced factor approach allows for variance reduction and hybrid techniques for the most difficult quantities to obtain through Monte Carlo