## wpiea2019052

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### Theory: portfolio management and determinants of currency composition
- Central banks are highly risk-averse; bulk of reserves invested in safe assets.
- Common practice: portfolio optimization with mean-variance approach; objectives: “Safety, liquidity, and profitability” (safety and liquidity primary).
- Two-tranche structure commonly used:
  - Liquidity tranche: finances day-to-day FX needs; often exclusively T-bills and time deposits.
  - Investment tranche: pursues highest return subject to risk constraints; relevant for large reserve holdings.
- Empirical prevalence: two-thirds of central banks in a BIS survey had two (or more) separate tranches; tranche sizes vary (some < 10 percent of reserves in liquidity tranche, others > half).
- Modeling approach advocated:
  - Combine classical mean-variance for investment tranche with asset-liability framework for liquidity tranche.
  - Asset-liability framework can deliver closed-form solution to facilitate parameter estimation and counterfactuals.
- Determinants emphasized:
  - Currency composition of imports invoicing matters if reserves insure access to goods and services.
  - Currency composition of short-term external debt matters if reserves hedge access to financial markets.
  - Shift toward using domestic currency as numeraire affects measured risk and optimal allocations.
- Focus on exchange rate risk: exchange rate movement is much more volatile than interest rate movement; framework can be extended to include interest-rate risk.

### Reduced-form regression: specification, data, and key findings
- Regression specification:
  - x_it = c_i + β_1 imports invoicing_it + β_2 average return_it + β_3 return volatility_it + ε_it
  - Fixed-effect transformed form described.
- Definitions:
  - x_it: share of reserve currency i (EUR, JPY, GBP, USD).
  - imports invoicing_it: share of currency i in imports invoicing.
  - average return_it: currency i’s average return rates using SDR as numeraire in past 7 years up to each quarter.
  - return volatility_it: volatility of return rates using SDR as numeraire in past 7 years up to each quarter.
- Sample and data:
  - Main sample: 22 emerging and developing countries.
  - Time period: 2006 Q1 to 2014 Q4.
  - Selection criteria: FX reserves > USD10bn in 2017 or included in imports invoicing dataset in Gopinath (2016).
  - Aggregate of 22 countries accounted for 88 percent of total FX reserve holdings of emerging and developing economies in 2015 Q1.
- Reduced-form regression key results (selected):
  - Column (1): imports invoicing = 0.622 ∗∗∗ (4.90); average return = 47.60 ∗∗ (2.15); return volatility = -3.602 (-1.46); N = 144; within R-square = 0.2459.
  - Column (2): ST external debt denomination = 0.413 ∗∗∗ (11.20); average return = 97.10 ∗∗∗ (6.06); return volatility = 1.383 (0.69); N = 144; within R-square = 0.5374.
  - Column (3): imports invoicing and ST external debt denomination = 0.549 ∗∗∗ (9.96); average return = 64.26 ∗∗∗ (3.76); return volatility = -0.318 (-0.15); N = 144; within R-square = 0.4859.
  - Column (4) (adds 2-year government bond bid-ask spread): imports and ST external debt = 0.550 ∗∗∗ (9.98); average return = 69.51 ∗∗∗ (3.87); return volatility = -1.714 (-0.67); government bond spread = -9.588 (-0.94); N = 144; within R-square = 0.4892.
- Key reduced-form findings:
  - Currency compositions of imports invoicing and short-term external debt have significant effects on FX reserve currency composition.
  - Average return rates have significant positive effects, consistent with an investment tranche motive.
  - Return volatility effects are not statistically significant.
  - Government bond spread is not statistically significant; interpretation: government bonds of EUR, JPY, GBP, USD issuers are sufficiently liquid for reserve buyers.

### Portfolio choice model (two-tranche) — structure and implications
- Two-tranche definitions:
  - Liquidity tranche size: L (covers FX obligations related to imports and short-term external debt).
  - Investment tranche size: A − L when A > L; if A ≤ L the whole portfolio is liquidity tranche.
- Notation and assumptions:
  - y_i: share of obligations in currency i.
  - R_i: growth rate of value of currency i obligations (exchange rate growth using domestic currency as numeraire).
  - Benchmark: asset returns assumed zero; exchange rates drive asset values.
  - Central bank chooses x_Li (liquidity tranche shares) and x_Ii (investment tranche shares).
- Liquidity tranche optimization (mean-variance form):
  - min_{x_L} 1/2 (x_L − y)^T Ω (x_L − y) − λ_L m^T (x_L − y)  s.t. e^T x_L = 1
  - Solution: x_L = y + λ_L(Ω^{-1} m − (e^T Ω^{-1} m)/(e^T Ω^{-1} e) Ω^{-1} e)
  - If λ_L = 0, x_L = y.
- Investment tranche optimization:
  - min_{x_I} 1/2 x_I^T Ω x_I − λ_I m^T x_I  s.t. e^T x_I = 1
  - Solution: x_I = Ω^{-1} e (e^T Ω^{-1} e)^{-1} + λ_I(Ω^{-1} m − (e^T Ω^{-1} m)/(e^T Ω^{-1} e) Ω^{-1} e)
- Aggregate portfolio when A > L:
  - x = x_I + (L/A)(x_L − x_I)
- FX obligations approximation:
  - Ly = aT + bD where T = imports invoicing in foreign currencies, D = short-term external debt in foreign currencies; foreign currencies dominated by EUR, JPY, GBP, USD.
- Aggregation: x_aggregate is function of (A, Ω, m, T, D, a, b, λ_I, λ_L).

