3. Estimation techniques

The above models increase the number of explanatory factors according to theoretical considerations. In our analysis, we test whether there is statistical evidence for the proposed risk factors to be significantly evaluated by the market. As before, we acknowledge that there might be inefficiencies present in the market. Similar to stock markets, we then assume an approximate factor structure to hold in the bond markets. As before, we then test for significance using the Fama/MacBeth approach. The data used consist of European Zero Coupon Curves estimated by ICAP and provided by Thomson Reuters. We then determine whether risk factors are significant for every time scale and not only on an aggregate level. Similar to our analysis with regards to the stock markets, we find that the significance of the risk factors varies with different time scales. By reconstructing the time series using the significant time scales only, we concentrate on a relatively small number of wavelet functions. We then investigate the scaled and significantly evaluated risk factors for their ability to help forecast the term structure of interest rates. In our analysis, we can only detect four significantly evaluated

Structural models based on the idea of Merton result in theoretical credit spreads that significantly deviate from observable corporate bond markets spread [17]. The models can only explain a limited proportion of corporate bond market spreads even if tax asymmetries, liquidity, and conversion options are considered. This empirical finding is referred to as the credit spread puzzle [18]. Similarities between equity and corporate bond market's risk have long been recognized and risk factors similar to those applied in stock markets are included in the analysis of corporate bond spreads, for example see [2]. The set of explanatory variables is enriched by other researchers to also account for market inefficiencies. For example, it can be assumed that there are limits to arbitrage which combined with noise leads to predictable deviations of market prices from the asset's fundamental value [19]. A solution could be a dynamic model with dispersed information in which noisy investors only learn about fundamental information with a time delay in order to solve the puzzle. Furthermore, it can be assumed that market participants develop habit formation [20]. Other researchers find that there are higher spreads for bonds for which analysts' forecasts are more diverse, i.e., that higher risk premiums are present for bonds where there is higher disagreement [21, 22]. Furthermore, the necessity to analyze varying frequency behavior in the data has been documented for credit markets, for example see [23]. In contrast to the stock and bond market, we do not impose Ross' approximate factor structure, but instead we use Merton's approach to postulate a straightforward relationship between credit spreads and risk factors that influence the corporate's ability to pay back its debt and credit spreads on corporate bond markets in general (fundamental factors). If the purpose is to analyze corporate bond markets jointly, the

To estimate the proportion of credit spreads (cs) explained by risk factors, Eq. (6) has to be

cst ¼ a þ b ðxtÞ þ ut (6)

risk factors for the term structure of interest rates [16].

assumption of Ross's factor structure would become necessary.

with ut being a white noise error term, and xt being the risk factors.

2.3. Corporate bonds

72 Wavelet Theory and Its Applications

analyzed econometrically.

Wavelet analysis estimates the frequency structure of a time series and in addition to that it keeps the information when an event of the time series takes place. This way an event can be localized in the time domain with regards to its time of occurrence although frequencies are analyzed as well. The functions at the heart of our analyses are wavelets. In contrast to co-sine functions (waves), wavelets are not defined over the entire time axis but have limited support. In order to achieve the ability to analyze relationships for different time periods, the wavelets are moved over the time axis and at the various scales the support is accordingly. By doing so it is possible to allow for changing regime shifts and the problem of parameter constancy is less severe which removes the necessity to eliminate extreme market moves from a purely statistical point of view. The length (width) of a wavelet on a certain scale represents an investment period of interest. The maximal overlap discrete wavelet transform (MODWT) increases the support of the dilated wavelet with increasing scale, thereby increasing the investment period. The advantage of this form of discrete wavelet transform is that it can be applied to any number of observations of the time series of interest.

