Diagnosing Bubbles in Fixed Income

Markets

Masterthesis

In Collaboration with the Chair of Entrepreneurial Risks, ETH Zürich

Examiners Name: Prof. Dr. Didier Sornette

Supervisor: Dr. Peter Cauwels

Autor: Tuncay Michael Irmak

Burgdorf

Matrikel Nr.: 10-212-769

Altenbergstrasse 90

3013 Bern

Bern October 31, 2015

Acknowledgments

I would like to thank prof. Didier Sornette for accepting my proposal and making his

resources available for me and giving me the opportunity to work on such an interesting

topic. Also I would like to thank dr. Peter Cauwels for all our valuable discussions and

for his great supervision. For the opportunity to write my thesis at an other university,

I would like to thank the University of Bern and the ETH for their �exibility. Since this

may be the end of my basic education, I would like to thank my parents for all the support

since my �rst day at school.

I

Executive Summary

As observed in the recent sovereign debt crisis, �nancial bubbles can cause huge losses for

the individual investor and for global economic. Based on the current knowledge in the

�eld, this master thesis 103 sovereign and corporate CDS spread indices have been scanned

for bubbles. This has been executed with the Log Periodic Power Law model referring

to the methodology of the Financial Crisis Observatory (FCO) of the ETH Zürich. The

model has been used for calculating the two bubble indicators DS LPPL con�dence and

DS LPPL trust. These are used at the FCO as a bubble warn system and are daily

calculated as well as uploaded for a huge number of di�erent asset classes. In this thesis,

the past performance of these indicators have been tested and used for the interpretation

of the present indicator values. With the accurate calibration of the model the indicators

were able to predict a reverse in the growth rate at over 60% of the 13'283 observations

in the past. This result is exceptional. Based on these �ndings, two positive bubbles

today (17.08.15) for sovereign 10 years Dutch and German and two negative bubbles for 5

years corporate US Metals/Mining and US Manufacturing CDS spread indices have been

diagnosed.

II

Contents

1 Introduction 1

2 Fixed Income Securities 4

2.1 Fixed Income Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Clean versus dirty prices . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Financial Bubbles 13

3.1 E�cient Market Hypothesis and Financial Bubbles . . . . . . . . . . . . . 14

3.2 Innovation and Speculation . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Positive Feedback Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Faster than exponential growth . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Log-Periodic Power Law Model 21

4.1 The LPPL model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Deriving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Bubble Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Previous Academic research on bubbles in �xed income markets . . . . . . 28

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Data & Methodology 30

5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2.1 DS LPPL Con�dence and Trust . . . . . . . . . . . . . . . . . . . . 34

5.2.2 Historical Indicator Testing . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Results & Conclusion 41

6.1 Past Perfomance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.2 Bubble Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

III

6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 Summary 47

IV

List of Figures

1 US 10 Year Treasury Bond index: Yield to Maturity . . . . . . . . . . . . . 2

2 S&P 500 stock market versus US GDP . . . . . . . . . . . . . . . . . . . . 14

3 Growth Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Portugal CDS spread: short time window . . . . . . . . . . . . . . . . . . . 35

5 Portugal CDS spread: medium time window . . . . . . . . . . . . . . . . . 35

6 Portugal CDS spread: large time window . . . . . . . . . . . . . . . . . . . 36

7 Good Prediction: Sovereign Italy CDS spreads, short time window (50;100) 38

8 Good Prediction: Sovereign Spain medium window (150;250) . . . . . . . . 38

9 Wrong Prediction: Corporate US Containers long window (250;500) . . . . 39

10 Wrong Prediction: Corporate Us Real Estate Investment Trust short win-

dow (50;100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

List of Tables

1 Filtering Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Corporate CDS-Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Corporate Sub-CDS-Indices . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Sovereign CDS-Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Prediction Quality: All time window sets . . . . . . . . . . . . . . . . . . . 41

6 Prediction Quality: small time window set . . . . . . . . . . . . . . . . . . 42

7 Prediction Quality: medium time window set . . . . . . . . . . . . . . . . . 42

8 Prediction Quality: long time window set . . . . . . . . . . . . . . . . . . . 43

9 Negative Bubble Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Positive Bubble Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

V

List of Abbreviations

LPPL Log Periodic Power Law

FCO Financial Crisis Observatory

YTM Yield to Maturity

OTC Over the Counter

CDS Credit Default Swap

GDP Gross Domestic Product

GIPSI Greece, Italy, Portugal, Spain and Italy

VI

1 Introduction

In 2000 the prices of the technology stocks of the NASDAQ Composite index rapidly

increased. At that time high �ying tech stocks were sold at an excess of 100-times earnings.

In March 2000 the bubble crashed, resulting in decreasing prices so the previous mentioned

high �yers where sold in 2003 for less than 20% of their price in 2000.[The Wall Street

Journal (2010] Similar to the dotcom bubble, US stock and housing prices crashed after a

fast increase in 2007. This resulted in a global �nancial crisis. To stem the depression after

2007, policy makers implemented lower interest rates and quantitative easing.[Sornette

D. and Cauwels P. (2012)] Lower interest rates make borrowing cheaper and increase

the expectation of future growth. Also, the discount rates for cash generating assets

are getting lower. Both e�ects combined should boost stock prices.[?] The reaction of

central banks to depressions with low interest rates remained the same in the last thirty

years. Sornette D and Cauwels P. (2012) are criticising in their paper The Illusion of the

Perpetual Money Machine the hope or illusion of policy makers, that by creating money,

real wealth will be created. They argue that it is this policy which sets the start for a

new speculative bubble. This raises the question where the next bubble is or will expand

and when it will burst.

Today one may be expanding in the �xed income market.[?] Financial newspapers with

headlines like "Is the bond market a bubble?" [The Wall Street Journal (2010)] or"The

Great American Bond Bubble" [Garth S. (2013), p. 18] re�ect the fear of a bond bubble

which can burst. They argue that because of the risk aversion of the investors, caused

by the recent �nancial crisis, as also because of the extraordinary policy by central banks

and the hunt for yields of investors, sovereign and corporate �xed income markets are

in danger of being or becoming a �nancial bubble. [?] One example therefore are 10

years US treasury bonds. From 1985 until today1, a secular trend of decreasing US 10

years treasury bond yields can be observed. This is visualized in the �gure (??).[TR

Datastream, 17.08.15] Declining interest rates have fuelled a three decade bull market in

�xed income. Bubble diagnosis is crucial.

117.08.2015

1

01/01/80 01/01/90 01/01/00 01/01/10 01/01/201

2

3

4

5

6

7

8

9

10

11

Yield to Maturity

Figure 1: US 10 Year Treasury Bond index: Yield to Maturity

While in other bubble researches the fundamental value of an asset is estimated using

valuation techniques and then compared with the market price, this work takes a step

back and looks at a lower resolution picture. It assumes that the price of an asset follows

a Log Periodic Power Law LPPL with a faster then exponential growth, caused by posi-

tive feedback mechanisms of investors.[?] This approach was introduced by Johansen A.,

Ledoit O. and Sornette D. and was applied successfully to a large variety of �nancial bub-

bles and crashes.[Sornette D. et al. (2013)] After the past �nancial crisis, a FCO! (FCO!)

was set up by the chair of Entrepreneurial Risks of the ETH Zürich. There a huge range

of di�erent assets is scanned on a daily basis to diagnose bubbles and their crashing dates

with the LPPL method.[?]2 The methodology of the FCO for detecting a bubble with

the LPPL approach has been documented in the recent research paper by Sornette D.

and Cauwels P. (2014). The current research applies that methodology to answer the

question: whether a bubble is expanding today in �xed income markets.

For this, the work is divided in two parts. In part one, �rst �xed income securities and

their price dependence on di�erent factors will be described. Further, the reasons and

the e�ects of �nancial bubbles will be shown based on the framework of Sornette D. and

Cauwels P. (2014). In the end of the �rst part the Log Periodic Power Law model, which

will be used in the second part to scan timeseries for bubbles, will be introduced. In the

second part �rst the data and methodology which will be used for the following research

will be explained. Next, the results of the research will be presented and the conclusions

based on them will be made. In the end, the thesis will be summarised and the use of

2http://www.er.ethz.ch/�nancial-crisis-observatory.html

2

this research will be explained.

3

2 Fixed Income Securities

Before pricing anomalies can be diagnosed in a market, it is important to understand

the underlying pricing mechanics of the assets and on what they depend. By doing

so, incorrectly diagnosed abnormal pricing behaviour, which is actually caused by the

fundamental mechanics of the asset, can be avoided. This chapter will show how �xed

income securities are rationally priced. First it will be shown that the price of a bond

depends on the discount rate of its future cash �ows. Then, the interest rate risk and the

credit risk, which both determine the discount rate, will be de�ned. In the end of this

chapter, the best known methods to estimate the credit risk of a bond will be summarized.

2.1 Fixed Income Fundamentals

2.1.1 Discount Rates

A Bonds present value is determined by all discounted future cash�ows using an appropri-

ate discount rate. The value of a Bond (P ) can be de�ned with the following equation:[?]

P =T∑t=1

Ct

(1 + y)t(1)

were:

• Ct = cash �ow in period t

• t = number of periods to each payment

• T = number of periods to �nal maturity

• y = discount rate per period

With a given discount rate, the value of di�erent types of bonds can be calculated. Take

for example a normal coupon bond with ten annual payments or a zero coupon bond

with one payment at maturity. In both types at the end of maturity the principal gets

repaid by a de�ned amount called the face value in addition to the coupon payment.[?] By

knowing the market price, the coupon payments and the face value, the YTM! (YTM!)

can be calculated. It is the quotient of the sum of all cash�ows and the purchase price of

the bond. The yield to maturity shows the return of a bond which is hold until maturity.

A higher yield to maturity indicates a lower bond price and vica versa. This is a useful

tool for comparing bonds with each other.[?] 3 Depending on the coupon payments and

3http://www.investopedia.com/terms/y/yieldtomaturity.asp, 03.06.15

4

the discount rate, the face value of the bond can be higher, lower or equal the bond price.

If the coupon rate exactly matches the discount rate, the present value and so the price

have to be the same as the face value of the bond. In this case the bond is said to be at

par value. If the coupon is higher than the discount rate, the price of the bond must be

higher than its face value, characterising a premium bond. If it is lower than the discount

rate, the face value has to be higher than the price of the bond, meaning that this is

a discount bond.[?] The discount factor and the bond price are negatively non linearly

related to each other. This means that an increase in the discount rate causes a decrease

in the bond price. The discount rate of a bond is determined by the interest rate - and

the credit risk. For risk management purposes it is important to understand in what way

and how strongly both factors in�uence the bond price. In chapters (??) and (??) both

factors and their in�uence on the bond price will be de�ned.

