## _wp0897 - 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6

## Source details

**Canonical URL:** [_wp0897 - 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0897.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0897.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0897.pdf.json)

---

### Introduction
- Lebanon’s core macroeconomic vulnerability: 177 percent of GDP in 2006 (public debt).
- Bank deposits: 283 percent of GDP in 2007.
- Key historical shocks noted: assassination of former prime minister Hariri in 2005; conflict with Israel in 2006.
- Conceptual definition: debt sustainability = government’s ability to pursue its fiscal policy stance into the future without threatening solvency and by operating within its intertemporal budget constraint.
- Empirical debt-threshold literature cited as implying crises for Lebanon, but political and practical considerations shape targets.
- Debt-dynamics caveats:
  - Debt-to-GDP dynamics are stochastic, depending on GDP growth, interest rates, and exchange rate.
  - High debt and rollover needs create rollover/liquidity risk; assigning probabilities requires distribution of liquidity shocks.
- Lebanese specifics:
  - Debt is essentially backed by short-term deposits → large rollover risk.
  - Dollarization of government debt around 50 percent (since 1994).
  - De facto peg to the U.S. dollar since 1998 lowered borrowing costs but increased balance-sheet vulnerabilities.
  - Spread between two-year domestic T-bills and Eurobonds averaged 3.6 percent over 1998–2007 (data until end-of-month October 2007 used).
  - Standard deviation of Lebanon’s short-term interest rate (annualized) = 1.2 percent per year (second lowest in group examined).
  - Historical episodes of financial stress since 1993 tied to depositor confidence; central bank financing and expectations of foreign financial assistance helped restore confidence.

### Lebanon’s debt dynamics (historical context and key figures)
- Post-civil war (1991) government debt ≈ 50 percent of GDP; rose through 1990s.
- Primary deficit in 1992–97 averaged 7.8 percent of GDP.
- Real GDP growth averaged 5.6 percent (annual average in 1994–97).
- Real interest on Treasury bills averaged 7 percent (annual average 1994–97).
- Fiscal consolidation began in 1997; debt ratio continued rising until 2001 due to growth weakness.
- Stabilization until 2004 aided by GDP pickup, rising primary surplus, and lower borrowing costs (including soft financing from Paris II, November 2002).
- Since 2004, a 10 percentage point increase in the debt ratio driven by:
  - economic slowdown related to political instability and 2006 conflict with Israel;
  - increases in the interest bill (rising spreads and maturing of Paris II zero interest credits);
  - fiscal impact of higher oil prices.
- Eurobonds largely held by domestic banks as counterpart to foreign currency deposits; international investors marginal.

### Methodology (analytical framework)
- Core debt accumulation equation (equation (1)):
  - d_{t+1} = d_t + (r_t - g_t) d_t - p_t
  - d = debt-to-GDP ratio; r = effective real interest rate on the debt; g = real GDP growth; p = primary surplus-to-GDP ratio.
- One-off privatization receipts assumed: 30.8 percent of GDP in 2008 (primary fiscal balance adjusted for associated loss of revenue).
- Confidence intervals for d_{2008–12} constructed via simulation methods (Garcia and Rigobon, 2004; Mendoza and Oviedo, 2004; Hostland and Karam, 2005) with simplifications:
  - (i) limited imposed structure on the data;
  - (ii) abstract from endogeneity of fiscal policy (no fiscal policy reaction to shocks).
- Two-step methodology:
  1. Extraction of covariance structure of shocks.
  2. Monte Carlo simulation feeding shocks into equation (1).
- Covariance extracted from historical monthly data over 1998–2007.
- Monte Carlo: 10,000 iterations over 2008–12.
- Distinctive approach: apply stochastic techniques to a normative (adjustment) scenario (IMF, 2007 / Paris III adjustment strategy presented January 25, 2007) rather than to an unchanged-policies baseline.
- For effective interest rate r:
  - Innovations to dollar financing costs extracted from secondary market Eurobond yields.
  - Domestic currency borrowing innovations extracted from local currency deposit rates (banks are main buyers of T-bills; authorities plan greater interest rate flexibility).

