There is rich literature around modeling and forecasting the term structure of government bond yields and interest rate swap rates by decomposing the yield curve into the level, slope, and curvature components. Our paper extends that methodology to swap spreads, defined as the difference between swap rates and corresponding Treasury yields of the same maturity. This is because swap spreads convey critical information about financial markets, over and above the information conveyed by swap rates. We build 28 models, each generating baseline and stress scenario forecasts of swap rates for a specific currency. Although there is a common theme across the models, country-specific idiosyncrasies in swap markets mean that we have to be flexible in our choice and interpretation of drivers. To our knowledge, this is the first attempt to model and forecast the term structure of swap spreads across a range of currencies using a principle component decomposition.
Interest swap contracts are the most common derivative by which financial intermediaries hedge against interest rate risk in the over-the-counter credit universe. Consequently, reliable swap models and forecasts have become an integral part of bank stress tests, such as the Federal Reserve's annual Comprehensive Capital Analysis and Review (CCAR) exercise. We consider swap spreads – defined as the difference between the fixed rate paid in a swap contract and the Treasury yield of the same maturity – to be the most natural approach to modeling swap contracts. This differential captures the economic price of paying the fixed rate in a swap contract. To model swap contracts, we draw from the literature on Treasury yields. Existing studies of government bond yields1 across a large range of countries employ a principle component analysis (PCA) approach to decompose a yield curve into the level, slope, and curvature components. Not only does this reduce dimensionality, the first two components have a nice interpretation. While the level component shifts the yield curve up or down, the slope component rotates the curve, making it steeper or flatter. It is the slope component that, by loading differently on the short and long tenors2, captures the difference in movements in the short and long ends of the curve. With the level and slope components usually explaining more than 90% of the variation in rates, the curvature is usually left out of the model. Although this works very well for government bond yields and swap rates, it does not explain the movements in swap spreads.
Put simply, while a PCA on swap rates can decompose the changes in swap rates, it cannot identify how much of this change is due to changes in the underlying government bond yield, and how much of it is due to changes in the swap spreads. The latter conveys important information about the state of financial markets, over and above the information conveyed by swap rates. Given the importance of swap spreads, we develop a model to forecast swap spreads for the 28 currencies listed in Appendix 1. These spreads, when added to the corresponding government bond yields, produce the swap rates.
The rest of this paper is split broadly into three sections. First we summarize some of the factors affecting swap spreads, as discussed in the literature so far. Next, we lay out our modeling methodology, our forecasts of swap spreads and rates, and key limitations of the models. Finally, we present several examples of how banking clients use our forecasts.
Per the existing literature, the factors affecting the term structure of swap spreads are liquidity, default risk, the level and slope of the yield curve, and the demand and supply of government debt securities (Sun, Sundaresan, and Wang, 1993; Choudhry, 2008; Kobor, Shi, and Zelenko, 2005).
More than one study has shown that liquidity is the more important determinant of swap spreads, with credit or default risk having second-order impacts (Huang and Neftci, 2003; Grinblatt, 2001; Feldhütter and Lando, 2008). The reasoning is that the current industry practice – with both parties netting out all existing swap positions and imposing collateral against each other based on daily net mark-to-market values of all open positions – has essentially removed the risk of default by a counterparty.
The higher liquidity of the government securities generates a convenience yield, or liquidity premium, which is lost to an investor wishing to receive fixed payments in a swap agreement (Grinblatt, 2001). In other words, the liquidity premium is the markup an investor demands to participate in the less liquid interbank lending markets. This is measured, for instance, by the Treasury-Eurodollar (TED3 ) spread, which represents the difference between a short-term money market rate such as the London Interbank Offered Rate (Libor) and a Treasury bill of matching maturity. Meanwhile, deteriorating liquidity in the market for government securities – measured as the spread between the general collateral (GC) repo rate and the yields on highly liquid short-term government debt (He, 2000; Li, 2004; Liu, Longstaff and Mandell, 2006) – chips away at this liquidity premium and results in narrower swap spreads. This phenomenon is most obvious in the post-financial crisis times when strict bank regulations around leverage and capital requirements have caused banks to shy away from exchanging government securities in the repo market. As a result, repo rates have risen and the shrinking Libor-to-repo-rates spreads have pulled down swap spreads. In the US and several other economies such as the UK, the shrinking spread has turned swap spreads for longer maturities negative by eroding arbitrage opportunities.