2.1.2 Clean versus dirty prices

When bonds are priced, the number of days since the last coupon payment and the

remaining days to the next coupon payment in�uence the price of the bond. For example

if two bonds are compared, that only di�er in the number of remaining days to the next

coupon payment, the bond with the closer next coupon payment will be preferred and

priced higher.[?] The premium on a bond which is closer to the next coupon payment

is due to the accrued interest. The price accounts for the interest in the coupon of the

bond which has been recognized but not paid out yet.[?] The accrued interest (AI) can

be calculated with following equation:[?]

AI = C · At

AT

(2)

were:

• At := Actual number of days since last coupon

• AT := Actual number of days between last and next coupon

Bonds are quoted in dirty and clean prices. In contrast to the clean price, the dirty

price includes the accrued interest. When searching for bond price anomalies the accrued

interest premium on a bond represents is a fundamental pricing mechanism and should

not be included in the analysis. This is why the clean price should be used in this research.

5

2.2 Interest Rate Risk

The �rst component of the discount rate on a bond's cash-�ows is the risk free interest

rate. It is mainly driven by central bank policies and in�ation expectations. In a study

of Eugene F. Fama (1975), the US one to six month treasury bills return was compared

with the in�ation rate in the same period from 1953 to 1971. It has been reported, that

the market correctly used the information of past interest rates to predict the future

in�ation rates by setting the YTM of the bond as least as high as the future in�ation

rate. In theory, the risk free rate of return represents an investment with zero risk. This

makes it to be the minimum return for any risk bearing investment. In practice, no risk

free interest rate exists because every investment bears some risk.[?]4 As a proxy for the

risk free interest rate, mostly 3 months government bonds of countries with lower risk of

defaults are taken. Every bond's price depends on the risk free interest rate. Interestingly

due the huge variation of types of bonds, the price can react di�erently to interest rate

changes. For example a change in interest rates on a zero bond with maturity in ten years

has a higher impact on its price than for a bond with annual coupon payments which also

matures in ten years. For hedging purposes the sensitivity of a bond's price to interest

rate movements has to be measured. For this, the two measures duration and convexity

are used which will be explained in the following chapter.

The price to interest rate sensitivity of a bond can be approximated through the following

Taylor expansion:[?]

∆P =dP

dy∆y +

1

2

d2P

dy2∆y2...(infinite) (3)

where:

• ∆P = Change in the Bond Price

• ∆y = Change in the risk free interest rate

When the interest rate changes are small, the quadratic term also gets very small. This

is why in practice investors concentrate mostly on the �rst and second term in equation

(??) with respect to interest rate changes. In order to calculate the �rst derivative of the

bonds price with respect to interest rate changes the so called Macaulay Duration (D) is

used. It measures how much time on average is needed to receive all cash �ows from a

bond and is de�ned as: [?]

D =T∑t=1

tCt/(1 + y)t∑Ct(1 + y)t

(4)

4http://www.investopedia.com/terms/r/risk-freerate.asp, 03.06.15.

6

By modifying (D) the modi�ed duration (D∗) is obtained. It represents the �rst derivative

with respect to interest rate changes.[Joridon P. (2009)]

dP

dy= D∗ = − D

(1 + y)· P (5)

With (D∗) the impact of small interest rate changes on the price of a bond can be cal-

culated. By doing so, the price - interest rate relationship is assumed to be linear. To

correct for this non-linearity, the convexity of a bond can be added when interest rate

changes are large. It is de�ned with the following equation:[?]

C =T∑t=1

t · (1 + t) · Ct

(1 + y)t+2· 1

p(6)

With the modi�ed duration D∗ the e�ect of interest rate changes on the price are linearly

approximated. Because for larger interest rate changes the price-interest rate function

becomes more curved, the linear approximation through the modi�ed duration underesti-

mates the price increase when the interest rate decreases strongly and overestimates the

price decrease when the interest rate increases strongly. In order to make the approxi-

mation more accurate, the convexity needs to be added in such a case.[?] "All else equal,

duration and convexity both increase for longer maturities, lower coupons, and lower [risk

free interest rate]".[Jorion P. (2009), p.19] For the further research it is important to note

that the price reaction of a bond on interest rates can vary because of a variation of the

duration and convexity of a bond. Consequently it would be wrong to de�ne a pricing

anomaly resulting from a change in interest rates, which is caused due to a change in

the duration and convexity of a bond. Therefore, in our research, we will use constant

maturities.

7

2.3 Credit Risk

The credit risk of a bond is captured in the di�erence between its yield and the risk

free interest rate. This is de�ned as credit spread. It is the spread which compensates

investors for bearing the credit risk of the bond.

While the market risk is determined by the volatility of the price of an asset, the credit

risk is de�ned by the risk of default of the credit issuer.[?] The credit risk is driven by the

probability of default (PD) of the credit issuer, by the market value of the claim on the

counterparty which is called credit exposure (CE) and by the recovery rate (f). Based

on these drivers, the expected loss (EL) of a bond can be calculated with the following

equation:[?]

EL = PD · CE · (1− f) (7)

While the calculation of the expected loss with given default probabilities and recovery

rates is simple, the estimation of it gets more complex when the probability of default

and the recovery rate are unknown. For that case, di�erent methods based on corporate

balance sheets ratios, debt and equity market prices or CDS spreads can be used and will

be overviewed in this section.

Since the risk of default of a sovereign or corporate bond is coupled to the �nancial

situation of the credit issuer, the credit risk can be estimated by analysing �nancial ratios

of the issuer's balance sheet. In a study of Wosnitza J. and Leker J. (2014) following

�nancial ratios were used for the estimation of the credit risk of corporate bonds:[?]

• Cash �ow return on investment = CashflowTotalassets

• Net debt ratio =Financialliabilities−CashCapitalemployed

• Interest coverage = EBITDAInterestexpenses

• Equity ratio = EquityTotalassets

• Liability turnover ratio: = AccountspayableNetsales

• Debt repayment capacity = Financialliabilities−CashCashflow

For sovereign bonds, �nancial ratios like debt to GDP ratios can be used. Having a

dataset including the assumed ratios on which the credit risk depends, the probability of

default of a bond can be estimated through di�erent regression analyses. There are two

main problems regarding this method. First, the availability of the needed information is

often restricted. If for example the credit risk of a corporate bond is estimated through

8

�nancial ratios based on the yearly balance sheet of the corporate, the data may not

represent the actual situation of the company. Second, the assumed set of the chosen

ratios may be incomplete or wrong and may not capture all relevant information. If, for

example a company has a new CEO with di�erent risk preferences, this may have an

impact on the credit risk of the issued bonds of the company. This is not captured in the

balance sheet of the company. In contrast, market prices are more frequently updated

and should capture, in an e�cient market, all available information concerning the risk of

an asset. For this, the most intuitive method is to derive the credit risk from bond market

prices. In a risk neutral world, the price of a bond has to be equal to the expected cash

�ows discounted with the risk-free interest rate(i). Assuming two possible outcomes of a

zero bond with a face value (fv) and a recovery rate (f) which matures in one year, the

relationship between the probability of default (PD), the recovery rate and the current

price can be described as follow:

P∗ = (1− PD) · fv

(1 + i)+ PD · fv · f

(1 + i)(8)

When solving (??) for (PD) we get the following equation for the probability of default:

PD =1

1− f·[1− 1 + i

1 + y

](9)

Were (y) is the implied yield of the bond derived from the bond price and the face value of

the bond. With this approach the probability of default (PD) can be derived from bond

market prices by assuming a recovery rate (f). Bonds are mainly traded onOTC! (OTC!)

markets. As result, bond market price data is rare in most countries. The reasons for the

lacking data are that most countries have not developed a bond market and that bond

markets are illiquid which can result in distorted prices. An alternative method for this

is the Merton approach. This uses equity market prices instead of bond market prices

to estimate the credit risk of a bond. It links the risk of default of a company with the

variability of the value of its assets, which is implied by the variability of the value of the

company's equity. By assuming that the debt of the company is represented by a single

zero bond, the face value of the bond only gets fully paid out if the market value of the

assets of the �rm is as least as high as the face value of the bond at maturity. If the

market value of its assets at maturity of the bond is smaller than the face value of the

bond, the bond holders can not be fully paid out and su�er a lost. If it is higher, the

di�erence belongs to the equity holders. With this framework the value of the �rm's debt

can be replicated by adding a risk-free bond and a short put option on the value of the

assets with a strike equal the face value of the debt. The value of the short put can be

9

calculated with the Black Scholes formula by knowing the volatility of the value of the

�rm's assets.[?]

The problem with the Merton model is that it assumes that a company only can default at

the maturity of its debt and that it holds only one class of debt. Both assumptions are far

away from reality. When di�erent classes of debt with di�erent seniorities are introduced,

they have to be implemented in the model which makes it more complex.[Altman E. et

al. (2004)]

Another, more direct way to estimate the credit risk of a bond through the market is to

look at Credit Default Swaps CDS! (CDS!). A Credit Default Swap is a contract where

the buyer pays a premium to the seller in exchange for a payment if a credit event occurs.

A credit event is in general de�ned as bankruptcies or violations on a bond indentures.[?]

5 The premium payment for the credit risk insurance is mostly an annual payment and

is refered to as CDS spread. It is mostly used as quotation of a CDS contract.[?] In a

risk neutral world the value (V) of a CDS contract is determined as the di�erence of the

present value of the expected payment when a credit event occurs and the present value

of the expected CDS spread payments as long as no credit event occurs. This relation

can be described with following equation:

V =

(T∑t=1

kt(1− f) ·N · 1

(1 + i)t

)− s

(T∑t=1

St−1 ·N ·1

(1 + i)t

)(10)

where:

• N := nominal amount

• St−1 := survival probability until the year t

• kt = St−1dt := marginal default probability

• s := credit spread

By knowing the market price of the contract (V), the CDS spread and by assuming a

recovery rate (f), the implied default probability until year T can be derived when solving

(??) for kt and then adding up all kt's until year T.

