Back Matter

### APPENDIX I. Sovereign Debt Dynamics

Debt-to-GDP dynamics takes into account that the government issues both bonds denominated in local currency and bonds denominated in foreign currency, as well as it borrows from bilateral and/or multilateral lenders (which we call for simplicity official debt). Thus, the stock of debt at the end of period t, denoted by Dt equals the sum of the debt denominated in domestic currency,${D}_{t}^{d}$; the debt denominated in foreign currency (typically US dollars),${D}_{t}^{f}$, converted in domestic currency at the existing nominal exchange rate, et (units of domestic currency per unit of foreign currency); and the official debt (also denominated in US dollars)${D}_{t}^{o}:{D}_{t}={D}_{t}^{d}+{e}_{t}{D}_{t}^{f}+{e}_{t}{D}_{t}^{o}$. This relationship can be converted in terms of ratios to GDP by dividing both sides by nominal GDP (expressed in local currency), Yt:

$\begin{array}{cc}{d}_{t}={d}_{t}^{d}+{d}_{t}^{f}+{d}_{t}^{o},& \phantom{\rule{7.0em}{0ex}}& \left({A}.1\end{array}\right)$

where${d}_{t}^{d}=\frac{{D}_{t}^{d}}{{Y}_{t}},{d}_{t}^{f}=\frac{{e}_{t}{D}_{t}^{f}}{{Y}_{t}},{d}_{t}^{o}=\frac{{e}_{t}{D}_{t}^{o}}{{Y}_{t}}$.

The dynamics of the domestic-currency-denominated debt is described by the government’s flow budget identity${D}_{t}^{d}=\left(1+{i}_{t}\right){D}_{t-1}^{d}-{\alpha }_{t}{B}_{t}$, where it is the interest rate on local-currency denominated government debt whose maturity is at time t, Bt is the primary balance, and αt ∈[0,1] is the fraction of primary balance used to service the domestic debt exposure. The primary balance is defined as Bt = TtGt, where Tt is the revenue collected and Gt represents the non-interest public spending. The budget identity can be expressed in terms of ratios to GDP by dividing both sides by Yt:

$\frac{{D}_{t}^{d}}{{Y}_{t}}=\frac{{Y}_{t-1}}{{Y}_{t}}\frac{{D}_{t-1}^{d}}{{Y}_{t-1}}\left(1+{i}_{t}\right)-{\alpha }_{t}\frac{{B}_{t}}{{Y}_{t}},$
${d}_{t}^{d}=\frac{1+{i}_{t}}{1+{y}_{t}}{d}_{t-1}-{\alpha }_{t}{b}_{t},$

where yt is growth rate of nominal GDP and bt is the ratio of primary balance to GDP.

Deflating the nominal growth rate by the inflation rate, πt, the flow of budget identity can be expressed in terms or real growth rate, gt:

$\begin{array}{cc}{d}_{t}^{d}=\frac{\left(1+{i}_{t}\right)}{\left(1+{\pi }_{t}\right)\left(1+{g}_{t}\right)}{d}_{t-1}-{\alpha }_{t}.& \phantom{\rule{7.0em}{0ex}}& \left({A}.2\right)\end{array}$

Analogously, the dynamics of the foreign-currency-denominated debt is described by the following first-order difference equation:${D}_{t}^{f}=\left(1+{i}_{t}^{f}\right){D}_{t-1}^{d}-{\beta }_{t}\frac{{B}_{t}}{{e}_{t}}$, where${i}_{t}^{f}$is the nominal interest rate paid on government bonds denominated in foreign currency whose maturity is at time t, and βt ∈ [0,1] is the fraction of primary balance used to service the foreign commercial debt exposure. Multiplying both sides by et and dividing through by Yt yields the following:

