ńņš. 3 |

u

52 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

1. General market risk: This risk category comprises price changes of a

ļ¬nancial instrument, which are caused by changes of the general mar-

ket situation. General market conditions in the interest rate sector are

characterized by the shape and the moves of benchmark yield curves,

which are usually constructed from several benchmark instruments. The

benchmark instruments are chosen in such a way so that they allow for a

representative view on present market conditions in a particular market

sector.

2. Residual risk: Residual risk characterizes the fact that the actual price

of a given ļ¬nancial instrument can change in a way diļ¬erent from the

changes of the market benchmark (however, abrupt changes which are

caused by events in the sphere of the obligor are excluded from this risk

category). These price changes cannot be accounted for by the volatility

of the market benchmark. Residual risk is contained in the day-to-day

price variation of a given instrument relative to the market benchmark

and, thus, can be observed continuously in time. Residual risk is also

called idiosyncratic risk.

3. Event risk: Abrupt price changes of a given ļ¬nancial instrument relative

to the benchmark, which signiļ¬cantly exceed the continuously observable

price changes due to the latter two risk categories, are called event risk.

Such price jumps are usually caused by events in the sphere of the obligor.

They are observed infrequently and irregularly.

Residual risk and event risk form the two components of so-called speciļ¬c price

risk or speciļ¬c risk ā” a term used in documents on banking regulation, Bank for

International Settlements (1998a), Bank for International Settlements (1998b)

ā” and characterize the contribution of the individual risk of a given ļ¬nancial

instrument to its overall risk.

The distinction between general market risk and residual risk is not unique but

depends on the choice of the benchmark curve, which is used in the analysis

of general market risk: The market for interest rate products in a given cur-

rency has a substructure (market-sectors), which is reļ¬‚ected by product-speciļ¬c

(swaps, bonds, etc.), industry-speciļ¬c (bank, ļ¬nancial institution, retail com-

pany, etc.) and rating-speciļ¬c (AAA, AA, A, BBB, etc.) yield curves. For the

most liquid markets (USD, EUR, JPY), data for these sub-markets is available

from commercial data providers like Bloomberg. Moreover, there are addi-

tional inļ¬‚uencing factors like collateral, ļ¬nancial restrictions etc., which give

3.3 Descriptive Statistics of Yield Spread Time Series 53

rise to further variants of the yield curves mentioned above. Presently, however,

hardly any standardized data on these factors is available from data providers.

The larger the universe of benchmark curves a bank uses for modeling its

interest risk, the smaller is the residual risk. A bank, which e.g. only uses

product-speciļ¬c yield curves but neglects the inļ¬‚uence of industry- and rating-

speciļ¬c eļ¬ects in modelling its general market risk, can expect speciļ¬c price

risk to be signiļ¬cantly larger than in a bank which includes these inļ¬‚uences

in modeling general market risk. The diļ¬erence is due to the consideration of

product-, industry- and rating-speciļ¬c spreads over the benchmark curve for

(almost) riskless government bonds. This leads to the question, whether the

risk of a spread change, the spread risk, should be interpreted as part of the

general market risk or as part of the speciļ¬c risk. The uncertainty is due to

the fact that it is hard to deļ¬ne what a market-sector is. The deļ¬nition of

benchmark curves for the analysis of general market risk depends, however,

critically on the market sectors identiļ¬ed.

We will not further pursue this question in the following but will instead inves-

tigate some properties of this spread risk and draw conclusions for modeling

spread risk within internal risk models. We restrict ourselves to the continuous

changes of the yield curves and the spreads, respectively, and do not discuss

event risk. In this contribution diļ¬erent methods for the quantiļ¬cation of the

risk of a ļ¬ctive USD zero bond are analyzed. Our investigation is based on

time series of daily market yields of US treasury bonds and US bonds (banks

and industry) of diļ¬erent credit quality (rating) and time to maturity.

3.3 Descriptive Statistics of Yield Spread Time

Series

Before we start modeling the interest rate and spread risk we will investigate

some of the descriptive statistics of the spread time series. Our investigations

are based on commercially available yield curve histories. The Bloomberg

dataset we use in this investigation consists of daily yield data for US trea-

sury bonds as well as for bonds issued by banks and ļ¬nancial institutions with

ratings AAA, AA+/AA, A+, A, Aā’ (we use the Standard & Poorā˜s naming

convention) and for corporate/industry bonds with ratings AAA, AA, AAā’,

A+, A, Aā’, BBB+, BBB, BBBā’, BB+, BB, BBā’, B+, B, Bā’. The data we

use for the industry sector covers the time interval from March 09 1992 to June

08 2000 and corresponds to 2147 observations. The data for banks/ļ¬nancial

54 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

institutions covers the interval from March 09 1992 to September 14 1999 and

corresponds to 1955 observations. We use yields for 3 and 6 month (3M, 6M)

as well as 1, 2, 3, 4, 5, 7, and 10 year maturities (1Y, 2Y, 3Y, 4Y, 5Y, 7Y, 10Y).

Each yield curve is based on information on the prices of a set of representative

bonds with diļ¬erent maturities. The yield curve, of course, depends on the

choice of bonds. Yields are option-adjusted but not corrected for coupon pay-

ments. The yields for the chosen maturities are constructed by Bloombergā™s

interpolation algorithm for yield curves. We use the USD treasury curve as a

benchmark for riskless rates and calculate yield spreads relative to the bench-

mark curve for the diļ¬erent rating categories and the two industries. We correct

the data history for obvious ļ¬‚aws using complementary information from other

data sources. Some parts of our analysis in this section can be compared with

the results given in Kiesel, Perraudin and Taylor (1999).

3.3.1 Data Analysis with XploRe

We store the time series of the diļ¬erent yield curves in individual ļ¬les. The ļ¬le

names, the corresponding industries and ratings and the names of the matrices

used in the XploRe code are listed in Table 3.2. Each ļ¬le contains data for

the maturities 3M to 10Y in columns 4 to 12. XploRe creates matrices from

the data listed in column 4 of Table 3.2 and produces summary statistics for

the diļ¬erent yield curves. As example ļ¬les the data sets for US treasury and

industry bonds with rating AAA are provided. The output of the summarize

command for the INAAA curve is given in Table 3.1.

