Investigation of the Market-Implied Sovereign Default Probabilities

Contents

1 Introduction
1.1 Problem formulation
1.2 Delimitations
1.3 Methodology

2 Theory: Credit Default Swaps
2.1 The contract and its terms
2.2 Corporate vs. sovereign CDS
2.3 Inferring default probabilities from CDS spreads
2.3.1 Default probabilities and the hazard rate λ
2.3.2 The Arbitrage-free model

3 Application of market-implied default probabilities
3.1 Default probabilities in five years for European sovereigns
3.2 Historical defaults and market-implied default probabilities

4 Analysis of CDS spreads
4.1 Greece’s, Germany’s, France’s and Italy’s spread evolution
4.2 Determinants of sovereign CDS spreads
4.3 Global risk factors
4.3.1 Stock market volatility index: VIX
4.3.2 The risk-free rate
4.3.3 Foreign Exchange rate, EUR/USD
4.4 Country specific factors
4.4.1 Local stock market
4.4.2 Domestic inflation rate
4.4.3 Unemployment Rate
4.5 Expected e↵ects of explanatory variables on the spreads

5 Regression results
5.1 Greece’s CDS spread determinants
5.2 Germany’s CDS spread determinants
5.3 Italy’s CDS spread determinants
5.4 France’s CDS spread determinants
5.5 Predicted vs. actual spreads

6 Conclusion

7 Appendix

Executive summary

This thesis investigates the default probabilities of market-implied sovereign default probabilities by using credit default swap (CDS) spreads and conducts an analysis of the spreads’ determinants.

Building on the no-arbitrage pricing model of CDS, the probabilities for Greece defaulting within the next five years from today is approximately 75%, assuming a recovery rate of 25%. Against economical intuition, the results provide a positive correlation between the recovery rate and CDS spreads.

A comparison of Greece’s default in 2012 with its default probabilities indicates that spreads closer to 2012 are much higher, hence imply higher default probabilities and therefore capture the default accurately. Though, spreads taken exactly T years before the default provide vague, i.e. very low default probabilities.

The regression analysis provides di↵erent results for the determinants of spreads of either country, though in neither model all variables are statisti- cally significant individually, e.g. Greece’s spreads can solely be explained by the country specific factor, the unemployment rate. The unexpected negative e↵ects of FX on the spreads could be an indicator of a cracked Euro-zone.

Further, the analysis of today’s actual vs. the regression’s predicted spread indicates an overestimation of the predicted spreads of Greece, Italy, Germany and France, where Greece’s absolute di↵erence between the actual and predicted spread by far is the biggest (approx. 7000 BPS). Hence, Greece also seems to be very likely to default from statistical view. Assuming that the statistical model is correct and always overestimates the spreads by a factor of 4.4, then results suggest a potential statistical arbitrage possibility where the arbitrageur would go long in the French spreads and short in the German spreads.

1 Introduction

The bankruptcy of Lehman Brothers and the subsequent financial crisis triggered an increased interest in securing the exposure against credit risk, i.e. the default of a company or sovereign. The crisis has caused many companies to default, which has increased focus on credit risk. Especially after the Greece default in 2012, there has been an increased demand in securing the exposure against sovereign default.

Credit rating agencies, such as Standard & Poor and Moody’s, rate the creditworthiness of di↵erent corporate bonds as well as sovereigns on basis of both of credit derivatives and expert judgement. Thus, the credit rating agencies conduct estimates of ratings by using both quantitative and qualitative data. However, it is possible to compute the ”raw” default probabilities for both corporates and sovereigns by using spreads of Credit Default Swaps (CDS). This CDS-implied default probability is market-implied because it can be computed directly from the CDS spread that can be observed in the market.

A CDS is a contract where the credit risk on the reference entity is transferred from the contract buyer, who has an exposure on assets of the reference entity, to the contract seller. The CDS spreads from which the market-implied default probabilities can be implied, measure the annnual premium paid by the contract buyer in a no-arbitrage model.

As it seems possible to calculate the probability for e.g. Greece defaulting within the next five years using today’s 5-year CDS spread, these market-implied probabilities can be compared to the real credit event of e.g. Greece in 2012. Similiary, the spreads of all the sovereigns that have not defaulted yet, can be compared with their market-implied default probabilities. By conducting the comparison with the countries’ default or survival and their respective default probability, it can be seen, whether the spreads do measure credit events appropriately. If they do, the results should imply high default probabilities up to five years prior to the default of the sovereign that has defaulted (assuming a CDS with five years to maturity). Conversely, low default probabilities should arise for sovereigns that have not defaulted yet.

