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Direct Calibration of Maturity Adjustment Formulae from Average Cumulative Issuer-Weighted

Authors: Dmitry Petrov Postgraduate Student, Lomonosov Moscow State University (Moscow, Russia) Analyst, EGAR Technology Inc. e-mail: dapetroff@gmail.com mobile: +7-910-407-91-79

Michael Pomazanov

PhD

Vice-Director of Credit Risk Management, Bank Zenit e-mail: m.pomazanov@zenit.ru tel. +7-495-937-07-37

Corporate Default Rates, Compared withBasel II Recommendations.

Associate Professor, State University Higher School of Economics (Moscow, Russia)

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Abstract Of late years there is considerable progress in the development of credit risk models. Revised Framework on International Convergence of Capital Measurement and Capital Standards (2004) raised the standards of risk management on the new high level. At the same time the problem of theoretical investigation of probability of default time structure (and consequently maturity dependence of capital requirement, expected loss, etc.) rests actual. Basel Committee recommends to perform maturity adjustment in capital requirement. By its sense this adjustment is a penalty for exceeding one year maturity. However the direct procedure of receiving of proposed maturity adjustment rests undisclosed. In this article we propose a method of calculation of maturity adjustment directly from open data. In addition analytical expressions revealing probability of default time structure are received. The character of our results is close enough to Basel proposal. However it was discovered that for low probabilities of default (high ratings) and maturities of 2-3 year there may exist underestimation of risk capital up to 50%. Key words: Maturity adjustment, Capital Requirement, Basel II, Probability of default, PD time structure.

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The unknown credit losses the bank will suffer can be represented by two components: expected loss (EL) and unexpected loss (UL). While a bank can forecast the average level of EL and manage them, UL are peak losses that exceed expected levels. Capital is needed to cover the risks of such peak losses, and therefore it has a loss-absorbing function. In June 2004, the Basel Committee issued a Revised Framework on International Convergence of Capital Measurement and Capital Standards (hereinafter Basel II) (see Basel Committee on Banking Supervision, 2004). In this document Basel II Internal Ratings-Based (IRB) approach was introduced. This approach is built on the following risk parameters: Probability of Default (PD), Loss Given Default (LGD) and Exposure at Default (EAD). Under advanced IRB (AIRB) approach, institutions are allowed to use their own internal models for base parameters of credit risk as primary inputs to the Capital Requirement (Capital at Risk or CAR) calculation. Banks generally employ a one-year planning horizon. The majority of well known portfolio models (CreditPortfolioView, CreditRisk+, CreditPortfolioManager, Credit Metrics, etc.) agree in fact, that the value of the credit portfolio is only observed with respect to a predefined time horizon (typically one year) that is consequently equals to time horizon in Basel II. In fact this time horizon generally does not correspond with the actual maturity of the loans in credit portfolio. It is obvious that long term loans are riskier. With respect to a three-year term loan, for example, the one-year horizon could mean that more than two-thirds of the credit risk is potentially ignored. So maturity becomes one of the important risk parameters and we need to make adjustment in PD and CAR to account this fact. Particularly, this is valid for Default Mode (DM) models like the one of Basel II. Basel Committee proposes maturity adjustment (Basel Committee on Banking Supervision, 2005), but there is no available detailed explication for this result. Thereby we consistently receive term structure of cumulative PD and maturity adjustment for capital requirement on base of open data published by rating ageneses and compare it with Basel II proposal. Our results are close to Basel II recommendation. They can make the process of economic capital allocation for long-term loans more clear. The topic of maturity effects is rater popular and widely discussed in recent literature. Number of authors worked on multi-horizon economic capital allocation on base of Mark to Market (MTM) paradigm (see Kalkbrener and Overbeck, 2002, Barco, 2004, Grundke, 2003). In these models changes in portfolio value are caused by changes in credit spreads which in their turn strongly depend on credit rating migration. Transition probabilities are normally assembled into the matrix form called a transition probability matrix. The transition probability matrix is convenient for describing the behavior of a Markov chain because multi-step transition probabilities are easily obtained. Though the Markov assumption for PD time dependence is not proved there is much extant literature and many texts on Markov chains and their applications for maturity effects (see, for example, Jarrow et al., 1997, Inamura, 2006, Frydman & Schuermann, 2005). One of the latest works is the article by Bluhm &Overbeck (2007) where Markov assumption is not rejected but is adopted by dropping the homogeneity assumption with Non-Homogeneity Continuous-Time Markov Chains (NHCTMCs). For models on base of DM paradigm there exists few literature analyzing account of long risk horizons. For example, Gurtler and Heithecker (2005) propose two approaches (“Capital for One Period” and “Capital to Maturity”) to calculate economic capital adjustment on base of rating ageneses data and also compare it with Basel Committee recommendations. Capital Requirement Calculation The Basel Risk Weight Functions used for the derivation of supervisory capital charges for UL are based on a specific model developed by the Basel Committee on Banking Supervision (2005). In the bottom of this model lie the results of Merton (1974) and Vasicek (2002).

