The document discusses determining the efficient frontier for CDS portfolio optimization. It describes identifying acceptable long and short CDS trades, choosing risk and return measures like conditional VaR, estimating an initial efficient frontier through random searches, and using an optimization algorithm to improve the frontier. Generalizations and limitations of the approach are noted, such as dependency on accurate risk measures like probability of default estimates.

1. Vallabh Muralikrishnan
Determining the Efficient Frontier for Quantitative Analyst
CDS Portfolios BMO Capital Markets
Hans J.H. Tuenter
Mathematical Finance
Program,
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2. Objectives
• Positive EVA
• Minimize Tail Risk
• Maximize Expected Return
• Manage Return on Capital
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3. Optimization Strategy
4. Use optimization algorithm to improve
1. Identify acceptable trades
the efficient frontier
2. Choose risk-return measures 5. Select desired level of risk and return
3. Estimate the efficient frontier 6. Back Test performance of portfolio
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4. Identify Universe of Trades
LONGS SHORTS
Acceptable Credits Acceptable Credits
Liquid Notional and Tenors Liquid Notional and Tenors
Best EVA Trade per Credit Best EVA Trade per Credit
Using only 200 swaps, one can create 2200 = 1.6 x 1060 portfolios!!!
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5. Choose Risk-Return Measures
Several options: RAROC, RORC, EVA, Historical MTM, VaR
In this study:
• Risk: Conditional VaR (1 year horizon)
• Return: Spread × Notional
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6. Conditional Value-at-Risk
Loss distribution generated by one-factor Gaussian copula model using
correlation estimates from KMV
CVaR calculated using Monte-Carlo simulation
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7. Estimate the Efficient Frontier
• The efficient frontier of CDS
portfolios is discrete because it is
difficult to meaningfully
interpolate between portfolios.
• A random search of several
thousand portfolios can provide
an estimate of the efficient
frontier.
• The green line represents the
non-dominated portfolios from
this search. It represents the
portfolios with the best risk-
return trade-off.
INITIAL ESTIMATE
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8. Improve the Frontier with Optimization
RANDOM SEARCH
OPTIMIZATION ALGORITHM
Starting from the initial estimate, an optimization algorithm can identify more/better
portfolios than continuing a random search.
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9. Generalizations
This optimization approach presented here can be customized in many ways
Choice of trade universe
• Longs only; shorts only; other assets;
Choice of Risk-Return measures
• VaR, Economic Capital
Change Optimization algorithm
• Genetic Search
Discussion Points
Mathematical Optimization models can give you results that are only as good as the risk
measures used.
• There are a lot more long positions than short positions in the CDS universe identified
in this study. Does this mean that the capital measure to calculate EVA is wrong?
• Portfolio risk measures depend on estimates of PD, LGD, and asset value
correlations. If the measures are not accurate, your portfolios will be suboptimal. For
example, consider PD estimates of Lehman Brothers, one month before they
defaulted. Does this mean the PD estimate was wrong or that we were just unlucky?
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10. Acknowledgements and References
The work presented here was developed jointly with prof. Hans J.H. Tuenter from the
Mathematical Finance Program at the University of Toronto.
The authors would like to acknowledge Ulf Lagercrantz (VP, BMO Capital Markets) for
his help in developing the algorithm to identify the list of potential longs and shorts.
Further Reading:
• Vallabh Muralikrishnan, “Optimization by Simulated Annealing”, GARP Risk Review, 42:45 – 48,
June/July 2008.
• Hans J.H. Tuenter, “Minimum L1-distance Projection onto the Boundary of a Convex Set”, The
Journal of Optimization Theory and Applications, 112(2):441 – 445, February 2002.
• Gunter Löffler and Peter N. Posch, “Credit Risk Modeling using Excel and VBA”, Wiley Finance.
pg 119 – 146, 2007.
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