### Estimation approach and parameter estimates
- Estimation strategy:
  - For each quarter t and central bank l, Ω_lt and m_lt estimated using latest seven-year sample up to t; numeraire = domestic currency.
  - Construct vector u_i(Λ_t, a, b) from aggregates to map model to predicted aggregate shares.
  - Reduced-form linking equation: x_it = c_i + β x(Λ_t,a,b,λ_I,λ_L)_aggregate,i + ε_it.
  - Fixed-effect transformation and GMM estimator with orthogonality conditions used.
  - Two-step estimation: 1) search for (a*, b*) minimizing sample moments (closed-form β*, λ_I*, λ_L* used); 2) form W_n and re-minimize to obtain ˆθ = (ˆa, ˆb, ˆβ, ˆλ_I, ˆλ_L).
- Estimation results (Table 2):
  - a: 2.6101 ∗∗∗ (3.1603)
  - b: 2.3853 ∗∗∗ (2.7436)
  - β: 0.5086 ∗∗∗ (7.0743)
  - λ_I: 0.0419 ∗∗∗ (5.9580)
  - λ_L: -0.0017 (-0.5094)
  - Significance: ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
- Interpretation:
  - a and b positive and statistically significant: imports invoicing and short-term external debt exposures matter quantitatively for reserve currency shares.
  - β = 0.5086 ∗∗∗: model-implied aggregate component transmits partially (β < 1) into observed shares—consistent with central banks’ inertia.
  - λ_I = 0.0419 ∗∗∗: investment tranche responds to expected returns.
  - λ_L not statistically different from zero: liquidity tranche does not target returns.

### Quantitative findings on tranche sizes and dynamics
- Model fit: fitted series capture trends and fluctuations of EUR and USD shares for 22 countries (2006 Q1–2014 Q4) (Figure 7 described).
- Aggregate tranche estimates:
  - L_aggregate liquidity = ∑_{l∈J1} e^T(aT_l + bD_l) + ∑_{l∈J2} A_l
  - L_aggregate investment = ∑_{l∈J1} (A_l − e^T(aT_l + bD_l))
- Dynamics:
  - Compared to 2006 and 2007, absolute and relative sizes of aggregate investment tranche increased significantly afterwards.
  - After 2008, absolute size of developing countries’ aggregate investment tranche was stable; both absolute and relative sizes of liquidity tranche increased due to growing imports invoicing and short-term external debt in reserve currencies.
- Optimal tranche composition movements:
  - Between 2007 Q1 and 2008 Q1: optimal euro share increased and optimal dollar share declined in investment tranche (consistent with returns movements); trend reversed after 2008.
- Estimated currency-specific fixed terms (c_i):
  - c_EUR = 0.1814
  - c_JPY = 0.0070
  - c_GBP = 0.0590
  - c_USD = 0.2439
  - Interpretation: dollar and euro enjoy greater privilege compared with yen and pound; model explains most euro-dollar share difference though residual c_i differences remain.