PD =T∑t=1

kt (11)

Since both represent the credit risk premium, the CDS spread should approximately be

equal to the yield spread of the underlying bond. The advantages of CDS spreads are

5http://www.investopedia.com/terms/credit-event.asp, 24,9,15

10

at �rst the task of de�ning a "risk free interest rate" for the yield spread is skipped and

second that CDS markets are more liquid because they are not cash founded.[?] The

problem with using CDS spreads in order to analyse credit risks of bonds is that the

default events in a CDS contract are precisely de�ned. Consequently not all possible

default events of a bond may be insured in the CDS contract. As a result, depending on

the contract, the credit risk represented in the CDS spread, may not fully represent the

credit risk of the underlying bond.[?]

This section highlights that although the de�nition of the expected loss and so the credit

risk is simple, an appropriate estimation of the probability of default and the recovery

rate is quite challenging. The presented methods are simplistic and represent the basic

estimation methods for credit risks. Another mentionable source for the estimation of

credit risks would be external ratings from rating agencies. All of the methods have

advantages and disadvantages among each other and should be chosen individually for

di�erent types of analyses.

2.4 Summary

Due to their �xed payments, their maturity and their seniority, �xed income securities

strongly di�er from stocks. In the further research we are going to examine whether the

credit risk is over or undervalued in credit markets. Besides to the credit risk, the price

and so the implied yield of a bond depends on fundamental and mechanical factors. As

we are interested in the estimated credit risk of a bond, it is important to understand the

mechanics underlying �xed income securities. On these base the changes in the price/yield

of a bond can be interpreted correctly regarding the credit risk. For this propose the

standard fundamental valuation model has been introduced. Linked to this, the interest

rate risk with the two measurements duration and convexity were presented. With more

complex securities, for example with di�erent coupon date windows which are coupled to

the in�ation rate, the fundamental valuation and the duration/convexity calculation get

more complex. It has been shown that the di�erence of the risk free interest rate and the

yield of a bond is caused by the credit risk of the bond. The credit spread is determined

by the probability of default and its recovery rate. Finally, di�erent credit risk estimation

methods based on di�erent sources have been introduced. Some of them like the Merton

model are getting complex when the underlying bonds get more complex. Based on this

analysis, we will use CDS spreads in our research because of following reasons. First, they

represent the credit risk without the need of deducting a risk free interest rate. Second,

11

because of their �xed maturity and third because of their higher data availability.

12

3 Financial Bubbles

In most research papers, bubbles are de�ned as the part of the price movement which

is unexplainable through the fundamental value of the underlying asset.[?] Based on this

de�nition, bubbles have been tried to be diagnosed by comparing market prices with

estimated fundamental values. This poses the problem that the fundamental value should

re�ect all future returns. For this purpose, assumptions about the future are necessary.

Small variations in these assumptions can have huge e�ects on the estimated fundamental

value. Sornette D. and Cauwels P. (2014) show that e�ect with an example of the Gordon

Shapiro model: "[...] by simply reducing the expected total return from 8 percent to

6 percent, the expected stock price doubles to 5000, for the same dividend and growth

expectations." [Sornette D. and Cauwels P. (2014), p.6]

Because of the complexity and error potential of valuing �nancial assets, the previously

mentioned �nancial bubble de�nition is a�rmed in the paper of Sornette D. and Cauwels

P. (2014) but not used as de�nition for the empirical research. The authors also criticise

that through the comparison of market and fundamental values, a bubble may be indeed

detected, but the ending of a bubble may be left unexplained.

While the previous chapter characterises the asset and its market of our interest, this

section explains the market behaviour which we de�ne as a bubble. In the following

section, the circumstances for the start of a bubble will be characterised. After this, it

will be explained how these circumstances a�ect human psychology, resulting in irrational

market behaviour. After exploring the reasons for bubble development, their e�ect on the

price or yield movement and how this is used in the research will be explained.

13

3.1 E�cient Market Hypothesis and Financial Bubbles

Especially in �nancial markets, assets are priced through the interaction of investors. An

asset is valued based on the given information about the asset. If an investor values an

asset higher than the actual market price, he is going to buy or hold it. Reversely, if he

values an asset lower than the actual market price, he is going to sell it. The market price

is the aggregate of all investment decisions based on all information available. It should

re�ect the best estimation of its fundamental value. This theory is called the e�cient

market hypothesis from Fama E. (1970). In general, a bubble is said occur when the price

of an asset is far above or below its fundamental value. According to that de�nition, the

appearance of a bubble is in con�ict with the e�cient market theory. Inconsistent with

the theory as well is the book irrational exuberance of Shiller R. (2006). He showed that

markets can exhibit irrational exuberance. Both, Fama E. (1970) and Shiller R. (2006)

were honored with the Nobel Memorial Prize in Economic Sciences for their work.[?] This

raises the question whether asset prices can or cannot move irrationally away from their

fundamental value.

Linked to this question Sornette D. and Cauwels P. (2012) have compared the real growth

of the US Stock market with the US GDP! (GDP!) growth from 1952 to 2012. This has

been visualized in �gure (??)[Sornette D. and Cauwels P. (2012)].

Figure 2: S&P 500 stock market versus US GDP

They argued that due to the link between an investors' return in the form of a dividend

yield or a capital increase and the generation of pro�t, the aggregate return on all �nancial

investments can not grow faster than the real economy. Regarding this hypothesis two

14

things need to be taken in consideration. First of all, stock prices also capture future

growth expectations and can thereby temporary accelerate faster than the present pro�ts

of a company. Second, pro�ts can come from other countries with o�shore investments.

According to the authors, these e�ects can account for small di�erences between real

stock market return and real GDP growth. Larger di�erences can be explained through

the existence of a bubble.

Furthermore they observed that in the long run the US GDP and the US stock market

grew at a similar rate. Simultaneously the US stock market had higher returns than

the US GDP growth in shorter terms.Connecting these results with the work of Fama E.

(1970) and Shiller R. (2006), it can be concluded that normally as well as in the long

term markets are e�cient, but situations can be observed where market exhibit irrational

exuberance and bubbles can be diagnosed.

3.2 Innovation and Speculation

As discussed in the previous chapter, markets can be ine�cient. This chapter explains

why business cycles may provide the required setting that are leading to such irrational

market behaviour. Later it will be explained how, when issues with uncertain outcomes

appear, humans tend to imitate others rather than creating an individual solution. Capital

markets are by nature uncertain. Business can be described in a boom bust cycle. This

is caused by the dynamic interaction between innovation and speculation. "As a wave of

new innovations �ourishes, a boom sets in."[Sornette D. and Peter Cauwels (2014), p.5]

Companies with high future growth perspectives because of innovations expectations, are

created. They are generally cash poor and need to be �nanced. In consequence the wave of

innovations only can �uctuate through �nancial intermediaries that move cash from cash

generating companies to cash absorbing companies. As a consequence, periods of high

innovation are accompanied by a growth in the size of the �nancial sector. As discussed in

chapter (??) a higher di�erence between the growth of the pro�ts in the real economy and

the �nancial sector can be explained through a bubble. In our case where new innovative

companies are created, the di�erence between real and �nancial growth can be caused by

expected growth perspectives. However these perspectives are highly uncertain. It is this

uncertainty which provides room for speculations and bubbles. Once a bubble starts to

develop, the price of the underlying assets moves far away from its fundamental value. At

a certain point the overreach of the �nancing industry causes a collapse and puts an end

to the cycle. A new wave of innovations can restart the process.[Sornette D. and Peter

15

Cauwels (2014] Besides new innovations, di�erent other circumstances like new market

conditions can leave room for uncertainty and speculations.

16

3.3 Positive Feedback Mechanism

As discussed in chapters (??),(??) low interest rate policy and waves of innovation can

cause bubbles. However, it has to be pointed out that by de�nition low interest rates and

high future earnings expectations through innovations, increase the fundamental value

of a �nancial asset. The question of interest deals with the fact why the increasing or

decreasing price can overreact in such new situations in the form of a bubble.

This is where the human psychology plays a major role. In a �rst stage smart money �ows

into new opportunity or expectation with a good story about terri�c future prospects. In-

vestors value this new opportunity, which leads to a �rst price appreciation.[citeS14] This

�rst price movement re�ects the estimated value of the asset by the investors. After that,

the behaviour of the investors starts to behave di�erently. While the �rst change in the

price was caused by the valuation of the asset by the investors, the investors now start to

value the �nancial asset based on its past performance without a fundamental valuation.

By that the price moves far away from its fundamental value. Sornette D. and Cauwels

P. (2014) describe this phenomenon as follow: "At some point, demand goes up as the

price increases, and the price goes up as the demand increases. This is called positive

feedback mechanism, which fuels a spiralling growth away from equilibrium." [Sornette D.

and Cauwels P. (2014) , p.7] There are several positive feedback mechanisms which can

be classi�ed into technical/rational and behavioural. The previously mentioned situation

can be addressed to the second group. Due to the di�culties of valuing investment op-

portunities with such high uncertainty concerning future outcomes, investors use simple

heuristics in order to solve this problem with imitation and herding behaviour. It is proved

that imitation is the optimum investment strategy in situations of uncertainty until the

bubble bursts. This is because the price results from the aggregate polling of decisions

and can be used as proxy for the aggregate sentiment of the investors. Imitation and

herding behaviour evolved hundred years ago when the use of simple heuristics provided

survival advantages.[Sornette D. and Cauwels P. (2014)] Sornette D. (2009) also explains

how the imitation behaviour depends on the hierarchical structure of social networks. He

explains that with a larger cluster of investors which follow the same strategy, the danger

of a bubble to burst increases.[?]

One example for rational/ technical positive feedback mechanisms is option hedging.

When we want to hedge the risk of selling a call option we have to buy more of the

underlying asset if the price goes up. This is a positive feedback reaction to the price

movement of the asset.[?]

17

3.4 Faster than exponential growth

While the previous chapter explained behavioural positive feedback mechanisms as the

underlying reason for �nancial bubbles, this chapter characterises the price dynamics when

positive feedback is involved. In case of a developing bubble this characterisation of the

price movement sets the basic assumption for later research. By means of this assumption,

it will be also explained why bubbles do not continue to grow limitless. In the following

chapter the growth rate of the price of an asset will be de�ned as unsustainable when a

bubble is developing. For this purpose, �rst an example for sustainable without limited

resources and an example of a logistic growth process will be presented. Next, the growth

rate when a bubble is developing will be characterised and categorised as unsustainable.

