Debt Sustainability Risk Analysis with Analytica c

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1 1 Debt Sustainability Risk Analysis with Analytica c Eduardo Ley & Ngoc-Bich Tran We present a user-friendly toolkit for Debt-Sustainability Risk Analysis (DSRA) which provides useful indicators to identify public-debt vulnerabilities and risks. Using a minimal set of parameters and initial values, the prototypical DSA toolkit calculates, among other indicators, the path of debt-to-gdp and a debt-service (cashflow) profile i.e., addressing liquidity and rollover risk issues using Monte Carlo simulations. In this note, we will present the required set of inputs, and show different debt indicators generated by the toolkit. The DSRA Analytica toolkit allows the user to easily produce scenarios under different sets of assumptions and conduct Monte-Carlo risk analysis e.g., assessing risks related to the exchange rate, to inflation and to GDP growth. The user enters a country s (or subnational entity s) initial conditions, values for baseline projections, etc using the free Analytica Player 1 to produce a number of simulations for variables of interest, under different stochastic scenarios. Moreover, the Analytica toolkit produces a clear visual representation of the model graphically displaying the relationships among the variables. I. DSA MODULE: REQUIRED INPUTS The user-specified input values can either be initial conditions, or series of variables values. Time subscripts take the values t = 0,..., T. Values for the initial period shall be denoted with a 0 subscript, e.g., d 0. In this note, series of values will be denoted within brackets e.g., {g t } T t=0 or simply {g t}. See Section III for a refresher of the basic algebra of DSA, and details on the notation used here. Analytica s interface is used to input values for the initial conditions, and to enter baseline values using the Edit Table button (figure 1). 1 The Player can be downloaded from However, a licensed version of the software is required to modify the module i.e., the structure of the model. Also, the toolkit could easily be adapted for external-debt analysis which is isomorphic to public-debt analysis. A. Macroeconomic variables GDP real growth rate, {g t } Initial GDP level (in local currency), P 0 Y 0 Domestic Inflation rate, {π h t } Foreign Inflation rate, {π f t } Share of tradable sector in GDP, {β f t } Exchange rate (local currency per foreign currency), {e t } Domestic Real interest rate on initial debt, r h 0 Foreign Real interest rate on initial debt, r f 0 Domestic Real interest rate on new debt, r h Foreign Real interest rate on new debt, r f Remarks If a series of projected real interest rates is available, the user can use the alternative input Real interest rate (baseline) as the baseline projection for the real interest rate, while setting to zero all the aforementioned real interest rate variables (initial and new, foreign and domestic). B. Budget variables Ratio of primary balance to GDP, {b t } Initial interest payments (in local currency), id 0 C. Debt variables Initial debt-to-gdp ratio for domestic debt, d h 0 Maturity profile for the initial stock of domestic debt, {A h t } Initial stock of foreign-currency debt (in foreign currency), D f 0

2 stock of debt: {A t /D t }. D. Probability distributions for stochastic shocks Analytica allows the user to choose from a wide family of probability distributions e.g., Normal, Gamma, triangular, etc for any of the variables model. Here we use that capability for specifying stochastic shocks to the fundamental macroeconomic variables i.e., growth, inflation, interest rates, etc see figure 2. However, in the prototypical DSA module, for simplicity, we only use the Normal distribution. The analyst could instead use other distributions. Thus, stochastic variables are specified using the formulation: x t = x + ɛ t, ɛ t N(0, σ 2 ) with x being the mean of the variable of interest e.g., the inflation rate, the growth rate, the real interest rate and the exchange rate. Since these are expressed as percentages, attention must be paid to use appropriate units in the shock specification e.g., a normally-distributed shock with a standard deviation of one percentage point will be specified as: ɛ t N(0, σ 2 = = ) Fig. 1. End-user interface Maturity profile for the initial stock of foreign debt, {A f t } Share of foreign debt in newly-issued debt, {γ t } Maturity of newly-issued debt Remarks The debt maturity profile requires information on the share of the initial stock of debt that due in 1,..., T years. Thus, the user specify the amortizations due in t, A t, divided by the current Fig. 2. Specification of the probability distributions for stochastic shocks As noted, the user can change the parameters (mean and variance) of the Normal distribution functions used above (figure 3), and even the distributions themselves choosing from a wide family of densities. or even bootstrap shocks from empirical distributions obtained, e.g., from WEO data. 2 2 To set up a boostrapping strategy, the user would need to use a licensed version of the software. The user has to create an index (in the model Data Index ) with the same number of observations than the empirical data (in the model Data ). The Resample Data command will generate a bootstrapped sample that has the size set by the user for the Monte Carlo sampling. See also example in 2