Once the default probabilities are computed from the CDS spreads, and these are compared to an actual default and survivals, a remaining question is what determines the CDS spreads from which we compute the market-implied default probabilities. To answer this a regression analysis helps to determine the significance of di↵erent explanatory variables. Using the estimates of the regression, it is possible to conduct a comparison to the countries’ actual default probabilties and conclude whether the regression model over- or underestimates defaults.

1.1 Problem formulation

The thesis describes the elements of a CDS and its valuation with the no-arbitrage model that results in a formula for CDS-implied default probabilities. By applying the traded CDS spreads to this formular, CDS-implied probabilities for nine European sovereigns defaulting in five years from now will be computed.

A comparison of Greece’s default in 2012 with its historical spreads will be conducted in order to analyze which spreads provide the best prediction of the default in 2012. Further, the determinants of CDS spreads of four countries will be analyzed by using linear multiple regression. With the results of the regression, we will compare today’s predicted spread with its actual spread.

1.2 Delimitations

The thesis investigates the default probabilities of nine European countries from the Euro zone as well as the determinants of CDS spreads of four of these in order to contrast extremes. CDS spreads from other countries might be a↵ected by other determinants, though this is beyond the scope of this thesis, so not all potential determinants of the spreads will be analyzed.

Besides, counterparty risk on the CDS contracts, e.g. the buyer of the CDS contract will not receive the promised insurance if the counterparty defaults at the same time as the reference entity defaults, will not be taken into account, though this may reduce the value of the insurance for the protection buyer. Such a valuation adjustment is known as Credit Valuation Adjustment which is beyond the scope of this thesis, so we will assume that there is no counterparty risk in the contracts.

Further, the regression analysis’ assumptions will not be checked, though the consequences of violations of the assumptions will be considered.

1.3 Methodology

The rest of the thesis proceeds as follows: Section 2 develops the theory behind CDS and the arbitrage-free model with the aim of computing market-implied default probabilities. These probabilities are empirically compared to a real default in section 3. Section 4 analyzes the determinants of the CDS spreads, where section 5 provides the results of a regression analysis and compares model’s predicted spread of today with its actual. We conclude in section 6 and additional material can be found in the appendix.

2 Theory: Credit Default Swaps

2.1 The contract and its terms

A CDS is a type of credit derivative, whose payo↵ depends on the whether a third party, i.e. either a company (for corporate CDS) or a country (for sovereign CDS), defaults or not. It is a contract between a default protection buyer and a default protection seller to provide insurance against the risk of default by a third party, the so-called reference entity. The reference entity however, is not part of the contract. The contract gives protection until some specified maturity date T or until default, whichever occurs first. It also specifies the time for the periodic payments of the buyer, which usually are made in arrears quarterly. The buyer of the contract takes a long position in the contract, whereas the protection seller takes the corresponding short position. The buyer owns one or more assets of the reference entity, so the buyer has an exposure to these assets of the reference entity. In return to the insurance against the credit risk, which the protection buyer receives, the protection buyer makes the periodic payments, which is known as the premium leg, until the end of the contract or until a default occurs - whichever event occurs first, will stop the payments and the contract will cease (Hull, 2012, pp. 547-548).

The payment of the protection seller to the protection buyer to compensate for the loss in case of default is called the protection leg. The protection leg is equal to the di↵erence between par and the price of the cheapest to deliver asset of the reference entity on the face value of the protection (O’Kane and Turnbull, 2003, p. 3).

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Figure 1: The CDS contract (Hull, 2012, p. 549)

The notional principal is the total face value of the bonds that can be sold. The total premium, represented by the cost the protection buyer has to pay to the seller, as a percentage of the notional principal, that the buyer annually pays to the protection seller is known as the CDS Spread and is measured in basis points (BPS), where one basis point equals 0.01% (Hull, 2012, pp. 547-554). As the spread increases, the value of the contract increases for the protection buyer, because the protection buyer could unwind the position at a higher spread level (https://www.markit.com/news/Credit%20Indices%20Primer. pdf, 2008, p. 6).