Contents of summ

Minimum Maximum Mean Median Std.Error

----------------------------------------------------------------

3M 3.13 6.93 5.0952 5.44 0.95896

6M 3.28 7.16 5.2646 5.58 0.98476

1Y 3.59 7.79 5.5148 5.75 0.95457

2Y 4.03 8.05 5.8175 5.95 0.86897

3Y 4.4 8.14 6.0431 6.1 0.79523

4Y 4.65 8.21 6.2141 6.23 0.74613

5Y 4.61 8.26 6.3466 6.36 0.72282

7Y 4.75 8.3 6.5246 6.52 0.69877

10Y 4.87 8.36 6.6962 6.7 0.69854

Table 3.1. Output of for the curve.

summarize INAAA

XFGsummary.xpl

The long term means are of particular interest. Therefore, we summarize them

in Table 3.3. In order to get an impression of the development of the treasury

3.3 Descriptive Statistics of Yield Spread Time Series 55

Industry Rating File Name Matrix Name

Government riskless USTF USTF

Industry AAA INAAA INAAA

Industry AA INAA2.DAT INAA2

Industry AA- INAA3.DAT INAA3

Industry A+ INA1.DAT INA1

Industry A INA2.DAT INA2

Industry A- INA3.DAT INA3

Industry BBB+ INBBB1.DAT INBBB1

Industry BBB INBBB2.DAT INBBB2

Industry BBB- INBBB3.DAT INBBB3

Industry BB+ INBB1.DAT INBB1

Industry BB INBB2.DAT INBB2

Industry BB- INBB3.DAT INBB3

Industry B+ INB1.DAT INB1

Industry B INB2.DAT INB2

Industry B- INB3.DAT INB3

Bank AAA BNAAA.DAT BNAAA

Bank AA+/AA BNAA12.DAT BNAA12

Bank A+ BNA1.DAT BNA1

Bank A BNA2.DAT BNA2

Bank A- BNA3.DAT BNA3

Table 3.2. Data variables

yields in time, we plot the time series for the USTF 3M, 1Y, 2Y, 5Y, and 10Y

yields. The results are displayed in Figure 3.1, XFGtreasury.xpl. The

averaged yields within the observation period are displayed in Figure 3.2 for

USTF, INAAA, INBBB2, INBB2 and INB2, XFGyields.xpl.

In the next step we calculate spreads relative to the treasury curve by sub-

tracting the treasury curve from the rating-speciļ¬c yield curves and store them

to variables SINAAA, SINAA2, etc. For illustrative purposes we display time

series of the 1Y, 2Y, 3Y, 5Y, 7Y, and 10Y spreads for the curves INAAA, INA2,

INBBB2, INBB2, INB2 in Figure 3.3, XFGseries.xpl.

We run the summary statistics to obtain information on the mean spreads.

Our results, which can also be obtained with the mean command, are collected

in Table 3.4, XFGmeans.xpl.

56 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

USTF 4.73 4.92 5.16 5.50 5.71 5.89 6.00 6.19 6.33

INAAA 5.10 5.26 5.51 5.82 6.04 6.21 6.35 6.52 6.70

INAA2 5.19 5.37 5.59 5.87 6.08 6.26 6.39 6.59 6.76

INAA3 5.25 - 5.64 5.92 6.13 6.30 6.43 6.63 6.81

INA1 5.32 5.50 5.71 5.99 6.20 6.38 6.51 6.73 6.90

INA2 5.37 5.55 5.76 6.03 6.27 6.47 6.61 6.83 7.00

INA3 - - 5.84 6.12 6.34 6.54 6.69 6.91 7.09

INBBB1 5.54 5.73 5.94 6.21 6.44 6.63 6.78 7.02 7.19

INBBB2 5.65 5.83 6.03 6.31 6.54 6.72 6.86 7.10 7.27

INBBB3 5.83 5.98 6.19 6.45 6.69 6.88 7.03 7.29 7.52

INBB1 6.33 6.48 6.67 6.92 7.13 7.29 7.44 7.71 7.97

INBB2 6.56 6.74 6.95 7.24 7.50 7.74 7.97 8.34 8.69

INBB3 6.98 7.17 7.41 7.71 7.99 8.23 8.46 8.79 9.06

INB1 7.32 7.53 7.79 8.09 8.35 8.61 8.82 9.13 9.39

INB2 7.80 7.96 8.21 8.54 8.83 9.12 9.37 9.68 9.96

INB3 8.47 8.69 8.97 9.33 9.60 9.89 10.13 10.45 10.74

BNAAA 5.05 5.22 5.45 5.76 5.99 6.20 6.36 6.60 6.79

BNAA12 5.14 5.30 5.52 5.83 6.06 6.27 6.45 6.68 6.87

BNA1 5.22 5.41 5.63 5.94 6.19 6.39 6.55 6.80 7.00

BNA2 5.28 5.47 5.68 5.99 6.24 6.45 6.61 6.88 7.07

BNA3 5.36 5.54 5.76 6.07 6.32 6.52 6.68 6.94 7.13

Table 3.3. Long term mean for diļ¬erent USD yield curves

Now we calculate the 1-day spread changes from the observed yields and store

them to variables DASIN01AAA, etc. We run the descriptive routine to cal-

culate the ļ¬rst four moments of the distribution of absolute spread changes.

Volatility as well as skewness and kurtosis for selected curves are displayed in

Tables 3.5, 3.6 and 3.7.

XFGchange.xpl

For the variable DASIN01AAA[,12] (the 10 year AAA spreads) we demonstrate

the output of the descriptive command in Table 3.8.

Finally we calculate 1-day relative spread changes and run the descriptive

command. The results for the estimates of volatility, skewness and kurtosis are

summarized in Tables 3.9, 3.10 and 3.11. XFGrelchange.xpl

3.3 Descriptive Statistics of Yield Spread Time Series 57

US Treasury Yields (3M, 1Y, 2Y, 5Y, 10Y)

8

7

6

Yield in %

5

4

3

0 500 1000 1500 2000

Day

Figure 3.1. US Treasury Yields. XFGtreasury.xpl

Yields for Different Risk Levels

8

Average Yield in %

7

6

5

5 10

Time to Maturity in Years

Figure 3.2. Averaged Yields. XFGyields.xpl

58 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

1Y-Spread (AAA, A2, BBB2, BB2, B2) 3Y-Spread (AAA, A2, BBB2, BB2, B2) 7Y-Spread (AAA, A2, BBB2, BB2, B2)

5

5

5

4

4

4

3

3

Spread in %

Spread in %

Spread in %

3

2

2

2

1

1

1

0

0

0

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Day*E2 Day*E2 Day*E2

2Y-Spread (AAA, A2, BBB2, BB2, B2) 5Y-Spread (AAA, A2, BBB2, BB2, B2) 10Y-Spread (AAA, A2, BBB2, BB2, B2)

6

5

5

4

4

4

3

Spread in %

Spread in %

Spread in %

3

3

2

2

2

1

1

1

0

0

0

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Day*E2 Day*E2 Day*E2

Figure 3.3. Credit Spreads. XFGseries.xpl

3.3.2 Discussion of Results

Time Development of Yields and Spreads: The time development of US trea-

sury yields displayed in Figure 3.1 indicates that the yield curve was steeper

at the beginning of the observation period and ļ¬‚attened in the second half.