3 Fig. 3. Choosing a probability distributions for stochastic shocks II. OUTPUT OF THE DSA TOOLKIT The toolkit produces indicators related to debt sustainability and liquidity while allowing for risk analysis. In the following sections, terms between quotation marks refer to the terminology used in the Analytica s User Guide. Fig. 4. Debt indicators A. Debt sustainability and cashflow indicators The toolkit produces projections for the following indicators (figure 4): The debt-to-gdp ratio, {d t } Nominal debt (i.e., not normalized by GDP), {D t } The share of foreign debt in total debt, {α f t } The debt service with detail on interest and principal payments; {i t D t }, {A t } The cashflow profile (as % of GDP) Clicking on the desired indicator brings a popup Result window, displaying either a graph that can be easily copied and pasted into any document (figure 5), or a table of values that can be exported to an excel file (figure 6). In order to get a visual representation of the underlying debt sustainability analysis model, one can double-click on the button Model located in the user interface. The resulting representation is shown in figure 7. Outcomes for indicators that are not displayed in the standard Output group can be obtained by 3 Fig. 5. Fig. 6. Graph for the debt-to-gdp indicator Table of values for the debt-to-gdp

4 Fig. 7. Visual representation of the debt sustainability analysis model 1) selecting the variable of interest in the visual model representation e.g., the variable Primary balance, and by 2) clicking on the Result button (see figure 8). visual model representation (figure 9). Fig. 9. Variable definition Fig. 8. Result button B. Risk Analysis Note further that the definition of a given variable corresponding to the cell s formula in an excel file, and a detailed description thereof can be displayed by double-clicking on the variable in the Analytica allows to take into account the uncertainty surrounding the projection of fundamental macroeconomic variables such as the exchange rate, the GDP growth rate, the real interest rate and the 4

5 inflation rate by doing a number of Monte-Carlo simulations (figure 10) drawing shocks from specific probability distributions. Fig. 10. Choosing the number of Monte Carlo simulations The prototypical DSA toolkit specifies Normal distributions for simplicity, but, as noted above, the user could make better choices for the distributions or even boostrap the shocks from, e.g., WEO historical values. The Result window displays either deterministic projections (for example for the debt-to-gdp ratio) or stochastic projections, in the case of a risk analysis. The choice of the type of projection to be displayed in the Result window is made using the Uncertainty views button (figure 11). Statistics displays a table of descriptive statistics including the minimum, maximum, median, mean and the standard deviation estimated from the generated sample paths. Probability Bands Probability Density displays the empirical probability density function of the variables. Cumulative Probability Sample shows an array of the random values from the distribution generated by the Monte Carlo sampling procedure. We will focus here on risk analysis using the probability bands and the cumulative probability views. The probability bands display the percentiles (by default the 5%, 25%, 50%, 75% and 95%) computed from the N Monte-Carlo simulations (figure 12). The interpretation of the probability bands is as follows: in 75% of the cases, the projected value of the debt-to-gdp ratio lies below the limit indicated by the 75% band on the vertical axes (figure 12). Let x t be the debt-to-gdp ratio for year t and ˆF t be its empirical cumulative distribution function, calculated for each year t. ˆFt is thus and the probability bands indicates the 5%, 25% etc. percentiles of ˆF t thus the 5% percentile is given by ˆF t (0.05). Hence the area lying between two bands, say the 25% and 75% probability bands, gives a range of values that can be taken by the debt-to-gdp ratio for each year, with a given probability in this case, with a probability of 50%, as 75% 25% = 50%. Fig. 11. Uncertainty view The Mid value selection refers to a deterministic projection, whereas the six following items are the output of stochastic simulations: Mean value 5 Fig. 12. Probability bands for the debt-to-gdp indicator

