Credit Default Swaps for Dummies: Part V - Correlation and Synthetic CDOs

In Part I of Credit Default Swaps for Dummies we explained single name CDS.
In Part II of Credit Default Swaps for Dummies we explained CDS indices.
In Part III of Credit Default Swaps for Dummies we explained asset securitization.
In Part IV of Credit Default Swaps for Dummies we explained synthetic CDOs.
In Part V of Credit Default Swaps for Dummies we will demonstrate how correlations between defaults affect the pricing of synthetic CDOs.

Pricing a single name CDS is fairly straightforward.  Pricing CDOs and synthetic CDOs turns out to be very difficult as it involves not only modelling the probability of default for each name but also the correlation between these defaults.  Rather than trying to explain risk neutral pricing, Poisson processes, and copulas a simple example should be able to demonstrate where the issue lies.  Let’s look at a very simple example which will demonstrate how sensitive the pricing of synthetic CDOs are to assumptions about correlation. 

Assume we are asked to insure 100 USD of company A's debt for one year.  And assume that we know that the probability that A defaults on her debt over the next year is 50% and if she defaults it will be a full default with 0 USD recovered.  Assume that you are willing to insure her for fair value.  How much would you require to be paid to write the insurance?   There is a 50% chance that A will not default in which case you pay 0 USD.  There is also a 50% of A defaulting and you end up having to pay 100 USD.  So the premium that you would require to write this insurance would be 50% * 0 USD + 50% * 100 USD = 50 USD.  That is just a very simple CDS pricing model.

Now assume that we are asked to insure 100 USD of company B's debt for one year.  Assume we know that the probability that B defaults on his debt over the next year is 50% and if he defaults it will be a full default with 0 USD recovered.  And assume that you are willing to insure him for fair value.  How much would you require to be paid to write the insurance?   Again you would require to be paid a premium of 50% * 0 USD + 50% * 100 USD = 50 USD.

Now assume you are asked to create an index CDS which is made up of 100 USD of company A and 100 USD of company B.  And assume that you are willing to insure for fair value.  How much would you require to be paid to write insurance on this basket?   The index CDS is just the sum of the two individual CDSs above.  So you would require to be a paid a premium of 50 USD to insure company A's part of the basket and 50 USD to insure company B's part of the basket or 50 USD + 50 USD = 100 USD to insure this basket.  So good so far.

Now let’s create a CDO.  Let’s create an underlying pool of 100 USD of A and 100 USD of B.  Structure it with two tranches; senior and junior.   The first 100 USD of defaults will go to the junior tranche.  Any defaults above 100 USD will go to the senior tranche.  Now how much premium would you require to agree to write the junior or senior tranche?  It turns out that it is very dependent on what you assume about the correlations in the defaults. Let's look at a few extreme cases;

Correlation in Defaults = 100%

First assume that there is a 100% correlation between the defaults of A and B.  So if A defaults then B defaults, and if B defaults then A defaults.  Let’s look at the possible outcomes and their associated probabilities

(1)    50 % chance:   A and B both default.  Loss to the pool (if it did occur) = 200 USD.  The junior tranche absorbs 100 USD of losses and the senior tranche absorbs 100 USD of losses.

(2)    50% chance:  neither A nor B defaults.  Loss to the pool = 0 USD.  Neither the senior nor the junior tranche absorbs any losses.

(3)    0% chance:  A defaults and B does not default.  Loss to the pool (if it did occur) 100 USD.  The junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

(4)    0% chance:  B defaults and A does not default.  Loss to the pool (if it did occur) 100 USD.  The junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

So what is the fair value premium to write each of the two tranches?  To calculate the expected loss from writing insurance on one of the tranches calculate ProbabilityOf(case 1)*Payout(if case 1 occurs)+ProbabilityOf(case 2)*Payout(if case 2 occurs)+ProbabilityOf(case 3)*Payout(if case 3 occurs)+ProbabilityOf(case 4)*Payout(if case 4 occurs).  So the expected loss from writing the junior tranche is 50% * 100 USD + 50% * 0 USD + 0% * 100 USD + 0% * 100 USD = 50 USD.  Similarly the expected loss from writing the senior tranche is 50% * 100 USD + 50% * 0 USD + 0% * 100 USD + 0% * 100 USD = 50 USD.   So if the correlation in defaults is 100% then you would require 50 USD to write either the senior or junior tranches.  This may look a bit strange but the first to default structure results in the two tranches senior and junior paying out equally in two situations - if either both A and B default or if neither A nor B default.  By our  assumption about correlations A defaults if and only if B defaults.  Hence senior and junior tranches have the same expected payouts.

