The way monoline insurers estimate the FAS 157 credit value adjustments (CVA) on their ABS CDO insurance portfolios vastly overstates the benefits. We propose a simple method that is more accurate, especially when the counterparty default risk is high. The counterparty default recovery rate is also a critical input.Credit Value Adjustment, Super Senior ABS CDO, Monoline insurer

Li, Hui

English

Research Papers in Economics

CVA calculation for CDS on super senior ABS CDO 
Hui Li 1 
August 2008 
 
ABSTRACT 
 
The way monoline insurers estimate the FAS 157 credit value adjustments (CVA) 
on their ABS CDO insurance portfolios vastly overstates the benefits. We 
propose a simple method that is more accurate, especially when the counterparty 
default risk is high. The counterparty default recovery rate is also a critical input. 
 
 
Counterparty credit risk is an important topic today with credit crunch affecting 
increasing number of financial firms. Firms holding super senior piece of ABS CDO 
would normally hedge their position with a monoline insurer in the form of credit 
default swap (CDS). Under the stress market conditions for monoline insurers, the 
effectiveness of the hedge has to be reevaluated based on the credit quality of the 
counterparty. Credit value adjustment (CVA) is precisely the measurement of 
counterparty credit risk on OTC derivative transactions.  Under the new FAS 157 
requirement, counterparties can also adjust their liabilities based on their own credit 
quality, which leads to the CVA benefits shown on monoline insurers’ quarterly reports 
when their credit spreads widened a lot due to downgrading below AAA rating. 
 
In the usual framework for credit value adjustment calculation, potential future 
exposure has to be simulated before the adjustment can be calculated (see reference [1]). 
With all the structural complexities of ABS CDOs, it is not realistic to carry out the 
simulation in order to calculate CVA. One simple way people would use is to adjust the 
discount rate curve by the counterparty CDS spread curve. This method is reasonable 
only when the counterparty CDS spread is small. It is the market convention to quote 
spread for IG names, but quote price for HY names, because using CDS spread adjusted 
discount curve is no longer accurate for HY names. The recent mind boggling results of 
monoline insurers’ quarterly reports on CVA benefits have proven that this simple 
method has to be improved for stressed counterparties. Here we propose a simplified 
approach with reasonable accuracy even for stressed names, which will make the 
calculation easy to implement in the ABS CDO valuation frame work. 
 
We make a few assumptions about the model as follows.  First, most of the CDS 
negative basis trades on super senior CDOs were entered at very low spread, which will 
have a negative mark-to-market value for the protection seller in the foreseeable future. 
So the counterparty risk will be unilateral on the protection seller, or the monoline 
insurers. Second, we assume the credit quality of the counterparty is independent of the 
ABS CDO collateral performance. In reality, this assumption is not necessarily true, in 
the light of the fact that monoline insurers have been dragged down by the MTM losses 
                                                      
1  Email: hui.li@aig.com. The views expressed are the author’s own, not those of AIG. due to their subprime exposures. Assuming all the stress has already been reflected in 
the CDS spread curves, this is still reasonable. Third, we assume recovery rate R  is 
constant, thus independent of counterparty default or market conditions. This is 
currently the standard market practice, especially R  is normally set to 40%. 
 
Under these three assumptions, the standard formula for CVA (see reference [1]) is 
 
∫ − =
T
t dPD t EE R CVA
0
) , 0 ( ) ( ) 1 (     (1) 
 
where  is the risk-neutral discounted expected exposure given by  ) (t EE
 
)] ( [ ) (
0 t E
B
B
E t EE
t
Q =       ( 2 )  
 
where  is the net present value of future cash flows valued at time  and is 
independent of counterparty default,  is the future value of one unit of base currency 
invested in the money market account.  is the risk-neutral probability of 
counterparty default between time 0 and t with 
) (t E t
t B
) , 0 ( t PD
0 ) 0 , 0 ( = PD .  can be calibrated 
from the term structure of counterparty CDS spreads. The calibration is based on the 
assumption of 40% recovery rate for corporate bond. 
) , 0 ( t PD
 
Normally firms would have a cash flow model or a correlation model to value their ABS 
CDO insurance portfolio. We assume essentially  is the sum of risk-neutral 
discounted expected future cash flows at or after time t under the CDS contract. So 
is the risk-neutral discounted value of the expected cash flow at time t. We 
have  
) (t EE
) (t dEE −
 
) ) ( )( , 0 ( ) 1 ( ) 0 ( ) 1 (
) ) ( ))( , 0 ( 1 ( ) 1 (
) ) ( )( , 0 ( ) 1 (
) ( ) , 0 ( ) 1 ( ) , 0 ( ) ( ) 1 (
0
0
0
0
0
∫
∫
∫
∫
− − − − =
− − − =
− − =
− − − =
T
T
T
T
T
t dEE t PS R EE R
t dEE t PS R
t dEE t PD R
t dEE t PD R t PD t EE R CVA
   (3) 
 
where we choose T to be slightly after the maturity of the deal so that the exposure at T 
will always be zero, and  is the survival probability between time 0 and time t 
with  .   is the current value of the ABS CDO insurance 
portfolio without counterparty default risk. 
) , 0 ( t PS
) , 0 ( 1 ) , 0 ( t PD t PS − = ) 0 ( EE
 An intuitive way to understand equation (3) is to separate the future cash flows into two 
parts, the loss and the recovery. The recovery part will be independent of counterparty 
default risk, while the loss part is contingent on counterparty default before the time of 
the future cash flows.  One easy way to implement the CVA calculation is to multiply 
the discount factors by  *  and rerun the CDS on ABS CDO valuation. To 
facilitate the calculation, it is required to separate discount curve from the forward Libor 
rate calculation. 
) 1 ( R − ) , 0 ( t PD
 
Unlike the interest rate swaps, credit default swaps could terminate earlier when there is 
a credit event. In fact, there are several forms of CDS contracts: cash settlement, physical 
settlement and Pay-As-You-Go.  The first two will terminate when credit event happens, 
but the latter has the same maturity as the underlying ABS/CDO bond. Monoline 
insurers normally have the Pay-As-You-Go settlement on their insurance contracts. The 
method described above is suitable for this case. Since the timing of credit event could 
be scenario dependent and the counterparty risk no longer exists after settlement of 
default loss, the above framework has to be expanded to handle the first two cases. 
 
