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Stability of correlation between credit Stability of correlation between credit Stability of correlation between credit Stability of correlation between credit

and market risk over different holding and market risk over different holding and market risk over different holding and market risk over different holding

periods periods periods periods

Frederik Deblauwe and Ha Le

University of Antwerp Management school 't Brantijser Sint-Jacobsmarkt 9-13 2000 Antwerpen

Under supervision of Prof. Jan Annaert and Prof. Marc De Ceuster

Abstract

This paper uses a pair-wise test and the Jennrich test to determine the ex-

ante optimal stability over different holding periods and using a different

frequency of data. We find that, using a pair-wise test, generally the monthly

calculated log returns are more stable than the weekly or daily. Further, the

test pointed out that holding periods over one month are more stable than

over 2 or 4 months. Jennrich test confirms the non-homogeneity of

correlation matrices for all holding periods and all data frequency.

Nevertheless, using Bootstrap and GARCH(1,1) to generate new data sets

did weaken the confirmation of Jennrich test. Bootstrap monthly data are

likely to provide more stable correlation matrices over longer holding

periods.

Keywords: credit risk, market risk, correlations, stability, Jennrich test,

pair-wise test

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1. Introduction

Academics have been writing tremendous amounts of papers on market or credit risk.

Straight, market risk comprises many macro-economic and other risks which could

affect prices of financial instruments, usually categorised in three types: interest rate

risk, currency risk, and other price risk (IFRS, IG32). Credit risk, on the other hand,

consists of default risk, credit migration risk, exposure uncertainty risk and recovery

risk. Many models have been thought out to reliably control market and credit risk

individually. While there is a consensus that both types of risk should not be

considered separately, literature devoted to the combination of market and credit risks

is scarce. (Kafetzaki, 2001)

Correlations are regularly used for portfolio optimisation or diversification (asset

allocation and risk measurement), and hedges (hedge ratio). Unless if the correlation

between market and credit risks equals one, banks should not simply add up the two

types of risks from their portfolio to measure the global risk. We show that the

correlation usually fluctuates around zero, hence a simple sum methodology proves

insufficient.

In literature, correlations between domestic and international stocks have frequently

been studied. Those correlations are usually lower than inter-domestic correlations,

making it more interesting to diversify. Examples of this type of studies can be found

in Grubel (1968), Levy and Sarnat (1970) and Solnik (1974).

The general aim of this paper is to check the stability of correlation between credit

and market risk over different holding periods. Seeing that the holding period can

have a significant influence on the correlation, we would like to specify which would

be the most stable holding period. This would facilitate the selection of a quick ex

ante optimal investment strategy without the need of using more advanced models.

For the same purpose, we will investigate whether high or low frequency data

preferably should be used to calculate returns.

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In the literature, various models are presented to determine the stability of correlations

or covariances. Generally, two approaches can be considered: the asset level and the

portfolio level. The former is examined through a pair-wise test on the individual

correlations, the latter uses a matrix test which investigates the portfolio on the whole.

We decided to look at both levels, and for the asset level we generated correlation

statistics which we evaluated on hand of two bootstrap methods. The large number of

indices obliged us to make a selection of correlation, so as to keep it surveyable. For

the portfolio level, we used the Jennrich test which is considered the most appropriate

to investigate correlation stability. The large quantity of indices formed no hindrance

in this case.

From the pair-wise test, it became apparent that the computation of monthly log

returns are preferable over daily or weekly log returns to maintain stability. Further,

the 1 month holding period seemed to be more stable than 2 or 4 month holding

periods.

Jennrich test with original data entirely rejected the null hypothesis of homogeneity of

correlation matrices over all different horizons for monthly, weekly and daily data.

However, when generating new data with Bootstrap and GARCH(1,1) model, the

results of Jennrich changed, showing consistence with the pair-wise test for the data

frequency aspect. However, the opposite was noticed for the horizon holding aspect:

shorter holding period resulted in less stable correlation matrices.

The paper consists of six parts. After the introduction, we will give a literature

overview in section 2 of the most used stability tests. Section 3 will discuss the data

and spread construction, look at the descriptive statistics of the data, and give the

entire correlation matrix of the different spreads. Section 4 conducts the pair-wise test

on asset level, and evaluates descriptive statistics of the correlations through a simple

bootstrap method and a GARCH(1,1) bootstrap. Section 5 reports on the Jennrich test,

providing conclusions on the stability of the correlation matrix on the whole.

Bootstrap method is also included in this section to check the robustness of the results.

Lastly, we will make the main conclusions in section 6.

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2. Literature review

In the literature, many statistical approaches exist in order to assess the inter-temporal

stability of correlation. However, Philippatos (1983) remarks that many studies

committed statistical inaccuracies because of the use of questionable methodologies,

dissimilar sample periods, and variable differencing intervals. This makes it

sometimes difficult to compare results of various papers.

Generally, two main approaches exist to determine the stability of the correlations

between different variables over time. The first category includes the pair-wise tests.

This is a fairly straightforward test which looks at the stability of the correlations

separately over time. An example of a pair-wise test is the normal distribution test.

Problems using this test are encountered when you have to make conclusions based on

higher dimension matrix, or correlations between many variables.

For the study of correlation between the various credit and market risk variables, we

thus looked into a second category with models which were able to consider the

matrix on a whole. Possible methods were the principal component analysis, cluster

analysis, the box test, and the Jennrich test. In the next section, we will give a brief

overview of the most generally used methods, and motivate why we eventually

choose to follow the Jennrich method.

Normal Distribution test

The normal distribution test is a simple statistical test to verify whether the correlation

from the first period is the same as the second period, hence resolving the stability

issue. If the null-hypothesis – the difference of the correlation of the second and the

first period equals zero – is not rejected, then stability would be alleged. The normal

distribution test was used by Haney and Lloyd (1978), Watson (1980), and

Maldonado and Sounders (1981).

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Regrettably, this is an example of a pair-wise method, which will determine for every

correlation individually if it is stable over time or not. It is restricted to separate

coefficients and cannot make conclusions of the entire matrix. When considering

many variables, this technique is therefore less opportune.

Principal component analysis

The principal component analysis is a special case of Factor Analysis, often used to

analyse common movement in several series. It is a fairly complex technique that

explores interrelationships among vectors which is caused by common (economical)

factors. The resulting component correlations indicate the relation between the factor

and the original variables. They can then be compared along the variables and it can

be seen if the same risk factors show up.

This technique was used by various academics, such as Philippatos(1983),

Spyros(1974), and Ripley(1973). The main critique, however, is that they conclude

the correlations to be stable, only based on the causal observations that the common

factors across two sub-periods are similar (Cheung and Ho, 1991). It can therefore be

disputed if this really is to be considered a test on stability.

Cluster analysis

A cluster analysis or numerical taxonomy, is a method to partition a dataset into

different subsets with common traits. Numerical taxonomy is used in various

disciplines, such as biology, geology, and psychology. In economics, a type of

clustering, called hierarchical clustering, is often used. Panton (1976) points out that

using this method, the n entities are firstly treated as a separate cluster. Based on a

defined measure, exempli gratia correlation, the two most similar entities are bundled

into a single cluster. This process is repeated until the desired number of clusters is

reached. This can be illustrated through the use of a dendrogram, as shown in figure 1.