### Size of the liquidity tranche: interaction tests and implications
- Hypothesis 1 tested: larger relative liquidity tranche size amplifies effect of imports invoicing and short-term external debt on reserve currency composition.
  - Result: interaction terms positive and statistically significant (Columns (1), (3), (4) in Table 3).
- Hypothesis 2 tested: larger relative liquidity tranche size reduces effect of exchange rate returns on reserve currency composition.
  - Result: interaction terms negative and statistically significant (Columns (1)-(4) in Table 3).
- Selected regression results (Table 3):
  - imports invoicing: 0.631 ∗∗∗ (5.06)
  - ST external debt denomination: 0.383 ∗∗∗ (7.23)
  - imports and ST external debt: 0.500 ∗∗∗ (8.26) and 0.500 ∗∗∗ (8.24)
  - average return across columns: 658.0 ∗∗∗ (3.00); 455.8 ∗∗ (2.57); 551.0 ∗∗∗ (3.05); 590.2 ∗∗∗ (2.83)
  - Interaction terms with rel liquidity size:
    - imports invoicing × rel liquidity size: 0.0913 ∗ (1.74)
    - imports and ST external debt × rel liquidity size: 0.0726 ∗ (1.77) and 0.0709 ∗ (1.71)
    - average return × rel liquidity size: -717.1 ∗∗∗ (-2.82); -416.4 ∗∗ (-2.03); -566.4 ∗∗∗ (-2.71); -614.6 ∗∗ (-2.50)
  - Sample size and fit: N = 144; within R-square: Column (1) 0.3024, Column (2) 0.5544, Column (3) 0.5254, Column (4) 0.5259.
- RMB-specific liquidity and convertibility measurements:
  - Liquidity of government bond markets measured by mean daily bid-ask spread (one-year sample up to each quarter).
  - Ease of convertibility measured by bid-ask spread of reserve currencies versus domestic currency (mean daily bid-ask spread; RMB onshore versus domestic currencies of China, Saudi Arabia, Russia, India, Brazil averaged).
  - Observations: China’s government bonds are less liquid than other reserve currency issuers’ bonds; liquidity improved since first half of 2018; RMB (onshore) transaction cost is above those of other reserve currencies in most times.

### The RMB as a reserve currency — scenarios and quantitative benchmarks
- Baseline expectation:
  - China will gradually enhance capital account convertibility.
  - RMB share in central banks’ FX reserves expected to increase from current level of around 2 percent.
  - Over 60 central banks or monetary authorities hold RMB reserves (Yi, 2018).
- Scenario methodology:
  - For each country excluding China, set RMB’s expected return, variance and covariances with EUR, JPY, GBP, USD equal to average of SDR currencies’ corresponding parameters; RMB fixed term set to average of SDR currencies’ estimated fixed terms.
  - Analysis applied to 2014 Q4 with Ωt, mt, Tt, Dt augmented to include RMB.
- Scenario 1 — larger share of RMB in imports invoicing and short-term external debt:
  - If share of RMB in imports invoicing and short-term external debt = 50 percent, then:
    - RMB share in FX reserves of developing countries including China would be around 15 percent.
    - RMB share in FX reserves of developing countries excluding China would be around 35 percent.
- Scenario 2 — RMB as a reference currency (stronger co-movement; smaller variance):
  - Vary RMB return between 0.1×10^−5 and 10×10^−5 and average variance between 1.7×10^−5 and 24×10^−5 (Figure 13 described).
  - Finding: more prominent reference-currency status (smaller variance) can lead to higher RMB share; effect larger when RMB has relatively higher return.
- Relevant numeric benchmarks and averages (estimated using 2007 Q4 to 2014 Q4):
  - Current RMB share in central bank FX reserves: around 2 percent.
  - Over 60 central banks hold RMB reserves.
  - Return ranges used in Figure 13: 0.1×10^−5 to 10×10^−5.
  - Variance ranges used in Figure 13: 1.7×10^−5 to 24×10^−5.
  - Average returns (sample including China):
    - EUR: 7.18×10^−5
    - JPY: 5.44×10^−5
    - GBP: 9.26×10^−5
    - USD: 12.54×10^−5
  - Average variances (sample including China):
    - EUR: 5.23×10^−5
    - JPY: 40.71×10^−5
    - GBP: 5.98×10^−5
    - USD: 4.97×10^−5

*Source: wpiea2019052 — IMF working paper excerpts (Theory; 3.4 Evidence from a reduced-form regression; 6.2; 7.2).*

### 2.1    Theory .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### 2.1    Theory

### Portfolio management practices of central banks
- Central banks are in general highly risk-averse, with the bulk of their reserves invested in safe assets.
- Central banks often follow a portfolio optimization strategy with the mean-variance portfolio diversification approach being a popular one.
- “Safety, liquidity, and profitability” are generally accepted as the objectives of reserve management, with safety and liquidity being the primary goals.
- Central banks often create a “liquidity tranche” portfolio and an “investment portfolio”:
  - The liquidity tranche is designed to finance day-to-day FX needs and often includes exclusively T-bills and time deposits.
  - The investment portfolio pursues the highest return subject to risk constraints and is especially relevant for countries with large reserve holdings.
- Two-thirds of the central banks responding to a BIS survey had established two (or more) separate tranches.
- The size of each tranche differs by central bank: in some cases, less than 10 percent of reserves are allocated to the liquidity tranche, while in others, over half of reserves are held in this tranche.