$\frac{{e}_{t}{D}_{t}^{f}}{{Y}_{t}}=\frac{{e}_{t}}{{e}_{t-1}}\frac{{Y}_{t-1}}{{Y}_{t}}\frac{{e}_{t-1}{D}_{t-1}^{f}}{{Y}_{t-1}}\left(1+{i}_{t}^{f}\right)-{\beta }_{t}\frac{{B}_{t}}{{Y}_{t}},$
${d}_{t}^{f}=\frac{\left(1+{ϵ}_{t}\right)\left(1+{i}_{t}^{f}\right)}{\left(1+{y}_{t}\right)}{d}_{t-1}^{f}-{\beta }_{t}{b}_{t},$

where ϵt represents the nominal exchange rate depreciation. By deflating both the nominal growth factor and the nominal exchange rate factor by the domestic inflation rate, the above expression can expressed in terms of the real growth rate gt, the real exchange rate depreciation ηt, and the real interest rate paid on foreign commercial debt$\left(1+{i}_{t}^{f}\right)/\left(1+{\pi }_{t}^{*}\right)$:

$\begin{array}{cc}{d}_{t}^{f}=\frac{\left(1+{\eta }_{t}\right)\left(1+{i}_{t}^{f}\right)}{\left(1+{g}_{t}\right)\left(1+{\pi }_{t}^{*}\right)}{d}_{t-1}^{f}-{\beta }_{t}{b}_{t}.& \phantom{\rule{7.0em}{0ex}}& \left({A}.3\right)\end{array}$

The dynamics of the official debt-to-GDP ratio is analogous to the one above. A substantial difference is that typically the interest rate applied by bilateral or multilateral lenders,${i}_{t}^{o}$, is lower than the market rate:

$\begin{array}{cc}{d}_{t}^{o}=\frac{\left(1+{\eta }_{t}\right)\left(1+{i}_{t}^{o}\right)}{\left(1+{g}_{t}\right)\left(1+{\pi }_{t}^{*}\right)}{d}_{t-1}^{o}-\left(1-{\alpha }_{t}-{\beta }_{t}\right){b}_{t}.& \phantom{\rule{7.0em}{0ex}}& \left({A}.4\end{array}\right)$

Another difference across different components of sovereign debt is that the average maturity of each class of debt differs. We take these differences into account when we compute the interest rate paid in every period. Let md, mf, and mo be the average maturities of domestic, foreign and official debt respectively,the one-period government bond rate for the aforementioned classes of sovereign debt, then:

$\begin{array}{ccc}{i}_{t}^{d}=\frac{1}{{m}^{d}}\sum _{n=1}^{{m}^{d}}{\iota }_{t-n}^{d},& {i}_{t}^{f}=\frac{1}{{m}^{f}}\sum _{n=1}^{{m}^{f}}{\iota }_{t-n}^{f},& {i}_{t}^{o}=\frac{1}{{m}^{o}}\sum _{n=1}^{{m}^{o}}{\iota }_{t-n}^{o}.\end{array}$

The nominal interest rate paid on one-period commercial bonds denominated in foreign currency,${\iota }_{t}^{f}$, can be computed as the sum of an international risk-free rate,${i}_{t}^{*}$, and a spread${\sigma }_{t}^{f}$representing the country’s sovereign risk:${\iota }_{t}^{f}={i}_{t}^{*}+{\sigma }_{t}^{f}$. Similarly,${\iota }_{t}^{o}={i}_{t}^{*}+{\sigma }_{t}^{o}$, where typically the spread paid to multilateral and bilateral lenders,${\sigma }_{t}^{o}$, is lower than the one paid to international financial markets,${\sigma }_{t}^{f}$.

Finally, as information on the fraction of primary balance servicing each component of debt is not available, we assume that at time t the fraction of primary balance that services a certain component of debt is proportional to the existing share of debt of a particular type over the total:

$\begin{array}{cc}{\alpha }_{t}=\frac{{d}_{t-1}^{d}}{{d}_{t-1}},& {\beta }_{t}=\frac{{d}_{t-1}^{f}}{{d}_{t-1}}.\end{array}$

Summing up equation (A.2) to (A.4) yields the initial equation (A.1).

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We are thankful to Abdel Senhadji and Emanuele Baldacci for insightful comments and suggestions. We also thank all other participants at the seminar presentations in the Fiscal Affairs Department at the International Monetary Fund.

See, for example, Kaminsky and Reinhart (2000).

This would be the case, for example, with a binding debt ceiling rule.

Notice that as the framework is set in terms of primary spending, automatic stabilizers on revenues and interest expenditures are allowed to play in full.