Wavelets (ψj, k and ϕJ,k) when multiplied with their respective coefficients at a certain level "j" or "J" are called atoms Dj,k and SJ,k (i.e., dj,k\*ψj,k = Dj,k and sJ,k\*ϕJ,k = SJ,k) with ψj,k and ϕJ,k being the wavelet and scaling functions at level "j" or "J" and "k" indicating the location of the wavelet on the time axis. The sum of all atoms SJ,k(t) and Dj,k(t) over all locations on the time axis k = 1, …, <sup>n</sup> <sup>2</sup><sup>j</sup> at a certain level "j" or "J" are given by Eqs. (7) and (8).

$$\mathbf{S}\_{\mathbf{l}}(\mathbf{t}) = \phi\_{\mathbf{l},k} \text{ at level } \mathbf{l} \tag{7}$$

$$\mathbf{D}\_{\mathbf{j}}(\mathbf{t}) = \sum\_{k=1}^{\frac{n}{2^{\circ}}} d\_{j,k} \boldsymbol{\psi}\_{j,k} \forall \mathbf{j} = 1, \ldots, \mathbf{j} \tag{8}$$

Defining the importance of information to be valid for a specific time period only, the time series are decomposed into their respective resolutions in time (time scales). The time series are then approximated using only parts of the coefficients and their respective wavelets, i.e., the multiresolution decomposition is applied to the time series which are then in turn reconstructed using only the significant portions at the various scales.

The wavelets used in the analysis are "symmlets." Those wavelets are best suited for the analyses because their characteristics are closest to the functions used in the classical Fourier analysis in that they are symmetric and do not contain jumps. This makes most sense if our goal is to analyze the time series in the time and frequency domain. As co-sine functions, the chosen wavelets should not require an interpretation in itself. In that sense, those wavelets are the most "neutral" functions so that no other wavelet functions are considered that would require additional explanations. Our goal is to be able to allow for an analysis on different scales but we would like to keep as much structure of the original time series as possible. The decomposition of the data is done by identifying significant wavelets at certain scales, i.e., wavelets with a specific support on the time axis. The search for significant wavelets is then repeated on the next higher scale (lower frequency). With each increase of the wavelets´ widths a new scale is defined. The number of scales used in this analysis equals four (i.e., J = 4) which is a direct result of the number of observations available. For an explanation of how many levels are recommendable, see [24]. Level "j" wavelet coefficients are associated with periods [2j , 2j+1]. The sums of all atoms at all levels—one to four—result in the original time series.

We perform the regression analysis at each level. Asset returns are regressed on risk factors at different time scales, i.e., the factor pricing equations are estimated at every time scale 1, …, J using the reconstructed time series as outlined before (see Eqs. (9) and (10)):

$$(\mathbf{c}\mathbf{s}\_l)\begin{bmatrix}\mathbf{d}\_l\end{bmatrix} = \mathbf{a} + \mathbf{b}\begin{pmatrix}\mathbf{x}\_l\end{pmatrix}\begin{bmatrix}\mathbf{d}\_l\end{bmatrix} + u\_l \text{ for all } \mathbf{d1} \text{ to } \mathbf{d4} \tag{9}$$

$$\mathbf{a}\left(\mathbf{c}\mathbf{s}\_{l}\right)\mathbf{s}\_{4}\mathbf{l} = \mathbf{a} + \mathbf{b}\left(\mathbf{x}\_{l}\right)\left|\mathbf{s}\_{4}\right| + \boldsymbol{\mu}\_{l}\text{ s4} \tag{10}$$

by low liquidity we do not distinguish the high yield index with regards to time to maturity. As explanatory variables, the level, slope and volatility of the bond markets are calculated from the monthly time series for European government term structures of interest rates available from the same source. Data with regards to the stock index Dax are included in the analysis to capture risk characteristics present in the stock markets. Volatility for the stock market is calculated from that time series. Data with regards to European corporate default probabilities are taken from Moody's. Monthly 1-year and 5-year default rates for European investment grade and Caa-C rated companies are available from January 2000 to April 2012. Due to data availability and quality of the data, the 5-year default rates are combined with 1-

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75

We decompose each time series using the maximal overlap discrete wavelet transform (MODWT), i.e. the time series European credit spreads, Government Yields, Slopes, Volatilities, monthly return of Dax, volatility of monthly return of Dax, and Moody's default rates for European Investment Grade Corporates and CCC-Lower-Rated Corporates are decomposed to their respective time and frequency domain components as explained in section 3. Calculating the volatility from the monthly return data, the number of monthly observations we are left to be able to use is 132 (January 2001 to April 2012). As a result of the number of observations the number is set to four. The MODWT estimates the wavelet coefficients "d1" to "d4" and

The decomposition of the time series and the amount of variation explained with Crystals (sum of wavelets and their estimated coefficients at levels j = 1, …, 4) are summarized in Table 1.