Sustainable growth is independent from any kind of internal or external resources. This

means that a sustainable growth can persist forever. As an example, the proportional

growth of capital can be taken, which is visualised as the red line in �gure (3). Capital

grows in general exponentially due to the mechanism of compounded interest rates. In

contrast, a logistic growth process is not independent resources. As consequence it can

reach a steady state or reverse itself. For example, when a pair of sheep is put on an island

with a limited carrying capacity, the population will �rst grow due to almost limitless food

resources. Later, after a period of decelerating growth, the sheep population reaches its

critical stage where the available resources are not su�cient to maintain the population.

When this point is reached, the growth rate slows down due to its negative feedback

of competition for the scarce food. After these negative feedback mechanisms growth

will eventually stop and an equilibrium between resources and sheep population will be

reached. This represents a growth process which depends on external resources and can

be classi�ed as logistic growth. The example is visualised in �gure (??) as the green line.[?]

18

40 45 50 55 60 65 70 75 800

2000

4000

6000

8000

10000

12000

14000

16000

18000

exponentialhyperboliclogistic

Figure 3: Growth Dynamics

When we are looking at the price movement in the event of a bubble, we mostly

observe a strong growth in the price up to a certain critical point, before the bubble

bursts and the growth rate reverses itself. This dynamic of the growth rate in a bubble

�ts to the character of an unsustainable growth. In order to predict crashes in such a

bubble scenario, it is essential to know on what the growth rate is depending on. We have

discussed earlier that positive feedback is responsible for the growth when a bubble is

expanding. While stock prices normally are supposed to grow exponential, it is assumed

that the growth rate is no longer constant and grows autonomously when positive feedback

is involved . This is called hyperbolic growth (faster than exponential). In contrast to our

sheep example the stability of the growth rate is independent of exogenous factors like

resources. The growth rate itself causes the collapse of the system. At a certain point the

growth rate becomes large enough so the price hits a wall and the model breaks down.

This is also shown in �gure (??). In physics this is called a singularity. It has been shown

that beyond a critical point the hyperbolic process has no solution. As a result a change of

regime is unavoidable. Where other models ensure the presence of an equilibrium, the key

point of this methodology is the nonexistence of a solution for predicting the breakdown

of a bubble. Assuming that a bubble expands due to positive feedback which causes a

hyperbolic growth makes it possible to diagnose bubbles in time series. The detection of a

price movement with a a hyperbolic trajectory can be seen as a warning for an expanding

bubble. [?]

It has to be noted that the price in a bubble regime will not follow the smooth path of

19

a hyperbolic power law. It will be instead a�ected by bursts of volatility resulting in

subsequent restarts of the bubble. This is caused by the structure of the market and the

interaction between value traders and trend followers. Sornette D. and Cauwels P. (2014)

describe this dynamics as oscillations with increasing frequency and decreasing amplitude

over the time span of the bubble.

Taken together, these �ndings characterise the price dynamics which we de�ne as bubble.

For the present research the time series will be scanned for this speci�c characteristics

and bubbles will be diagnosed in the case they match these characteristics.

3.5 Summary

Mostly bubbles are de�ned as the part of the price movement which is unexplainable on

the basis of the fundamental value of the underlying asset.[?] Using this de�nition, bub-

bles have been tried to be diagnosed by comparing estimated fundamental values with

the market prices. This procedure has a huge error potential. That is why for the later

research bubbles are explained and characterised based on the framework of the earlier

chapters as follow:

A bubble starts with a new opportunity or technology with a good story about future

prospects. Because of the uncertainty about the future outcomes investors start to imitate

and herd. This is leading to a price movement followed by hyperbolic course with a �nite

time singularity caused by positive feedback mechanism.[?] This price movement is charac-

terized trough oscillations with increasing frequency and decreasing amplitude.[?] "During

a bubble, the market has changed structurally and entered a completely new regime, which

is entirely driven by sentiment and no longer re�ects any real underlying value." [Sornette

D. and Cauwels P. (2014), p.7] When the bubble is in�ating, the whole construct gets

more and more fragile and can collapse on the smallest shocks.[?]

This de�nition changes the point of view on bubbles. Assuming that a hyperbolic growth

is caused by a positive feedback mechanism, the di�cult task of valuing an asset to diag-

nose over and undervaluation is skipped, and the bubble diagnose is maid on the basis of

market dynamic analysis. Principally a bubble is diagnosed when a hyperbolic growth,

decorated by oscillations with increasing frequency and decreasing amplitude, is found.[?]

This section provided the basic framework for the LPPL model which will be further

explained in the next section and be used for the present research.

20

4 Log-Periodic Power Law Model

While the previous section described the origin and the e�ect of �nancial bubbles, the

following section presents the model and its parameters which are supposed to capture

these e�ects. After that, the model will be derived and it will be shown how the model

is �tted to time series. Consequently the two bubble indicators used by the FCO DS

LPPLS con�dence and DS LPPLS trust will be introduced. These also will be used in

the present research. At the end of this section earlier researches in credit markets with

the LPPL model will be summarized.

4.1 The LPPL model

Based on the de�nition of the previous section, the model, which should represent the price

dynamics on bubbles and which is used to diagnose bubbles in time series is called the log-

periodic power law. It combines the accelerating oscillations (Log Periodic) and the faster

than exponential growth of the price (Power Law) and can be written as follow:[Sornette

D. and Cauwels P. (2014)]

lnE([p(t)]) = A+B(tc − t)m + C(tc − t)m cos( ωln(tc − t)− Φ) (12)

were:

• ln(Ep(t)) := expected log Price

• tc := critical time (date of termination of the bubble and transition in a new regime)

• A := expected log price at the peak when the end of the bubble is reached at tc

• B := amplitude of the power law acceleration

• C := amplitude of the log-periodic oscillations

• m := degree of super exponential growth

• ω := scaling ratio of the temporal hierarchy of oscillations

• Φ := time scale of the oscillations

The model has three components representing a bubble. The �rst A+B(tc − t)m han-

dles the hyperbolic power law. For m < 1:"[...]when price growth becomes unsustainable,

and at tc the growth rate becomes in�nite." [Sornette D. and Cauwels P. (2014), p.15] The

21

second term C(tc − t)m controls the amplitude of the oscillations. It drops to zero at the

critical time tc. The third term ωln(tc − t)− φ , models the frequency of the oscillations.

They become in�nite at tc. How the model is derived and how it is used to diagnose

bubbles will be explained in the next chapter.

4.2 Deriving the model

Referring to the work of Johansen A. and Sornette D. (2000), in this section the JLS LPPL

model will be derived. The foundation of the JLS LPPL model lies on the seminal work

of Blanchard O. and Mark Watson M. (1982). The JLS model starts with the assumption

that the dynamics of a bubble can be described with the following stochastic equation

with drift and jump:[Sornette D. et al. (2013)]

dp

p= µ(t)dt+ σdW − κdj (13)

where

• p(t) := stock market bubble price

• µ(t) := drift or trend

• dW := increment of a Wiener process (with zero mean and unit variance)

• dj := discontinuous jump such that j = 0 before crash and j = 1 after crash

• κ := loss amplitude associated with the occurrence of a crash

• σ := volatility

To predict when the bubble will burst, it is important to understand the dynamics of the

jumps dj. They are governed by a crash hazard rate h(t). The crash hazard rate is the

probability of the crash to occur between t and t + dt. If the crash did not occur up to

the time t we have Et[dj] = 1 · h(t)dt + 0 x (1 − h(t)dt) which brings us to the following

equation:

Et[dj] = h(t)dt (14)

The model assumes that noise traders exhibit collective herding behaviours that may

destabilize the market. For this the model accounts for the aggregate e�ect of noise

traders and assumes the following hazard rate:

22

h(t) = B′(tc − t)m−1 + C ′(tc − t)m−1cos(ωln(tc − t)− Φ′) (15)

The equation (??) accounts for the accelerating panic punctuating growth of the bubble

and de�nes a hyperbolic power law ending in a �nite time singularity. Both are the result

of positive feedback mechanisms which e�ect the growth rate. With the non-arbitrage

condition, the unconditional expectation Et[dp] of the price increment must be 0 . This

results in the following equation:

Et

[dp

p

]= κh(t)dt (16)

The equation (??) balances the expected return in a time interval dt, with the probability

of a loss κ in the same time interval being equal to the hazard rate h(t). Substituating

expression(??) for h(t) and integrating results in called Log Periodic Power Law (LPPL):

lnE[p(t)] = A+B(tc − t)m + C(tc − t)mcos(ωln(tc − t)− Φ) (17)

with:

• tc := critical time (date of termination of the bubble and transition in a new regime)

• A = lnE(tc)

• B = -κ C ′/m

• C = = κC ′/√m2 + ω2

• Φ = Φ′ + π/2

Note that the LPPL expression only describes the average price dynamics until the critical

time tc. It does not specify what happens beyond tc. This means that the bubble does

not necessarily have to burst after the critical time. The critical time tc is an estimation

and gives the most probable time when the regime could change. Since the precise value

of tc is not known the estimation of tc can be written as:

testimatedc = tc + ε (18)

with

• tc = the real critical time

• ε = error term

23

4.3 Fitting Procedure

To diagnose LPPL structures and bubbles, the time series have to be �tted into the

model. In a next step the estimated sets of parameters have to be interpreted, whether

they represent a bubble or not. In the following, the �tting procedure which is used in

the research, will be presented. It has been introduced by Filimonov V. and Sornette D.

(2013) and is used at the Financial Crisis Observatory.

The LPPL model has 4 non linear parameters (tc,m, ω, φ) and 3 linear parameters (A,B,C)

They should be chosen with the goal to minimize the di�erence between the predicted

values of the model ln(p) and the real value ln(p). This represents a minimization prob-

lem with 3 linear and 4 non linear parameters which have to be found. To decrease

the complexity of this task, equation(??) is rewritten. For this, two new parameters are

introduced:

C1 = Ccosφ, C2 = Csinφ (19)

and the equation(??) can be rewritten as:

lnE[p(t)] = A+B(tc−t)m+C(tc−t)mcos(ωln(tc−t)−Φ)+C2(tc−t)msin(ωln(tc−t)) (20)

By doing so, the model (??) has now 3 non linear (tc, ω,m) and 4 linear parameters

(A,B,C1, C2). To estimate the parameters which are �tted to the time series the least

squares method with the following cost function (??) is used.