Deteriorating credit quality will increase swap spreads through an increase in interbank lending rates to which the floating leg of the swap payments is benchmarked. Since the floating rate payer has to pay a higher variable rate, he demands a higher fixed rate to be made even. This is the default premium built into swap spreads.
The spread between the uncollateralized interbank lending rate and a riskless short-term lending rate is taken as a proxy for credit or default risk. In the US swap market, this rate is usually taken as the difference between the three-month Libor and the three-month GC repo rate (Liu, Longstaff, and Mandell, 2006). Spreads of corporate bond yields over matching maturity government bond yields are also used as a measure of perceived default risk among investors (Brown, Harlow, and Smith, 1994; Duffie and Singleton, 1997; Sun, Sundaresan, and Wang, 1993).
The shape of the yield curve encapsulates information regarding the current and expected future states of the economy. In general, swap spreads are higher when the level of interest rate is higher, although this relationship typically breaks down for maturities greater than 10 years. Since the government bond yield curve is upward-sloping, the term structure of the swap spreads is hump-shaped, rising upward from the short to medium maturities and sloping downward thereafter. The slope of the government bond yield curve, or the term premium, carries information regarding the future path of interest rates. A positively sloped yield curve means investors expect higher future interest rates, so floating rates go up, narrowing swap spreads (Choudhry, 2008; Kobor, Shi, and Zelenko, 2005; Huang and Neftci, 2003).
Excess demand for government debt depresses government bond yields, increasing swap spreads. Quantitative easing (QE) monetary accommodation measures in many countries in the wake of the financial crisis and flight to quality for safe currencies have this effect. On the other hand, high corporate debt issuance in the low interest rate environment and the increased demand to be the fixed-leg receiver in a swap have brought down swap spreads (Friedman, 2016).
We model and forecast swap spreads using the two-step approach for modeling and forecasting swap rates described in Licari, Loiseau-Aslanidi, and Suarez-Lledo (2013). Our dataset contains monthly swap rates and government bond yields data, by maturity, from Thomson Reuters for 28 currencies. We also have the history and forecasts for various macroeconomic variables produced from Moody's Analytics structural econometric country models. In step 1, we decompose the swap spreads for each currency into the level and slope components and forecast these components using the forecasts of macroeconomic factors. The first two components of the PCA, on average, cumulatively explain more than 90% of the variation in swap spreads. For the typical country, the first component loads on all maturities with the same sign, shifting the swap spreads up or down across the yield curve. The second component loads on short and long maturities with the opposite sign rotating the yield curve for swap spreads. In step 2, we forecast the swap spread for a given currency and maturity from the forecasts of the PCA components. Finally, we add the forecasts of the swap spreads to our forecasts of the government bond yields of the equivalent maturities to obtain the swap rates.
The idiosyncrasies in the interest rate swaps market across currencies mean that the PCA components on the swap spreads do not have the same interpretation across currencies, unlike the PCA components on government yields or swap rates. That, in combination with data restrictions, is why we have kept flexible our choice of drivers for the two components to approximate the theoretical drivers listed in Section 1. For instance, since we do not have data on repo or call rates for all 28 currencies in our study, necessary to measure the cost of the hedge, we employ, where available, the spread of the three-month interbank lending rate and the yield on the three-month highly liquid government debt as a proxy for the liquidity premiums in swap spreads. This is consistent with many studies, including Grinblatt (2001).