However, an inverse shape of the yield curve occurred hardly ever. The long

term average of the US treasury yield curve, the lowest curve in Figure 3.2,

also has an upward sloping shape.

The time development of the spreads over US treasury yields displayed in Fig-

ure 3.3 is diļ¬erent for diļ¬erent credit qualities. While there is a large variation

of spreads for the speculative grades, the variation in the investment grade sec-

tor is much smaller. A remarkable feature is the signiļ¬cant spread increase for

all credit qualities in the last quarter of the observation period which coincides

with the emerging market crises in the late 90s. The term structure of the long

term averages of the rating-speciļ¬c yield curves is also normal. The spreads

over the benchmark curve increase with decreasing credit quality.

Mean Spread: The term structure of the long term averages of the rating-

speciļ¬c yield curves, which is displayed in Figure 3.3, is normal (see also Ta-

ble 3.4). The spreads over the benchmark curve increase with decreasing credit

quality. For long maturities the mean spreads are larger than for intermediate

maturities as expected. However, for short maturities the mean spreads are

3.3 Descriptive Statistics of Yield Spread Time Series 59

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

INAAA 36 35 35 31 33 31 35 33 37

INAA2 45 45 43 37 37 36 40 39 44

INAA3 52 - 48 42 42 40 44 44 49

INA1 58 58 55 49 49 49 52 53 57

INA2 63 63 60 53 56 57 62 64 68

INA3 - - 68 62 63 64 69 72 76

INBBB1 81 82 78 71 72 74 79 83 86

INBBB2 91 91 87 80 82 82 87 90 94

INBBB3 110 106 103 95 98 98 104 110 119

INBB1 160 156 151 142 141 140 145 151 164

INBB2 183 182 179 173 179 185 197 215 236

INBB3 225 225 225 221 228 233 247 259 273

INB1 259 261 263 259 264 271 282 294 306

INB2 306 304 305 304 311 322 336 348 363

INB3 373 377 380 382 389 400 413 425 441

BNAAA 41 39 38 33 35 35 41 43 47

BNAA12 50 47 45 40 42 42 49 52 56

BNA1 57 59 57 52 54 54 59 64 68

BNA2 64 65 62 57 59 60 65 71 75

BNA3 72 72 70 65 67 67 72 76 81

Table 3.4. Mean spread in basis points p.a.

larger compared with intermediate maturities.

Volatility: The results for the volatility for absolute 1-day spread changes in

basis points p.a. are listed in Table 3.5. From short to intermediate maturities

the volatilities decrease. For long maturities a slight volatility increase can be

observed compared to intermediate maturities. For equal maturities volatility

is constant over the investment grade ratings, while for worse credit qualities a

signiļ¬cant increase in absolute volatility can be observed. Volatility for relative

spread changes is much larger for short maturities than for intermediate and

long maturities. As in the case of absolute spread changes, a slight volatility

increase exists for the transition from intermediate to long maturities. Since

absolute spreads increase more strongly with decreasing credit quality than

absolute spread volatility, relative spread volatility decreases with decreasing

credit quality (see Table 3.9).

Skewness: The results for absolute 1-day changes (see Table 3.6) are all close to

zero, which indicates that the distribution of changes is almost symmetric. The

corresponding distribution of relative changes should have a positive skewness,

60 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

INAAA 4.1 3.5 3.3 2.3 2.4 2.2 2.1 2.2 2.5

INAA2 4.0 3.5 3.3 2.3 2.4 2.2 2.2 2.2 2.5

INAA3 4.0 - 3.3 2.2 2.3 2.2 2.2 2.2 2.5

INA1 4.0 3.7 3.3 2.3 2.4 2.2 2.2 2.2 2.6

INA2 4.1 3.7 3.3 2.4 2.4 2.1 2.2 2.3 2.5

INA3 - - 3.4 2.4 2.4 2.2 2.2 2.3 2.6

INBBB1 4.2 3.6 3.2 2.3 2.3 2.2 2.1 2.3 2.6

INBBB2 4.0 3.5 3.4 2.3 2.4 2.1 2.2 2.3 2.6

INBBB3 4.2 3.6 3.5 2.4 2.5 2.2 2.3 2.5 2.9

INBB1 4.8 4.4 4.1 3.3 3.3 3.1 3.1 3.9 3.4

INBB2 4.9 4.6 4.5 3.8 3.8 3.8 3.7 4.3 4.0

INBB3 5.5 5.1 4.9 4.3 4.4 4.2 4.1 4.7 4.3

INB1 6.0 5.2 4.9 4.5 4.5 4.4 4.4 4.9 4.6

INB2 5.6 5.2 5.2 4.8 4.9 4.8 4.8 5.3 4.9

INB3 5.8 6.1 6.4 5.1 5.2 5.1 5.1 5.7 5.3

BNAAA 3.9 3.5 3.3 2.5 2.5 2.3 2.2 2.3 2.6

BNAA12 5.4 3.6 3.3 2.4 2.3 2.2 2.1 2.3 2.6

BNA1 4.1 3.7 3.2 2.1 2.2 2.1 2.0 2.2 2.6

BNA2 3.8 3.5 3.1 2.3 2.2 2.0 2.1 2.2 2.5

BNA3 3.8 3.5 3.2 2.2 2.2 2.1 2.1 2.2 2.5

Table 3.5. volatility for absolute spread changes in basis points p.a.

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 10Y

INAAA 0.1 0.0 -0.1 0.6 0.5 0.0 -0.5 0.6

INAA2 0.0 -0.2 0.0 0.4 0.5 -0.1 -0.2 0.3

INA2 0.0 -0.3 0.1 0.2 0.4 0.1 -0.1 0.4

INBBB2 0.2 0.0 0.2 1.0 1.1 0.5 0.5 0.9

INBB2 -0.2 -0.5 -0.4 -0.3 0.3 0.5 0.4 -0.3

Table 3.6. Skewness for absolute 1-day spread changes (in Ļ 3 ).

which is indeed the conclusion from the results in Table 3.10.

Kurtosis: The absolute 1-day changes lead to a kurtosis, which is signiļ¬cantly

larger than 3 (see Table 3.6). Thus, the distribution of absolute changes is

leptokurtic. There is no signiļ¬cant dependence on credit quality or maturity.

The distribution of relative 1-day changes is also leptokurtic (see Table 3.10).

The deviation from normality increases with decreasing credit quality and de-

creasing maturity.

