Historical Value at Risk models applied to Italian ﬂoating rate government bonds

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Historical Value at Risk models

applied to Italian floating rate

government bonds

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Master’s thesis Econometrics, Operations Research and Actuarial Studies

Supervisors:

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Historical Value at Risk models applied to Italian

floating rate government bonds

Nora R.W. Ottink

Abstract

Historical simulation Value at Risk (HVaR) uses the nonparametric histori-cal distribution of risk factors to estimate extreme percentiles of the profit and loss distribution. Simple (equally weighted) HVaR as well as age and volatility weighted approaches of HVaR are applied to Italian floating rate government bonds. Risk factors of these floaters are derived from observed market prices. EWMA and (multivariate) GARCH models are used in the volatility weighted approach to translate historical shocks in risk factors to current volatility levels. The backtest results show that volatility weighted approaches are most accu-rate: they have the correct exceedance rate and VaR exceedances are serially independent.

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Preface

This thesis is the result of my graduation project in order to obtain the master’s degree in Econometrics. The research is conducted during an internship at ING trading market risk management. I would like to thank the following people for their help.

First of all, I would like to express my thanks to Paul Bekker for his supervision throughout the project. His suggestions and comments were very valuable. I enjoyed the discussions about doing research and convincing the scientific community of your ideas. I also would like to thank my second supervisor Harry Trentelman for his efforts. Second, I would like to thank all colleagues at ING for giving me a great time. I appreciate your enthousiasm about my project and the willingness to help me. During my internship I learned a lot about financial markets and risk management problems. I am especially grateful to Simon den Hertog, Antoine van der Ploeg and Norbert Hari. Simon gave me the opportunity to carry out this research. He got me up to speed with the topic and introduced me to the right people. Antoine and Norbert were always willing to help me with any technical questions.

Finally, I am thankful to Jan for his useful comments on my thesis and most of all for always being there for me, and to my parents for their love and support.

Nora Ottink

Amsterdam, July 2009

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Chapter 1 Introduction

Value at Risk (VaR) has become the standard measure of market risk in financial institutions such as banks. The main reason for its popularity is that it shows the total risk of a portfolio in a single number. Internally, banks use VaR as risk management tool by setting VaR limits for individual activities. Externally, regulators impose capital requirements on banks based on their VaR values. Therefore, accurately estimating VaR is of great importance. VaR estimates the maximum expected loss over a certain period with a certain confidence level. That is, it estimates low quantiles of the profit and loss (P&L) distribution of an instrument or portfolio. In a trading environment, 1-day VaR figures at 99% confidence levels are widely used.

Different approaches can be used to estimate VaR. Manganelli and Engle (1998) give an overview of VaR models used in finance. In so-called parametric VaR models the underlying distribution of risk factors is specified. Risk factors can be interest rates, exchange rates, equity prices etcetera. In practice, it is often assumed that these risk factors are (multivariately) normally distributed. Unfortunately, this assumption is often not justified. Financial time series are usually fat-tailed, which means they have a number of extreme observations that is larger than is likely under the assumption of normality. Therefore, incorrectly assuming normality leads to underestimation of the number of extreme losses. The historical simulation approach does not have this issue since it relies on the non-parametric historical distribution of risk factors. Correlation

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CHAPTER 1. INTRODUCTION 2

between market variables, and non-normal behaviour such as fat tails are automatically incorporated in the estimate. We apply different versions of this Historical Value at Risk (HVaR).

The idea behind HVaR is to draw scenarios from the historical distribution of relevant risk factors and assess the influence of their behaviour on the price of an instrument. To obtain an estimate for the profit and loss (P&L) distribution we take a set of recent historical changes, or shocks, in the market variables. The first P&L scenario assumes that tomorrow’s shock in the market variable will be the same as the first shock in this set. Each historical shock is applied to the current value of that market variable and the instrument is priced with this hypothetical value of the market variable. The difference with the current price of the instrument is a P&L scenario. The set of used historical shocks is a rolling window, each day the window is moved forward one observation.

While the concept of HVaR is very intuitive, it has two major drawbacks. First, extreme percentiles of the distribution are hard to estimate with few data. Thus, our window of historical shocks should be large enough to avoid estimation problems. Second, the implicit assumption is that within the window the observations have the same distribution. This does not allow for time-varying volatility. Thus, the window should not be too large to avoid including observations outside the current volatility cluster. Clearly, satisfying both requirements is not trivial.

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CHAPTER 1. INTRODUCTION 3

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Chapter 2 Problem formulation

In this chapter we describe the research questions more precisely and set out the scope and methodology of the research.

As mentioned in the introduction, the focus of this research will be on historical simulation Value at Risk (HVaR) of floating rate government bonds. Reasons for this topic are the following. First, HVaR is becoming the standard VaR method within (Dutch) banks. Furthermore, prices of Italian floating rate government bonds were very volatile during the financial crisis of 2008, even though floaters are known to be relatively stable compared to fixed rate bonds. This is a good reason to study this particular instrument and estimate its risk under different HVaR approaches. In particular, we will give an answer to the following research question:

Which approach of calculating historical simulation Value at Risk performs best on Italian floating rate government bonds in terms of accuracy?

To this end, we start with a theoretical part where we introduce the relevant notions, models, and methods, and examine their properties. The following topics will be discussed:

• the different approaches of historical simulation VaR available and their (dis)advantages; • how we test and compare the performance of different HVaR approaches;

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CHAPTER 2. PROBLEM FORMULATION 5

• which models we use to estimate conditional volatility and correlation of risk factors;

• what are the risk factors of floating rate notes;

• how we calculate these risk factors from market data. The theoretical part will cover the chapters 3, 4 and 5.

This theory is applied in Chapter 6, the empirical part. We answer the research question by performing a backtest on Italian floating rate government bonds. First, we calculate the values of the relevant risk factors from market prices. Second, we fit different conditional volatility and correlation models to the data. Third, for each day in the sample period, we calculate HVaR under different approaches and check whether the observed loss exceeded the VaR or not. With this information, we can test the results on properties a good VaR method should have. Furthermore, we look at the precision of the estimates by constructing 95% confidence bounds for VaR.

We restrict the research to Italian floating rate government bonds, so we do not include any other financial instruments or indices. Furthermore, we only consider VaR methods within the class of historical simulation approaches. This does not imply other (parametric) VaR methods or non-VaR methods might not work well.

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Chapter 3 Historical simulation Value at Risk

In the first section of this chapter we explain how to estimate historical simulation Value at Risk (HVaR) in the simplest way. Two potential improvements are discussed in Section 3.2. For a more extensive discussion we refer to the literature, e.g. Man-ganelli and Engle (1998), Holton (1998) or Hull (2006, Chap. 18). In Section 3.3 we discuss several tests that can be used to compare and evaluate the performance of VaR approaches in terms of accuracy.

3.1 Simple HVaR

In this section we give a definition of VaR and discuss the implementation of the historical simulation approach.

The 100 · x% 1-day VaR is defined as

P rt(∆Pt+1≥ −V aRx) = x

where Pt is the value of the instrument or portfolio at time t (measured in days) and

∆Pt+1 = Pt+1 − Pt, tomorrow’s profit or loss (P&L). We focus on 1-day VaR figures

at 95% and 99% confidence levels, so either x = 0.95 or x = 0.99. Today’s value Pt is

known, but at time t, Pt+1 is a random variable. It depends on a set of J underlying

random variables, its risk factors. These factors can be interest rates, exchange rates, equity prices etcetera. Given a set of realizations of these J factors and a price function

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CHAPTER 3. HISTORICAL SIMULATION VALUE AT RISK 7

f , the value of the instrument at time t is

Pt = f (vt1, . . . , vtJ) J ≥ 1

where vtj is the value of the jth risk factor at time t.

