The scaling behaviour of credit spread returns in an NIG

framework

Author: Hidde Hovenkamp

Academic Supervisor: Svetlana Borovkova (VU)

Professional Supervisor: Sidney Leever (RiskQuest)

February 2, 2015

Abstract

Under Solvency II the internal modeling of yearly VaR provides significant problems caused by the lack

of data at this frequency. As a result, adequate scaling of results estimated on a lower data frequency is

essential for risk management. This paper advocates the use of the normal inverse Gaussian as distribution

to model credit spread risk. Its attractive scaling properties can best be supported by a scaling factor that

accounts for short-run autocorrelations and long memory in the data. For this purpose, scaling factors

based on ARFIMA models as well as Hurst exponent estimation can be used. Empirical analysis on the

scaling of daily to monthly spread return data has proven the superiority of these scaling factors over the

commonly used square-root-of-time. Backtesting monthly VaR with historical losses supports this result.

Scaling factors based on Hurst exponent estimation slightly outperform those using the autocorrelation

function of ARFIMA models.

Student number: 2451936

Contents

1 Introduction 4

2 Literature review 5

2.1 Credit spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The normal inverse Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Generalized hyperbolic distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Re-parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Time scaling: theoretical considerations 9

3.1 Square-root-of-time rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 General scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Modeling the underlying process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.2 AR(1) plus GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3.3 ARFIMA(p,d,q) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Rescaled range analysis and Hurst exponent estimation . . . . . . . . . . . . . . . . . . . . . 16

3.5 Empirical findings for H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Data 20

4.1 Description of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Autocorrelation and volatility clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Fitting the distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3.1 Subsamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Empirical validation of scaling factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Results 29

5.1 ARFIMA(p,d,q) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Hurst coefficient estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 Scaling the distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6 Backtesting 41

7 Conclusion 44

7.1 Summary of main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2

A Simulation results for AR(1) and AR(1) plus GARCH(1,1) 52

A.1 AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.2 AR(1) plus GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

B Additional figures and tables 58

3

1 Introduction

Internal modeling under Solvency II has become increasingly important for the risk management of insurance

companies. Within this framework we distinguish market risk from underwriting risk (life and non-life), credit

risk, operational risk and liquidity risk. For most large insurance corporations, the market risk module is

most important, under which we find equity risk, interest rate risk and credit spread risk. While equity risk

and interest rate risk have been studied extensively, credit spread risk is a relatively new field of investigation.

Within the Solvency framework, credit spread risk and credit risk are two separate modules. Credit risk,

comprised of default- and settlement risk, is the risk of a change in value due to actual credit losses deviating

from expected credit losses due to the failure to meet contractual obligations (Solvency (2007)). Credit risk

can arise on issuers of securities, debtors and intermediaries to whom the company has exposure. Traditional

credit risk models look at probabilities of default, recovery rates and interaction effects between probabilities

of default and exposure for example. Since credit risk concerns securities not traded on the market, a

separate market risk module is associated with securities that contain credit risk but have a market price:

credit spread risk (or spread risk in short). This is defined as the risk of a change in value due to a deviation

of the actual market price of credit risk from the expected price of credit risk (Solvency (2007)). In general,

all fixed-income assets that are sensitive to changes in credit spreads fall under spread risk.

Credit spreads can be decomposed into several components: credit risk premium, illiquity risk premium

and residual premium De Jong and Driessen (2012). The credit risk premium is further divided into a default

risk, which is based on the current credit rating, and a migration risk, which stems from expected losses due

to possible downgrades. The illiqudity risk premium is demanded by investors for not being able to sell large

amounts of an asset, wile the residual spread encompasses other effects (e.g. double taxation in the US).

Credit spread risk can be thought of as the overarching risk associated with changes in credit spread resulting

from any of these underlying factors.

To model movements in credit spreads for risk management purposes, two methods can be distiguished:

bottom-up and top-down. In the former, the distribution of spread shocks is based on analysis of each

component of the credit spread, using similar techniques as in the credit risk module. The top-down method,

on the other hand, determines the distribution of changes in the full credit spread based on time series analysis

of market indices, representative of asset classes held in the portfolio of a company. While the litature on

bottom-up type models is extensive, not much has been written about modeling the credit spread directly

using a top-down approach.

By examining the time varying dynamics as well as distributional properties of credit spreads, spread

risk can effectively be modeled through a top-down approach. A detailled discussion of the modeling of

credit spreads over time will be given in section 2. While the normal distribution has long been suggested

as too simple for the use in risk management, the Students t-distribution was often put forward as the best

alternative for the distribution of most risk factors. More recently, semi-heavy and heavy tailed distributions

as well as extreme value theory have been employed to account for extreme scenarios such as the financial crisis

4

of 2008. The semi-heavy tailed class of normal inverse Gaussian (NIG) distributions has been proposed as

an alternative to the Students t-distribution. This variance-mean mixture of a normal and inverse Gaussian

(IG) distribution has several characteristics attractive for risk management modeling purposes. The main

being that it can be scaled over time. While most studies using the NIG distribution have focused on equity

risk and interest rate risk, the NIG distribution has not often been suggested in relation to the modeling of

credit spread risk.

Besides finding the appropriate distribution for the risk factors, another difficult issue within the Sol-

vency II framework is the time horizon. To determine the Solvency Capital Requirement (SCR), insurance

companies are required to calculate one-year 99.5% (or 1-in-200) Value-at-Risk (VaR). The problem with

yearly VaR is that there is not enough yearly data points to come even close to a proper dataset from which

a distribution can be estimated. While for equity risk, yearly data may be available for as much as 50 years

(which is still very little), such time series of credit spreads do not exist. Consequently, the modeling of risk

factors has to involve scaling from a higher data frequency to obtain yearly estimates. While the square-

root-of-time provides a theoretically simple solution to this issue, the assumptions on which this rule relies

are hardly ever met in practice.

This paper contributes to the gap in the literature on modeling credit spread risk by dealing with the

use of the NIG distribution and examining empirical solutions to the problem of time scaling. It will use

concepts more familiar in the context of equity risk and interest rate risk and see whether these can be

applied for credit spread risk modeling purposes. The paper is structured as follows. Section 2 evaluates the

current literature on credit spreads and the use of normal inverse Gaussian distribution for risk management

purposes. Section 3 discusses the issue of time scaling from a theoretical perspective. Section 4 elaborates on

the data set and methodology used in this research. Section 5 provides a detailed discussion of the results.

In section 6 the results are backtested against historical losses. In section 7 the main findings of this paper

are summarized and suggestions for further research are given.

2 Literature review

2.1 Credit spreads

Two main theoretical approaches to modeling credit risk can be distinguished in the literature. The structural

approach, first developed in the influential paper by Merton (1974) looks at debt as a contingent claim written

on the assets of the firm. The firm value is modeled through a stochastic process, from which the value of

risky debt is subsequently derived. Another well-known extension of Mertons model is the structural model

proposed by Longstaff and Schwartz (1995). They allow for stochastic interest rates that are described by

the Vasicek model and default occurs when the firms asset value declines to a prespecified level. In case of

default, bondholders recover a constant fraction of the principal and coupon.

However, these models have been criticized because empirical defaults occur too infrequently to be con-

5

sistent with model predictions (Fuss and Rindler (2011)). Credit spreads implied by structural models have

also been shown to be much lower than those actually observed in the financial markets (Huang and Huang

(2012)). Zhou (1997) provides a solution to this problem by modeling the evolution of the firm through a

jump-diffusion process. He proposes this new structural approach, because the standard diffusion approach

does not capture the basic features of credit risk well. He concludes that by using a jump-diffusion approach,

the size of corporate credit spreads in the market can be matched and various shapes of yield curves can be

generated, including downward-sloping, flat and hump-shaped instead of merely upward-shaped curves for

the standard diffusion approach.

The reduced-form approach directly models the default process of risky debt, by explicitly modeling

its underlying factors such as the risk-free rate and the recovery rate in case of default. It makes use of

stochastic processes similar to those used in the modeling of the riskless term structure to model the default

probability. For example, Jacobs and Li (2008) use a two-factor affine model to describe the joint dynamics

of the instantaneous default probability and the volatility of the default probability. Other papers using this

approach include Jarrow and Turnbull (1995) and Jarrow et al. (1997). Reduced form models may also use a

rating-based approach where default is attained through gradual changes in credit rating driven by a Markov

transition matrix (Della Ratta and Urga (2005)).

Another section of the literature uses an empirical approach by examining the underlying factors which

are able to explain the behavior of credit spreads. These works concentrate on the use of econometric models

and inputs such as interest rates, inflation, taxation, liquidity and implied volatility. Especially the relation

between interest rates and credit spreads has been studied extensively. For example, Longstaff and Schwartz

(1995) find evidence of a strong negative relation between changes in credit spreads and interest rates. Neal

et al. (2000) on the other hand, find little evidence of effects of interest rates on callable bonds. Many more

studies can be included in this list, but the ambiguity of the results indicicate a consensus on the relationship

between interest rates and credit spreads is yet to be found. In addition to the behaviour of credit spreads

itself, the behavior and time-varying dynamics of the volatility of credit spreads is studied, using GARCH-like

models (e.g. see Pedrosa and Roll (1998)). Alizadeh and Gabrielsen (2013) extend such techniques to examine

the dynamic behavior of higher moments of credit spreads such as skewness and kurtosis. They, together

with Heston and Nandi (2000)Tahani and Ecole des hautes etudes commerciales (Montreal(2000), claim

incorporating higher moments in the modeling of credit spreads can greatly improve results for pricing and

risk management purposes. By using a Threshold GARCH (TGARCH) model, or GJR-GARCH, Alizadeh

and Gabrielsen (2013) create an assymetric response of volatility to positive and negative shocks. The idea

is that large negative squared returns have a stronger effect on volatility than positive ones.

