## _wp07293

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### I. Introduction
- Objective: test econometrically whether portfolio rebalancing is the dominant dynamic allocation strategy using IMF aggregate COFER data for 1999–2005 (published on IMF website at end-March 2007; study uses actual data through 2005).
- Motivation:
  - Rising interest in currency composition of foreign exchange reserves (COFER) reflects rapid reserve growth and concerns about large, abrupt shifts in reserve currency composition possibly pressuring exchange rates.
  - Two dynamic strategies (Truman and Wong, 2006):
    - Market trend strategy: sells a currency when it depreciates and buys when it appreciates (trend-enhancing).
    - Portfolio rebalancing (stabilizing diversification): buys a currency when it depreciates and sells when it appreciates (offsets currency movements).

### II. Portfolio rebalancing — theory and implications
- Purpose:
  - Return portfolio toward originally chosen optimal allocation when differing returns cause allocations to drift.
- Performance conditions:
  - Rebalancing performs best in volatile, mean-reverting, relatively trendless markets; underperforms in long trending markets.
- Practical features:
  - Implementation guided by rebalancing frequency and deviation thresholds; costs and benefits influence degree/speed of restoration.
- Empirical implication (qualitative):
  - If a currency appreciates (Δds>0) and rebalancing is implemented:
    - Δdval>0 (valuation effect) but Δdqty<0 (dollars sold) → observed fall in dollar’s real share.
  - If a currency depreciates (Δds<0) and rebalancing is implemented:
    - Δdval<0 but Δdqty>0 (dollars bought) → observed rise in dollar’s real share.

### III. Empirical model and data
- Decomposition (SDR numeraire):
  - Δd = Δdqty + Δdval, where Δdqty arises from quantity changes (ΔD, ΔE, ΔP, ΔY, ΔF) and Δdval from valuation/exchange rate changes (Δds, Δes, Δps, Δys, Δfs).
- Key equations:
  - Eqn 1: d = D*ds/(D*ds + E*es + P*ps + Y*ys + F*fs) = D*ds/(T)
  - Eqn 2: total differentiation yields decomposition into Δdqty and Δdval.
  - Eqn 3 (price changes zero): Δdqty = (1/T)*{(1-d)*ds*ΔD – d*[es*ΔE + ps*ΔP + ys*ΔY + fs*ΔF]}
  - Eqn 4 (quantity changes zero): Δdval = (1/T)*{(1-d)*Δds*D – d*[E*Δes + P*Δps + Y*Δys + F*Δfs]}
- Empirical regressions (changes in percentage points of real shares regressed on percentage changes in SDR exchange rates):
  - Δdqty = c1 + c2*Δds + c3*Δes + c4*Δps + c5*Δys + c6*Δfs + e1
  - Δeqty = c7 + c8*Δds + c9*Δes + c10*Δps + c11*Δys + c12*Δfs + e2
  - Δpqty = c13 + c14*Δds + c15*Δes + c16*Δps + c16*Δys + c17*Δfs + e3
  - Δyqty = c18 + c19*Δds + c20*Δes + c21*Δps + c22*Δys + c23*Δfs + e4
  - Δsqty = c24 + c25*Δds + c26*Δes + c27*Δps + c28*Δys + c29*Δfs + e5
- Expected coefficient signs under rebalancing:
  - Own-currency exchange rate coefficient negative: c2<0, c9<0, c16<0, c22<0, c29<0.
  - Other-currency exchange rate coefficients positive.
- Data:
  - COFER quarterly data for 1999Q1–2005Q4.
  - Panel breakdowns referenced: Panel 1 (millions of SDRs); Panel 2 (excludes “Other Currencies”); Panel 3 (real COFER data with exchange rates fixed at 1999Q1 levels); Panel 4 (end-period SDR exchange rates).
  - Dollar share evolution:
    - ~72–73 percent in 1999Q1–2002Q1;
    - 70 percent in 2002Q2;
    - 68 percent in 2005Q4.
  - Sub-period movements:
    - 1999Q1–2002Q1: dollar appreciates by 9 percent and its real share falls from 72 percent to 69 percent (consistent with rebalancing).
    - 2002Q2–2005Q4: dollar depreciates by 7 percent and its real share rises from 69 percent to 70 percent (partial offset).
    - Whole period 1999Q1–2005Q4: dollar depreciates by 5 percent while its real share falls from 72 percent to 70 percent.
- Estimation method:
  - Seemingly Unrelated Regressions (SUR) to account for contemporaneous error correlations since Δdqty + Δeqty + Δpqty + Δyqty + Δsqty = 0.
  - Sum of coefficients of each exchange rate change across equations constrained to zero (e.g., c2 + c8 + c14 + c19 + c25 = 0).
  - Swiss franc equation exchange rate coefficients set as residuals (negative sum of others).
  - Dummy variable for addition of new COFER reporters in 2003Q4.
  - Autoregressive (AR) terms added to certain equations to remove autocorrelation.