The default by the reference entity is called a credit event, which is a crucial term in the contract since the contractual definition of a credit event, i.e. default can a↵ect the default probabilities, because a wider definition implies a higher default probability, vice versa. The International Swaps and Derivatives Association (ISDA) defines in ”2014 ISDA Credit Derivatives Definitions and Related Material” in Article IV a credit event as one of the following six situations (https://www.iosco.org/library/pubdocs/pdf/ IOSCOPD385.pdf, 2012, p. 13 and http://credit-deriv.com/isdadefinitions.htm):

1. Bankruptcy: Bankruptcy occurs whenever the reference entity has filed for relief under bankruptcy law (or equivalent law). Further, actions undertaken by the reference entity, such as a board meeting or meeting of shareholders where a filing of a liquidiation petition is considered, can be seen as bankruptcy and therefore trigger a credit event under ISDA definitions.
2. Obligation Acceleration: Obligation acceleration happens when the obligation becomes due and payable before it normally would have been due and payable as a result of a default by the reference entity.
3. Obligation Default: An obligation default means that the obligation becomes capable of being declared due and payable before such obligation normally would have been capable of being so declared, as a result of a default by the reference entity.
4. Failure to Pay: Failure to pay occurs whenever the reference entity fails to make, where and when due, any payments under one or more obligations. If grace periods are applicable in the trading documentation, these are taken into account.
5. Repudiation/ Moratorium: A Repudiation or Moratorium occurs whenever the reference entity or another governmental authority disaffirms, disclaims or likewise challenges the validity of the obligation. To be able to determine such an event and avoid misunderstandings, a default requirement threshold is specified.
6. Restructuring: Restructuring is a situation where the configuration of debt obliga- tions is changed so that the credit holder is unfavorably a↵ected by e.g. a reduction in the prinicipal amount or interest payable under the obligation, a postponement of payments and change of the currency. Again, a default threshold is specified to avoid any doubts.

If one of these events occurs, the protection buyer obtains the right to sell bonds issued by the reference entity for their face value. In this case, the settlement involves either cash or physical delivery. In case of cash settlement - which is most commonly used - some days after the credit event, an ISDA-organized auction process determines the mid-market value of the cheapest deliverable bond. If however, the contract specifies physical delivery, the contract buyer obtains the right to sell bonds issued by the reference entity for their face value (Hull, 2012, p. 549).

Another crucial term for a CDS contract is the recovery rate R on a bond because it impacts the expected payo↵. The recovery rate of a bond is the value of the bond’s market value a few days after a credit event, as a percentage of its face value (Hull, 2012, p. 523) which implies a loss given default (LGD) of 1 − R. From the valuation of CDS it will later be possible to infer default probabilities which incorporate the recovery rate. Without an assumption of the recovery rate that is assumed to be 25% for sovereigns in this thesis, it would not be possible to compute default probabilities. The impact on the assumption of the recovery rate on default probabilities will later in section 3.1 be analyzed.

CDS are traded on the over-the-counter market and categorized as single-name credit derivative as its payo↵ depends on the creditworthiness of a single company or a single sovereign (Hull, 2012, p. 547).

Even though a CDS contract seems to be like an insurance, there are di↵erences be- tween a CDS and an insurance. It seems obvious that an insurance contingent event is based on other things than a credit event (e.g. theft). Hence, di↵erent events lead to an in- surance payout, whereas only occurance of one of the six strict definitions of a credit event leads to a CDS payout (Augustin et al., 2014, p. 8). However, a regular insurance company might also specify its insurance contingent events explicitly, though the definitions are dif- ferent to a CDS payout due to the nature of the contract. A more crucial di↵erence is that for an insurance payout, the insurance buyer does need to incur an actual loss (e.g. loss of a wallet), whereas the buyer of a CDS contract does not need to incur an actual loss on the reference entity. Since sovereign CDS insure against a default of a sovereign, one must not incur an actual loss on that country in order to be eligible for a CDS payout. Lastly, CDS contracts on a company or a sovereign entity’s debt are not regulated by insurance regulators, so a CDS contract buyer does not need to own the underlying debt to be able to buy the provided insurance, whereas one needs to own the insured item in a ”normal” insurance (https://www.markit.com/news/Credit%20Indices%20Primer.pdf, 2008, p. 4). However, some features remain equivalent, such as a contract between a protection buyer and seller, the periodic payments and a potential payout.