### Construction of the variance-covariance matrix of shocks
- Five monthly data series used to form Σ:
  1. 12-month moving average of the growth rate of the Coincident Indicator (proxy for GDP growth)
  2. Domestic Lebanese pound (LL) deposit rate
  3. Domestic U.S. dollar deposit rate
  4. Five-year Eurobond rate
  5. Three-month U.S. dollar LIBOR rate
- Shock definition: first difference in the monthly variable: e_m = x_m - x_{m-1} (equation (2)).
- Modeling as first differences consistent with unit roots in interest rate series and autocorrelation in the coincident indicator.
- Innovations capture empirical covariance relationships; innovations to LIBOR affect Lebanese rates and GDP growth via covariance.

### Monte Carlo simulation (overview)
- Using Σ, Monte Carlo simulates monthly shocks and feeds them into equation (1) to construct confidence intervals for d over 2008–12.
- Simulation specifics: 10,000 iterations; different assumptions on how shocks enter the debt equation (temporary vs. permanent interest rate shocks).

### Simulation design and shock generation
- Monthly shocks drawn i.i.d. for five variables over 2008–12.
- Shocks jointly-normally distributed with mean zero and variance-covariance matrix Σ.
- 10,000 Monte Carlo repetitions (steps repeated 10,000 times) to build empirical confidence intervals.

### Aggregation of monthly shocks into annual innovations
- Growth shock (annual): εg_t = Σ_{m=1}^{12} εg_{tm}.
- Effective LL (local currency) interest rate annual construction:
  - T-bills assumed issued evenly through the year with two-year maturity.
  - Monthly LL deposit-rate shocks weighted so innovations in 2008 impact effective LL rates with weight ½ in 2008; by end-2009 all local currency debt exposed.
  - Persistence via carryover to following period.
- Effective dollar (Eurobond) rate annual construction:
  - Eurobonds assumed issued evenly with 5-year maturity.
  - Effective dollar rate is a five-year moving average of marginal Eurobond rate shocks.
  - Monthly indices m = –11, –23, –35, –47 correspond to January of year t–1, t–2, t–3, t–4.
  - Only by 2012 is the effective rate fully exposed to market shocks → persistence over four years.
- Effective real interest rate r:
  - r_t = 0.5 × εLL_t + 0.5 × εEURO_t (degree of dollarization assumed unchanged at around 50 percent).
- Primary surplus shock response:
  - p_t^ε = 0.2 × εg_t (derived from revenue elasticity and averages; see Appendix).

### Assumptions about shock duration and processes (two alternatives)
- Assumption 1 (temporary shocks):
  - For t = 2008–12:
    - g_t = g_t^scenario + εg_t
    - p_t = p_t^scenario + εp_t
    - r_t = r_t^scenario + εr_t
- Assumption 2 (permanent interest rate shocks; growth shocks temporary):
  - Growth and primary surplus shocks temporary.
  - Interest rate shocks permanent: realized r_t depends on past values and accumulated past shocks after 2008, implying accumulation of past shocks in r_t.

### Debt path calculation and one-off adjustments
- Use equation (1), initial d in 2007, and realized r, g, p (from assumptions) to compute d_t over 2008–12.
- Two one-off deductions to d_t included in 2008 for privatization receipts (30.8 percent of GDP).

### Numerical implementation details and constraints
- Floor on real rate: –2 percent (consistent with nominal rates non-negative and scenario’s constant inflation of 2 percent).
- Annual shocks derived by summing monthly shocks; moving-average and carry-over increase persistence and variance over time.

### Summary statistics and key volatility measures (1998–2007)
- Monthly and annualized standard deviations (In percent), Table 2:
  - Growth: Monthly 0.62; Annualized 2.13
  - LL deposit: Monthly 0.35; Annualized 1.21
  - U.S. $ deposit: Monthly 0.33; Annualized 1.13
  - Eurobond: Monthly 0.58; Annualized 2.00
  - LIBOR: Monthly 0.39; Annualized 1.37
- Table 1 excerpt (Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007; annualized, in percent):
  - Argentina 46.3
  - Brazil 8.7
  - Chile 4.4
  - Colombia 9.6
  - Dominican Republic 10.4
  - El Salvador 7.8
  - Mexico 7.0
  - Venezuela, Rep. Bol. 22.1
  - Bulgaria 10.5
  - Hungary 2.5
  - Poland 3.7
  - Turkey 163.7
  - Egypt 3.0
  - Jordan 2.1
  - Lebanon 1.2
  - Morocco 2.2
  - South Africa 2.6
  - Tunisia 0.9
  - China, P.R.: Hong Kong 5.8
  - India 74.2
  - Indonesia 19.7
  - Korea 2.8
  - Malaysia 1.7
  - Pakistan 9.0
  - Philippines 2.8
  - Singapore 2.1
  - Sri Lanka 10.5
  - Thailand 5.4
- Note: All rates are deposit rates, except Pakistan and Tunisia (money market rates only available). Bulgaria calculated over 1999–2007 given a large spike in 1998. Data source: International Financial Statistics.