3.3 Descriptive Statistics of Yield Spread Time Series 61

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 10Y

INAAA 12.7 6.0 8.1 10.1 16.8 9.1 11.2 12.8

INAA2 10.5 6.4 7.8 10.1 15.8 7.8 9.5 10.0

INA2 13.5 8.5 9.2 12.3 18.2 8.2 9.4 9.8

INBBB2 13.7 7.0 9.9 14.5 21.8 10.5 13.9 14.7

INBB2 11.2 13.0 11.0 15.8 12.3 13.2 11.0 11.3

Table 3.7. Kurtosis for absolute spread changes (in Ļ 4 ).

=========================================================

Variable 10Y

=========================================================

Mean 0.000354147

Std.Error 0.0253712 Variance 0.000643697

Minimum -0.18 Maximum 0.2

Range 0.38

Lowest cases Highest cases

1284: -0.18 1246: 0.14

1572: -0.14 1283: 0.14

1241: -0.13 2110: 0.19

1857: -0.11 1062: 0.19

598: -0.1 2056: 0.2

Median 0

25% Quartile -0.01 75% Quartile 0.01

Skewness 0.609321 Kurtosis 9.83974

Observations 2146

Distinct observations 75

Total number of {-Inf,Inf,NaN} 0

=========================================================

Table 3.8. Output of descriptive for the 10 years AAA spread.

We visualize symmetry and leptokursis of the distribution of absolute spread

changes for the INAAA 10Y data in Figure 3.4, where we plot the empirical dis-

tribution of absolute spreads around the mean spread in an averaged shifted

histogram and the normal distribution with the variance estimated from his-

torical data.

XFGdist.xpl

62 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y

INAAA 36.0 19.2 15.5 8.9 8.4 8.0 6.4 7.8 10.4

INAA2 23.5 13.1 11.2 7.2 7.4 6.4 5.8 6.2 7.6

INAA3 13.4 - 9.0 5.8 6.2 5.3 5.0 5.8 6.4

INA1 13.9 9.2 7.7 5.7 5.6 4.7 4.5 4.6 5.7

INA2 11.5 8.1 7.1 5.1 4.9 4.3 4.0 4.0 4.5

INA3 - - 6.4 4.6 4.3 3.8 3.5 3.5 4.1

INBBB1 8.1 6.0 5.4 3.9 3.7 3.3 3.0 3.2 3.8

INBBB2 7.0 5.3 5.0 3.3 3.3 2.9 2.8 2.9 3.3

INBBB3 5.7 4.7 4.4 3.2 3.0 2.7 2.5 2.6 2.9

INBB1 4.3 3.8 3.4 2.5 2.4 2.2 2.1 2.5 2.2

INBB2 3.7 3.3 3.0 2.2 2.1 2.0 1.8 2.0 1.7

INBB3 3.2 2.8 2.5 2.0 1.9 1.8 1.6 1.8 1.5

INB1 3.0 2.4 2.1 1.7 1.7 1.6 1.5 1.6 1.5

INB2 2.3 2.1 1.9 1.6 1.6 1.5 1.4 1.5 1.3

INB3 1.8 2.2 2.3 1.3 1.3 1.2 1.2 1.3 1.1

BNAAA 37.0 36.6 16.9 9.8 9.0 8.2 6.1 5.9 6.5

BNAA12 22.8 9.7 8.3 7.0 6.3 5.8 4.6 4.8 5.5

BNA1 36.6 10.1 7.9 5.6 4.8 4.4 3.8 3.9 4.4

BNA2 17.8 8.0 6.6 4.5 4.1 3.6 3.4 3.3 3.7

BNA3 9.9 6.9 5.6 3.7 3.6 3.3 3.1 3.1 3.4

Table 3.9. Volatility for relative spread changes in %

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 10Y

INAAA 2.3 4.6 4.3 2.2 2.3 2.1 0.6 4.6

INAA2 5.4 2.6 3.7 1.6 2.0 0.6 0.8 1.8

INA2 7.6 1.5 1.2 0.9 1.6 0.8 0.9 0.8

INBBB2 5.5 0.7 0.8 0.8 1.4 0.8 0.7 0.8

INBB2 0.8 0.4 0.6 0.3 0.4 0.5 0.3 -0.2

Table 3.10. Skewness for relative spread changes (in Ļ 3 ).

We note that by construction the area below both curves is normalized to

one. We calculate the 1%, 10%, 90% and 99% quantiles of the spread distribu-

tion with the quantile command. Those quantiles are popular in market risk

management. For the data used to generate Figure 3.4 the results are 0.30%,

0.35%, 0.40%, and 0.45%, respectively. The corresponding quantiles of the

plotted normal distribution are 0.31%, 0.34%, 0.41%, 0.43%. The diļ¬erences

are less obvious than the diļ¬erence in the shape of the distributions. However,

in a portfolio with diļ¬erent ļ¬nancial instruments, which is exposed to diļ¬erent

3.4 Historical Simulation and Value at Risk 63

Curve 3M 6M 1Y 2Y 3Y 4Y 5Y 10Y

INAAA 200.7 54.1 60.1 27.8 28.3 33.9 16.8 69.3

INAA2 185.3 29.5 60.5 22.1 27.4 11.0 17.5 23.0

INA2 131.1 22.1 18.0 13.9 26.5 16.4 18.5 13.9

INBBB2 107.1 13.9 16.9 12.0 20.0 14.0 16.6 16.7

INBB2 16.3 11.9 12.9 12.4 11.0 10.1 10.2 12.0

Table 3.11. Kurtosis for relative spread changes (in Ļ 4 ).

Historical vs. Normal Distribution

30

Density Function

20

10

0

0.2 0.3 0.4 0.5

Absolute Spread Change

Figure 3.4. Historical distribution and estimated normal distribution.

XFGdist.xpl

risk factors with diļ¬erent correlations, the diļ¬erence in the shape of the distri-

bution can play an important role. That is why a simple variance-covariance

approach, J.P. Morgan (1996) and Kiesel et al. (1999), seems not adequate to

capture spread risk.

3.4 Historical Simulation and Value at Risk

We investigate the behavior of a ļ¬ctive zero-bond of a given credit quality

with principal 1 USD, which matures after T years. In all simulations t = 0

denotes the beginning and t = T the end of the lifetime of the zero-bond. The

starting point of the simulation is denoted by t0 , the end by t1 . The observation

64 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

period, i.e., the time window investigated, consists of N ā„ 1 trading days and

the holding period of h ā„ 1 trading days. The conļ¬dence level for the VaR is

Ī± ā [0, 1]. At each point in time 0 ā¤ t ā¤ t1 the risky yields Ri (t) (full yield

curve) and the riskless treasury yields Bi (t) (benchmark curve) for any time to

maturity 0 < T1 < Ā· Ā· Ā· < Tn are contained in our data set for 1 ā¤ i ā¤ n, where

n is the number of diļ¬erent maturities. The corresponding spreads are deļ¬ned

by Si (t) = Ri (t) ā’ Bi (t) for 1 ā¤ i ā¤ n.