The traditional literature on credit spread modeling focuses on stationarity versus non-stationarity of

credit spreads. For example, Kiesel et al. (2001) suggest that credit spreads are driven by a combination of a

stationary and random walk component and claim spread risk is in fact the most important risk component

of high quality portfolios. However, Della Ratta and Urga (2005) argue we should also look at the degree

6

of dependence. They investigate whether credit spreads are short- or long-term memory processes using

a fractional Brownian motion framework. The degree of dependence of credit spreads is relevant because

it strongly influences the scaling behavior. Batten et al. (2002) investigate the long-term dependence and

scaling behavior of Australian Eurobonds credit spreads and find a negative long-term dependence. This

implies that positive spread returns will follow negative spread returns, and negative follow positive. As a

consequence, the square root of time rule for scaling volatility is inappropriate, which will be explained in

detail later in this paper.

Reviewing the various types of models and methods used to examine credit spreads is useful for this paper

in two ways. First, to properly understand any distributional properties of credit spreads as well as scaling

behaviour of volatility or higher moments one must be familiar with the data generating process of such

credit spreads. Second, by modeling credit spreads appropriately simulation techniques can be employed to

compute empirical scaling factors. Before we move to elaborate on these and other techniques for calculating

scaling factors, let us first discuss the hypothesized distribution of credit spread returns: the normal inverse

Gaussian (NIG).

2.2 The normal inverse Gaussian distribution

The normal inverse Gaussian (NIG) distribution is defined as the variance-mean mixture of a normal distribu-

tion with the inverse Gaussian (IG) as the mixing distribution. This class of continuous distributions was first

introduced by Barndorff-Nielsen (1977) and has become increasingly popular in finance, particularly for risk

management purposes. Further relevant references of the NIG distribution from a risk analysis perspective

include Barndorff-Nielsen (1997), Barndorff-Nielsen and Prause (2001) and Venter and de Jongh (2002).

The NIG distribution is able to model both symmetric and assymetric distributions, with long tails in

both directions using only four parameters. The tail behavior has been classified as semi-heavy tailed, so it

may not be able to deal with fully heavy tails but generally fits well to financial data. Moreover, another

very attractive property is that the sums of NIG distributed random variables with the same parameters are

again NIG distributed. In other words, it is closed under convolution. This property proves very useful in

the time scaling of risk and is not met by many other distributions, such as the commonly used Students

t-distribution Spadafora et al. (2014). Spadafora et al. (2014) show that only when lies below the critical

value ( = 3.14) is it possible to scale the Students t-distribution.

The NIG distribution can be parametrized in many ways, but the most common specification is the one

with (, , , ) parameters. This parametrization will be referred to as standard throughout this paper. The

parameter space is given by

0 || , < , > 0

The distribution is symmetric around if = 0. In the standard parametrization, the sum of two NIG

distributed variables is NIG distributed with (, , , ) = (, , 1 + 2, 1 + 2). The NIG distribution has

7

the following density

fNIG(x;, , , ) = e22

pi2 + (x )2K1(

2 + (x )2)e(x)

where K1 is the modified Bessel function of third order and index 1. In general, we denote the integral

notation of the modified Bessel function of third order as

K(x) =1

2

0

t1e12x(t+t

1)dt, x > 0

The moment generating function of NIG(, , , ) is given by

M(u;, , , ) = e[(22)

(2(+u)2)]+u

All moments of the distribution thus have explicit expressions. In particular, the mean, variance, skewness

and excess kurtosis are

1 = +

2 =2

3

3 =3

()

4 =3(1 + 42/2)

()

where =

(2 2). For some purposes, instead of the classical skewness and kurtosis values, it is usefulto work with steepness and asymmetry parameters and defined by

= (1 + )1/2, =

The domain of variation for (,) is the NIG shape triangle

0 < < 1, 1 < < 1,

Distributions with = 0 are symmetric and the normal and Cauchy distributions can be found for (,)

near (0, 0) and (0, 1). Barndorff-Nielsen and Prause conclude that in practice values of for daily financial

return series lie between 0.6 and 0.9 Barndorff-Nielsen and Prause (2001). This deviation from zero strongly

indicates non-normality.

8

2.2.1 Generalized hyperbolic distribution

To better understand the characteristics of the NIG distribution it is useful to see its relation with other

distributions within the more general class of generalized hyperbolic (GH) distributions. The GH distribution

is obtain by mixing normal with a generalized inverse Gaussian (GIG) as the mixing distribution and is a five

parameter class of distributions with (, , , , ) as standard parametrization. We will not go into much

detail on the exact density and characteristics of the GH class of distributions, but figure 11 in the appendix

shows how to obtain the NIG distributions as well as its relation to other well-known distributions. Using

the standard parametrization, the NIG distribution is defined as a GH distribution with fixed parameter

= 12 . From the figure it also evident that the normal and Cauchy distributions are indeed special casesof the NIG distribution.

2.2.2 Re-parametrization

As mentioned earlier, several parametrizations are used in the literature. In this paper we will generally

use the standard (, , , , ), but one other is relevant to discuss briefly. It has been shown by Breymann

and Luthi (2013) that switching to parameters (, , , , ) can be very useful for optimization purposes. It

becomes much easier and faster to fit the distribution to empirical data, because this parametrization does

not necessitate additional constraints to eliminate the redundant degree of freedom (Breymann and Luthi

(2013)). Therefore, it is relevant to elaborate on how to switch from parameters (, , , , ) to the standard

(, , , , ).

In the NIG case, when = 12 , then = = . From there, we can obtain the standard parametersusing the following mapping:

=

1

2( + (

)2)

=

2

=2

while stays the same. From the closure under convolution property of the NIG distribution we know that

the sum of S NIG distributed random variables are distributed as NIG(, , S, S) assuming the same and

parameters. For the alternative parametrization this translates into the sum of S NIG distributed variables

being distributed as NIG(S, S,S, S). This will prove very useful for re-scaling the distribution to a

lower data frequency.

3 Time scaling: theoretical considerations

In risk management, a lack of data is often a problem for determining the distribution of risk factors.

Especially within the Solvency II framework, where insurance companies have to compute the one-year

9

99.5% Value at Risk (VaR), data frequency becomes an issue. There is not nearly enough yearly data to

compute such a VaR, since this would mean at least 200 data points have to be available. To overcome this

problem, various solutions have been proposed. Most common are: the use of overlapping data and scaling

results from higher frequency data.

The use of overlapping data is generally seen as an invalid method as it greatly increases the autocorre-

lation in the data and makes the results very difficult to interpret (Harri and Brorsen (1998)). The second

option is to make use of higher frequency data. For example monthly data can used to then scale the results

for yearly VaR. Although this sounds relatively simple, in practice it turns out to be rather complicated. In

the next section, various methods for scaling the data will be discussed in detail. We will start with the most

commonly used square-root-of-time rule and explain why this is fact inappropriate in most cases.

3.1 Square-root-of-time rule

A common rule of thumb in risk management, borrowed from the time scaling of volatility, is the square-

root-of-time rule (SRTR) according to which financial risk is scaled by the length of the time interval. This

is similar to the Black-Scholes option pricing formula where a t-period volatility is given by t. Let us first

briefly explain where the SRTR comes from, before elaborating on the assumptions it is based on and why

it most often does not hold in practice.

If we use the example of the sum of k daily returns, where one return is denoted as Xi, the variance over

this horizon k is defined as

2(k) = V ar(

ki=1

Xi) = Cov(

ki=1

Xi,

kj=1

Xj) =

ki=1

kj=1

ijij

To go from this expression to the SRTR we have to make two assumptions. First, by assuming X is serially

uncorrelated, the sum of all ij will equal k. In other words, ij = 1 for all i 6= j. Second, under theassumption of stationary variance, i.e. V ar(Xk) =

2 for all k, the expression above simplifies to

(k) =

ki=1

kj=1

ijij =2 k =

k

where is the constant one-day volatility and k 1. Hence the time scaling factor under the SRTR isdefined as

S(k) =k (1)

McNeil et al. (2005) provide a more detailled explanation of these concepts and the consequences for value

at risk (VaR) and expected shortfall (ES).

While the SRTR is often used in practice and even advocated by regulators, it leans heavily on the

10

assumption of independent and identically (i.i.d.) distributed returns as well as normality of returns (Wang

et al. (2011)). These assumptions are not met in empirical financial returns and numerous stylized facts are

in conflict with these properties. Various studies have attempted to identify how these different effects bias

the approximation of the SRTR.

First, dependence in asset returns is often present, both in levels and higher moments. As Wang et al.

