A Model of Safe Asset Determination

Four features of the safe assets we actually observe: their concentration in a few sovereigns, their high valuations relative to comparable fundamentals, the persistence of those valuations as fundamentals deteriorate, and their tendency to appreciate during turmoil. The opening paragraph lays these out: US government debt is “the premier example of a global safe asset” and German debt “occupies a similar position as the safe asset within Europe”; both “appear to have high valuations relative to the debt of other countries with similar fundamentals, measured in terms of debt or deficit to income ratios”; “as fundamentals in the US and Germany have deteriorated, these high valuations have persisted”; and “during times of turmoil, the value of these countries’ bonds rise relative to the value of other countries’ bonds in a flight-to-quality” (Section 1). The flight-to-quality claim is quantified in Section 3.6: the authors compute that between September 12 and the close on September 15, 2008 the value of outstanding US government debt “rose by just over $70bn,” and that from September 1 to December 31, 2008 the value of the debt outstanding as of September 1 “rose in value by around $210bn” – over a period in which “the expected liabilities of the US government likely rose by several trillion dollars.”

A two-period, two-country model with a continuum of homogeneous risk-neutral investors, in which a country defaults if and only if its fiscal surplus plus the proceeds of its bond sale fall short of the debt it owes. Each investor is endowed with one unit of the numeraire good at date 0, the aggregate measure of investors (and hence of demand) is 1+f, and investors place market orders so that “the aggregate investor demand does not depend on the equilibrium price” (Section 2.1) – a modeling choice made to sidestep the rational-expectations-equilibrium problem of investors inferring information from prices. The large country’s debt size is normalized to s1 = 1 and the small country’s is s in (0,1], so aggregate supply is 1+s. Writing the fiscal surplus as s_i·theta_i, country i has s_i·theta_i + s_i·p_i available against obligations of s_i, so default occurs if and only if p_i < 1 - theta_i (equation 1), with zero recovery in the baseline. The authors are explicit that the absence of any other storage technology “is important to the analysis,” and they relax it in Section 3.4. They also note the model “features a multiple equilibrium crisis, in the sense of Calvo [9] and Cole and Kehoe [12]”: if investors expect others to stay away, the price is low, default becomes likely, and the expectation is self-confirming.

The surplus available to roll debt over, which the authors read flexibly – as a fiscal surplus, as reserves in the foreign-currency-debt case, as central-bank resources in the domestic-currency case, or even as reputational default costs. For most of the paper theta_i is “the country’s fiscal surplus, which then increases the funds available to the country to roll over its debt.” But “for the case of foreign currency denominated debt, theta_i can include both the fiscal surplus and the foreign reserves of the country,” and for domestic-currency debt it can include “resources the central bank may be willing to provide to forestall a rollover crisis,” resources “limited by central bank concerns over inflation or a devalued exchange rate.” Finally theta_i “can also be interpreted to include reputational costs associated with defaulting on debts, in which case the default equation … can be read as one where default is driven by unwillingness-to-pay” (Section 2.1).

Below the participation threshold investors’ actions are complements, because the bond pays nothing unless enough investors show up; above the threshold they are substitutes, because a fixed supply of bonds bid up by extra demand yields a lower return – and debt size governs how strong the substitution force is. “Investor actions are complements – as more investors invest in a country’s bonds, other investors are incentivized to follow suit,” while “once the number of investors who invest in the bonds exceeds the threshold required to roll over debts, then investor actions become substitutes. Beyond the threshold, more demand for the bond that is in fixed supply drives up the bond price, leading to lower returns” (Section 1). Figure 1 makes the size interaction concrete: at full participation the return on the large country’s bonds is 1/(1+f) while the small country’s is s/(1+f), “because country 2 has a small bond issue and hence an increase in demand for country 2 bonds increases the bond price (decreases return) more than the same increase in demand for country 1 bonds.” The authors position this against the global-games literature, noting that unlike models such as Rochet and Vives, actions here “can also be strategic substitutes,” that the substitution effect is stronger than in Goldstein and Pauzner and can support multiple equilibria, and that the resulting non-monotone “oscillating” equilibrium “is new and a contribution to the global games literature” (Section 1).

The threshold is delta(s,z) = -((1-s)/(1+s))·z + (-s ln s)/(1+s), where z = log((1+f)/(1-theta)); the first term is the large country’s liquidity advantage and the second its rollover-risk disadvantage.* The threshold is pinned down by indifference for the marginal investor receiving signal delta*, and the authors label the two components directly in equation (12) as “liquidity, (-)” and “rollover risk, (+).” They explain that “z measures aggregate funding conditions, which is greater if either more aggregate funds f are available or there is a higher aggregate fundamental theta. The ‘savings glut’ which many have argued to characterize the world economy for the last decade is a case of high z,” and that “the benefit term is modulated by the aggregate funding condition z” (Section 2.2). Proposition 2 establishes that this threshold equilibrium “is the unique equilibrium within the monotone strategy space.”