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Assume that a loan defaults if the value of the borrower's assets at the loan maturity T falls below the contractual value B of its obligations payable. Let A be the value of borrower’s assets, described by the process: dA Adt Adxµ σ= + were asset value at T can be represented as:

( ) ( ) 21log log2

A T A T T T Xµ σ σ= + − +

where X is a standard normal variable. The probability of default on risk horizon T ( )TPD is than ( ) [ ] ( )TPD A T B X c N c= < = < = P P where

21log log

2B A T T

cT

µ σ

σ

− − +=

( )N is a cumulative normal distribution function. The variable X is standard normal, and can therefore be represented as 1X Y Zρ ρ= + − where Y, Z are mutually independent standard normal variables. The variable Y can be interpreted as a common factor, such as an economic index, over the interval (0, T). Then ρ represents correlation of a borrower with state of the economy. The term Y ρ is the company’s

exposure to the common factor and the term 1iZ ρ− represents the company specific risk. We will evaluate the probability of default as the expectation over the common factor Y of the conditional probability given Y. This can be interpreted as assuming various scenarios for the economy, determining the probability of default under each scenario, and then weighting each scenario by its likelihood. When the common factor is fixed, the conditional probability of default ( )pd is

( ) ( ) [ ]

( ) ( )1 1

| | 1 |

.1 1 1

T T

pd Y A T B Y X c Y Y Z c Y

N PD Y N PD Yc YZ Z N

ρ ρ

ρ ρρρ ρ ρ

− −

= < = ≤ = + − ≤ = − −−

= ≤ = ≤ = − − −

P P P

P P

The quantity pd(Y) provides the company default probability under the given scenario. The unconditional default probability PDT is the average of the conditional probabilities over the scenarios. So we have the worst scenario when the common factor takes the worst magnitude. Y is a standard normal variable, so this magnitude is given by ( )1N α−− with some confidence level α (Basel Committee recommends 0.999α = ). Then the worst conditional probability of default is

( ) ( ) ( )1 1

.1

TN PD Npd N

α ρα

ρ

− − += −

Under this worst scenario we will have the most serious loss and the capital requirement for a loan (worst loss – expected loss) is then given by

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( )

( )( ) ( )1 1

Capital Requirement , , , , Worst Loss- Expected Loss =

=

.1

T

T

TT

PD EAD LGD

EAD LGD pd EAD LGD PD

N PD NEAD LGD N PD

α ρ

α

α ρ

ρ

− −

=

⋅ ⋅ − ⋅ ⋅ =

+ = ⋅ ⋅ − −

Or in a compact form ( ) ( )Capital Requirement , , , , , , ,TPD EAD LGD EAD LGD FDaR Tα ρ α ρ= ⋅ ⋅ (1)

where ( ) ( ) ( )1 1

, , TN PD NFDaR T N

α ρα ρ

ρ

− − +=

.

Figure 1 illustrates the dependence of capital requirement on probability of default for one year maturity. .

