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?

Monday, August 03, 2009 , By Vallabh Muralikrishnan and Hans J.H. Tuenter

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Since Harry Markowitz published his seminal work on portfolio optimization in 1952, it has become standard practice in the asset management industry to construct portfolios that are "optimal" in some sense. Indeed, it is natural for portfolio managers to want to maximize return for a certain level of risk that they assume.

In this article, we propose a general framework for portfolio 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 optimizing equity portfolios, its limitations become apparent when applied to financial instruments with multiple characteristics, such as swaps. The structure of CDS, for example, distinguishes 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 payoff 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 creates 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.

Acceptable Trades

The 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 notional and tenor that would never be considered in practice. For example, we might decide to remove auto manufacturers 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 potential CDS trades. Although it is theoretically possible to write a CDS contract for any notional and tenor combination, 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" tenors, 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 reference entity, notional and tenor that are deemed undesirable by other idiosyncratic constraints. (This would be the responsibility of the PM.)

Figure 2 illustrates the aforementioned process of restricting the universe of theoretical CDS trades to an actionable list of discrete 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 after 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 evaluate 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 calculate 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 Measures

Before 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 characteristics of the assets in his universe.

On April 8th, 2009, the new Standardized North American 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 payment to make up for the difference in value between market spreads and the fixed premium. The structure of this contract means that the protection buyer is effectively paying the protection seller a percentage of the notional as an annual premium. For the sake of simplicity, we choose to 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 notional for shorts is negative by convention. This makes intuitive 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 credit portfolios is provided and extensively discussed by Löffler and Posch [2007], we refer the reader to their book for further 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 Portfolios

Having 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. Figure 3 illustrates our initial estimate of the efficient frontier.

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 frontier. 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 distinction, 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 represents a departure from the traditional Euclidean L2-norm.

The rationale for this is that the standard L2-norm is computationally much more demanding than the L1-norm. Under 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 optimization algorithm was able to identify many more and better portfolios on the efficient frontier than a simple random search. Figure 4 illustrates the results of using the SA algorithm, while Figure 5 illustrates the results of a simple random search.

Figure 4: Efficient Frontier Estimate Using Simulated Annealing Algorithm for 5000 Iterations

Figure 5: Efficient Frontier Estimate Using Random Search for 5000 Iterations

Back Testing

The 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 optimization 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 portfolios and an additional five inefficient portfolios to compare their average historical performance. The results are presented 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 methodology. 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.

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