Assume for a scenario the credit event happens at time  , then the CVA for that 
scenario is calculated as follows 
0 t
         
  )) ( ( ) , 0 ( ) 1 ( ) , 0 ( ) ( ) 1 (
) , 0 ( ) ( ) 1 (
0
0
0
0
0
0 0
0
0
0
0
t E
B
B
d t PD R t PD t E
B
B
R
t dPD t E
B
B
R CVA
t
t t
t
t
∫
∫
− − − =
− =
   (4) 
Assume C(0,t) is the cumulative cash flow from time 0 to time t. Then  
   ∫ ∫ = =
T
t s
T
t s
t
t t
s dC
B
B
s dC
B
B
B
B
t E
B
B
) , 0 ( ) , 0 ( ) (
0 0 0     (5) 
and 
 
)) , 0 ( ) , 0 ( ) , 0 ( ) , 0 ( )( 1 ( ) 0 ( ) 1 (
) , 0 ( ) , 0 ( ) 1 ( ) , 0 ( ) , 0 ( ) 1 (
0
0
0
0
0
0
0
0
0
0
0
0 0
t C d
B
B
t PS t C d
B
B
t PS R E R
t dC
B
B
t PD R t C d
B
B
t PD R CVA
T
t t
t
t
t
t T
t t
∫ ∫
∫ ∫
+ − − − =
− + − =
(6)  
 
This suggests that, for that scenario, the default probability curve should be set to 
constant after the credit event and then be multiplied to the discount curve. So we have 
to make the adjustment to discount curve based on the timing of credit event in each 
scenario. The final CVA value will be the average over all scenarios. 
 
The method currently used by the monoline insurers is equivalent to setting the recovery 
rate to zero when calibrating the CDS default probability and calculating the CVA. The 
market standard recovery rate assumption is 40% when quoting the CDS spreads. When CDS spreads are not high, CVA valuation is not very sensitive to the recovery rate 
assumption. However, when CDS spreads are wide, the recovery rate assumption is very 
important. A quick numerical evaluation on a sample portfolio of ABS CDO insurance 
portfolio reveals that, when counterparty CDS spreads are high (over 2000bps), which is 
typical for the monoline insurers, the difference between the two methods (recovery rate 
0% vs 40%) is huge (mostly over 100%). Of course, actual results may vary depending 
on the underlying CDO portfolios. But it is safe to say that the monoline insurers’ 
quarterly results are hugely overstated compared with using 40% recovery rate. 
 
Before making any conclusion, we would like to review the assumptions underlying our 
method and see if they need revision to be more precise. The assumption of negative 
potential future exposures for the CDS insurance on super senior ABS CDO is 
reasonable, since the insurance premium is so low that market spread will not be lower in 
foreseeable future. The rationale is that if the premium on the deal is very low then the 
method can be used. Next, as mentioned before, the assumption of independence between 
counterparty default and the CDS valuation is questionable. For single name CDS, there 
is discussion in the literature on how to add correlation to CVA calculation (see reference 
[2]). In the case of ABS CDO affected by the mortgage crisis, we can assume the 
correlation is through a systematic factor, like the housing price appreciation (HPA). If 
the CDO valuation is driven by HPA scenarios imbedded in a cash flow model, this 
should not be difficult to implement.  The third assumption is also debatable. Research on 
historical data suggests recovery rate is negatively correlated with default rate. Recent 
work has also demonstrated that stochastic recovery rate model might be helpful for 
calibrating the Gaussian copula model on the super senior tranches of CDX and iTraxx 
indices (see reference [3]). It is also arguable to use the same recovery rate for both 
corporate bonds and OTC derivative contracts. 
 
In this note, we showed a more accurate method to calculate CVA for CDS on super 
senior ABS CDO, especially for stressed counterparties. The way is to adjust the discount 
curve by counterparty default probability multiplied by the loss given default. The key 
point is that, when counterparty credit spread is wide, CVA calculation is very sensitive 
to recovery rate assumption. Adjusting the discount curve by the counterparty credit 
spread curve is equivalent to assuming zero recovery rate, which could vastly overstate 
the CVA benefits for stressed counterparties. The fair value CVA benefits reported by the 
monoline insurers in their quarterly reports should be taken with a grain of salt. 
 
 
References 
 
[1] M. Pykhtin and S. Zhu, “A Guide to Modeling Counterparty Credit Risk”, GARP 
Risk Review, July/August 2007 Issue 37 p16; 
[2] D. Brigo and K. Chourdakis, “Counterparty Risk for Credit Default Swaps: Impact of 
spread volatility and default correlation”, FitchSolutions, May 2008; 
[3] S. Amraoui and S. Hitier, “Optimal Stochastic Recovery for Base Correlation”, June 
2008. 

CVA calculation for CDS on super senior ABS CDO

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