Relation between Credit Spread and Default Probability

[Yan He]

I have a question when trying to find the continuous function between credit spread and probability. Say, we have about 50 issuers bond and CDO in a portfolio. And I can get the credit spread for each of the issuer. I try to get a continuous function or a curve, if given spread, then I can calculate default probability. From the rating of each issuer, I get the default probability.

After plot the default and spread, I did regression analysis using Excel. I ended up with two functions: one is a 3rd degree polinomial with very small coefficients, the other is an exponential function. The residual for both are all small enough, but R-square of exponential is much higher...like .975. But regression from MATLAB shows poly graph fits better.

With only spread as one independent, did I get the correct function ? Or any good idea about how can I get the good curve ? :-k:-k:-k

Thanks a lot. :smt024

 

[naeron]

Commenting on your correlation number which is very high I can say that in markets like today the cross sector correlation go high. The analysis you did will only tell you the information about the correlation trade. For e.g. you can say that correlation is very high right now and in the near future it can go high further so you set a trade in the underlying so that they move in calculated correlation way.
Other thing that you can do is to do a correlation analysis with the 15 day lag and 30 day lag .. this will give some predictive power to the relations you will get in the securities..

Feel free to comment on my thoughts..
Nalin Aeron

 

[John]

I have a question when trying to find the continuous function between credit spread and probability. Say, we have about 50 issuers bond and CDO in a portfolio. And I can get the credit spread for each of the issuer. I try to get a continuous function or a curve, if given spread, then I can calculate default probability. From the rating of each issuer, I get the default probability.

After plot the default and spread, I did regression analysis using Excel. I ended up with two functions: one is a 3rd degree polinomial with very small coefficients, the other is an exponential function. The residual for both are all small enough, but R-square of exponential is much higher...like .975. But regression from MATLAB shows poly graph fits better.

With only spread as one independent, did I get the correct function ? Or any good idea about how can I get the good curve ? :-k:-k:-k

Thanks a lot. :smt024

is the credit spread you are referring the credit default swap spread? if so then there's already a functional relationship binding probabily of default (cumulative probability curve) and credit default swap spread, and it is through this equation, PV premium leg = PV contingent payment leg. So if you have either info (i.e. cumulative default probability curve or term structure of cds spread), you are derive the other one. Note that you also need the yield curve to discount the cash flows.

 

[RussianMike]

where do u get default probs from? Bond prices? I'm thinking about CDS of ABS single-name tranches

 

[John]

where do u get default probs from? Bond prices? I'm thinking about CDS of ABS single-name tranches

If there's CDS of ABS traded, then you have a 'quoted' and traded credit spread (the best measure credit risk). Assuming a recovery rate, you can back out the default probability for that traded maturity. See the attached JPM paper.

You can also get it from cash bonds, in which case you get an 'implied' or artificial credit spread over the benchmark curve (either treasury yield or swap curve) at the maturity point you want. This is not as good a measure, since typically you have a mismatch of maturity between your bond and the benchmark, so interpolation is needed.

 

[fixedincomenyc]

I have a question when trying to find the continuous function between credit spread and probability. Say, we have about 50 issuers bond and CDO in a portfolio. And I can get the credit spread for each of the issuer. I try to get a continuous function or a curve, if given spread, then I can calculate default probability. From the rating of each issuer, I get the default probability.

After plot the default and spread, I did regression analysis using Excel. I ended up with two functions: one is a 3rd degree polinomial with very small coefficients, the other is an exponential function. The residual for both are all small enough, but R-square of exponential is much higher...like .975. But regression from MATLAB shows poly graph fits better.

With only spread as one independent, did I get the correct function ? Or any good idea about how can I get the good curve ? :-k:-k:-k

Thanks a lot. :smt024

The way your problem description is stated, I would think you were trying to derive a FTD-pricer of sorts, for a n-th to default structure. Which would make sense because the premium/spread pricing is a coupla probability to default calculation, and is similiar to what you have described. However, your correlation calcs don't make much sense to me. Are you trying to calculate a tranche premium, and relate its changes to the underlying credits? If you are, then we are talking about a "credit delta" on the tranche delta, which really is a concept that borrows PV01 delta from equities but applies it to the fixed income space. Tranche MTM to Credit MTM for a 0.01 bps move. Obviously this approach would have transactional costs due to rebalancing, so you would use a "average delta" instead of each of the credits (i.e. the appropriate index CDX/ABX/CDOx, etc.)

 

[fixedincomenyc]

where do u get default probs from? Bond prices? I'm thinking about CDS of ABS single-name tranches

Mike,

The major rating agencies: fitch, moodies, and s&p publish the default probabilities you speak of based on their proprietary models. They look at peer rating class, then evaluate historical credit-default loss experience. BBB is something like below investment grade is something like 4% in 10 years, and tripple-A is 1% in 10 years. Plenty of literature is avail at the rating agency sites for issuers/cdos.

 

[Yan He]

The way your problem description is stated, I would think you were trying to derive a FTD-pricer of sorts, for a n-th to default structure. Which would make sense because the premium/spread pricing is a coupla probability to default calculation, and is similiar to what you have described. However, your correlation calcs don't make much sense to me. Are you trying to calculate a tranche premium, and relate its changes to the underlying credits? If you are, then we are talking about a "credit delta" on the tranche delta, which really is a concept that borrows PV01 delta from equities but applies it to the fixed income space. Tranche MTM to Credit MTM for a 0.01 bps move. Obviously this approach would have transactional costs due to rebalancing, so you would use a "average delta" instead of each of the credits (i.e. the appropriate index CDX/ABX/CDOx, etc.)

Hi, Fixed-income, thank you for the reply. I am actually just kinda of exploring a relation ( not simply correlation) between spread and default probability. From trader's report, I am given rating and spread of each issuer, but rating itself wouldn't make too much sense. I try to get sth. that can kind of predicting the default prob. from spread. (DV01 and CS01 are also given)

As long as "credit delta", can you give me more detail? Thanks.