In a similar vein, we do not have data on corporate bond spreads for the majority of currencies in our sample. Since such credit risk is priced into short-term interbank lending rates, we instead use the spread of the three-month interbank rate over the three-month government bond yield also as a proxy for bank default risk. By the same token, stock market volatility is generally procyclical with financial market uncertainty or stress. We use the Chicago Board Options Exchange (CBOE) Volatility Index (VIX) – a measure of implied volatility in US equity markets – and the rolling standard deviations of country-specific stock indexes as an additional signal of the default premiums in swap spreads. Some studies have shown that the default premium does not have a term structure (Liu, Longstaff, and Mandell, 2006). This is likely not true in a stress scenario because, in the event of a financial market crisis, the risk of default on near-term obligations is higher. This is one explanation of why during stress, short-term swap spreads generally jump more than their longer-term counterparts. In many cases, this results in a brief period of inversion in the swap spread term structure.
As a final example, we use the term premium, or the slope of the bond yield curve, and the growth rates of GDP and stock prices as proxies for expectations of future interest rates. A steepening yield curve is usually accompanied by tightening swap spreads (Choudhry, 2008; Kobor, Shi, and Zelenko, 2005; Huang and Neftci, 2003). The reverse is also observed in most countries in our sample: As the yield curve flattens and eventually inverts in the run-up to a recession, swap spreads rise. The increase in spreads may also be due to increasing default risk and eroding liquidity in financial markets. Since spreads rise more for shorter maturities, the yield curve for swap spreads flattens, and sometimes inverts. With central banks yanking down the short end of the bond yield curve and long-term yields responding with a lag, the term premium widens. Swap spreads decline, with spreads on short maturities narrowing more than spreads on long maturities.
As an illustration of the identification obstacles in modeling swap spreads, Figure 1 displays the swap spreads for a typical economic region – the United Kingdom.4
In analogy to other jurisdictions, UK swap spreads gradually increased in the buildup to the 2008 financial crisis, and quickly collapsed during its apex, as theory predicts: Tighter monetary policy drove up short-term interest rates after 2005 and the flattening term premium prompted higher swap spreads. By the time the crisis reached its peak in September 2008, central bank responses had, of course, already been drastic. In the US, the Federal Reserve as pack leader cut its policy rate targets dramatically from over 5% to 0.25% within the course of a single year. The Bank of England and the European Central Bank followed suit. In such an environment, the inertia of long-term yields caused a rapid steepening of the term premium, and swap spreads fell sharply in response. In the years since, they have recovered somewhat, more so on the short end of the swap curve, while several longer tenors have lingered near or in negative territory, substantially below the shorter tenors.
Despite similarities, there are differences across countries, illustrative of the futility of "one-size-fits-all" attempts of swap spread modeling. First, the timing of rise and decline in spreads varied across countries, perhaps reflecting different response times of central banks and dealers to the crisis. Second, most tenors witnessed a gradual creep-up in spreads before 2008, but some short-term spreads, especially those on UK one-year swaps, shot up abruptly before falling. Meanwhile, at the long end of the swap curve, some spreads had virtually no buildup and rather led the common decline. Post-crisis dynamics exhibit marked differences across countries. For instance, UK short-term swap spreads showed a second rise during the European sovereign debt crisis in 2012, unlike others such as the US. Finally, while long-term spreads have been stubbornly negative in the UK since 2010, similar to those in the US, others such as the European common currency area have had a more solid recovery.
The main takeaway from Figure 1 is that simple liquidity considerations alone are insufficient to adequately describe the behavior of swap spreads across mature economies, let alone all countries in our broader dataset. In our view, their dynamics inadequately explain the sharp rise in short-term spreads in the pretext of the financial crisis observed in some countries and the buildup in spreads during the European sovereign debt crisis. The panic aspect of these events rather points toward counterparty risk. More puzzling, perhaps, the strong inversion of swap spreads across maturities and persistent, negative long-term swap spreads suggest the presence of unexploited arbitrage opportunities. Increased regulation motivating end-of-quarter bond sell-offs by banks and large-scale QE-induced tightness of the repo market, resulting in costlier and thus unprofitable hedges, are the most likely explanations for reduced dealer appetite to participate in such agreements.
For all practical purposes it is impossible to disentangle the specific characteristics of each individual country to develop stress test forecasts. We therefore develop a simple yet flexible approach of modeling swaps for a large cross section of countries, bearing some analogy to the widespread Nelson-Siegel method for Treasury yields. In the following, we will illustrate this approach using the United Kingdom as an example5.