### IV. Econometric results (1999Q1–2005Q4)
- Regression setup:
  - Dependent variables: changes in real shares (percentage points).
  - Explanatory variables: percentage changes in SDR exchange rates; estimated coefficients measured in percentage points.
  - Observations: 27 (quarters) per equation.
- Key statistically significant exchange-rate impacts (probability value < 0.05):
  - SDR/dollar → dollar real share: -0.269 (standard error 0.088; probability value 0.003).
  - SDR/dollar → euro real share: 0.243 (standard error 0.079; probability value 0.003).
  - SDR/pound → Japanese yen real share: 0.054 (standard error 0.015; probability value 0.001).
  - SDR/pound → Swiss franc real share: 0.018 (standard error 0.066; probability value 0.003).
  - SDR/swiss franc → Japanese yen real share: 0.025 (standard error 0.012; probability value 0.048).
  - SDR/swiss franc → Swiss franc real share: 0.010 (standard error 0.056; probability value 0.004).
- Interpretation of major coefficients:
  - Dollar appreciation associated with sales of dollars: decline of 0.27 percentage points in dollar’s real share per one percentage point appreciation.
  - Offset largely via purchases of euros: increase of 0.24 percentage points in euro’s real share per one percentage point dollar appreciation → suggests rebalancing primarily via dollar–euro switches.
  - Yen purchases (0.03 points in three-currency example) help close gap but not always statistically significant (probability value 0.179 for some estimates).
  - Pound and Swiss franc impacts are smaller, reflecting lesser reserve importance; some coefficients significant but lacking clear offsetting sales.
  - One anomaly: franc appreciation associated with purchases of the franc (increase of 0.01 point), contrary to rebalancing expectation.
  - Many other exchange rate coefficients insignificant though many have expected signs.
- Constants and dynamics:
  - Dummy for 2003Q4 new reporters is highly significant.
  - AR terms mostly significant where included.
  - Constants:
    - Euro equation constant: 0.30 percentage points (significant) → trend toward euro.
    - Yen equation constant: -0.17 percentage points (significant) → trend away from yen.
    - Dollar constant: -0.09 (not statistically significant; probability value 0.216).
- Goodness of fit (R-Squared by equation):
  - Dollar: 0.615
  - Euro: 0.714
  - Japanese yen: 0.656
  - Pound sterling: 0.373
  - Swiss franc: 0.516

### V. Conclusions — substantive findings and implications
- Main finding:
  - Econometric results suggest portfolio rebalancing is likely the dominant dynamic allocation strategy in reserve management for 1999–2005.
  - Reserve managers appear to purchase depreciating currencies and sell appreciating currencies, consistent with stabilizing diversification.
- Implications:
  - Rebalancing transactions tend to offset market trend movements in currency exchange rates, reducing the likelihood that reserve diversifications create pressure and disrupt exchange markets.
  - Rebalancing helps explain relative stability of currency shares over long periods:
    - Largest one-quarter shift observed: two percentage point decline in dollar’s share in 2002Q2; only one such shift for dollar in period.
  - Reserve portfolio revisions appear implemented gradually or determinants of optimal portfolios did not change abruptly during study period.
- Specific empirical insights:
  - Rebalancing dominated by switches between the dollar and the euro (coefficients for dollar/euro switches in response to dollar exchange rate changes considerably larger than those among minor currencies).
  - Rebalancing appears demarcated:
    - Valuation changes in major currencies → switches primarily between major currencies.
    - Valuation changes in minor currencies → switches primarily among minor currencies.
  - Trend shifts independent of exchange rate effects:
    - Trend into euros at expense of yen (positive euro constant, negative yen constant); dollar constant negative but statistically insignificant.
- Caveat:
  - Longer-term analysis may be affected by changes in desired optimal allocations; rebalancing over short intervals may be obscured by gradual changes in optimal shares over longer intervals.