The risk factors are well explained by coarse scales (low frequencies, e.g., "s4"). The only variable that has different features is the return of the DAX index. In that case the high

EMU corporates all maturities spread 0.3 0.4 2.2 5.5 91.5 EMU corporates 1–3 year maturity spread 0.4 0.6 2.7 7.2 89 EMU corporates 3–5 year maturity spread 0.3 0.5 2.4 5.6 91.2 Euro high yield spread 0.4 1 2.1 4.6 92 DAX return 46 25 12 8 8 DAX volatility 0.1 0.4 2.4 3 94.1 Euro government 10-year yield 0.1 0.1 0.3 0.4 99 Euro government yield curve slope 0.4 0.5 1.4 1.7 96 Euro government yield volatility 0.1 0.3 1.9 4.4 93.2 European investment grade default rates 0.1 0.2 0.3 1.7 97.7 European high yield default rates 0.5 0.7 1.6 6.8 90.3

"d1" "d2" "d3" "d4" "s4"

year default rates.

4.2. Wavelet analysis

"s4" scaling coefficients.

Table 1. Variation of the time series explained by crystals (in %).

The proportion explained by the risk factors is therefore estimated at each time scale.

#### 4. Empirical analysis

#### 4.1. The data

The credit index data used in this analysis are taken from Bank of America/Merrill Lynch. We use monthly OAS spreads of corporate bond indexes for the time period January 2000 to January 2013. We analyze EMU corporates in the rating category BBB-A (all EMU Corporates). The analysis is performed by using the indexes for various times to maturities 1–3, 3–5 in case of investment grade corporates. The differentiation is necessary to address the phenomenon that short maturities of investment grade corporate bonds depict a higher extend of the credit spread puzzle. For the Euro high yield market we use the Euro high yield index which contains firms with credit ratings BB and lower. Due to concerns with regards to biases caused by low liquidity we do not distinguish the high yield index with regards to time to maturity. As explanatory variables, the level, slope and volatility of the bond markets are calculated from the monthly time series for European government term structures of interest rates available from the same source. Data with regards to the stock index Dax are included in the analysis to capture risk characteristics present in the stock markets. Volatility for the stock market is calculated from that time series. Data with regards to European corporate default probabilities are taken from Moody's. Monthly 1-year and 5-year default rates for European investment grade and Caa-C rated companies are available from January 2000 to April 2012. Due to data availability and quality of the data, the 5-year default rates are combined with 1 year default rates.

#### 4.2. Wavelet analysis

Defining the importance of information to be valid for a specific time period only, the time series are decomposed into their respective resolutions in time (time scales). The time series are then approximated using only parts of the coefficients and their respective wavelets, i.e., the multiresolution decomposition is applied to the time series which are then in turn

The wavelets used in the analysis are "symmlets." Those wavelets are best suited for the analyses because their characteristics are closest to the functions used in the classical Fourier analysis in that they are symmetric and do not contain jumps. This makes most sense if our goal is to analyze the time series in the time and frequency domain. As co-sine functions, the chosen wavelets should not require an interpretation in itself. In that sense, those wavelets are the most "neutral" functions so that no other wavelet functions are considered that would require additional explanations. Our goal is to be able to allow for an analysis on different scales but we would like to keep as much structure of the original time series as possible. The decomposition of the data is done by identifying significant wavelets at certain scales, i.e., wavelets with a specific support on the time axis. The search for significant wavelets is then repeated on the next higher scale (lower frequency). With each increase of the wavelets´ widths a new scale is defined. The number of scales used in this analysis equals four (i.e., J = 4) which is a direct result of the number of observations available. For an explanation of how many levels are recommendable, see [24]. Level "j" wavelet coefficients are associated with periods

, 2j+1]. The sums of all atoms at all levels—one to four—result in the original time series.

