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ince Harry Markowitz published his seminal work on portfolio optimization in 1952, it has become standard practice in the asset manage-ment industry to construct portfolios that are “optimal” in some sense. Indeed, it is natural for portfolio managers to want to maximize re-

turn for a certain level of risk that they assume. In this article, we propose a general framework for portfo-

lio optimization and show how it can be used to determine the efficient frontier in the credit default swap (CDS) market. We also discuss the limitations of portfolio optimization.

Although the Markowitz framework is useful in optimiz-ing equity portfolios, its limitations become apparent when applied to financial instruments with multiple characteris-tics, such as swaps. The structure of CDS, for example, dis-tinguishes them from basic asset classes such as stocks and bonds. A CDS is a credit derivative contract between two counterparties, whereby one party makes periodic payments to the other for a pre-defined time period and receives a pay-off when a third party defaults within that time frame. The former party receives credit protection and is said to be short the credit while the other party provides credit protection and is said to be long the credit. The third party is known as the reference entity.

Each CDS contract specifies a notional amount for which the protection is sold and a term over which the protection is provided. The aforementioned features of CDS contracts provide unique modeling challenges for a portfolio manager (PM) trying to construct efficient portfolios of CDS.

For example, under the Markowitz framework, a PM cre-

A Framework for Portfolio Optimization

What steps can a portfolio manager take to construct optimal risk/return portfolios, and how can portfolio optimization be used to determine the

efficient frontier in the credit default swap market?By VallaBh Muralikrishnan and hans J.h. TuenTer

ates different portfolios by allocating the available capital amongst the assets in his investment universe. Given that CDS contracts specify a notional, a tenor and an underlying reference entity, the PM not only has to allocate his capital (by selecting the notional) but also has to decide on the tenor of each trade. Therefore, a new framework is required to construct optimal portfolios of CDS.

We propose to first reduce the investment universe of CDS to discrete trades, and to then use a combination of random sampling and optimization algorithms to identify portfolios of CDS with very good (if not optimal) risk-return profiles. The general procedure is summarized in Figure 1 (see below).

Figure 1: A Framework to Determine and Test the Efficient Frontier of CDS Portfolios

Acceptable TradesThe first step in our framework is to identify the universe of CDS from which to construct optimal portfolios. This requires specifying the notional (positive notional for longs and negative notional for shorts), the tenor and the reference entity for every swap that the portfolio manager is willing to trade.

For this task, we use the knowledge of domain experts (credit analysts, traders, and portfolio managers) to pre-screen, vet and eliminate any credits and combinations of no-tional and tenor that would never be considered in practice. For example, we might decide to remove auto manufactur-ers from our universe of potential CDS trades, if we do not want exposure to the automotive sector. Several constraints can be incorporated during this step of the framework. Our goal is to develop a discrete list of actionable CDS trades from which to construct optimal CDS portfolios.

This is achieved via the following steps:1. Dividing the universe of potential CDS trades into long

and short positions.2. Further reducing the list of potential long and short

trades by eliminating reference entities that are deemed undesirable. For example, it might be undesirable to take a long position in American automakers, if their default seems imminent. Likewise, it might be undesirable to take a short position on a particular company, if the risk of default is judged to be unlikely or distant.

3. Using liquidity constraints to discretize the list of po-tential CDS trades. Although it is theoretically possible to write a CDS contract for any notional and tenor combina-tion, most combinations are unlikely in practice. Most CDS contracts trade with a “round number” notional, such as $10 million or $20 million, and for “round number” ten-ors, such as 1 year or 5 years. Therefore, in practice, a PM can only consider trades with specific notional and tenors. Implementing liquidity constraints leads to a list of specific CDS trades, which the PM can choose to either execute or not.

4. Going through the list of potential trades generated by step 3, and then further eliminating any combination of ref-erence entity, notional and tenor that are deemed undesir-able by other idiosyncratic constraints. (This would be the responsibility of the PM.)

Figure 2 (above, right) illustrates the aforementioned pro-cess of restricting the universe of theoretical CDS trades to an actionable list of discrete trades.

Figure 2: Restricting the Theoretical Universe of Possible CDS Trades

It is important to note that overly stringent constraints can render the portfolio optimization trivial. For example, if only 10 distinct trades remain in the investment universe af-ter applying a given set of constraints, a PM can only create 1,024 (i.e., 210) portfolios. In such a case, it is easy to evalu-ate each and every portfolio, calculate the risk and return, and select the particular portfolio with the greatest level of return for an acceptable level of risk.

