I was going through this semi-mathematical paper by semi-academicians on valuation of Credit Default Swaps. The paper was published by, none other than, Lehman Brothers. With the benefit of hindsight it was even more interesting how these guys valued this instrument in 2003-2004 when it was just starting its meteoric rise in the financial markets. The CDS markets (total notional - I'll touch upon that in this post) increased from ~2 Trillion US $ to about 45 Trillion US $ in the early part of 2008). Consider the fact that this is about three times the size of the US GDP and you get an idea of how huge the market was. So why was everyone running after this?
What is a CDS?
There are plenty of resources that would explain in much detail what a CDS is. However, I'll touch upon it keeping in mind the valuation aspect - which will form the meaty part of this post.
So assume that you are a small bank who lends out mortgage to some guy. However you are not really sure about the creditworthiness and think that he might not honor the payments in the future. So you go to a say AIG. The conversation goes something like this -
You - Hey! I am not really sure if the guy will pay up. Can you help?
AIG - Sure we can! If you pay us a small premium (usually called a CDS spread in finance parlance) we will make sure that you get your loan amount back.
You - Great! But don't you think it is risky for you too. Why do you want to do this?
AIG - Oh we think that the house prices are going to go up. In case of a credit event (this is the term used to denote say a default or bankruptcy of the home owner) we can always sell off the houses quickly and recover the money.
Now let us look at how this affects You. The protection buyer.
1. The first impact is that it tells you that you don't really need to worry about the credit-worthiness of the home owner. In case of a default you are going to receive the money anyway. You are willing to pay a small premium to AIG for this because it is more than off-set by the commissions and the interest that you will receive - how?
The interest charged on homes is typically 4-5% higher compared to the premium you pay to AIG. Even if you receive interest payments for half the maturity of the loan - you are good to go.
2. The thought that house prices will go up in future means that you are willing to make loans that are under-collateralized. The bank usually demands a collateral of say 120k for a loan of 100k. However, since the house itself is the collateral and you think that its prices will shoot up in future you are willing to lend much more - based on the wrong assumption that the increased house price will cover the loan amount fully. The number of sub-prime loans increase!
The way it affects AIG - specifically its incentive structure is as follows.
1. It thinks that housing prices will continue to go up. This simply means that the chances of defaults - or the probability of credit event is extremely low. It then makes perfect sense for them to write as many CDS as possible. The funny thing about CDS is that anyone can buy it for any underlying asset. This would mean that people who think that housing prices will tumble (and there would be multiple credit event) can go and buy these CDS from AIG. These are not protection buyers but speculators or as they were called euphemistically - Investors.
2. It assumes that the houses can be sold immediately. If this were the case the losses should be reduced since the recovery rate (the % of asset vale recovered on sale - the asset value is the notional amount - the amount insured for) would be higher. However, this was again proven wrong as a contagion suddenly clogged the housing market and the asset recovery took more than an year!
The valuation model
The model given by the two guys at Lehman Brothers is quite mathematical in nature. However, I will try to capture the essence of the model and focus more on the assumptions - that I think were severely misplaced.
The model has two parts -
1. The valuation is done from the point of view of the protection buyer.
The protection buyer pays a premium and receives a big payoff in case of a default or a credit event. Now the idea is that a net payment of 100 - Recovery rate is made to the protection buyer. This can happen anytime during the life of the CDS. So the key is to estimate when will the default occur.
The basic idea is to assume a continuous time period in which the probability of default is given by some function of the hazard rate. This is defined as follows -
The probability of the credit event (say the party defaults) in a given small time period dt is given by k(t)dt - where k is the hazard rate. How is the hazard rate calculated?
The amount of premium that the CDS buyer pays is indicative of the likelihood of default. A high premium simply means that the seller of the protection thinks that the mortgage buyer has a high probability of default. In a liquid market these spreads are quoted for different maturities and this can be used to get the value of k. For the purpose of this post we assume it as give. However, the exact computation can be seen here. This can be now used to calculate the probability of default for any time period - say in the next 1 year. Let us call it Q.
The overall value of the swap - in terms of the likelihood of default then becomes -
Integrate [(1-R) * Q * Discount factor *(1-k)ds] over the maturity of the CDS
The protection seller pays a net amount of (1-R) in case of a default. The entity has survived till a given time (Q) and could default in the next small time period (1-k). Integrating this over the maturity period gives the expected cash outflow.