In the following subsections 3.4.1 to 3.4.5 we specify diļ¬erent variants of the

historical simulation method which we use for estimating the distribution of

losses from the zero-bond position. The estimate for the distribution of losses

can then be used to calculate the quantile-based risk measure Value-at-Risk.

The variants diļ¬er in the choice of risk factors, i.e., in our case the compo-

nents of the historical yield time series. In Section 3.6 we describe how the

VaR estimation is carried out with XploRe commands provided that the loss

distribution has been estimated by means of one of the methods introduced

and can be used as an input variable.

3.4.1 Risk Factor: Full Yield

1. Basic Historical Simulation:

We consider a historical simulation, where the risk factors are given by the

full yield curve, Ri (t) for i = 1, . . . , n. The yield R(t, T ā’ t) at time t0 ā¤

t ā¤ t1 for the remaining time to maturity T ā’ t is determined by means of

linear interpolation from the adjacent values Ri (t) = R(t, Ti ) and Ri+1 (t) =

R(t, Ti+1 ) with Ti ā¤ T ā’ t < Ti+1 (for reasons of simplicity we do not consider

remaining times to maturity T ā’ t < T1 and T ā’ t > Tn ):

[Ti+1 ā’ (T ā’ t)]Ri (t) + [(T ā’ t) ā’ Ti ]Ri+1 (t)

R(t, T ā’ t) = . (3.1)

Ti+1 ā’ Ti

The present value of the bond P V (t) at time t can be obtained by discounting,

1

t 0 ā¤ t ā¤ t1 .

P V (t) = , (3.2)

T ā’t

1 + R(t, T ā’ t)

In the historical simulation the relative risk factor changes

Ri t ā’ k/N ā’ Ri t ā’ (k + h)/N

(k)

0 ā¤ k ā¤ N ā’ 1,

āi (t) = , (3.3)

Ri t ā’ (k + h)/N

3.4 Historical Simulation and Value at Risk 65

are calculated for t0 ā¤ t ā¤ t1 and each 1 ā¤ i ā¤ n. Thus, for each scenario k we

obtain a new ļ¬ctive yield curve at time t + h, which can be determined from

the observed yields and the risk factor changes,

(k) (k)

1 ā¤ i ā¤ n,

Ri (t + h) = Ri (t) 1 + āi (t) , (3.4)

by means of linear interpolation. This procedure implies that the distribution of

risk factor changes is stationary between tā’(N ā’1+h)/N and t. Each scenario

corresponds to a drawing from an identical and independent distribution, which

can be related to an i.i.d. random variable Īµi (t) with variance one via

āi (t) = Ļi Īµi (t). (3.5)

This assumption implies homoscedasticity of the volatility of the risk factors,

i.e., a constant volatility level within the observation period. If this were not the

case, diļ¬erent drawings would originate from diļ¬erent underlying distributions.

Consequently, a sequence of historically observed risk factor changes could not

be used for estimating the future loss distribution.

In analogy to (3.1) for time t + h and remaining time to maturity T ā’ t one

obtains

(k) (k)

[Ti+1 ā’ (T ā’ t)]Ri (t) + [(T ā’ t) ā’ Ti ]Ri+1 (t)

(k)

(t + h, T ā’ t) =

R

Ti+1 ā’ Ti

for the yield. With (3.2) we obtain a new ļ¬ctive present value at time t + h:

1

P V (k) (t + h) = . (3.6)

T ā’t

R(k) (t + h, T ā’ t)

1+

In this equation we neglected the eļ¬ect of the shortening of the time to maturity

in the transition from t to t + h on the present value. Such an approximation

should be reļ¬ned for ļ¬nancial instruments whose time to maturity/time to

expiration is of the order of h, which is not relevant for the constellations

investigated in the following.

Now the ļ¬ctive present value P V (k) (t + h) is compared with the present value

for unchanged yield R(t + h, T ā’ t) = R(t, T ā’ t) for each scenario k (here the

remaining time to maturity is not changed, either).

1

P V (t + h) = . (3.7)

T ā’t

1 + R(t + h, T ā’ t)

66 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

The loss occurring is

L(k) (t + h) = P V (t + h) ā’ P V (k) (t + h) 0 ā¤ k ā¤ N ā’ 1, (3.8)

i.e., losses in the economic sense are positive while proļ¬ts are negative. The

VaR is the loss which is not exceeded with a probability Ī± and is estimated as

the [(1 ā’ Ī±)N + 1]-th-largest value in the set

{L(k) (t + h) | 0 ā¤ k ā¤ N ā’ 1}.

This is the (1 ā’ Ī±)-quantile of the corresponding empirical distribution.

2. Mean Adjustment:

A reļ¬ned historical simulation includes an adjustment for the average of those

relative changes in the observation period which are used for generating the

scenarios according to (3.3). If for ļ¬xed 1 ā¤ i ā¤ n the average of relative

(k)

changes āi (t) is diļ¬erent from 0, a trend is projected from the past to the

future in the generation of ļ¬ctive yields in (3.4). Thus the relative changes are

(k) (k)

corrected for the mean by replacing the relative change āi (t) with āi (t) ā’

āi (t) for 1 ā¤ i ā¤ n in (3.4):

N ā’1

1 (k)

āi (t) = āi (t), (3.9)

N

k=0

This mean correction is presented in Hull (1998).

3. Volatility Updating:

An important variant of historical simulation uses volatility updating Hull

(1998). At each point in time t the exponentially weighted volatility of rel-

ative historical changes is estimated for t0 ā¤ t ā¤ t1 by

N ā’1

2

(k)

2

Ī³ k āi (t)

= (1 ā’ Ī³) 1 ā¤ i ā¤ n.

Ļi (t) , (3.10)

k=0

The parameter Ī³ ā [0, 1] is a decay factor, which must be calibrated to generate

a best ļ¬t to empirical data. The recursion formula

2

(0)

2 2

Ļi (t) = (1 ā’ Ī³)Ļi (t ā’ 1/N ) + Ī³ āi (t) 1 ā¤ i ā¤ n,

, (3.11)

is valid for t0 ā¤ t ā¤ t1 . The idea of volatility updating consists in adjusting the

historical risk factor changes to the present volatility level. This is achieved by

3.4 Historical Simulation and Value at Risk 67

a renormalization of the relative risk factor changes from (3.3) with the corre-

sponding estimation of volatility for the observation day and a multiplication

with the estimate for the volatility valid at time t. Thus, we calculate the

quantity

(k)

āi (t)

(k)

= Ļi (t) Ā· 0 ā¤ k ā¤ N ā’ 1.