We perform the regression analysis at each level. Asset returns are regressed on risk factors at different time scales, i.e., the factor pricing equations are estimated at every time scale 1, …, J

The credit index data used in this analysis are taken from Bank of America/Merrill Lynch. We use monthly OAS spreads of corporate bond indexes for the time period January 2000 to January 2013. We analyze EMU corporates in the rating category BBB-A (all EMU Corporates). The analysis is performed by using the indexes for various times to maturities 1–3, 3–5 in case of investment grade corporates. The differentiation is necessary to address the phenomenon that short maturities of investment grade corporate bonds depict a higher extend of the credit spread puzzle. For the Euro high yield market we use the Euro high yield index which contains firms with credit ratings BB and lower. Due to concerns with regards to biases caused

<sup>þ</sup> ut for all d1 to d4 (9)

ð Þ cst ½ �¼ s4 a þ b xð Þ<sup>t</sup> ½ �þ s4 ut s4 (10)

using the reconstructed time series as outlined before (see Eqs. (9) and (10)):

<sup>¼</sup> <sup>a</sup> <sup>þ</sup> b xð Þ<sup>t</sup> dj

The proportion explained by the risk factors is therefore estimated at each time scale.

ð Þ cst dj

4. Empirical analysis

74 Wavelet Theory and Its Applications

4.1. The data

reconstructed using only the significant portions at the various scales.

[2j

We decompose each time series using the maximal overlap discrete wavelet transform (MODWT), i.e. the time series European credit spreads, Government Yields, Slopes, Volatilities, monthly return of Dax, volatility of monthly return of Dax, and Moody's default rates for European Investment Grade Corporates and CCC-Lower-Rated Corporates are decomposed to their respective time and frequency domain components as explained in section 3. Calculating the volatility from the monthly return data, the number of monthly observations we are left to be able to use is 132 (January 2001 to April 2012). As a result of the number of observations the number is set to four. The MODWT estimates the wavelet coefficients "d1" to "d4" and "s4" scaling coefficients.

The decomposition of the time series and the amount of variation explained with Crystals (sum of wavelets and their estimated coefficients at levels j = 1, …, 4) are summarized in Table 1.

The risk factors are well explained by coarse scales (low frequencies, e.g., "s4"). The only variable that has different features is the return of the DAX index. In that case the high


Table 1. Variation of the time series explained by crystals (in %).

frequencies contribute the most in explaining the variation of the time series. The other variables are best explained by time scales ranging from "d4" to "s4," whereas the return of the DAX is best explained by time scales "d1" to "d3."

At each scale "j" the coefficients are associated with time periods [2j , 2j+1]. The decomposition of the monthly data allows us to extract components of the data that prevail in the medium or long term. At the highest frequency of the monthly data—at scale "d1"—coefficients approximate reactions to information for the time period of 2–4 months. At scale two, three, and four, the respective time periods are 4–8 months, 8–16 months, 16–32 months. We associate the scales "d1," "d2," and "d3" with the medium term (short medium term equals 2–4 months, medium term 4–8 months, and longer medium term 8–16 months). The remaining two scales at the lower frequencies represent long-term behavior (1.3–2.6 years and longer). Extracting the components of the data that are influential in the medium or long term allows us to detect patterns that can be a result of different investment behavior or different information used in forming expectations, i.e., we are able to allow for inefficiencies in the credit market as outlined above.