2.2 Corporate vs. sovereign CDS

After having described the elements of a CDS contract and defined a credit event section 2.1, the focus for the remainder is sovereign rather than corporate CDS since the empirical part of this thesis only takes sovereigns into account, computes their default probabilities and later detects the determinants of their spreads. However, there are several di↵erences between corporate and sovereign CDS contracts, which will be presented in the following.

Firstly, the interpretations of a credit event (i.e. default) are ambiguous since corporate credit events are bankruptcy, failure to pay and sometimes restructering. For sovereigns however, bankruptcy is replaced by repudiation or moratorium, which occurs whenever the reference entity repudiates one or more relevant obligations or declares a moratorium. Additionally, sovereign reference entitities normally trade with complete restructuring, whereas corporate CDS normally trade with modified modified restructuring or no re- structuring which indicates no maturity limitation on deliverable obligations (Augustin et al., 2014, p. 100).

Besides the credit event interpretation another di↵erence between corporate and sovereign CDS is that sovereign reference entities are characterized by less concentrated trading for 5-year CDS contracts.

Yet another di↵erence between the two types of CDS contracts is the relative impor- tance of a currency denomination of the contract for sovereign reference entities. If a sovereign defaults, there is a high risk of currency depreciation or re-denomination. Con- sider the example of the U.S. defaulting, so the payout in a foreign currency, e.g. Euro, would be more likely to be higher than the U.S. dollar payout (Augustin et al., 2014, pp. 100-101).

2.3 Inferring default probabilities from CDS spreads

Both the CDS contract and the major di↵erences between sovereign and corporate CDS have been explained so far. Now, we move on to the arbitrage-free model to infer the default probabilities from the traded CDS spreads. This is done firstly, by explaining crucial terms that are used to infer the default probabilities from CDS spreads. Secondly, these are applied to the arbitrage-free pricing model.

2.3.1 Default probabilities and the hazard rate λ

When pricing a CDS, a few terms must be explained prior to calculations. Initially, we need to distinguish between di↵erent types of probabilities. The unconditional default probability is the probability of defaulting during a certain year, say 5, as seen at time 0. The corresponding survival probability is the probability that the bond will survive until the end of the year 5 is (100% − unconditional def ault probability). If however, we want to know the default probability during a year conditional on no earlier default, then this is the unconditional default probability divided with the survival probability. This depicts a conditional default probability (with the condition of no earlier default) and is known as the hazard rate λ (Hull, 2012, p. 522). Thus, λ represents the conditional default probability, whose condition is no earlier default.

If we now take a time period other than one year into consideration, we get the time period of length Δt. Then the hazard rate λ(t) at time t is defined such that λ(t)Δt is the default probability between time t and t + Δt conditional on no earlier default.

Further, let V(t) be the cumulative probability of surviving to time t, then the conditional probability of default between time t and t + Δt is¹

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Multiplying by - V(t), we get

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Di↵erentiating with respect to t gives

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Multiplying by d(t) gives the di↵erential equation

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This is a di↵erential equation, which has solution²

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where λ (T) is the average hazard rate between time 0 and T, thus represents the average default probability conditional on no earlier default (Hull, 2012, pp. 522-523).

Having found the cumulative survival probability to time T, it is easy to find the default probability by time T, Q(T), since Q(T) = 1 - V(T) = 100% - V(T), so we insert the survival probability V(T) from (1) which gives the probability of default by time t³:

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So we see that the probability of default between 0 and T depends on the average hazard rate at time T, λ(T) and T. However, the conditional default probability still depends on the unconditional default probability which still is unknown.⁴ Thus, it is still not yet possible to determine the default probability because of the unknown unconditional default probability. A solution for this is to apply the default probabilities on the arbitragefree model that incorporates the CDS spreads, which is done the next section.

2.3.2 The Arbitrage-free model

So far, we have seen how to infer the default probabilities by using the conditional default probability λ. Though, it is not possible to infer default probabilities from these formulas yet, because the calculated probabilities depend on another conditional default probability, which also is unknown. However, this section provides a solution for this problem by using the principals of no-arbitrage.