2. From the point of view of the protection seller - the cash inflow is the premium that he receives over the life of the CDS -until there is a default.
The value is simply - Spread (in bp) * Total number of payments* Probability of survival in the time period
Equating the above two equations will give us the unknown - Spread (in bp) that would be charged by AIG. To put things in perspective -
1. The spread will decrease as Recovery rate increases. The more amount of money AIG thinks it will be able to recover - the less it would charge to its clients.
2. The spread will decrease as the survival probability increases. This also makes sense. The propensity to charge increases as the underlying becomes more risky. To put some numbers to it - The spread on CDS sold on Spanish debt as the underlying was around 2000 basis points (a very high value compared to the usual values of 100-150 bp). It then becomes a gamble not whether the country would default but on how soon it would default.
Now on to the assumptions.
1. The recovery time - The time it takes for the asset to be recovered is assumed to be at a maximum of 72 days. The model itself was built on the assumption that this could happen in no time. The assumption leads to a severe underestimation of the CDS spread. In 2008 as the systemic risk skyrocketed the whole housing market suddenly dried up. Asset recovery became extremely time consuming ( about an year). Worse still - this was happening when house prices were depreciating rapidly.
2. The asset value - The authors have done some simulations in which the recovery amount is estimated to be about 70 - 80 %. When Rob Shiller talked about irrational exuberance - he talked about the biases that we have developed about stock prices always moving upward. Housing was thought to be a killer investment and again these recovery rates were affected by the enormous expenses that the banks faced in liquidating the houses.
3. Speculation - The model has been built primarily considering two counter-parties. But in 2008 these CDS were sold to investors far far away in different countries. These investors (buyers of CDS) included Hedge Funds who merely would have got huge pay-offs if the housing market crashed. Imagine the enormous amount of pressure it puts on the house prices. A decline in the house prices would create immense benefits to these investor by triggering a credit event. This is something that has been completely ignored.
4. The hazard rate - The hazard rates have been calculated based on CDS spreads quoted for varying maturities. But in an environment where the whole CDS market is in a disarray and the quotes beyond an year are not even available in the market how does one value these instruments?
The 4 points that were made bring out one important thing. CDS as an instrument is like a market clearing tool. By transferring risk from people who don't need it to people who can afford it - it brings some order in the financial markets. The problems come when these risk intermediaries become greedy. Imagine if your salary was based on the number of CDS you could sell to investors. Would you really care to evaluate the likelihood of defaults or would your focus be just on volumes. The distorted incentive structures are the key issues.
Derivatives are not weapons of mass destruction in themselves. They become dangerous when they are in the hands of wrong people.
1. The recovery time - The time it takes for the asset to be recovered is assumed to be at a maximum of 72 days. The model itself was built on the assumption that this could happen in no time. The assumption leads to a severe underestimation of the CDS spread. In 2008 as the systemic risk skyrocketed the whole housing market suddenly dried up. Asset recovery became extremely time consuming ( about an year). Worse still - this was happening when house prices were depreciating rapidly.
2. The asset value - The authors have done some simulations in which the recovery amount is estimated to be about 70 - 80 %. When Rob Shiller talked about irrational exuberance - he talked about the biases that we have developed about stock prices always moving upward. Housing was thought to be a killer investment and again these recovery rates were affected by the enormous expenses that the banks faced in liquidating the houses.
3. Speculation - The model has been built primarily considering two counter-parties. But in 2008 these CDS were sold to investors far far away in different countries. These investors (buyers of CDS) included Hedge Funds who merely would have got huge pay-offs if the housing market crashed. Imagine the enormous amount of pressure it puts on the house prices. A decline in the house prices would create immense benefits to these investor by triggering a credit event. This is something that has been completely ignored.
4. The hazard rate - The hazard rates have been calculated based on CDS spreads quoted for varying maturities. But in an environment where the whole CDS market is in a disarray and the quotes beyond an year are not even available in the market how does one value these instruments?
The 4 points that were made bring out one important thing. CDS as an instrument is like a market clearing tool. By transferring risk from people who don't need it to people who can afford it - it brings some order in the financial markets. The problems come when these risk intermediaries become greedy. Imagine if your salary was based on the number of CDS you could sell to investors. Would you really care to evaluate the likelihood of defaults or would your focus be just on volumes. The distorted incentive structures are the key issues.
Derivatives are not weapons of mass destruction in themselves. They become dangerous when they are in the hands of wrong people.
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