No-Arbitrage pricing of GDP-Linked bonds

Abstract

We develop a novel term-structure model for pricing GDP-linked bonds, hypothetical securities with cash-flows indexed to the level of U.S. GDP. For this purpose, we rely on a term-structure model of equity yields estimated using the prices of dividend swaps, which we assume span GDP growth. Our approach provides a novel way of estimating the relative cost of conventional and GDP-linked bonds, as well as measuring more general market-based expectations of (and risks around) GDP growth. Our model predicts that U.S. GDP-linked bonds would typically have yields lower than those on conventional Treasury bonds with the same maturity in our sample from 2010 to 2017. Positive expected future GDP growth lowers the yield on GDP-linked bonds relative to conventional bonds, which typically more than offsets the estimated GDP risk premium demanded by investors for holding GDP risk.

Introduction

General government debt is projected to increase to 96% of global GDP in 2020, up from 80% in 2012 (IMF, 2020). As of 2019, 300 billion US dollars of sovereign debt were in default, including 50 billion US dollars issued by advanced economies (Beers et al., 2020). In this context, debt instruments with repayments tied to the evolution of a country’s GDP (“GDP-linked bonds”) offer various benefits to both sovereign issuers and investors. Thus, such instruments could constitute a valuable addition to the global financial architecture.

Having a portion of their debt in the form of GDP-linked bonds is attractive to sovereigns for two main reasons. First, it increases fiscal flexibility. The basic feature of such instruments is that debt repayment obligations automatically fall in “bad times” (that is, when the economy is growing relatively slowly, which tends to coincide with periods of relatively low tax revenues). This provides greater fiscal policy space in the face of adverse shocks. Second, the countercyclical nature of GDP-linked bond repayments increases debt sustainability. This in turn could reduce risk premiums on the country’s sovereign bonds more generally, making overall financing cheaper. Moreover, from the point of view of investors, GDP-linked bonds provide a more direct way of gaining exposure to economic growth than at present (Borensztein and Mauro, 2004).

GDP-linked bonds also offer wider benefits to the global financial system. Most obviously, by increasing debt sustainability, they should bring benefits of greater financial stability. In addition, they should promote more precise pricing of sovereign risk and help generate market-based expectations of GDP growth, which could aid planning and risk-management by governments and corporates.

Interest in GDP-linked debt as a way of promoting financial stability is not new. The case for issuing GDP-linked bonds was originally put forward by Shiller (1998), but never taken to practice in its original form.¹ Beginning with Borensztein and Mauro (2004), there has been a renewed interest in both policy and academic circles in exploring the desirability of issuing growth-linked debt. Borensztein and Mauro (2004) discuss potential obstacles and suggest a number of fixes to attempt to jump-start a market. Chamon and Mauro (2006) put the focus on pricing, and discuss broader implications for sovereign debt structures and default probabilities. Following the global financial crisis (GFC) and the ensuing increase in global debt levels, there was a further wave of interest in GDP-linked bonds as a way of promoting financial stability. Benford et al. (2016) and Blanchard et al. (2016) study the desirability of such instruments in a post-GFC world, and CEPR (2018) compiles arguments for and challenges faced by a market for GDP-linked debt from the point of view of economists, lawyers and investors.² More recently, the acceleration in the increase of global debt levels following the outbreak of the Covid-19 pandemic has brought the debate on GDP-linked debt back into the forefront (see United Nations, 2020).

One of the main obstacles to jump-starting a market for GDP-linked debt, as highlighted by the references described above (see, for example, Borensztein and Mauro, 2004), is the difficulty of pricing GDP-linked instruments in general, and quantifying the GDP risk premium in particular. Under a GDP-linked debt contract, a greater share of the risk of weak growth outcomes is borne by the holders of these bonds than by holders of conventional bonds. As a consequence, issuers of GDP-linked bonds would likely need to pay an additional risk premium relative to conventional government bonds in order to compensate investors for the fact that cash-flows from the bond would be relatively small in bad times (when they are valued the most). However, it is not straightforward to quantify this risk premium in the absence of an existing market for GDP-linked bonds.