(2011) illustrate, the SRTR tends to understate the scaling factor and hence the tail-risk when returns follow

a persistent pattern (i.e. momentum is present), while it overstates this risk for returns with mean-reverting

behavior. In similar fashion, volatility clustering is found present in returns of most financial assets. Under

the dynamic setup introduced by Engle (1982) and Bollerslev (1986) it has been demonstrated that the

k-day estimate scaled by the SRTR yield overestimates of the variability of long-horizon volatility. Diebold

et al. (1997) show, using the GARCH(1,1) volatility process, that while temporal aggregation should reduce

volatility fluctuations, scaling by the SRTR amplifies them.

In addition to serial correlation and volatility clustering effects, non-normality of financial returns also

affects scaling with the SRTR. Although allowing for dynamic dependence in the conditional variance partially

contributes to the leptokurtic nature of the distribution, as Wang et al. (2011) mention, these GARCH effects

alone are not enough to explain the excess kurtosis often present in return series. On the one hand this has led

to studies using Students t- or other heavy tailed distributions in their empirical GARCH modeling. On the

other hand, researchers have turned to models that generate disregularities. Merton (1976) first introduced

a jump diffusion model that created discontinuous paths. Yet it was only until the work of Danielsson and

Zigrand (2006) that it became evident how underlying jumps influence the SRTR approximation of longer

horizon tails risks. They showed that the SRTR underestimates the time-aggregrated VaR and this downward

bias increases with the time horizon, caused by the existence of negative jumps. Wang et al. (2011) corectly

question whether this downward-bias would switch direction or just become negligible if the jump process

was not confied to negative price jumps only.

Although it is clear that various underlying effects influence the SRTR and give rise to biases in its

scaling, it is unclear what the overal effect of these influencing factors combined is. It could well be that a

negative bias coming from jumps is offset by a positive bias resulting from momentum in the return series.

Nevertheless, it is clearly the case that the square-root-of-time rule should be used with caution at best.

Therefore, let us now turn to a variety of alternatives for determining the appropriate scaling factor in case

of failure to meet some of the underlying assumptions on which the validity of the SRTR is based.

3.2 General scaling function

Let us start with a more general formula of the scaling of volatility, where the assumption of ij = 1 for all

i 6= j is dropped. Rab and Warnung (2010) show that instead of the square-root-of-time rule an alternativescaling factor for the volatlity can be constructed. This scaling factor corrects for all relevant autocorrelations,

11

making use of the autocorrelation function (acf). This scaling factor is defined as

S(k) =

k + k1i=1

2(k i)(i) (2)

where S is a funtion of k, which is the length of the scaling window and (i) the acf at time i. It is immediately

evident that the SRTR is a special case, where (i) = 0 for all i. Then equation 2 reduces to equation 1.

3.3 Modeling the underlying process

To compute a scaling factor that accounts for autocorrelation in the data, we need the acf for our spread

return data. This means that we have to make certain assumptions about the data generating process that

drive these spread returns. We can specify a model to approximate this process, from which we can extract

the acf and compute S(k) using equation 2.

To use this approach, we have to chose a model specification that accurately describes our daily spread

return data. In the next subsections, we will elaborate on three type of model specifications which we will

investigate further. First, when we only want to account for the autocorrelation in the data we can use a

simple AR(1) model fitted to the spread return data. Second, when also accounting for volatility clustering

an AR(1) plus GARCH(1,1) model can be used. Third, we investigate a class of fractal processes to describe

our spread return data. These models are called ARFIMA(p,d,q) models.

3.3.1 AR(1)

When we only have to deal with autocorrelation, we can use an auto-regressive process or order 1 (AR(1)),

which looks as follows

xt+1 = xt + t+1

with || < 1 for a stationary process x. When there is no volatility clustering, then the innovations N(0, 2). Since the autocorrelation function of an AR(1) process is given by (i) = i, the scaling factor S(k)

becomes

S(k) =

k + k1i=1

2(k i)i (3)

3.3.2 AR(1) plus GARCH(1,1)

When we include volatility clustering in the model, we use a GARCH(1,1) process to model the variance

2t+1 = + 12t + 1

2t

12

The theoretical scaling factor for the AR(1) plus GARCH(1,1) model does not change, as the GARCH process

does not influence the autocorrelation function. However, it is interesting to use Monte Carlo simulation

techniques to compare the results with the theoretical scaling factor. This can be done by simulating daily

observations, generating monthly observations from these (using k = 22) and the comparing the sample

variance for the two. The simulated scaling factor then becomes

S(k) =

m2(k)2d

. (4)

3.3.3 ARFIMA(p,d,q) models

A theoretically more sound way of modeling credit spreads is the class of so-called fractal processes, shown by

for example Della Ratta and Urga (2005). Let us start with the simplest one. The fractional Brownian motion

(fBm) BH(t) is a Gaussian process with zero mean, stationary increments, variance E[BH(t)BH(t)] = t2H

and covariance E[BH(t)BH(s)] =12 (t

2H + s2H |t s|2H).Depending on the value for the parameter H, the fBm has independent increments (H = 12 ), positive

covariance between two increments over non-overlapping time intervals ( 12 < H 1) or negative covariancebetween increments (0 < H < 12 ). In the case of

12 < H < 1 we say the process has long memory.

It is worth noting that the fBm is part of a more general class of processes, called self-similar processes.

A centered stochastic process Xt is said to be statistically self-similar with Hurst coefficient H, if it has the

same distribution as aHXat, for all a. The autocorrelation function of a self-similar process, and hence of

the fBm, is given by

(j) =1

2[(j + 1)2H 2j2H + (j 1)2H ]

The simplest long memory model is fractional white noise, which is defined as

yt = (1 L)dt

where L is the lag operator. E(t) = 0, E(2t ) =

2t , E(ts) = 0 for s 6= t and d = H 12 is the fractional

difference parameter. Let yt I(d). For d = 0, Yt = t and process is serially uncorrelated, while if d > 0the process has long memory and is mean square summable. Yt is stationary for all d 12 .A more general class of processes that contains fractional white noise as a particular case is the Autore-

gressive Fractionally Integrated Moving Average (ARFIMA) model first introduced by Granger and Joyeux

(1980). The ARFIMA(p,d,q) process is defined as

(L)(1 L)d(yt ) = (L)t

13

where (L) and (L) involve autoregressive and moving average coefficients of order p and q respectively and

t is a white noise process. The roots of (L) and (L) lie outside the unit circle. A fractional white noise

process, or fractional Brownian motion, is essentially equivalent to an ARFIMA(0,d,0) process. ARFIMA

processes are covariance stationary for 12 < d < 12 , mean reverting for d < 1 and weakly correlated ford = 0. For d 12 , these process have infinite variance, but it is more common in the literature to imposeinitial value conditions such that yt has changing, but finite, variance.

While the autocorrelation function of the ARFIMA(0,d,0) or fBm was a relatively simple formula, the

acf for an ARFIMA(p,d,q) is much more complex. Deriving the acf is beyond the scope of this paper, so we

will use the results as shown in the book on long memory by Palma (2007). He uses the following relation

between the auto-covariance function and the autocorrelation function

(h) =(h)

(0)

and deduces that

(h) = 2q

i=q

pj=1

(i) j C(d, p+ i h, j)

with

i =

[j

pi=1

(1 ij)m6=j

(j m)]1

and

C(d, h, ) =0(h)

2[2p(h) + (h) 1]

where (h) = F (d + h, 1, 1 d + h, ) and F (a, b, c, x) is the Gaussian hypergeometric function (see Palma(2007)). Though seemingly complex, the use of this autocorrelation function will prove very useful later in

this paper, when we want to compute scaling factors based on an ARFIMA process. We will again make us

of the theoretical scaling formula from equation 2 and plug in the acf as described above.

Let us now elaborate on methods for estimating the value of d. We distinguish two parametric methods:

the Whittle estimator and the exact maximum likelihood (EML) estimator proposed by Sowell (1992). As

Della Ratta and Urga (2005) rightly note, both these procedures are applicable to stationary ARFIMA

models, yet many financial time series lie on the border of being non-stationary. However, since we are

interested in the spread return series - the first difference of the original spread series - this problem will most

likely be less of an issue in our case.

The parametric procedure, first proposed by Whittle (1953), leads to an estimation of d which we will

14

denote as dW . The estimator is based on the periodogram and involves the following function

Q() =

pipi

I()

fX(, )d

where fX(, ) is the known spectral density at frequency , I() the periodogram function and denotes

the vector of unknown parameters, including d as well as the AR and MA coefficients. The Whittle estimator

is the value of that minimizes the function Q(). Reisen et al. (2001) show that dW is strongly consistent,

assymptotically normally distributed and assymptotically efficient.

Sowell (1992) derived the exact maximum likelihood estimator for ARFIMA processes with normally

distributed innovations. However, this approach is computationally demanding because with each iteration

of the likelihood a T-dimensional covariance matrix has to be inverted, where each element is a nonlinear

fucntion of hypergeometric functions. It also requires all roots of the autoregressive polynomial to be distinct

and for the theoretical mean parameters to either be zero or known. The EML estimates of d, herafter

denoted as dE , are assymptotically normally distributed, making it possible to test hypotheses on d.