[Figure 1 about here] Basel Maturity Adjustment Basel II capital requirement formula includes a component responsible for maturity (maturity adjustment). It is noted that this adjustment follows from the regression of the output of the KMV Portfolio ManagerTM

.. Maturity adjustment is linear, changes for maturities from 1 to 5 years and has the following from:

( ) ( )( )

1 2.5Maturity Adjustment Basel II ,

1 1.5T b PD

b PD+ − ⋅

=− ⋅

(2)

( ) ( )( )20.11852 0.05478logb PD PD= −

where PD is one-year probability of default. Figures 2a and 2b illustrate the dependence of Basel II maturity adjustment on one-year probability of default for fixed maturity and on maturity for fixed one-year PD consequently.

[Figures 2a and 2b about here]

PD time structure From cumulative default rates published by major rating agencies, such as Fitch Ratings (2006), Moody’s (2006), Standard & Poor’s (2007) directly follows that probability of default increases with the increase of risk horizon (Table 1, Figure 4). So we need to perform an adjustment in one-year PD if we want to take into account maturity longer than a year. Consequently we have to make adjustments in capital requirement working with such maturities. This adjustment is equivalent to a penalty for excess of one-year risk horizon. In this article we based on the statistical data provided by Moody’s (2006) (see Table 1).

[Table 1 and Figure 3 about here]

There are some potential errors in this data (see Credit MetricsTM Technical Document, 1997):

• Output cumulative default likelihoods violate proper rank order. For instance, presented table shows that AAAs have defaulted more often at the 10-year horizon than have AAs. This is true also for B1 and Ba3 ratings.

• Limited historical observation yields “granularity” in estimates. For instance, the AAA row in the table is supported by limited firm-years worth of observation. In 1997 it was

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only 1,658 firm-years. This is enough to yield a “resolution” of 0.06% (i.e., only probabilities in increments of 0.06% – or 1/1658 – are possible).

This lack of resolution may erroneously suggest that some probabilities are identically zero. For instance, if there were truly a 0.01% chance of AAA default, then we would have to watch about for another 80 years before there would be a 50% chance of tabulating a non-zero AAA default probability. In spite of these slight errors we suppose that presented statistical data reflects rather well the time structure of probability of default except, probably, several first ratings for the reasons mentioned earlier. Firstly we fit Moody’s cumulative probabilities for every rating n with 3 parameters ( ,nPD ,na nb ) special function:

( )( )

( )( )

( )( )

( )( )

, , ,

1- exp - 1- exp - 1- exp - 1 exp100 1- exp - 1- exp - 1- exp - 100

T n n n

n n n nn

n n n n

PD F PD a b T

T a T a T b bPDa a b b

= =

⋅ ⋅ ⋅ − − = ⋅ + − ⋅ ⋅

(3)

Fitting function was chosen to satisfy several essential properties:

• For the maturity of one year parameter nPD is equivalent to one year probability of default taken in percents

• For zero maturity TPD equals zero • Function have an asymptotic for large terms (parameter na should be grater than nb ).

This property follows from the notion that with time companies either default rather fast or attain higher ratings. So with time we have some kind of stabilization.

• Function have a change in convexity (for low probabilities of default we have concavity, for high - salience). This property follows from notion that companies with high rating pass several lower ratings before default. So there exist some initial period where cumulative probability of default doesn’t grow very fast (concavity). Companies with low ratings can come to default rather fast so we can’t observe such effect and cumulative probability of default grows immediately (salience).

Of cause, proposed function is not unique, but it shows very good fitting results (see Table 2 and Figure 4).

[Figure 4 about here] In the Table 2 the set of received data is presented: 3 parameters for every rating, one-year probabilities of default corresponding to ratings. R-square shows that proposed function precisely takes into account particularities of used data.

[Table 2 about here] Heretofore we used probabilities of default which correspond to discrete ratings. But PD is continuous by its nature. So we need to pass from discrete ratings (and corresponding one-year PDs) to continuous default probabilities. To do that we smoothed the PDn parameter, which has the sense of one year probability of default. Linear dependence was established between numeric ratings and natural logarithm of PDn (see Figure 5). Quality of this approximation is rather high: R square equals 0.974. ( ) ( )Rating exp 0.561 Numeric Rating 6.307 ,PDA = ⋅ − (4)

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where PDA is approximation for PDn parameter.