The challenge of pricing GDP-linked debt has spurred a literature on its own. However, previous attempts to quantify the GDP risk premium have significant limitations.³ Some studies combine a model of the time-series properties of GDP with assumptions about investors’ preferences, and solve for the required risk premium. For example, Barr et al. (2014) assume that investors have constant relative risk aversion. However, such standard utility functions cannot explain the magnitude and dynamics of the risk premiums of existing assets, so it is not clear why we should attach great weight to their predictions for risk premiums on assets that do not currently exist. Other studies (including Borensztein, Mauro, 2004, Kamstra, Shiller, 2009 and Bowman and Naylor, 2016) have adopted an approach based on the Capital Asset Pricing Model (CAPM), and use the “beta” of observed GDP growth with respect to returns on the market portfolio as a proxy for the “beta” of GDP-linked bonds. Unfortunately, it is by no means obvious that returns on hypothetical GDP-linked bonds would have the same covariance with returns on the market portfolio as GDP growth. Moreover, none of the studies listed above consider how the GDP risk premium may vary with the maturity of the GDP-linked bond, which is a crucial practical question for prospective issuers.

In this paper, we move this literature forward by developing a novel asset pricing approach to estimate U.S. nominal GDP (NGDP) risk premiums using the prices of existing assets. Specifically, we build a no-arbitrage affine term structure model (ATSM) of equity yields derived from the prices of S&P 500 dividend swaps, which we assume span the priced component of GDP growth, hence allowing us to estimate predicted yields of bonds with payoffs linked to nominal GDP. Our approach avoids the limitations of previous studies: it allows a much more flexible specification of investors’ preferences, which is known to match risk premiums on existing assets; it avoids any assumptions about the relative covariances of GDP growth and returns on GDP-linked bonds with the market portfolio; and it allows us to study how the nominal GDP risk premium varies with the maturity of the bond. The key assumption underpinning our analysis is that nominal GDP and dividend growth are spanned by the cross section of the term structure of equity yields. The spanning assumption requires that—in the absence of measurement error—we can invert nominal GDP growth and dividend growth from the term structure of equity yields. In consequence, we can back out the risk-neutral dynamics of nominal GDP growth and dividends from observed equity yields—which allows us to compute predicted yields on GDP-linked bonds. Hence, a second output from our model is a set of market-implied expectations of (and risks around) GDP growth at various horizons.

In our framework, the spread (or “breakeven”) of the yield on a conventional bond minus the yield on a GDP-linked bond of the same maturity is affected by two main factors. First, if GDP growth is expected to be positive, this pushes up the terminal cash-flow on a GDP-linked bond, raises the current price, and lowers the current yield relative to conventional bonds. Second, if GDP growth is expected to be relatively low in bad times (that is, when discount factors are relatively high), investors will demand an additional risk premium for bearing GDP risk. This GDP risk premium pushes down the current prices of GDP-linked bonds and raises the yields relative to conventional bonds; that is, the GDP risk premium component of the breakeven is likely to be negative (and hence increase the yields of GDP-linked bonds).

We find that our model can fit nominal GDP growth precisely, while also reproducing the broad movements in equity yields. Using the output from the model we can then compute the price of GDP-linked bonds. We find that the unconditional average term structure of GDP risk premiums is indeed negative and increase monotonically with maturity. At relatively short maturities between 6 months and 2 years the average premiums are in the region of -7 percent to -4 percent, but increase to -1 percent at the 10-year maturity. That is, the increase in the yield of GDP-linked bonds owing to GDP risk premia decreases with maturity. This seems intuitive if we consider the relatively low persistence of shocks to NGDP growth, which should hence be less important over the lifespan of bonds with longer maturities.

This paper is related to the study developed contemporaneously by Mouabbi et al. (2020). That study uses a consumption-based macro-finance model with habits. The main difference between their and our study is that our discount factor takes a more flexible form that is known to be successful in pricing existing assets such as conventional government bonds. Moreover, they only consider the aggregate stock market index as a target, missing the information in the term structure of dividends.

The remainder of this paper is structured as follows. In Section 2, we describe the basic structure of the hypothetical GDP-linked bonds we consider and provide some basic intuition for the difference in yields between GDP-linked and conventional government bonds. In Section 3, we present a joint no-arbitrage term structure model of GDP-linked bond and equity yields. In Section 4, we explain how we estimate this joint model in the absence of observed GDP-linked bond yields. In Section 5 we present our main results. In Section 6, we summarize our conclusions.