To get a better understanding of ARFIMA processes, figure 1 shows 1000 simulated observation for three

different ARFIMA specifications. The first is a pure fractional Brownian motion with d = 0.4. The second is

an ARFIMA(1,d,1) process with d = 0.4 and positive autoregressive and moving average parameters, which

shows momentum behaviour. The third model is an ARFIMA(1,d,1) process with d = 0.4 and negative

autoregressive and moving average parameters, which shows mean reverting behaviour.

Figure 1: 1000 simulated observations from ARFIMA models with Gaussian white noise N(0,1). The first series isan ARFIMA(0,d,0) model with d = 0.4. The second series is an ARFIMA(1,d,1) model with d = 0.4, 1 = 0.6 and1 = 0.3. The third series is an ARFIMA(1,d,1) model with d = 0.4, 1 = 0.6 and 1 = 0.3.

A more direct approach to calculating the scaling factor assumes an fBM as the data generating process for

15

credit spreads and uses some of the properties without actually estimating the parameters. From equation 2

we know that the theoretical scaling rule is given by

S(k) =

k + k1i=1

2(k i)(i)

where (i) is the autocorrelation function. Plugging in the autocorrelation function of the fBM we get

S(k) =

k + k1i=1

(k i)[(i+ 1)2H 2i2H + (i 1)2H] (5)with k the lenghth of the scaling horizon, as before. Now all that remains is to estimate the Hurst exponent

H. This can be done directly from estimating the ARFIMA(0,d,0) process on the data and using the relation

H = d+ 0.5. Another group of methods uses different techniques to estimate H directly from the data. We

will elaborate on these methods in the next section.

3.4 Rescaled range analysis and Hurst exponent estimation

Rescaled range analysis was first introduced by Hurst (1951) while studying the statistical properties of the

Nile. He expressed the absolute displacement in terms of rescaled cumulative deviations from the mean and

defined time as the number of data points used. The classical rescaled range statistic is defined as

Qn =1

n

[max

kj=1

(Xj Xn)minkj=1

(Xj Xn)]

(6)

where n is the usual (ML) standard deviation estimator. The first term is the maximum of the partial sums

of the first k deviations of Xj from the mean and is always nonnegative. The second term is the minimum over

this same sequence and is always nonpositive. The difference, the range, is therefore always nonnegative:

Q 0. We refer to the rescaled range statistisc as Qn, but it is also commonly known as (R/S). To avoidconfusion with the scaling factor S we will use the term Qn.

The scaling exponent of Qn = c nH is now referred to as the Hurst exponent and gives us informationon the presence of long-range correlations in time series. If the data is completely independent H will be

12 . By computing the values of the rescaled range Qn for different values of n, we can estimate the Hurst

exponent Couillard and Davison (2005). This is done through a simple ordinary least squares regression:

log(Qn) log(c) + H log(n). Barunik and Kristoufek (2010) show that Qn is biased in small samples.Couillard and Davison (2005) Mandelbrot and Wallis (1968) and Mandelbrot and Taqqu (1979) demonstrate

the superiority of rescaled range analysis to more convential methods of determining long-range dependence

such as the anaysis of autocorrelations or variance ratios. Monte Carlo simulation studies show that the

Qn statistic can still detect long-range dependence in highly non-Gaussian time series with large skewness

16

and kurtosis (Mandelbrot and Wallis (1968)). This property is especially useful for our purposes, since

we are hypothesizing credit spread returns to be NIG distributed which deviates strongly from the normal

distribution.

One problem with rescaled range analysis is the stationarity assumption as explained in Couillard and

Davison (2005). The test statistic assumes that the underlying process remains the same throughout the

process. To test the validity of this assumption and the effect on the estimation of H, Couillard and Davison

(2005) propose to divide the data set in both overlapping and non-overlapping subperiods. The Hurst

exponents of the subsamples can then be compared to the exponent of the entire sample to see if it is

constant through time. In this sense there is a trade-off between the amount of data needed to properly

estimate H and the stationarity assumption.

Another major shortcoming of rescaled range analysis is the sensitivity to short-range correlations (Couil-

lard and Davison (2005)). Although the ratio of logQnlogn converges to12 in the limit, this fraction will deviate

from this value in the short run (Lo (1989)). One way to account for this bias is to use a the modified Qn

statistic introduced by Lo (1989). Lo proposes to use the following modification:

Qn =1

n(q)

[max

kj=1

(Xj Xn)minkj=1

(Xj Xn)]

(7)

where the only difference with the traditional Qn lies in the denominator:

n(q) =

2x + 2 qj=1

j(q)j , j(q) = 1 1q + 1

, q < n

where 2x and j are the usual sample variance and autocovariance estimators of X. If X is subject to

short range dependence, the variance of the partial sum is not simply the sum of the variances of the

individual terms, but also includes autocovariances (Lo (1989)). Therefore, the esimator n(q) also includes

the weighted autocovariances up to lag q, where the weights ensure a positive n2(q). Determining the

appropriate truncation lag q must de done with some consideration of the data. Teverovsky et al. (1999)

made a thorough investigation of the modified Qn statistic and found that as the lag q increased the statistic

had a strong bias towards accepting the null hypothesis of no long-range correlations. In this paper use a

data-driven optimal value of q 1.

In a paper published a few years after Lo (1989), Moody et al. (1996) claim that Los modified rescaled

range statistic Qn is itself biased and introduces other problems, causing distortion of the Hurst exponents.

They propose another variation of the statistic that corrects for mean bias in the range R, but does not suffer

from the short term biases that Los modification introduces. Experiments on simulated random walk, AR(1)

and high-frequency exchange rate data support their claims Moody et al. (1996). Moody and Wu argue for

1q =[( 3N2

13 ( 2

12 )23]Teverovsky et al. (1999)

17

replacing the biased rescaling factor of Lo by an unbiased estimate of the variance, resulting in

Qn =1

n(q)

[max

kj=1

(Xj Xn)minkj=1

(Xj Xn)]

(8)

with

n(q) =

[1 + 2 qj=1

j(q)N jN2

] 1N 1

t0+Nt=t0+1

(Xt Xn)2 + 2N

qj=1

j(q)

t0+Nt=t0+1

(Xt Xn)(Xtj Xn)

where j(q) is the same weighting function as defined by Lo. Xt and Xn are the return process X at time t

and the mean of X respectively (like before).

In addition to these modifications, Annis and Lloyd (1976) developed a modified version of the Qn statistic

that corrects the small smaple bias of the original statistic. Peters (1994) later denoted that the Anis and

LLoyds corrected version is again more difficult to implement for large n, as the correction includes a gamma

function, which becomes computationally intensive. An approximating version, which circumvents the use

of the gamma function, is proposed by Peters, to be used in case of samples larger than approximately 300.

The adjustments in both these version cause the standard deviation to scale at a slower rate than the range

for small values of n. Hence, the rescaled range will scale at a faster rate (H will be greater than 0.5) when

n is small. As such, we can define the Anis-LLoyd-Peters corrected expected Qn statistic as

E(Qn) =(n 12 )

n1r=1

(nr)r

n 12pi(9)

where n is the length of the subperiods. We have set this to 50 following Peters (1994).

So now we have elaborated on types of rescaled range analysis, which can yields us Hursts original Qn,

Los modified Qn, Moody and Wus slightly differently modified Qn and finally Anis-Lloyd-Peters expected

statistic E(Qn). From these statistics we can compute an estimated value for the Hurst exponent H, as has

been explained before. However, there is another method for estimating H, which is called the generalized

Hurst exponent (GHE) approach.

This method was recently re-explored for analysis of financial time series by Di Matteo et al. (2003) and

is based on scaling of the q-th order moment of the increments of the process X(t). The statistic is defined

as

Kq() =

Tt=0 |X(t+ )X(t)|q

(T + 1) (10)

for time series of length T. The statistic scales as Kq() c qH(q). Barunik and Kristoufek explain that thecase of q = 2 is especially relevant for the purpose of long-range dependence detection, as K2() is connected

to the scaling of the autocorrelation function of the increments Barunik and Kristoufek (2010). Therefore,

18

we can estimate H(2) using this approach, which will be comparable to estimates of H using rescaled range

analysis. The case of H(2) is extremely relevant for the purposes of this paper as it is directly related to

the scaling of increments of a process over time. As such the scaling of credit spread changes from a daily

to a monthly range will surely benefit from the results of the GHE approach. For q=1, H(1) characterizes

the absolute deviations of the process Di Matteo et al. (2003). Following Di Matteo et al. (2003)Di Matteo

(2007) we choose = 19.

Barunik and Kristoufek (2010) have conducted an elaborate comparitive study between various approaches

of computing the Hurst exponent as have been explained in the last sections. They conclude that rescaled

range analysis together with generalized Hurst exponent (GHE) approach are most robust to heavy tails in

the underlying process. Di Matteo (2007) claims in his paper that the GHE method is in fact more robust

to outliers than the rescaled range analysis approach. This paper will further investigate these results by

comparing the estimated Hurst exponents in its performance for time scaling credit spread returns. These

five Hurst exponents estimates will be denoted as: HH , HLo, HMW , HALP and HGHE .

3.5 Empirical findings for H

Before we move to our empirical analysis it is important to review empirical results in the literature for

Hurst exponent estimation. This helps understand and interpret the results and provides a framework for

comparison.

