ASSEt mANAgEmENt                                                                                                          ...
ASSEt mANAgEmENt                                                                                                          ...
ASSEt mANAgEmENt      Extensions and Limitations of Optimization      The aforementioned framework can be extended and cus...
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A Framework For Portfolio Optimization


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A paper I wrote on how to approach credit portfolio optimization with Credit Default Swaps

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A Framework For Portfolio Optimization

  1. 1. ASSEt mANAgEmENt ASSEt mANAgEmENt A Framework for Acceptable Trades Figure 2: Restricting the Theoretical Universe of The first step in our framework is to identify the universe Possible CDS Trades of CDS from which to construct optimal portfolios. This Portfolio Optimization requires specifying the notional (positive notional for longs Longs Shorts and negative notional for shorts), the tenor and the reference entity for every swap that the portfolio manager is willing to Acceptable Credits Acceptable Credits trade. Liquid Notional and Tenors Liquid Notional and Tenors For this task, we use the knowledge of domain experts What steps can a portfolio manager take to construct optimal risk/return (credit analysts, traders, and portfolio managers) to pre- Other Constraints Other Constraints portfolios, and how can portfolio optimization be used to determine the screen, vet and eliminate any credits and combinations of no- tional and tenor that would never be considered in practice. efficient frontier in the credit default swap market? For example, we might decide to remove auto manufactur- ers from our universe of potential CDS trades, if we do not By VallaBh Muralikrishnan and hans J.h. TuenTer want exposure to the automotive sector. Several constraints S can be incorporated during this step of the framework. Our ates different portfolios by allocating the available capital goal is to develop a discrete list of actionable CDS trades It is important to note that overly stringent constraints ince Harry Markowitz published his seminal amongst the assets in his investment universe. Given that from which to construct optimal CDS portfolios. can render the portfolio optimization trivial. For example, if work on portfolio optimization in 1952, it has CDS contracts specify a notional, a tenor and an underlying This is achieved via the following steps: only 10 distinct trades remain in the investment universe af- become standard practice in the asset manage- reference entity, the PM not only has to allocate his capital 1. Dividing the universe of potential CDS trades into long ter applying a given set of constraints, a PM can only create ment industry to construct portfolios that are (by selecting the notional) but also has to decide on the tenor and short positions. 1,024 (i.e., 210) portfolios. In such a case, it is easy to evalu- “optimal” in some sense. Indeed, it is natural of each trade. Therefore, a new framework is required to 2. Further reducing the list of potential long and short ate each and every portfolio, calculate the risk and return, for portfolio managers to want to maximize re- construct optimal portfolios of CDS. trades by eliminating reference entities that are deemed and select the particular portfolio with the greatest level of turn for a certain level of risk that they assume. We propose to first reduce the investment universe of undesirable. For example, it might be undesirable to take a return for an acceptable level of risk. In this article, we propose a general framework for portfo- CDS to discrete trades, and to then use a combination of long position in American automakers, if their default seems In practice, however, it is common to have at least a few lio optimization and show how it can be used to determine random sampling and optimization algorithms to identify imminent. Likewise, it might be undesirable to take a short hundred CDS trades from which to construct portfolios. With the efficient frontier in the credit default swap (CDS) market. portfolios of CDS with very good (if not optimal) risk-return position on a particular company, if the risk of default is only 200 trades, a PM can construct 2200 (approximately 1.6 We also discuss the limitations of portfolio optimization. profiles. The general procedure is summarized in Figure 1 judged to be unlikely or distant. × 1060) portfolios. It is obvious that it is impossible to calcu- Although the Markowitz framework is useful in optimiz- (see below). 3. Using liquidity constraints to discretize the list of po- late the risk and return characteristics for each one of these ing equity portfolios, its limitations become apparent when tential CDS trades. Although it is theoretically possible to portfolios. Therefore, we suggest using search algorithms to applied to financial instruments with multiple characteris- write a CDS contract for any notional and tenor combina- estimate the efficient frontier of CDS portfolios. tics, such as swaps. The structure of CDS, for example, dis- Figure 1: A Framework to Determine and Test the tion, most combinations are unlikely in practice. Most CDS tinguishes them from basic asset classes such as stocks and Efficient Frontier of CDS Portfolios contracts trade with a “round number” notional, such as Deciding on Risk and Return Measures bonds. A CDS is a credit derivative contract between two $10 million or $20 million, and for “round number” ten- Before we can begin our optimization process, we must counterparties, whereby one party makes periodic payments ors, such as 1 year or 5 years. Therefore, in practice, a