### Correlations among shocks (1998–2007) — selected entries (Table 3, In percent)
- Growth with LL deposit: –0.08
- Growth with U.S. $ deposit: –0.13
- Growth with Eurobond: –0.14
- Growth with LIBOR: –0.23
- LL deposit with U.S. $ deposit: 0.74
- LL deposit with Eurobond: 0.54
- LL deposit with LIBOR: 0.64
- U.S. $ deposit with Eurobond: 0.51
- U.S. $ deposit with LIBOR: 0.88
- Eurobond with LIBOR: 0.45
- Interpretation:
  - Interest rate shocks negatively correlated with growth shocks.
  - Interest rate shocks positively correlated among themselves; short rates show highest mutual correlations.

### Simulation distributions and volatility evolution (selected standard deviations)
- Simulated g distributions (standard deviations for 2008–2012): 2.98, 3.01, 2.97, 2.97, 3.00 percent (respectively).
- Simulated r distributions under temporary shocks (standard deviations for 2008–2012): 0.50, 1.41, 1.77, 2.24, 2.80 percent (respectively).
- Simulated r distributions under permanent shocks (standard deviations for 2008–2012): 0.50, 1.79, 3.15, 4.63, 6.25 percent (respectively).
- Simulated d distributions under temporary shocks (standard deviations for 2008–2012): 5.70, 7.71, 9.63, 11.79, 14.26 percent (respectively).
- Simulated d distributions under permanent shocks (standard deviations for 2008–2012): 5.70, 7.88, 11.01, 16.08, 23.43 percent (respectively).
- Observations:
  - Variance of r and d increases over time under both assumptions due to moving-average components and accumulation of shocks.
  - Under permanent shocks, distributions skew strongly (r hits –2 percent floor more frequently), producing left-skewed non-normal distributions in later years.

### Fan-chart results and probabilistic interpretation
- Under temporary shocks:
  - Less than a 5 percent probability that the debt ratio will fail to decline by 2012 (95th percentile corresponds to a nearly stable debt ratio).
- Under permanent shocks:
  - Probability that the debt ratio will begin rising again by 2012 ≈ 25 percent (75th percentile begins rising around 2012).
- Assessment: the “true” stochastic process likely between these two extremes (some permanence but some mean reversion).

### Conclusions and limitations
- Contribution: stochastic simulation provides probabilistic dimension to debt sustainability by incorporating stochastic GDP growth and interest-rate shocks estimated from 1998–2007.
- Main conclusion for Lebanon: under the adjustment scenario, the authorities’ adjustment effort has a reasonable probability of placing the debt ratio on a steady downward path over the medium term despite adverse shocks.
- Major limitations:
  - Abstracts from endogenous interactions between primary balance, growth, and interest rates; fiscal policy predetermined and does not react to shocks.
  - Does not allow for adverse fiscal shocks (e.g., political instability, realization of contingent liabilities) nor for fiscal policy responses to counteract shocks.
  - Data limitations and the break in policy framework justify the simpler analytical framework chosen.

