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By guest blogger Dr. Sebastian Ceria, CEO at Axioma, Inc.

If only...

As we have been vividly reminded of late, hedging is anything but easy. Indeed, it is one of the toughest challenges in all of risk management.

But it is also a challenge that everyone faces, regardless of whether one is a quant or a fundamental manager.

The concept is deceptively simple. Given a portfolio, reduce or neutralize that portfolio’s future risk and/or risk exposures by buying securities that are likely to behave in a manner opposite to those of the portfolio. To put it another way, the hedging process essentially involves finding securities that have negative correlations to the risk that you want to hedge.

For example, say your holdings have too much corporate debt. You would want to buy securities that will behave negatively to corporate debt. Hence, if the risk of corporate debt increases, your hedge kicks in and mitigates the overall risk increase.

Building a hedge can only be accomplished by a combination of history—looking at the past as an indicator of future behavior—plus a model whose job it is to forecast the behavior of your securities in the future.

We need three things to get started: an existing portfolio with risks we want to hedge, a risk model and a portfolio optimizer. The risk model serves as a proxy for the future behavior of those securities we want to hedge. The optimizer is used to build a basket of stocks that, when added to the existing portfolio, will reduce the undesired risks.

But here’s where things get “interesting.”

Axioma, for instance, offers multiple risk models—a short-term model, a medium-term model, a fundamental model and a statistical model. Depending on the model you choose, the result of the hedge could be quite different. Say you want to build a hedge for a month. Should you use a statistical model or a fundamental model? Should you use the short-term or medium-term model? Perhaps you want to build a hedge that works for three months. Now which model do you choose?

Let’s look at an example. Suppose we have constructed a low volatility portfolio by equi-weighting the 200 names in the Russell 1000 with the lowest Volatility score as measured by the Volatility factor in Axioma’s Medium Horizon, Fundamental Factor Risk Model. As with many low volatility/low beta portfolios, this portfolio has easily out-performed the Russell 1000 since 2000.

Suppose now, that we become even more risk averse and wonder if we can reduce the risk of this portfolio even further without degrading its performance significantly. We still want to have at least the same very low exposure to the Volatility factor – that is the bet we intend to make. But, all other things being equal and choosing only stocks within the Russell 1000 universe, can we reduce risk even further?

The answer, of course, is yes, but with other changes in the portfolio’s performance. Table 1 summarizes the performance of three portfolios: the Russell 1000, the original portfolio (equi-weighted, low 200 names), and the hedged portfolio¹.

Table 1. Performance statistics for three portfolios, 1998 to 2012.

As designed, the realized risk of the hedged portfolio (9.9%) was less than that of the original portfolio (11.7%), both of which were less than the benchmark. The predicted beta of the hedged portfolio was substantially less than that of the original portfolio, indicating that the Volatility exposure of the hedged portfolio was even deeper than that of the original portfolio.

However, as often happens with hedged portfolios, the total return decreased relative to the original portfolio. As shown in Fig. 1, the hedged portfolio had similar returns as the original portfolio, but its performance lagged during the bull market of 2003 to 2007 and also in late 2010. This occurred because the hedged portfolio’s beta was lower than the original portfolio.

Figure 1. Cumulative returns for three portfolios, 1998 to 2012.

Although the total return of the hedged portfolio took a hit, its performance during the worst market downturns was better than the original portfolio. Figure 2 shows the maximum 30-day drawdown for the three portfolios.

Figure 2. Cumulative returns for three portfolios, 1998 to 2012.

Both the original and hedged portfolios had significantly less drawdown than the Russell 1000 in 2000 and again in 2007. The hedged portfolio, however, outperformed the original portfolio during the worst of the crisis in 2007 and 2008, with drawdowns that were 5% less than the original portfolios. So, as advertised, the hedge worked to protect the portfolio from the worst market downturns.

For quant practitioners who want to take their understanding of hedging strategies to higher levels, the Advanced Risk and Portfolio Management Bootcamp, August 13-18 in New York, is a great opportunity. The use of advanced mathematics and sophisticated analytics, such as stress testing, will be among the topics discussed in this intensive week-long program.