Before turning into arbitrage-free pricing, refresh from section 2.1 that the recovery rate R is determined as the underlying’s market value after a default of a company or sovereign, as a percent of its face value. The recovery rates are strongly negatively cor- related with default rates, i.e. high default probabilities imply low recovery rates to be more likely, which is a doubly negative e↵ect. Likewise lower default probabilities imply higher recovery rates (Hull, 2012, pp. 523-524). Since the recovery rate is not a market observable input - unlike the CDS spreads and interest rates, we assume for the remainder of the thesis the recovery rate to be 25% as this is the most common used recovery rate for sovereign CDS in practice (Augustin et al., 2014, p. 112). Though, we will try to relax this assumption and compute default probabilities with recovery rate other than 25% in section 3.1.

Now, let us move on to the arbitrage-free valuation of a CDS to infer default probabilities from traded CDS spreads. To get the market-implied default probabilities, the value of a CDS is calculated under the risk-neutral (Q) measure, equivalent to most theories assumption in finance. As the risk free rate we use the German government bond yield rates as these seem to be the best proxy for the risk free rate in praxis. This will be explained further in section 4.3.2, where the determinants of the spreads will be described. However, for now we describe r simply as the risk free rate.

In order to compute the arbitrage-free default probabilities, we need to consider the expected payments of the two sides of the contract, the buyer’s premium leg and the seller’s protection leg. The expected value of the premium leg ppremium(T ) paid by the protection buyer until default or until maturity, whichever occurs first, is the present value of the premium leg’s expected value that the buyer needs to pay over the lifetime of the contract. Mathematically it is⁵

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The protection buyer has a short position in the premium leg, since he has to make the periodic payments to the seller. The protection seller takes the corresponding long position in the premium leg.

For the value of the protection leg which is the payment made by the protection seller in case of default, the expected value needs to be adjusted for the value after default, which the previously described recovery rate implements. So the expected value of the protection leg is⁶

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Here, the protection seller takes a short position in the protection leg since the seller needs to give the buyer money in case of default. The buyer takes the corresponding long position in the protection leg as he receives the amount if a credit event occurs.

Since CDS, like other derivative contracts, are assets in zero net supply because pro- tection buyer and seller have the same number of contracts outstanding, we can conduct arbitrage-free pricing. Under the assumption of arbitrage-free pricing, it would hold that the expected present value of both legs initially is equal (Augustin et al., 2014, pp. 29-30), i.e.,

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Plugging in the values achieved from equation (3) and (4) into (5) gives

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which means that the expected present value of the value of the premium leg equals the expected present value of the protection leg.

Extracting the spread s(T) from the expected value of the premium leg and (1 − R) from the expected value of the protection leg, we get

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Next, we assume constant interest rates r. If we now further assume constant hazard rates λ(T)⌘ λ, we can extract the hazard rate λ(T) first from the integral and then from the expected payments EQ0 . This results in a simplified equation, such that⁷

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where λ is a constant hazard rate.

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O’Kane and Turnbull (2003, p. 12) suggest, that this assumption of the hazard rate being a piecewise flat function of maturity time is reasonable, since it is only possible to get one piece of information about the term structure of hazard rates, given only one data point.

However, if we do not assume constant hazard rates, i.e. we need to take hazard rates that change deterministically over time into consideration, the average hazard rate λ from 0 to T can be used as an approximation in equation (8) instead. This gives the following approximation

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It is possible to observe the CDS spreads for di↵erent maturities in the market (e.g. via Bloomberg) and we assume the recovery rate R to be25%. Hence, there is only one unknown, the hazard rate λ, which now can be calculated as equations (8) and (9) indicate. Having found the approximation of the average hazard rate λ by equation (9), it is now possible to compute the default probabilities by inserting equation (9) into equation (2) which gives

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Finally, we are now able to compute the market-implied default probabilities by using CDS spreads quoted in the market and assuming a recovery rate of 25%. Though, there is still remaining uncertainty because of R.

3 Application of market-implied default probabilities

3.1 Default probabilities in five years for European sovereigns

By the no-arbitrage model presented in section 2.2, we found the default probability, whose variables are the CDS spread, the maturity of the CDS and a recovery rate. All of these are known, except for the recovery rate, which is assumed to be 25%. Thus, we can now conduct the computations of CDS-implied default probabilities on di↵erent sovereigns. We will then change the recovery rate to see if the assumption of 25% a↵ects the probabilities. Because the default probability relies on the principles of no-arbitrage, we assume no-arbitrage from now on.