In practice, however, it is common to have at least a few hundred CDS trades from which to construct portfolios. With only 200 trades, a PM can construct 2200 (approximately 1.6 × 1060) portfolios. It is obvious that it is impossible to calcu-late the risk and return characteristics for each one of these portfolios. Therefore, we suggest using search algorithms to estimate the efficient frontier of CDS portfolios.

Deciding on Risk and Return MeasuresBefore we can begin our optimization process, we must choose measures of portfolio risk and return. This choice will be driven by the objectives of the PM and the charac-teristics of the assets in his universe.

On April 8th, 2009, the new Standardized North Ameri-can Contract (SNAC) for CDS came into effect. Under this contract, the protection buyer pays a fixed, annual premium (100 basis points [bps] for investment-grade names and 500 bps for non-investment grade names) and a lump sum pay-ment to make up for the difference in value between mar-ket spreads and the fixed premium. The structure of this contract means that the protection buyer is effectively pay-ing the protection seller a percentage of the notional as an annual premium. For the sake of simplicity, we choose to

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Identify acceptable trades Use optimization algorithim to improve the efficient frontier1 4

Choose risk-return measures

Select desired level of risk and return2 5

Estimate efficient frontier Back Test performance of portfolio3 6

Longs

Liquid Notional and Tenors

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ShortsAcceptable Credits

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Acceptable Credits

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measure return simply as the annual premium on a trade, as follows:

Annual Premium = Market Spread × Notional

This premium is the annual cost (or gain) for taking a short (or long) position. Note that under this measure, shorts are considered to have “negative return,” because the no-tional for shorts is negative by convention. This makes intui-tive sense because the portfolio manager would be paying the premium. It is also important to remember that short positions reduce the credit risk of the portfolio by hedging against default events.

Credit events are rare, and therefore losses and gains on CDS positions are subject to extreme events. Consequently, many portfolio managers have an interest in managing the risk in the tail of the credit loss distribution, and expected tail loss — also called conditional value-at-risk (CVaR) — will be a more appropriate risk measure than standard VaR.

In our example, we calculate CVaR through Monte Carlo simulation. (As the methodology to calculate CVaR for cred-it portfolios is provided and extensively discussed by Löffler and Posch [2007], we refer the reader to their book for fur-ther details.) For our purposes, we use a one-factor Gaussian copula model to simulate a portfolio loss distribution and take the expected shortfall at the 99th percentile as a risk measure. The choice of 99th percentile as the threshold for our risk measure is meant to be an illustrative number; the PM is free to choose any measure of risk that he or she wishes.

It must also be noted that in order to estimate portfolio risk using this model, we require estimates of probability of default, loss-given default and default correlations for each trade in a portfolio. In this study, we have used proprietary estimates of these parameters to calculate portfolio risk.

Generating an Initial Set of PortfoliosHaving chosen a measure of risk and return, we calculate an initial estimate of the efficient frontier. As an illustration, we randomly selected a set of portfolios and calculated the risk and return metrics for each one. The initial estimate of the efficient frontier is the set of portfolios that have risk-return metrics that are not dominated by any other portfolios. Fig-ure 3 (see above, right) illustrates our initial estimate of the efficient frontier.

Figure 3: The Initial Estimate of the Efficient Fron-tier for a Randomly Generated Set of Portfolios

The red line in Figure 3 is the (upper) convex hull of all

the points representing the various portfolios, and the red dots represent those portfolios that are on the efficient fron-tier. In the classic Markowitz setting, any linear combination of neighboring portfolios that are on the efficient frontier also constitutes an efficient portfolio.

However, as mentioned in the introduction, this is not necessarily the case for a CDS portfolio. To make this dis-tinction, all the portfolios that do not lie on the convex hull of the efficient portfolios, and do not have a dominating portfolio, have been colored green and are connected by a step function that represents a discretized efficient frontier.

Improving the Initial Efficient Frontier Before we can improve our initial estimate of the efficient frontier, we must choose a distance measure to determine how far any particular portfolio is from the efficient frontier. For this purpose, we have chosen the L1-norm. This repre-sents a departure from the traditional Euclidean L2-norm.

The rationale for this is that the standard L2-norm is com-putationally much more demanding than the L1-norm. Un-der the L1-norm, the distance from an interior point to the convex hull can easily be determined as the minimum of a set of univariate projections (see Tuenter [2002] for details). This saving in computational time is extremely important, as it allows one to evaluate that many more portfolios in the simulated annealing approach, and thus arrive at a far better ultimate solution.