In HVaR, we draw scenarios from the historical non-parametric distribution of the risk factors to obtain an estimate of the P&L distribution of the instrument. To calculate HVaR, we take the I most recent daily historical shocks in all risk factors. So, for i = 1, . . . , I and j = 1, . . . , J we calculate

∆vij = vt+1−i,j− vt−i,j

where ∆v1j = vtj− vt−1,j is the most recent shock.1 The result is the I × J matrix of

risk factor shocks ∆V.

One by one, the sets of J shocks are applied to the current values of the risk factors to obtain I scenarios. Since in each scenario the J shocks occured on the same day, correlation is automatically incorporated. Tomorrow’s value of the instrument in the ith scenario is

ˆ

Pt+1,i= f (vt1+ ∆vi1, . . . , vtJ + ∆viJ), i = 1, . . . , I (3.1)

and the ith P&L scenario is

∆ ˆPt+1,i= ˆPt+1,i− Pt, i = 1, . . . , I.

Next, we sort the estimated P&L vector, ∆ ˆPt+1 (with length I), and take the negative

of the 100(1 − x)% worst value as our 1-day 100 · x% VaR. When the desired quantile is between two adjacent points, we use linear interpolation.

Note that we can easily distinguish between different sources of risk. Suppose we would like to estimate the risk associated with the jth risk factor. To calculate ˆP_t+1,ij ,

1_{We will be dealing with bond prices where underlying risk factors are interest rates for which we}

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we shock only the jth variable in (3.1) and keep the other J − 1 risk factors constant. Furthermore, if we want to calculate interest rate risk (or exchange rate risk), we shock all interest rate (or exchange rate) risk factors and keep the others constant.

3.2 Weighted HVaR

So far we discussed the simplest HVaR approach where each scenario is equally impor-tant. In this section we discuss potential improvements of this approach.

In simple HVaR, the choice of window length I is subject to competing demands. We want to use as much data as possible to be able to estimate VaR precisely, so I should be large. At the same time we want to use data that are iid. This assumption is likely to be violated if we incorporate relatively old data, for example because volatility has changed. In Section 4.1 we show that heteroskedasticity (non-constant volatility) increases the kurtosis of a time series. If we wrongly assume data are iid, we mistakenly take non-constant volatility for fat tails (leptokurtosis). This makes simple HVaR tend to overstate the effect of leptokurtosis.

3.2.1 Age weighted HVaR

A pragmatic solution is the age weighted approach of Boudoukh et al. (1998). Their idea is intuitively appealing, since in this approach yesterday’s shock is more important than a shock one year ago. Exponentially declining weights are applied to the scenarios, where the weight of scenario i is

ωi =

1 − λ 1 − λIλ

i−1_, _{i = 1, . . . , I}

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CHAPTER 3. HISTORICAL SIMULATION VALUE AT RISK 9 scenarios we get            ∆ ˆPt+1,1 ω1 .. . ... ∆ ˆPt+1,i ωi .. . ... ∆ ˆPt+1,I ωI            →            ∆ ˆPt+1,[1] ω[1] .. . ... ∆ ˆPt+1,[i] ω[i] .. . ... ∆ ˆPt+1,[I] ω[I]           

where, using order statistics, ∆ ˆPt+1,[i] is the ith worst P&L scenario and ω[i] is the

weight corresponding to the ith worst P&L scenario. Next, starting with the lowest P&L, sum the weights until (1 − x) is reached. That is, we solve

k

X

i=1

ω[i] = 1 − x

for k. The VaR estimate is the negative of the P&L corresponding to the last weight in this sum:

V aRx = −∆ ˆPt+1,[k].

If there is no integer k for which the equation exactly holds, we find the smallest k for which

k

X

i=1

ω[i] ≥ 1 − x

and then linearly interpolate ∆ ˆPt+1,[k−1]and ∆ ˆPt+1,[k]. Note that as λ → 1, the weights

converge to 1/I (by l’Hopital’s rule) and we are back in the case of simple HVaR, where all scenarios are equally weighted.

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Figure 3.1: Weights in simple and age weighted HVaR with λ = 0.97 and I = 260

3.2.2 Volatility weighted HVaR

Another, theoretically more appealing improvement of simple HVaR is the volatility weighted approach suggested by Hull and White (1998). Volatilities of market variables tend to cluster, so if today’s volatility is high, we expect a high volatility tomorrow as well. If current markets are very volatile while they were relatively calm in the past, the shocks observed in the past probably understate the shocks we expect to see in the near future. Instead of assuming that all observed shocks are iid, the underlying assumption in the volatility weighted approach is that standardized shocks are iid. The observed shocks are scaled by the ratio of the current volatility to the volatility at the time the shock occured. The ith scaled shock in risk factor j is

∆v0_ij = σ1j σij

∆vij, i = 1, . . . , I, j = 1, . . . , J. (3.2)

where σij is the historical estimate of the daily volatility and σ1j is the current volatility

estimate. Volatility updating schemes such as GARCH and EWMA can be used to obtain these volatility estimates. These univariate models will be discussed in the Sec-tion 4.1. To estimate volatility weighted HVaR, the volatility adjusted shocks instead of the observed shocks are used to calculate the P&L vector.

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shocks is

∆v_i0 = Σ1/2₁ Σ−1/2_i ∆vi, i = 1, . . . , I (3.3)

where Σi is the variance-covariance matrix on day i and Σ1 is the most recent

variance-covariance matrix. With Σ−1/2_i we denote the matrix square root of the inverse of Σi. For this, we need a multivariate model that estimates conditional volatilities and

covariances simultaneously. Multivariate GARCH models such as dynamic conditional correlation (DCC) can be used. We discuss the properties of this particular model in Section 4.2.

3.2.3 Differences between age and volatility weighted HVaR

An important difference between the age and volatility weighted approaches is the reaction to a large positive shock. Age weighted HVaR will not change due to this observation since it only takes into account observations in the lower tail. To the contrary, volatility weighted HVaR will increase. Market volatility increases due to this positive shock so all shocks are scaled upwards, also the negative ones, which become more negative.

Another difference is the range of HVaR. In age weighted (and simple) HVaR the VaR estimate will always be less than or equal to the negative of the worst observed P&L. This is not necessarily the case in volatility weighted HVaR. Since scaled shocks are used instead of observed shocks the VaR estimate can be much bigger than the negative of the worst observed P&L.

3.3 HVaR performance tests

In the previous sections, we introduced simple, age weighted and volatility weighted HVaR. In this section we discuss what properties a good VaR model should have and how we statistically test the HVaR approaches on these properties.

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every 100 days. If the VaR estimate is exceeded we speak of a tail event. The sec-ond property is serial independence of tail events: the probability of having a VaR exceedance tomorrow should not depend on whether or not we have a VaR exceedance today. This means that the VaR estimate avoids clustering of tail events, so-called bunching. If both properties hold, tail events are serially independent and the proba-bility of a tail event equals 1 − x, the VaR model has correct conditional coverage. A VaR model that has correct conditional coverage is called accurate.

In Section 6.3 we test the HVaR approaches on accuracy. Thus, we test them on unconditional coverage, serial independence and conditional coverage. In the remainder of this section we give the relevant test statistics as mentioned by Kupiec (1995), Lopez (1999) and Boudoukh et al. (1998).

3.3.1 Correct unconditional coverage

Under the assumption that the VaR estimate is accurate, tail events can be modeled as independent draws from a binomial distribution with parameters (1 − x) and N . The first property that a VaR estimate should have, is that it has correct unconditional coverage. That is, the exceedance rate of the model should not be significantly different from the desired exceedance rate 1 − x. Let X be the true, unknown confidence level of the HVaR model, so (1 − X) is the true exceedance rate.