PM choose measures of portfolio risk and return. This choice to the other for a pre-defined time period and receives a pay- 1 Identify acceptable trades 4 Use optimization algorithim to improve the efficient frontier can only consider trades with specific notional and tenors. will be driven by the objectives of the PM and the charac- off when a third party defaults within that time frame. The Implementing liquidity constraints leads to a list of specific teristics of the assets in his universe. former party receives credit protection and is said to be short CDS trades, which the PM can choose to either execute or On April 8th, 2009, the new Standardized North Ameri- the credit while the other party provides credit protection not. can Contract (SNAC) for CDS came into effect. Under this and is said to be long the credit. The third party is known as Choose risk-return Select desired level of risk 4. Going through the list of potential trades generated by contract, the protection buyer pays a fixed, annual premium the reference entity. 2 5 measures and return step 3, and then further eliminating any combination of ref- (100 basis points [bps] for investment-grade names and 500 Each CDS contract specifies a notional amount for which erence entity, notional and tenor that are deemed undesir- bps for non-investment grade names) and a lump sum pay- the protection is sold and a term over which the protection able by other idiosyncratic constraints. (This would be the ment to make up for the difference in value between mar- is provided. The aforementioned features of CDS contracts responsibility of the PM.) ket spreads and the fixed premium. The structure of this provide unique modeling challenges for a portfolio manager Back Test performance of Figure 2 (above, right) illustrates the aforementioned pro- contract means that the protection buyer is effectively pay- (PM) trying to construct efficient portfolios of CDS. 3 Estimate efficient frontier 6 portfolio cess of restricting the universe of theoretical CDS trades to ing the protection seller a percentage of the notional as an For example, under the Markowitz framework, a PM cre- an actionable list of discrete trades. annual premium. For the sake of simplicity, we choose to38 RISK PROFESSIONAL A U G U S T 2 0 0 9 A U G U S T 2 0 0 9 RISK PROFESSIONAL 39
  2. 2. ASSEt mANAgEmENt ASSEt mANAgEmENt measure return simply as the annual premium on a trade, Figure 3: The Initial Estimate of the Efficient Fron- use a simulated annealing (SA) algorithm1 to improve on the Back Testing as follows: tier for a Randomly Generated Set of Portfolios initial efficient frontier, as described earlier. As a benchmark, The optimization framework we have presented is clear and we also continued a random search beginning from the same relatively straightforward to implement. But how does it fare Annual Premium = Market Spread × Notional Efficient Frontier initial estimate. in practice? To validate our process, we ran the optimiza- 240 - After just 5000 iterations, one can see that the optimi- tion process (through January 8th, 2008) and searched 6000 This premium is the annual cost (or gain) for taking a zation algorithm was able to identify many more and bet- portfolios. Of these 6000, 20 portfolios were identified as Expected Annual Revenue ($MM) 220 - short (or long) position. Note that under this measure, shorts ter portfolios on the efficient frontier than a simple random optimal. We randomly selected five of these optimal portfo- are considered to have “negative return,” because the no- 200 - search. Figure 4 illustrates the results of using the SA al- lios and an additional five inefficient portfolios to compare tional for shorts is negative by convention. This makes intui- 180 - gorithm, while Figure 5 illustrates the results of a simple their average historical performance. The results are pre- tive sense because the portfolio manager would be paying random search. sented in Figure 6 (below). the premium. It is also important to remember that short 160 - positions reduce the credit risk of the portfolio by hedging 140 - Figure 4: Efficient Frontier Estimate Using Simu- Figure 6: Comparing the Average P&L of Optimal against default events. lated Annealing Algorithm for 5000 Iterations and Suboptimal Portfolios 120 - Credit events are rare, and therefore losses and gains on | | | | | | | | | | | CDS positions are subject to extreme events. Consequently, 80 90 100 110 120 130 140 150 160 170 180 Efficient Frontier Comparison of Average P&L of Optimal and Suboptimal Portfolios Annual Average Loss Beyond 99% ($MM) many portfolio managers have an interest in managing the 240 - 50 - risk in the tail of the credit loss distribution, and expected tail The red line in Figure 3 is the (upper) convex hull of all Expected Annual Revenue ($MM) loss — also called conditional value-at-risk (CVaR) — will be a 220 - 0- the points representing the various portfolios, and the red more appropriate risk measure than standard VaR. Profit and Loss in $M dots represent those portfolios that are on the efficient fron- 200 - In our example, we calculate CVaR through Monte Carlo -50 - tier. In the classic Markowitz setting, any linear combination 180 - simulation. (As the methodology to calculate CVaR for cred- of neighboring portfolios that are on the efficient frontier -100 - it portfolios is provided and extensively discussed by Löffler also constitutes an efficient portfolio. 