Starting with the initial estimate presented in Figure 3, we

use a simulated annealing (SA) algorithm1 to improve on the initial efficient frontier, as described earlier. As a benchmark, we also continued a random search beginning from the same initial estimate.

After just 5000 iterations, one can see that the optimi-zation algorithm was able to identify many more and bet-ter portfolios on the efficient frontier than a simple random search. Figure 4 illustrates the results of using the SA al-gorithm, while Figure 5 illustrates the results of a simple random search.

Figure 4: Efficient Frontier Estimate Using Simu-lated Annealing Algorithm for 5000 Iterations

Figure 5: Efficient Frontier Estimate Using Ran-dom Search for 5000 Iterations

Back TestingThe optimization framework we have presented is clear and relatively straightforward to implement. But how does it fare in practice? To validate our process, we ran the optimiza-tion process (through January 8th, 2008) and searched 6000 portfolios. Of these 6000, 20 portfolios were identified as optimal. We randomly selected five of these optimal portfo-lios and an additional five inefficient portfolios to compare their average historical performance. The results are pre-sented in Figure 6 (below).

Figure 6: Comparing the Average P&L of Optimal and Suboptimal Portfolios

For this particular comparison, we kept the portfolios static in order to highlight the relative performance and to demonstrate the potential improvements of our methodol-ogy. Of course, more exhaustive back-testing procedures are possible, but this simple approach serves to illustrate a few interesting points.

In our particular test, it is clear that the optimal portfolios consistently outperform the suboptimal portfolios over the entire time horizon that was considered. However, it is also clear that both portfolios largely traced the same systemic market movements.

In our case, this phenomenon is due to extremely high correlations witnessed in the credit markets during the un-certain year of 2008. Nevertheless, this highlights some of the limitations of portfolio optimization. Clearly, portfolio optimization is not a panacea that can immunize managers from systemic market movements.

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Extensions and Limitations of OptimizationThe aforementioned framework can be extended and cus-tomized in many ways. For example, the PM can use this framework to construct optimal portfolios using any instru-ments as long as he or she can calculate an appropriate mea-sure of portfolio risk and return. The choice of risk and re-turn measure will be driven by the specific aims of the user. The PM also has the flexibility to customize the algorithm used to drive the search for the efficient frontier.

So, is optimization the solution to portfolio management with credit derivatives? It is not. This is because the result of optimization will only be as good as the accuracy of the risk and return measures used. For example, if your measure of portfolio risk underestimates the real risk of the portfolio, a PM could be exposed to catastrophic losses. Therefore, the real challenge of portfolio management is to develop appro-priate measures of risk and return.

Once these choices are made, we have shown how port-folio optimization can be implemented as a straightforward mathematical exercise. The skill of a portfolio manager will be measured by his or her choice of risk and return mea-sures.

ad_RiskProf_outlines2_RMRR_7-15-09.indd 1 7/15/09 6:06:52 PM

REFERENCESAcerbi, C., C. Nordio and C. Sirtori. “Expected

Shortfall as a Tool for Financial Risk Management.” Working paper (2001). See http://www.gloriamundi.org.

Casey, O. “The CDS Big Bang: Understanding the Changes to the Global CDS Contract and North American Conventions,” The Markit Magazine, Spring 2009 (60-66).

Löffler, G., and P.N. Posch. Credit Risk Modeling Using Excel and VBA, Wiley Finance, 2007 (119 – 146).

Markowitz, H. “Portfolio Selection,” The Journal of Finance, March 1952 (77 – 91).

Tuenter, H.J.H. “Minimum L1-distance Projection onto the Boundary of a Convex Set.” The Journal of Optimization Theory and Applications, February 2002 (441 – 445).

Hans J.H. Tuenter (PhD) is a senior model developer in the energy markets division at Ontario Power Generation and an adjunct professor at the University of Toronto, where he teaches a workshop on energy markets in the mathematical finance program. He can be reached at hans.tuenter@opg.com.

Vallabh Muralikrishnan is a graduate assistant in credit and debt markets re-search at the Salomon Center for the Study of Financial Institutions. At the time of writing this article, he was an associate in the asset portfolio management group at BMO Capital Markets. He can be reached at vm692@nyu.edu.

The authors would like to thank Ulf Lagercrantz (vice president, BMO Capital Markets) for his help in providing the data on which the implementation of this framework was tested.

FOOTNOTES1. Simulated annealing is a search algorithm that

can efficiently search the vast universe of potential portfolios for optimal combinations of risk and return. For more details on this algorithm, see V. Muralikrish-nan’s article, “Optimization by Simulated Annealing” (GARP Risk Review, June/July 2008, pgs. 45-48).

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