[Figure 5 about here] To receive continuous dependency of cumulative default probabilities from one-year PD and maturity we also need to smooth other two parameter ( a ,n nb ). From the analysis of dependence of parameters on PDA the following two fitting functions were proposed:

af depends on two parameters ( aα and aβ ):

( )( ) expa a af x xα β= ⋅ ⋅

bf depends on three parameters ( bα , bβ , bγ ):

( )( )2( ) expb b b bf x xα β γ= ⋅ − ⋅ + .

Approximation a, b of parameters an and nb gives the following results: ( ) ( )0.080 exp 0.639 ln(100 )a a PD PD= = ⋅ ⋅ ⋅ (5)

( ) ( )( )( )21.278 exp 0.293 ln 100 0.938 .b b PD PD= = ⋅ − ⋅ ⋅ − (6)

Constraint on a and b ( a have to be grater then b ) is fulfilled. From (3), (5) and (6) follows the formula which gives probability of default (PDT) for every one-year default probability (PD) and maturity (T in years) (see Figure 6): ( ) ( )( ), , , .TPD F PD a PD b PD T= (7)

[Figure 6 about here]

Maturity Adjustment Now, when the dependence of probability of default PDT for every maturity is known we can construct maturity adjustment for capital requirement in a following way:

( ) ( )

( )( )( )

Capital Requerement , , , ,Maturity Adjustment , =

Capital Requerement , , , ,

, ,1, ,

T

T

PD EAD LGDPD T

PD EAD LGD

FDaR T PDFDaR PD

α ρα ρ

α ρα ρ

=

−=

−

where TPD is calculated from (5).

[Figure 7 about here] Maturity adjustment does not depend strongly on correlation coefficient ρ (see Figure 7). For example, it can be taken in from proposed in Basel II:

( )( ) ( )( )( )( ) ( )( )

1

1

0.12 1 exp 50 / 1 exp 50

0.24 1 1 exp 50 / 1 exp 50 .

year

year

PD

PD

ρ −

−

= × − − × − − +

+ × − − − × − −

(8)

Dependence of maturity adjustment on confidence level α is rather strong, particularly for low probabilities of default (see Figure 8). But under Basel Committee recommendation we work with high confidence levels ( 0.999α = or even 0.9999α = ).

[Figure 8 about here]

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Figure 9 illustrates received maturity adjustment and Basel II maturity adjustment for several maturities, so it is possible to compare them.

[Figure 9 about here] Though the character of maturity adjustments is close enough, there is a difference in Basel II proposal and our results (see Figure 9). Received adjustment is higher for small probabilities of default (high ratings) and for maturities about 2, 3 years. It also reduces faster with the increase of one-year PD. Higher level of adjustment for high ratings is partly explainable and follows from the dependence of capital requirement on default probability (see Figure 1). For small probabilities the slope of the curve is grater then for large probabilities, so the same change in probability (with time) gives the greater change in capital requirement. But at the moment there is no complete explanation of difference between these two adjustments. If we had more exact information about methodology and data used for Basel II maturity adjustment it seems to us to be possible to explain other disagreements. Conclusion In this article the dependence of default probability on time was continuously parameterized using data provided by Moody’s. This approach gives results expressed analytically. It corresponds well with statistical data. Time structure of PD allows to receive maturity adjustment (or penalty for excess of one year maturity) for capital requirement. It was shown that the character of Basel II AIRB approach maturity adjustment function can be explained rather well from open statistical data. However from received results follows that there exist possible underestimate of risk fixed by Basel II maturity adjustment function. It is shown that penalty is higher for assets with good rating (investment grade) and maturities about 2 years. So that possible underestimate may be up to 50%. Reference:

1. Barco M. (2004), Bringing Credit Portfolio Modeling to Maturity, Risk 17(1), pages 86-90.

2. Basel Committee on Banking Supervision (2004), International Convergence of Capital Measurement and Capital Standards, Bank for International Settlements, June.