F (Tc,m, ω,A,B,C1, C2) =N∑i=1

[ln(p(ti))− (A+B(tc − t)m

+ C(tc − t)mcos(ωln(tc − t)− Φ) + C2(Tc − t)msin(ωln(tc − t))]2 (21)

Next, the 4 linear parameters are slaved to the 3 non linear parameters. This is done by

using the LU decomposition algorithm to solve the following �rst order condition:N

∑fi

∑gi

∑hi∑

fi∑f 2i

∑figi

∑fihi∑

gi∑figi

∑g2i

∑gihi∑

hi∑fihi

∑gihi

∑h2i

·A

B

C1

C2

=

∑yi∑yifi∑yigi∑yihi

(22)

where:

• yi = ln(pt)

24

• fi = (tc − ti)m

• gi = (tc − ti)mcos(ωln(tc − t)

• hi = (tc − ti)msin(ωln(tc − t)

As a result, the minimization problem (??) and the cost function (??) can be written as

follows:

{tc, m, ω} = arg mintc,m,ω

F1(tc,m, ω) (23)

F1(tc,m, ω) = arg minA,B,C1,C2

(24)

This procedure decreases the complexity of the problem. While before, the problem

has been de�ned as one with 4 non linear parameters, a solution only could have been

found with metaheuristics. With the work of Filimonov V. and Sornette D. (2013)

the non linear parameters are reduced from four to three. Because of that, rigorous

search methods like the nonlinear least squares algorithm or the Nelder-Mead simplex

method are now su�cient.[?] When estimating the parameters, the following restrictions

are made:[Sornette D. et al (2015)]

1. 0 < m < 2

2. 1 < ω < 50

3. t2 − 0.2dt < tc < t2 + 0.2dt

These restrictions narrow down the search space for the model but do not represent the

ranges which are de�ned as a bubble. In a second step, the estimated set of parameters

is analysed and checked whether they match the ranges of parameters of our bubble def-

inition which will be explained in the next chapter.

25

4.4 Bubble Diagnosis

The following chapter presents the two bubble indicators DS LPPL Con�dence and DS

LPPL Trust. They are used as bubble alarm indices with the goal to predict bubbles ex-

ante. The indicators are used to monitor the bubble risk of di�erent world stock markets,

commodities, US sectors and US �rms at the FCO.[Sornette D. et al (2015)]

For the bubble diagnosis and the calculation of the indicators, the following search space

and �ltering condition is used by the FCO:[Sornette D. et al (2015)]

Item Notation Search Space Filtering condition

3 nonlinear parameters m [0, 2] [0.01, 0.99]

ω [1, 50] [2, 25]

tc [t2 - 0.2dt [t2 - 0.05,dt

, t2+0.2dt] t2+0.1dt]

Number of oscillations 12 ln|

tc−1t2−t1 | - [2.5, +∞]

Damping m|B|ω|C| - [1, +∞]

Relative error pt−pt

pt- [0,0.2]

Table 1: Filtering Conditions

When the time series are �tted to the LPPL model, the starting point t1 and the end

point t2 have to be set. The point t2 represents a �ctitious "present" up to which the

data is recorded. Then the model is �tted to the time interval [t1, t2]. If the obtained

parameters match the �ltering condition of table 1, a bubble is diagnosed. The problem

with this is that the diagnosis of a bubble in t2 can strongly depend on the starting point

t1. It could be that for example no bubble is diagnosed in t2 with a given t1 but when t1

is set one year earlier, a bubble could be diagnosed in the same t2. Because of this, the

indicators are calculated based on shrinking time windows twindow with a �xed t2 and an

increasing t1. For this, a minimum (twindow,min) and a maximum (twindow,max)will be set

for every t2. Additionally the step-size of the increase in t1 has to be de�ned. By doing so,

the model is �tted totwindow,max−twindow,min

stepsizetime windows with the search space conditions

of table 1, for a �xed t2. The obtained parameters of each window are afterwards used

for the calculation of bubble indicators. These can be interpreted as bubble warning

indicators at t2. They are calculated and interpreted as follows:[Sornette D. et al (2015)]

26

DS LPPL Con�dence

The DS LPPL Con�dence is the number of time windows for which the LPPL calibration satis�es

the �ltering condition in Table 1, divided by the number of time windows. The indicator shows

the percentage of time windows that are diagnosed as LPPL bubble structure in the interval form

the minimum time window to the maximum time window. Sornette D. et. al. (2015) describe

the DS LPPL con�dence indicator as follow: "It measures the sensitivity of the observed bubble

pattern to the time scale dt. A large value indicates that the LPPLS pattern is present at most

scales and is thus more reliable. A small value signals a possible fragility of the signal since it is

present only in a few time windows" [ Sornette D. et. al. (2015), p.5 ] By using the indicator to

scan time series for bubbles, the risk of data snooping to diagnose a bubble gets smaller.

DS LPPL Trust

The DS LPPL Trust is calculated in two steps. First, the residuals of the LPPL �t of each time

window are resampled 100 times and then re-added to the LPPL structure. This give us 100

synthetic price time series for each window. Second, the percentage of the 100 synthetic price

time series which satisfy the �ltering condition 2 in table 1 is calculated. This is done for all time

windows. Finally the median of the calculated values in step 2 is taken as the DS LPPL Trust.

It measures the dependency of the calibrated LPPLS model on the realisation of the residuals.

A smaller value means that the LPPL pattern is strongly depended on the realisation of the

residual and a higher value means that the pattern is almost independent on the realisation of

the residuals. Sornette D. et al. (2015) describe the indicator as follow:"It measures how closely

the theoretical LPPLS model matches the empirical price time series, 0 being a bad and 1 being

a perfect match. As a rule of thumb, a value of DS LPPLS Trust larger than 5% indicates that

the price process is not sustainable and there is a substantial risk for a critical transition to oc-

cur." [Sornette D et al. (2015), p.5]

These indicators are used to detect LPPL structures in time series. They make the analysis

more robust and reliable regarding the variating time and residuals dependency. They do not

provide estimated ending points of the bubble. The conditions in table 1 are gathered from

empirical evidence of previous bubble researches.[Sornette D. et al (2015)] Both indicators as

well as the �ltering conditions from table 1 will be used in the later research.

27

4.5 Previous Academic research on bubbles in �xed income mar-

kets

This section presents 3 previous research results in credit markets with the LPPL model.

In the research of Wosnitza J. and Leker J. (2014) they �rst compare the discriminative power

of the estimated probability of defaults derived from balance sheet ratios against the implied

default probability derived from market data. For the estimation of the implied default prob-

ability by the market, stock prices and CDS spreads were used. The result of this comparison

was that the implied probability of default derived from CDS spreads outperformed both, the

implied probability of default estimated through stock market prices and the estimated proba-

bility of default through balance sheet ratios. In other words: the CDS market estimated the

credit risk better than the estimation based on balance sheet ratios. One reason for this may

be that balance sheets are mostly published once a year. Because of that, the estimation of

the probability of default based on a balance sheet may not re�ect the actual situation of the

company. In contrast, CDS market data information was updated daily. Based on this result

the authors argue that it would be rational for creditors to base their investment decisions based

on the probability of default derived from CDS spreads as follow: "[...] in other words to imi-

tate the aggregate opinion of the market participants." [ Wosnitza J. and Leker J. (2014), p.440 ]

Linking this �nding with the LPPL model where imitating investors causing LPPL structures in

�nancial time series, the authors conclude that this makes it reasonable to scan CDS time series

for LPPL structures. This has been done in the same research for governmental CDS spreads of

Cyprus, Greece, Ireland, Portugal and Spain from 2009 to 2011. The result was that the CDS

spread trajectories of Cyprus and Ireland exhibited the clearest LPPL structures. Greek and

Portuguese CDS spreads development could not adequately be described as a LPPL structure, a

LPPL structure in Spanish CDS spreads in this time period was out of question according to the

authors. In a other research Wosnitza J. and Denz C. (2013) analyse the daily spread trajecto-

ries of senior CDS spread with 1-year maturities of 40 international banks from June 2007 until

approximately April 2009. During this period they found LPPL structures in the CDS spread

movement for all banks. They also reported di�erent LPPL parameters for banks with high

and low credit quality. According to the authors these di�erences characterise the behaviour of

investors when banks are at risk of insolvency. Based on the results it is explained that the CDS

market is characterized by smaller values of ω than for example stock markets. In the end the

authors support the hypothesis that herding behaviour among investors can limit the re�nancing

options of banks which can end up in insolvency. In the research of Cauwels P. et al. (2009) the

LPPL model has been used to detect bubbles in the major CDS indices (Itraxx and CDX) in

2009. By comparing the implicit probability of default of 18.7% of the crossover itraxx index on

the 15th of december 2008 and the CDX high yield index with an implied probability of default

28

of 23.6% with the highest real default rate of US speculative grade issuers of 15.4% in the great

depression, the research for a bubble with the LPPL model was justi�ed and necessary.[Cauwels

P. et al. (2009)] In the analysis strong LPPL patterns for several indices were found and the �rst

week of February was de�ned as critical point. This means that the implied default rate by the

market did not re�ect any longer its fundamental value. Instead it was caused by imitation and

herd behaviour which led to an overvalued cost of protection.

In the research of Wosnitza J. and Leker J. (2014) CDS spreads are described as the best data

source for the estimation of the default probability. For that reason it has been argued that CDS

markets are predestined to exhibit LPPL patterns. In consistence LPPL structures have been

found in every research. In conclusion LPPL patterns in CDS spread trajectories are possible

and plausible.

4.6 Summary

This section presented the LPPL model and the calibration of its parameters. It has been

explained that the model predicts the end of the bubble regime with no speci�cation about

future regimes beyond that point. This means that it is unclear if the bubble will burst or if the

price will follow another trajectory. This is important to know when the model gets tested by

using historical bond examples. Afterwards the two bubble risk indicators DS LPPL con�dence

and DS LPPL trust which were introduced and are used by the Financial Crisis Observatory,

have been presented. It had become clear that these indicators make the analysis of time series

for LPPL patterns more robust and reliable. At the end previous researches in credit markets

with the LPPL model have been presented. They all had in common that they have used CDS

spreads to detect LPPL patterns in credit markets. This section provided the basics for the

methodology used in the present research.

29

5 Data & Methodology

We are interested to investigate using the LPPL model, whether bubbles are developing in �xed

income markets. The diagnosis will be made, based on the two indicators DS LPPL con�dence

and DS LPPL trust. For a better interpretation of the actual bubble indicator values, the past

performance of the indicators, on the same time series which will be analysed for present bubbles,

will be evaluated. This will be done by comparing the time before and after the bubble alarm.

