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A Framework for Portfolio Optimization
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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

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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