The underlying data are taken from Markit over the sample period from January 2003 until March 2015 for nine European sovereign countries, all of those from the Euro zone. Though, all spreads of the CDS contracts are denominated in USD in the raw data which we later convert into BPS. Data are daily-trading days, so they represent the daily CDS-spreads for 12 years. As a restructering clause only ”Complete Restruc- turing” (CR) is taken into account, because this definition of credit event includes the broadest range of credit events. This also means that more events can trigger a pay- ment to the contract buyer which implies that the CDS protection is more valuable than it is in a more narrow definition (https://www.markit.com/news/Credit\%20Indices\ %20Primer.pdf,2008,p.28). The spreads span over maturities of 1, 2, 3, 5, 7 and 10 years. Since the CDS with maturity of five years (from now on denoted as 5-Y CDS) is the most liquid and most used, the 5-Y CDS is primary used in this thesis. However, default probabilities for CDS spreads with maturity other than five are shown in Table

13 in the appendix. In order to compute the default probabilities Q(T ) of the selected countries in T = 5 years, the most recent CDS spread, i.e. March 30, 2015, is used. Taking today’s (March 30, 2015) spread of each country and plugging it to formula (10), we find the default probabilities of each country defaulting by T = 5 years and assuming a recovery rate of 25%. The results for all nine countries are presented in Table 1.

As an example, consider Germany’s 5-Y CDS that was traded at a spread of 0.001523091 on March 30, 2015, so T = 5, s(5) = 0.001523091 and R = 25%. Plugging in these numbers into equation (10) gives the following approximation

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So the market-implied probability of default for Germany within the next five years is 1.01% as of 30/03/2015, assuming R = 25%.

The striking observation of the results in Table 1 is that Greece’s default probability is very high, reaching 74.68%. Greece is also the only country that has a default probability significantly higher than 10%. Also, all of the so called PIGS (Portugal, Italy, Greece, Spain) countries have a higher default probability than any of the other countries, as their default probability is between 5.15% (Spain) and 74.68% (Greece). The remaining countries have a notably lower default probability, rising from very low 1.01% for Germany to 3.04% for Ireland.

The conclusions partly provide an answer on the main question of this thesis, since we found the CDS-implied probability for Greece defaulting within the next five years (74.68%) from today’s date, assuming R = 25%. From this observation, it is very likely that Greece will default again, taking into account the assumptions made, i.e. the recovery rate and all else equal.

Table 1: Market-implied default probabilities of 9 countries with T = 5, R = 25%, s is today’s (30/3/15) spread and Q(T = 5) is equation (10)

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As we assumed the recovery rate to be 25%, this factor could change our results. Table 2 shows the country’s default probabilities with di↵erent recovery rates, ranging from 1% to 90%. However, it is not realistic to assume either of these extremes, because the recovery rate depicts the underlying’s market value after a default, which rarely is neither R = 1% nor R = 90%.⁸ Though, for the analysis of default probabilities, we can check to what extend the default probabilities depend on the recovery rate, and if some of the results di↵er from the former results that assumed a recovery rate of 25%. The results are summarized in Table 2. Again, Q(T) is the default probability from equation (10), measured in %.

Table 2: Default probabilities for 9 countries with T = 5 and di↵erent recovery rates

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On one hand, the results are suprising because the negative correlation between de- fault probabilities and recovery rates, as described in section 2.3.2, does not hold. One can see, that with increasing recovery rate, the default probabilities also increase rather than decrease, as they should with the negative correlation proposed by Hull. Mathematically this can be explained, though. If R increases, the denominator (1 - R) decreases, which increases the fraction and hence increases the default probability Q(T). However, econom- ically it does not make sense that higher recovery rates imply higher default probabilities as the results indicate. The reason for this is that a higher recovery rate means less loss given default (LGD) (since LGD = 1 − R) which should imply a lower default probability.

On the other hand, even though di↵erent recovery rates do change the default probabil- ities according to Table 2, the relative ranking of each of the country’s default probability remains the same with di↵erent recovery rates. Thus, with either recovery rate Greece will still be most likely to default whereas Germany remains the least likely to default, ceteris paribus.

Especially the change from 50% to 70% in the recovery rate changes the default prob- ability more than an increase in R at a lower rate. This is true for all countries. Assuming di↵erent recovery rates still leads to the same conclusion of very high probabilities for Greece defaulting within the next five years. When R is very low (1%) Greece’s default probability is still 64.68%. Increasing R to 90%, it is approx. 100%, so a default within the next five years seems to be very likely with this R. Thus, R has positive correlation to the default probability and therefore does change the likelihood of a default. However, with either recovery rate Greece seems to be very likely to default again.