In the following section the data and the methodology which are used for the analysis will be

presented

5.1 Data

The criteria for the chosen data-type were that they should re�ect the credit risk estimation of

the market and be independent of other risk factors or fundamental pricing mechanisms. Due

to the advantages of CDS spreads over bond yield spreads as discussed in the chapters (??) (??)

CDS spreads Indices were selected for the research except for the analyse of the US 10 years

government bonds index. For the US 10 years government bond index, the yield to maturity was

taken due for two reasons: First there were not enough data available for the required time period

of the analysis. Second, the yield of the US 10 years government bond index already represented

the credit risk of the government to default without the need of deducting a risk free interest

rate. Indices were taken to cover markets and to �x the maturity of the CDS spreads and for the

ten year government bond. For the analysis the daily mid spread close values were taken. Those

values describe the average value of the ask and bid price at the end of each trading day. The

indices represent CDS contracts for senior bonds. Senior bonds are less risky then sub ordinated

bonds since they get repaid �rst when the credit issuer defaults. For the analysis Corporate and

sovereign CDS spread indices were selected.

The data was gathered from Thomson Reuters Datastream and Thomson Reuters Eikon. The

corporate CDS indices summarise di�erent corporate CDS markets sectors and sub sectors in

north America and Europe. The indices were equally weighted and re�ect the average mid spread

of the given index constituents. Their index members were chosen based on the most liquid term

(5 year CDS). In order to ensure that the indices re�ect the most liquid CDS market, the mem-

bers of the indices were rebalanced every six month. The di�erent sector and sub-sector indices

are presented in table (??) and table (??). The di�erence in number of members is due to their

diverse availability. The Sovereign CDS spread indices represent the CDS spreads on ten year

senior government bonds. They were chosen with the goal to re�ect the most liquid sovereign

CDS markets. Additionally the CDS spreads of the GIPSI! (GIPSI!) bonds were taken for

the analysis of the past performance of the bubble warn indicators. The chosen sovereign CDS

spread indices are presented in table (??).

30

North America / Europe

Banks

Consumer Goods

Electric Power

Energy Company

Manufacturing

Other Financial

Service Company

Telephone

Transportation

Table 2: Corporate CDS-Indices

31

North America Europe

Automotive Manufacturer Automotive Manufacturer

Banking Banking

Beverage/Bottling Beverage/Bottling

Building Products Building Products

Cable/Media Cable/Media

Chemicals Chemicals

Conglomerate/Diversi�ed Mfg Conglomerate/Diversi�ed Mfg

Consumer Products Electronics

Containers Financial � Other

Electric Utility Mid Quality Leisure

Electronics Metals/Mining

Financial � Other Oil & Gas

Food Processors Service � other

Healthcare Facilities Telecommunications

Healthcare Supply Utility Other

Home Builders

Industrials-Other

Information Technology

Leisure

Life Insurance

Lodging

Machinery

Metals/Mining

Oil & Gas

Oil�eld Machinery/Services

Pharmaceuticals

Property and Casualty Insurance

Publishing

Railroads

Real Estate Investment Trust

Restaurants

Retail Stores � Food/Drugs

Retails Stores � Other

Service � other

Telecommunications

Textiles/Apparel/Shoes

Tobacco

Transportation � Other

Utility Other

Vehicle Parts

Table 3: Corporate Sub-CDS-Indices32

Present Research Historical Examples

Switzerland Portugal

Australia Greece

Canada Italy

China Ireland

Germany Spain

Finland

Italy

Australia

France

Ireland

Italy

Turkiy

Russia

Netherlands

Portugal

United States of America

Spain

Indonesia

Sweden

Norway

Denmark

New Zealand

India

Belgium

Slovakia

Malaysia

Table 4: Sovereign CDS-Indices

33

5.2 Methodology

5.2.1 DS LPPL Con�dence and Trust

For the �tting procedure and the calculation of both indicators, the methods presented in chap-

ters (??) (??) have been used with the same search space and the same �ltering conditions as

presented in table (1). As the time period from the start of a bubble to the end can vary, three

di�erent sets of time windows have been used for the calculation of the indicators. All of them

move from the minimum time window twindow,min to the maximum time window twindow,max

in daily steps twindow,max. The �rst set has a minimum time window of twindow,min = 50 days

and a max. window of twindow,max = 100. This means that for the calculation of the bubble

warning indicators at t2, 50 time windows were analysed. The second has a minimum time

window of twindow,min = 150 and a max. window of twindow,max = 250, and the third has a a

minimum time window of twindow,min = 250 and a max. window of twindow,max = 500. For the

presentation of the results and for the later discussion these sets were named as small(50;100) - ,

medium(150;250) - and large (250;500) time window set. This di�erentiation in the time window

sets has been made in order to diagnose di�erent bubbles with respect to the length of the time

period of the bubble. With the de�ned time window sets, for every time series, three di�erent DS

LPPLS con�dence and DS LPPL trust values have been calculated based on the di�erent time

window sets. The �gures (??) - (??) show the indicator values which where calculated based on

these di�erent time window sets for the same time series of Portuguese CDS spreads.6

6Thomson Reuters Datastream

34

01/01/08 01/01/10 01/01/12 01/01/1440

310.325

580.65

850.975

1121.3

01/01/08 01/01/10 01/01/12 01/01/14

0

0.2

0.4

0.6

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 4: Portugal CDS spread: short time window

01/01/08 01/01/10 01/01/12 01/01/1440

310.325

580.65

850.975

1121.3

01/01/08 01/01/10 01/01/12 01/01/14

−0.2

0

0.2

0.4

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 5: Portugal CDS spread: medium time window

35

01/01/08 01/01/10 01/01/12 01/01/1440

310.325

580.65

850.975

1121.3

01/01/08 01/01/10 01/01/12 01/01/14

−0.4

−0.2

0

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 6: Portugal CDS spread: large time window

36

5.2.2 Historical Indicator Testing

In order to evaluate the performance of the indicators in the past, the dates t2 for the calculation

had to be set �rst. For this, each day in the past of each time series, starting at day twindow,min

until the end of the time series, have been set as t2. This gave us for each time series three daily

sets of bubble indicators. Next, the past bubble warning indicator values and their dates had to

be de�ned and tested. For this, the daily sum of the indicator values which were higher than

20% where taken for the performance evaluation. By doing so, around 30'000 data points have

been de�ned for the analysis.

These indicate a past LPPL pattern in the CDS spreads and predict a change of regime in the

future. Based on that, the alarms were evaluated to check whether their indication of a change in

the CDS spread trajectory was correct. For this the CDS spreads trajectory has been compared

before and after the date of the alarm by using the average daily growth rate before and after the

alarm. An average daily growth which changed the direction after the alarm was interpreted as a

good a prediction where no change in direction was interpreted as a bad prediction. A change in

the direction of the growth rate is de�ned as a change from a positive to a negative growth and

vica versa. The time window used to calculate the average daily growth rate before and after

the alarm was set equal to the average of the minimum and maximum time window of the time

window set which has been used for the calculation of the analysed indicators. Two examples

for a bubble alarm which were categorized as good predictions and two examples which were

categorized as wrong predictions are visualized in the �gures (??) - (??).

37

01/01/13 01/01/14 01/01/15 01/01/16114.29

196.9543

01/01/13 01/01/14 01/01/15 01/01/16−0.5

−0.25

0

0.25

0.5

DS LPPL ConfidenceDS LPPL Trust

01/01/13 01/01/14 01/01/15 01/01/16−0.5

−0.25

0

0.25

0.5

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 7: Good Prediction: Sovereign Italy CDS spreads, short time window (50;100)

01/01/13 01/01/14 01/01/15 01/01/1683.39

173.67

01/01/13 01/01/14 01/01/15 01/01/16−0.5

−0.25

0

0.25

0.5

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 8: Good Prediction: Sovereign Spain medium window (150;250)

38

01/01/12 01/01/13 01/01/14 01/01/15 01/01/1698.707

158.8932

219.0795

01/01/12 01/01/13 01/01/14 01/01/15 01/01/16−0.5

−0.25

0

0.25

0.5

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 9: Wrong Prediction: Corporate US Containers long window (250;500)

01/04/12 01/07/12 01/10/12 01/01/13 01/04/13

210.824

01/04/12 01/07/12 01/10/12 01/01/13 01/04/13−0.5

−0.25

0

0.25

0.5

DS LPPL ConfidenceDS LPPL TrustSpread MidDS LPPL ConfidenceDS LPPL Trust

Figure 10: Wrong Prediction: Corporate Us Real Estate Investment Trust short window

(50;100)

Based on the performance evaluation of the historical indicator values, the actual indicators

of the same time series have been interpreted. In order to make the interpretation more robust,

the average DS LPPL trust and DS LPPL con�dence values of the last 30 days have been taken

for the analysis.

39

5.3 Summary

With exception of the US treasury bond index, the analysis has been made based on CDS

spread indices. The main reasons for this approach are that CDS spreads re�ect the credit risk

estimation of the market without the need of deducting a risk free interest rate and also the fact

that CDS have a constant maturity. The analysis has been made on 103 time series of sovereign

and corporate CDS spread indices. For the analysis the DS LPPL con�dence and the DS LPPL

trust indicators have been used. They have been calculated based on di�erent time window sets.

The research has been divided into a past performance evaluation of the indicators and into a

analysis of the current situation. For the past performance evaluation the indicator values have

been calculated for each day in the past. In the following, the sums of the indicator values which

was higher than 20% in the past have been used for the evaluation of their predictive power in

the future. For this, it has been analysed whether the growth rate has reversed after the bubble

alarm as predicted from the indicator or not. This method has been designed to capture the

future predicted change of regime of the LPPL model. It has to be criticised that a reverse in the

growth rate stands just for one type of a change in regime but not for all types in general. Other

types could be decreasing or stabilising growth rates after the alarm. These are not captured

with this method. Another point which can be made is the time period after the alarm that is

used for the analysis. It has the same length as the time period which is used for the diagnosis

of the bubble. This implies the assumption that the change of regime after the alarm can be

captured in the same time window on which the LPPL pattern can be diagnosed before the

alarm. The problem with that is that the LPPL model has no solution after its critical point.