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Figure 1: Example of a dendrogram to visualise clustering (Source: Pantom, 1976)

These dendrograms can reproduce cophenetic values. Panton (1976) defines the

cophenetic value for pair i-j as “the level of association (correlation) where stems

from the two entities meet on the dendrogram”. Correlation of the cophenetic values

over time will now be analysed to determine stability-issue. While this is clearly a

unique approach, this is not a real test on stability, and due to the categorisation,

arbitrariness will unavoidably find its way in the results.

Timeseries

Timeseries can be used to find out if the correlations follow a random walk. If so, one

should be very cautious in concluding the correlation is stable over a specific period.

Ex-ante, models would be unlikely to be very successful. In its easiest form, the

timeseries could look as follows (Maldonado, 1981):

tijtijt e+= −1βρρ

With:

E(et) =0;

Cov (et, et-1) = 0; and

E(et, et) = σ²

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When after conducting the regression, β appears to be insignificantly different from

zero, no random walk would be perceived and stability could be assumed. This is

offcourse premature, and therefore more advanced models were sought out. The Box-

Jenkins test is an example of an ARIMA model which enables a multivariate approach

in forecasting the correlation.

Maldonado (1981) uses autocorrelation of annual timeseries no further than the

fourth-order lag, because of constraints of degrees of freedom. The null-hypothesis

that the autocorrelation equals zero can then be rejected or not for a 5% acceptance

level. If it cannot be rejected, instability could again be assumed. The Box M test has

been used by various researchers such as Merit and Merit (1989) and Cheung and Ho

(1991).

A big advantage of the Box-Jenkins test is that it compares entire matrices, instead of

pairs. Nevertheless, the Box-Jenkins test is only applicable for covariance matrices

and not for correlations. Conclusions drawn from covariance matrices do not

necessarily apply to correlation matrices. For instance, when in the second period the

variances and covariances have proportionally increased, the correlation matrices are

the same and perfectly stable, but the covariance matrix has changed. The box test

would not reject the null-hypothesis that two covariance matrices are not the same,

whereas the correlations are.

Tang (1995) claimed having found a way to by-pass this shortcoming by assuming the

data is multivariate normal. However, it is shown by Annaert e.a. (2006) that this

approach is erroneous, if the data is not normal, the power, as well as the size and

specification of the test will be affected.

Jennrich-test

Jennrich-test is an asymptotic χ ² test for the equality of two correlation matrices

(Jennrich, 1970). The test can also be used to test the equality of two covariance

matrices. The extending formula of Jennrich (1970) statistic is applied to joint-test for

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the homogeneity of a set of correlation matrices. Although there are some methods to

test for the stability of correlation matrices of different distributions (see Tang, 1995,

1995B), it is advisable to use Jennrich test (Annaert et al, 2006) that takes into

account the whole correlation matrix, not only the individual correlation in the matrix.

In this study, in addition to testing the pair-wise correlation, we also wanted to check

the stability of the whole correlation matrix for market risk and credit risk. We

therefore used the Jennrich test for this purpose. The details of this test will be

discussed in the methodology section below.

3. Data and descriptive statistics

Data

In our study, the data was captured from Fortis daily database for 10 years from the

2nd January 1997 to 31st December 2006. The return indices for risky investment of

Fortis are in both EUR and USD, including returns on EUR Mid cap equities, EUR

Large cap equities, USD Mid cap equities and USD Large cap equities. The indices

for EUR New real estate, USD Commodities, USD Hedge funds, JPY Japanese

Equities were also taken for the purpose of risky asset diversification.

To illustrate the returns on riskless investment that was assumed to be the investment

in deposits with Fortis over different time to maturities, the interest rates for 1 month,

1 year, 3 years, 10 years, 30 years in EUR and USD were obtained. Also, the FX rate

EUR/USD was taken for the whole period, though the EUR usage was effective from

2000. Here the exchange rate for the period before 2000 was based on the calculation

of Fortis.

The indices on the investment portfolios in different credit rating categories focused

on the pool of A- rating corporations marked by “Credit 1”, on Fortis Bank, Fortis

Group and Fortis Insurance which were all equal and above AA-ratings and marked

“Credit 2”, “Credit 3”, “Credit 4” respectively. Therefore it is expected that the return

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differences between these categories are not large. These portfolios are held from

three to five years by Fortis.

The return computed was the log return for all return indices. For the interest rate, the

return was the difference between interest rate at time t and interest at time (t-1). We

took into account all the daily, weekly (five days) and monthly (22 days) returns. For

weekly and monthly returns, the latest values were taken first and the redundant

values would be the oldest ones with regards to the time series.

In total, we have three indices for risky assets in EUR, five series of interest rate in

EUR for five maturities, one series of FX rate (USD/EUR – amount of USD/1 EUR),

and four indices for portfolios in different credit rating groups. For investment in

USD, we have four indices for risky assets, five series of interest rate all of which will

be converted with FX rate for the purpose of considering the FX risk.

Market risk and Credit risk construction

In the literature, market risk reflects the impact of many market factors on the

fluctuation of prices of financial instruments, including macro-economic factors like

changes in government’s relevant policies (in interest rate, FX controlling, required

reserve ratios…), economic crisis, industrial movements… Commonly, the prices of

financial instruments are driven by the six main components: volatility, spread,

commodity, FX, interest rate, and equity. In this study, we examined the three factors

similarly to Barnhill & Maxwell (2002). They include the Market Risk Spread, FX

Risk Spread (FX spread) and Interest Rate Risk Spread (Interest rate spread).

Market Risk Spread implies the overall risk premium of an asset class, represented by

the difference between the returns on risky assets and the returns on riskless assets.

The more risky assets, the higher risk premium required. FX Spread exhibits the effect

of FX rates on the returns on foreign currency investment, calculated by the changes

in FX rate over time. This effect can be positive or negative. Interest rate spread or

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the changes in the slope of the term structure are measured by the difference between

long-term interest rate and short term interest rate.

Because our study used the data provided by Fortis, the returns on riskless assets was

assumed to be identical to the one month interest rate or the short term interest rate.

Therefore, the Market Risk Spread was computed by taking the difference between

the log returns on risky assets1, and the return on one month interest rate in EUR or

USD respectively:

Market Risk Spread = Log return on risky asset – Return on one month interest rate

Interest rate spread was obtained by subtracting the yield on one month interest rate

from the yields on longer terms of 1 year, 3 years, 10 years, 30 years interest rates.

Interest rate spread=Yield on long term interest rate – Yield on 1 month interest rate

In order to compare the FX risk for USD assets in terms of EUR, FX spread is

computed by taking the difference between the log returns on USD risky or riskless

assets and the log returns on FX rate, as per the below formula:

FX spread_USD = Rt – RFX _t

In which:

Rt is the log return on USD assets (risky assets or riskless assets)

RFX_t is the log return on FX being the logarithm of the outcome of FX rate

day t divided by FX rate day (t-1)

RFX_t = ln 1)-(t rate FX

rate(t) FX

Intuitively, the credit risk is defined as potential loss due to change in credit quality or

default of a counter party (Hull-6th edition; Annaert et al, 2000). The four main

sources of credit risks are Default risk, Credit migration risk, Exposure uncertainty

risk and Recovery risk.

1 Including following indices: EUR/USD Mid cap equities, EUR/USD Large cap equities, the EUR

New real estate, USD Commodities, USD Hedge funds, JPY Japanese Equities

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Default risk comprises the risk one encounters when the firm would default to meet

their obligations. Credit migration risk refers to the risk that a company would change

in rating, hence implying a different portfolio value. Exposure uncertainty risk

considers all the specific characteristics loans and bond contracts can have, such as

pre-payment options, puttable bonds, etc. Lastly, Recovery risk considers the

uncertain value of the bond when the bond defaults.

