Defaults probability

We set pt as the probability that a bond will default in year t, and is the probability of default up to year t, while is the expected recovery rate on the bond should it default. The default probability is assumed to fluctuate over time, while the recovery rate remains constant. Therefore, the probability that the bond will not have defaulted up to the beginning of year t is given by ... 

Generally, the theoretical default spread is almost exactly proportional to the default probability, assuming a constant default probability. Generally, however, the default probability is not constant over time, nor do we expect it to be. In Figure 8.3, we show the theoretical default spread for triple-B-rated bonds of various maturities, where the default probability rises from 0.2% to 1 % over time. The longer dated bonds, therefore, have a higher aimual default risk and so their theoretical default spread is higher. Note that after around 20 years the expected default probability is constant at 1%, so the required yield premium is also fairly constant. 

Default probabilities are not known with certainty, and credit rating agencies suggest that higher risk bonds have more uncertain default probabilities. The agencies publish default rates for each rating category (which are used in credit... 

Portfolio managers must also take account of a further relationship between default risk and interest-rate risk. That is, if two corporate bonds have the same duration but one bond has a higher default probability, it essentially has a shorter duration because there is a greater chance that it will experience premature cash flows, in the event of default. 

This means that an investor who holds bonds that carry an element of default risk should in theory take this default risk into consideration when calculating the duration of his or her portfolio. In practical terms this only has an effect with unrated or junk bonds, which have default probabilities much greater than 1%. Figure 8.5 shows how the theoretical duration of a bond decreases as its assumed default probability increases. 

FIGURE 8.5 Duration of a 30-year bond relative to default probability. 

In order to solve the probability of default, reduced-form models adopt a different approach. They are mainly based on debt prices rather than equity prices. In fact, they do not take into account the fundamentals of the firm and the default event is determined as an exogenous process without considering the underlying asset movements. In addition, the models are mainly based oti X t), that is the default intensity as a function of time. In particular, these models use the decomposition of the risky rate (risk-free rate and risk premium) in order to determine the default probabilities, recovery rates and debt values. Although structural models have the advantage to foUow a reliable measure of credit risk, that is the firm value, reduced-form approach overcomes the Umitatimi in which the balance sheet is not the unique indicator of the default prediction. 

Fons (1994) studied the term structure of credit risk based on the historical default probabilities, ratings and recovery rates. In fact, he proposes a bond pricing model in which the value depends on the probability of default and average recovery rate. The model presented is based on the following assumptions ... 

Fitch uses the combination of the LTV and the affordability measure for a loan in order to arrive at a base case default probability for any particular borrower in a particular rating test. In the United Kingdom, the income multiple has traditionally been used as the measure of loan affordability, and Fitch places loans in one of five classifications based on this measure (Exhibit 11.4). 

The base case default probability will then depend on the rating being considered and the LTV of the loan. Exhibit 11.5 shows the Fitch default probability assumptions for loans where the income multiple is between 2.75 and 3.00 for various ratings tests. 

The initial default probability assumption is then adjusted (usually increased) to take into account any other important features of the loan. These factors include ... 

Mortgage Type Interest-only mortgages that are not linked to a repayment vehicle will have the base case default probability increased by a factor up to 1.33, dependent on the time to maturity of the loan. 

Equity Withdrawal Loans taken out to refinance an existing mortgage will not be penalised unless the borrower uses the opportunity to withdraw equity from the property when the default probability will be increased by 1.10-1.25x. 

Servicer Quality There is qualitative judgement based on the quality of the underwriting and servicing processes and systems. This can increase or decrease the final default probability assumption. 

It is often widely commentated that LTV is not an indicator of default probability. However, we would argue that a borrower with a 50% equity stake at risk from a potential forced sale of a property would have a greater incentive to maintain debt service payments than if the same borrower had an equity stake of, say, 20%, and so although it may not be the most important influence, the LTV of a loan could be expected to have some influence on the default rate. 

The risk analysis for CDOs performed by potential investors is necessarily different to that undertaken for other securitised asset classes. For CDOs, the three main factors to consider are default probabilities, default correlations and recovery rates. Analysts make assumptions about each of these with regard to individual reference assets, usually with recourse to historical data. We introduce each factor in turn. 

The correlation between assets in a specified portfolio is an important aspect in CDO risk analysis. Challenges exist in terms of determining what precise correlation values to use these can be correlation between default probabilities, correlation between timing of default, and correlation between spreads. The diversity score value of a portfolio attempts to measure and encapsulate these concepts by way of simplification. The higher the score, presumably the less correlated the default likelihood of each asset becomes. 

Since credit default swaps are written on the reference entities, their pricing provide information on the default probabilities of the issuer and are not subject to liquidity premia that can be present in the credit spreads of the credit risky bonds. Therefore, the term structure of credit default swap spreads for a particular issuer is used to determine the cumulative default probability of the issuer. 

Default can take place randomly over time, and the default probability can be determined using the risk-neutral transition matrix. 

A hazard rate function may be determined from the term structure of credit. The hazard rate function has its foundation in statistics and may be linked to the instantaneous default probability. 

If the time of default for a reference credit is l, then let us denote the cumulative (risk neutral) default probabilities for the reference credit as... 

The reader may notice that the value of the example default swap is close to zero this is not coincidental and does warrant some explanation. It is often heard in the market that default swap spreads are representative of default probabilities—it is clear from our example that the hazard rate, X, equals 5.00%, but the premium of the default swap is only 4.00%. The reason for this discrepancy is not complex and resnlts directly from our assumption of the recovery value, R = 20%. 

Finally, apart from these methods using direct market inputs to calculate hazard rates, a variety of analytics firms attempt to predict default probabilities by examining the fundamentals of a company. We will return to this topic in the last section of this chapter. 

It is important to note that this expected default probability does not say anything about potential correlations among the 100 credits. It is still merely a starting point for assessing the overall risk of the portfolio. Other inputs are required to reach our goal— including the principal correlation proxy for this model diversity score. 

Our portfolio, which we assume to be relatively clumped within a few industries, has a diversity score of 49. It is expected to behave as if it were composed of 49 equal-weighted, uncorrelated assets, each of which has the same expected default probability of approximately 0.97%. When defaults occur, we will assume each increment of default is 1/49 of the portfolio rather than 1/100. 

This section examines a few commercially available software packages and analytic tools designed to mitigate risk in the increasingly innovative credit derivative market. It reviews CDS data providers, examines analytic programs designed to provide expected default probabilities and theoretical prices, and highlights applications intended to simplify CDO investments. ... 

We will now turn our attention to sophisticated risk management tools. These tools are critical for companies involved in the credit derivative market. The following products are designed to produce default probabilities, the fundamental building block for effective risk management. 

Kamakura s KRIS-cr platform is a risk management system specially focused on ensuring that companies are compliant with the new Basel II Capital Accord. KRIS-cr relies on multiple models to provide daily output of default probabilities. Their current models inclnde ... 

Robert Jarrow and Stuart Turnbull, Pricing Derivatives on Financial Securities Subject to Default Risk, Journal of Finance 50, no. 1 (1995), pp. 53-86. Kamakura Risk Information Services Credit Risk Overview Kamakura s Press Release, Kamakura Launches Basel II Default Probability Service and Announces First Client, October 31, 2002. 

Calculate weighted average default probability of the portfolio from the individual rating tiers. 

Impute the rating of the portfolio from its average default probability. 

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