2. Factor models in finance

Factor models have always been of great interest to explain price movements on all major capital markets. If risk factors can be identified that are significantly evaluated by the market, that information is valuable for the purpose of general management, determining fair values of firms, asset management, finance, and controlling.

#### 2.1. Stock markets

sensitivities toward these risk factors. Similarities between equity and corporate bond markets' risks have long been recognized and risk factors similar to those applied in stock markets were included in the analysis of bond markets, and corporate bond spreads, for example, see [2]. Other empirical analyses present models for the simultaneous pricing of stock and bond returns [3]. Generally speaking, it has long been recognized that capital markets have similar characteristics [4]. Cutler et al. formulated four important characteristics of data concerning returns in the stock, bond, foreign exchange, and other capital markets. Using monthly return data, there appears to be a positive first-order auto-correlation from 1 month to the next. This does, however, change if the time horizon is medium or even long term. In those cases, the auto-correlation becomes negative. Finally, fundamental factors explain capital market movements significantly in the medium and/or long term which can be explained by allowing for capital markets' inefficiencies, for example, they postulate that the positive 1 month autocorrelation of data could be cause investors who only learn about relevant risk factors with a time lag. In addition to that also traders acting on the basis of technical analyses can cause a positive auto-correlation in the very short run. The negative medium, or long term, auto-correlation is then a direct result from misperceptions that are corrected on those time scales. In addition to those explanations, we consider that market participants have different objectives and therefore also different time horizons for their investments. Arbitrageurs seek to exploit mispricing in nanoseconds. Day-traders want to use knowledge derived from technical analysis on a daily or weekly basis. Although asset and wealth managers can represent investors with all sorts of investment horizons their performance is evaluated at least every month. To summarize, it is highly unlikely that the data generating process is the same for all investment horizons which is the reason why we apply wavelet analysis to allow for discrepancies at different time

We apply wavelet analysis to shed light on the applicability of factor models for stock, bond, and corporate bond markets. For this purpose, we shortly summarize our respective findings for stock and bond markets. We then present a detailed, exemplary, new analysis for European corporate bond markets and present general ideas why the use of wavelet analysis improves

The wavelet decomposition we apply allows us to specifically distinguish short, medium, and long run periods and at the same time it is possible to investigate if information from past continues to be of importance for the following time period. There is little information about the frequency content of data if no frequency analysis is performed. The frequency analysis, however, is not able to maintain information about the time location of events. In our empirical analysis of these models, explanatory variables are selected according to general considerations which fundamental variable influence the capital markets and proceed by assuming that the identified k factors contain the important information, so that we assume an approximate factor structure to hold. We investigate if averaging over various time periods veils the fact that the risk factors are of importance in explaining capital markets' asset returns for certain time scales only, i.e., we investigate if risk factors are especially powerful in explaining asset returns at certain time horizons. For that purpose, we decompose asset returns and risk factors into their time-scale components using the maximal overlap discrete wavelet transform

horizons.

68 Wavelet Theory and Its Applications

on the applicability of factor models in practice.

One of the most important and general approaches to explain price movements on stock markets is the arbitrage pricing theory (APT) developed by Ross [1]. The advantage of the APT is its generality. Various factor models can be derived and require different estimation and testing techniques. A detailed overview of the various possibilities for factor models is given in [7]. The factor models can be distinguished according to the origin of the factors. Statistical factors can be derived from applying factor analysis. Factors can also be determined in advance—derived from theoretical considerations—and observable data of macro-economic variables can be investigated for being risk factors. Since the purpose is to identify risk factors and not to derive fair prices for financial derivatives, the relationship between asset prices and risk factors is restricted to be approximately linear [7].

Ross develops his theory in the context of neo-classical assumptions concerning capital markets without frictions. He assumes that investors differ in their opinion of the exact distribution of the risk factors, however they all agree on a linear k-factor structure. The main assumption is the following: the return at the end of the period is determined by the return that was expected at the beginning of the investment period (μ<sup>i</sup> ) but also by the returns of the common risk factors (λ~k). The importance of the risk factors for an asset i depends on how sensitive the asset is with regards to the k risk factors (bik). Those sensitivities are called factor loadings. Last but not least there is a white noise error variable (~Ei). The k factors are common factors, i.e., every asset reacts to the development of these factors.

$$
\tilde{r}\_i = \mu\_i + b\_{i1}\tilde{\lambda}\_1 + \dots + b\_{ik}\tilde{\lambda}\_k + \tilde{\epsilon}\_i \qquad \forall i = 1, \dots, n \tag{1}
$$

with T = numbers of observations, λ<sup>k</sup> = arithmetic mean of λbkt, and s(λk) = standard deviation

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

http://dx.doi.org/10.5772/intechopen.74725

71

Wavelet analysis is then applied to decompose the risk factors and asset returns. The test for significantly evaluated risk factors is not only performed on an aggregate level but also at different time scales that allow information to be of relevance for certain time periods only. Furthermore, we also apply wavelet to distinguish expected and unexpected components of the risk factors. This approach results in the identification of risk factors that remain significant over longer time periods, the problem of parameter constancy is therefore mitigated as well.