Due to the data limitation, we focused our study only on Credit migration risk

measured by the difference of log returns from bonds of four credit ratings (“Credit

1”, “Credit 2”, “Credit 3” and “Credit 4”) and the yield on Fortis’s 3 year interest rate

which is the proxy for the government bond yield on the same maturity. The selection

of this maturity is due to the fact that all the four portfolios on “Credit 1” , “Credit 2”,

“Credit 3” and “Credit 4” are held for 3 to 5 years.

Credit Migration Risk = Log return on Portfolio – Yield on 3 year interest rate.

The portfolios are for “Credit 1” , “Credit 2”, “Credit 3” and “Credit 4” ratings. It is

expected that the Credit migration risk for the portfolio of lower rating (Credit 1) will

be higher than for the portfolio of higher rating (Credit 2, Credit 3, Credit 4) because

higher credit risk , higher risk premium is required.

Descriptive statistics of the market and credit spreads

Table 1 Panel A shows the descriptive statistics for all the market risks and credit

risks for daily data for the whole original data samples. As could have been expected

that higher risky investment requires larger risk premium, the spread for the risky

assets are significantly higher than the interest rate spread. The mean of the spread for

EUR risky assets are all larger than 3 basis points (bp) while the biggest mean for the

spread of EUR riskless assets is only 0,01 basis point. For example, the market risk

spread of Mid cap equities_EUR is 4,34 bp whereas the spread for yield on 1 year

interest rate is 0,01 bp. The three other spreads for yield on 3 year, 10 year and 30

year interest rate are all negative. Meanwhile, the volatility of the Market spread is at

least five times the volatility of the Interest rate spread. In addition, the mean of FX

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risk spread for USD risky assets are much bigger than the for the USD riskless assets.

Notably, the effect of FX rate makes the return on USD riskless assets become

negative. The FX spread for yield on 1 month interest rate in USD is -0,03bp. The

mean of all FX risk spreads for USD riskless assets is monotonically more negative

when the term to maturity increases. The same trend is applied to EUR interest rate

spreads. Regarding Credit risk spreads, the means are not significant different

between each other although the mean for Credit 1 is higher than for the others

because Credit 1 is an index of a portfolio of a pool of A-rating corporations while

Credit 2, Credit 3 and Credit 4 are indices of portfolios of AA-ratings.

The kurtosis coefficients of all spreads are significantly higher than 3, implying that

these series distributions have fatter tails than normal distributions. For risk

management purposes, this means that more extreme changes would effectively occur

than expected. In general, the Market spreads, FX risk spreads for risky assets and the

Credit spread are left-skewed with negative skewness coefficients while these series

for riskless assets are right-skewed.

Taking into account the auto-correlation of spread series, the three first order auto-

correlations are not significantly different from zero except for the first order auto-

correlation of Interest rate spreads and of FX risk spread for USD 1 month interest

rate. Small auto-correlations exhibit the variable independence in each series meaning

that it is likely expected to have a high risk spread following a low risk spread.

The characters of insignificant auto-correlations and fat tails of spread series are

expected to influence the exactness of testing for the stability of correlation matrices

in the later part – low auto-correlations will increase the correctness while fat tail

distribution will reduce the correctness of the test results (Annaert et al., 2006).

The descriptive statistics for weekly and monthly data are presented in Table 1 under

panel B and C in the Appendix. In general, the same stylised facts as daily data show.

However the kurtosis and skewness coefficients for monthly data are much less than

for daily and weekly data, 26 out of 40 skewness and excess kurtosis coefficients for

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monthly data are not significant from zero at 5% confident level. The monthly data

clearly shows more characteristics of normal distribution than daily and weekly data.

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Table 1 Panel A: Descriptive Statistics for daily data-Market risk & Credit risk

spreads

Mean (in bp)

Std (in bp) Skew Kurtosis

First Order

Autocorrel

2nd Order Autocorrel

3rd Order Autocorrel

Market risk spread

Mid cap equities_EUR 4,34 94,25 -0,73* 7,07* 0,08* 0,02 0,02

New real estate_EUR 4,87 54,46 -0,94* 12,09* 0,03 0,02 0,03

Large cap equity_EUR 3,10 135,64 -0,17* 5,75* 0,01 -0,02 -0,05*

Interest rate spread

Interest rate 1 year_EUR 0,01 14,92 0,49* 18,04* -0,36* -0,09* -0,07*

Interest rate 3 year_EUR -0,03 11,50 0,53* 4,88* -0,31* -0,08* -0,09*



FX spread_risky asset

Mid cap equities_USD 2,81 142,82 -0,20* 4,80* 0,04* -0,04* 0,01

Large cap equities_USD 2,68 121,89 -0,06 6,13* 0,00 -0,03 -0,02

Commodities_USD 1,54 142,12 -0,15* 4,38* -0,03 -0,01 0,01

Hedge funds_USD 2,92 73,93 1,06* 23,75* 0,03 -0,01 0,02

FX spread_riskless asset

Interest rate 1 month_USD -0,03 335,41 -0,56* 4,88* -0,31* -0,07* -0,09*

Interest rate 1 year_USD -0,41 281,45 0,34* 215,44* 0,06* -0,12* -0,07*

Interest rate 3 year_USD -0,89 184,84 0,19* 8,95* 0,01 -0,01 -0,02

Interest rate 10 year_USD -1,15 138,15 0,08 5,70* 0,01 0,00 -0,04*

Interest rate 30 year_USD -1,30 131,08 0,42* 63,44* -0,14* 0,00 -0,02

Credit risk spread

Credit 1_EUR 0,25 5,62 -0,37* 9,40* 0,06* 0,13* 0,06*

Credit 2_EUR 0,19 4,98 -0,47* 7,04* -0,02 0,13* 0,07*

Credit 3_EUR 0,19 4,92 -0,47* 7,13* -0,03 0,12* 0,07*

Credit 4_EUR 0,22 4,96 -0,30* 9,19* -0,07* 0,07* 0,02 Calculations are based upon the data provided by Fortis. The daily indices, FX rates and interest rates

are captured from 2nd January 1997 to 31st December 2006. Market risk spread is the difference

between risky asset log returns and 1 month interest rates. Interest rate spread computed by subtracting

the 1 month interest rates from the longer term interest rates (1 year, 3 year, 10 year and 30 years). FX

risk spread is taken into account the FX rate multiplied by the asset log returns. Credit risk spread is the

difference between the returns on portfolios and the 3 year interest rate- a proxy for comparable

government bond yield. Credit 1 is a portfolio in a pool of A-rating corporations and Credit 2,3,4 are

the portfolios in Fortis bank, Fortis Corporation, Fortis Insurance respectively which are all equal and

above AA-rating (FORTIS).

* : significant at a 5% confidence level

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Correlation matrix of Market risks and Credit risks

The correlations between market risk and credit risks are presented in the Table 2. The

correlation matrix is computed from the original data. It can be seen from the table

that for the market risk spread of daily data, the spreads on Mid cap equities have the

highest correlations with the spreads on Credit risks (0,136). The correlation

monotonically reduces from Credit 1 to Credit 4 and from Mid cap equities to New

real Estate, Large Cap equities with only one exception of correlation between Large

cap equity and Credit 2.