Correlation in Defaults =  -100%

Now let’s assume that there is a -100% correlation between the defaults of A and B.  So if A defaults then B does not default, and if B defaults then A does not default.  Let’s look at the possible outcomes and their associated probabilities

(1)    0 % chance:   A and B both default.  Loss to the pool (if it did occur) = 200 USD.  The junior tranche absorbs 100 USD of losses and the senior tranche absorbs 100 USD of losses.

(2)    0% chance:  neither A nor B defaults.  Loss to the pool (if it did occur) = 0 USD.  Neither the senior nor the junior tranche absorbs any losses.

(3)    50% chance:  A defaults and B does not default.  Loss to the pool (if it did occur) 100 USD.  The junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

(4)    50% chance:  B defaults and A does not default.  Loss to the pool (if it did occur) 100 USD.  The junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

So what is the fair value premium to write each of the two tranches?   The expected loss from writing the junior tranche is 0% * 100 USD + 0% * 0 USD + 50% * 100 USD + 50% * 100 USD = 100 USD.   Similarly the expected loss from writing the senior tranche is 0% * 100 USD + 0% * 0 USD + 50% * 0 USD + 50% * 0 USD = 0 USD.  So if the correlation in defaults is -100% then you would require 100 USD to write the junior tranche but 0 USD to write the senior tranche.  This looks a bit strange but in this case the defaults never reach up to the senior tranche so he never ends up paying out.

Correlation in Defaults =  0%

Now let’s assume that there is a 0% correlation between the defaults of A and B.  So the probabilities of A and B defaulting are independent- meaning that A defaulting tells you nothing about whether B defaulted and visa-versa.  Let’s look at the possible outcomes and their associated probabilities

(1)    25 % chance:   A and B both default.  Loss to the pool (if it did occur) = 200 USD.  The junior tranche absorbs 100 USD of losses and the senior tranche absorbs 100USD of losses.

(2)    25% chance:  neither A nor B defaults.  Loss to the pool (if it did occur) = 0 USD.  Neither the senior nor the junior tranche absorbs any losses.

(3)    25% chance:  A defaults and B does not default.  Loss to the pool (if it did occur) 100 USD.  Junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

(4)    25% chance:  B defaults and A does not default.  Loss to the pool (if it did occur) 100 USD.  Junior tranche absorbs 100 USD in losses.  The senior tranche absorbs 0 USD in losses.

So now what is the fair value premium to write each of the two tranches?  The expected loss from writing the junior tranche is 25% * 100 USD + 25% * 0 USD + 25% * 100 USD + 25% * 100 USD = 75 USD.  Similarly the expected loss from writing the senior tranche is 25% * 100 USD + 25% * 0 USD + 25% * 0 USD + 25% * 0 USD = 25 USD.  So if the correlation in defaults is 0% then you would require 75 USD to write the junior tranche and 25 USD to write the senior tranche.

Summary of the above results

Correlation in Defaults	Junior Tranche Required Premia	Senior Tranche Required Premia
100%	50	50
0	75	25
-100%	100	0

In each of the above cases the probability of A defaulting was 50% and the probability of B defaulting was 50% but the pricing of the two tranches of the CDO turned out to be very different depending on what we assumed about the correlations in the defaults.  Obviously this was a very contrived example but it demonstrates why correlations matter.  The problem really comes in determining the correlations among defaults.  Accurate probabilities of default can be difficult to predict but correlations between events of default are even more difficult to predict especially when you are dealing with small probabilities to start with.

Now that you are an expert on Credit Default Swaps and synthetic CDOs lets return to the original topic - how JPMorgan lost billions see here here here

For more on these topics see

• Understanding Credit Derivatives and Related Instruments - Antulio Bonfim
• The Credit Default Swap Basis - Moorad Choudhry
• CQF Module 5
• Markit Credit Indices Primer
• Structures Credit Products - Moorad Choudhry
• Credit Default Swaps - Wikipedia
• Collateralized Debt Obligations - Wikipedia