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(1999), Bundesaufsichtsamt f¨r das Kreditwesen (2001).
u
52 3 Quanti¬cation of Spread Risk by Means of Historical Simulation


1. General market risk: This risk category comprises price changes of a
¬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 ¬nancial instrument can change in a way di¬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 ¬nancial instrument relative
to the benchmark, which signi¬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 speci¬c price
risk or speci¬c risk ” a term used in documents on banking regulation, Bank for
International Settlements (1998a), Bank for International Settlements (1998b)
” and characterize the contribution of the individual risk of a given ¬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 re¬‚ected by product-speci¬c
(swaps, bonds, etc.), industry-speci¬c (bank, ¬nancial institution, retail com-
pany, etc.) and rating-speci¬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 in¬‚uencing factors like collateral, ¬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-speci¬c yield curves but neglects the in¬‚uence of industry- and rating-
speci¬c e¬ects in modelling its general market risk, can expect speci¬c price
risk to be signi¬cantly larger than in a bank which includes these in¬‚uences
in modeling general market risk. The di¬erence is due to the consideration of
product-, industry- and rating-speci¬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 speci¬c risk. The uncertainty is due to
the fact that it is hard to de¬ne what a market-sector is. The de¬nition of
benchmark curves for the analysis of general market risk depends, however,
critically on the market sectors identi¬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 di¬erent methods for the quanti¬cation of the
risk of a ¬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 di¬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 ¬nancial institutions with
ratings AAA, AA+/AA, A+, A, A’ (we use the Standard & Poor˜s naming
convention) and for corporate/industry bonds with ratings AAA, AA, AA’,
A+, A, A’, BBB+, BBB, BBB’, BB+, BB, BB’, B+, B, B’. 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/¬nancial
54 3 Quanti¬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 di¬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 Bloomberg™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 di¬erent rating categories and the two industries. We correct
the data history for obvious ¬‚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 di¬erent yield curves in individual ¬les. The ¬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 ¬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 di¬erent yield curves. As example ¬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-speci¬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 Quanti¬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 di¬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 ¬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 Quanti¬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 ¬‚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 di¬erent for di¬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 signi¬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-speci¬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-
speci¬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
signi¬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 Quanti¬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 σ 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 signi¬cantly
larger than 3 (see Table 3.6). Thus, the distribution of absolute changes is
leptokurtic. There is no signi¬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 σ 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 Quanti¬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 σ 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 di¬erences
are less obvious than the di¬erence in the shape of the distributions. However,
in a portfolio with di¬erent ¬nancial instruments, which is exposed to di¬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 σ 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 di¬erent correlations, the di¬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 ¬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 Quanti¬cation of Spread Risk by Means of Historical Simulation


period, i.e., the time window investigated, consists of N ≥ 1 trading days and
the holding period of h ≥ 1 trading days. The con¬dence level for the VaR is
± ∈ [0, 1]. At each point in time 0 ¤ t ¤ 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 < · · · < Tn are contained in our data set for 1 ¤ i ¤ n, where
n is the number of di¬erent maturities. The corresponding spreads are de¬ned
by Si (t) = Ri (t) ’ Bi (t) for 1 ¤ i ¤ n.
In the following subsections 3.4.1 to 3.4.5 we specify di¬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 di¬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 ’ t) at time t0 ¤
t ¤ t1 for the remaining time to maturity T ’ 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 ¤ T ’ t < Ti+1 (for reasons of simplicity we do not consider
remaining times to maturity T ’ t < T1 and T ’ t > Tn ):

[Ti+1 ’ (T ’ t)]Ri (t) + [(T ’ t) ’ Ti ]Ri+1 (t)
R(t, T ’ t) = . (3.1)
Ti+1 ’ Ti

The present value of the bond P V (t) at time t can be obtained by discounting,
1
t 0 ¤ t ¤ t1 .
P V (t) = , (3.2)
T ’t
1 + R(t, T ’ t)
In the historical simulation the relative risk factor changes