Ī“i (t) , (3.12)

Ļi (t ā’ (k + h)/N )

In a situation, where risk factor volatility is heteroscedastic and, thus, the

process of risk factor changes is not stationary, volatility updating cures this

violation of the assumptions made in basic historical simulation, because the

process of re-scaled risk factor changes āi (t)/Ļi (t)) is stationary. For each k

these renormalized relative changes are used in analogy to (3.4) for the deter-

mination of ļ¬ctive scenarios:

(k) (k)

1 ā¤ i ā¤ n,

Ri (t + h) = Ri (t) 1 + Ī“i (t) , (3.13)

The other considerations concerning the VaR calculation in historical simula-

tion remain unchanged.

4. Volatility Updating and Mean Adjustment:

Within the volatility updating framework, we can also apply a correction for

the average change according to 3.4.1(2). For this purpose, we calculate the

average

N ā’1

1 (k)

Ī“ i (t) = Ī“i (t), (3.14)

N

k=0

(k) (k)

and use the adjusted relative risk factor change Ī“i (t) ā’ Ī“ i (t) instead of Ī“i (t)

in (3.13).

3.4.2 Risk Factor: Benchmark

In this subsection the risk factors are relative changes of the benchmark curve

instead of the full yield curve. This restriction is adequate for quantifying

general market risk, when there is no need to include spread risk. The risk

factors are the yields Bi (t) for i = 1, . . . , n. The yield B(t, T ā’ t) at time t for

68 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

remaining time to maturity T ā’ t is calculated similarly to (3.1) from adjacent

values by linear interpolation,

{Ti+1 ā’ (T ā’ t)}Bi (t) + {(T ā’ t) ā’ Ti }Bi+1 (t)

B(t, T ā’ t) = . (3.15)

Ti+1 ā’ Ti

The generation of scenarios and the interpolation of the ļ¬ctive benchmark curve

is carried out in analogy to the procedure for the full yield curve. We use

Bi t ā’ k/N ā’ Bi t ā’ (k + h)/N

(k)

0 ā¤ k ā¤ N ā’ 1,

āi (t) = , (3.16)

Bi t ā’ (k + h)/N

and

(k) (k)

1 ā¤ i ā¤ n.

Bi (t + h) = Bi (t) 1 + āi (t) , (3.17)

Linear interpolation yields

(k) (k)

{Ti+1 ā’ (T ā’ t)}Bi (t) + {(T ā’ t) ā’ Ti }Bi+1 (t)

(k)

(t + h, T ā’ t) =

B .

Ti+1 ā’ Ti

In the determination of the ļ¬ctive full yield we now assume that the spread

remains unchanged within the holding period. Thus, for the k-th scenario we

obtain the representation

R(k) (t + h, T ā’ t) = B (k) (t + h, T ā’ t) + S(t, T ā’ t), (3.18)

which is used for the calculation of a new ļ¬ctive present value and the corre-

sponding loss. With this choice of risk factors we can introduce an adjustment

for the average relative changes or/and volatility updating in complete analogy

to the four variants described in the preceding subsection.

3.4.3 Risk Factor: Spread over Benchmark Yield

When we take the view that risk is only caused by spread changes but not

by changes of the benchmark curve, we investigate the behavior of the spread

risk factors Si (t) for i = 1, . . . , n. The spread S(t, T ā’ t) at time t for time to

maturity T ā’ t is again obtained by linear interpolation. We now use

Si t ā’ k/N ā’ Si t ā’ (k + h)/N

(k)

0 ā¤ k ā¤ N ā’ 1,

āi (t) = , (3.19)

Si t ā’ (k + h)/N

3.4 Historical Simulation and Value at Risk 69

and

(k) (k)

1 ā¤ i ā¤ n.

Si (t + h) = Si (t) 1 + āi (t) , (3.20)

Here, linear interpolation yields

(k) (k)

{Ti+1 ā’ (T ā’ t)}Si (t) + {(T ā’ t) ā’ Ti }Si+1 (t)

(k)

(t + h, T ā’ t) =

S .

Ti+1 ā’ Ti

Thus, in the determination of the ļ¬ctive full yield the benchmark curve is

considered deterministic and the spread stochastic. This constellation is the

opposite of the constellation in the preceding subsection. For the k-th scenario

one obtains

R(k) (t + h, T ā’ t) = B(t, T ā’ t) + S (k) (t + h, T ā’ t). (3.21)

In this context we can also work with adjustment for average relative spread

changes and volatility updating.

3.4.4 Conservative Approach

In the conservative approach we assume full correlation between risk from the

benchmark curve and risk from the spread changes. In this worst case scenario

we add (ordered) losses, which are calculated as in the two preceding sections

from each scenario. From this loss distribution the VaR is determined.

3.4.5 Simultaneous Simulation

Finally, we consider simultaneous relative changes of the benchmark curve and

the spreads. For this purpose (3.18) and (3.21) are replaced with

R(k) (t + h, T ā’ t) = B (k) (t + h, T ā’ t) + S (k) (t, T ā’ t), (3.22)

where, again, corrections for average risk factor changes or/and volatility up-

dating can be added. We note that the use of relative risk factor changes

is the reason for diļ¬erent results of the variants in subsection 3.4.1 and this

subsection.

70 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

3.5 Mark-to-Model Backtesting

A backtesting procedure compares the VaR prediction with the observed loss.

In a mark-to-model backtesting the observed loss is determined by calculation

of the present value before and after consideration of the actually observed risk

factor changes. For t0 ā¤ t ā¤ t1 the present value at time t+h is calculated with

the yield R(t + h, T ā’ t), which is obtained from observed data for Ri (t + h)

by linear interpolation, according to

1

P V (t) = . (3.23)

T ā’t

1 + R(t + h, T ā’ t)

This corresponds to a loss L(t) = P V (t) ā’ P V (t + h), where, again, the short-

ening of the time to maturity is not taken into account.

The diļ¬erent frameworks for the VaR estimation can easily be integrated into

the backtesting procedure. When we, e.g., only consider changes of the bench-

mark curve, R(t+h, T ā’t) in (3.23) is replaced with B(t+h, T ā’t)+S(t, T ā’t).

On an average (1 ā’ Ī±) Ā· 100 per cent of the observed losses in a given time in-

terval should exceed the corresponding VaR (outliers). Thus, the percentage of

observed losses is a measure for the predictive power of historical simulation.