A likelihood ratio test can be used to test for unconditional coverage with null hypothesis

H₀uc: E(X) = x

against the alternative

H_Auc : E(X) 6= x.

We denote the number of tail events observed in the sample by n. The observed proportion of tail events in the sample, 1 − ˆX = n/N , is used as an estimator of E(X). The test statistic is (see Kupiec (1995))

LRuc = 2 log

(1 − ˆX)nXˆN −n− 2 log (1 − x)n_xN −n a ∼ χ2

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The asymptotical distribution of the test statistic under the null hypothesis is χ2 _with

one degree of freedom. If the hypothesis of correct unconditional coverage is rejected, we conclude that the used HVaR method under- or overestimates VaR. Note that if a VaR estimate has correct unconditional coverage it is still possible that the VaR estimate allows for bunching. Another test is needed to test this property. Christoffersen (1998) introduced a test of conditional coverage of a VaR estimate. This jointly tests for unconditional coverage and serial independence of tail events.

3.3.2 Correct conditional coverage

If tail events bunch up, the VaR estimate does not correctly adopt to changing circum-stances. If a tail event happens today, tomorrow’s VaR estimate has to adjust to make sure the probability of another tail event tomorrow is exactly (1 − x). Let Ix be the

indicator vector where Ix(t) = 1 if a tail event happened on day t and zero otherwise.

No bunching requires that the elements I(t) are serially independent.

At time t, we are either in state 1 (VaR exeendance) or 0 (no VaR exceedance). For i, j ∈ {0, 1} let Sij denote the number of observations in state j after having been in

state i on the previous day and πij = Sij

Sii+Sij are the corresponding probabilities. We

test the null hypothesis

H₀ind: π01 = π11

against the alternative

H_Aind: π016= π11.

Christoffersen (1998) derives the appropriate likelihood ratio test statistic. The likeli-hood function under the alternative is

LA= (1 − π01)S00π01S01(1 − π11)S10π11S11

while under the null hypothesis it equals

L0 = (1 − π)S00+S10πS01+S11

with π = S01+S11

N . This gives the likelihood ratio test statistic of serial independence as

LRind= 2 log LA− 2 log L0 a

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The test for conditional coverage tests jointly for unconditional coverage and serial independence. The null hypothesis is

H₀cc : π01= π11= 1 − x.

According to Christoffersen (1998), the test statistic is LRcc = LRuc+ LRind

a

∼ χ2 2.

3.3.3 Mean absolute percentage error

Furthermore, we perform a test for both unconditional coverage and bunching as sug-gested by Boudoukh et al. (1998). They calculate a rolling measure of the absolute percentage error. For each consecutive period of 100 days the difference between the number of observed tail events and the number of expected tail events 100(1 − x) is calculated. The Mean Absolute Percentage Error (MAPE) is the mean of the absolute differences. Let Jk be a vector of length T containing zeros except for 100 elements

that equal 1 starting at element k. Mape is given by

M apex = 1 N − 99 N −99 X k=1 |J0_kIx− 100(1 − x)|.

A lower Mape value indicates better performance.

In Section 6.3 we perform a backtest of the HVaR approaches. We evaluate the performance of these methods using the tests for unconditional coverage, serial inde-pendence, conditional coverage and Mape.

3.4 VaR confidence bounds

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Value at Risk is an extreme quantile of the (unknown) P&L distribution. The vector ∆ ˆPt+1 contains I drawings from the P&L probability distribution. The assumption

behind HVaR estimation is that these I drawings are independent and identically distributed. Note that this holds in the simple HVaR approach as well as in the volatility weighted approach. Although these approaches differ in the assumptions underlying the calculation of P&L scenarios, both assume the obtained I P&L scenarios are iid. This does not hold for age weighted HVaR, so the bootstrap algorithm of this section can not be applied to age weighted HVaR.

From the I iid observations we can easily calculate the HVaR estimate. To obtain a VaR confidence interval we randomly draw with replacement from the P&L scenarios. One bootstrap sample is obtained by independently drawing I times (with replacement) from the I iid scenarios. From this bootstrap sample we calculate the 100 · x% 1-day VaR. Doing this m times (m large) gives us m 100 · x% 1-day VaR estimates. Stated otherwise, it gives us an estimate of the distribution of 100 · x% 1-day VaR. From this sample of estimates we cut off the lowest and highest 2.5% to obtain a 95% confidence interval.

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Chapter 4 Volatility updating schemes

In the previous chapter the volatility weighted HVaR approach is discussed. If we scale each risk factor shock individually, as in (3.2), we need a univariate volatility up-dating scheme. Since volatilities tend to cluster, we want to estimate volatility levels conditional on recent data and update the volatility on a daily basis. To do this, we use both generalized autoregressive conditional heteroscedasticity (GARCH) and ex-ponentially weighted moving average (EWMA). These univariate models are discussed in Section 4.1. If we scale all risk factor shocks simultaneously, as in (3.3), we also incorporate conditional covariances. Several multivariate GARCH models have been introduced to estimate time-varying covariance matrices. We use the multivariate dy-namic conditional correlation (DCC) model introduced by Engle (2002). Its properties are discussed in Section 4.2.

4.1 Univariate models

We apply both the EWMA and GARCH models to calculate daily volatility estimates. In the EWMA model, today’s volatility estimate is

σ_t2 = (1 − λ)∆v_t−12 + λσ_t−12 , 0 < λ < 1. (4.1)

Today’s volatility is positively correlated with yesterday’s volatility and yesterday’s risk factor shock. Note that the bigger λ is, the more persistent high volatilities are.

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CHAPTER 4. VOLATILITY UPDATING SCHEMES 17

On the other hand, a bigger (1 − λ) means that the reaction of the volatility after a market shock will be bigger. Similar to Hull and White (1998), we use λ = 0.94.

4.1.1 GARCH

The GARCH(p, q) model suggested by Bollerslev (1986) is given by ∆vt = µ + t t = σtzt, zt iid ∼ (0, 1) (4.2) σ2_t = ω + q X i=1 αi∆vt−i2 + p X i=1 βiσt−i2 .

Under the restrictions ω > 0, αi ≥ 0, i = 1, . . . , q, βi ≥ 0, i = 1, . . . , p and

Pq

i=1αi+

Pp

i=1βi < 1 the conditional variances are positive and the variance process is stationary.

In practice, the GARCH(1,1) model is widely used. In this particular case the variance equation reduces to

σ2_t = ω + α∆v2_t−1+ βσ2_t−1. (4.3)

Maximum likelihood can be used to estimate the parameters. In order to do this, a distribution must be specified. The most generally applied assumption is that the distribution of z is standard normal. We will refer to this as Gaussian-GARCH, or g-GARCH. Even if the real distribution is not standard normal, maximum likelhood estimators are consistent (quasi-maximum likelihood) but obtained standard errors are not efficient. Bollerslev and Wooldridge (1992) derive robust standard errors in this case. Since financial time series are usually fat-tailed, the normality assumption is likely to be violated. As an alternative, a more heavy-tailed distribution such as the student’s t can be used. We will refer to this as t-GARCH.

Note that if we restrict the parameters to ω = 0 and α = 1 − β the GARCH(1,1) model reduces to EWMA. In contrast to the EWMA model, the GARCH model does not require the sum of the coefficients to equal one. If α + β < 1, the GARCH variance process is mean-reverting to variance level σ2 ₌ ω

1−α−β. If α + β ≥ 1, the variance

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An extension of the traditional GARCH model is the asymmetric GARCH model, also known as GJR-GARCH as it was introduced by Glosten, Jagannathan, and Runkle (1993). This model allows positive and negative shocks to have different impacts on the conditional variance. Let Iti = 1 if ∆vt−i < 0, then the variance equation of the

asymmetric GARCH model is

σ_t2 = ω + q X i=1 αi∆v2t−i+ o X i=1 γiIti∆v2t−i+ p X i=1 βiσ2t−i.

