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 treasury
bonds as well as for
bonds issued by banks and financial 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/financial 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 different maturities.
The yield curve, of
course, depends on the choice of bonds. Yields are option-adjusted but not
corrected for coupon payments. 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 benchmark curve for the
different rating categories and the two industries.
We correct the data history for obvious flaws using complementary information
from other data sources.
Some parts of our analysis in this section can be compared with
the results given in Kiesel et al. (1999).
We store the time series of the different yield curves in individual files.
The file names, the corresponding
industries and ratings and the names of the matrices used in the XploRe code
are listed in Table 3.2.
Each file 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 different
yield curves. As example files 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.
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The long term means are of particular interest. Therefore, we
summarize them in Table 3.3.
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In the next step we calculate spreads relative to the treasury
curve by subtracting the treasury curve from the rating-specific
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
.
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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 calculate the first 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.
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|
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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.
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|
|
Time Development of Yields and Spreads: The time development of US treasury yields displayed in Figure 3.1 indicates that the yield curve was steeper at the beginning of the observation period and flattened 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 Figure 3.3 is different for different credit qualities. While there is a large variation of spreads for the speculative grades, the variation in the investment grade sector is much smaller. A remarkable feature is the significant 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-specific 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-specific yield curves, which is displayed in Figure 3.3, is normal (see also Table 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 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 significant 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, which is indeed the conclusion from the results in Table 3.10.
Kurtosis: The absolute 1-day changes lead to a kurtosis, which is significantly larger than 3 (see Table 3.6). Thus, the distribution of absolute changes is leptokurtic. There is no significant 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 decreasing maturity.
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 distribution of absolute spreads around the mean spread in an averaged shifted histogram and the normal distribution with the variance estimated from historical data.
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 distribution 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 differences are less obvious than
the difference in the shape of the distributions. However, in a portfolio
with different financial instruments, which is exposed to different
risk factors with different correlations, the difference in the shape of
the distribution 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.