In a next step we regress the credit spreads on the explanatory variables on a scale-by-scale basis, i.e., we restrict features of the data to be of importance in the medium ("d1" to "d3") or long term ("d4" to "s4"). After decomposing the regression variables, we reconstruct the time series using features of the time series at the respective resolutions 1, …, 4 only, thereby restricting their variation to the respective time scale. On the other hand, it allows for the possibility that information from more than just the previous period continues to be of influence in explaining credit spreads. By analyzing the amount of the variation explained in a regression (R<sup>2</sup> ) of the decomposed data at various time scales, we can infer which of the above outlined possible expectation formations is significant in the medium and long run. Table 2 summarizes the regression results for regressing European investment grade and European high yield credit spreads on the explanatory variables when the data are decomposed, i.e., when the time series are reconstructed to represent behavior present at scales "d1" to "s4."

Determining significant components gives us insights into how long time periods are for processing information. For the short medium term (2–4 months), we find that the default rate is either not significant ("d1" for EMU Corporate all maturities, and "d1," and "d2" for EMU high yield) or even of negative influence. This is a strong indication that the fundamentals are influential for longer time periods only, but do not explain well the variation in investment grade credit spreads for shorter time periods. The credit spread puzzle therefore manifests itself if the data are analyzed on time scales and in that the default rate is not significant in explaining credit spreads at all at some time scales. At the time scales that carry most of the energy, the default rate is significantly positive in explaining the investment grade credit spreads, i.e., for time periods (1.3–2.6 years). For high yield spreads, the default rate is significant only for a longer time horizon (8 months and above). We find that the influence of other explanatory variables changes at the various time scales as well. At the coarsest scale "s4," we find all explanatory variables of significant influence for the credit spreads. However, at scale "d3" (i.e., for a time period of 8– 16 months), the variables that capture the volatilities in the stock and bond markets cannot be viewed as being significant variables. The volatility of the DAX, although of importance in the aggregate data, loses its significance for investment grade credit spreads on several scales. It

continues to be important in explaining the high yield spreads though (with the exception of "d3"), which is another indication for the fact that stock market characteristics are more influential in the high yield bond markets than in the investment grade bond markets. The R2 supports the fact that has to be performed allowing for inefficiencies in the markets. We find that the amount of the variation in credit spreads explained by the identified risk factors is highest for time horizons from 1.3 to 2.6 years and above. The R2 at these time horizons in case of the investment grade bonds is 85–98%. Similar results are achieved to the high yield spreads. For

Table 2. Regression results for the European investment grade and high yield credit spreads on explanatory variables

DAX DAXV Yield Yield slope Yield vola. Default rate R2

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d1 �275 2.3\* 40.9 �<sup>131</sup> <sup>890</sup> <sup>257</sup> 0.1 d2 �16 0.1\* �21.3\* 39.9\* �264.6\* �7079\* 0.53 d3 �418\* 0.03 �19.8 72.9\* 19.3 �10662\* 0.61 d4 �365.7\* 0.01 �77\* 21.7\* 323.4\* <sup>14632</sup>\* 0.85 s4 �2278\* 0.4\* �114\* 65.2\* �645.2\* <sup>5518</sup>\* 0.98

d1 1.1 �0.0 �52.2\* 46.2\* �18.1 �4219\* 0.48 d2 20.7 0.1\* �18.1\* <sup>57</sup>\* �317.2\* �8857\* 0.55 d3 �336\* 0.02 �34.7\* <sup>86</sup>\* 12.7 �6722\* 0.61 d4 �399\* 0.0 �96.5\* 36.8\* 368.3\* <sup>16641</sup>\* 0.85 s4 �2201\* 0.45\* �112\* 83.7\* �670.5\* <sup>5381</sup>\* 0.98

d1 �26 0.05 �42.9\* 24.8\* �53.1 �3471\* 0.44 d2 �27 0.1\* �29.9\* 40.6\* �243.5\* �7071\* 0.53 d3 �410\* 0.03 �23.4\* 76.1\* 14.2 �10330\* 0.64 d4 �348\* 0.0 �79.5\* 23.2\* 306.5\* <sup>14859</sup>\* 0.87 s4 �2610\* 0.4\* �111\* <sup>59</sup>\* �520\* <sup>4130</sup>\* 0.98

d1 �275 2.2\* 40.9 �<sup>132</sup> <sup>890</sup> <sup>257</sup> 0.09 d2 �792 2.3\* 109.9 �333.5 �1025 1029 0.12 d3 �6267\* 0.2 �71.1 700.7\* 117.9 <sup>2622</sup>\* 0.38 d4 �4765\* 1.3\* �135.9 �404.3\* <sup>5096</sup>\* <sup>4224</sup>\* 0.84 s4 �9471\* 2.8\* �411\* <sup>340</sup>\* �4762\* <sup>6988</sup>\* 0.96