Taking into account the FX risk, the correlations between market spread risks and

credit risks are sharply decreased, even some correlations become negative. For

instance, the correlation between market spread risk of Large cap equities and the

credit risk of Credit 1 is 0,090. When including FX risk this correlation goes down to

-0,001. However the weaken effect is not consistent over the four credit ratings. The

correlation between FX risks for risky assets and Credit risks obtains the largest value

for Credit 2 category instead of Credit 1 category (for FX risk of Mid cap equities:

correlation with Credit 1 is 0,010 and with Credit 2 is 0,026).

While the Market spread risks generally present the positive correlation with Credit

risks, the Interest risks oppositely show the negative correlations with Credit risks

except for the risk on 1 year interest rate. For examples, the Interest risk spread on 1

year interest rate has a correlation of 0,69 with Credit risk on Credit 2 category. These

numbers for 3 year, 10 year, 30 year interest rate are -0,062; -0,048; and -0,031

respectively.

The correlations between Credit risks and FX risks on riskless assets become

extremely negative. The magnitudes of these correlations are triple the correlations of

Interest rate risks and Credit risks. Notably, the FX risk for 1 year interest rate

becomes negatively correlated with Credit risks of all categories (these correlations

are : -0,130; -0,124; -0,136; -0,239). It could be inferred that FX has a strong impact

on the correlation between yield on riskless assets and returns on different credit

rating categories.

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In summary, for daily data, the correlations between Spread risks for EUR risky assets

and Credit risks are almost positive while the correlations between Interest rate risks

and Credit risks tend to be negative. Taking FX risk into account, the correlations for

Market risks of risky assets and Credit risks reduce significantly even though they are

still positive. Whereas FX risks make the correlations of Market risks of riskless

assets and Credit risk sharply increase in the magnitude representing in the absolute

value of the correlations. It is also observed that the highest correlation is between the

Spread risk for EUR Mid cap equities and Credit risk of Credit 1 category.

When the frequency of the data is reduced, a new trend of correlation matrix is

presented. In general the correlations of Market spread risks and Credit risks increase

significantly accordingly to the decrease of frequency. These correlations for weekly

data is nearly double than those for daily data but roughly a half than those for

monthly data. For example, these correlations for EUR Mid cap equities are 0,136 ;

0,280; 0,416 for daily, weekly and monthly data. Differently from daily data, the FX

log returns of weekly data just makes the correlations of Market risk for USD risky

assets with Credit risks slightly reduce. The correlation of Spread risk for Mid-cap-

equities _Eur and Credit risk of Credit 3 is 0,242, the correlation of FX risk for Mid-

cap-equities_USD and Credit risk of Credit 3 is 0,222.

For correlations between Interest rate risk, FX risk of riskless assets and Credit risks

there is no consistent movement from daily data to weekly and monthly data, such as

the change of correlations for Credit 1 with Interest risk are much less than that for the

3 other credit rating categories (Credit 2, Credit 3, Credit 4).

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Table 2: Correlation matrix of market and credit risks Daily Spread risk Interest risk Mid cap equities New real estate Large cap equity 1 year 3 year 10 year 30 year Credit risk Credit 1 0,136 0,110 0,090 0,036 -0,062 -0,052 -0,036 Credit 2 0,118 0,104 0,101 0,069 -0,062 -0,048 -0,031 Credit 3 0,109 0,100 0,091 0,060 -0,068 -0,052 -0,033 Credit 4 0,012 0,054 -0,025 -0,040 -0,119 -0,087 -0,050 FX risk for risky assets FX risk for riskless assets Mid cap equities Large cap equities Commodities Hedge funds 1 month 1 year 3 year 10 year 30 year Credit 1 0,010 -0,001 0,013 -0,049 -0,064 -0,130 -0,184 -0,199 -0,137 Credit 2 0,026 0,020 0,044 -0,014 -0,061 -0,124 -0,190 -0,188 -0,166 Credit 3 0,018 0,012 0,041 -0,022 -0,064 -0,136 -0,209 -0,205 -0,177 Credit 4 -0,069 -0,071 0,003 -0,098 -0,089 -0,239 -0,366 -0,344 -0,271 Weekly Spread risk Interest risk Mid cap equities New real estate Large cap equity 1 year 3 year 10 year 30 year Credit risk Credit 1 0,280 0,134 0,244 0,011 -0,033 -0,050 -0,035 Credit 2 0,250 0,141 0,247 -0,041 -0,098 -0,119 -0,105 Credit 3 0,242 0,138 0,240 -0,046 -0,107 -0,126 -0,107 Credit 4 0,121 0,083 0,118 -0,098 -0,199 -0,178 -0,115 FX risk for risky assets FX risk for riskless assets Mid cap equities Large cap equities Commodities Hedge funds 1 month 1 year 3 year 10 year 30 year Credit 1 0,206 0,164 -0,001 -0,003 -0,067 -0,056 -0,082 -0,145 -0,176 Credit 2 0,230 0,197 -0,013 0,025 -0,029 -0,084 -0,121 -0,164 -0,177 Credit 3 0,222 0,192 -0,017 0,018 -0,033 -0,095 -0,138 -0,178 -0,188 Credit 4 0,104 0,100 -0,061 -0,056 -0,074 -0,206 -0,317 -0,321 -0,290

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Monthly Spread risk Interest risk Mid cap equities New real estate Large cap equity 1 year 3 year 10 year 30 year Credit risk Credit 1 0,416 0,261 0,429 0,023 0,003 -0,029 -0,055 Credit 2 0,372 0,293 0,421 -0,107 -0,111 -0,138 -0,131 Credit 3 0,360 0,295 0,411 -0,120 -0,128 -0,148 -0,136 Credit 4 0,156 0,277 0,227 -0,256 -0,311 -0,255 -0,173 FX risk for risky assets FX risk for riskless assets Mid cap equities Large cap equities Commodities Hedge funds 1 month 1 year 3 year 10 year 30 year Credit 1 0,437 0,323 0,073 -0,012 -0,098 -0,070 0,019 -0,037 -0,106 Credit 2 0,474 0,403 0,043 0,013 -0,084 -0,162 -0,101 -0,137 -0,184 Credit 3 0,460 0,393 0,037 0,006 -0,090 -0,178 -0,124 -0,157 -0,202 Credit 4 0,228 0,207 -0,042 -0,085 -0,151 -0,359 -0,384 -0,383 -0,395

Table 2 gives the relation between the credit risk and the different types of market risk for every index as based on daily, weekly, and monthly natural logarithmic returns

Page | 19

4. Pair-wise test

Methodology

a) Correlation descriptives

The downside of a correlation matrix, as presented in table 2, is that it contains no

other statistical information than the correlation coefficients. Indications about

variance, kurtosis, and skewness could provide us a basic impression on the stability

over different holding periods. Further, there are a few pitfalls one should pay

attention to. Philippatos(1983) pointed out that some papers on stability focused –

unrightfully – on only a part of the correlation matrix, hence leading to biased results.

Next and more specific to our dataset, we only have ten years of observations and for

the hedge fund only monthly instead of daily observations were given.

Hence, we used a simplified approach which avoided those problems. First, we

computed the daily, weekly, and monthly logarithmic returns over a rolling window.

In other words, we calculated the log return over a give horizon; shifted one day and

recomputed, resulting in a series of log returns. As disadvantage, our returns include

overlap and are less independent. On the positive side, we now have a lot more

observations for the weekly and monthly returns.