### Modeling approach advocated in the paper
- The paper adopts a central bank reserve portfolio choice model that combines:
  - A classical mean-variance framework for the investment tranche.
  - An asset-liability framework for the liquidity tranche, appropriate where liability types differ (e.g., FX obligations from imports and debt payments).
- The asset-liability framework can deliver a closed-form solution, facilitating parameter estimation.
- The structural model is designed to quantify the importance of various factors influencing central banks’ currency-composition decisions and to enable counter-factual analyses and scenario-based forecasts.

### Determinants of currency composition highlighted
- The currency compositions of imports invoicing and short-term external debt can have a significant effect on the currency composition of FX reserves:
  - If reserves primarily provide insurance against loss of access to foreign goods and services, the currency composition of the import basket is relevant.
  - If reserves primarily hedge against loss of access to international financial markets, the currency composition of external liabilities is more relevant.
- Well-known rules of thumb (e.g., ratio of reserves to imports; ratio of reserves to short-term external liabilities) have been used to assess reserve adequacy.
- A shift toward using domestic currency as numeraire may have taken place and affects measured risk and optimal allocations.

### Evidence and implications from prior theory and empirical work
- Theoretical literature on internationalization of currencies includes:
  - Multiple equilibrium and network externality frameworks (Krugman, 1984; Kiyotaki et al., 1993; Rey, 2001).
  - Models linking reserve-asset status to relative fundamentals and relative debt sizes (He et al., 2018).
- Empirical findings from the literature:
  - Aggregate-level analyses find reserve-currency countries’ shares of global GDP and global trade have significant positive effects on currency shares in global FX reserves (Eichengreen, 1998).
  - Lagged depreciation and inflation (or exchange rate volatility) have negative and significant influence on currency share; income has a significantly positive influence; high persistence of currency shares (coefficient on lagged dependent variable between 0.85 and 0.96) (Chinn and Frankel, 2005).
  - Confidential country-level COFER-based studies find currency pegs, direction of trade, and currency of foreign debt are significant and robust determinants of reserve-currency composition (Dooley et al., 1989; Eichengreen and Mathieson, 2000).
  - Mean-variance frameworks modified for monetary authorities show optimum portfolios may differ from observed portfolios, with particular notes on euro weights (Papaioannou et al., 2006).
  - Two-tranche models (high risk aversion liquidity tranche; lower risk aversion investment tranche) are used to capture central bank behavior (Zhang et al., 2012).

### Use of domestic currency as numeraire — method and evidence
- Different numeraire choices imply different optimal allocations; if a domestic currency varies less against the dollar than other majors, a dollar-heavy reserve portfolio may pose less risk when measured in domestic currency.
- Empirical support exists for use of domestic currency as numeraire (examples: Australia; Chile).
- To quantify each country’s currency co-movement with main reserve currencies, the paper adopts the Frankel and Wei method:
  - ∆ ln
    X
    t
    CHF
    t
    =ρ
    1
    ∆ ln
    EUR
    t
    CHF
    t
    +ρ
    2
    ∆ ln
    JPY
    t
    CHF
    t
    +ρ
    3
    ∆ ln
    GBP
    t
    CHF
    t
    +ρ
    4
    ∆ ln
    USD
    t
    CHF
    t
    +α+ε
    t
  - X
    t
    CHF
    t
    is each country’s exchange rate versus Swiss Franc. ρ1–ρ4 are named the weights of euro zone, yen zone, pound zone, and dollar zone.
- A 10-year sample (2007 to September 2017) is used to estimate ρ1 to ρ4 for each country reporting to COFER.
- Results are consistent with the use of domestic currency as numeraire (see figures referenced for relationships between currency zone weights and reserve shares).

### Focus on exchange rate risk
- The paper focuses on exchange rate risk because exchange rate movement is much more volatile than interest rate movement; government bond yields are relatively stable.
- The framework can be extended to incorporate interest rate risk and other types of risks.