Many papers in the literature have studied financial time series through rescaled range analysis or other

methods of estimating the Hurst exponent. Most studies have focused on the scaling of stock returns so

we will briefly discuss these findings first. Domino (2011) finds values between 0.4 and 0.8 for the Warsaw

Stock Exchange. For the major Middle East and North African (MENA) stock markets Rejichi and Aloui

(2012) find values of H > 0.5 indicating long range dependence on all MENA markets investigated. Morales

et al. (2012) look at a dynamically calculated Hurst exponent for U.S. companies on the NYSE hit by the

financial crisis. They also find values around 0.5. In general, most papers find values for H of 12and up when

examining time series of stocks.

Now let us turn to the empirical findings of scaling behaviour of credit spreads. Although much less

has been written about credit spreads than stock returns a few studies are worth mentioning. McCarthy

et al. (2009) find strong evidence of long memory in yield spreads with H ranging between 0.85 and 1. They

look at the spread between AAA and BBB corporate bonds as well as between either and 10-year Treasury

bills. McCarthy et al made use of daily, weekly and monthly data, using two techniques: wavelet theory and

an approach building on the aggregated series as proposed byTaqqu and Teverovsky (1998). For the latter

approach, they find that the strongest evidence of long memory is for the weekly spread between AAA and

BBB, while the lowest estimate for H is found on the spread between BBB and 10-year Treasury bills. Based

on the wavelet method, the results indicate the strongest long memory for the monthly AAA to BBB spread

while the lowest coefficient is again the weekly spread between BBB and 10-year Treasury bills.

19

Batten et al. (2002) examined the volatility scaling of Australian Eurobond spreads by calculating the

scaling factor H based on implied volatilities for several multi-day horizons. Their data included the spread

between AAA Eurobonds and AA Eurobonds with different maturities (2,5,7 and 10 years) as well as the

spread between AA Eurobonds and A Eurobonds with different maturities. For all spread return series tested,

values of H lower than 12were estimated, indicating negative long-term dependence. In general, the estimated

scaling exponent increased for spreads with lower ratings. Batten et al. looked a scaling horizons of 5, 12,

22 and 252 days and found evidence that the Hurst exponent decreased when estimated on a longer time

horizon.

4 Data

4.1 Description of the data

To investigate the scaling behaviour and distributional properties of credit spread changes various time series

will be examined. This paper will focus on European corporate bond spread indices, where the spread is

defined as the option-adjusted spread (OAS) over the German government bond. All time series have been

obtained from Barclays Live. The series for the OAS over the german government bond runs from 18-05-2000

until 30-04-2014. Any empty data points have been removed from the series as these are associated with

days when the stock markets where closed and hence are irrelevant. The average trading days per year in

the data set was found to be 260 which we used to come to a monthly average of 26012 = 21.67 (which will be

rounded to 22).

We have chosen to examine corporate bond indices instead of individual bonds for several reasons. First,

each index incoporats numerous bonds a certain market segment, so the obtained results can be considered as

a more widespread pehonomenon (Martin et al. (2003);Della Ratta and Urga (2005)). If only a small number

of bonds would exhibit such behaviour it would probably not be noticable. A second, more practical reason is

that testing for long-range dependence and distribution fitting requires large samples, which are more readily

available for indices than for single bonds (Martin et al. (2003)). Third, the market for individual corporate

bonds is often illiquid and the consistency of the credit spread component of corporate yield is strongly

affected by liquidity constraints, so using indices overcomes this issue (Della Ratta and Urga (2005)).

All indices are part of the greater Barclays Euro-Aggregrate index which consists of bonds issued in the

euro and must be investment grade rated, fixed-rate securities with at least one year remaining to maturity

2. The mininimum outstanding amount for all bonds in the index is 300 million euro. All indices are

reviewed and rebalanced once a month, on the last calendar day of the month. The spread incidices have

been categorized on the basis of three characteristics: sector, rating and maturity. Three coporate sectors

are distinguished: financials, industrials and utility. In addition, the total corporate sector is considered. For

ratings we have only looked at investment grade and higher, which means we distinguish: AAA, AA, A and

2Index description Barclays Live

20

BBB. The ratings are determined by looking at the three main rating agencies (Moodys, Standard & Poors

and Fitch). At least two out of three ratings must be availabe and the lowest rating is taken. For maturity

the following indices are examined: 1 to 3 years, 3 to 5 years, 5 to 7 years, 7 to 10 years and 10 years and

more.

Table 1 shows summary statistics for all the bond indices used. The first columns of the table show

characteristics of the index and credit spread series (OAS) while the last four columns provides descriptives

of the associated credit spread returns (dOAS). The corporate sector index includes the largest number of

issuers, followed by 10+ years maturity and AAA rated. The biggest average spread is for 10+ years maturity,

followed by industrials and 7-10 years maturity. The table also gives the starting values (at 8-05-2000) and

end values (30-04-2014) of the spread series. If we look at the spread returns, it can be seen that the mean is

practically zero for all indices. In additon, the standard deviations of the returns are relatively comparable

between indices, except for two cases. First, it is worth noting that the 3-5years maturity series shows to be

less volatile than for example 1-3years while this would not be expected. Second, the 10+ years maturity

index has by far the largest standard deviation of all the series, as well as very small and large minimum

and maximum, indicating large volatility. Between the indices categorized by sector, the financials seem to

be most volatile with a significantly larger standard deviation and large minimum and maximum values (in

aboslute terms).

OAS (in basis pts) dOAS (in basis pts)Issuers Mean Start End Mean Std. Dev. Min. Max.

Corp. 1466 0.692 1.015 1.362 0.009 2.566 -33.655 23.195Fin. 623 0.581 1.111 1.496 0.015 3.700 -78.527 65.081Uti. 149 0.771 1.035 1.085 0.008 2.328 -29.379 26.261Indu. 694 0.940 0.903 1.247 -0.001 2.261 -26.237 22.683AAA 939 0.155 0.443 0.581 0.008 2.501 -41.897 25.508AA 796 0.269 0.631 0.680 0.010 2.428 -36.849 21.817A 679 0.274 0.734 0.723 0.013 2.411 -31.608 20.810BBB 583 0.330 0.780 0.709 0.013 2.488 -34.576 19.7371-3yrs 303 0.270 0.812 0.573 0.016 2.347 -23.294 21.5663-5yrs 727 0.150 0.178 0.248 0.001 1.461 -15.304 20.7645-7yrs 622 0.308 0.367 0.614 0.002 3.520 -53.381 40.7767-10yrs 900 0.826 0.848 1.265 0.001 3.846 -86.517 37.67210+yrs 1051 1.372 1.292 2.159 -0.002 6.401 -172.930 59.265

Table 1: Descriptive statistics for daily credit spread series and daily credit spread return series. OAS is option-adjusted-spread and dOAS is the return of option-adjusted-spread (absolute change)

The risk factor for credit spread risk can be modeled in three ways: absolute changes, relative changes or

log-changes. The log-change model is immediately ruled out because it is theoretically possible for spreads

to become negative, as they are measured against the german government bond yield or LIBOR curve. Some

corporate bond yields can temporarily be lower than this yield resulting in a negative spread. Although this

does not happen often there are a few cases present in our times series. The difference between the relative

change model and absolute change model is that in a relative change model the spreads shock will be higher

21

when spreads are high, while in an absolute change model the shocks are independent of the current spread

level. The latter makes capital requirements less cyclical and also avoids any problems that might occur

when spreads are almost zero. Therefore we will evaluate absolute changes in credit spreads. For simplicity

sake, these will henceforth be referred to as spread returns.

Figure 2 shows the time series plots of spread returns for the four different sectors. As can be seen from

the figure, there are clearly very volatile periods and much more quiet periods. Especially the financial crisis

in 2008 is very visible (starts around trading day 2000 in the sample) and volatility remains high until quite

recently. The differences in behavior between the indices of the four different sectors is limited.

Figure 2: Time series of credit spread returns (absolute changes) between May 2000 and April 2014 (in basis points).First panel: EU Corporate (Corp.) Second panel: EU Financials (Fin.) Third panel: EU Utility (Uti.) Fourth panel:EU Industrials (Indu.)

4.2 Autocorrelation and volatility clustering

To understand the scaling behaviour of credit spread returns two of the most important characteristics of

time series to examine are: autocorrelation and volatility clustering. Both of these concepts play a very

important role and shall now be elaborated upon.

22

Let us start with a plot of the sample autocorrelation function (acf) of the spread return series to get a

better understanding for the data. Figure 3 shows these plots for the return series categrorized by sector,

with up to 22 lags (one month). From the figure it can be seen that the all corporate index has strong

positive autocorrelation with values beteen 0.1 and 0.2 for the first 10 lags. This indicates momentum in the

spread return series and suggest some kind of autoregressive process might drive the returns. The utilities

and financials sectors also indicate quite strong positive autocorrelation, although the values ares slightly

lower. The industrials sector has the least autocorrelation, but still gives slightly positive values. For the

other nine spread return series, the sample acf plots can be found in figures 12 and 13 in the appendix.

To compare the relative size for all spread seris figure 4 shows a heatmap of the sample autocorrelation

coefficients. Table 19 in the appendix shows the corresponding values of these sample autocorrelation coeffi-

cients in a t able. We can see that the indices categrorized by sector have relatively high positive coefficients

while those categrorized by rating or maturity are generally lower or even negative. For all spread series

holds that as the lags increase the coefficients go down which is as expected. Interesting to note is that in the

heatmap we can see that lag 18 shows a significantly lower mostly negative coefficeints for almost all spread

series.