Next, we generated non-overlapping sub-datasets based on different holding periods

of 1, 2, and 4 months. We did not take longer holding periods, since our number of

sub-datasets would be much lower than 30, which is generally considered a minimum

to make statistical conclusions. Tang(1995b), however, used holding intervals ranging

from one to twelve months. While his dataset was larger, his 12-month holding

interval leaded only to 21 sub-datasets. Having too few sub-datasets could likely bias

the reader to conclude longer holding periods imply more stability. Indeed, in

extremum, only one sub-dataset would be used, thus leading to one correlation. The

variance would be zero and an incorrect conclusion would be that this is the optimal

holding period.

Page | 20

So as to keep the output table readable and within proportions, we made a selection of

basis variables on which we would compute correlation. For credit risk, we chose to

base it on the first credit index, since this index is more risky and presumed to be less

stable. As risk premium, we based this variable on the mid cap equities (EUR) and as

foreign exchange risk, we based this variable on the mid cap equities (USD). Mid cap

equity return indices were selected specifically, as they were the only index provided

in USD and EUR. This should enable us to better view the impact of the foreign

exchange risk. Finally, we computed the interest rate risk from the 1 year EUR

interest rate given that it is most close to our holding periods.

b) Simple bootstrapping

While the resulting table indeed gives descriptive statistics, we need to develop some

kind of measure to interpret them. One possibility would be to assume the returns to

be normally distributed. This, however, would be erroneous as became apparent from

the descriptive statistics (conferatur supra). Hence, we introduced a first bootstrap on

the data to obtain a distribution of our dataset.

For the bootstrap, we first scrambled the return indices and recalculated the

logreturns. Again, we generated correlations based on different holding periods, as

under section a. Finally, this procedure would be repeated 10,000 times and the results

were stored, resulting in a distribution of variance, mean, kurtosis, etc. The results of

the first part could now be positioned towards the distribution. As illustrated in figure

2, if the correlation descriptives are larger than the 99,5th percentile, we would

typically reject the null hypothesis of stability and assume instability.

Page | 21

Figure 2: Distribution of the standard deviation generated by the simple bootstrap

method (monthly return, 1 month horizon)

c) GARCH(1,1) bootstrapping

The first bootstrap method had the advantage it is fairly simple to conduct, and helps

reducing the dependence caused by the rolling window from the return calculation.

The downside is that the volatility structure over time is destroyed. This could mainly

be resolved by adopting a second bootstrap using the GARCH model.

We start by fitting a GARCH process to each return series. Every series of log returns,

yt, can be modelled as follows:

yt = m + et

with m is the unconditional average and et the residual at time t.

Further,

ttt zhe .=

with ht following a GARCH(1,1) process, and zt being the standardised residuals.

0.16 0.18 0.2 0.22 0.24 0.26 0.280

20

40

60

80

100

120

140

160

Reject H0 Don’t reject H0

Page | 22

For our bootstrap, we first needed to compute the standardised residuals, zt. The

formula is the following:

t

tt

h

ez

ˆ

ˆ=

With

êt = yt - m

12

1 ..ˆ−− ++= ttt heh βαω

As starting values for e0 and h0, we assumed the latter to be ω /(1-alpha-beta) and the

former equal to the first return, y0. The parameters α, β, and ω were estimated using

the maximum likelihood estimator:

max ∑=

−−

n

t t

tt

h

eh

0

2

ˆˆ

)ˆln(

After calculation, a series of zt is obtained, which are suitable to be scrambled, hence

initiating a bootstrap. Given the estimates for α, β, ω, and zt, it is now possible to

recompute the returns based on those parameters while preserving the main volatility

structure. After acquiring the returns, again, correlations were computed over the

different holding periods, based on our generated return series.

We repeated the process from the scrambling to the end for 10,000 times. We ended

up with a distribution of descriptives over the different holding periods. Like for the

first bootstrap, we would reject the null hypothesis of stability if the descriptive

statistics from the variance under section a are bigger than the 99,5th percentile of our

distribution.

Results

a) Correlation descriptives

Table 3 presents the resulting descriptive statistics. Since this is a pair-wise test, based

on only a selection of the economically most interesting values, we are able to make

conclusions on an asset-level instead of portfolio-level as will be the case in the

Page | 23

Jennrich test. Regarding the mean, it becomes apparent that the risk premium, the FX

risk, and the commodity risk are consistently positively correlated with the credit risk.

In other words, when one of those risks goes up, the credit risk increases as well. The

interest risk, conversely, is negatively correlated with the first credit risk rating,

regardless of the return calculation of the horizon. For instance, for the daily log

returns over 1 month, if the interest risk would increase by 1%, the credit risk would

on average decrease by 0,128%.

The standard deviation of the correlation between interest rate risk and credit risk is

generally higher than the other correlations. Nevertheless, conclusions on stability

will be drawn in a later paragraph. Table 3 gives the impression, though, that daily

returns are varying less than monthly returns, and the longer the holding period, again,

the higher the variance. As indicated before, the lowest number of intervals is 29, so

as to preserve the generally accepted statistical significance level which is assumed to

be around 30.

b) Simple bootstrapping

The distribution achieved by bootstrapping the correlation descriptives can now be

used to determine stability. Assuming that this distribution should be stable, we would

compare the correlation descriptives for standard deviation to the distribution’s 99,5th

percentile. Hence, table 4 designates the difference between the 99,5th percentile of

the bootstrap distribution and the previously calculated correlation descriptives. For a

positive outcome, we reject the null hypothesis of stability, and for a negative

outcome we do not reject it. Negative numbers are formatted in bold italic.

For monthly returns, the outcomes are always negative. Stability for weekly returns is

rejected in some cases, but for daily returns, the null hypothesis of stability was

rejected in the majority of the cases. Hence, based on this test, the first main

conclusion is that using monthly log returns are likely to be most stable across time.

Next, one can observe that – although the difference is only small – shorter holding

periods lead to more negative outcomes. For instance, for weekly log returns, all of

the outcomes are negative for the 1 month holding period, three for the 2 months, and

Page | 24

merely two for the 4 month holding period. Table 4 further gives information about

the significance of the mean, skewness, and kurtosis, but this does not give us more

insight in stability-issues and will therefore remain undiscussed.

c) GARCH(1,1) bootstrapping

The main critique on the simple bootstrapping method is that it destroys possible

volatility patterns. GARCH(1,1) is considered to capture most volatility of

relationships such as covariance and correlations. The technique has been used by

academics such as Andersen, Bollerslev, and Christoffersen (2005) and Hunter &

Simon (2005); ARCH was used by Sancetta(2003).

The results in table 5 are similar to those of table 4, but are generally more often

negative. The monthly log returns remain the most stable basis as compared to the

daily and weekly. However, now only eight negative outcomes can be noted for the

standard deviation, whereas table 4 had only three for the daily log returns. With

0,068, the weekly logreturns for the 4 month holding period from correlation 1 is the

only outcome where the correlation descriptives exceed the 99,5th percentile of the

GARCH(1,1) distribution leading to instability. The monthly returns are deemed

stable over all horizons. Consistent with table 4, we conclude that the shorter the

holding period, the more likely the null hypothesis of stability will not be rejected.

