'Seminal' paper tackles fault in fixed-income hedging

Challenge of hedging local and global risks solved with ‘flylets’ approach

Maths equations

Buy-side Awards 2016

Fixed-income traders typically hedge against parallel shifts, steepening and convexity in the rate curve, but know they will struggle to hedge more localised changes.

Their traditional approach, using principal component analysis (PCA) to measure these three main elements, tells them little about how individual points on the curve might change – as the result of a peak in corporate issuance, say, or another firm unwinding a big portfolio.

Methods that focus on local effects, meanwhile, would be in danger of missing the big picture of global changes to the curve. In their paper Flylets and invariant risk metrics, Dario Villani and Santhanam Nagarajan of Tudor Investment Corporation, and Kharen Musaelian of HiQu Capital, have come up with a solution to this long-standing problem.

"The brilliancy has been the ability to connect the dots between linear, non-linear and invariant measures of risk," says senior quant Raffaele Ghigliazza at hedge fund Wellington Management, based in Boston, Massachusetts. "It's going to be one of those seminal pieces of work of very long-lasting influence."


Villani, Nagarajan and Musaelian developed the idea while working together at Hutchin Hill Capital, a discretionary hedge fund based in New York, where they grappled with the shortcomings of existing fixed-income models.

"The issue with fixed-income, as opposed to equity, portfolios is you tend to use leverage that is much higher and much more varied across different strategies," says Musaelian.

Many models, including PCA, don't take this leverage into account. While a portfolio may look well hedged, if there happens to be more corporate issuance, or if someone decides to unwind their portfolio, a portfolio manager may find the correlation structure for rates breaks down.

Hedging a few of the principal components does not give you any quantitative indication of those idiosyncratic risks. "We realised that what you need is some sort of an alternative set of basic elements that are themselves not exposed to principal components, so they are sort of residual by construction."

Fixed-income portfolio managers analyse yield changes by breaking them down into a handful of risk factors weighted according to the variance of each factor. This is similar to how white light can be split into a range of colours. If the red component is dominant, the light will appear reddish; if the blue component is dominant, the light appears bluish.

These factors are the portfolio's principal components. They're constructed so the first few factors contribute a significant proportion of the total variance – usually over 90%. Because of this, although the total number of factors is large, managers will tend to consider only two or three.

"What this paper is outlining, I have never seen it anywhere," says Jeff Ziglar, a hedge fund portfolio manager. "For banks and large interest rate dealers, this is a huge, huge change. It will allow them to simplify their risk process and do it more accurately in real time."

What this paper is outlining, I have never seen it anywhere. For banks and large interest rate dealers, this is a huge, huge change. It will allow them to simplify their risk process and do it more accurately in real time
Jeff Ziglar, hedge fund portfolio manager

Before PCA became popular, if a portfolio manager wanted to hedge their interest rate products, they would rely on duration matching. This implicitly assumed the yield curve could only shift up or down without any change in shape – a parallel shift.

But such an approach ignores the possibility of the curve becoming steeper or curvier. Shifts in small, specific segments of the curve also occur, often the result of corporate hedging or portfolio unwinds involving particular maturities.

The first few principal components have an intuitive interpretation. The first factor roughly corresponds to parallel shifts, the second to steepening and the third to bowing – short and long maturities moving in one direction, while the maturities in the middle move the other way.

Portfolio managers have therefore typically hedged a given factor when it reached a pre-defined value. If the first factor had too high a variance, for example, the manager would want to hedge parallel shifts in the curve and would trade bond futures to adjust the duration of the portfolio to zero.

With the new 'flylet' framework, a chosen handful of these components is aggregated into a single risk metric incorporating convexity. The remaining components, representing the residual risk, are reformulated as flylets.

"These two risks - the first principal component and the second principal component – are not entirely independent of each other," said Musaelian. "And just viewing them in isolation as two separate numbers that you need to limit is not as productive as having one overall measure of risk within the portfolio."


Flylets represent the curvature at different points along the yield curve. For example, the risk due to maturities in the vicinity of the tenth point along the yield curve – nine years, 10 years and 11 years, say – will be measured using the tenth flylet. The name derives from products called flies. These comprise three assets, each with different maturities, where the shortest and longest maturity products are either both long or short, while the middle maturity product is the opposite.

While the single risk metric representing the first few principal components is affected by all points along the curve (albeit by varying amounts), the individual measures relating to each flylet are only affected by small and specific segments. The single risk metric is therefore termed a 'global' measure, while the flylet measures are 'local'.