*Source: wpiea2019052 - 2.1    Theory (excerpt).*

### 3.4    Evidence from a reduced-form regression

### 3.4 Evidence from a reduced-form regression

### Regression specification and data
- Regression model:
  - x_it = c_i + β_1 imports invoicing_it + β_2 average return_it + β_3 return volatility_it + ε_it
  - Fixed-effect transformed form: x_it − ̄x_i = β_1(imports invoicing_it −  ̄imports invoicing_i) + β_2(average return_it −  ̄average return_i) + β_3(return volatility_it −  ̄return volatility_i) + (ε_it − ̄ε_i)
- Definitions:
  - x_it: share of reserve currency i (EUR, JPY, GBP, or USD) in period t among the sample of emerging and developing countries’ FX reserves.
  - imports invoicing_it: share of currency i among these countries’ imports invoicing in period t.
  - average return_it: currency i’s average return rates using SDR as numeraire in the past 7 years up to each quarter.
  - return volatility_it: currency i’s volatility of return rates using SDR as numeraire in the past 7 years up to each quarter.
- Sample:
  - Main sample includes 22 emerging and developing countries.
  - Time period: 2006 Q1 to 2014 Q4.
  - Selection criteria: countries either had FX reserves exceeding USD10bn in 2017, or are included in the imports invoicing dataset in Gopinath (2016).
  - Aggregate of the 22 countries accounted for 88 percent of total FX reserve holdings of emerging and developing economies in 2015 Q1; COFER aggregate reserve currency composition is used as approximation.
- Data sources and approximations:
  - Imports invoicing average currency composition between 1999 and 2015 from Gopinath (2016); this average used to construct yearly imports invoicing in each currency.
  - Missing country-level shares approximated using currency shares from countries with available data.
  - Short-term external debt currency composition approximated by currency composition of total external debt or PPG debt from World Bank International Debt Statistics; where unavailable, short-term external debt-weighted averages of other countries’ currency shares are used.
  - Country-level short-term external debt data sourced from Haver.
  - Imports of goods: IMF DOTS; imports of services: UNCTAD.
  - Quarterly country-level reserve currency composition for cross-sectional analysis uses limited IMF COFER country coverage.

### Reduced-form regression results (Table 1 summary)
- Column (1): dependent variable explained with imports invoicing
  - imports invoicing: 0.622 ∗∗∗ (4.90)
  - average return: 47.60 ∗∗ (2.15)
  - return volatility: -3.602 (-1.46)
  - const: 0.117 ∗∗∗ (3.53)
  - N: 144
  - within R-square: 0.2459
- Column (2): dependent variable explained with ST external debt denomination
  - ST external debt denomination: 0.413 ∗∗∗ (11.20)
  - average return: 97.10 ∗∗∗ (6.06)
  - return volatility: 1.383 (0.69)
  - const: 0.154 ∗∗∗ (11.48)
  - N: 144
  - within R-square: 0.5374
- Column (3): weighted average of imports invoicing and ST external debt denomination
  - imports invoicing and ST external debt denomination: 0.549 ∗∗∗ (9.96)
  - average return: 64.26 ∗∗∗ (3.76)
  - return volatility: -0.318 (-0.15)
  - const: 0.115 ∗∗∗ (6.35)
  - N: 144
  - within R-square: 0.4859
- Column (4): adds 2-year government bond bid-ask spread as control
  - imports invoicing and ST external debt denomination: 0.550 ∗∗∗ (9.98)
  - average return: 69.51 ∗∗∗ (3.87)
  - return volatility: -1.714 (-0.67)
  - government bond spread: -9.588 (-0.94)
  - const: 0.123 ∗∗∗ (6.12)
  - N: 144
  - within R-square: 0.4892
- Significance notation:
  - ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
- Key reduced-form findings:
  - The currency compositions of imports invoicing and short-term external debt have significant effects on the currency composition of FX reserves.
  - Average return rates have significant positive effects, consistent with an investment tranche in central bank reserve portfolios.
  - Return volatility effects are not statistically significant, consistent with Chan, Peng, and Shu (2011).
  - Government bond spread is not statistically significant; authors interpret this as government bonds of the four main reserve currency issuers (EUR, JPY, GBP, USD) being sufficiently liquid that liquidity is not a major concern for reserve buyers, though a liquid government bond market may be a necessary condition for central banks to hold these bonds.

### Portfolio choice model (Section 5) — structure and implications
- Two-tranche structure:
  - Liquidity tranche size: L (covers FX obligations related to payments for imports and short-term external debt).
  - Investment tranche size: A − L when A > L; if A ≤ L the whole portfolio is liquidity tranche.
- Notation and main assumptions:
  - y_i: share of obligations in currency i.
  - R_i: growth rate of value of currency i obligations (exchange rate growth using domestic currency as numeraire).
  - Assets returns assumed zero in benchmark (so exchange rates drive asset values); justification: much larger FX risk compared to interest rate risk (Figures 5 and 6).
  - Central bank chooses currency shares x_Li and x_Ii for liquidity and investment tranches respectively.
- Liquidity tranche objective:
  - Maximize surplus subject to risk constraints; optimization problem reduces to a mean-variance form:
    - min_{x_L} 1/2 (x_L − y)^T Ω (x_L − y) − λ_L m^T (x_L − y)  s.t. e^T x_L = 1
  - Solution: x_L = y + λ_L(Ω^{-1} m − (e^T Ω^{-1} m)/(e^T Ω^{-1} e) Ω^{-1} e)
  - If λ_L = 0 (no return concern), x_L = y (liquidity tranche matches obligations).
- Investment tranche objective:
  - Maximize returns subject to risk constraints; mean-variance form:
    - min_{x_I} 1/2 x_I^T Ω x_I − λ_I m^T x_I  s.t. e^T x_I = 1
  - Solution: x_I = Ω^{-1} e (e^T Ω^{-1} e)^{-1} + λ_I(Ω^{-1} m − (e^T Ω^{-1} m)/(e^T Ω^{-1} e) Ω^{-1} e)
- Model implications:
  - When A > L, aggregate portfolio composition: x = x_I + (L/A)(x_L − x_I).
  - If returns are not a concern for liquidity tranche (λ_L = 0), liquidity tranche composition equals obligations’ currency composition.
  - FX obligations approximated as Ly = aT + bD where T = imports invoicing in foreign currencies, D = short-term external debt in foreign currencies, and foreign currencies dominated by EUR, JPY, GBP, USD.
  - Optimal currency composition expressions provided for central banks with and without investment tranches (l ∈ J1 where L < A; l ∈ J2 where L ≥ A).
  - Aggregation across central banks yields x_aggregate as a function of (A, Ω, m, T, D, a, b, λ_I, λ_L).