Page | 25

Table 3: Descriptive stability test

1 month holding period 2 months holding period 4 months holding period corr1* corr2** corr3*** corr4**** corr1* corr2** c orr3*** corr4**** corr1* corr2** corr3*** corr4**** daily returns mean 0,067 -0,128 0,068 0,043 0,087 -0,118 0,075 0,051 0,100 -0,093 0,076 0,053 std 0,249 0,303 0,241 0,218 0,184 0,264 0,188 0,143 0,149 0,234 0,162 0,089 skewness -0,257 -0,217 0,199 -0,152 -0,067 -0,420 0,319 -0,323 -0,456 -0,648 0,332 0,053 kurtosis 2,843 2,434 2,776 2,570 2,897 2,456 2,720 3,313 2,753 2,798 2,492 2,208 count 120 120 120 120 60 60 60 60 30 30 30 30 weekly returns mean 0,139 -0,187 0,126 0,010 0,184 -0,201 0,152 0,030 0,218 -0,166 0,156 0,038 std 0,360 0,403 0,368 0,365 0,256 0,304 0,275 0,273 0,174 0,246 0,219 0,159 skewness -0,137 0,212 -0,251 -0,181 -0,069 -0,084 -0,252 -0,241 0,308 -0,127 -0,482 -0,368 kurtosis 2,164 1,990 2,213 2,487 2,697 2,372 2,513 2,559 2,426 1,974 2,698 2,816 count 120 120 120 120 60 60 60 60 30 30 30 30 monthly returns mean 0,239 -0,178 0,152 0,073 0,282 -0,169 0,200 0,083 0,341 -0,113 0,175 0,092 std 0,486 0,479 0,517 0,487 0,411 0,432 0,475 0,494 0,304 0,375 0,329 0,374 skewness -0,616 0,369 -0,363 -0,224 -0,508 0,068 -0,611 -0,278 -0,321 -0,426 0,359 0,144 kurtosis 2,145 2,050 2,008 1,984 2,454 1,970 2,351 1,826 2,665 2,313 2,525 2,159 count 119 119 119 119 59 59 59 59 29 29 29 29 * correlation between the risk premium and the first credit risk rating ** correlation between the interest risk and the first credit risk rating *** correlation between the foreign exchange risk and the first credit risk rating ** ** correlation between the commodity risk and the first credit risk rating

Table 3 gives descriptive statistics on four selected correlations and compares them on base of frequency(daily, weekly, monthly) and holding period (1, 2, and 4 months)

Page | 26

Table 4: Difference between the 99.5th simple bootstrap percentile and the correlation descriptives

1 month holding period 2 months holding period 4 months holding period corr1* corr2** corr3*** corr4**** corr1* corr2** c orr3*** corr4**** corr1* corr2** corr3*** corr4**** daily returns mean 0,135 -0,268 -0,657 -0,762 0,155 -0,256 -0,653 -0,755 0,168 -0,229 -0,654 -0,754 std -0,032 0,033 0,087 0,075 -0,029 0,058 0,077 0,051 -0,016 0,075 0,080 0,022 skewness -0,905 -0,637 0,442 0,272 -0,956 -1,137 0,109 -0,367 -1,690 -1,705 -0,288 -0,490 kurtosis -1,008 -1,256 -10,317 -27,666 -1,853 -2,202 -6,783 -9,327 -3,002 -2,571 -5,196 -6,957 weekly returns mean 0,066 -0,140 -0,725 -0,805 0,067 -0,138 -0,728 -0,806 0,068 -0,136 -0,730 -0,806 std -0,136 -0,451 -0,023 -0,123 -0,027 -0,404 0,044 -0,062 0,055 -0,324 0,073 -0,028 skewness -0,283 0,007 0,594 0,770 -0,672 -0,388 0,073 0,189 -1,005 -0,815 -0,462 -0,388 kurtosis -3,924 -3,506 -12,146 -23,777 -4,826 -4,604 -9,266 -12,685 -5,339 -5,655 -8,333 -9,420 monthly returns mean 0,064 -0,141 -0,729 -0,806 0,067 -0,139 -0,730 -0,807 0,067 -0,138 -0,730 -0,807 std -0,245 -0,239 -0,132 -0,111 -0,198 -0,192 -0,103 -0,083 -0,159 -0,154 -0,080 -0,064 skewness -0,424 -0,595 0,312 0,381 -0,587 -0,830 -0,024 -0,003 -0,839 -1,169 -0,453 -0,473 kurtosis -3,486 -3,501 -10,554 -18,016 -4,368 -4,353 -8,222 -10,075 -5,341 -5,117 -7,358 -8,360 * correlation between the risk premium and the first credit risk rating ** correlation between the interest risk and the first credit risk rating *** correlation between the foreign exchange risk and the first credit risk rating ** ** correlation between the commodity risk and the first credit risk rating

Table 4 considers the difference between the 99.5th simple bootstrap percentile and the correlation descriptives. This can be used for stability testing: if the outcome for the

standard deviation is negative, the null hypothesis of stability is not rejected. If the outcome is positive, the null hypothesis is rejected. Negative numbers are formatted in

bold italic.

Page | 27

Table 5: Difference between the 99.5th simple bootstrap percentile and the correlation descriptives

1 month holding period 2 months holding period 4 months holding period corr1* corr2** corr3*** corr4**** corr1* corr2** c orr3*** corr4**** corr1* corr2** corr3*** corr4**** daily returns Mean -0,042 -0,130 -0,024 -0,026 -0,021 -0,121 -0,016 -0,014 -0,008 -0,099 -0,015 -0,010 Std -0,013 0,026 -0,016 -0,026 -0,016 0,052 -0,005 -0,040 -0,005 0,069 0,015 -0,052 Skewness -0,615 -0,650 -0,189 -0,556 -0,689 -1,144 -0,327 -0,990 -1,498 -1,766 -0,720 -1,013 Kurtosis -0,878 -1,116 -0,941 -1,230 -1,704 -2,137 -1,872 -1,513 -2,670 -2,741 -3,010 -3,426 weekly returns Mean -0,159 0,090 -0,172 -0,058 -0,156 0,080 -0,168 -0,058 -0,154 0,070 -0,165 -0,056 Std -0,111 -0,436 -0,128 -0,240 -0,008 -0,392 -0,044 -0,161 0,068 -0,315 -0,001 -0,112 Skewness -0,015 -0,261 -0,018 -0,136 -0,557 -0,726 -0,505 -0,580 -1,136 -1,257 -1,115 -1,077 Kurtosis -4,293 -3,919 -4,340 -4,366 -5,482 -5,745 -5,727 -5,989 -6,487 -6,946 -7,129 -7,683 monthly returns Mean -0,248 0,084 -0,197 -0,059 -0,243 0,086 -0,195 -0,058 -0,239 0,084 -0,194 -0,059 Std -0,237 -0,256 -0,240 -0,244 -0,183 -0,203 -0,184 -0,186 -0,149 -0,177 -0,148 -0,146 Skewness -0,066 -0,718 -0,118 -0,356 -0,495 -0,886 -0,340 -0,634 -1,035 -1,089 -0,738 -0,944 Kurtosis -3,758 -3,889 -4,251 -3,328 -5,046 -8,129 -6,785 -4,701 -6,288 -11,445 -8,440 -6,048 * correlation between the risk premium and the first credit risk rating ** correlation between the interest risk and the first credit risk rating *** correlation between the foreign exchange risk and the first credit risk rating ** ** correlation between the commodity risk and the first credit risk rating

Table 5 considers the difference between the 99.5th GARCH(1,1) bootstrap percentile and the correlation descriptives. This can be used for stability testing: if the outcome for

the standard deviation is negative, the null hypothesis of stability is not rejected. If the outcome is positive, the null hypothesis is rejected. Negative numbers are formatted in

bold italic.