That better aggregate funding always lowers the threshold, that the large country is favored for every small-country size exactly when z is at least 1, and that the large country is safest over the widest range of fundamentals when its competitor is smallest. Proposition 1’s three parts state: delta*(s,z) “is decreasing in the aggregate funding conditions z,” so “country 1’s bonds can be the safe asset for worse values of country 1 fundamentals” when theta or f is higher; “delta*(s,z) <= 0 for all s in (0,1], if and only if z >= 1”; and “lim_{s->0} delta*(s,z) = inf delta*(s,z) = -z < 0.” The figures make the contrast vivid. At z = 1 the threshold is always negative and rises monotonically in s. At z = 0.2, by contrast, “for medium levels of s (around 0.4), investors are concerned that there will not be enough demand for the large country bonds, exposing the large country to rollover risk. As a result, investors coordinate investment into the small country’s debt. Note that this may be the case even if the small country has worse fundamentals” (Section 3.1). The authors’ summary: “the large country’s debt size is an unambiguous advantage only when the aggregate funding conditions are strong.”

Because the coordination motive makes investors chase whichever bond they expect others to judge safer, so in the model the relatively better country takes all the savings and the other takes none. The authors contrast this with a frictionless benchmark in which two countries’ bonds are priced as p_i = E[min(theta_i,1)]/(1+R*), so that “bond prices depend on fundamentals, but not particularly on relative fundamentals.” In their model, by contrast, “if country-i has the better fundamentals (relative to the equilibrium threshold delta*), it attracts all the savings so that p_i = 1 + f and p_{-i} = 0” (Section 3.2). The empirical payoff they claim is the persistence puzzle: “despite deteriorating US fiscal conditions, US Treasury bond prices have continued to be high: In short, all countries’ fiscal conditions have deteriorated along with the US, so that US debt has maintained and perhaps strengthened its safe asset status,” and “the Bund has retained/enhanced its value because of the deteriorating general European fiscal conditions.”

The pre-World-War-I dominance of the UK consol despite weaker fundamentals, its later displacement by US debt, and Japan’s ability to carry a very large debt without a rollover crisis. On the consol: “Despite the fact that the GDP of the US had caught up to the GDP of the UK by 1870, the UK consol bond was the premier safe asset. This seems even more puzzling, as in 1890, the US had a lower Debt/GDP ratio than the UK (0.10 for US versus 0.43 for UK).” The authors’ explanation is float: “In 1890, the absolute amount of UK Debt was about 4.3 times the size of US Debt, and the higher float of UK debt was perhaps one reason that the UK attracted safe asset demand during a period when its fundamentals were likely worse.” The reversal follows from the size mechanism turning against the incumbent – UK Debt/GDP reaching 1.40 by 1920 from 0.43 in 1870, and the US-to-UK debt ratio reaching 0.46 by 1920 from 0.23 in 1870 – so “as the UK debt size grew, size turned from a liquidity advantage to a rollover risk concern” while the rise of US debt as a liquid alternative pulled investors across (Section 3.3). On Japan, the s -> 0 limit offers “one perspective on why Japan has been able to sustain a large debt without suffering a rollover crisis”: domestic investors “eschewing foreign alternative investments” leave investors nowhere else to go, and “if this home bias in investment disappeared, then Japanese debt may no longer be safe” (Section 3.1).

Yes, and the authors state it as a suggestion the model generates rather than as an estimated effect. In the high-savings regime, raising the large country’s debt size lowers delta* and thereby “decreases the safety threshold of the smaller country.” Linking that to data, “from 2007Q4 to 2009Q4, the supply of US Treasury bonds increased by $2.7 trillion (the money stock increased another $1.3 trillion). Our model suggests that this increase should have hurt the safety of other country’s debts. That is, our model suggests a causal link from the increase in US Treasury bond supply/Fed QE and the eruption of the European sovereign debt crisis in 2010. Intuitively, the expansion of US debt supply created safe ‘parking spots’ for funds that may otherwise have been invested in European sovereign debt” (Section 3.3). The load-bearing verb here is the paper’s own: the model suggests a causal link; no empirical test of it is offered in this paper.