We will refer to this model as the GARCH(p, o, q) model, where o represents the order of asymmetric innovations. The restriction ω > 0, αi ≥ 0, i = 1, . . . , q, and βi ≥ 0, i =

1, . . . , p still apply. Furthermore, it is required that αi + γi > 0, i = 1, . . . max(q, o),

and Pq

i=1αi + 0.5

Po

i=1γi +

Pp

i=1βi < 1. The GARCH(p, o, q) model coincides with

the GARCH(p, q) model if o = 0.

Another variation on the traditional GARCH model is the absolute GARCH model introduced by Taylor (1986) and Schwert (1989). In this case, conditional standard deviation is a function of past absolute shocks. By writing the model in terms of con-ditional scale instead of concon-ditional standard deviation, the usual GARCH restrictions ensure positivity and stationarity. Conditional scale is defined by

νt = Et−1|t| = σtE|zt|.

For example, if the zt are normally distributed, we have E|zt| =p2/π. The volatility

equation of the absolute GARCH model in terms of conditional scale is given by

νt = ω + q X i=1 αi|∆vt−i| + p X i=1 βiνt−i, σt = νt E|zt|.

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4.1.2 Kurtosis of a GARCH process

To estimate VaR, we model the tail of the P&L distribution. Kurtosis is a quantity that is closely related to the tail of a distribution. Kurtosis, the standardized fourth moment, is a measure of peakedness of the probability distribution. High kurtosis means most of the variance is due to a few extreme observations. The excess kurtosis is defined as the kurtosis minus 3, the kurtosis of the normal distribution. A process is heavy-tailed, or leptokurtic, if the excess kurtosis is positive. Bollerslev (1986) shows that the normal GARCH(1,1) process is heavy-tailed: if 3α2 _{+ 2αβ + β}2 _{< 1 the}

(unconditional) fourth order moment of t exists and the excess kurtosis equals

κ = E( 4 t) (E(2 t))2 − 3 = 6α 2 1 − 3α2_{− 2αβ − β}2 > 0.

By assumption, this is positive and we conclude that the GARCH(1,1) process is heavy-tailed. Stated otherwise, volatility clustering increases the kurtosis of a time series.

Bai, Russell, and Tiao (2003) show that this result holds in general for a GARCH(p, q) process, also if z is not normally distributed. They derive that the kurtosis of a se-ries depends positively on the kurtosis of z and the kurtosis induced by time-varying volatility. This implies that if a series with time-varying volatility is assumed to be iid, the leptokurtosis of the distribution is overstated. This is a disadvantage of simple HVaR and suggests the volatility weigted approach is more suitable. We come back to this issue in Chapter 6.

In Chapter 6 we apply EWMA, g-GARCH and t-GARCH models to shocks in risk factors of floating rate government bonds. The obtained time-volatilities are used to estimate volatility weighted HVaR as suggested by Hull and White (1998) and discussed in Section 3.2.

4.2 Multivariate models

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approach of Duffie and Pan (1997). J risk factor shocks can be scaled simultaneously as in (3.3). Note the difference with (3.2) where only individual volatilities are used so univariate GARCH models suffice.

In multivariate modeling of risk factor shocks we assume ∆vt = µ + t t = Σ 1/2 t zt zt iid ∼ N (0, I). (4.4)

So residuals are conditionally multivariate normal with time-varying covariance matrix Σt of size J × J . This is more general than the univariate GARCH(p, q) model (4.2)

where only individual residuals are modeled as being iid with mean 0 and conditional volatility σt. The challenge is to estimate Σt.

Two properties are required: the estimated covariance matrices should be positive semi-definite and the number of parameters should be limited. The second requirement is not trivial. Several models have been introduced where the number of parameters to be estimated grows polynomially in the number of risk factors included. These models are not parsimonious.

4.2.1 Correlation models

An interesting class of models is that of correlation models. The conditional covariance matrix is decomposed in the correlation matrix and a diagonal matrix containing the conditional variances: Σt= D 1/2 ΣtRtD 1/2 Σt

where Rt is the J × J matrix of conditional correlations and DA is the J × J matrix

containing the diagonal elements of matrix A. Thus, D1/2_Σ

t is the matrix with the

conditional volatilities on the diagonal.

One of the parsimoneous models within this class is the constant correlation model of Bollerslev (1990). In this model it is assumed that Rt= R, the correlation matrix is

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Assuming correlation matrix R does not depend on time greatly reduces the number of parameters. However, research (e.g. Engle and Sheppard (2001)) shows that cor-relations between asset returns are not constant over time. Therefore, the model is extended to allow for time-varying correlations.

4.2.2 Dynamic Conditional Correlation

The dynamic conditional correlation (DCC) model of Engle (2002) is one of these models. This model has several advantages. First, it is parsimoneous, the number of parameters grows linearly with the number of risk factors included. Second, it is a flexible model. Different GARCH models are allowed for different risk factors and the correlations are not restricted to being constant. Third, it is attractive from a computational point of view since parameters can be estimated in two steps. In the first step, we fit univariate GARCH models to the risk factors to estimate volatility parameters. In the second step we use standardized residuals from the first step to estimate the parameters of the correlation matrix. The advantage is that this does not require nonlinear optimization of a large number of parameters at the same time.

In the DCC model the correlation matrix Rt is modelled by

Rt = Q∗−1t QtQ∗−1t

Qt = (1 − a − b) ¯Q + aztz0t+ bQt−1 (4.5)

Q∗_t = diag(√q11,t, . . . ,

√ qJ J,t)

where ¯Q is the unconditional covariance matrix of the standardized residuals of the first step. The covariance matrix is positive definite under the usual GARCH restrictions and a, b > 0 and a + b < 1.

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Chapter 5 Floating rate notes

This chapter starts with introducing several concepts and definitions of a bond market. In Section 5.2 we derive a pricing formula for default-free floating rate notes (floaters). Section 5.3 is devoted to analysing the risk factors underlying defaultable floaters. In the last part we discuss how to extract the risk factors from market prices of floaters.

5.1 Bond market definitions

This section summarizes the results from Bj¨ork (2004, Chap. 20). We define a zero coupon bond as a contract that guarantees the holder 1 monetary unit at maturity, risk-free. The price at time t of the zero coupon bond maturing at time T is denoted as pt(T ). From now on we refer to such a bond as a T -bond. Its time to maturity

is τ = T − t. We assume pt(t) = 1 to avoid arbitrage. Furthermore, we assume zero

coupon bonds are traded for all maturities T > 0.

Spot and forward interest rates can be extracted from this market with bond prices. The extraction procedure is demonstrated with the construction in Table 5.1, which costs nothing at time t. At time T1 1 is invested and at T2 the construction pays

pt(T1)/pt(T2). So, it gives us a deterministic rate of return, or risk-free interest rate

over the future period [T1, T2]. This is defined as the forward rate. The continuously

compounded forward rate over the period [T1, T2] is defined as the solution to the

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CHAPTER 5. FLOATING RATE NOTES 24

equation

exp(rc_t(T1, T2)(T2− T1)) =

pt(T1)

pt(T2)

and the simple forward rate solves

1 + r_ts(T1, T2)(T2− T1) =

pt(T1)

pt(T2)

.

If t = T1 we are not dealing with forward rates but with spot rates. Solving the above

equations leads to the following interest rate definitions.