Ramon Verastegui, head of flow strategy and solutions in the Americas at Societe Generale, says: "One of the good things about flylets is they also represent the local curvature of the yield curve. The problem is you have convexity and curvature in the curve that is not that easy to model. In this case, this is something they're giving you as part of the risk measure. They're giving you a combination of global and local measures."

Each principal component is a collective representation of the entire curve. However, Musaelian explains: "You want to find also a local measure. And flylets are basically a local measure of risk because they are portfolios of three adjacent swaps that are neutral with respect to the first two principal components." A flylet, therefore, "allows you to have a measure of risk not captured by principal components in a way that is local, hedgable and understandable".

Suppose that, instead of using flylets, a portfolio manager used a greater number of principal components. In trying to hedge a five-year swap, the manager would need to use rate products with a wider range of maturities. "It would just ripple through the entire maturity structure, and it doesn't make sense intuitively," says Musaelian.

"It would essentially force you to hedge some local butterfly around a five-year maturity with a butterfly in the long end based on some spurious estimate of correlation. Whereas, if you just insist on hedging exclusively locally, you're completely missing the very salient fact that there's a high correlation across maturities, and you can really hedge most of your interest rate risk with just the 10-year maturity."

Meeting of minds

The authors met in 2004 when Musaelian hired Villani for the proprietary positioning business at JP Morgan. There they managed a structured credit portfolio together until 2006, when Musaelian moved on to become co-head of global strategic risk at Merrill Lynch, where Villani later joined him. Nagarajan met Villani during a spell at Deutsche Bank in 2009, and all three teamed up at BlueCrest Capital Management in 2011.

Villani is global head of portfolio strategy and risk at Tudor and teaches finance at Princeton University. Nagarajan is Tudor's portfolio oversight manager. Musaelian, who earned a Physics Olympiad gold medal while growing up in the Soviet Union, is launching his own hedge fund, HiQu Capital.

Practitioners say the flylet approach will allow them to combine linear and non-linear rate products, such as swaps and options, in a single portfolio. Current practice is to run linear books and non-linear books separately.

Nick Riley, Head of BlueCrest's emerging markets fund, says: "If you're managing swaps and swaptions or bond options together – which a large number of the portfolio managers here would do – it gives you a very simple risk metric for the aggregate portfolio."

"It exposes risks that cannot be captured by Greeks; it gives the ability to capture a more complete picture of fixed-income exposure," says a senior quant at a US hedge fund.

Wellington's Ghigliazza says: "In our business, we lose money because of tail events, so having something that incorporates those in a unified measure is extremely important. It has not been implemented at Wellington yet, but it will be."

Sharing ideas is a good thing for the industry, and it's good for the institution, as well – to the extent that it shows the institution being on the cutting edge
Kharen Musaelian, HiQu Capital

According to one senior quant at a hedge fund, the method could be applied to asset classes beyond fixed income – "everywhere there is a term structure, like in commodities". And Musaelian agrees: "In credit, you could apply it to credit default swap curves, especially index curves, as they have a similar structure to rate curves."

Ziglar says: "When you have a big commodity book, you have this real big second-order risk. That is why commodity funds tend to blow up a lot more and they tend to make a lot of money when they work. But if you are able to actually measure [the second-order risk] more accurately, that will smooth out the returns at a higher Sharpe Ratio or higher return-to-risk ratio."

Tudor has already implemented the flylet methodology – and doing so should be straightforward for others, too, according to Villani, using standard linear algebra libraries. The firm, which has $10 billion under management, has recently been recruiting mathematicians and scientists in an effort to apply more quantitative analysis to the trading process. Unlike quant funds, however, the alpha generation will remain discretionary.

Villani says: "From the point of view of discretionary trading, applying quantitative analysis has become indispensable both for risk management and identifying opportunities... The future of finance is in bringing together the rigour of data analysis and a practical understanding of economics and business – the science and art of human decision-making."

What do the authors plan to research next?

"A lot of focus is on the way to enhance alpha through systematic targeting of volatility and low-rank representation of complex trading books," Villani says.

Alpha generation is a closely kept secret for many hedge funds, but risk management is far from being a zero-sum game. "In the end, we want everyone to do well," says Nagarajan. Musaelian adds: "Sharing ideas is a good thing for the industry, and it's good for the institution, as well – to the extent that it shows the institution being on the cutting edge."

Image: Shutterstock

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