The difficulty in forecasting the exact recovery rate is also a discussion in current news. For example, Kathimerini claims that the European Central Bank (ECB) is cur- rently working with three scenarios concerning the recovery rate. Kathimerini calls the loss given default, (1 − R), a ”haircut”. According to Kathimerini, the three scenarios that the ECB is working with are recovery rates of 56%, 35% and 20%, so the ”hair- cuts” are 44%, 65% and 80% (Kathimerini, 2015, http://ekathimerini.com/4dcgi/_w_ articles_wsite1_1_03/05/2015_549660). These recovery rates would give default prob- abilities of approximately 90%, 80% and 73% respectively, given the results from Table 2. It can be seen that not even the ECB is sure about which recovery rate to use, which also leads to default probabilities for Greece that range from about 73% to 90%, given today’s (March, 30 2015) spread for a 5-Y CDS contract.

3.2 Historical defaults and market-implied default probabilities

So far we calculated the default probabilities for di↵erent sovereigns defaulting in five years from now. For this, we looked into today’s price and received market-implied probabil- ities telling us how likely the sovereigns are to default by time T = 5 (thus 2020) from now (2015). The question is now whether these CDS-implied probabilities are accurate predictors for future sovereign defaults. If not, this could rise the question whether the only input variable (the CDS spreads) - besides the assumed constant recovery rate - have been overall under- or overestimated. To see whether the market-implied default probabilities empirically did give a good approximation to defaults, we can look in the past and compare sovereign defaults with their default probabilities, i.e. spreads, up to T years before the default. We also compare non-defaulted sovereigns with their respective default probabilities.

In order to compare the CDS-implied probabilities with the actual outcome (default or no default), we need to find out if any of the nine countries has defaulted within our timeframe. Since our data includes CDS spreads from 2003 - 2015 for six di↵erent maturities (1, 2, 3, 5, 7 and 10 years), we should only look at defaults between 2004 and 2015. For example, the only probabilities we can get for 2004, are the ones obtained from the 1-Y CDS from 2003, because the 1-Y CDS spread can be used to calculate the probability that the countries will default within one year from 2003, hence in 2004. From the 10-Y CDS from 2003, it is possible to calculate the probability of default by 2013. Thus, is it not possible to determine the default probability for 2003 even if we have CDS spreads for 2003.⁹

The only of the nine analyzed sovereigns that has defaulted in the period between 2004 and 2015 is Greece: On March, 9 2012, the ISDA declared a restructuring credit event in respect of the debt of the Hellenic Republic. This triggered an auction to be held with the aim of determining the amount the holders of the contracts would be paid (http: //www.wsj.com/articles/SB10001424052970204603004577270542625035960).

Since Greece defaulted in 2012, the question is therefore whether the arbitrage-free pricing model gives a relatively high default probability for Greece on March, 9 2012, since a high default probability is correlated to high spreads. To answer this, we need to look whether our data for all six maturities are sufficient. As we need the default year minus the maturity of the CDS spread, i.e. 2012 - T, data we disregard the 10-Y CDS, because of the data restriction (data start from 2003). For the other CDS with maturities of 1, 2, 3, 5 and 7 years, it is still possible to compute the default probabilities for March, 9 2012 and are calculated as follows.

For Greece’s CDS spread with maturity of T = 1, the spread from March, 9 2011 is needed (its spread is 0.117419). Following the principles of section 3.1, where the spread is plugged into equation (10), the probability of default for Greece between March 9, 2011 and its default date, March, 9 2012 can be calculated. Thus,

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¹Hull, 2012, pp. 522-523

²The computations behind the solution of the di↵erential equation are behind the scope of this thesis.

³Hull, 2012, p. 523

⁴Remember, that λ is the conditional default probability which is calculated as the unconditional default probability over the survival probability.

⁵Wu, 2009, p. 10

⁶Wu, 2009, p. 10

⁷Wu, 2009, pp. 9-10

⁸Equivalently, the loss given default is rarely neither 1 − 1% = 99% nor 1 − 90% = 10%.

⁹Of course it would be possible to imply the default probability with the 10-Y CDS spread by using the spread from March, 9 2002. However, as one of the limitations in this thesis is the data that only covers spreads from 2003, this is excluded.