Figure 3: Sample autocorrelation function with confidence bounds for credit spread returns up to 22 lags, categorizedby sector. Top left panel: Corp. Top right panel: Fin. Bottom left panel: Uti. Bottom right panel: Indu. Blue linesindicate

To formally test for serial correlation in the spread return series we have conducted the Ljung-Box test, where

we have tested for 11, 22, 44 and ln(N) lags (N = 3468, ln(N) 8). For all 13 spread return series the

23

Figure 4: Heatmap of sample autocorrelation coefficients for all credit spread return series up to 22 lags. The colorbarright of the figure shows the size of the autocorrelation coefficient. Spread return series 1 to 13 are: Corporate,Financials, Utility, Industial, AAA, AA, A, BBB, 1-3yrs, 3-5yrs, 5-7yrs, 7-10yrs and 10+yrs.

test statistic indicates siginificant evidence of serial correlation for all four lag lengths with p < 0.001. So in

line with indications from the ACF plots and the heatmap with sample autocorrelation coefficients discussed

earlier we can conclude that the spread return series are strongly autocorrelated over time.

Table 20 in the appendix shows an example of the autocorrelation matrix for a specific spread return

series up to 11 lags. If there had been no autocorrelation at all the matrix would only consists of ones on

the diagonal. In this case it is obvious that the square root of the sum of the autocorrelation matrix would

be equal to

11 3.32 and the square-root-of-time would hold. In this example, however, there is largepositive autocorrelation which results in the square root of the sum of the autocorrelation matrix equal to

approximately 5.62. Hence the positive autocorrelation is reflected in the scaling factor being larger than

what the square-root-of-time rule would give us.

Last, we use the Ljung-Box test to detect volatility clustering. We evaluate the squared spread returns

and test for 11, 22, 44 and ln(N) lags (N = 3468, ln(N) 8). For all 13 spread return series the teststatistic indicates siginificant evidence of serial correlation for all four lag lengths, with p < 0.001. So it can

be concluded that there is definite evidence of volatility clustering in our spread return data.

24

4.3 Fitting the distribution

To estimate the parameters of the normal inverse Gaussian (NIG) probabililty density function (pdf) we use

maximum likelihood with the BFGS algorithm. Since we are dealing with univariate series this procedure is

not too complex and performs well. Especially since we are fitting the distribution using the parametrization

that is shown in the literature to converge properly as explained in section 2.

Corp. 0.101 (0.009) 0.015 (0.017) -0.100 (0.135) 0.723 (0.049)Fin. 0.051 (0.006) 0.008 (0.017) -0.095 (0.261) 0.737 (0.068)Uti. 0.107 (0.009) 0.011 (0.015) -0.057 (0.126) 0.638 (0.045)Indu. 0.152 (0.013) 0.021 (0.019) -0.106 (0.092) 0.770 (0.043)AAA 0.087 (0.007) 0.007 (0.014) -0.041 (0.154) 0.605 (0.047)AA 0.066 (0.005) 0.005 (0.012) -0.028 (0.196) 0.514 (0.049)A 0.060 (0.005) 0.004 (0.011) -0.023 (0.214) 0.495 (0.050)BBB 0.046 (0.004) 0.005 (0.010) -0.034 (0.279) 0.447 (0.054)1-3yrs 0.046 (0.004) 0.003 (0.009) -0.012 (0.283) 0.412 (0.052)3-5yrs 0.109 (0.005) 0.006 (0.007) -0.015 (0.121) 0.285 (0.028)5-7yrs 0.026 (0.004) 0.001 (0.011) -0.020 (0.561) 0.474 (0.073)7-10yrs 0.034 (0.004) 0.004 (0.014) -0.072 (0.403) 0.609 (0.074)10+yrs 0.038 (0.007) 0.003 (0.032) -0.113 (0.354) 1.378 (0.108)

Table 2: Estimated parameters of NIG distribution for all 13 daily spread return series based on maximum likelihoodprocedure. Standard error estimates based on

diag(H1) are shown in parentheses (H is the Hessian matrix)

Figure 5: Empirical pdf from the data, NIG pdf, normal pdf and Students t pdf for two credit spread returns series:5-7yrs maturity (left panel) and BBB rating (right panel). The NIG, normal and Students t have been fitted to thedata using a maximum likelihood procedure

Figure 5 shows a comparison between three theoretical distributions and the empirical distribution for two

example spread series (5-7yrs maturity and BBB rated). The parameters for the normal distribution are:

1 = 0.013, 1 = 2.411, 2 = 0.002, 2 = 6.401. For Students t they are: 1 = 0.022, 1 = 0.454, 1 =0.962, 2 = 0.104, 2 = 1.424, 2 = 1.260. Clearly the normal distribution does not give an accurate

25

description of reality, when compared to the empirical distribution which was to be expected. Both the fitted

Students t and NIG pdf are very close to the empirical pdf and seemed to indicate a good fit. A similar

picture is found for the other 11 spread return series that are examined in this paper so we can conclude that

the Students t-distribution and NIG distribution are a big improvement over the normal distribution. For

the NIG distribution, table 2 shows the parameter estimates for the daily spread return data. The standard

errors indicated in parentheses indicate that ML generally was able to fit the distribution well for and ,

while and gave more problems. The interpretation for is that daily spread returns are expected not to

be significantly different from zero. Table 21 in the appendix shows the parameter estimates when the NIG is

fitted on the monthly spread return data using ML. Here the standard errors are very high for all parameters

except which suggests the (other) parameters can probably not be trusted. This is likely to be caused by

the small number of monthly data points (Nm = 163). Especially a complicated four parameter distribution

such as the NIG needs more observations to properly be fitted to the data. All in all, these problems provides

more support for the need to investigate a way to construct a distribution for monthly spreads from scaling

a daily distribution.

To provide a more formal investigation of the accuracy of the theoretical distribution we have performed

the Kolmogorov-Smirnov (K-S) test. The K-S test is a nonparametric test of the null hypothesis that

the population cdf of the data is equal to the hypothesized cdf. Table 3 shows the p-values for the three

hypothesized distributions discussed earlier. It is again immediately clear that the normal distribution does

not suffice, with p < 0.001 for all 13 series. If we compare the outcome for the NIG and Students t-

distribution we can draw very similar conclusions. Assuming = 0.05, we find a different result only for the

index of the utility sector, where the fitted Students t-distribution is rejected while the NIG distribution

is not. Although close, the p-values for the NIG distribution are, on average, slightly higher than for the

Students t, indicating a slightly better fit.

Interesting to note is that for all 13 spread series the estimated parameter lies far below the critical

value posed in Spadafora et al. (2014). They show that if lies below the critical value of 3.41, the Students

t-distribution does not scale in time. Since this is clearly the case for our spread return series this provides

additional reason to prefer the NIG distribution over the commonly used Students t, as the NIG distribution

does scale in time.

4.3.1 Subsamples

We fit the distribution on subperiods of the sample to compare the results for the estimated parameters.

We follow Eberlein et al. (2003) in taking 500 datapoints for parameter estimation. This means the total

daily sample, which includes 3468 data points, has been split up into seven periods of which the first six

include 500 data points and the last includes 468. Figure 6 shows the plots of the NIG pdfs fitted on all of

these seven subsamples and compared to the original NIG pdf fitted to the entire sample. As an example

the all corporate index serie is shown. From the figure it can be seen that the pdfs differ substantially when

26

NIG Students t Normal

Corp. 0.221 0.068 0.000**Fin. 0.040* 0.034* 0.000**Uti. 0.185 0.037* 0.000**Indu. 0.461 0.065 0.000**AAA 0.104 0.101 0.000**AA 0.005** 0.020* 0.000**A 0.001** 0.011* 0.000**BBB 0.000** 0.000** 0.000**1-3yrs 0.000** 0.001** 0.000 **3-5yrs 0.078 0.085 0.000**5-7yrs 0.000** 0.009** 0.000**7-10yrs 0.018* 0.026* 0.000**10+yrs 0.045* 0.005** 0.000**

Table 3: p-values for K-S test for NIG, Students t and normal distribution shown for all 13 spread return series.** indicates significance at the 5% level, * indicates significance at the 1% level. The K-S test shows whether thepopulation cdf of the data is equal to the hypothesized cdf of the three fitted distributions

estimated on different parts of the sample. The original pdf, estimated on the entire sample is shown in

black and compares well with those estimated on period 1, 4 and 7. For periods 2 and 3, the kurtosis of the

distribution is much higher, which indicates many more observations were found around the mean, which

in turn is in line with a more quiet period and relatively low daily changes in credit spread. On the other

hand, period 5 and 6 indeed show a much flatter distribution, indicating a much more volatile period with

positive and negative extreme spread changes. This period is associated with the financial crisis of 2008 and

afterwards.

Figure 6: NIG pdf fitted to 7 subperiods of the sample compared to the pdf fitted on the entire sample. Eachsubperiod comprises of 500 observations, so P1 comprises of the first 500 observations, P2 the second 500 observationsetc. Results are shown for spread return series of Corp.