EMU corporates all maturities credit spread

EMU corporates maturity 1–3 years credit spread

EMU corporates maturity 3–5 years credit spread

EMU corporates high yield credit spread

Indicates significance at a 5% confidence level.

using reconstructed time.

\*

We therefore conclude that if information from the fundamental risk factors is allowed to be of influencing longer time periods (1.3–2.6 years and above), then the variables from Eqs (9) and (10)

shorter time periods, the amount of variation explained is much lower.

Empirical Support for Fundamental, Factor Models Explaining Major Capital Markets Using Wavelets http://dx.doi.org/10.5772/intechopen.74725 77


\* Indicates significance at a 5% confidence level.

frequencies contribute the most in explaining the variation of the time series. The other variables are best explained by time scales ranging from "d4" to "s4," whereas the return of the

the monthly data allows us to extract components of the data that prevail in the medium or long term. At the highest frequency of the monthly data—at scale "d1"—coefficients approximate reactions to information for the time period of 2–4 months. At scale two, three, and four, the respective time periods are 4–8 months, 8–16 months, 16–32 months. We associate the scales "d1," "d2," and "d3" with the medium term (short medium term equals 2–4 months, medium term 4–8 months, and longer medium term 8–16 months). The remaining two scales at the lower frequencies represent long-term behavior (1.3–2.6 years and longer). Extracting the components of the data that are influential in the medium or long term allows us to detect patterns that can be a result of different investment behavior or different information used in forming expectations,

In a next step we regress the credit spreads on the explanatory variables on a scale-by-scale basis, i.e., we restrict features of the data to be of importance in the medium ("d1" to "d3") or long term ("d4" to "s4"). After decomposing the regression variables, we reconstruct the time series using features of the time series at the respective resolutions 1, …, 4 only, thereby restricting their variation to the respective time scale. On the other hand, it allows for the possibility that information from more than just the previous period continues to be of influence in explaining credit spreads. By analyzing the amount of the variation explained in a

outlined possible expectation formations is significant in the medium and long run. Table 2 summarizes the regression results for regressing European investment grade and European high yield credit spreads on the explanatory variables when the data are decomposed, i.e., when the time series are reconstructed to represent behavior present at scales "d1" to "s4."

Determining significant components gives us insights into how long time periods are for processing information. For the short medium term (2–4 months), we find that the default rate is either not significant ("d1" for EMU Corporate all maturities, and "d1," and "d2" for EMU high yield) or even of negative influence. This is a strong indication that the fundamentals are influential for longer time periods only, but do not explain well the variation in investment grade credit spreads for shorter time periods. The credit spread puzzle therefore manifests itself if the data are analyzed on time scales and in that the default rate is not significant in explaining credit spreads at all at some time scales. At the time scales that carry most of the energy, the default rate is significantly positive in explaining the investment grade credit spreads, i.e., for time periods (1.3–2.6 years). For high yield spreads, the default rate is significant only for a longer time horizon (8 months and above). We find that the influence of other explanatory variables changes at the various time scales as well. At the coarsest scale "s4," we find all explanatory variables of significant influence for the credit spreads. However, at scale "d3" (i.e., for a time period of 8– 16 months), the variables that capture the volatilities in the stock and bond markets cannot be viewed as being significant variables. The volatility of the DAX, although of importance in the aggregate data, loses its significance for investment grade credit spreads on several scales. It

) of the decomposed data at various time scales, we can infer which of the above

, 2j+1]. The decomposition of

DAX is best explained by time scales "d1" to "d3."