They can be absorbed by redefining the pool of savings chasing rollover-risky debt, and shrinking their supply raises their price – which is what the model says about sub-zero Swiss and Danish yields. Section 3.4 introduces a quantity of bonds whose issuers can commit to repay, shows that arbitrage between them and the sovereign bonds pins down an adjusted savings pool, and concludes that “the model can be interpreted as one where alternative savings vehicles do exist, but their supplies are such that substantially most of the world’s safe asset needs must still be satisfied by debt that is subject to rollover risk.” That quantitative premise is defended in a footnote: Swiss government debt was $127bn in early 2015 against central-bank liabilities near $500bn, Danish government debt was $155bn, world central-bank gold holdings were roughly $1.2tn and “largely backing for government liabilities, rather than privately investable gold,” US gold ETFs were $39bn – against a total supply of Treasury bonds plus central bank liabilities “over $16tn.” On the negative yields: “Denmark and Switzerland have recently restricted their supplies of safe bonds. The result has been that the prices of their bonds have risen, with interest rates in both countries falling below zero. We can also see this in our model.”

A non-monotone equilibrium in which an investor’s willingness to buy country 1’s bond alternates as its signal rises, approximating a mixed strategy as noise vanishes; its payoff is that both countries’ bonds can be safe at once, which monotone strategies cannot deliver. Under oscillating strategies, “agents invest in country 2 for sufficiently low signals. If the signal is slightly above an endogenous threshold delta_L, agents then invest in country 1, but go back to investing in country 2 for higher signals, oscillating back and forth,” stopping above a second threshold delta_H (Section 3.5). The driver is the substitution force: in the region where both countries are safe, the signal is no longer payoff-relevant, “so oscillation leads to investment in exactly the proportions that equalize equilibrium returns.” The authors note the non-monotonicity “occurs only in the region where both countries are safe,” and defend the construction on economic grounds: “Though seemingly exotic, it is interesting that equilibria with such non-monotone strategies lead to the economically plausible situation that both countries’ debts may be safe when z is high. This possibility cannot emerge in the case of monotone strategies in which one country always survives and one country always defaults.” Proposition 3 confirms that Proposition 1’s qualitative results survive, with one substantive change: in the oscillating equilibrium “improved aggregate funding conditions makes both countries safer,” whereas under monotone thresholds a higher z helps the large country at the small country’s expense.

It needs positive recovery in default; with recovery, cash-in-the-market pricing links the two bonds so that a deterioration in aggregate fundamentals pushes funds out of the weak bond and into the safe one, raising its price. In the zero-recovery baseline the safe bond’s price is (1+f)/s_i regardless of shocks, which the authors call a “stark result” that “does not allow us to derive predictions for the beta” (Section 3.6). Introducing recovery 0 < l_i < 1 creates “a strong strategic substitution force that pushes investors to buy the defaulting country’s debt if nobody else does so,” since an infinitesimal investor alone in a defaulting bond with positive recovery “would earn an unbounded return” – which is why threshold strategies fail and the analysis uses oscillating strategies here too. In the region where country 1 is safe, the prices are p1 = (1+f)/(1+l2·s) and p2 = l2(1+f)/(1+l2·s), so “when the recovery of country 2 decreases, p2 drops and p1 increases.” Hence “country 1’s bonds gain when aggregate fundamentals deteriorate, which makes it a negative beta asset, while country 2’s bonds lose,” and Figure 3 shows that “the higher the country 1’s relative fundamental, the more negative the beta of its bonds.” Two simplifications are stated: recovery is assumed not to depend on the relative fundamental, and the beta is computed in an appendix under an illustrative parameterization.

That the welfare gain is unambiguous only once the common bond is a large enough share of total issuance, and that small common-bond programs can make the small country less safe rather than more. Countries issue a common bond of size alpha(1+s) alongside individual bonds of size (1-alpha)s_i, in a two-stage auction; return equalization pins the common bond price at (1+f)/(1+s) and the funds drawn into it at alpha(1+f), independent of the distribution of the relative fundamental (Section 4.2). Proposition 4 gives two equilibria: a threshold “minimum joint safety” equilibrium existing for alpha in [0, alpha*] with alpha* = exp(-z)(1+s) and delta*(alpha*) = 0, and an oscillating “maximum joint safety” equilibrium existing for alpha in [alpha_HL, 1] with alpha_HL < alpha*, so the two overlap. The normative conclusion is stated sharply: “increases in common bond issuance, i.e., increases in alpha, only create Pareto gains (when gains are thought of in terms of increasing country safety) when alpha > alpha*. In this case, increases in alpha raise the safety of both country 1 and country 2. For alpha < alpha* and in the minimum safety equilibrium, a greater alpha reduces safety of one country while increasing safety of the other country. Thus, small steps towards a fiscal union could be worse than no step” (Section 4.1). The mechanism behind the counterintuitive part is distributional: common bonds lower both countries’ default thresholds, but “the small country receives proportionally less common bonds proceeds,” so for s near zero “almost all the common bond proceeds and thus the rollover risk reduction accrue to the large country” and “introducing common bonds hurts, rather than enhances, the safety of the small country” (Section 4.2). The authors add that they “are unaware of other similar models or formal analysis of this issue.”

