# Debt Strategy

A classic question in economics is: How to design an optimal debt strategy? This is a problem confronted by the treasury of any government or any big corporation. A debt strategy must determine an amount of debt issuances and its maturity structure. Economics offers a number of theories that guide that design. Although each theory brings different insights, each theory is cast through a common exercise: the theory sets up a fictitious economic environment and presents an optimal debt-maturity profile for that environment. That solution is translated into policy principles that can be applied by practitioners. Current work by Professors Saki Bigio (UCLA), Juan Passadore (Einaudi Institute) and Galo Nuño (Bank of Spain) is a new take on that classic problem.

In their recent paper entitled “Optimal Maturity-Debt Management”, the researchers make two innovations to study that classic question. Their first innovation is conceptual. Their theory highlights the importance of liquidity frictions, the notion that abrupt adjustments in the debt of a given maturity can saturate the market for that debt. The consequent price impact is a common consideration by practitioners, but has been neglected by normative studies on debt management. The second innovation is technical. In the past, theories could only deal with problems where the strategy considers a choice among only a few number of bonds of one specific structure. Those specific bonds, known as consols, assume that bonds mature by retiring a constant fraction of the outstanding principle every period. Consols are rarely issued in practice. Instead, their study adapts techniques from other sciences to allow the study of the debt management problem where it is allowed to have bonds of any arbitrary number of maturities and any arbitrary cash-flow structure.

A principle that emerges from this theoretical study is that the problem can be studied as if the Government delegates the issuance problem to a continuum of subordinate traders. Each trader is in charge of the issuance of bonds of a single maturity. In this fictitious delegation, each trader then follows a simple rule. Each trader computes the internal valuation of bonds of the maturity that he manages. For that valuation, traders use discount provided by the Government. To determine his optimal issuance, each trader compares his valuation to the market price of that bond. Typically, prices and valuations will differ. The authors derive a simple guiding principle for the optimal issuance of bonds of a given maturity:

% issuance/GDP = liquidity coefficient ⋅ % value gap.

The principle states that in a given period, the issuances of a bond of a given maturity (relative to GDP) should equal the product of a value gap and a liquidity coefficient. The value gap is the difference between the market price and the internal valuation of a bond, as a percentage of the price. Think now of a trader. A trader would want to issue as much debt as possible whenever the price he receives is higher than his valuation (when there is a positive value gap). The formula says that there is a force that contains that desire: the liquidity cost that is produced by the price impact. Mathematically, this liquidity force appears as a coefficient that can be estimated with measures of bond-market liquidity and intermediation spreads. The higher the liquidity coefficient, the greater the issuances.

One piece is missing. For this delegation principal to deliver *the *optimal debt issuance, the authors show that the Government must assign his fictitious traders an appropriate discount factor. The authors develop a numerical algorithm to solve for that discount factor given information on expected tax revenues, and the paths of expected interest rates. At the moment, the authors are developing applications of their methodology, and anticipate that these will be relevant for government and corporate treasuries optimal policy discussions.