6 Example: In 90% of the cases, the debt-to-gdp ratio projected for 2012 takes on values lying between 52.5% and 56.5% (figure 12). The cumulative probability plots for each year the probability that the projected value for the debt-to- GDP (horizontal axis) is less than a given quantile (figure 13). Hence, it allows to address the issue of the probability or risk that the debt-to-gdp ratio may exceed, say 55%, in 4 years. Fig. 13. Cumulative probability of the debt-to-gdp indicator Example: The probability that the debt-to-gdp ratio in 2012 exceeds 55% is slightly above 30% (figure 13). III. DEBT SUSTAINABILITY FRAMEWORK This section follows the exposition in E. Ley (2009) Fiscal and External Sustainability (World Bank: unpublished), refer to that document for further details. A. Public Debt Dynamics Let D t denote the stock of government debt at the end of year t, let i t be the (average) nominal interest rate and let B t be the primary (i.e., non-interest) government balance. The government budget constraint implies that the change in debt stock is driven by the overall balance OB t i.e., the difference between total government revenues and expenditures: D t = D t 1 (B t i t D t 1 ) }{{} (1) Overall balance D t = OB t = (B t i t D t 1 ) (2) with the overall balance being given by the primary balance net of interest payments. Equation (2) always holds ex-post and states that when the government is running a surplus in the primary balance B t > 0, that surplus can be used to reduce the stock of existing debt. In contrast when the primary balance is negative, B t < 0, new debt has to be issued to finance the government s deficit. (For simplicity, seignorage is not explicitly addressed in this framework.) Cashflow Analysis The size of gross debt issuance (or withdrawals) is determined by the primary balance and the debt service. Debt service is defined as the sum of principal repayments (amortization), A t, and interest payments, i t D t 1. Equivalently, gross debt issuance is determined by the overall balance and amortizations. However, ss equation (2) shows, net change in debt, D, is determined solely by the overall balance. Rewriting equation (2), we obtain the debtdynamics equation: 6 D t = D t 1 OB t = (D t 1 A t ) + (A t + i t D t 1 B t ) (3) }{{}}{{} Existing debt New debt The size of the gross debt issuance is important for assessing rollover risk an appropriate cashflow analysis can flag important vulnerabilities.

7 Ratios to GDP Normalizing equation (1) by nominal GDP, P t Y t, as a measure of national income, provides a measure of the government s ability to service its debt obligations. The law of motion of the government s debt-to-gdp ratio is thus given by: D t = (1 + i t)d t 1 B t P t Y t P t Y t P t Y ( t ) (1 + i t ) Dt 1 = (1 + g t )(1 + π t ) P t 1 Y t 1 B t P t Y t where g t is the growth rate of real GDP and π t is the inflation rate (measured as the rate of change of the GDP deflator, P ). Using lowercase symbols to denote ratios to GDP, the dynamics of the debt-to-gdp ratio are: d t = = (1 + i t ) (1 + g t )(1 + π t ) d t 1 b t ( ) 1 + rt d t 1 b t (4) 1 + g t where the real interest rate is given by r = (i π) (1 + π). The change in the debt-to-gdp ratio is therefore given by ( ) rt g t d t = d t 1 b t (5) 1 + g t From (5) it is apparent that to stabilize the debtto-gdp ratio d t ( d t = 0), the real interest-growth differential (r t g t ) must be balanced by the primary surplus/deficit. Thus, when r t > g t, a primary surplus is required to achieve a non-explosive path for the debt-to-gdp ratio. B. Debt in Foreign Currency fraction of domestic-denominated debt as α h = 1 α f. In this context, movements in the exchange rate impact the nominal GDP dynamics through price changes in the tradable sector so that nominal GDP is expressed as: P Y = P h Y h + ep f Y f (7) where P h Y h is nominal output in the non-tradable sector and P f Y f is nominal output in the tradable sector. The dynamics of the debt-to-gdp ratio in an open economy is given by: d t = 1 + î t + ε t α f (1 + i f t ) (1 + g t )[1 + ˆπ t + ε t β f (1 + π f t )] d t 1 b t (8) with ε t = e t /e t 1 being the rate of depreciation of the local currency i.e., ε t > 0 means depreciation, î t = α h i h t + α f i f t being a weighted average of the domestic interest rate i h t and the foreign interest rate i f t, β f = Pt 1Y f t 1/P f t 1 Y t 1 being the share of the tradable sector, β h = 1 β f the share of the non-tradable sector, and ˆπ t = β h πt h + β f π f t being a weighted average of domestic inflation πt h and foreign inflation π f t. Particular attention must now be paid to exchange rate movements since they may affect the debt burden, especially during sharp exchange-rate adjustments. IV. FURTHER READING Analytica s User Guide, available at Burnside, G., editor (2005) Fiscal Sustainability in Theory and in Practice: A Handbook, Washington DC: The World Bank. Ley, E. (2009) Fiscal and External Sustainability (World Bank: unpublished), available at When governments can borrow in foreign currency, then government debt D has a domesticdenominated component D h (home), and a foreigndenominated component D f : D t = D h t + e t D f t (6) with e being the exchange rate (price of one unit of foreign currency in terms of domestic currency). Let us define the fraction of foreign-denominated debt as α f = e t 1 D f t 1/D t 1, and, reciprocally, the 7