3. Basel Committee on Banking Supervision (2005), An Explanatory Note on the Basel II IRB Risk Weight Functions, Bank for International Settlements, July.

4. Bluhm C., Overbeck L. (2007), Calibration of PD term structures: to be Markov or not to be, Risk 20(11), pages 98-103.

5. Credit MetricsTM Technical Document (1997), J.P. Morgan & Co. Incorporated, April. 6. Fitch Ratings (2006), Fitch Ratings Global Corporate Finance 1990–2005 Transition and

Default Study, Fitch Ratings Corporate Finance Credit Market Research, August. 7. Frydman H., Schuermann T. (2005), Credit Ratings Dynamics and Markov Mixture

Models, Working Paper, Wharton Financial Institutions. 8. Gordy M. B. (2003), A risk-factor model foundation for ratings-based bank capital rules.

Journal of Financial Intermediation 12, 199 – 232. 9. Grundke P. (2003), The Term Structure of Credit Spreads as a Determinant of the

Maturity Effect on Credit Risk Capital, Finance Letters 1(6), S. 4-9. 10. Gurtler M., Heithecker D. (2005), Multi-Period Defaults and Maturity Effects on

Economic Capital in a Ratings-Based Default-Mode Model, Finanz Wirtschaft Working

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Paper Series FW19V2/05, Braunschweig University of Technology Institute for Economics and Business Administration Department of Finance.

11. Inamura Y. (2006), Estimating Continuous Time Transition Matrices from Discretely Observed Data, Bank of Japan Working Paper Series 06, E07, April.

12. Jarrow R., Lando D., Turnbull S. (1997), A Marcov Model fro the Trem Structure of Credit Risk Spreads, Review of Financial Studies 10, pages 481-523.

13. Kalkbrener M., and Overbeck L. (2002), The Maturity Effect on Credit Risk Capital, Risk 14(7), pages 59-63.

14. Merton R.C. (1974), On the Pricing of Corporate Debt: the Risk Structure of Interest Rates, Journal of Finance 29, pages 449-470.

15. Moody’s (2006), Default and Recovery Rates of Corporate Bond Issuers 1920-2005, Moody’s Investor Service Global Credit Research, January.

16. Standard & Poor’s (2007), Annual 2006 Global Corporate Default Study and Ratings Transitions, S&P Global Fixed Income Research, January.

17. Vasicek O. (2002), Loan portfolio value. RISK, December 2002, 160 – 162.

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Table 1. Example of average cumulative corporate defaults rates for several ratings/years Rating Year 1 Year 2 Year 3 Year 4 … Year 16 Year 17 Year 18 Year 19 Year 20

Aaa 0.000 0.000 0.000 0.039 … 0.208 0.208 0.208 0.208 0.208 Aa1 0.000 0.000 0.000 0.110 … 0.941 0.941 0.941 0.941 0.941 Aa2 0.000 0.011 0.048 0.120 … 0.970 1.177 1.414 1.684 1.710

… … … … … … … … … … … B1 3.223 8.503 13.573 17.635 … 32.161 32.161 32.161 32.161 32.161 B2 5.457 12.067 17.141 21.057 … 29.598 29.680 29.756 29.756 29.756 B3 10.460 18.653 25.249 29.887 … 38.964 38.964 38.985 38.985 38.985 Caa-C 20.982 30.274 36.115 39.500 … 43.326 43.326 43.326 43.326 43.326 Note: see Moody's (2006), Exhibit 36

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Table 2. Fitting results Alphanumeric

