How should nations issue their public debt?
Since the global financial crisis of 2008, governments around the world have issued substantial amounts of debt. This increase has been even more dramatic on the eve of the global pandemic. Today, the stock of global debt is reaching levels not seen since World War II. However, there are many ways in which nations can issue debt: they can offer to pay in one year’s time, issuing Bills, or commit to a perpetual stream of payments, as did eighteenth-century Britain. Recently, Argentina and Mexico have issued 100-year bonds, whereas the US Treasury refrained from issuing 50-year debt. How should a nation issue debt? Economic theory, thus far, provides useful insights but I think there’s more to be done to take those guidelines to a practical use. In this note, I discuss what I think is missing in Economic theory and discuss some promising recent advances and use this as an excuse to describe some work I have contributed to in Bigio, Nuño, and Passadore, (2021, from now on BNP and all further references found in that paper).
What economic forces does Economic theory highlight? Two fields that seem separate but are intimately related provide guidelines to public debt management: public finance and international finance. In public finance papers, expenditures are typically exogenous, and the objective is to smooth taxes. In international finance, income is exogenous, and there is a desire to smooth expenditures. Aside from these superficial differences, papers in these fields provide prescriptions based on three economic forces: smoothing, insurance, and limited commitment.
No-arbitrage is at the heart of classic theories. Below, I argue that the essence of these theories’ prescriptions is the no-arbitrage pricing of government debt. No-arbitrage pricing means that, given an expected path (a probability tree really) of expenditures and taxes, financial markets will price debt such that it is impossible to make money, trading bonds of different maturity. Not even in a probabilistic sense if we correctly adjust for risk.
If a classic paper that I reference below provides a prescription for debt-maturity management, it is based on an arbitrage-free pricing argument.
What is missing in classic theories? A short answer: liquidity. What do I mean by this? A world with liquidity frictions is where there are different markets for debt are issued at different horizons, and arbitrage across those markets is not possible. Why different markets? Well, take the case of pension funds, the largest source of savings in an economy. Typically, pension funds want to guarantee a cash flow many years down the road and have an inelastic demand for long-term debt. A venture-capital fund or private equity firm, on the contrary, will want access to cash in the short run to exploit investment opportunities. Life and casualty insurance companies will want to guarantee a constant stream. These funds want a different risk exposure to the price changes in bond markets. Why the lack of arbitrage? Managers need time to study markets and have limited cognitive ability, for starters. Furthermore, to curb agency frictions, the fund’s mandates typically inhibit their shorting capacity and limit their speed of response. As a result, we live in a world where short and long-term debt, and in between, there are potentially different markets.
In a world where liquidity frictions are prevalent, the relative quantities of one debt or the other affect prices. Ralph Koijen of Chicago Booth and Motohiro Yogo of Princeton are developing a fantastic asset-pricing agenda exploiting that quantities matter for pricing. The implication of Koijen and Yogo’s agenda for governments is that if they issue too much debt in one maturity, they will impact its price, even if they plan the same path of fiscal deficits.
The debt-management offices of most nations are aware of this price impact brought about by liquidity and act upon it. They are evidently concerned about liquidity because they issue at all points of the yield curve. Economic theory does not provide a cohesive framework that helps design their debt-management strategies. This is something that we can fix.
Evidence of Liquidity in Debt Markets. At a minimum, if I want to argue that liquidity matters, I should point to some evidence. It turns out that the evidence is ample. For example, we can find survey evidence in [Vayanos and Wang 2013] or [Darrell Duffie’s AFA Presidential Adress in, 2010]. Earlier work includes [Cammack 1991], [Duffe 1996], [Spindt and Stolz 1992], [Fleming 2002], [Krishnamurthy and Vising Jorgensen, 2012], and many others papers discussed in BNP.
A piece of evidence in BNP. Here is a concrete piece of evidence from BNP that I find fascinating. To produce BNP, we obtained information of all the debt auctions in Spain. Spain is an ideal laboratory to measure liquidity costs. It has an active secondary market from which we can observe market prices and it is small compared to Europe, so its issuances have little impact on the whole Euro area deficit and yields. Yet, we find consistent deviations from primary and secondary prices that increase with a bond auction size. The figure below shows reports the coefficients of the impact of issuances as % of GDP on the auction markup between the primary and secondary markets. I’m sure similar deviations can be found for many other countries.