• The continuously compounded forward rate for [T1, T2] contracted at t is defined

as rc_t(T1, T2) = 1 T2− T1 lnpt(T1) pt(T2) . (5.1)

• The continuously compounded spot rate for the period [t, T ] is defined as r_tc(T ) = −1

τ ln pt(T ). (5.2)

• The simple forward rate for [T1, T2] contracted at t is defined as

r_ts(T1, T2) = 1 T2− T1 pt(T1) pt(T2) − 1 . (5.3)

• The simple spot rate for the period [t, T ] is defined as rs_t(T ) = 1 τ 1 pt(T ) − 1 . (5.4)

For fixed t, the function that gives rc_t(T ) for different T is called the (continuously compounded zero coupon) yield curve. We can use this function to discount future cash flows to today. From (5.2) follows that the price and yield of a zero coupon bond are inversely related. The higher the price, the lower the yield and vice versa.

As the name suggests, fixed rate bonds pay a coupon c which is fixed in advance. Let T0 be the issue date. At each coupon date Tj (j = 1, 2, . . . , J ) the coupon is paid.

At the maturity date TJ the principal K is repaid to the bondholder together with

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CHAPTER 5. FLOATING RATE NOTES 25 Table 5.1: A portfolio constructed at time t paying a riskless rate of interest over the future time interval [T1, T2]

time action cash flow

t sell 1 T1-bond pt(T1)

t buy pt(T1)

pt(T2) T2-bonds −pt(T1)

T1 T1-bond matures -1

T2 T2-bonds mature p_pt_t(T_(T1₂)₎

value can be given in terms of prices or continuously compounded yields of zero coupon bonds. The price of a fixed rate bond equals the sum of all discounted cash flows

Pt = Kpt(TJ) + J X j=1 c · pt(Tj) = K exp (−r_tc(TJ)τJ) + J X j=1 c exp (−rc_t(Tj)τj) . (5.5)

In this expression T1 should be interpreted as the date of the next coupon, not as the

date the bond pays its first coupon. Otherwise, (5.5) has no meaning when t > T1.

The sum of all discounted cash flows in (5.5) is the so-called dirty price of a bond. Part of this value is interest that has accumulated since the issue date or previous interest payment. This accrued interest is calculated as the part of the coupon period that has already passed times the next coupon.1 To this end, we define δ = Tj − Tj−1 as the

time between two coupon dates. Once the accrued interest is subtracted from the dirty price, we obtain the clean price of the bond

Pt= K exp (−rct(TJ)τJ) + J X j=1 c exp (−rc_t(Tj)τj) − δ − τ1 δ c.

If the clean price of the bond changes this is due to economic factors like changing interest rates. The value of the dirty price also changes every day since the time that has passed since the previous interest payment changes.

1_{Theoretically, calculating accrued interest linearly creates arbitrage opportunities. However, in}

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Note that the inverse relation between price and yield of zero coupon bonds also holds for fixed rate bonds. So, the value of a fixed rate bond decreases if market interest rates rise.

5.2 Default-free floating rate notes

In the previous section we introduced several concepts relevant to bond markets. Now, we use these definitions to derive the value of a default-free floater. A floating rate note, or floater, differs from a fixed rate bond in that the coupon is reset every coupon period rather than it is fixed in advance. The resetting is linked to a financial benchmark, the reference rate. This is often a market interest rate such as Euribor or a T-bill rate. With a default-free floater we mean a floater that does not suffer from any issuer specific default risk or product specific liquidity risk. For example, consider a default-free floater maturing in 2 years, with principal K = 1, paying semi-annual coupons. For simplicity, assume the bond is issued today, so T0 = t. In general, the jth coupon

equals

cj = rTsj−1(Tj)K(Tj − Tj−1), j = 1, . . . , J. (5.6)

In this specific case, Tj− Tj−1 = 1₂ (year) and K = 1. Since the first coupon is given by

today’s T1-spot rate, it equals c1 = 1₂rts(T1). The other coupons are not known because

they depend on future interest rates. Immediately after c1 is paid, c2 is fixed at the

current interest rate and is paid out at T2. The coupons c3 and c4 are determined in a

similar way. At T4, also the principal is repaid to the bondholder.

5.2.1 Pricing formula

Since future coupons are uncertain, we can not use the fixed rate bond price formula (5.5) to value a floater. Fortunately, there is another way to determine the value of a default-free floater. The following derivation is taken from Bj¨ork (2004, Chap. 20). We assume coupons are equally spaced in time, so Tj − Tj−1 = δ. Without loss of

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Let T0 be the issue date of the floater. Below, we show that the value of a

default-free floater at time t ≤ T0 is

Pt= pt(T0). (5.7)

Note that if t = T0, the value of the floater is 1. This implies that the value of a

default-free floater equals par at reset dates.

We show (5.7) holds by replicating the coupons with zero coupon bonds to determine the coupons’ value at time t. The value of the jth coupon is given in (5.6). Using the definition of the simple spot rate, (5.4), and assuming equally spaced coupons and a unit principal, we find

cj = 1 δ 1 pTj−1(Tj) − 1 1 · δ = 1 pTj−1(Tj) − 1, j = 1, . . . , J.

For each j ∈ {1, . . . , J }, we can determine the value of cj today by replicating it. By

definition, 1 at time Tj equals pt(Tj) at time t. Taking the steps shown in Table 5.2

exactly replicates coupon at the cost of pt(Tj) − pt(Tj−1) today. Thus, the value today

of obtaining cj at Tj is

pt(Tj−1) − pt(Tj), j = 1, . . . , J.

As the value of the floater is the sum of today’s value of all future coupons this is given by Pt = pt(TJ) + J X j=1 pt(Tj−1) − pt(Tj) = pt(T0).

All terms cancel against each other except for one. We have shown that a default-free floater can be viewed as a zero coupon bond maturing at the issue date.

Note that the result in (5.7) is not restricted to valuing a floater only before its issue date. Consider a floater between two coupon dates. The next coupon c1 is

already known since it is fixed at the previous coupon date. We can treat it as c1 zero

coupon bonds. All remaining (unknown) cash flows can be treated as a floater issued at time T1. Together, the value at time t ≤ T1 is

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CHAPTER 5. FLOATING RATE NOTES 28 Table 5.2: Replicating the jth coupon of a default-free floater

time action cash flow

t sell 1 Tj-bond and buy 1 Tj−1-bond pt(Tj) − pt(Tj−1)

Tj−1 Tj−1-bond matures 1

Tj−1 buy _p 1

Tj−1(Tj) Tj-bonds -1

Tj Tj-bonds mature _p 1

Tj−1(Tj) − 1

Similar to the discussion in the previous section, this is the so-called dirty price of the floater. The clean price of the floater is

Pt= (1 + c1)pt(T1) −

δ − τ1

δ c1.

From now on, when we refer to the value of a floater, we refer to its clean price. We can also replicate a future coupon using the forward rate (which is a risk-free return over a future period). In (5.6), we substitute the future spot rate by the simple forward rate (5.3). This way we get ˆctj, a forecast of the coupon ctj which we call the

forward rate forecast. The value at time t of this jth forecasted coupon is ˆ ctjpt(Tj) = δrst(Tj−1, Tj)pt(Tj) = δ1 δ pt(Tj−1) pt(Tj) − 1 pt(Tj) = pt(Tj−1) − pt(Tj), j = 1, . . . , J.

This is exactly equal to the previously derived value where we used the spot rate. This is not surprising, since spot rates and forward rates are directly related. In the remainder of this thesis we forecast future coupons using forward rates.

5.2.2 Interest rate risk

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a floater is relatively small since shifts in yields after the next coupon payment do not influence the price.