3.6 VaR Estimation and Backtesting with XploRe

In this section we explain, how a VaR can be calculated and a backtesting can

be implemented with the help of XploRe routines. We present numerical results

for the diļ¬erent yield curves. The VaR estimation is carried out with the help

of the VaRest command. The VaRest command calculates a VaR for historical

simulation, if one speciļ¬es the method parameter as āEDFā (empirical distri-

bution function). However, one has to be careful when specifying the sequence

of asset returns which are used as input for the estimation procedure. If one

calculates zero-bond returns from relative risk factor changes (interest rates or

spreads) the complete empirical distribution of the proļ¬ts and losses must be

estimated anew for each day from the N relative risk factor changes, because

the proļ¬t/loss observations are not identical with the risk factor changes.

For each day the N proļ¬t/loss observations generated with one of the methods

described in subsections 3.4.1 to 3.4.5 are stored to a new row in an array PL.

The actual proļ¬t and loss data from a mark-to-model calculation for holding

3.6 VaR Estimation and Backtesting with XploRe 71

period h are stored to a one-column-vector MMPL. It is not possible to use a

continuous sequence of proļ¬t/loss data with overlapping time windows for the

VaR estimation. Instead the VaRest command must be called separately for

each day. The consequence is that the data the VaRest command operates

on consists of a row of N + 1 numbers: N proļ¬t/loss values contained in the

vector (PL[t,])ā™, which has one column and N rows followed by the actual

mark-to-model proļ¬t or loss MMPL[t,1] within holding period h in the last row.

The procedure is implemented in the quantlet XFGpl which can be downloaded

from quantlet download page of this book.

VaR timeplot

15

10

5

returns*E-2

0

-5

-10

5 10 15

time*E2

Figure 3.5. VaR time plot basic historical simulation.

XFGtimeseries.xpl

The result is displayed for the INAAA curve in Figures. 3.5 (basic historical

simulation) and 3.6 (historical simulation with volatility updating). The time

plots allow for a quick detection of violations of the VaR prediction. A striking

feature in the basic historical simulation with the full yield curve as risk fac-

tor is the platform-shaped VaR prediction, while with volatility updating the

VaR prediction decays exponentially after the occurrence of peak events in the

market data. This is a consequence of the exponentially weighted historical

72 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

VaR timeplot

15

10

5

returns*E-2

0

-5

-10

-15

5 10 15

time*E2

Figure 3.6. VaR time plot historical simulation with volatility updating.

XFGtimeseries2.xpl

volatility in the scenarios. The peak VaR values are much larger for volatility

updating than for the basic historical simulation.

In order to ļ¬nd out, which framework for VaR estimation has the best predictive

power, we count the number of violations of the VaR prediction and divide it

by the number of actually observed losses. We use the 99% quantile, for which

we would expect an violation rate of 1% for an optimal VaR estimator. The

history used for the drawings of the scenarios consists of N = 250 days, and the

holding period is h = 1 day. For the volatility updating we use a decay factor of

Ī³ = 0.94, J.P. Morgan (1996). For the simulation we assume that the synthetic

zero-bond has a remaining time to maturity of 10 years at the beginning of

the simulations. For the calculation of the ļ¬rst scenario of a basic historical

simulation N + h ā’ 1 observations are required. A historical simulation with

volatility updating requires 2(N + h ā’ 1) observations preceding the trading

day the ļ¬rst scenario refers to. In order to allow for a comparison between

diļ¬erent methods for the VaR calculation, the beginning of the simulations

is t0 = [2(N + h ā’ 1)/N ]. With these simulation parameters we obtain 1646

3.7 P-P Plots 73

observations for a zero-bond in the industry sector and 1454 observations for a

zero-bond in the banking sector.

In Tables 3.12 to 3.14 we list the percentage of violations for all yield curves

and the four variants of historical simulation V1 to V4 (V1 = Basic Historical

Simulation; V2 = Basic Historical Simulation with Mean Adjustment; V3 =

Historical Simulation with Mean Adjustment; V4 = Historical Simulation with

Volatility Updating and Mean Adjustment). In the last row we display the

average of the violations of all curves. Table 3.12 contains the results for the

simulation with relative changes of the full yield curves and of the yield spreads

over the benchmark curve as risk factors. In Table 3.13 the risk factors are

changes of the benchmark curves. The violations in the conservative approach

and in the simultaneous simulation of relative spread and benchmark changes

are listed in Table 3.14.

XFGexc.xpl

3.7 P-P Plots

The evaluation of the predictive power across all possible conļ¬dence levels

Ī± ā [0, 1] can be carried out with the help of a transformation of the empirical

distribution {L(k) | 0 ā¤ k ā¤ N ā’ 1}. If F is the true distribution function

of the loss L within the holding period h, then the random quantity F (L) is

(approximately) uniformly distributed on [0, 1]. Therefore we check the values

Fe L(t) for t0 ā¤ t ā¤ t1 , where Fe is the empirical distribution. If the prediction

quality of the model is adequate, these values should not diļ¬er signiļ¬cantly from

a sample with size 250 (t1 ā’ t0 + 1) from a uniform distribution on [0, 1].

The P-P plot of the transformed distribution against the uniform distribution

(which represents the distribution function of the transformed empirical distri-

bution) should therefore be located as closely to the main diagonal as possible.

The mean squared deviation from the uniform distribution (MSD) summed

over all quantile levels can serve as an indicator of the predictive power of a

quantile-based risk measure like VaR. The XFGpp.xpl quantlet creates a P-P

plot and calculates the MSD indicator.

74 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

Full yield Spread curve

Curve V1 V2 V3 V4 V1 V2 V3 V4

INAAA 1,34 1,34 1,09 1,28 1,34 1,34 1,34 1,34

INAA2 1,34 1,22 1,22 1,22 1,46 1,52 1,22 1,22

INAA3 1,15 1,22 1,15 1,15 1,09 1,09 0,85 0,91

INA1 1,09 1,09 1,46 1,52 1,40 1,46 1,03 1,09

INA2 1,28 1,28 1,28 1,28 1,15 1,15 0,91 0,91

INA3 1,22 1,22 1,15 1,22 1,15 1,22 1,09 1,15

INBBB1 1,28 1,22 1,09 1,15 1,46 1,46 1,40 1,40

INBBB2 1,09 1,15 0,91 0,91 1,28 1,28 0,91 0,91

INBBB3 1,15 1,15 1,09 1,09 1,34 1,34 1,46 1,52

INBB1 1,34 1,28 1,03 1,03 1,28 1,28 0,97 0,97

INBB2 1,22 1,22 1,22 1,34 1,22 1,22 1,09 1,09

INBB3 1,34 1,28 1,28 1,22 1,09 1,28 1,09 1,09

INB1 1,40 1,40 1,34 1,34 1,52 1,46 1,09 1,03

INB2 1,52 1,46 1,28 1,28 1,34 1,40 1,15 1,15

INB3 1,40 1,40 1,15 1,15 1,46 1,34 1,09 1,15

BNAAA 1,24 1,38 1,10 1,10 0,89 0,89 1,03 1,31

BNAA1/2 1,38 1,24 1,31 1,31 1,03 1,10 1,38 1,38

BNA1 1,03 1,03 1,10 1,17 1,03 1,10 1,24 1,24

BNA2 1,24 1,31 1,24 1,17 0,76 0,83 1,03 1,03

BNA3 1,31 1,24 1,17 1,10 1,03 1,10 1,24 1,17

Average 1,27 1,25 1,18 1,20 1,22 1,24 1,13 1,15

Table 3.12. Violations full yield and spread curve (in %)