Page | 28

5. The Jennrich test

Methodology

The main idea of Jennrich-test is to check the homogeneity of a set of correlation

matrices. In the above part of pair-wise test, we did focus on some specific

correlations between selected market risks and credit risks. In this part, we intended to

extend the test to cover all correlations in a matrix and check the stability of all

correlation matrices obtained over different horizons for different data frequencies.

Once again, our purpose is to find a horizon (1 up till 12 months) corresponding to a

data frequency (daily, weekly, monthly) that will provide the most stability for

correlation matrices. Despite the fact of small sample size in this part, we wanted to

include horizons more than 4 months to check if longer holding period can provide

better stability for correlation matrix. Similarly to pair-wise test, we then tried to

check the robustness of the results by using the bootstrap method and GARCH (1,1)

model for generating new data sets.

The Jennrich statistics is calculated as following (Jennrich, 1970).

Let R1, R2, ……, Rk be sample correlation matrices based on sample size n1, n2, … nk,

from k, p-variate, normal populations which have the same, but unknown, correlation

matrix. Let:

n= n1 + n2 +… + nk ,

R = (n1x R1 + n2x R2 +… + nkx Rk)/n = ( r ij),

S = (δ ij + r ij x r ij)

Zi = n i x R -1 x (Ri - R ).

δ ij denotes Kronecker’s delta such that δ ij = 1 if i = j and zero else;

R -1 is the inverse matrix of R

r ij is the element ij of R and r ij is the element ij of R -1

The Jennrich statistic χ ² then is computed as follow:

∑=

−−=k

iiii ZdgSZdgZtr

1

122 ))()(')(2

1(χ

Page | 29

Where tr(.) denotes the trace-function and dg(Zi) make the main diagonal of Zi

become a column vector.

The χ ² has an asymptotic χ ² distribution with 2

1)-(p . p . 1)-(k degrees of freedom.

The null hypothesis that all k populations have the same correlation matrix is rejected

when χ ² has a significant value. To prove the significance of χ ², we calculated the

inverse of chi-square cdf specified by the same degree of freedom 2

1)-(p . p . 1)-(k for

the corresponding probability of 99%. It is expected to observe the values greater than

this inverse value only 1% of the time by chance. Therefore, if the χ ² statistic

obtained from Jennrich test is bigger than this inverse value, the χ ² statistic is to be

considered significant at 1% confident level. We can then reject the null hypothesis

that the whole correlation matrices are homogeneity at 1% significant level.

Results

The whole data sample was divided into different intervals having the same horizon.

For daily and weekly return, the considered horizons were from 1 month, 2 months,

… up to 12 months while for monthly return, these numbers were from 3 months, 4

months, … up to 12 months to ensure non-singular correlation matrix. The correlation

matrices will be then computed for each interval. We applied the Jennrich test to

check the homogeneity of all correlation matrices obtained. The results of Jennrich

test are shown in the table 6.

Page | 30

Table 6: Jennrich statistics to determine stability of daily, weekly, and monthly log

returns over a holdout period from 1-12 months

Whole Correlation Matrices

Horizon Daily Weekly Monthly

DoF Chi_2 Inv_value DoF Chi_2 Inv_value DoF Chi_2 Inv_value

1 month 22610 47214 23108 24890 41194 25412

2 month 11210 31139 11561 12350 18024 12719

3 month 7410 24351 7696,1 8170 12308 8470,3 7410 12800 7696,1

4 month 5510 21652 5757,1 6080 9397,8 6339,5 5510 8120,9 5757,1

5 month 4370 20264 4590,4 4750 7946,4 4979,7 4370 5863,4 4590,4

6 month 3610 18433 3810,6 3990 6820,7 4200,8 3610 4881,8 3810,6

7 month 3040 17431 3224,3 3230 6288,2 3419,9 3040 3971,8 3224,3

8 month 2660 16382 2832,6 2850 5555,1 3028,6 2660 3461,5 2832,6

9 month 2280 15601 2440 2470 5102,7 2636,4 2280 2903,8 2440

10 month 2090 15076 2243,3 2280 4742,6 2440 2090 2655,9 2243,3

11 month 1710 14022 1849 2090 4638,1 2243,3 1710 2276 1849

12 month 1710 14578 1849 1900 4599,3 2046,3 1710 2180,8 1849

Calculations of Jennrich statistics (Chi_2) are based on 2656 original observations of daily data, 531

observations of weekly data and 120 observations of monthly data. The horizons are 1 month, 2months

…12 months for daily and weekly data and 3 months to 12 months for monthly data. DoF denotes the

degree of freedom and Inv_value denotes the inverse value of chi-square cdf specified by the degree of

freedom DoF correspondingly.

Obviously, all χ² statistics obtained are larger than the inverse value of the same

degree of freedom. Furthermore, calculating the p-value for the chi-square values

obtained, we found that all the chi-square values are significant (all p-values are zero).

Accordingly, the null hypothesis of the homogeneity of all correlation matrices is

rejected entirely. Based on this result, we can induce that the correlation matrix of

market risk and credit risk are unstable overtime. For daily data, weekly data or

monthly data, no effective horizons (1 month, 2 months,…,12 months) can generate

the stable correlation matrix.

Next, we looked into the robustness of the results. Using the bootstrap method

together with the GARCH (1,1) model mentioned in the above pair-wise test, the

Page | 31

procedures from data scrambling to end-results were repeated 1000 times for daily

and weekly data and 10000 times for monthly data. The bootstrap data results are

exhibited in Table 7

P-value for Bootstrap data

Horizon (months) Daily data Weekly data (10-10) Monthly data (10-10)

1 0 0 2 0 0 3 0 0 0 4 0 0 0 5 0 0,000000001 0 6 0 0 0,000001395 7 0 0,000037306 0,045578522 8 0 0,000006425 4,326700937* 9 0 0,081508986 463,405416339**

10 0 0,000277818 3393,095854670** 11 0 0,000105209 143984,849204333** 12 0 0,002513322 112875,128158141**

Asterisk * denotes the significance of p-value from zero at 5% confidence level and **

denotes the significance at 1% confidence level. For daily and weekly data, the bootstrap

is run 1000 times, for monthly data is 10000 times.

The results obtained for daily data are entirely identical to the original data’s results

with all p-values equal zero, the null hypothesis of stability for all correlation matrices

for daily is strongly rejected. For bootstrap weekly and monthly data, there are some

differences with the original data. The holding periods bigger than 4 months contain

some positive p-values. The highest mean of p-values is of the 9 month horizon for

weekly data, and 11 month horizon for monthly data. While no significant p-value is

observed with weekly data, monthly data provides four means of p-values that are

significant from zero at 1% confidence level and one significant value at 5%

confidence level. This finding is consistent with the results from the above Pair wise

test that confirm the more stability of correlation matrix for monthly data in

comparison with weekly and daily data. However all the chi-square statistics obtained

for weekly data are significant at 1% confidence level leading to the rejection of the

null hypothesis. There are only 2 out of 10,000 bootstrap monthly data sets over 11

Page | 32

month and 12 month horizons having χ² not significant2. Therefore, we can not

induce that monthly data will provide the stability for correlation matrices. Opposite

to Pair-wise test, the results from Jennrich test with Bootstrap data show that the

longer horizon, the more homogeneity of correlations matrices.