76 Wavelet Theory and Its Applications

regression (R<sup>2</sup>

At each scale "j" the coefficients are associated with time periods [2j

i.e., we are able to allow for inefficiencies in the credit market as outlined above.

Table 2. Regression results for the European investment grade and high yield credit spreads on explanatory variables using reconstructed time.

continues to be important in explaining the high yield spreads though (with the exception of "d3"), which is another indication for the fact that stock market characteristics are more influential in the high yield bond markets than in the investment grade bond markets. The R2 supports the fact that has to be performed allowing for inefficiencies in the markets. We find that the amount of the variation in credit spreads explained by the identified risk factors is highest for time horizons from 1.3 to 2.6 years and above. The R2 at these time horizons in case of the investment grade bonds is 85–98%. Similar results are achieved to the high yield spreads. For shorter time periods, the amount of variation explained is much lower.

We therefore conclude that if information from the fundamental risk factors is allowed to be of influencing longer time periods (1.3–2.6 years and above), then the variables from Eqs (9) and (10) are significantly linked and the amount of variation explained is high. This means if we allow information from the previous 1.3–2.6 years at scales "d4"and "s4" to be relevant, the proportion of credit spreads explained by risk factors is higher. At the short horizon, technical trading is perceived to be the most important influence in forming expectations; therefore, the insignificance of the default rate to explain credit spreads for shorter time period is in line with previous results and market data.

Author details

Michaela M. Kiermeier

Darmstadt, Germany

References

Address all correspondence to: michaela.kiermeier@h-da.de

341-360. DOI: 10.1016/0022-0531(76)90046-6

of Finance Working Paper Series FIN-03-013. 2003

traders. American Economic Review. 1990;80:63-68

Using Wavelets. New York: New York University; 1996

Reserve Bank of St. Louis, Working Paper 2005-050A. 2005

Journal of Empirical Finance. 2010;17:867-894

Florence: European University Institute; 1998

Econometrics. 2003;7:1-18

Economy. 1973;83:607-636

Econometrics. 2006;130:337-364

Department of Economics and Business Administration, University of Applied Sciences,

Empirical Support for Fundamental, Factor Models Explaining Major Capital Markets Using Wavelets

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[1] Ross S. The arbitrage theory of capital asset pricing. Journal of Economic Theory. 1976;13:

[2] Huang JZ, Kong W. Explaining Credit Spread Changes: Some New Evidence from Option-Adjusted Spreads of Bond Indexes, NYU Stern School of Business, Department

[3] Bekaert G, Engstrom E, Grenadier SR. Stock and bond returns with moody investors.

[4] Cutler DM, Poterba JM, Summers LH. Speculative dynamics and the role of feedback

[5] Ramsey JB, Lampart C. The Decomposition of Economic Relationships by Time Scale

[6] Kim S, Haueck IF. The relationship between financial variables and real economic activity: Evidence from spectral and wavelet analysis. Studies in Nonlinear Dynamics and

[7] Kiermeier MM. Essays on wavelet analysis and the arbitrage pricing theory [thesis].

[8] Raihan S, Wen Y, Zeng B. Wavelet: A New Tool for Business Cycle Analysis. The Federal

[9] Gallegati M, Gallegati M, Ramsey JB, Semmler W. The US wage Phillips curve across frequencies and over time. Oxford Bulletins of Economics and Statistics. 2011;73:489-508

[10] Gencay R, Selçuk R, Whitcher B. An Introduction to Wavelets and Other Filtering

[11] Fama EF, McBeth J. Risk, return, and equilibrium: Empirical tests. Journal of Political

[12] Diebold FX, Li C. Forecasting the term structure of government bond yields. Journal of

Methods in Finance and Economics. Philadelphia: Academic Press; 2009

We conclude that aggregating over time scales "d1" to "s4" results in misleading interpretations of the influence of the various risk factors in explaining credit spreads. Only at time scales that represent medium terms, the default rate is of significant, positive influence. The amount of variation explainable with the fundamental risk factors is highest at that time scales. This supports the fact that fundamental considerations are more important in longer time periods and that inefficiencies in the credit markets are present at shorter time periods.