Ri t ’ k/N ’ Ri t ’ (k + h)/N
(k)
0 ¤ k ¤ N ’ 1,
∆i (t) = , (3.3)
Ri t ’ (k + h)/N
3.4 Historical Simulation and Value at Risk 65


are calculated for t0 ¤ t ¤ t1 and each 1 ¤ i ¤ n. Thus, for each scenario k we
obtain a new ¬ctive yield curve at time t + h, which can be determined from
the observed yields and the risk factor changes,
(k) (k)
1 ¤ i ¤ n,
Ri (t + h) = Ri (t) 1 + ∆i (t) , (3.4)

by means of linear interpolation. This procedure implies that the distribution of
risk factor changes is stationary between t’(N ’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 µi (t) with variance one via

∆i (t) = σi µi (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, di¬erent drawings would originate from di¬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 ’ t one
obtains
(k) (k)
[Ti+1 ’ (T ’ t)]Ri (t) + [(T ’ t) ’ Ti ]Ri+1 (t)
(k)
(t + h, T ’ t) =
R
Ti+1 ’ Ti

for the yield. With (3.2) we obtain a new ¬ctive present value at time t + h:
1
P V (k) (t + h) = . (3.6)
T ’t
R(k) (t + h, T ’ t)
1+

In this equation we neglected the e¬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 re¬ned for ¬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 ¬ctive present value P V (k) (t + h) is compared with the present value
for unchanged yield R(t + h, T ’ t) = R(t, T ’ t) for each scenario k (here the
remaining time to maturity is not changed, either).
1
P V (t + h) = . (3.7)
T ’t
1 + R(t + h, T ’ t)
66 3 Quanti¬cation of Spread Risk by Means of Historical Simulation


The loss occurring is

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

i.e., losses in the economic sense are positive while pro¬ts are negative. The
VaR is the loss which is not exceeded with a probability ± and is estimated as
the [(1 ’ ±)N + 1]-th-largest value in the set

{L(k) (t + h) | 0 ¤ k ¤ N ’ 1}.

This is the (1 ’ ±)-quantile of the corresponding empirical distribution.
2. Mean Adjustment:
A re¬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 ¬xed 1 ¤ i ¤ n the average of relative
(k)
changes ∆i (t) is di¬erent from 0, a trend is projected from the past to the
future in the generation of ¬ctive yields in (3.4). Thus the relative changes are
(k) (k)
corrected for the mean by replacing the relative change ∆i (t) with ∆i (t) ’
∆i (t) for 1 ¤ i ¤ n in (3.4):
N ’1
1 (k)
∆i (t) = ∆i (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 ¤ t ¤ t1 by
N ’1
2
(k)
2
γ k ∆i (t)
= (1 ’ γ) 1 ¤ i ¤ n.
σi (t) , (3.10)
k=0

The parameter γ ∈ [0, 1] is a decay factor, which must be calibrated to generate
a best ¬t to empirical data. The recursion formula
2
(0)
2 2
σi (t) = (1 ’ γ)σi (t ’ 1/N ) + γ ∆i (t) 1 ¤ i ¤ n,
, (3.11)

is valid for t0 ¤ t ¤ 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)
∆i (t)
(k)
= σi (t) · 0 ¤ k ¤ N ’ 1.
δi (t) , (3.12)
σi (t ’ (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 ∆i (t)/σi (t)) is stationary. For each k
these renormalized relative changes are used in analogy to (3.4) for the deter-
mination of ¬ctive scenarios:

(k) (k)
1 ¤ i ¤ n,
Ri (t + h) = Ri (t) 1 + δi (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 ’1
1 (k)
δ i (t) = δi (t), (3.14)
N
k=0
(k) (k)
and use the adjusted relative risk factor change δi (t) ’ δ i (t) instead of δi (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 ’ t) at time t for
68 3 Quanti¬cation of Spread Risk by Means of Historical Simulation


remaining time to maturity T ’ t is calculated similarly to (3.1) from adjacent
values by linear interpolation,

{Ti+1 ’ (T ’ t)}Bi (t) + {(T ’ t) ’ Ti }Bi+1 (t)
B(t, T ’ t) = . (3.15)
Ti+1 ’ Ti