Asset pricing models with liquidity frictions: a starting point. A few influential frameworks provide modeling tools to incorporate liquidity frictions. In Vayanos and Vila (Econometrica 2020, VV) investors demand bonds of only a specific maturity (a habitat) and arbitrageurs intermediate across maturities. In Duffie, Garleanu and Pedersen, (Econometrica 2005, DGP) bonds are held by investors that face high discounts and need time to reallocate bonds. A recent paper by Lester, Kargar, and Weill (2021) modify DGP to study the inventory holdings — -in DGP, bond portfolios are binary. The common theme in these papers in that the larger an issuance, the lower the price.
Although frameworks for asset pricing exist, as noted in conclusion in VV, there has not been much work done on how nations should issue debt if they face these liquidity frictions.
A technical challenge. By now, my claim should be clear: we should study debt problems where the government internalizes its liquidity frictions. In the presence of liquidity costs, because we abandon arbitrage-free pricing, the number and types of bonds that the government can issue is no innocuous — -this is also true in incomplete market models. Here’s the problem. Suppose we want to construct a realistic model where the government only issues 30-year bonds; We need at least 30 state variables because a 30-year bond becomes a 29-year bond the following year, a 28-year bond the year after, and so on. By contrast, a bond that matures by 5-per cent every year is still a bond that matures by 5-per cent the year after its issuance. Thus, the number of variables in a realistic problem blows up quickly. For that reason, due to this curse of dimensionality, qualitative analysis is often relegated to highly stylized models, and quantitative models allow for a small number of decaying perpetuities. In practice, governments simultaneously issue in many maturities, and perpetuities are a rarity. What can we do from here?
Bigio, Nuño, and Passadore. In BNP, we allow the government to issue an arbitrary number of finite-life bonds, an exercise hitherto not carried out. The approach is to make restrictive assumptions on shocks but allows for an analytic characterization and a quantitative evaluation of optimal debt management with a realistic debt structure.
The analysis in BNP uncovers a general principle that I think applies with much greater generality. Debt problems can be studied through limited arbitrage. We can think of multiple artificial traders in charge of issuing debt of the corresponding maturity.
Each trader must apply a simple “limited-arb” strategy:
Issuance/GDP at Maturity τ= Inverse Liquidity Coefficient x Value Gap
The strategy follows from a condition that equates marginal auction revenues to an internal debt valuation. The rule states that optimal issuances of a bond of τ-maturity equal the ratio of a value gap over the liquidity coefficient of that maturity. The value gap is the proportional difference between the secondary-market price and a domestic valuation. The liquidity coefficient is bar in a given maturity of the picture above.
With liquidity costs, the government’s discount rate does not equal the market’s short-term rate, hence the “limited-arb” term. Thus, the domestic valuation is a counterfactual bond price, computed using the government’s discount rate instead of the short-term rate. A positive value gap indicates the desire to arbitrage the difference between market prices and domestic valuations by the artificial traders.
The liquidity coefficient modulates the willingness to arbitrage in a given maturity. As a result of this limited arb, it is optimal to issue in all maturities at all times, as most nations do. Since the domestic discount factor rate must be internally consistent at an optimum, a single equilibrium variable summarizes a problem with infinite control variables.
I should note that the domestic discount rate is the solution to a fixed-point problem: A conjectured expenditure path maps to a domestic discount rate. This discount rate generates an issuance path via the optimal rule. Ultimately, the issuance path must be consistent with the expenditure path’s debt services. The conjectured expenditure path must coincide with the actual expenditure path obtained, applying the issuance rule at the optimum. This logic underlies the computational algorithm in BNP.
Back to the Classics. It is a valuable exercise to return to some classic work and speculate about how liquidity considerations modify classic prescriptions (or the lack of them). As I noted above, the prescriptions in the classic papers build three economic forces: smoothing, insurance, and limited commitment.
What does liquidity mean for smoothing? I guess modern public finance literature starts with Barro (1979). Barro showed that absent risk considerations, a government will smooth taxes by equating their marginal distortions over time. Tax smoothing is akin to the consumption smoothing that prevails in international finance. In both instances, smoothing prescribes a path for the national debt — which depends on initial debt — but no prescriptions for maturity. Underlying these results is the property that the government’s discount rates coincide with the path of short-term rates. With liquidity costs, this is no longer the case. BNP provides a formal proposition for the effect of smoothing on debt maturity.