27

4.4 Empirical validation of scaling factor

Ultimately, the aim of this paper is to find an appropriate method for scaling the NIG ditribution, accounting

for non-zero autocorrelation in credit spread returns data. To evaluate the results we are looking at the scaling

of daily spread returns to monthly spread returns, since data is available for both horizons. In section 2 the

closure under convolution property of the NIG distribution is discussed and it is concluded that the sum

of NIG distributed random variables is NIG distributed. In particular, the parameters and remain

unchanged while and sum up. When we apply this property to scale an NIG distributed random variable

over time this means we would expect and to be multiplied by a factor t. All these theoretical properties

of course assume indepence (i.e. no autocorrelation).

To examine the empirical validity of these theoretical concepts and specifically applied to spread returns

data we are examining, a Monte Carlo simulation has been employed. Based on the daily spread return data

of our 13 indices we first fitted the NIG distribution as before. Table 2 shows the estimated parameters.

Then, using these estimated parameters, 10,000 daily spread returns were simulated, from which monthly

series have been constructed (assuming 22 trading days this resulted in 454 months or 38 years of data).

Instead of using the real daily data, we use simulated daily data because these are now independent. The

monthly series computed from the simulated data were used to again fit the NIG distribution. Now we can

compare the parameters estimated on the simulated monthly with the original parameters to evaluate the

scaling factor. As such, we have repeated the above explained exercise 100 times, effectively constructing 100

samples of 10,000 data points from which 100 scaling factors are computed. Table 4 shows the mean and

standard deviation of the computed scaling factors for the four NIG parameters.

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.

Corp. 23.027 4.771 25.402 18.639 1.084 0.300 1.200 0.955Fin. 22.104 3.066 22.509 14.658 1.037 0.215 0.953 0.730Uti. 23.267 4.641 26.956 27.406 1.102 0.270 1.266 1.275Indu. 25.666 16.859 44.095 170.971 1.592 4.285 3.985 25.893AAA 22.678 3.215 21.130 28.054 1.065 0.203 1.058 1.287AA 22.467 3.001 29.109 34.579 1.046 0.233 1.237 1.478A 22.811 2.923 25.509 45.624 1.078 0.222 1.004 1.654BBB 22.485 2.477 25.119 21.985 1.057 0.218 1.255 1.0801-3yrs 21.967 2.468 24.667 50.243 1.042 0.222 0.919 1.5883-5yrs 22.479 2.726 24.060 32.134 1.080 0.234 1.188 1.9955-7yrs 22.391 1.998 23.070 40.659 1.083 0.213 1.459 2.8437-10yrs 22.272 2.530 21.364 12.466 1.045 0.184 0.905 0.88510+yrs 23.138 3.263 24.572 23.395 1.077 0.205 1.026 1.235

Table 4: Simulated scaling factors for NIG parameters using estimated parameters for 13 spread return series asbasis. A total of 100 samples where simulated, each containing 10,000 simulated daily spreads. To compute monthlydata 22 trading days per month were assumed. The simulated scaling factors is then computed by dividing the meanof the parameter for the simulated monthly sample with the mean parameter for the daily sample

On average, the results of the simulation experiment are compliant with the theoretical expectations. The

table shows that and scale with approximately 22 for most of the indices, while the scaling factor

28

for and lies close to 1. However, the industrial index deviates quite strongly from the theoretical value,

especially for and . The standard deviations associated with the scaling factors for this index are extremely

high, indicating untrustworthy results. An interesting result that applies to all indices is that the standard

deviations for the scaling of are much lower than for , although the means are relatively close. One

explanation for this could be that the values for lie very close to zero. Consequently, the possible error

in parameter estimation for monthly data can be blown up easily as the scaling factor is calculated through

division with the daily parameter (which is very small for the case of ). This phenomenon seems to be

reflected in the high standard deviaton for the scaling of . If we compare the results for the scaling of

and , we also see a larger standard deviation for on average. Again this could be the cause of parameters

for being generally smaller, casuing a bigger chance of errors.

The overall results from simulation are quite satisfactory, apart from the results for the industrial index.

Besides the fact that the standard deviation for is much higher than for , it might theoretically make sense

to constrain = 0 in any case, building on the assumption that credit spread changes should be roughly

zero in the long run. Taking both these considerations into account, we believe the empirical scaling of

should be leading in the validation of the various scaling factors computed in the next section. This result is

supported by the standard errors for estimating when fitting the NIG distribution to both daily and mothly

spread returns since these are the relative lowest.

5 Results

To calculate the scaling factors two different approaches will be investigated in this section. First, we will find

an appropriate model for credit spread returns from which we can calculate the scaling factor for scaling daily

to monthly returns. Second, rescaled range analysis and Hurst coefficient estimation will be used to calculate

scaling factors in a more direct manner. Third, these scaling factors will be used to construct monthly NIG

distributions which can then be compared with the real monthly distributions of credit spread returns.

5.1 ARFIMA(p,d,q) models

Initially, we started with two very simple models for credit spread returns: an AR(1) and an AR(1) plus

GARCH(1,1). However, the results immediately showed a bad fit with the data of spread returns and the

scaling factors computed from these models did not work very well. Therefore, the elaborate evaluation

of these models is shown in appendix A. A much better way of modeling credit spread returns is to use

ARFIMA(p,d,q) models which assumes an underlying fractal process. Let us therefore move to these results

directly.

For each of the 13 first differenced spread series, we estimate the long memory parameter d for nine

combinations of ARFIMA(p,d,q) models where p and q are between 0 and 2 and we choose the model which

minimises the Akaike Information Criterion (AIC). In general we implemented the Whittle estimator for

29

computing the parameters estimates as this proved to be computationally faster than the EML estimator of

Sowell. In addition, the EML estimator had more difficulties converging to the global minimum. However, for

some series the Whittle estimator did not converge in which case we used the EML estimator as alternative.

Table 5 shows the AIC values for the nine specifications of ARFIMA models ranging from a fractional

Brownian motion untill an ARFIMA(2,d,2) process. The parameters of the model were estimated using

the Whittle algorithm. The results show that for most spread return series the best model includes both

autoregressive and moving average terms. Interestingly though, for the 7-10 years maturity the best fit

is in fact the fBm. Overall, ARFIMA(1,d,2), ARFIMA(2,d,1) and ARFIMA(2,d,2) perform equally well,

providing the best fit to three series each. The ARFIMA(1,d,1) specifcation also performs well providing

the best fit to two series. Although including more AR and MA components in the model seems to lead

to a better specification, there are also problems associated with a more coplex model as the chance of

non-convergence increases with complexity. While the five simplest models converge for all time series, the

remaining four experience problems. Moreover, the difference between AIC values are extremely small for

most spread return series, indicating that the simpler models also suffice. Consequently, one could argue it

would in fact be better to go for a more parsimonious model with a low risk of convergence problems at a

small cost of accuracy of the fit. If we compare the models with just AR or just MA components, the pure

AR models outperform the MA models, especially when including two lags.

(0,0) (0,1) (0,2) (1,0) (2,0) (1,1) (1,2) (2,1) (2,2)

Corp. 2.824 2.811 3.262 2.812 2.812 2.806 2.807 2.807 2.806Fin. 3.592 3.778 3.982 3.572 3.572 3.567 3.568 3.567 3.567Uti. 2.694 2.696 3.221 2.680 2.680 2.874 2.679 2.678 2.881Indu. 2.586 2.572 3.150 2.576 2.572 2.563 2.563 2.563 2.564AAA 2.843 2.847 3.020 2.841 2.838 2.831 2.829 2.836AA 2.787 3.000 2.930 2.787 2.784 2.778 2.776 2.785A 2.775 3.016 2.888 2.775 2.774 2.768BBB 2.838 3.073 2.947 2.837 2.836 3.094 3.182 2.832 2.8331-3yrs 2.719 2.745 2.835 2.718 2.718 3.226 3.533 2.7183-5yrs 1.760 1.755 1.749 1.756 1.753 1.746 1.746 1.7475-7yrs 3.525 3.526 3.648 3.512 3.509 3.511 - 3.509 3.5087-10yrs 3.696 3.701 3.834 3.696 3.696 3.872 3.814 3.70210+yrs 4.718 4.718 4.737 4.715 4.713 6.359 4.858 4.708

Table 5: AIC values for nine ARFIMA(p,d,q) specifications with p and q between 0 and 2. Column headers representthe p and q values of the ARMA part. When no value is shown, the Whittle estimation algorithm did not converge.Bold indicates the lowest AIC value for a particular serie

For completeness the AIC values for the nine ARFIMA specifications based on the exact maximum likelihood

algorithm of Sowell are shown in table 23 in the appendix. For two series the EML algorithm converged while

the Whittle algorithm did not, but for the remaining seven models the algorithm did not converge either,

which means no parameters estimates could be found for those combinations of series and model specification.

Just like for the Whittle algorithm, all models including just AR or MA lags did converge. The best fit was

generally found for the ARFIMA(2,d,2) and ARFIMA(2,d,1) specifications, but interesting to note is that

30

for three spread series the ARFIMA(0,d,0) in fact proved to be the best fit.