Note that interest rate risk decreases as we get closer to the next coupon date. The only risk factor is rc₁ in pt(T1) = exp(−r1cτ1). If we take the limit τ1 → 0, shocks in

rc

1 do not change the value of this expression. Therefore, interest rate risk converges

to zero. Stated otherwise, just before the coupon is paid and the next coupon is fixed interest rate risk does not exist.

Immediately after the fixing of the next coupon, interest rate risk reappears. Shocks in rc

1 have a relatively big impact on the price of the floater because τ1 = δ, which is

the maximum value of τ1.

5.3 Defaultable floating rate notes

In the previous section we showed that the value of a default-free floater only depends on the τ1-yield and introduced the concept of interest rate risk. In practice, bonds are

never default-free and investors demand higher yields than the risk-free yield. In the coming part we focus on defaultable floaters and introduce the concept of credit risk. The premium on top of the risk-free rate is a compensation for issuer specific default risk and product specific liquidity risk. Default risk is the risk the bond issuer does not pay its coupons or repay the principal. Liquidity risk is the risk that the bond can not be sold quickly enough in the market to prevent a loss.

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Eu-CHAPTER 5. FLOATING RATE NOTES 30

rozone countries and Germany. But they also derived that liquidity risk has reduced since the start of the EMU. A factor contributing to default risk is for example the debt-GDP ratio of a country. If a countries debt is small relative to total EU debt this is a source of liquidity risk. Countries with a larger share of total EU debt pay lower rates than countries with a smaller share. From now on, we refer to the risk premium investors demand on top of the risk-free yield as the credit spread.

5.3.1 Credit Spread

Issuers of floaters pay a fixed premium on top of the reference rate to compensate for the default and liquidity risk, the so-called quoted margin m. For example, the coupon equals the reference rate plus 30 bp. In general, the jth coupon equals

cj = (rsTj−1(Tj) + m)δK, j = 1, . . . , J.

The quoted margin does not necessarily equal the credit spread of this specific instru-ment. The credit spread is not directly observable but we can derive it from market prices. To this end we derive the implied floater yields r_t∗. That is, we want to find the combinations (T, r_t∗(T )) such that for all traded floaters of a specific issuer

Pt = c1exp (−rt∗(T1)τ1) + K exp (−r∗t(TJ)τJ) + J X j=2 ˆ ctjexp (−r∗t(Tj)τj) − δ − τ1 δ c1 (5.8)

where Ptis the observed clean market price and ˆctjis the forward rate forecast of coupon

j. The last term in the expression is the accrued interest that has to be subtracted from the discounted value of all future coupons to equal the clean market price. Since the issuer pays a quoted margin on top of the reference rate, the forecasted jth coupon is

ˆ

ctj = (rst(Tj−1, Tj) + m) δK, j = 1, . . . , J. (5.9)

In Section 5.4 we explain how to obtain the implied yield curve from a set of market prices. The implied yields r∗ will be higher than the risk-free rates since it also incor-porates default and liquidity risk. The difference between the implied yield curve and risk-free yield curve is the implied credit spread s∗:

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This is the extra yield investors demand for the risk of this specific instrument of this issuer. If the risk assessment made by investors differs from the quoted margin the issuer offers (s∗ differs from m) the price of the floater is not equal to par at reset dates.

5.3.2 Interest rate and credit risk

In this chapter, we have discussed two sources of risk that influence the price of a floater: interest rate risk in Section 5.2 and credit risk in this section. With interest rate risk we mean a potential change in the price of the instrument because of a change in the yield curve. Credit risk is a potential change in the price caused by a change in the implied credit spreads. In Section 5.2 we showed that the only source of interest rate risk is the τ1-risk-free yield where τ1 is the remaining time until the next coupon date.

The sources of credit risk on the other hand, are the τj-credit spreads for j = 1, . . . , J .

To give an example of the impact of both sources of risk, consider a floater paying semi-annual coupons, priced at par with 4 years and 9 months until maturity. Assume the risk-free yield curve is flat at 3.5% and the credit spreads and quoted margin equal 30 bp so the implied yield curve is flat at 3.8%. A parallel shift of 10 bp in the risk-free curve causes a -0.03% change in the price of the floater, while a parallel shift in the spread curve of 10 bp causes a -0.41% change in the price. The difference is caused by the fact that if market interest rates rise, future coupons rise as well. Only the change in the τ1-yield matters since this coupon is already fixed while the discount rate

varies. However, if the credit spread increases, coupons do not change since the quoted margin is fixed. A shift in the spread curve directly leads to a decrease in the price of the floater. This example illustrates that the impact of shifts in the risk-free curve is relatively small while the impact of shifts in the credit spreads is substantial.

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j = 1, . . . , J . The calculation of the daily absolute shocks in these risk factors is straightforward. These shocks are then applied to the current levels of the risk factors to obtain the P&L vector from which quantiles can be estimated. We distinguish between interest rate risk, credit risk and total risk.

The interest rate risk scenarios are calculated by shocking only the τ1-yield. Note

that we first discount all future coupons to time T1. If the credit spreads equal the

quoted margin, this equals K, the principal. Next, we discount this value and the next coupon c1 to today using the shocked τ1-yield. The ith hypothetical value of the floater

tomorrow due to interest rate changes is ˆ P_t+1,iIRR = " c1+ K exp (−rt∗(T1, TJ)(TJ − T1)) + J X j=2 ˆ ctjexp (−r∗t(T1, Tj)(Tj − T1)) # · exp (−(r_t∗(T1) + ∆ri(T1))τ1) − δ − τ1 δ c1, i = 1, . . . , I. (5.10)

The term r∗_t(T1, Tj) for j = 2, . . . , J is the time t discount rate between T1 and Tj.

Given r_t∗(T1) and r∗t(Tj) it is uniquely determined since to avoid arbitrage we should

have

exp(−r_t∗(T1, Tj)(Tj− T1)) = exp(−rt∗(TJ)τJ) exp(rt∗(T1)τ1).

Solving this shows r∗_t(T1, Tj) =

1 Tj − T1

(r_t∗(Tj)τj − rt∗(T1)τ1) , j = 2, . . . , J.

The credit risk scenarios are calculated by shocking all implied spreads. The ith hypothetical value of the floater due to credit spread changes is

ˆ

P_t+1,iCR = c1exp (−(rt∗(T1) + ∆s∗i(T1))τ1) + K exp (−(rt∗(TJ) + ∆s∗i(TJ))τJ)

+ J X j=2 ˆ ctjexp (−(r∗t(Tj) + ∆s∗i(Tj))τj) − δ − τ1 δ c1, i = 1, . . . , I. (5.11)

Third, in the total risk scenarios all risk factors are shocked simultaneously. We discount future cash flows in two steps, similar to interest rate scenarios. In the first step we discount all future coupons to T1 using the shocked implied credit spreads. To

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exp (−(r_t∗(T1, Tj) + ∆s∗i(T1, Tj))(Tj − T1)) = exp (−(r∗t(Tj) + ∆s∗i(Tj))τj)

· exp ((r∗_t(T1) + ∆s∗i(T1))τ1) , i = 1, . . . , I, j = 1, . . . , J.

Next, we discount this ‘T1-value’ and the next coupon to today using the shocked

τ1-risk-free yield and the shocked τ1-credit spread. This leads to the ith scenario

ˆ P_t+1,iT OT = [c1+ K exp (−(rt∗(T1, TJ) + ∆s∗i(T1, TJ))(TJ − T1)) + J X j=2 ˆ ctjexp (−(rt∗(T1, Tj) + ∆s∗i(T1, Tj))(Tj− T1))] · exp (−(r∗_t(T1) + ∆ri(T1) + ∆s∗i(T1))τ1) − δ − τ1 δ c1, i = 1, . . . , I. (5.12)

In all cases, for i = 1, . . . , I the ith P&L scenario is the difference with the current value of the floater. The above discussion shows the necessity of deriving the implied yield curve from a set of market prices of different maturities. In the next section we will discuss how to do this and what assumptions have been made.