We begin by extracting the joint variation in UK swap spreads across all available tenors with sufficient coverage by obtaining their first and second principal component scores. Figure 2 captures the dynamics of the level and slope components, while Figure 3 reports their factor loadings.
The level component captures the joint movement of swap spreads over time in analogy to the bond yields. This conclusion roots in the observation that the factor loadings of all tenors have positive signs and are of similar magnitude.
The slope component captures different dynamic behavior of the short and long ends of the swap curve. Lower tenors have positive loadings, while higher tenors have negative loadings. This "rotating" pattern is typical across countries. Meanwhile, in the UK, the first principal component accounts for about 61% of the joint variation in UK swap spreads, while the second principal component accounts for about 31%.
In an effort to identify drivers for the level and slope of UK swap spreads, two features are suggestive:
- The level component rose during the stress periods of 2008 and 2012, suggesting that spreads uniformly widened for some time. In a similar vein, the level component fell in the post-crisis period of low-interest-rate monetary policy. This is consistent with the notion that all tenors react roughly to higher yield levels and counterparty default risk. Natural driver candidates for the level are, thus, yield levels and financial stress measures such as stock market volatility or the TED spread.
- A sharp increase in the slope component around 2008 coincided with inversion and a sizable drop in the 30-year swap spread, dipping well into negative territory. The rise in the term premium due to the Bank of England's emergency response during the crisis is a consistent explanation of the narrowing and inversion of the longer terms; however, the strength and persistence of the drop for tenors above the 10-year swap spreads hint that other factors, such as financial market frictions, were likely at play here as well.
To model the principal components for the UK we flexibly choose variables from the driver pool identified by the literature in the Section 1. We lean toward interest rate measures and financial stress as drivers of the first component and lean toward term premium and liquidity measures as drivers of the second principal component score. Employing the model selection process developed in Licari, Loiseau-Aslanidi, and Suarez-Lledo (2013), we choose an optimal set of drivers based on consistency with theory, in- and out-of-sample performance, and forecast consistency. The method of estimation is linear generalized least squares, relying on Newey-West errors to address issues of autocorrelation and heteroskedasticity.
Figure 4 plots the PCA level component and its drivers. As expected, the level component falls alongside interest rates in the UK money market and follows their lower trajectory post-crisis. Similarly, financial stress, captured by stock market volatility and defined as a rolling window standard deviation of stock market returns, captures the upward shift in swap spreads past 2008 and again during the European debt crisis that began in 2009. The final driver, which we call global equity factor (GEF), is the first principal component score of US, UK, Japanese, and German stock indexes. Stronger market performance is negatively associated with swap spreads.
The slope component and its drivers are illustrated in Figure 5. Our method identifies as optimal drivers the TED spread as a measure of the liquidity premium, UK stock market volatility, and its monetary policy rate. Higher liquidity premiums and market volatility push up the slope component, resulting in larger spreads of shorter tenors relative to longer ones, causing compression or, in the case of the UK, deeper inversion of the swap spreads, consistent with a steeper term premium. Similarly, stronger inversion kicked in when the Bank of England lowered its policy rate.
It should be noted that stock market volatility appears as driver for both components. This is typical for the trade-offs we face in obtaining a consistent and feasible model approach for such a large cross section of countries. In practice, allowing the same set of variables to drive both components substantially improves model fit and forecast consistency, despite added "theoretical murkiness" in interpreting the PCA components' meaning. From country to country, the exact association of the macroeconomic drivers with the level and slope component can vary significantly as a result of vast international differences in financial market development, liquidity, regulation, and monetary policy. As a pragmatic approach, we therefore select drivers based on best fit and forecast consistency from the broader pool of relevant variables suggested by theory, without attempting to identify exactly if a specific driver captures liquidity, credit risk, or another underlying determinant.