Curve V1 V2 V3 V4

INAAA, INAA2, INAA3, INA1, INA2, 1,52 1,28 1,22 1,15

INA3, INBBB1, INBBB2, INBBB3,

INBB1, INBB2, INBB3, INB1, INB2,

INB3

BNAAA, BNAA1/2, BNA1, BNA2, BNA3 1,72 1,44 1,17 1,10

Average 1,57 1,32 1,20 1,14

Table 3.13. Violations benchmark curve (in %)

3.8 Q-Q Plots

With a quantile plot (Q-Q plot) it is possible to visualize whether an ordered

sample is distributed according to a given distribution function. If, e.g., a

sample is normally distributed, the plot of the empirical quantiles vs. the

3.9 Discussion of Simulation Results 75

conservative approach simultaneous simulation

Curve V1 V2 V3 V4 V1 V2 V3 V4

INAAA 0,24 0,24 0,30 0,30 1,22 1,28 0,97 1,03

INAA2 0,24 0,30 0,36 0,30 1,22 1,28 1,03 1,15

INAA3 0,43 0,36 0,30 0,30 1,22 1,15 1,09 1,09

INA1 0,36 0,43 0,55 0,55 1,03 1,03 1,03 1,09

INA2 0,49 0,43 0,49 0,49 1,34 1,28 0,97 0,97

INA3 0,30 0,36 0,30 0,30 1,22 1,15 1,09 1,09

INBBB1 0,43 0,49 0,36 0,36 1,09 1,09 1,03 1,03

INBBB2 0,49 0,49 0,30 0,30 1,03 1,03 0,85 0,79

INBBB3 0,30 0,30 0,36 0,36 1,15 1,22 1,03 1,03

INBB1 0,36 0,30 0,43 0,43 1,34 1,34 1,03 0,97

INBB2 0,43 0,36 0,43 0,43 1,40 1,34 1,15 1,09

INBB3 0,30 0,30 0,36 0,36 1,15 1,15 0,91 0,91

INB1 0,43 0,43 0,43 0,43 1,34 1,34 0,91 0,97

INB2 0,30 0,30 0,30 0,30 1,34 1,34 0,97 1,03

INB3 0,30 0,30 0,36 0,30 1,46 1,40 1,22 1,22

BNAAA 0,62 0,62 0,48 0,48 1,31 1,31 1,10 1,03

BNAA1/2 0,55 0,55 0,55 0,48 1,24 1,31 1,10 1,17

BNA1 0,62 0,62 0,55 0,55 0,96 1,03 1,10 1,17

BNA2 0,55 0,62 0,69 0,69 0,89 1,96 1,03 1,03

BNA3 0,55 0,55 0,28 0,28 1,38 1,31 1,03 1,10

Average 0,41 0,42 0,41 0,40 1,22 1,22 1,03 1,05

Table 3.14. Violations in the conservative approach and simultaneous

simulation(in %)

quantiles of a normal distribution should result in an approximately linear

plot. Q-Q plots vs. a normal distribution can be generated with the following

command:

VaRqqplot(matrix(N,1)|MMPL,VaR,opt)

3.9 Discussion of Simulation Results

In Figure 3.7 the P-P plots for the historical simulation with the full yield curve

(INAAA) as risk factor are displayed for the diļ¬erent variants of the simulation.

From the P-P plots it is apparent that mean adjustment signiļ¬cantly improves

the predictive power in particular for intermediate conļ¬dence levels (i.e., for

small risk factor changes).

76 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

Basic Simulation Mean Adjustment

1

1

Empirical Distribution

Empirical Distribution

0.5

0.5

0

0

0 0.5 1 0 0.5 1

Uniform Distribution Uniform Distribution

Volatility Updating Volatility Updating & Mean Adjustment

1

1

Empirical Distribution

Empirical Distribution

0.5

0.5

0

0

0 0.5 1 0 0.5 1

Uniform Distribution Uniform Distribution

Figure 3.7. P-P Plots variants of the simulation. XFGpp.xpl

Figure 3.8 displays the P-P plots for the same data set and the basic historical

simulation with diļ¬erent choices of risk factors. A striking feature is the poor

predictive power for a model with the spread as risk factor. Moreover, the

over-estimation of the risk in the conservative approach is clearly reļ¬‚ected by

a sine-shaped function, which is superposed on the ideal diagonal function.

In Figs. 3.9 and 3.10 we show the Q-Q plots for basic historic simulation and

volatility updating using the INAAA data set and the full yield curve as risk

factors. A striking feature of all Q-Q plots is the deviation from linearity (and,

thus, normality) for extreme quantiles. This observation corresponds to the

leptokurtic distributions of time series of market data changes (e.g. spread

changes as discussed in section 3.3.2).

3.9 Discussion of Simulation Results 77

Benchmark Curve Spread Curve

1

1

Empirical Distribution

Empirical Distribution

0.5

0.5

0

0

0 0.5 1 0 0.5 1

Uniform Distribution Uniform Distribution

Conservative Approach Simultaneous Simulation

1

1

Empirical Distribution

Empirical Distribution

0.5

0.5

0

0

0 0.5 1 0 0.5 1

Uniform Distribution Uniform Distribution

Figure 3.8. P-P Plots choice of risk factors. XFGpp.xpl

3.9.1 Risk Factor: Full Yield

The results in Table 3.12 indicate a small under-estimation of the actually

observed losses. While volatility updating leads to a reduction of violations,

this eļ¬ect is not clearly recognizable for the mean adjustment. The positive

results for volatility updating are also reļ¬‚ected in the corresponding mean

squared deviations in Table 3.15. Compared with the basic simulation, the

model quality can be improved. There is also a positive eļ¬ect of the mean

adjustment.

78 3 Quantiļ¬cation of Spread Risk by Means of Historical Simulation

VaR reliability plot

4

2

L/VaR quantiles

0

-2

-4

-4 -2 0 2 4

normal quantiles

Figure 3.9. Q-Q Plot for basic historical simulation.

3.9.2 Risk Factor: Benchmark

The results for the number of violations in Table 3.13 and the mean squared

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