### Appendix: derivation of primary surplus response to GDP shocks
- Government revenues T assumed: T = T̄ Y^α, with α = revenue-income elasticity.
- Primary surplus-to-output ratio p = T̄ Y^{α−1} − E/Y.
- Linearization/discrete approximation yields: Δp ≈ φ × Δy, where φ ≡ (α − t), t = T/Y.
- Using average values over projection period: p = 0.05; t = 0.25; α = 1 → φ = 0.2.
- Resulting specification used in simulations:
  - p_t^ε = 0.2 × εg_t

*Source: _wp0897 - 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6 (PDF content provided).*

### 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6

### _wp0897 - 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6

### Introduction
- Lebanon’s core macroeconomic vulnerability is a large public debt overhang: 177 percent of GDP in 2006.
- Domestic commercial banks have financed the government by tapping a vast pool of expatriate and regional investors; bank deposits reached 283 percent of GDP in 2007.
- Lebanon sustained large debt-to-GDP ratios despite shocks (assassination of former prime minister Hariri in 2005; conflict with Israel in 2006) because markets absorbed new debt in an environment of ample global and regional liquidity.
- Market willingness to hold Lebanon’s debt is not indefinite and cannot be divorced from debt sustainability considerations.
- Debt sustainability is conceptually defined by the government’s ability to pursue its fiscal policy stance into the future without threatening solvency and by operating within its intertemporal budget constraint.
- Empirical debt-threshold approaches (Reinhart, Rogoff, and Savastano (2003); Manasse, Roubini, and Schimmelpfennig (2005)) would have implied crises for Lebanon long ago; political and practical considerations often guide targets where explicit debt targets exist.
- Debt dynamics approach: government is operating within its budget constraint if expected fiscal policy keeps the debt-to-GDP ratio on a stable (or declining) path, but this approach has shortcomings:
  - Debt-to-GDP dynamics are stochastic, depending on GDP growth, interest rates, and exchange rate—factors outside government control.
  - High debt levels and rollover needs create risk that liquidity shocks will unravel into a debt crisis; without the probability distribution of liquidity shocks, assigning a probability is impossible.
- Lebanon’s debt is essentially backed by short-term deposits, creating large potential rollover risk despite historically stable investor base.

### Lebanon’s Debt Dynamics (historical context and key figures)
- Post-civil war (1991) government debt was around 50 percent of GDP; it rose through the 1990s as reconstruction costs outpaced revenue.
- Primary deficit in 1992–97 averaged 7.8 percent of GDP.
- Real GDP growth averaged 5.6 percent (annual average in 1994–97).
- Real interest on Treasury bills averaged 7 percent (annual average 1994–97).
- Fiscal consolidation began in 1997; growth weakness kept the debt ratio rising until 2001 despite improvements in the primary balance.
- Stabilization until 2004 aided by pickup in GDP growth, gradual increase in primary fiscal surplus, and lowering of borrowing costs (including soft financing from Paris II, November 2002).
- Since 2004, a 10 percentage point increase in the debt ratio was driven by: economic slowdown related to political instability and 2006 conflict with Israel; increases in the interest bill (rising spreads and maturing of Paris II zero interest credits); and fiscal impact of higher oil prices.
- Dollarization of government debt grew since 1994 and stands at around 50 percent.
- Move to de facto peg to the U.S. dollar in 1998 helped reduce borrowing costs and lower effective interest rate on government debt but increased balance sheet vulnerabilities.
- Spread between two-year domestic T-bills and Eurobonds (averaging five-year maturity) averaged 3.6 percent over 1998–2007 (data until end-of-month October 2007 used to compute this average).
- Eurobonds, though issued internationally, are mostly held by domestic banks as counterpart to foreign currency deposits; international investors play a marginal role.
- Lebanon shows lower pass-through of international interest rate changes compared with other emerging markets (see Poddar and others (2006)).
- Standard deviation of Lebanon’s short-term interest rate (annualized) is 1.2 percent per year—the second lowest in the group examined—reflecting low volatility of market interest rates and the central bank’s ability to manage interest rates amid volatile capital flows.
- Despite stability, Lebanon experienced periods of financial stress since 1993, chiefly tied to depositor confidence declines triggered by domestic political tensions and relations with Israel; these episodes led to central bank financing and international reserve pressures but did not trigger full banking, balance of payments, and debt crises. Financial responses and expectations of foreign financial assistance in 2002–03 and 2006–07 helped restore confidence.