This means that the time period of the new regime is also not de�ned. Knowing this, it could

be possible that the bubble bursts in a shorter time period and then later, the same bubble

as diagnosed before would develop within the analysed time period. This could have the e�ect

that no reverse in average daily growth rate could have been measured while a reverse would

have been diagnosed in a shorter time period. A solution for this would be to de�ne di�erent

time periods for the analysis after the peak. All in all the method for the evaluation for the

historical performance of the indicators does not capture all types of changes in regimes but

provides robust results due to its simplicity.

For the bubble diagnosis of present CDS indices, the average and mean trust values of the last

30 days have been taken and interpreted based on the results of the past indicator performance.

40

6 Results & Conclusion

6.1 Past Perfomance Evaluation

The performance of the indicators for all time windows and for each time window set separately

is summarized in the tables(??)-(??). The column % correct predictions describes in percentage

how many times the alarm was able to predict a change from a positive to a negative or from a

negative to a positive growth rate on average. The performance of the indicators have also been

di�erentiated for di�erent average indicators values.

Table (??) summarizes the performance of the indicators for all time window sets. In 54.7%

of the cases, the alarms predicted a change in the growth rate correctly. With higher average

indicator values the percentage of correct predictions also increased in general. All of the di�erent

thresholds for the di�erent average indicator values predicted in more than 50% of the cases

correctly a reverse in the growth rate.

Average Indicator Value Obervations % correct predictions

10 % - 15 % 9335 50.69%

15 % - 25 % 11560 53.97%

25 % - 40 % 7833 59.23%

40 % - 70 % 2692 58.17%

70 % - 100 % 106 66.03%

10 % - 100 % 31526 54.7%

Table 5: Prediction Quality: All time window sets

41

Table (??) summarizes the performance of the small time window set. In 46.28% of the cases,

the alarms predicted a change in the growth rate correctly. Interestingly, no dependency of the

percentage of correct predictions on the average indicator value can be observed.

Averag Indicator Value obervations % correct predictions

10 % - 15 % 3502 46.45%

15 % - 25 % 3460 46.24%

25 % - 40 % 1522 45.92%

40 % - 70 % 332 46.68%

70 % - 100 % 2 0%

10 % - 100 % 8818 46.28%

Table 6: Prediction Quality: small time window set

Table (??) summarizes the performance of the medium time window set. In 45.06% of

the cases, the alarms predicted a change in the growth rate correctly. The percentage of correct

predictions increased with higher average indicator values up to 40%. After it started to decrease

with higher average indicator values.

Averag Indicator Value obervations % correct predictions

10 % - 15 % 2715 44.67%

15 % - 25 % 3408 45.53%

25 % - 40 % 2359 48.74%

40 % - 70 % 902 36.03%

70 % - 100 % 41 17.07%

10 % - 100 % 9425 45.06%

Table 7: Prediction Quality: medium time window set

42

Table (??) summarizes the performance of the long time window set. In 67.14% of the cases,

the alarms predicted a change in the growth rate correctly. The percentage of correct predictions

increased with higher average indicator values from 60.67% to 100%.

Averag Indicator Value obervations % correct predictions

10 % - 15 % 3118 60.67%

15 % - 25 % 4692 65.79%

25 % - 40 % 3952 70.62%

40 % - 70 % 1458 74.48%

70 % - 100 % 63 100%

10 % - 100 % 13283 67.14%

Table 8: Prediction Quality: long time window set

When comparing the performance of the average indicators on di�erent time window sets

whether they have predicted correctly a reverse in the growth rate or not, the results can be

summarized as follow: All of them predicted in more than 44% of the cases correctly a reverse

in the growth rate. The indicators with the large time window set had the best performance

with over 67.14% correct predictions. With higher average indicator values the performance of

the indicators increased with the large time window set. This statement can not be made for

the other time window sets.

43

6.2 Bubble Diagnosis

For the actual diagnosis of bubbles the mean and median indicator values of the last 30 trading

days7 of each timeseries have been calculated. The highest positive and negative bubble alarms

for each time window set are shown in the tables (??) and (??). It has to be noted that because

the presented indicator values refer to CDS spreads, when a positive bubble is diagnosed in the

spreads that this is approximately equal to a negative bubble in the bond prices. As already

explained, this is due to the negative price/yield relationship. By looking at both tables it can be

concluded that according to the indicators many CDS spread trajectories in the past, followed a

LPPL bubble pattern. There are di�erent sovereign and corporate indices for which the indicators

give bubble warnings. Because of the past performance evaluation of the indicators on di�erent

time window sets, the bubble warnings, based on the large time window set, are de�ned as most

accurate. Because of that the following bubble diagnosis is based on the indicators which were

calculated with the large time window set.

The highest positive alarm signals have been found in the CDS spreads of sovereign 10 year senior

unsecured Dutch and German bonds. The highest negative bubble alarms have been found for

the US metals mining and the US Manufacturing CDS spread indices. For these indicators the

chance for a reversal in the growth rate in the next 375 trading days is estimated at over 60%

and for the Dutch CDS spreads even over 70%

For the other indices which were calculated based on shorter windows, with a average indicator

value over 15% or under -15% the probability of a reversal in the growth rate is estimated at

over 44%. This means that there is a higher probability that the trend continues.

Index Win.Set mean (tr) mean (co) median (tr) median (co) mean(tr + co)

Netherlands large -33.85% -51.88% -35% -50.59% -42.86%

Germany -11.96% -26.34% -12% -27.88% -19.15%

Greece -23.8% -1.58% -20% -1.19% -12.69%

US: ConsumerGoods -11.62% -11.87% -11.5% -11.55% -11.75%

US: HomeBuilders -3.54% -16.66% -4% -18.32% -10.1%

US: BuildingProducts -10.87% -6.67% -11% -6.77% -8.77%

Switzerland -10.67% -4.17% -9% -4.38% -7.42%

Netherlands medium -8.88% -24.09% -8% -17.64% -16.49%

United States of America -3.7% -29.03% -3.5% -31.37% -16.37%

US: HealthcareFacilities -2.62% -23.59% -2.5% -27.45% -13.11%

Switzerland -7.72% -7.4% -8.5% -7.84% -7.56%

Germany -1.29% -13.02% -1% -11.76% -7.16%

Netherlands small -5.17% -24.08% -5% -20.79% -14.62%

United States of America -5.22% -19.7% -5% -20.79% -12.46%

Greece -2.58% -21.81% -2% -23.76% -12.19%

Table 9: Negative Bubble Alarms

707.07.15 - 17.08.15

44

Index Win.Set mean (tr) mean (co) median (tr) median (co) mean(tr + co)

US: MetalsMining large 35.91% 5.08% 31% 5.57% 20.5%

US: Manufacturing 7.77% 26.46% 6% 27.09% 17.11%

US: Publishing 17.24% 2.68% 16% 0.39% 9.96%

US: Serviceother 2.79% 9.22% 1.5% 3.58% 6%

US: Railroads 2.54% 8.49% 2% 9.96% 5.52%

Turkiy 7.33% 3.02% 8% 1.59% 5.17%

US: MetalsMining medium 18.5% 61.03% 18.5% 62.74% 39.76%

Norway, Kingdom of (Government) 9.14% 42.82% 9% 45.09% 25.98%

US: RealEstateInvestmentTrust 15.82% 26.43% 16% 29.41% 21.13%

US: Railroads 14.32% 22.32% 16% 21.56% 18.32%

US: Manufacturing 5.79% 26.43% 3% 31.37% 16.11%

US: Publishing 4.82% 22.39% 0% 0% 13.6%

US: Airline 1.01% 13.78% 1% 13.72% 7.4%

US: MetalsMining small 6.67% 36.76% 7% 39.6% 21.71%

US: Railroads 17.19% 18.49% 16% 14.85% 17.84%

US: HealthcareSupply 7.27% 24.33% 7% 22.77% 15.8%

US: Manufacturing 5.98% 24.04% 5% 22.77% 15.01%

Norway, Kingdom of (Government) 6.51% 21.75% 6.5% 17.82% 14.13%

US: RealEstateInvestmentTrust 10.01% 12.8% 11.5% 9.9% 11.41%

Canada 6.96% 13.41% 6% 5.94% 10.19%

Eu:Service � other 9.67% 10.66% 3% 0.99% 10.17%

Eu:Conglomerate/Diversi�ed Mfg 7.58% 11.88% 0% 0% 9.73%

Eu:Service Company 7.5% 9.96% 3% 0.99% 8.73%

US: Serviceother 2.5% 13.82% 1% 11.88% 8.16%

US: OilGas 3.79% 11.59% 2% 8.91% 7.69%

India 5.51% 9.58% 4% 3.96% 7.54%

US: EnergyComp 4.88% 10.09% 3% 12.87% 7.48%

US: EnergyComp 4.88% 10.09% 3% 12.87% 7.48%

Belgium 4.64% 10.09% 4% 8.91% 7.36%

Eu:Cable/Media 4.24% 10.38% 4% 4.95% 7.31%

Eu:Building Products 7.95% 6.64% 7% 2.97% 7.29%

Eu:Manufacturing 3.32% 8.68% 1.5% 2.97% 6%

Table 10: Positive Bubble Alarms

45

6.3 Conclusion

As seen in the results of the performance evaluation,n while the average indicator values with

the small and medium time window sets predicted in under 47% of the cases a correct reverse in

the growth rate, the average indicator values with the large time window set predicted it in over

67% of the cases correctly. Also interesting was, that while the prediction quality increased with

higher average indicator values which were calculated with the large time window set, this was

not always the case in the analysis with the other time window sets. Because of these facts it is

concluded that bubbles in �xed income markets mainly develop in longer time periods between

250 and 500 trading days. Since the shorter time windows give a good indication that the trend

will persists, they can be de�ned as good indicators of momentum. Another questions is, whether

these 67% correct predictions with the large time window set is a good result or not. Since the

direction of the growth rate in the future can stay the same or reverse itself, two outcomes are

possible. So when we �ip a coin, the chance to correctly predict the outcome is also 50%. The

large time window set preformed better than the �ip of a coin. Based on that the performance

of the indicators which were calculated with the large time window set is evaluated as good and

useful. It has to be noted that, with the comparison of the performance of the indicators with

the �ip of a coin, it is assumed that the probability of a reverse in the average daily growth rate

itself and the probability of no reverse over a longer time period is distributed equally. This has

not to be right. For example in the case of a developing bubble, growth rates follow a trend

over longer periods until the bubble bursts and the growth rate reverses itself at one point of

time. This may make a reverse of the growth rate direction less probable than 50%. This is what

we see in the probabilities that were calculated using shorter time windows. For a more precise

conclusion of these results, the same performance test should be done by picking random dates

in the same time series and then analyse how many times these random dates could predict a

reverse of the growth rate. After the performance of the random dates should be compared with

the performance of the bubble indicators. This could be a good follow-up research topic

With the large time window set, the smallest average indicator values from 10 - 15% already

predicted in more than 60% correctly the growth rate reverses. This gives the conclusion, that

with already small bubble alarm indicators a change in the growth rate direction can be expected

with more than 60%. This is an important result because these smaller signals may provide an

earlier warning of a developing bubble than the higher indicator. For the large time window

set, the prediction quality of the average indicator value increased with higher average indicator

values. This increases the usefulness of both indicators even more. The highest indicator values

from 70% to 100% should be interpreted carefully because of their small number of observation

of 63. Further the indicators separately could be evaluated.

