This case study demonstrates two optimization setups for determining attachment points in step-up Collateralized Debt Obligation (CDO). The first setup minimizes outcome spread payments for the given CDO portfolio subject to a risk constraint assuring the rating of a tranche, single-period default probability constraints, and linear constraints on attachment points. The second setup simultaneously optimizes the CDO portfolio and credit tranching subject to a risk constraint assuring the rating of a tranche, a constraint on income spread payments, and a budget constraint.

This case study is done from the bank-originator point of view. CDO is a credit derivative based on defaults of a pool of assets. CDO makes available credit risk exposure to a broad set of investors. A common structure of CDO involves tranching or slicing the credit risk of the reference pool into different risk levels. The risk of loss on the reference portfolio is divided into tranches of increasing seniority. The losses first affect the equity (first loss) tranche, then the mezzanine tranche, and finally the senior and super senior tranches. The lower tranche boundary is called the attachment point, while the upper tranche boundary is called the detachment point. The payoff structure of a CDO is designed to offer risk/return profiles that are specifically targeted to investment restrictions of different investor groups. For instance, a CDO based on unrated or speculative-grade underlying portfolio enhances the credit rating of most of the notes to the high investment-grade ratings by concentrating the default risk in the first loss tranche. Investors invest in these notes, however, they may not be allowed to invest in the underlying assets themselves.

Here we discuss a structuring problem for a so-called step-up CDO. In the step-up CDO, the tranche attachments points are not constant, but change (typically increase) each time period.

In the considered step-up CDO contract, attachment points depend only on time, but do not depend on credit events. CDO tranche default occurs when cumulative collateral loss exceeds the tranche attachment point. A step-up CDO contract can be structured such that it exposes its holder to a pre-determined level of risk in each of the time periods until the maturity. These risk exposures are established at their specific levels by setting attachment points. CDO tranche valuation depends on estimates of risk exposure in all periods.

Given a pool of assets (CDO portfolio), the originator bank can maximize its profit by selling tranches (or credit default swaps on tranches) bearing as much risk as possible for the lowest spread payments that can be offered. Lower spread payments are possible if a certain credit rating is maintained. A target credit rating imposes constraints on the default risk that a tranche bears. The profit can also be maximized by selecting optimal CDO portfolio maximizing total income spread payments and simultaneously minimizing outcome spread payments.

This case study demonstrates how to optimize CDO portfolio and simultaneously calibrate attachment points to maximize profit (difference between income and outcome spread payments).

Problem 1

Simplified Problem Statement

Minimize Linear (minimize upper bound of spread payments over all periods for tranche m)

subject to

Prmulti_pen ≤ Const1 (risk constraint assuring rating of a tranche)

Box constraints (box constraints on attachment points)

where

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	0.9172416352
Solving Time (sec)	0.27

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

Problem 2

Simplified Problem Statement

Minimize Linear (minimize upper bound of spread payments over all periods for tranche m)

subject to

Linearmulti = Const2 (linear constraints)

Prmulti_pen ≤ Const3 (risk constraint assuring rating of a tranche)

Box constraints (box constraints on attachment points)

where

Linearmulti = Linear Multiple

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	1.119680810637
Solving Time (sec)	1.08

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

Problem 3

Simplified Problem Statement

Minimize PV (minimize PV of expected spread payments over all periods for tranche m)

subject to

Pr_pen ≤ Const4

Prmulti_pen ≤ Linear (default probabilities constraints at time period t)

Box constraints (box constraints on attachment points)

where

PV = Present Value

Pr_pen = Probability Exceeding Penalty for Loss

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	0.588263342701
Solving Time (sec)	0.71

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

Problem 4

Simplified Problem Statement

Minimize PV (minimize PV of expected spread payments over all periods for tranche m)

subject to

Prmulti_pen ≤ Const5 (risk constraint assuring rating of a tranche)

Box constraints (box constraints on attachment points)

where

PV = Present Value

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	0.583782987354
Solving Time (sec)	0.35

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

Problem 5

Simplified Problem Statement

Minimize PV (minimize PV of expected spread payments over all periods for tranche m)

subject to

Linearmulti = Const6 (linear constraints)

Prmulti_pen ≤ Const7 (risk constraint assuring rating of a tranche)

Box constraints (box constraints on attachment points)

where

PV = Present Value

Linearmulti = Linear Multiple

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	0.78188185221
Solving Time (sec)	0.18

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

Problem 6

Simplified Problem Statement

Minimize Linear (minimize upper bound of spread payments over all periods for tranche m)

subject to

Prmulti_pen ≤ Const8 (risk constraint assuring rating of a tranche)

Linear ≥ Const8 (constraint on income spread payments)

Linear = 1 (sum of weights constraint)

Box constraints (box constraints on attachment points)

where

Prmulti_pen = Probability Exceeding Penalty for Loss Multiple

Box constraints = constraints on individual decision variables

Mathematical Problem Statement

•	Formal Problem Statement

Problem dimension and solving time

Number of Variables	5
Number of Scenarios	10,000
Objective Value	1.005085900229
Solving Time (sec)	6.37

Solution in Run-File Environment

•	Description (Run-File)

Input Files to run CS:

•	Problem Statement (.txt file)

•	DATA (.zip file)

Output Files:

•	Output DATA (.zip file)

Solution in MATLAB Environment

Solved with PSG MATLAB function tbpsg_run (General (Text) Format of PSG in MATLAB):

•	Description (tbpsg_run)

Input Files to run CS:

•	MATLAB code (.txt file)

•	Data (.zip file with .m and .mat files)

References

[1] Burtschell X., Gregory, J., and J-P Laurent (2005): A Comparative Analysis of CDO Pricing Models, BNP-Paribas.

[2] Hull, J., and A. White. (2004): Valuation of a CDO and nth to Default CDS without Monte Carlo Simulation, Journal of Derivatives, Vol. 12, No. 2, pp. 8-23.