160 - and Posch [2007], we refer the reader to their book for fur- However, as mentioned in the introduction, this is not 140 - -150 - ther details.) For our purposes, we use a one-factor Gaussian necessarily the case for a CDS portfolio. To make this dis- copula model to simulate a portfolio loss distribution and tinction, all the portfolios that do not lie on the convex hull 120 - -200 - | | | | | | | take the expected shortfall at the 99th percentile as a risk | | | | | | | | | | | Jan Mar May Jul Sep Nov Dec of the efficient portfolios, and do not have a dominating 80 90 100 110 120 130 140 150 160 170 180 measure. The choice of 99th percentile as the threshold portfolio, have been colored green and are connected by a Annual Average Loss Beyond 99% ($MM) The Year 2008 for our risk measure is meant to be an illustrative number; step function that represents a discretized efficient frontier. the PM is free to choose any measure of risk that he or she For this particular comparison, we kept the portfolios wishes. Improving the Initial Efficient Frontier Figure 5: Efficient Frontier Estimate Using Ran- static in order to highlight the relative performance and to It must also be noted that in order to estimate portfolio Before we can improve our initial estimate of the efficient dom Search for 5000 Iterations demonstrate the potential improvements of our methodol- risk using this model, we require estimates of probability of frontier, we must choose a distance measure to determine ogy. Of course, more exhaustive back-testing procedures are default, loss-given default and default correlations for each how far any particular portfolio is from the efficient frontier. Efficient Frontier possible, but this simple approach serves to illustrate a few trade in a portfolio. In this study, we have used proprietary For this purpose, we have chosen the L1-norm. This repre- 240 - interesting points. estimates of these parameters to calculate portfolio risk. sents a departure from the traditional Euclidean L2-norm. In our particular test, it is clear that the optimal portfolios Expected Annual Revenue ($MM) 220 - The rationale for this is that the standard L2-norm is com- consistently outperform the suboptimal portfolios over the Generating an Initial Set of Portfolios putationally much more demanding than the L1-norm. Un- 200 - entire time horizon that was considered. However, it is also Having chosen a measure of risk and return, we calculate an der the L1-norm, the distance from an interior point to the clear that both portfolios largely traced the same systemic initial estimate of the efficient frontier. As an illustration, we 180 - convex hull can easily be determined as the minimum of a market movements. randomly selected a set of portfolios and calculated the risk set of univariate projections (see Tuenter [2002] for details). 160 - In our case, this phenomenon is due to extremely high and return metrics for each one. The initial estimate of the This saving in computational time is extremely important, correlations witnessed in the credit markets during the un- efficient frontier is the set of portfolios that have risk-return 140 - as it allows one to evaluate that many more portfolios in the certain year of 2008. Nevertheless, this highlights some of metrics that are not dominated by any other portfolios. Fig- simulated annealing approach, and thus arrive at a far better 120 - the limitations of portfolio optimization. Clearly, portfolio ure 3 (see above, right) illustrates our initial estimate of the ultimate solution. | 80 | 90 | 100 | 110 | 120 | 130 | 140 | 150 | 160 | | 170 180 optimization is not a panacea that can immunize managers efficient frontier. Starting with the initial estimate presented in Figure 3, we Annual Average Loss Beyond 99% ($MM) from systemic market movements.40 RISK PROFESSIONAL A U G U S T 2 0 0 9 A U G U S T 2 0 0 9 RISK PROFESSIONAL 41
  3. 3. ASSEt mANAgEmENt Extensions and Limitations of Optimization The aforementioned framework can be extended and cus- FOOTNOTES tomized in many ways. For example, the PM can use this 1. Simulated annealing is a search algorithm that framework to construct optimal portfolios using any instru- can efficiently search the vast universe of potential ments as long as he or she can calculate an appropriate mea- portfolios for optimal combinations of risk and return. sure of portfolio risk and return. The choice of risk and re- For more details on this algorithm, see V. Muralikrish- turn measure will be driven by the specific aims of the user. nan’s article, “Optimization by Simulated Annealing” The PM also has the flexibility to customize the algorithm (GARP Risk Review, June/July 2008, pgs. 45-48). used to drive the search for the efficient frontier. So, is optimization the solution to portfolio management 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 with credit derivatives? It is not. This is because the result of Toronto, where he teaches a workshop on energy markets in the mathematical finance optimization will only be as good as the accuracy of the risk program. He can be reached at and return measures used. For example, if your measure of Vallabh Muralikrishnan is a graduate assistant in credit and debt markets re- portfolio risk underestimates the real risk of the portfolio, a search at the Salomon Center for the Study of Financial Institutions. At the time of PM could be exposed to catastrophic losses. Therefore, the writing this article, he was an associate in the asset portfolio management group at real challenge of portfolio management is to develop appro- BMO Capital Markets. He can be reached at priate measures of risk and return. The authors would like to thank Ulf Lagercrantz (vice president, BMO Capital Once these choices are made, we have shown how port- Markets) for his help in providing the data on which the implementation of this framework was tested. 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. REFERENCES Acerbi, 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).42 RISK PROFESSIONAL A U G U S T 2 0 0 9
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