5.4 Bootstrapping a yield curve

Yield curves are necessary input in the calculation of the HVaR of a floater. From a set of market prices of the same instrument with different maturities we can construct an implied yield curve. We apply an iterative procedure known as the bootstrap method for constructing a yield curve (see e.g. Hull (2006, Chap. 4) or Luenberger (1998, Chap. 4)) with linear interpolation and extrapolation. Other potential interpolation methods are discussed in Hagan and West (2006). Given a floater’s implied yield curve and the current risk-free curve, we know the implied credit spread curve which is the difference between the two.

The goal of the bootstrap method is to obtain the implied yield r∗ such that for each bond (5.8) holds. Of each floater we know its clean market price Pt, the next

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the forward rate forecasts of the other coupons ˆctj, j > 1 using (5.9). Without loss of

generality, we assume K = 1 and δ = Tj− Tj−1= 1₂ for simplicity.

We start with sorting the floaters by their time to maturity. For a bond with only one coupon remaining (J = 1) calculating its yield is straightforward since (5.8) reduces to

Pt= (1 + c1) exp (−r∗t(T1)τ1) −

δ − τ1

δ c1

and we can solve this directly for r∗_t(T1). This information can be used for the other

floaters.

Suppose the next bond has two coupon dates remaining where T1 coincides with

the coupon date of the previous floater. The equation we want to solve is Pt= c1exp (−rt∗(T1)τ1) + (1 + ˆct2) exp (−r∗t(T2)τ2) −

δ − τ1

δ c1.

From the previous bond we know r∗_t(T1) so the only remaining unknown is rt∗(T2) which

is easily calculated.

5.4.1 Interpolation and extrapolation

It is not always the case that coupon dates exactly coincide. For example, suppose the third bond has three coupons remaining, where the first two fall exactly between the previous coupon dates as shown in the first graph of Figure 5.1. The plus-signs correspond to yields we obtained from the previous bonds. We need to apply some interpolation rule to determine r∗_t(T1) and r∗t(T2). It is common practice to assume the

yield curve is flat prior to the first point (see e.g. Hull (2006, Chap. 4)). Furthermore, we assume it is linear between two consecutive points. This means we fix r∗_t(T1) and

r∗_t(T2) as is shown in the second graph of Figure 5.1. This leaves r∗t(T3) as the only

remaining unknown.

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Figure 5.1: Linear interpolation of a yield curve

calculate r_t∗(T4) and r∗t(T5) simultaneously. That is, we assume these yields are given

by

r_t∗(T4) = r∗t(T3) + a(T4− T3)

and

r∗_t(T5) = rt∗(T3) + a(T5− T3).

Inserting this into (5.8) leaves one remaining unknown: the slope coefficient a. Once this parameter is determined, the yields are given as is shown in the second graph in Figure 5.2.

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This procedure is continued for all bonds and all points (T, r_t∗(T )) together form the implied yield curve of this instrument on day t. A last assumption we make is that the curve is flat for maturities exceeding the largest maturity of the bonds.2

We apply the bootstrap procedure outlined in this section to Italian floating rate government bonds. We give the results in Section 6.1.

2_{By construction, the yield curve is piecewise linear.} _{This is easy to implement but has the}

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Chapter 6 Empirical results

In this chapter we apply the HVaR methods of Chapter 3 to Italian floating rate government bonds, so-called CCTs (Certificati di Credito del Tesoro). A CCT is a floater issued by the Italian government, paying semi-annual coupons. The reference rate is the 6 month Italian T-bill rate with a quoted margin of 30 bp. The maximum maturity is seven years.

We start with a description of the data we use in Section 6.1. From historical market prices the implied yield curves are bootstrapped following the procedure described in Section 5.4. This leads to a set of daily implied credit spread curves. To calculate the volatility weighted HVaR, we need a volatility updating scheme such as GARCH or EWMA (see Chapter 4). The estimation results of these models are discussed in Section 6.2. In Section 6.3 we compare the results of different HVaR approaches (simple, age weighted and volatility weighted) and test them on accuracy using the tests described in Section 3.3. Finally, in Section 6.4 we construct VaR confidence bounds.

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CHAPTER 6. EMPIRICAL RESULTS 38

6.1 Data

6.1.1 Market data

We use a dataset containing 1002 daily market prices of all CCTs traded between July 20, 2005 and June 12, 2009 obtained from Bloomberg.1 _{Information on past coupons}

is obtained from Bloomberg as well. In Table A.1 a list of all CCTs issued is shown. Six issues that are regarded illiquid are removed from the sample containing 27 CCTs in total. These six issues have substantially lower issued amounts compared to the others (less than e5,000,000,000). Few prices are quoted and/or the bid-ask spread is substantially higher than on the other issues. Furthermore, ten days before maturity of a CCT, this particular CCT is removed from the bootstrap sample.2

In Figure 6.1 the price process of the CCT maturing in November, 2011 is shown. This is the clean price of the CCT as explained in Section 5.1. The coupon reset dates of this CCT are May 1st and November 1st. Clearly, the price is not necessarily equal to par at reset dates as we would expect from a floater. What stands out is the sharp drop of the price in October, 2008 followed by a sharp increase in December. The drop happened at the height of the financial crisis. Several potential explanations exist. First, investors became much more risk averse during the credit crunch and demanded higher yields on the more risky government bonds. Second, liquidity declined which gave investors an incentive to move to more liquid bonds such as German government bonds.

To forecast coupons according to (5.9) the forward curve of the Italian T-bill rate is needed. T-bills have a maximum maturity of two years. Unfortunately, no spot or

1_{The pricing source we use is Bloomberg Generic. For more information on the Bloomberg package,}

see www.bloomberg.com.

2_{Liquidity is low since the bid-ask spread can never be regained. Keeping a CCT in the sample}

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Figure 6.1: Price of CCT maturing in November, 2011 (source: Bloomberg)

forward rates for longer maturities are available so we need a proxy. In Figure 6.2 the 6 month Italian T-bill rate is shown with two money market rates, the 6 month Euribor and 6 month Eonia (European OverNight Index Average). The 6 month Eonia rate is a swap rate with the Eonia (overnight) rate as the floating leg. As is shown in Figure 6.2 the 6 month Italian T-bill rate moves close to the 6 month Eonia rate. In the sample period, the spread between the two rates is on average very close to zero. The spread with Euribor on the contrary, increased substantially since the 2007 credit crunch. Therefore, we use the Eonia forward curve to forecast coupons. From Bloomberg we obtained Eonia swap curves with maturities up to ten years for the relevant period; July 20, 2005 until June 12, 2009. The Eonia curve is also used as the risk-free curve. This means we use Eonia to split the CCT implied yield curve into an interest rate part and a credit spread.

6.1.2 CCT Credit Spread descriptives

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Figure 6.2: Three money market rates: the 6 month Italian T-bill rate, the 6 month Euribor and the 6 month Eonia (source: Bloomberg)

and their difference, the credit spread.