Like a tournament with two regimes: a self-defeating rat race when countries are similarly sized and funding is plentiful, and a stabilizing “top dog” configuration when they are sufficiently asymmetric. Countries have natural sizes set by local conditions (“countries with a higher GDP will naturally have a larger stock of debt outstanding”) and pay an increasing, convex adjustment cost with zero marginal cost at zero adjustment, which is why any incentive to deviate bites (Section 5.1). With the threshold rewritten in terms of chosen sizes (equation 32) and zero adjustment cost, the symmetric solution is S(z) = exp(z-1). In Region A – similar natural sizes – “both countries will increase the size of their debts. This is driven by the safe asset effect: there is an externality whereby the larger country has a better safe asset position, and countries compete to become the safe asset by increasing their debt sizes. Of course, this competition to gain safe asset status is ultimately self-defeating. … The Nash equilibrium results in countries increasing debt sizes beyond their natural sizes in a rat race, while both would be better off and save adjustment costs if they coordinate not to expand” (Section 5.3). In Regions B and B’, “country 1 is the top dog and not worried about losing its safe asset status, and is primarily concerned about rollover risk,” so it contracts while the small country expands. The authors are careful to call the ranking suggestive rather than proved: “Our investigations are suggestive that asymmetry leads to better outcomes than symmetry” (Section 6). Region A “expands unambiguously as the aggregate funding condition z rises,” so better funding conditions draw countries into the race.

The pre-crisis expansion of US agency debt and of euro-area sovereign debt, both of which the authors say ended badly; and, since the world has plausibly moved from the symmetric to the top-dog region, an incentive for the US to shrink its debt. On the agencies: “In the US, the government agencies, Fannie Mae and Freddie Mac, initiated a program (‘Benchmark Notes’) in 1998 whose purpose was to offer debt that could compete with US Treasury bonds as a large and liquid savings vehicle. Of course, the expansion of such agency debt stocks ultimately resulted in the bailouts of Fannie Mae and Freddie Mac by the US government, suggesting that welfare would have been improved without these programs, or if the US Treasury had coordinated the debt sizes of the Agencies along with that of the rest of the federal government.” On Europe: “the expansion of sovereign debt after the formation of the Euro can similarly be seen as a rat race to serve as the safe asset within Europe. This rat race has also ended badly” (Section 5.4). And prospectively: “Suppose that the world economy was in region A prior to the financial crisis, but has since transitioned to region B, where US government debt is the top dog. … The model suggests that the US then has an incentive to shrink its debt.”

Either a large relative deterioration in US fundamentals, the emergence of a comparably large rival issue, or an end to the high-savings regime – and only the last of these makes a small, high-fundamentals issuer like the Bund the likely successor. “If the world continues in the high savings regime, the US will only be displaced if another country can offer a large debt size and/or good relative fundamentals. This seems unlikely in the foreseeable future. On the other hand, if the world switches to the low savings regime, it is possible that US Treasury bonds become unsafe, and another country debt with a smaller debt size and good fundamentals, such as the German Bund, becomes the dominant safe asset” (Section 3.3). On the rival-issue route, the introduction names Eurobonds as “the only possibility,” while noting “there is considerable uncertainty whether such bonds will exist and will have better fundamentals than the US debt.”

As supplying the piece those literatures leave out: which asset becomes the store of value. On the store-of-value tradition, “Samuelson [41] presents an overlapping generation model where money serves as a store of value … Diamond [14] presents a related model but where government debt satisfies the store of value role. In this class of models, there is a need for a store of value, but the models do not offer guidance on which asset will be the store of value. For example, it is money in Samuelson [41] and government debt in Diamond [14]. In our model, the store of value determination is endogenous” (Section 1). On safe-asset shortages: “We presume that there is a macroeconomic shortage of safe assets, and our model endogenously determines the characteristics of government debt supply that satisfies the safe asset demand.” On liquidity, the paper distinguishes its channel from search-based accounts: in Vayanos and Weill a larger float aids liquidity by making trading partners easier to find, whereas “in our model, the coordination element is through rollover risk, which interacts with debt float/liquidity,” and the authors note the search mechanism “could as well apply to risky assets as to safe assets,” which is why they focus on rollover risk instead.