6. Conclusion

There is a wide consensus that market risk and credit risk should not be considered

separately, however, this is not reflected in existing literature where academics wrote

many papers on just one of both risks. Further, many papers use different holding

periods, what makes it difficult to compare their results. This paper studies the

stability of the correlation of both risks over different holding periods and using

different frequencies of data. This way, ex-ante optimal calculations could be obtained

without the need of more different models.

In the literature, two kinds of stability checking are perceptible focussing on either the

asset, or the portfolio level. The asset level consists of pair-wise tests where

correlations are studied individually over time. The normal distribution test is a pair-

wise test in one of its most basic forms. We opted for a more progressed form of pair-

wise testing by generating correlation statistics and comparing them to two bootstrap

generated distributions: a simple bootstrap and a GARCH(1,1) bootstrap to maintain

the volatility structure. As our dataset contained many variables, we had to make a

selection of the most interesting ones. The portfolio tests, on the other hand,

comprised all the risks of our dataset. The types of portfolio tests are manifold,

ranging from cluster tests to Box tests or Jennrich tests. We chose the latter, as it

appeared to be most suitable for correlations.

Using the pair-wise test we found that correlations generally tend to be more stable

when calculated for monthly log returns instead of taking the daily or weekly

frequency. As for the holding period, a 1 month time window appeared to be more

stable than the 2 or 4 month holding periods. 2 The detailed results for bootstrap data are provided upon request

Page | 33

From the Jennrich test with original data we conclude that there is no horizon for all

daily, weekly and monthly data that provides the homogeneity of correlation matrices.

Fortunately, when relaxing the normality constraints by bootstrap data, we obtained

results which are quite consistent with pair-wise test, showing that monthly data will

generate more stable correlation matrices than weekly and daily data. Nevertheless,

longer holding periods (9 months for weekly data and 11 months for monthly data)

are likely to be more stable than shorter holding period.

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8. Appendix

Table 1 Panel B: Descriptive Statistics for weekly data of Market risk and Credit risk spreads

Mean (in bp)

Std (in bp) Skew Kurtosis 1st Order

Autocorrel 2nd Order Autocorrel


Market risk spread

Mid cap equities_EUR 12,27 210,06 -1,12* 7,03* 0,03 -0,02 0,11*

New real estate_EUR 14,53 119,03 -1,16* 7,97* 0,03 0,02 0,03

Large cap equity_EUR 6,35 269,50 -0,73* 5,50* -0,10* 0,02 0,08


Interest rate 1 year_EUR -1,40 18,35 -0,88* 12,19* 0,33* 0,18* 0,10*





Mid cap equities_USD 7,62 288,49 -0,58* 4,60* -0,01 -0,01 0,05

Large cap equities_USD 5,37 237,33 -0,21* 4,71* -0,05 0,01 0,08

Commodities_USD 5,34 286,09 -0,40* 3,82* 0,02 0,05 0,01

Hedge funds_USD 11,79 144,47 1,20* 10,35* -0,04 0,00 0,03


Interest rate 1 month_USD 22,20 432,91 -0,27* 4,96* 0,45* 0,30* 0,21*

Interest rate 1 year_USD -9,42 511,99 1,00* 65,86* -0,03 -0,06 0,07

Interest rate 3 year_USD -12,21 348,76 0,10 5,18* -0,01 0,02 0,09*

Interest rate 10 year_USD -11,31 265,41 -0,01 4,11* -0,01 0,01 0,01

Interest rate 30 year_USD -4,97 246,28 1,59* 20,53* -0,01 0,03 0,02

Credit risk spread

Credit 1_EUR 0,75 12,70 -1,19* 9,58* 0,21* 0,10* -0,03

Credit 2_EUR 0,60 10,52 -0,51* 4,68* 0,20* 0,17* 0,09*

Credit 3_EUR 0,63 10,28 -0,48* 4,54* 0,20* 0,17* 0,09* Credit 4_EUR 0,90 9,11 -0,20 3,40 0,08 0,10 0,09*

Cfr. Table 1 Panel C

Page | 38

Table 1 Panel C : Descriptive Statistics for monthly data of Market risk and Credit risk

spreads

Mean (in bp)

Std (in bp) Skew Kurtosis 1st Order

Autocorrel 2nd Order Autocorrel


Market risk spread

Mid cap equities_EUR 90,23 508,80 -0,69* 4,15* 0,13 0,15 -0,06

New real estate_EUR 98,30 256,14 -0,43 3,56 0,06 -0,08 -0,17

Large cap equity_EUR 62,00 576,77 -0,68* 4,53* 0,04 0,04 -0,09


Interest rate 1 year_EUR 1,16 25,81 0,56* 4,84* -0,27* 0,20* -0,10

Interest rate 3 year_EUR -0,79 27,84 0,29 4,03* -0,14 0,07 0,02

Interest rate 10 year_EUR -2,28 28,68 0,00 3,64 -0,16 0,14 0,05

Interest rate 30 year_EUR -3,56 28,95 -0,10 3,56 -0,20* 0,18 0,03


Mid cap equities_USD 56,71 656,04 -0,45* 3,21* 0,00 0,12 -0,24*

Large cap equities_USD 55,22 523,93 -0,21 3,45 0,01 0,07 -0,09

Commodities_USD 31,53 632,50 -0,12 2,65 0,10 0,06 0,02

Hedge funds_USD 63,13 344,76 -0,11 3,71 0,12 0,04 0,04


Interest rate 1 month_USD 1,29 863,74 -0,64* 4,56* 0,02 0,35* 0,19*

Interest rate 1 year_USD -5,29 1.151,10 1,07* 19,24* -0,11 0,01 0,10

Interest rate 3 year_USD -24,42 820,31 0,11 3,67 0,06 -0,02 0,08

Interest rate 10 year_USD -32,48 600,78 0,11 3,71 0,02 -0,05 0,05

Interest rate 30 year_USD -39,61 485,09 -0,26 3,68 0,06 -0,12 0,02

Credit risk spread

Credit 1_EUR 6,05 36,43 0,04 4,76* 0,08 0,00 0,09

Credit 2_EUR 4,44 30,23 0,09 3,55 0,08 0,02 0,02

Credit 3_EUR 4,50 29,34 0,09 3,50 0,08 0,03 0,03 Credit 4_EUR 5,08 24,15 -0,21 3,17 0,17 0,11 0,16

Calculations are based on Fortis data for the period 2 Jan 1997 to 31 Dec 2006. Weekly spreads are

constructed by taking difference between log return day t and day (t+5), totally there are 531

observations for weekly data. Monthly spreads are built by taking difference between log return day t

and day (t+22), totally 120 observations for monthly data. Market risk spread is the difference between

risky asset log returns and 1 month interest rates. Interest rate spread computed by subtracting the 1

month interest rates from the longer term interest rates (1 year, 3 year, 10 year and 30 years). FX risk

Page | 39

spread is taken into account the FX rate multiplied by the asset log returns. Credit risk spread is the

difference between the returns on portfolios and the 3 year interest rate- a proxy for comparable

government bond yield. Credit 1 is a portfolio in a pool of A-rating corporations and Credit 2,3,4 are

the portfolios in Fortis bank, Fortis Corporation, Fortis Insurance which are all equal and above AA-

rating (FORTIS).

* : significant at a 5% confidence level

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