However, if rules are more generally defined as institutional mechanisms aimed at supporting fiscal credibility and discipline, then the proposed framework could be considered a rule.

Balassone and Kumar (2007) show, for a sample of developed and developing countries, that the overall balance behavior over the business cycle is asymmetric, as these tend be relatively more negative in bad times than they are positive in good times.

For example, mechanisms can be introduced for an ex-post compensation for miscalculations about the state of the cycle, as for example in Switzerland. Also, a broader set of variables could be used to determine structural vs. cyclical components of revenues or expenditures, in addition to GDP. Discretional expenditure could also be included, provided this introduction is attached to an over-the-cycle mechanism compensatory mechanism. See Bornhorst et. al. (2011).

This is not referring to the technical aspects and multiple methodologies available to identify cycle and trend. It refers to the fact that available methodologies are weak in identifying the trend position at end-sample points, which results in large practical miscalculations that can undermine credibility, even when the methodology remains unchanged. See Bornhorst et. al. (2011).

For example, the IMF’s 1998 World Economic Outlook show that, for a sample of Asian countries, exchange rates and commodity prices are more important in assessing the cyclical components of revenues and expenditures than GDP fluctuations.

On the other extreme of the spectrum, simple debt rules may not be credible either, as they might be difficult to comply with under tail risks. For example, a simple commitment to reduce public debt within a specified deterministic path (for example, a debt ceiling) might prove politically and socially difficult as it implies a commitment to adjust negative shocks in full within the current year, forcing a strongly pro-cyclical fiscal stance. This difficulty can undermine their credibility.

Individual-country forecast variance decompositions are derived from a two-lag six-variable VAR featuring annual observations of the output growth rate, a short-term interest rate, the growth rate of the real exchange rate, the change in net capital inflows, the total revenue growth and the primary expenditure growth, respectively.

We abstract from cross foreign currency parity changes. This is not a significant simplification as most countries’ foreign currency debt used in the sample are denominated in US dollars, if not exclusively so.

Examples of works along these lines are, among others, Garcia and Rigobon (2005), Celasun, Debrun and Ostry (2006), Penalver and and Twaites (2006), Tanner and Samake (2008).

The simulation results need to be taken with caution. Any VAR approach to policy simulation is inherently subject to the Lucas critique. In fact, when policies change, the behavior of market participants might change as well, possibly affecting the estimated coefficients. This limitation can be addressed using a general equilibrium model.

For the purposes of the simulations we also use assumptions on the US Federal Funds rate, the Libor on the U.S. dollar and U.S. inflation in line with the WEO (2010) and we fit the domestic consumer price index to a simple pass-through equation in order to project the domestic inflation rate.

Also, the simulations under the structural balance rule are based on the same long-term GDP growth and real interest rate assumptions.

The primary expenditure under a structural balance rule is calculated as structural revenues minus the primary surplus that targets the same public debt to GDP target as in the framework proposal, for comparability. Structural revenues are calculated as the simulated revenues times the output gap (computed as the ratio of the GDP trend using the Hoddrick Prescott filter with lambda = 100 divided by the GDP level).

The calculations of the CV reported in Table 3 exclude the growth rate of primary expenditures in 2011 as, for both fiscal policy choices, it captures the initial consolidation required to set public debt on a sustainable path.

Notice that this is not a horse race between the two methodologies, as the results are conditional on the set of parameter choices for each.

These correlations include the initial adjustment for the 2011 budgets reported in Table 2.

Notice, that the simulation experiment also shows that the average correlation between revenue and GDP growth rates for this sample of countries is 88 percent under a structural balance rule, very high. However, this is only the result of a simplistic identification strategy of cyclical revenues, based only of output gap measures with a unit revenue elasticity as explained above.

In a strict sense, there are other factors that can explain the residual increase in debt related to debt management, such as pre-financing, or changes in government deposits. The high magnitude and consistent positive value of this discrepancy in all cases, however, suggests that debt takeovers or deficits inherited from other public sector activities and bodies outside the budget are important.

Public Debt Targeting An Application to the Caribbean
Author: Mr. Alejandro D Guerson and Mr. Giovanni Melina