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 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).


3.3.1 Data Analysis with XploRe

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 7310 summarize command for the INAAA curve is given in Table 3.1.


Table: Output of 7315 summarize for the INAAA curve. 7318 XFGsummary.xpl
\begin{table}\begin{center}{\scriptsize\begin{verbatim}Contents of summMini...
... 0.69877
10Y 4.87 8.36 6.6962 6.7 0.69854\end{verbatim}}
\end{center}\end{table}



Table 3.2: Data variables
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


The long term means are of particular interest. Therefore, we summarize them in Table 3.3.

Table 3.3: Long term mean for different USD yield curves
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


In order to get an impression of the development of the treasury 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, 7327 XFGtreasury.xpl . The averaged yields within the observation period are displayed in Figure 3.2 for USTF , INAAA , INBBB2, INBB2 and INB2, 7334 XFGyields.xpl .

Figure: US Treasury Yields. 7338 XFGtreasury.xpl
\includegraphics[width=1.2\defpicwidth]{d1.ps}

Figure: Averaged Yields. 7342 XFGyields.xpl
\includegraphics[width=1.2\defpicwidth]{dmean.ps}

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, 7347 XFGseries.xpl .

Figure: Credit Spreads. 7351 XFGseries.xpl
\includegraphics[width=1.4\defpicwidth]{d2.ps}

We run the summary statistics to obtain information on the mean spreads. Our results, which can also be obtained with the 7354 mean command, are collected in Table 3.4, 7357 XFGmeans.xpl .


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


Now we calculate the 1-day spread changes from the observed yields and store them to variables DASIN01AAA, etc. We run the 7360 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.


7364 XFGchange.xpl


Table 3.5: volatility for absolute spread changes in basis points p.a.
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 3.6: Skewness for absolute 1-day spread changes (in $ {\sigma }^3$).
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 3.7: Kurtosis for absolute spread changes (in $ \sigma ^4$).
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


For the variable DASIN01AAA[,12] (the 10 year AAA spreads) we demonstrate the output of the 7369 descriptive command in Table 3.8.


Table: Output of 7372 descriptive for the 10 years AAA spread.
\begin{table}\begin{center}
{\scriptsize\begin{verbatim}======================...
...=========================================\end{verbatim}}
\end{center}\end{table}


Finally we calculate 1-day relative spread changes and run the 7375 descriptive command. The results for the estimates of volatility, skewness and kurtosis are summarized in Tables 3.9, 3.10 and 3.11.

7379 XFGrelchange.xpl


Table 3.9: Volatility for relative spread changes in %
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 3.10: Skewness for relative spread changes (in $ \sigma ^3$).
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 3.11: Kurtosis for relative spread changes (in $ \sigma ^4$).
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



3.3.2 Discussion of Results

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.


7824 XFGdist.xpl

Figure: Historical distribution and estimated normal distribution. 7830 XFGdist.xpl
\includegraphics[width=1.4\defpicwidth]{d3.ps}

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 7833 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.