46

7 Summary

In this research �xed income markets have been analysed for developing bubbles. For this pur-

pose, �xed income securities and their market have been characterised. It has become clear that

the pricing depends next to the credit risk of the issuer, on other fundamental factors. Di�erent

ways to estimate the credit risk have been introduced. Due to practical aspects, CDS spreads

have been chosen for the analysis. After that, �nancial bubbles and their general de�nition have

been described. It has been argued that the diagnosis of �nancial bubbles by comparing market

with fundamental values is a complex task with a huge error potential. Based on this knowlege,

the de�nition of Sornette D. and Cauwels P. (2014) has been introduced. Furthermore it has

been argued that the main advantage of their de�nition of bubbles is that they can be diagnosed

by scanning the price pattern for a hyperbolic growth decorated with oscillations. Assuming

that this is caused by positive feedback mechanisms of the investors. Next the Log Periodic

Power Law model has been introduced. It is supposed to represent the price pattern of a bub-

ble. The procedure how to �t the model to the data and criteria for diagnosing a bubble based

on the model �tting has been described and de�ned. Linked to this the two bubble warning

indicators DS LPPL con�dence and DS LPPL trust have been introduced. Both are making

the diagnose more robust concerning the de�ned time period of the bubble and the distribution

of the residuals. After that, the data which should be analysed have been described. 103 time

series of corporate and sovereign CDS spread indices have been gathered from Thomson Reuters

Datastream and Thomson Reuters Eikon. They have been scanned for bubbles with the two

previously mentioned indicators with di�erent time window sets. For a better interpretation

of the actual indicator values, the past performance of the indicators has been analysed. The

indicators, calculated with the large time window set showed the best performance with over

60% correct predictions of a reverse in the growth rate. This result has been interpreted as very

useful. The results based on the lower time window sets performed worse than 50%, which was

regarded as good indicators for momentum. Based on these �ndings, it has been concluded that

the time period for a bubble in �xed income markets is between 250 and 500 trading days. In the

analysis of the actual indicator values positive and negative bubble warning signals have been

found for sovereign and corporate CDS spread indices.

Both indicators together have been calculated and were used for the �rst time in CDS markets.

The indicators provide a warning system for �nancial bubbles. The more it can be proved that

their warning signals are accurate, the more investors start to rely on them. In an optimistic

scenario early warning indicators could prevent investors to continue imitating the price move-

ment of the asset, which could stop a developing bubble. In this scenario huge economic losses

could be prevented. The past performance evaluation of this thesis makes the indicators more

reliable and with that this scenario comes a step closer.

47

References

Altman E., Resti A. and Sironi A. (2004). Default Recovery Rates in Credit Risk Modelling: A

Review of the Literature and Empirical Evidence, Economic Notes by Banca Monte

dei Paschi di Siena SpA, Vol. 33, No. 2, 183�208

(http://people.stern.nyu.edu/ealtman/Review1.pdf)

Blanchard O. and Mark Watson M. (1982). Bubbles, Rational Expectations and Financial

Markets. National Bureau of Economic Research. Working Paper No. 945.

(http://www.nber.org/papers/w0945)

Cauwels P., Bastiansen K. and Younis A. (2009). Modelling Flash: Will CDS spreads tumble in

February? Fortis Bank Investment Research, 27.01.2009.

Coudert V. and Gex M. (2010). Credit default swap and bond markets: which leads the other?.

Banque de France: Financial Stability Review No. 14 , 161-167.

(http://www.banque-france.fr)

Du�e D. and Singleton J. (2003). Credit risk: pricing, measurement, and management. Prince-

ton series in �nance.

Eatwell J.(1987). The New Palgrave: a Dictionary of Economics.

Fama E. (1975). Short-Term Interest Rates as Predictors of In�ation. The American Economic

Review Vol. 65 No. 3, 269-282.

(http://teach.business.uq.edu.au/courses/FINM6905/�les/module-

2/readings/Fama%20Real%20Rate.pdf)

Fama E. (1970). E�cient Capital Markets: A Review of Theory and Empirical Work. Journal

of Fiance, Vol. 25, Issue 2, 383-417.

(http://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1970.tb00518.x/abstract)

Filimonov V. and Sornette D. (2013). A stable and Robust Calibration Scheme of the Log Pe-

riodic Power Law Model. Physica A Vol. 392, 3698�3707.

(http://arxiv.org/abs/1108.0099)

48

Garber P. (1990). Famous First Bubbles. The Journal of Economic Perspectives Vol. 4 No. 2,

35-54.

(http://grizzly.la.psu.edu/ bickes/garber.pdf)

Garth S. (2013). Is the bond markets a bubble: Money Management February 28, 2013, 18-20.

Jacobsson E. (2009). How to predict crashes in �nancial markets with the Log-Periodic Power

Law, Mathematical Statistics Stockholm University.

(http://www2.math.su.se/matstat/reports/serieb/2009/rep7/report.pdf)

Jorion P. (2009). Financial Risk Manager Handbook, 5th Edition. John Wiley & Sons, Inc.,

Hoboken, New Jersey.

Johansen A. and Sornette D. (2000). Evaluation of the quantitative prediction of a trend rever-

sal on the japanese stock market in 1999, International Journal of Modern Physics C,

Vol. 11, Issue 02, 359-364.

(http://arxiv.org/abs/cond-mat/0002059)

Packer F. and Zhu H. (2005). Contractual Terms and CDS Pricing. BIS Quarterly Review,

01.03.2005.

Petr G. and Dean F. (2013). Everything you always wanted to know about log-periodic power

laws for bubble modeling but were afraid to ask. The European Journal of Finance Vol.

19 No.5, 366-391.

(http://www.tandfonline.com/doi/abs/10.1080/1351847X.2011.601657)

Shiller, R.(2006). Irrational Exuberrance. Crown Business 2. edition.

Siegel J.(2003). What Is an Asset Price Bubble? An Operational De�nition. European Financial

Management, Vol. 9, No. 1, 2003, 11 � 24.

(http://papers.ssrn.com/sol3/papers.cfm?abstract_id=388239)

Sornette D. (2009). Why Stock Markets Crash: Critical Events in Complex Financial Systems.

Princeton University Press.

49

Sornette D. and Cauwels P. (2014). The Illusion of the Perpetual Money Machine and what it

bodes for the future, Risks 2, 103-131.

(http://ssrn.com/abstract=2191509)

Sornette D. and Cauwels P. (2015). Managing risks in a creepy world, Journal of Risk Manage-

ment in Financial Institutions Vol. 8, 83-108.

(http://ssrn.com/abstract=2388739)

Sornette D. and Cauwels P. (2015). Financial bubbles: mechanisms and diagnostics Review of

Behavioral Economics Vol. 2 No.3.

(http://ssrn.com/abstract=2423790)

Sornette D., Woodard R., Yan W. and Zhou W. (2013). Clari�cations to Questions and Criti-

cisms on the Johansen-Ledoit-Sornette Bubble Model, Physica A Vol. 392 No. 19,

4417-4428.

(http://arxiv.org/abs/1107.3171)

Sornette D., Demos G., Zhang Q., Cauwels P., Filimonov V. and Zhang Q. (2015). Real-time

prediction and post-mortem analysis of the Shanghai 2015 stock market bubble and

crash, Journal of Investment Strategies Vol.4 No.4, 77-95.

(http://ssrn.com/abstract=2647354)

Veronesi P. (2010). Fixed Income Securities: Valuation, Risk and Risk Management, Wiley: 1

edition (January 12, 2010).

Wosnitza J. and Denz C. (2013). Liquidity crisis detection: An application of log-periodic power

law structures to default prediction. Physica A Vol. 392, 3666-3681.

(https://www.uni-muenster.de

/imperia/md/content/physik_ap/denz/publikationen/2013/07_wosnitza.pdf)

Wosnitza J. and Leker J. (2014). Why credit risk markets are predestined for exhibiting log-

periodic power law structures. Physica A Vol. 393, 427-449.

(http://www.sciencedirect.com/science/article/pii/S0378437113008224)

Financial Crisis Observatory. http://www.er.ethz.ch/�nancial-crisis-

observatory/description.html, 04.08.15.

50

Investopedia (a) . http://www.investopedia.com/terms/a/accruedinterest.asp, 03.06.15.

Investopedia (b) . http://www.investopedia.com/terms/r/risk-freerate.asp, 03.06.15.

Investopedia (c) . http://www.investopedia.com/terms/credit-event.asp, 24.09.15.

Investopedia . http://www.investopedia.com/terms/y/yieldtomaturity.asp, 03.06.15.

The Wall Street Journal (2010) . Jeremy Siegel and Jeremy Schwartz, The Great American

Bond Bubble: http://www.wsj.com/articles/SB10001424052748704407804575425384002846058,

08.07.15.

Wikipedia (2015) . https://en.wikipedia.org/wiki/Metaheuristic, 03.06.15.

51

Selbständigkeitserklärung

Ich erkläre hiermit, dass ich diese Arbeit selbstständig verfasst und keine anderen als die angegebe-

nen Quellen benutzt habe. Alle Stellen, die wörtlich oder sinngemäss aus Quellen entnommen

wurden, habe ich als solche gekennzeichnet. Mir ist bekannt, dass andernfalls der Senat gemäss

Artikel 36 Absatz 1 Buchstabe o des Gesetzes vom 5. September 1996 über die Universität zum

Entzug des aufgrund dieser Arbeit verliehenen Titels berechtigt ist.

Datum:

Unterschrift:

52