### Methodology (analytical framework)
- Core debt accumulation equation used:
  - d_{t+1} = d_t + (r_t - g_t) d_t - p_t  (presented as equation (1) in the source)
  - Where d is debt-to-GDP ratio, r is effective real interest rate on the debt, g is real GDP growth, and p is primary surplus-to-GDP ratio.
- One-off privatization receipts of 30.8 percent of GDP are assumed to come in 2008 (primary fiscal balance adjusted to reflect associated loss of revenue from privatized enterprises).
- Confidence intervals for debt-to-GDP ratio over 2008–12 constructed via simulation methods used in debt sustainability literature (Garcia and Rigobon, 2004; Mendoza and Oviedo, 2004; Hostland and Karam, 2005), with simplifications:
  - (i) limited imposed structure on the data;
  - (ii) abstract from endogeneity of fiscal policy (no fiscal policy reaction to shocks modeled).
- Two-step methodology:
  - Extraction of covariance structure of shocks.
  - Monte Carlo simulation feeding shocks into debt accumulation equation (1).
- Covariance of shocks to r, g, and p extracted from historical monthly data over 1998–2007.
- Monte Carlo simulation runs 10,000 iterations over forecast period 2008–12 to construct confidence intervals around a “central” scenario.
- Distinctive approach: stochastic techniques applied to a normative (adjustment) scenario rather than to the baseline (unchanged policies) scenario because the baseline is already unsustainable; central scenario equals the independently projected adjustment path from Lebanon—Staff Report for the 2007 Article IV Consultation (IMF, 2007) / the Paris III adjustment strategy presented January 25, 2007.
- Central scenario is not extrapolated from time series properties or an endogenous policy reaction function, requiring a different method to identify innovations and construct the variance-covariance matrix.
- For the effective interest rate on debt (r), innovations are derived from shocks to market rates at which debt is refinanced in local currency and foreign currency (U.S. dollar).
  - Innovations to dollar financing costs extracted from secondary market Eurobond yields.
  - Domestic currency borrowing innovations are extracted from local currency deposit rates (due to limited volatility in T-bill rates and lack of functioning secondary market).
  - Rationale: (i) banks as main buyers of T-bills—government returns determined by banks’ cost of funds; (ii) authorities plan greater interest rate flexibility moving forward so market pressures should be reflected more immediately in T-bill market.

### Construction of the Variance-Covariance Matrix of Shocks
- Shocks to r and g derived from the variance-covariance matrix (Σ) of shocks in five monthly data series:
  1. 12-month moving average of the growth rate of the Coincident Indicator (proxy for GDP growth)
  2. Domestic Lebanese pound (LL) deposit rate
  3. Domestic U.S. dollar deposit rate
  4. Five-year Eurobond rate
  5. Three-month U.S. dollar LIBOR rate
- Definition of a shock: first difference in the monthly variable; for variable x, shock e at month m is e_m = x_m - x_{m-1} (equation (2) in the source).
- Modeling shocks as first differences (random walk) is consistent with unit roots not being rejected in interest rate series and with strong autocorrelation in the coincident indicator due to its moving-average construction.
- Innovations in the five series reflect underlying shocks to Lebanese and world economies; the approach remains silent on structural transmission mechanisms but captures empirical covariance relationships.
- Innovations to U.S. dollar interest rate affect Lebanese interest rates and GDP growth via their covariance with those series.

### Monte Carlo Simulation (overview)
- Using Σ, Monte Carlo simulation constructs confidence intervals for debt-to-GDP ratio over 2008–12 by feeding simulated shocks into the debt accumulation equation (1).
- Simulation specifics provided in the source: 10,000 iterations over forecast period 2008–12; different assumptions made concerning how shocks enter the debt equation (details continue in subsequent sections of the source).

### Key statistics and table excerpt (Table 1: Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007; annualized, in percent)
- Argentina 46.3
- Brazil 8.7
- Chile 4.4
- Colombia 9.6
- Dominican Republic 10.4
- El Salvador 7.8
- Mexico 7.0
- Venezuela, Rep. Bol. 22.1
- Bulgaria 10.5
- Hungary 2.5
- Poland 3.7
- Turkey 163.7
- Egypt 3.0
- Jordan 2.1
- Lebanon 1.2
- Morocco 2.2
- South Africa 2.6
- Tunisia 0.9
- China, P.R.: Hong Kong 5.8
- India 74.2
- Indonesia 19.7
- Korea 2.8
- Malaysia 1.7
- Pakistan 9.0
- Philippines 2.8
- Singapore 2.1
- Sri Lanka 10.5
- Thailand 5.4
- Notes: All rates are deposit rates, except for Pakistan and Tunisia (money market rates only available). Bulgaria's standard deviation is calculated over 1999–2007 given a large spike in 1998. Data source: International Financial Statistics.