This approach reduces the variance of the estimated means of the risk premiums. Furthermore, it shows that only certain scale information of the risk factors remains important over time. We find that this approach improves on the findings which fundamental factors are significant in explaining stock market returns. For a detailed derivation of the estimation equations and the results in which fundamental factors are significantly evaluated in the stock market, see [7].

The models to explain the term structure of interest rates have been of interest to researchers for a long time. The models differ in the purpose they are built for. In our analysis, we assume that the data generating process for term structure of interest rates can be expressed as an approximate factor model as in the previous section. Those types of models are especially meaningful if the task at hand is to forecast future term structures of interest rates. The models that generate good forecasts and are equally satisfying from a theoretical, arbitrage-free viewpoint have been developed, for example, see [12–14]. The risk factors are found to represent information with regards to the level, slope, and curvature of the term structure of interest rates. We find that in this market too, for the same reasons as before, an analysis on an aggregate level can be misleading so that we perform our analysis on a scale-by-scale basis. We then apply the procedure of Fama/MacBeth to test for significance of risk factors. The Nelson-Siegel model approximates the actual yield curve observed in the market on any

specific date t for zero rates y with maturity τ through the following Eq. (5):

<sup>1</sup> � <sup>e</sup>�γτ γτ � � <sup>þ</sup> <sup>β</sup>2<sup>t</sup>

The respective βi's can be viewed as dynamic factors that represent short-, medium-, and longterm behavior [12]. The factors level (β0), slope (β1), curvature (β2), and γ the mean reversion rate are then identified as risk factors. The models parameters are then estimated by assuming

The dynamic generalized Nelson-Siegel [14] embeds the Nelson-Siegel approach in an arbitrage-free setting. In order to ensure the absence of arbitrage, the number of risk factors

<sup>1</sup> � <sup>e</sup>�γτ γτ � <sup>e</sup>

�γτ � � (5)

of the monthly estimates λbkt.

2.2. Term structure of interest rates

yt

with β0<sup>t</sup>

, β1<sup>t</sup> , β2<sup>t</sup>

has to be increased to five.

ð Þ¼ τ β0<sup>t</sup> þ β1<sup>t</sup>

, and γ as model parameters [15].

an autoregressive, dynamic data generating process for the factors.

with r~<sup>i</sup> = realization of the random variable asset i's asset return at the end of the investment period; μ<sup>i</sup> = expectation of asset i's return at the beginning of the investment period; bik = factor loading of asset i's return in relation to the risk factor k's realized end-of-period return; λ~<sup>k</sup> = realization of the random variable risk factor k's end of period return; and ~E<sup>i</sup> = realization of the random variable asset i's idiosyncratic risk.

In matrix notation this becomes Eq. (2):

$$
\begin{array}{ccccc}
\tilde{r} & = & \mu & & \text{ $B$ } & \bar{\lambda} & \text{ $\bar{\varepsilon}$ } \\
(n\ge 1) & = & (n\ge 1) & (n\ge 1) & (\text{k}\ge 1) & (n\ge 1)
\end{array}
\tag{2}
$$

In this economy, systematic risk is represented through unexpected changes of common risk factor returns. Ross assumes that idiosyncratic risk is diversifiable and that there are no arbitrage opportunities. It is then possible to derive a relationship between asset i's expected return and the factor loadings multiplied by the risk premiums of the k risk factors (λ1, …, λk). The exact APT equation is given by Eq. (3).

$$
\lambda\_i \mu\_i = \lambda\_0 + b\_{i1}\lambda\_1 + \dots + b\_{ik}\lambda\_k \qquad \forall i = 1, \dots, n \tag{3}
$$

This is the APT equation which we use in the empirical analysis to identify statistically significant risk factors. Without idiosyncratic risk, Eq. (3) is an immediate result arising from the absence of arbitrage opportunities, because a riskless portfolio is then simply a combination of assets such that the portfolio is insensitive with regards to the risk of the risk factors and therefore orthogonal to the column space of the B-matrix.

The factor models based on the APT can be summarized by four different model types according to the different ways to choose risk factors. They can be macro-economic, fundamental, statistical or non-linear. Once the factors are determined the asset returns sensitivities toward them must be estimated. In the second step, the estimated sensitivities are incorporated in a cross-section regression and the risk premiums are estimated.