### Estimation approach (Section 6) and results (Table 2)
- Estimation strategy:
  - For each quarter t and central bank l, Ω_lt and m_lt estimated using latest seven-year sample up to quarter t.
  - Numeraire assumption: each central bank uses its domestic currency.
  - Define vector u_i(Λ_t, a, b) (9×1) constructed from aggregates across central banks and used to map model parameters into predicted aggregate currency shares.
  - Reduced-form linking equation: x_it = c_i + β x(Λ_t,a,b,λ_I,λ_L)_aggregate,i + ε_it
  - Fixed-effect transformation and GMM estimator used with orthogonality conditions:
    - E[(u_it(a_0,b_0) − ̄u_i(a_0,b_0))(ε_it − ̄ε_i)] = 0 and E[ε_it − ̄ε_i] = 0
  - Two-step estimation:
    - Step 1: search for (a*, b*) minimizing identity-weighted sample moments; closed-form expressions for β*(a,b), λ_I*(a,b), λ_L*(a,b) reduce computational burden.
    - Step 2: form W_n from Step 1 residuals and re-minimize to obtain ˆθ = (ˆa, ˆb, ˆβ, ˆλ_I, ˆλ_L).
- Estimation results (Table 2):
  - a: 2.6101 ∗∗∗ (3.1603)
  - b: 2.3853 ∗∗∗ (2.7436)
  - β: 0.5086 ∗∗∗ (7.0743)
  - λ_I: 0.0419 ∗∗∗ (5.9580)
  - λ_L: -0.0017 (-0.5094)
  - t statistics in parentheses; ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
- Interpretation of estimation results:
  - Estimates imply positive and statistically significant coefficients on a and b, indicating that imports invoicing and short-term external debt exposure matter quantitatively for reserve currency shares.
  - Estimated β = 0.5086 ∗∗∗ indicates that the time-varying model-implied aggregate component is transmitted partially (β < 1) into observed reserve currency shares, consistent with central banks’ reluctance to rapidly adjust currency compositions (inertia and preference for stable reserve composition).
  - λ_I = 0.0419 ∗∗∗ is positive and statistically significant, indicating investment tranche responds to expected returns; λ_L is not statistically different from zero (cannot reject λ_L = 0), consistent with liquidity tranche not targeting returns.

### Quantitative findings, fitted series, and tranche sizes
- Model fit:
  - Figure 7 (described): actual and fitted EUR and USD shares in the 22 countries’ FX reserves (2006 Q1–2014 Q4) — fitted shares capture trends and fluctuations of actual shares.
- Estimated tranche sizes (aggregate):
  - L_aggregate liquidity = ∑_{l∈J1} e^T(aT_l + bD_l) + ∑_{l∈J2} A_l
  - L_aggregate investment = ∑_{l∈J1} (A_l − e^T(aT_l + bD_l))
- Dynamics and patterns:
  - Compared to 2006 and 2007, the absolute and relative sizes of the aggregate investment tranche increased significantly afterwards.
  - After 2008, despite growth in aggregate reserve portfolio, absolute size of the developing countries’ aggregate investment tranche was stable, and both absolute and relative sizes of the liquidity tranche increased, driven by growing imports invoicing and short-term external debt denominated in reserve currencies.
- Optimal tranche compositions:
  - Expressions provided for x_aggregate liquidity and x_aggregate investment as functions of model parameters and estimated Ω, m.
  - Between 2007 Q1 and 2008 Q1, optimal euro share increased and optimal dollar share declined in the investment tranche, in line with increasing euro returns and decreasing dollar returns in that period; trend reversed after 2008.
- Estimated currency-specific fixed terms (c_i) (from footnote description):
  - c_EUR = 0.1814
  - c_JPY = 0.0070
  - c_GBP = 0.0590
  - c_USD = 0.2439
  - Authors note dollar and euro enjoy greater privilege compared with yen and pound; model explains most of observed euro-dollar share difference though residual c_i differences remain.