The generation of scenarios and the interpolation of the ¬ctive benchmark curve
is carried out in analogy to the procedure for the full yield curve. We use

Bi t ’ k/N ’ Bi t ’ (k + h)/N
(k)
0 ¤ k ¤ N ’ 1,
∆i (t) = , (3.16)
Bi t ’ (k + h)/N

and
(k) (k)
1 ¤ i ¤ n.
Bi (t + h) = Bi (t) 1 + ∆i (t) , (3.17)

Linear interpolation yields
(k) (k)
{Ti+1 ’ (T ’ t)}Bi (t) + {(T ’ t) ’ Ti }Bi+1 (t)
(k)
(t + h, T ’ t) =
B .
Ti+1 ’ Ti

In the determination of the ¬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 ’ t) = B (k) (t + h, T ’ t) + S(t, T ’ t), (3.18)

which is used for the calculation of a new ¬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 ’ t) at time t for time to
maturity T ’ t is again obtained by linear interpolation. We now use

Si t ’ k/N ’ Si t ’ (k + h)/N
(k)
0 ¤ k ¤ N ’ 1,
∆i (t) = , (3.19)
Si t ’ (k + h)/N
3.4 Historical Simulation and Value at Risk 69


and
(k) (k)
1 ¤ i ¤ n.
Si (t + h) = Si (t) 1 + ∆i (t) , (3.20)
Here, linear interpolation yields
(k) (k)
{Ti+1 ’ (T ’ t)}Si (t) + {(T ’ t) ’ Ti }Si+1 (t)
(k)
(t + h, T ’ t) =
S .
Ti+1 ’ Ti

Thus, in the determination of the ¬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 ’ t) = B(t, T ’ t) + S (k) (t + h, T ’ 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 ’ t) = B (k) (t + h, T ’ t) + S (k) (t, T ’ 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 di¬erent results of the variants in subsection 3.4.1 and this
subsection.
70 3 Quanti¬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 ¤ t ¤ t1 the present value at time t+h is calculated with
the yield R(t + h, T ’ t), which is obtained from observed data for Ri (t + h)
by linear interpolation, according to
1
P V (t) = . (3.23)
T ’t
1 + R(t + h, T ’ t)

This corresponds to a loss L(t) = P V (t) ’ P V (t + h), where, again, the short-
ening of the time to maturity is not taken into account.
The di¬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 ’t) in (3.23) is replaced with B(t+h, T ’t)+S(t, T ’t).
On an average (1 ’ ±) · 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 di¬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 speci¬es the method parameter as ”EDF” (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 pro¬ts and losses must be
estimated anew for each day from the N relative risk factor changes, because
the pro¬t/loss observations are not identical with the risk factor changes.
For each day the N pro¬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 pro¬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 pro¬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 pro¬t/loss values contained in the
vector (PL[t,])™, which has one column and N rows followed by the actual
mark-to-model pro¬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 Quanti¬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 ¬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
γ = 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 ¬rst scenario of a basic historical
simulation N + h ’ 1 observations are required. A historical simulation with
volatility updating requires 2(N + h ’ 1) observations preceding the trading
day the ¬rst scenario refers to. In order to allow for a comparison between
di¬erent methods for the VaR calculation, the beginning of the simulations
is t0 = [2(N + h ’ 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 con¬dence levels
± ∈ [0, 1] can be carried out with the help of a transformation of the empirical
distribution {L(k) | 0 ¤ k ¤ N ’ 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 ¤ t ¤ t1 , where Fe is the empirical distribution. If the prediction
quality of the model is adequate, these values should not di¬er signi¬cantly from
a sample with size 250 (t1 ’ 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 Quanti¬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 di¬erent variants of the simulation.
From the P-P plots it is apparent that mean adjustment signi¬cantly improves
the predictive power in particular for intermediate con¬dence levels (i.e., for
small risk factor changes).
76 3 Quanti¬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 di¬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 re¬‚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 e¬ect is not clearly recognizable for the mean adjustment. The positive
results for volatility updating are also re¬‚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 e¬ect of the mean
adjustment.
78 3 Quanti¬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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