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¹ The hedged portfolio was constructed to minimize total portfolio risk by selecting stocks in the Russell 1000 and maintaining an exposure to Axioma’s Volatility factor that was at least as negative as the original portfolio. Asset holdings were long only, with a maximum of 3% for any security. Monthly rebalancing between 12/31/97 and 5/31/12.

Principal component analysis is widely used in fixed income risk management and pricing to estimate the effects of yield curve movements by transforming them into a reduced set of factors. When the analysis is done on spot rates, the factors have generally been described as level, steepness or slope, and curvature. But why? And is there a mathematical or economic reason for this observation?

Let’s now take a more critical look at the PCA decomposition of spot and forward rates. This analysis is based in part on a paper by Ilias Lekkos. In his paper, Lekkos presents a simple scenario. Divide the forward curve up into a collection of representative forwards, and then suppose that there is no correlation between each of these forwards (and hence the forward curve by extension), so that each is driven by an independent factor. If we simulated a bunch of forward curves out of this zero correlation factor structure, built the spot curves associated to each, and then performed a PCA on the spot curves, what would the PC loading structure look like?

Would there be correlation in the derived spot curves even though there was none present in the forward curves from which they were born?

The answer, it turns out, is a resounding yes.

I performed a similar analysis to Lekkos. In addition to replicating his findings I try to explain why the level, slope, and curvatures are important factors even if they appear to be a statistical construct of the no arbitrage averaging process that relates spots to forwards.

I started by taking 251 independent sets of 30 observations drawn from a uniform distribution between 0 and 1 to represent 251 independent one-year forward rates (any distribution would have given the same results). I then created a 30-year spot curve from these forwards for each of the 251 observations and applied principal component analysis to the z-scored returns of the spot rates. The results are reported in the following figures:

The first three factors explain about 87% of the total variance of the spot curves that are calculated from randomly generated forwards and the pattern of the factor loadings looks similar to what many expect the term structure to exhibit as drivers of its covariance: level, slope, and curvature.

To confirm this, I performed principal component analysis on one year of historical returns of the spot curve as of April 29, 2011. The spot curve here consists of 30 points with annual intervals. The results are summarized in the following figures:


The interpretation is almost identical between the two cases. So it appears that the factors that we observe in historical data are just a statistical construct of the no arbitrage averaging process that relates spots to forwards.

Next I repeated the same historical analysis but instead of using the spot curves I transformed the data into a forward curve with annual intervals by bootstrapping the spot rates into one-year forward rates. The results of the PCA on the one-year forward rate returns are presented below:

The first three factors still explain a large part of the variance but the pattern of their loadings is different. The results still exhibit curvature and slope but the level driver seems to have disappeared by the bootstrapping process.

Level and slope do have an economic interpretation because changes in those drivers for the spot curve imply changes in the forward curve. However, even if the first factor in the spot curve PCA looks like a level shift and explains the most variance, it does not seem to have the biggest impact on the forward curve. Its impact may have been exaggerated by the averaging process that defines the relationship between spots and forwards.

To illustrate this further, take a look at what happens to the spot and forward curves when the factor loadings for the first three factors are shifted. To do this, I took the PC loadings matrix I got from the PCA on the spot curve and applied shocks to the first three factors. Each of the factors was shocked by an amount equal to one unit of the explained variance for that factor. I then took a look at the resulting spot curve and the corresponding forward curve.

As we can see, the forward curve responds more to shifts in the second and third factor which look like a combination of slope and curvature (hump).

To complete the analysis I took a look at the forward curves generated by shifts to the forward PCA. Here again the results look like changes in slope and curvature. 

Forward rates are easier to interpret economically because they reflect future interest rate expectations. Most of their variation is probably due to slope and curvature changes. As for spot rates, a change in their slope and curvature seem to have a bigger impact on forward rates than a change in their level. The level factor is very likely the result of the mathematical relationship between spots and forwards.

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