After some transformation, Fama/MacBeth derived an OLS-estimator for risk premiums at every point in time λb<sup>t</sup> ¼ Bb<sup>k</sup> ' Bbk � ��<sup>1</sup> Bb' <sup>k</sup>~rt for all t = 1, …, T in a cross-section regression [11]. This results in a time series of estimated risk premiums λb<sup>t</sup> to which they apply a test statistic that is t-distributed and that allows to test for significantly evaluated risk factors, see Eq. (4).

$$t\left(\overline{\lambda}\_{k}\right) = \frac{\overline{\lambda}\_{k} \* T^{\ddagger}}{s\left(\widehat{\lambda}\_{k}\right)}\tag{4}$$

with T = numbers of observations, λ<sup>k</sup> = arithmetic mean of λbkt, and s(λk) = standard deviation of the monthly estimates λbkt.

Wavelet analysis is then applied to decompose the risk factors and asset returns. The test for significantly evaluated risk factors is not only performed on an aggregate level but also at different time scales that allow information to be of relevance for certain time periods only. Furthermore, we also apply wavelet to distinguish expected and unexpected components of the risk factors. This approach results in the identification of risk factors that remain significant over longer time periods, the problem of parameter constancy is therefore mitigated as well.

This approach reduces the variance of the estimated means of the risk premiums. Furthermore, it shows that only certain scale information of the risk factors remains important over time. We find that this approach improves on the findings which fundamental factors are significant in explaining stock market returns. For a detailed derivation of the estimation equations and the results in which fundamental factors are significantly evaluated in the stock market, see [7].

#### 2.2. Term structure of interest rates

is with regards to the k risk factors (bik). Those sensitivities are called factor loadings. Last but not least there is a white noise error variable (~Ei). The k factors are common factors, i.e., every

with r~<sup>i</sup> = realization of the random variable asset i's asset return at the end of the investment period; μ<sup>i</sup> = expectation of asset i's return at the beginning of the investment period; bik = factor loading of asset i's return in relation to the risk factor k's realized end-of-period return; λ~<sup>k</sup> = realization of the random variable risk factor k's end of period return; and ~E<sup>i</sup> = realization of the

B

ð Þ nxk ∗ λ~

In this economy, systematic risk is represented through unexpected changes of common risk factor returns. Ross assumes that idiosyncratic risk is diversifiable and that there are no arbitrage opportunities. It is then possible to derive a relationship between asset i's expected return and the factor loadings multiplied by the risk premiums of the k risk factors (λ1, …, λk).

This is the APT equation which we use in the empirical analysis to identify statistically significant risk factors. Without idiosyncratic risk, Eq. (3) is an immediate result arising from the absence of arbitrage opportunities, because a riskless portfolio is then simply a combination of assets such that the portfolio is insensitive with regards to the risk of the risk factors and

The factor models based on the APT can be summarized by four different model types according to the different ways to choose risk factors. They can be macro-economic, fundamental, statistical or non-linear. Once the factors are determined the asset returns sensitivities toward them must be estimated. In the second step, the estimated sensitivities are incorporated

After some transformation, Fama/MacBeth derived an OLS-estimator for risk premiums at

results in a time series of estimated risk premiums λb<sup>t</sup> to which they apply a test statistic that is

� � <sup>¼</sup> <sup>λ</sup>k∗<sup>T</sup>

s λb k 1 2

t-distributed and that allows to test for significantly evaluated risk factors, see Eq. (4).

t λ<sup>k</sup>

ð Þ kx1

þ

μ<sup>i</sup> ¼ λ<sup>0</sup> þ bi1λ<sup>1</sup> þ … þ bikλ<sup>k</sup> ∀i ¼ 1, …, n (3)

<sup>k</sup>~rt for all t = 1, …, T in a cross-section regression [11]. This

� � (4)

~ε

(2)

ð Þ nx1

<sup>r</sup>~<sup>i</sup> <sup>¼</sup> <sup>μ</sup><sup>i</sup> <sup>þ</sup> bi1λ~<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> bikλ~<sup>k</sup> <sup>þ</sup> <sup>~</sup>E<sup>i</sup> <sup>∀</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup>, …, n (1)

asset reacts to the development of these factors.

70 Wavelet Theory and Its Applications

random variable asset i's idiosyncratic risk.

The exact APT equation is given by Eq. (3).

every point in time λb<sup>t</sup> ¼ Bb<sup>k</sup>

~r ð Þ nx<sup>1</sup> <sup>¼</sup> <sup>μ</sup>

therefore orthogonal to the column space of the B-matrix.