From the start of the sample period, July 2005, CCT and Eonia rates steadily increased until they reached their peak at the beginning of the fall of 2008. This is the height of the financial crisis. In October 2008, the rates started to decline sharply, as the ECB lowered interest rates in several steps. The largest decline is in the shorter maturities. In the first years of the sample period the credit spreads are relatively small and stable. Note however, that the credit spread slightly increases with maturity. Starting in the fall of 2008, the credit spreads for all maturities increased. This is most clear at the 5 year credit spread. While the 5 year Eonia rate dropped, the 5 year CCT rate only partly followed. At the same time, credit spreads became much more volatile. This is very clear from the first graph in Figure 6.4 where daily shocks in the 18 month credit spread are shown. Note that a positive shock in this rate has a negative impact on the price of a CCT. The big positive shocks in the spread in October 2008 coincide with the sharp drop in the price of a CCT as shown in Figure 6.1.

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Figure 6.3: CCT, Eonia and Credit spread rates for different maturities

distribution, these are much higher. We perform the Jarque-Bera test for normality and reject the null hypothesis of a normal distribution. Clearly, we are dealing with heavy-tailed, or leptokurtic distributions. Part of this leptokurtosis might be due to volatility clustering as mentioned in Section 4.1. Removing this clustering would then lower the kurtosis. This will be further discussed in the next section. The Ljung-Box test (see e.g. Gouri´eroux (1997, Chap. 2)) is performed to test for autocorrelation up to 5 lags and up to 20 lags, referring to one week and one month. For all series of shocks at 5 and 20 lags, the hypothesis of zero autocorrelation is rejected.

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Figure 6.4: Daily shocks in 18m credit spread - over time and in histogram the price of the CCT did not change substantially this caused a jump in the 6 month implied CCT yield. This is a weakness of the model. Disregarding this particular jump, the maximum shock in the 6 month credit spread is in line with the maximum of the other series.

The skewness of the 6 month and 18 month credit spread shocks is not equal to zero. However, the GARCH models discussed in Chapter 4 assume shocks are from a symmetrical distribution. The only exception is the GARCH(p, o, q) model. We come back to this in the next section where we estimate different GARCH models to these series.

6.2 Estimation results of volatility updating schemes

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CHAPTER 6. EMPIRICAL RESULTS 43 Table 6.1: Descriptive statistics of daily shocks (in bp) in 6 month, 18 month and 5 year credit spreads

6 Month 18 Month 5 Year

mean 0.016 0.025 0.057 max 42.11 26.34 21.30 min -21.53 -20.80 -21.56 std 3.31 1.95 2.16 skewness 2.89 1.95 -0.16 kurtosis 43.88 57.92 36.30 JB 71,106 (0.00) 126,423 (0.00) 46,246 (0.00) LB(5) 38 (0.00) 79 (0.00) 110 (0.00) LB(20) 49 (0.00) 141 (0.00) 188 (0.00)

Note: p-values of test statistics are between parentheses.

variants within this class of models are discussed in Chapter 4. We discuss the es-timation results of univariate models in Subsection 6.2.1 and move to multivariate estimation results in Subsection 6.2.2.

6.2.1 Univariate results

Univariate GARCH models are discussed in Section 4.1. We use these models to model the volatility of risk factor shocks of Italian floaters. A distribution must be specified to estimate the parameters. First, we assume standardized shocks are normally dis-tributed. We refer to this as Gaussian-GARCH, or g-GARCH. Second, we assume standardized shocks follow a t-distribution. We refer to this as t-GARCH. Since we are dealing with fat-tailed series, the t-distribution seems more appropriate. Parameters are estimated using the Matlab toolbox created by Sheppard.3 We assume the con-ditional mean in (4.2) satisfies µ = 0. This seems reasonable because we are dealing

3_{On www.kevinsheppard.com/wiki/Category:MFE the Oxford MFE Toolbox for estimating}

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with daily shocks and the means in Table 6.1 are small. We also fit an EWMA model to the series with λ = 0.94 as in (4.1).

Tabel 6.2 reports the estimated parameters for the 18 month credit spread shocks of several GARCH models. Results for the 6 month and 5 year series are similar, but are not shown. The reported standard errors are robust standard errors as in Bollerslev and Wooldridge (1992). In the case of t-GARCH we also report the estimated degrees of freedom ν. Note that the t-distribution converges to the normal distribution if ν → ∞. In the lower part of Table 6.2 we report the loglikelihood and give the value of the expression Pq

i=1αi + 0.5

Po

i=1γi +

Pp

i=1βi. This is restricted to be less than

one. Furthermore, we report the kurtosis and the results of the Ljung-Box tests for the standardized residuals ˆ zt= ∆vt ˆ σt .

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original series.

In Figure 6.5 the estimated annualized daily volatilities of the 18 month credit spread are shown for each day. Note that the GARCH(1,1) volatilities fluctuate more than the EWMA volatilities; they react stronger to a market shock and volatility is less persistent as is in line with what we expect from the parameters in Table 6.2. If the volatility fluctuates more, we also expect the volatility weighted HVaR to fluctuate more. Furthermore, in periods where volatility increases, such as in October 2008, we expect volatility weighted HVaR to be higher than simple HVaR since passed shocks are scaled upward (where negative shocks become more negative). We will come back to this in Section 6.3.

Figure 6.5: Estimated annualized daily volatilities of 18 month daily credit spread shocks

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GARCH(1,1) model might be less sensitive to the extreme shocks in 2008. Estimation results are shown in Table 6.2. Unfortunately, the parameters sum to one in all cases. The upper boundary constraint is still restrictive and adding parameters to the model did not help. The richer models are also not really succesful in removing more kurtosis and autocorrelation from the series. Apparently, the variance process is not stationary. The graph of the daily shocks in Figure 6.4 indeed does not look like a typical stationary process, the shocks in the period June 2008 until March 2009 are really extreme.

We fit a g-GARCH(1,1) and t-GARCH(1,1) model to the series excluding all data after June 1, 2008 to check the hypothesis that these extreme shocks cause the sum of the parameters to equal one. The parameter estimates are shown in the last two columns of Table 6.2. Indeed, the GARCH(1,1) models seem to fit the data well. The sum constraint is no longer restrictive but the sum of the parameters is still high, indicating that variance is very persistent. Clearly, the usual GARCH(1,1) models are not able to capture the big change in volatility.

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series is clearly not trivial. In the backtests we use 260 days to estimate GARCH parameters. We motivate this choice of the rolling window length in Appendix B.

Historical Value at Risk models applied to Italian ﬂoating rate government bonds

Historical Value at Risk models

applied to Italian floating rate

government bonds

Historical Value at Risk models applied to Italian

floating rate government bonds

Nora R.W. Ottink

Preface

Contents

Chapter 1

Introduction

Chapter 2

Problem formulation

Chapter 3

Historical simulation Value at Risk

3.1

Simple HVaR

3.2

Weighted HVaR

3.2.1

Age weighted HVaR

3.2.2

Volatility weighted HVaR

3.2.3

Differences between age and volatility weighted HVaR

3.3

HVaR performance tests

3.3.1

Correct unconditional coverage

3.3.2

Correct conditional coverage

3.3.3

Mean absolute percentage error

3.4

VaR confidence bounds

Chapter 4

Volatility updating schemes

4.1

Univariate models

4.1.1

GARCH

4.1.2

Kurtosis of a GARCH process

4.2

Multivariate models

4.2.1

Correlation models

4.2.2

Dynamic Conditional Correlation

Chapter 5

Floating rate notes

5.1

Bond market definitions

5.2

Default-free floating rate notes

5.2.1

Pricing formula

5.2.2

Interest rate risk

5.3

Defaultable floating rate notes

5.3.1

Credit Spread

5.3.2

Interest rate and credit risk

5.4

Bootstrapping a yield curve

5.4.1

Interpolation and extrapolation

Chapter 6

Empirical results

6.1

Data

6.1.1

Market data

6.1.2

CCT Credit Spread descriptives

6.2

Estimation results of volatility updating schemes

6.2.1