Figure 6 displays our scenario forecasts for UK tenor swap spreads and the corresponding swap rates for the mid-cycle update6 of the Federal Reserve's Comprehensive Capital Analysis and Review baseline, adverse, and severely adverse scenarios. The predicted behavior of the swap spreads during stress is consistent with their direction and magnitude during the financial crises of the past 10 years. Likewise, their long-term behavior matches recent history, maintaining inversion between shorter and longer tenors. Turning to swap rates, we predict that a gradual normalization of interest rates will increase spreads on all tenors in the baseline, while short terms only gradually increase in the severely adverse scenarios, matched with a flat 10-year and a falling 30-year swap rate. We do not expect a drop of swap rates comparable to that in 2008 in the severely adverse scenario, since the Bank of England no longer has the same room to cut rates, pushing up against the zero bound. Once more, this feature is common for many countries in the sample.
We do not have a good way to model the changes in the relative supply of government debt in the wake of the financial crisis. Neither do we have a good proxy for liquidity in the government debt market for non-US countries. As an example, Lekkos and Milas (2001) state that the illiquid nature of the three-month Treasury bill in the UK makes the generally accepted three-month Libor to the three-month Treasury bill spread an inefficient measure of liquidity in the UK financial markets. These country-specific idiosyncratic elements are hard to capture in a model. Put differently, swap spreads do not respond to the same set of macroeconomic factors in the same way across currencies. The hard-to-capture cross-currency idiosyncrasies render futile the attempt to come up with a common interpretation of what drives swap spreads. In fact, to our knowledge, there have been very few papers that have tried to explain swap spreads globally. Choudhry (2008) is an exception.
Interest rate swap spreads are a key benchmark for pricing and hedging in the large universe of fixed-income securities. Therefore, financial institutions dealing with these securities require swap rate forecasts for various currencies under the baseline and alternative scenarios to satisfy a gamut of needs, including:
- Stress testing portfolios
- Calculating expected long-term losses to meet accounting needs
- Generating portfolio value projections for business-as-usual purposes
- Forecasting value of bonds denominated in non-USD currency
- Estimating market value of traded assets
- Valuing cash flows contingent on future interest rates
Interest rate swaps, frequently used for hedging against interest rate risks, are an invaluable tool in financial institutions' risk management repertoire. We model swap rates for various currencies using the principal components of the term structure of the swap spreads and macroeconomic indicators. The linkages with the macroeconomy allow us to produce forecasts of swap rates under different macroeconomic scenarios. While the models are able to capture the idiosyncratic behaviors in most swap markets, the "never-before-seen" feature of current global interest rates makes predicting the future direction of swap rates difficult.
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Li, Xiaofei. "Decomposing the Default Risk and Liquidity Components of Interest Rate Swap Spreads." York University Working Paper. Jan. 2004.
Licari, Juan M., Olga Loiseau-Aslanidi, and Jose Suarez-Lledo. "Modelling and Stressing the Interest Rates Swap Curve." Moody's Analytics whitepaper. Sept. 2013.
Liu, Jun, Francis A. Longstaff, and Ravit E. Mandell. "The Market Price of Risk in Interest Rate Swaps: The Roles of Default and Liquidity Risks." Journal of Business, vol. 79, no. 5, 2006, pp. 2,337-2,359.
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Appendix 1: List of Currencies
Hong Kong dollar
New Zealand dollar
South African rand
UK pound sterling
Director – Economic Research
Sohini Chowdhury is a director at Moody’s Analytics. She leads client projects related to stress testing and other regulatory requirements, and oversees the modeling of market risk instruments among the US staff. She has a PhD in economics from Purdue University, and a master’s in applied statistics from West Chester University. Before joining Moody’s, Sohini taught at the University of Cincinnati.
Assistant Director, Economist
Martin is an economist within Moody’s Analytics. He covers financial market risk, as well as US state and local economies. Before joining Moody’s Analytics, he served as associate professor of economics at Pacific Lutheran University in Tacoma, Washington. Martin has published on financial market development and informal economies and has conducted local impact studies and forecasts. He has a doctorate and a master’s degree from the University of Wisconsin – Milwaukee and completed his undergraduate work at the University of Potsdam and the Ludwig-Maximilians University in Munich, Germany.
Examines the role of disruptive technologies in the financial sector and how firms can improve their practices to remain competitive.
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