' Bbk � ��<sup>1</sup>

in a cross-section regression and the risk premiums are estimated.

Bb'

ð Þ nx1 þ

In matrix notation this becomes Eq. (2):

The models to explain the term structure of interest rates have been of interest to researchers for a long time. The models differ in the purpose they are built for. In our analysis, we assume that the data generating process for term structure of interest rates can be expressed as an approximate factor model as in the previous section. Those types of models are especially meaningful if the task at hand is to forecast future term structures of interest rates. The models that generate good forecasts and are equally satisfying from a theoretical, arbitrage-free viewpoint have been developed, for example, see [12–14]. The risk factors are found to represent information with regards to the level, slope, and curvature of the term structure of interest rates. We find that in this market too, for the same reasons as before, an analysis on an aggregate level can be misleading so that we perform our analysis on a scale-by-scale basis. We then apply the procedure of Fama/MacBeth to test for significance of risk factors. The Nelson-Siegel model approximates the actual yield curve observed in the market on any specific date t for zero rates y with maturity τ through the following Eq. (5):

$$y\_t(\tau) = \beta\_{0t} + \beta\_{1t} \left(\frac{1 - e^{-\gamma \tau}}{\gamma \tau}\right) + \beta\_{2t} \left(\frac{1 - e^{-\gamma \tau}}{\gamma \tau} - e^{-\gamma \tau}\right) \tag{5}$$

with β0<sup>t</sup> , β1<sup>t</sup> , β2<sup>t</sup> , and γ as model parameters [15].

The respective βi's can be viewed as dynamic factors that represent short-, medium-, and longterm behavior [12]. The factors level (β0), slope (β1), curvature (β2), and γ the mean reversion rate are then identified as risk factors. The models parameters are then estimated by assuming an autoregressive, dynamic data generating process for the factors.

The dynamic generalized Nelson-Siegel [14] embeds the Nelson-Siegel approach in an arbitrage-free setting. In order to ensure the absence of arbitrage, the number of risk factors has to be increased to five.

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

Risk factors represent risks arising from the possibility to default, term structure of interest rates, equity markets, liquidity from mutual funds, and business cycle. Huang and Kong find that for B (BB) rated corporate bonds approx. 68% (61%) of the variation in credit spreads can be explained by respective risk factors [2]. For investment grade bonds however they find that the proportion explained is much lower. Inefficiencies can lead to a higher proportion being explained by the models, for example see [19, 21]. Again we want to analyze the data at different time horizons and simultaneously allow for inefficiencies such as delayed learning about relevant information or other forms of feedback, or technical trading and account for

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

http://dx.doi.org/10.5772/intechopen.74725

73

We decompose the data with wavelet analysis. We then test for significantly evaluated risk factors on a scale-by-scale basis, we find that only four factors can be viewed as significantly

In the following section, we describe the respective analysis in detail for the European corpo-

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

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

SJðÞ¼ t ϕJ, <sup>k</sup> at level J (7)

dj, <sup>k</sup>ψj, <sup>k</sup>∀j ¼ 1, …, J (8)

<sup>2</sup><sup>j</sup> at a certain level "j" or "J" are given by Eqs. (7) and (8).

n 2j

k¼1

DjðÞ¼ <sup>t</sup> <sup>X</sup>

different investment horizons of market participants.

number of observations of the time series of interest.

evaluated by the market [16].

3. Estimation techniques

rate bond market.

axis k = 1, …, <sup>n</sup>

#### 2.3. Corporate bonds

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 assumption of Ross's factor structure would become necessary.

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

$$\mathbf{c}\mathbf{s}\_t = \mathbf{a} \, + \mathbf{b} \, (\mathbf{x}\_t) + \mathbf{u}\_t \tag{6}$$

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

Risk factors represent risks arising from the possibility to default, term structure of interest rates, equity markets, liquidity from mutual funds, and business cycle. Huang and Kong find that for B (BB) rated corporate bonds approx. 68% (61%) of the variation in credit spreads can be explained by respective risk factors [2]. For investment grade bonds however they find that the proportion explained is much lower. Inefficiencies can lead to a higher proportion being explained by the models, for example see [19, 21]. Again we want to analyze the data at different time horizons and simultaneously allow for inefficiencies such as delayed learning about relevant information or other forms of feedback, or technical trading and account for different investment horizons of market participants.

We decompose the data with wavelet analysis. We then test for significantly evaluated risk factors on a scale-by-scale basis, we find that only four factors can be viewed as significantly evaluated by the market [16].

In the following section, we describe the respective analysis in detail for the European corporate bond market.