*Source: IMF working paper content (3.4 Evidence from a reduced-form regression; related tables and figures described).*

### 6.2    The  size  of  the  liquidity  tranche  included  in  the  reduced-

### 6.2 The size of the liquidity tranche included in the reduced-form analysis

### Hypotheses tested
- Hypothesis 1: The larger the relative liquidity tranche size, the larger the effect of the currency compositions of imports invoicing and short-term external debt on the currency composition of FX reserves.
  - Test: interact the relative size of liquidity tranche with:
    - the currency composition of imports invoicing,
    - the currency composition of short-term external debt,
    - the weighted average of the currency composition in imports invoicing and short-term external debt,
    and include these interaction terms as control variables in the regressions in Section 2.5.
  - Result: the sign of the interaction terms are positive and statistically significant (Columns (1), (3), and (4) in Table 3).

- Hypothesis 2: The larger the relative size of liquidity tranche, the smaller the effect of exchange rate returns on the currency composition of FX reserves.
  - Test: interact the relative size of liquidity tranche with exchange rate returns.
  - Result: the sign of the interaction terms are negative and statistically significant (Columns (1)-(4) in Table 3).

- Robustness: The coefficients of the interaction terms in Table 3 indicate that the estimated relative size of liquidity tranche and the estimates of the model’s parameters are robust.

### Key regression results (Table 3)
- Columns reported: (1), (2), (3), (4)
- Coefficients, significance, and t-statistics (t-statistics in parentheses; significance: ∗ p <0.10, ∗∗ p <0.05, ∗∗∗ p <0.01):

  - imports invoicing: 0.631 ∗∗∗ (5.06)
  - ST external debt denomination: 0.383 ∗∗∗ (7.23)
  - imports invoicing and ST external debt denomination:
    - 0.500 ∗∗∗ (8.26)
    - 0.500 ∗∗∗ (8.24)
  - average return:
    - Column (1): 658.0 ∗∗∗ (3.00)
    - Column (2): 455.8 ∗∗ (2.57)
    - Column (3): 551.0 ∗∗∗ (3.05)
    - Column (4): 590.2 ∗∗∗ (2.83)
  - return volatility:
    - Column (1): -1.547 (-0.62)
    - Column (2): 2.340 (1.15)
    - Column (3): 1.340 (0.65)
    - Column (4): 2.066 (0.73)
  - government bond spread (reported in one column): 4.384 (0.38)

  - Interaction terms with relative liquidity size:
    - imports invoicing × rel liquidity size: 0.0913 ∗ (1.74)
    - ST external debt denomination × rel liquidity size: 0.0307 (0.67)
    - imports and ST external debt × rel liquidity size:
      - 0.0726 ∗ (1.77)
      - 0.0709 ∗ (1.71)
    - average return × rel liquidity size:
      - Column (1): -717.1 ∗∗∗ (-2.82)
      - Column (2): -416.4 ∗∗ (-2.03)
      - Column (3): -566.4 ∗∗∗ (-2.71)
      - Column (4): -614.6 ∗∗ (-2.50)

  - Constant terms:
    - Column (1): 0.0883 ∗∗∗ (2.63)
    - Column (2): 0.151 ∗∗∗ (11.33)
    - Column (3): 0.105 ∗∗∗ (5.92)
    - Column (4): 0.101 ∗∗∗ (4.84)

  - Sample size and fit:
    - N: 144 (in each column)
    - within R-square:
      - Column (1): 0.3024
      - Column (2): 0.5544
      - Column (3): 0.5254
      - Column (4): 0.5259

### The RMB as reserve currency — liquidity and convertibility considerations
- When central banks decide whether to include a new currency in FX reserves, two important factors are:
  - the liquidity of that currency issuer’s government bond market, and
  - the ease of convertibility of that currency.

- Measurement approach used:
  - Liquidity of government bond markets: measured by the bid-ask spread of government bonds (mean over time of the daily bid-ask spread, using the latest one-year sample up to each quarter).
  - Ease of convertibility (transaction cost in the forex market): measured by the bid-ask spread of each reserve currency versus the domestic currency (mean over time of the daily bid-ask spread of EUR, JPY, GBP, USD, and RMB (onshore market) versus the domestic currency of China, Saudi Arabia, Russia, India, Brazil, averaged for these five large emerging market economies, using the latest one-year sample up to each quarter).