Rating Numeric Rating

PD one-year, % PDn a b

R2

Aaa 1 0.000 0.000 0,364 0,404 0.915 Aa1 2 0.000 0.000 0,012 0,019 0.899 Aa2 3 0.000 0.027 0,000 0,006 0.975 Aa3 4 0.019 0.010 0,019 0,030 0.983 A1 5 0.003 0.033 0,040 0,060 0.970 A2 6 0.026 0.112 0,000 0,004 0.965 A3 7 0.037 0.063 0,000 0,016 0.951 Baa1 8 0.166 0.230 0,002 0,002 0.978 Baa2 9 0.161 0.157 0,041 0,252 0.998 Baa3 10 0.335 0.538 0,103 0,584 0.993 Ba1 11 0.753 1.072 0,084 1,106 0.996 Ba2 12 0.780 1.472 0,101 0,714 0.995 Ba3 13 2.069 4.117 0,162 0,762 0.986 B1 14 3.223 5.928 0,209 0,864 0.978 B2 15 5.457 7.325 0,297 1,252 0.992 B3 16 10.460 11.430 0,355 1,226 0.996 Caa-C 17 20.982 19.970 0,619 0,619 0.998

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

PD, %

Cap

ital R

equi

rem

ent

Figure 1. Dependence of Basel II capital requirement on probability of deafault, EAD=1, LGD=1, maturity is 1 year.

10 12 14 16 18 20PD, %

Mat

urity

Adj

ustm

ent

2 years

1 2 3

0.5

1.5

2.0

2.5

3.03.5

4.0

4.55.0

0.0

1.0

1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Maturity, yearsM

atyr

ity A

djus

tmen

t

PD=0.1%

PD=1%

PD=10%

Figure 2b. Dependence of Basel II maturity adjustment onmaturity for fixed one year PDs (0.1%, 1%, 10%).

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Figure 2a. Dependance of Basel II maturity adjustmenton one-year probability od default for �xed maturity.(1) - maturity 2 years, (2) - 3 years, (3) - 5 years.

2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40

45

Years

PD, %

Aaa Aa A Baa

Ba

B

Figure 3. Average cumulative issue-weighted corporate default rates by whole letter rating,1983-2005,Moody’s data.

Figure 4. Cumulative default rates fitting

A. Ratings from Caa-C to Ba3

2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

2 4 6 8 10 12 14 16 18 20

5

10

15

20

25

30

35

40

45

PD, %

Maturity, years

PD, %

Maturity, years

B. Ratings from Ba2 to Baa2

2 4 6 8 10 12 14 16 18 20

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

PD, %

Maturity, years2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Maturity, years

PD, %

C. Ratings from Baa1 to A1 D. Ratings from Aa3 to Aaa

2 4 6 8 10 12 14 16

0

1

2

3

Numeric Rating

ln(P

Dn)

; ln(

PDA

)

Figure 5. Dependence of natural logarithm of PDn parameter (dots)and natural logarithm of PDA aproximation (line)on numeric rating.

02

46

810

0

5

10

15

200

5

10

15

20

25

30

35

40

45

50

Maturity, year

PD

T, %

One-year PD, %

Figure 6. Smoothed cumulative probability of default (Surface) compared tocumulative default rates (dots; Moody’s data).

0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.261

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8M

atur

ity A

djus

tmen

t

Figure 7. Dependance of maturity adjustment on correlation coe�cient (ρ).Maturity equals 2.5 years. For curve (1) one-year PD equals 0.003%, (2) - 0.1%,(3) - 0.5%, (4) - 1%, (5) - 5%, (6) - 6%.

12

3

4

5

6

0.99 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 11

1.5

2

2.5

3

3.5

Confidence Level

Mat

urity

Adj

ustm

ent

0.03%0.1%

0.5%

1%

5%

10%

Figure 8. Dependance fo maturity adjustment on con�dence levelfor several one-year PDs(0.03%, 0.1%, 0.5%, 1%, 5%, 10%).

0 5 10 15 200

0.5

1

1.5

2

2.5

3

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4.

1.6

1.8.

PD,%

Mat

urity

Adj

ustm

ent

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

PD,%

PD,%PD,%

Mat

urity

Adj

ustm

ent

Mat

urity

Adj

ustm

ent

Mat

urity

Adj

ustm

ent

Figure 9. Comparison of recieved maturity adjustment (black curves) withBasel II maturity adjustment (grey curves) for several maturities.

a. Maturity 2 years b. Maturity 3 years

c. Maturity 4 years d. Maturity 5 years