What does liquidity mean for insurance? Risk naturally opens the door to insurance. In open economies, portfolios of bonds with different maturities can reproduce insurance markets: a condition for this is a sufficient correlation between fiscal shocks and changes in the yield curve. In another public finance classic, Lucas and Stokey (1983) showed that if governments can commit to a fiscal program, a government with access to state-contingent debt will also attempt to smooth taxes over time.
Of course, government’s don’t buy tax insurance. However, extending what Duffie and Huang (1985) had done for household economics, Angeletos (2002) and Buera and Nicolini (2004) showed that a government might also implement the optimal fiscal policy in Stokey and Lucas, with bonds of different maturity. In both the international and public-finance setting, the optimal strategy is to issue long-term debt and hold short-term assets. These shocks that steepen the yield curve produce capital gains during adverse events.
Underlying this prescription is, again, the idea of arbitrage-free pricing. The capital gains that allow governments to insure against adverse states are priced fairly. Thus, it is as if the government pays an insurance premium to earn income when its tax receipts drop. Buera and Nicolini showed that, quantitatively, changes in the yield curve slope are so small that the prescribed gross debt positions to hedge fiscal risks are implausibly large. I think that in practice, governments understand that dramatic changes in their debt portfolios would bring about significant financial losses and instability. In this is why they don’t actively manage their portfolios this way. Liquidity frictions are behind the view, not the lack of enough bonds.
To that point, Aiyagari, Marcet, Sargent, and Seppala (2004) studied Stokey Lucas under incomplete markets. They showed that a government that cannot insure because it issues in one maturity only would self-insure by taking less partly by reducing its debt stock — as in the consumption literature [ChamberlainWilson, WangWangYang16]. Self-insurance produces more realistic debt levels but does not rationalize why governments issue many maturities. Liquidity costs make it impossible to restructure debt immediately, so debt flows are not as dramatic as those described by Buera and Nicolini. Because portfolios cannot be easily undone, there is an interaction with self-insurance. BNP shows what greater self-insurance implies for the debt profile.
What about limited commitment? Limited commitment produces the prescription that nations should issue debt in a way that limits their ability to take advantage of debt holders or taxpayers. This prescription comes in different flavors. In public finance, Lucas and Stokey, anticipated that a government that lacked commitment would ex-post attempt to modify the stochastic discount factor of its lenders to make capital gains. Debertoli, Nunes and Yared (2014) proved that conjecture and showed that a government could avoid this temptation by issuing perpetuities that prevent it from making capital gains by manipulating the households’ pricing of debt through taxation. Some recent discussion papers by John Cochrane, resonate with this result.
In international finance, the prescription is the exact opposite when the government is allowed to default. Although in principle, default allows for greater market completeness [a point nicely articulated by Bill Zame in his 1993 AER piece], it is affected by limited commitment in two ways. First, the government cannot pre-commit to default in specific states. Thus, when issuing more debt, the government must consider its effect on default premia [as noted in the classic Eaton (1981), Arellano (2008), and Amador and Aguiar (2013)]. Second, in terms of maturity choice, [Bulow and Roggoff (1988)] identified that long-term debt is prone to dilution, the idea that once the long-term debt is issued, the price of a new issuance does not internalize the increased default premia on past debt. Debt dilution is thus a force toward shorter maturities and Amador, Aguiar, Hopenhayn and Werning (2019) articulated nicely. Building on this intuition, many papers in international finance study the quantitative implications of combining the insurance and limited commitment forces. Yet again, I think these papers miss the realistic consideration that a nation’s debt office has to “fill in” all the yield curve points.
Forces and Valuations in BNP. We show that smoothing, insurance, and incentives appear in domestic valuations in BNP. The simple “limited-arb” rule allows us to characterize how these forces shape an ideal debt profile cleanly. Furthermore, we can decompose the contribution of each force in calibration that matches the distribution of debt maturity of a country we want to study. In the paper, we do this for Spain.
Recent work. I should point to recent papers along the lines of BNP.
One close cousin is Bhandari, Evans, Golosov, and Sargent’s “Managing Public Portfolios”. Another is Faraglia, Marcet , Oikonomou, and Scott (2018, Restud), which is the only other paper to study optimal maturity management with finite life bonds.
Both paper stress the role of frictions that prevent the quick adjustment of the debt profile.