*Source: _wp0897 - 1.     Standard Deviation of Changes in Monthly Real Short-Term Interest Rates, 1998–2007 .... 6 (PDF content provided).*

### 1. We randomly draw i.i.d. monthly shocks to the five variables over 2008–12.

### _wp0897 - 1. We randomly draw i.i.d. monthly shocks to the five variables over 2008–12.

### Simulation design and shock generation
- Monthly shocks are drawn i.i.d. for five variables over 2008–12.
- Shocks are assumed jointly-normally distributed with mean zero and variance-covariance matrix Σ.
- 10,000 Monte Carlo repetitions are performed (steps 1–4 repeated 10,000 times) to construct empirical confidence intervals for 2008–12.

### Aggregation of monthly shocks into annual innovations
- Growth shock (annual): total annual shock at year t = sum over monthly shocks m = 1..12: εg_t = Σ_{m=1}^{12} εg_{tm}.
- Effective interest rate on LL (local currency) debt (annual): constructed assuming T-bills issued evenly through the year and two-year maturity; monthly LL deposit-rate shocks are weighted so that innovations in 2008 impact the effective LL interest rates with a weight of ½ in 2008, and by end-2009 all local currency debt is exposed to interest rate shocks. This implies persistence because shocks carry over to the following period.
- Effective interest rate on dollar (Eurobond) debt (annual): constructed assuming Eurobonds issued evenly through the year with 5-year maturity; the effective dollar rate is a five-year moving average of marginal Eurobond rate shocks. Monthly indices m = –11, –23, –35, –47 correspond to January of year t–1, t–2, t–3, and t–4. Only by 2012 is the effective rate fully exposed to market shocks. This also implies persistence due to carry-over over four years.
- Effective real interest rate in Lebanon, r: weighted average of LL and U.S. dollar effective rates assuming degree of dollarization remains unchanged at its present level of around 50 percent:
  - r_t = 0.5 × εLL_t + 0.5 × εEURO_t
- Primary surplus shock (as ratio to GDP), p_t^ε: only allowed to respond to GDP shocks through tax revenue elasticity. Innovation to primary balance:
  - p_t^ε = 0.2 × εg_t

### Assumptions about shock duration and processes (two alternative assumptions)
- Assumption 1 (temporary shocks):
  - Shocks to growth, primary surplus, and interest rates are temporary: for t = 2008–12:
    - g_t = g_t^scenario + εg_t
    - p_t = p_t^scenario + εp_t
    - r_t = r_t^scenario + εr_t
  - Scenario values denoted r_t^scenario, g_t^scenario, p_t^scenario.

- Assumption 2 (permanent interest rate shocks; growth shocks temporary):
  - Growth and primary surplus shocks remain temporary.
  - Interest rate shocks are permanent (extreme case): realized interest rates at time t depend on past values and past shocks after 2008. For t = 2008–12 the process implies accumulation of past shocks in r_t.

### Debt path calculation and one-off adjustments
- Using equation (1), the value of d in 2007, and realized r, g, and p from either assumption (3) or (4), the path of debt-to-GDP ratio d_t is calculated over 2008–12.
- Two one-off deductions to d_t are included in 2008 due to privatization receipts.

### Numerical implementation details and constraints
- A floor of –2 percent is placed on the real rate consistent with:
  - (i) the assumption that nominal interest rates cannot be negative; and
  - (ii) the scenario’s assumed constant inflation rate of 2 percent.
- Annual shocks are derived by summing monthly shocks as specified above; moving-average and carry-over components introduce persistence and increase variance over time.