Best Fit Whittle dW HW Best Fit EML dE HE

Corp. ARFIMA(1,d,1) -0.053 0.447 ARFIMA(1,d,1) -0.053 0.447Fin. ARFIMA(2,d,2) -0.076 0.424 ARFIMA(2,d,2) -0.075 0.425Uti. ARFIMA(2,d,1) 0.028 0.528 ARFIMA(2,d,2) -0.015 0.485Indu. ARFIMA(1,d,2) -0.285 0.215 ARFIMA(2,d,1) -0.313 0.187AAA ARFIMA(2,d,1) -0.414 0.086 ARFIMA(2,d,2) 0.053 0.553AA ARFIMA(1,d,2) -0.224 0.276 ARFIMA(0,d,0) 0.025 0.525A ARFIMA(1,d,1) 0.261 0.761 ARFIMA(0,d,0) 0.007 0.507BBB ARFIMA(2,d,1) -0.228 0.272 ARFIMA(2,d,1) -0.222 0.2781-3yrs ARFIMA(2,d,2) -0.151 0.349 ARFIMA(2,d,1) -0.163 0.3373-5yrs ARFIMA(1,d,2) -0.294 0.206 ARFIMA(1,d,2) -0.294 0.2065-7yrs ARFIMA(2,d,2) -0.255 0.245 ARFIMA(2,d,2) -0.255 0.2457-10yrs ARFIMA(0,d,0) 0.078 0.578 ARFIMA(0,d,0) 0.078 0.57810+yrs ARFIMA(2,d,1) -0.325 0.175 ARFIMA(2,d,1) -0.325 0.175

Table 6: Estimates of long memory parameter d and corresponding H for the model that provided the best fitaccording to the Whittle alrgorith (left) and EML algorithm (right) for all spread return series

Table 6 shows the estimated fractional integration coefficient d and the associated Hurst exponent for the

two estimation procedures of the model that provided the best fit. First, we can notice that all values lies in

the desired range of 12 < d < 12 . However, the estimated values for d are almost all negative, especially fordW . The more AR and MA terms included in the model, accounting for the short term effects of the spread

returns, the lower the estimate of d seems to be. In that sense, the short term autocorrelation effects are

taking away from the long memory that might exist in the serie. Although this could be the case theoretically,

this is not what you would expect, since the associated Hurst exponents become much lower than 12 . Since

both algorithms had problems converging for ARFIMA models with several AR and MA lags included, we

could take caution with interpreting the results for such models. Although all the models depicted in the

table did converge, it may well be a local minimum rather than the global minimum, which would provide us

with incorrect results. If we examine the series for AAA rated, AA rated and A rated in specific, we notice

that the two algorithms proposed a very different model specification as best fit. As a result, the estimated

coeffients for d and H are also very different for these series. This strongly indicates untrusthworthiness of

the results.

Since we believe it is important to be sure the estimates produced by the two estimation algorithms are

correct, we propose an alternative approach to the best fit for determining the correct model specification.

We discard all the models for which either of the algorithms did not always converge, since this also indicates

possible incorrect result for those series for which it did converge. Instead, we look at the estimates provided

by the two algorithms and use those for which the results are very similar.

In reality this means we immediately discard all models including both AR and MA lags, since for

all these models some series did not converge. For ARFIMA(0,d,0), ARFIMA(1,d,0), ARFIMA(2,d,0),

ARFIMA(0,d,1), ARFIMA(0,d,2) the estimates for d using both algoritms are shown in table 7. In the

table, d00W indicates the estimated value for d in the ARFIMA(0,d,0) model specification based on the Whit-

31

tle algorithm. The results show that for ARFIMA(0,d,0), ARFIMA(1,d,0), ARFIMA(2,d,0) all estimates of

d are identical for both algorithms. For ARFIMA(0,d,1) roughly half of the series give identical values for d

coming from the two algorithms, but some are substantially different. Taking a closer look, we may notice

that several estimates are exactly 0.5, which is the upper boundary of d. This suggests that the algorithm

might not have converged to the true minimum, but instead stopped at the boundary level in order not to

violate the constraint. For ARFIMA(0,d,2) many more series indicate the same problem, for both estimation

algorithms. As mentioned before, we would rather choose a simpler model, for which we can trust the esti-

mates produced by the two algorithms, instead of a more complex model that might provide incorrect results.

Therefore, we decide to base our scaling factor on the ARFIMA(0,d,0), ARFIMA(1,d,0) and ARFIMA(2,d,0)

model specifications.

d00W d00E d

10W d

10E d

20W d

20E d

01W d

01E d

02W d

02E

Corp. 0.165 0.165 0.253 0.252 0.267 0.268 0.289 0.291 0.318 -0.122Fin. 0.122 0.122 0.234 0.233 0.251 0.253 0.287 0.500 0.289 -0.178Uti. 0.065 0.065 0.162 0.162 0.149 0.150 0.184 0.085 0.167 -0.017Indu. 0.148 0.148 0.225 0.224 0.280 0.281 0.285 0.288 0.377 -0.023AAA 0.053 0.053 0.010 0.010 0.046 0.047 0.007 0.007 0.075 0.500AA 0.026 0.025 -0.002 -0.002 0.033 0.033 -0.245 -0.245 0.500 0.500A 0.007 0.007 0.005 0.005 0.029 0.029 0.500 0.500 0.500 0.500BBB -0.004 -0.004 -0.036 -0.037 -0.013 -0.013 -0.038 0.500 -0.008 0.5001-3yrs -0.033 -0.033 -0.069 -0.069 -0.043 -0.042 -0.070 -0.146 -0.042 0.5003-5yrs -0.084 -0.084 -0.038 -0.038 0.001 0.001 -0.014 -0.014 0.102 0.1025-7yrs 0.063 0.063 -0.053 -0.053 -0.012 -0.012 0.073 0.073 0.500 0.5007-10yrs 0.078 0.078 0.094 0.094 0.111 0.111 0.130 0.130 -0.132 -0.13210+yrs 0.075 0.075 0.023 0.023 0.055 0.055 0.022 0.082 0.063 0.174

Table 7: Estimated values for long memory parameter d, based on Whittle and EML algorithms, for theARFIMA(0,d,0), ARFIMA(1,d,0), ARFIMA(2,d,0), ARFIMA(0,d,1), ARFIMA(0,d,2) specifications. d00W means theestimated value for d in the ARFIMA(0,d,0) model based on the Whittle algorithm

Now that we have established which results to trust, we can interpret the outcomes for our three models.

Table 7 shows us the estimates for d, from which we can see the value lie between 0 and 12 for most series.

This indicates long memory, even when short-term autoregressive effects are accounted for. Interestingly, the

estimated parameter for d increases - for several series - when one or two AR lags are included.

In table 8 we find the estimated parameters and significance for the three ARFIMA models we are evalu-

ating. The table shows the estimates based on the Whittle algorithm, as this proved to be computationally

faster, but results can be expected to be almost identical for the EML estimator, as we already saw in table

7. We note that practically all parameters are found significant at the 1% level, except for some model spec-

ifications for the AA, A and BBB rated series. In general, the ARFIMA(2,d,0) specification find significant

parameters estimates for all series, which indicates this model provides a good fit. The AR coefficents are

generally negative, when associated with a positive estimate for d. This could mean that the two terms are

interacting and possibly affecting the estimates for d and subsequently for H. For BBB rated and 1-3 years

maturity all three models provide negative estimates of d, indicating a short memory process. For 3-5 years

32

d00 H00 d10 H10 101 d20 H20 201

202

Corp. 0.165* 0.665 0.253* 0.753 -0.164* 0.268* 0.768 -0.181* -0.022*Fin. 0.122* 0.622 0.234* 0.734 -0.207* 0.253* 0.753 -0.228* -0.027*Uti. 0.065* 0.565 0.162* 0.662 -0.178* 0.150* 0.650 -0.164* 0.018*Indu. 0.148* 0.648 0.225* 0.725 -0.148* 0.281* 0.781 -0.212* -0.087*AAA 0.058* 0.553 0.010** 0.510 0.075* 0.047* 0.547 0.042* -0.064*AA 0.026* 0.526 -0.002 0.498 0.051* 0.033* 0.533 0.019* -0.064*A 0.007** 0.507 0.005 0.505 0.004 0.029* 0.529 -0.019* -0.043*BBB -0.004 0.496 -0.036* 0.464 0.057* -0.013** 0.487 0.035* -0.040*1-3yrs -0.033* 0.467 -0.069* 0.431 0.061* -0.042* 0.458 0.036* -0.041*3-5yrs -0.084* 0.416 -0.038* 0.462 -0.091* 0.001 0.501 -0.136* -0.071*5-7yrs 0.063* 0.563 -0.053* 0.447 0.186* -0.012* 0.488 0.155* -0.068*7-10yrs 0.078* 0.578 0.094* 0.594 -0.029* 0.111* 0.611 -0.046* -0.027*10+yrs 0.075* 0.575 0.023* 0.523 0.089* 0.055* 0.555 0.061* -0.053*

Table 8: Estimated parameters for ARFIMA(0,d,0), ARFIMA(1,d,0) and ARFIMA(2,d,0) based on the Whittlealgorithm. d is the fractional integration parameter, H the associated Hurst exponent (H = d+ 1

2) and i is the AR

coefficient for lag i. * indicates significance at the 1% level and ** indicates significance at the 5% level

maturity, 5-7 years maturity and AA rated index series the results are ambiguous between the three models.

For all other series the three ARFIMA specifications provide positive estimates of d, indicating long memory.

The easies