- Empirical observations (based on Figures referenced):
  - China’s government bonds are less liquid than other reserve currency issuers’ government bonds.
  - The liquidity of China’s government bonds has improved since the first half of 2018.
  - The transaction cost of RMB (onshore) is above those of other reserve currencies in most times.

*Source: wpiea2019052 — section 6.2 and beginning of section 7 (excerpt).*

### 7.2    The RMB as reserve currency under different scenarios

### 7.2    The RMB as reserve currency under different scenarios

### Expected baseline and approach
- It is expected that China will gradually enhance capital account convertibility, and the share of RMB in central banks’ FX reserves will further increase from the current level of around 2 percent.
- Over 60 central banks or monetary authorities hold RMB reserves (Yi, 2018).
- Method: for each country in the sample excluding China, the RMB’s expected return, variance and covariances with EUR, JPY, GBP, USD are set to be the average of SDR currencies’ corresponding parameters. The currency-specific fixed term of RMB in equation (2) is set to be the average of SDR currencies’ estimated fix terms.
- The analysis is performed for the last period in the sample, 2014 Q4, using equation (2) with Ωt, mt, Tt, Dt augmented to include RMB terms.

### Scenario 1 — Larger share of RMB in imports invoicing and short-term external debt
- Mechanism: an increase in RMB’s share in imports invoicing and short-term external debt increases RMB’s share in the liquidity tranche, thus raising RMB’s share in aggregate FX reserves.
- Quantitative scenario results (Figure 12):
  - If the share of RMB in imports invoicing and short-term external debt are 50 percent, then:
    - RMB’s share in FX reserves of developing countries including China would be around 15 percent.
    - RMB’s share in FX reserves of developing countries excluding China would be around 35 percent.

### Scenario 2 — RMB as a reference currency (stronger co-movement)
- Definition: a reference currency exhibits a high degree of co-movement with the domestic currencies of reserve holders.
- Approach: for each sample country excluding China, the variance of RMB’s exchange rate return is adjusted (and covariances with EUR, JPY, GBP, USD adjusted accordingly, holding correlation parameters the same). A smaller variance implies a more prominent reference-currency status.
- Quantitative exploration (Figure 13):
  - Estimated share of RMB in FX reserves is mapped versus:
    - the return of RMB varying between 0.1×10^−5 and 10×10^−5, and
    - the average variance of RMB among sample countries excluding China varying between 1.7×10^−5 and 24×10^−5.
  - Finding: a more prominent status as reference currency could lead to higher RMB share in FX reserves; the magnitude of the effect is more significant when RMB has a relatively higher return.

### Model framework and key empirical findings used in scenarios
- Central banks manage reserves in two or more tranches: a liquidity tranche (to satisfy FX liquidity needs) and an investment tranche (to pursue higher returns subject to risk constraints).
- The exchange rate risk and returns of RMB affect its share in aggregate FX reserves through its share in the investment tranche.
- From estimates in Section 6 and the model:
  - The investment tranche has a returns objective; the liquidity tranche does not.
  - The larger the relative size of the liquidity tranche, the more important the effect of imports invoicing and short-term external debt, and the smaller the effect of reserve currencies’ returns on currency composition of FX reserves.
  - Choice of numeraire is relevant for the investment tranche; currency composition of liquidity tranche is determined by currency composition of FX liquidity needs.
  - There is a tendency for central banks to use domestic currency as numeraire to measure the worth of reserve assets (assumption used in estimation).
- Temporal pattern: compared to 2006 and 2007 there has been a significant increase in absolute and relative sizes of the investment tranche in emerging and developing countries’ FX reserves. After 2008 the absolute size of the estimated investment tranche has been stable.

### Relevant numeric benchmarks and estimates cited
- Current level of RMB in central banks’ FX reserves: around 2 percent.
- Over 60 central banks or monetary authorities hold RMB reserves.
- Scenario where RMB share in imports invoicing and short-term external debt = 50 percent:
  - RMB share in FX reserves (developing countries including China): around 15 percent.
  - RMB share in FX reserves (developing countries excluding China): around 35 percent.
- Return ranges used in Figure 13: 0.1×10^−5 to 10×10^−5.
- Variance ranges used in Figure 13: 1.7×10^−5 to 24×10^−5.
- Average returns among sample countries including China (estimated using 2007 Q4 to 2014 Q4):
  - EUR: 7.18×10^−5
  - JPY: 5.44×10^−5
  - GBP: 9.26×10^−5
  - USD: 12.54×10^−5
- Average variances among sample countries including China (estimated using 2007 Q4 to 2014 Q4):
  - EUR: 5.23×10^−5
  - JPY: 40.71×10^−5
  - GBP: 5.98×10^−5
  - USD: 4.97×10^−5

*Source: wpiea2019052 - 7.2    The RMB as reserve currency under different scenarios.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019052.pdf_