### Summary statistics and key volatility measures (1998–2007)
- Monthly and annualized standard deviations of shocks (In percent), Table 2:
  - Growth: Monthly 0.62; Annualized 2.13
  - LL deposit: Monthly 0.35; Annualized 1.21
  - U.S. $ deposit: Monthly 0.33; Annualized 1.13
  - Eurobond: Monthly 0.58; Annualized 2.00
  - LIBOR: Monthly 0.39; Annualized 1.37
- Note: annual value calculated by assuming monthly innovations are uncorrelated over time and equals sqrt(12) times the standard deviation of the monthly series (table note).

### Correlations among shocks (1998–2007), Table 3 (In percent)
- Correlation matrix (selected entries):
  - Growth with LL deposit: –0.08
  - Growth with U.S. $ deposit: –0.13
  - Growth with Eurobond: –0.14
  - Growth with LIBOR: –0.23
  - LL deposit with U.S. $ deposit: 0.74
  - LL deposit with Eurobond: 0.54
  - LL deposit with LIBOR: 0.64
  - U.S. $ deposit with Eurobond: 0.51
  - U.S. $ deposit with LIBOR: 0.88
  - Eurobond with LIBOR: 0.45
- Interpretation: interest rate shocks negatively correlated with growth shocks; interest rate shocks positively correlated among themselves; short rates show highest mutual correlations.

### Simulation distributions and volatility evolution (selected standard deviations)
- Simulated g distributions (standard deviations for 2008–2012): 2.98, 3.01, 2.97, 2.97, 3.00 percent (respectively).
- Simulated r distributions under temporary shocks (standard deviations for 2008–2012): 0.50, 1.41, 1.77, 2.24, 2.80 percent (respectively).
- Simulated r distributions under permanent shocks (standard deviations for 2008–2012): 0.50, 1.79, 3.15, 4.63, 6.25 percent (respectively).
- Simulated d distributions under temporary shocks (standard deviations for 2008–2012): 5.70, 7.71, 9.63, 11.79, 14.26 percent (respectively).
- Simulated d distributions under permanent shocks (standard deviations for 2008–2012): 5.70, 7.88, 11.01, 16.08, 23.43 percent (respectively).
- Observations:
  - Variance of r and d distributions increases over time under both assumptions due to moving-average components and accumulation of shocks.
  - Under permanent shocks, distributions skew strongly (r hits –2 percent floor more frequently), producing left-skewed non-normal distributions for later years.

### Fan-chart results and probabilistic interpretation
- Fan charts present scenario and confidence intervals for d under temporary and permanent interest rate shocks.
- Under temporary shocks:
  - Less than a 5 percent probability that the debt ratio will fail to decline by 2012 (95th percentile corresponds to a nearly stable debt ratio).
- Under permanent shocks:
  - Probability that the debt ratio will begin rising again by 2012 is around 25 percent (75th percentile begins rising around 2012).
- The “true” stochastic process likely lies between these two extremes (some permanence but some mean reversion).

### Conclusions and limitations
- The stochastic simulation adds a probabilistic dimension to debt sustainability analysis by incorporating stochastic GDP growth and interest rate shocks estimated from 1998–2007 data.
- Main conclusion for Lebanon: under the adjustment scenario, authorities’ adjustment effort has a reasonable probability of placing the debt ratio on a steady downward path over the medium term despite adverse shocks.
- Major limitations:
  - The analysis abstracts from endogenous interactions between primary balance, growth, and interest rates; policy is predetermined and does not react endogenously to shocks.
  - Does not allow for adverse fiscal shocks (e.g., political instability, realization of contingent liabilities) nor for fiscal policy responses to counteract shocks.
  - Data limitations and the break in policy framework justify the simpler analytical framework chosen.

### Appendix: derivation of primary surplus response to GDP shocks
- Government revenues T assumed function of nominal GDP Y: T = T̄ Y^α, with α = revenue-income elasticity.
- Primary surplus-to-output ratio p = (T̄ Y^α − E)/Y = T̄ Y^{α−1} − E/Y.
- Linearization and discrete approximation yield:
  - Δp ≈ φ × Δy, where φ ≡ (α − t), t = T/Y.
- Using average values p = 0.05 and t = 0.25 over the projection period and α = 1, the factor φ = 0.2, giving:
  - p_t^ε = 0.2 × εg_t

*Source: _wp0897 - 1. We randomly draw i.i.d. monthly shocks to the five variables over 2008–12.*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0897.pdf_