### Appendix I — Empirical model (three-currency simplification)
- Model assumptions: only 3 currencies — dollar, euro, yen; d + e + y = 1; T = total reserves in SDRs.
- Three-equation specification:
  - Δdqty = c1 + c2*Δds + c3*Δes + c5*Δys + e1
  - Δeqty = c7 + c8*Δds + c9*Δes + c11*Δys + e2
  - Δyqty = c18 + c19*Δds + c20*Δes + c22*Δys + e4
- Interpretation example (empirical coefficients):
  - c2 = -0.27
  - c8 = 0.24
  - c19 = 0.03
  - A one percentage point appreciation in the dollar’s exchange rate implies:
    - 0.27 percentage point decline in the dollar’s real share,
    - 0.24 percentage point increase in the euro’s real share,
    - 0.03 percentage point increase in the yen’s real share.
- Analytical identities (Eqns 6–8):
  - Δdqty = (1/T)*{(1-d)*ds*ΔD – d*[es*ΔE + ys*ΔY ]}  (Eqn 6)
  - Δeqty = (1/T)*{(1-e)*es*ΔE – e*[ds*ΔD + ys*ΔY ]}  (Eqn 7)
  - Δyqty = (1/T)*{(1-y)*ys*ΔY – y*[ds*ΔD + es*ΔE ]}  (Eqn 8)
- Exchange constraint when selling dollars for euros and yen:
  - - ds*ΔD = es*ΔE + ys*ΔY    (Eqn 9)
- Numerical substitution example (proportions of total reserves):
  - - 0.0027 = (ds*ΔD)/T
  - 0.0024 = (es*ΔE)/T
  - 0.0003 = (ys*ΔY)/T
- Conversion to own-currency unit changes:
  - ΔD = -0.0027*T/ds
  - ΔE = 0.0024*T/es
  - ΔY = 0.0003*T/ys

*Source: _wp07293 (COFER data and regressions for 1999Q1–2005Q4).*

### 1.  COFER Data and Real Shares 1999–2005 .........................................................................14

### 1.  COFER Data and Real Shares 1999–2005 .........................................................................14

### COFER Data and Real Shares 1999–2005
- Section title as presented: "COFER Data and Real Shares 1999–2005"
- Page reference shown: 14

### 2.  Regression Results 1999Q1–2005Q4 .................................................................................17

### Regression Results 1999Q1–2005Q4
- Section title as presented: "Regression Results 1999Q1–2005Q4"
- Page reference shown: 17

### Appendix

### 1. Empirical Model
- Appendix title as presented: "Empirical Model"
- Page reference region indicated beginning at page 21

*Source: _wp07293 - 1.  COFER Data and Real Shares 1999–2005 .........................................................................14*

### Bibliography ...........................................................................................................

### _wp07293 - Bibliography

### I. Introduction
- Rising interest in currency composition of foreign exchange reserves (COFER) reflects rapid reserve growth and concerns about large, abrupt shifts in reserve currency composition possibly pressuring exchange rates.
- Two dynamic investment/diversification strategies (Truman and Wong, 2006):
  - Market trend strategy: sells a currency when it depreciates and buys when it appreciates (trend-enhancing).
  - Portfolio rebalancing (stabilizing diversification): buys a currency when it depreciates and sells when it appreciates (offsets currency movements).
- Paper objective: test econometrically whether portfolio rebalancing is the dominant strategy using IMF aggregate COFER data for 1999–2005 (published on IMF website at end-March 2007; study uses actual data through 2005).

### II. Portfolio Rebalancing — theory and implications
- Purpose: return portfolio toward originally chosen optimal allocation when assets’ differing returns cause allocations to drift.
- Rebalancing performs best in volatile, mean-reverting, relatively trendless markets; underperforms in long trending markets.
- Example: a portfolio optimally allocated one-third each to dollars, euros, and yen; dollar depreciates while euro and yen appreciate → rebalancing buys dollars and sells euro/yen to restore allocation.
- Critics: intuitive objection (e.g., “cutting the flowers and watering the weeds”), but defenders note buy-low/sell-high contrarian benefit.
- Practical implementation: guidelines on rebalancing frequency and deviation thresholds; costs and benefits influence degree/speed of restoration.

### III. Empirical model and data
- Disaggregation of change in a currency’s share (using SDR as numeraire):
  - Δd = Δdqty + Δdval, where Δdqty arises from quantity changes (ΔD, ΔE, ΔP, ΔY, ΔF) and Δdval from valuation/exchange rate changes (Δds, Δes, Δps, Δys, Δfs).
  - Eqn 1: d = D*ds/(D*ds + E*es + P*ps + Y*ys + F*fs) = D*ds/(T)
  - Eqn 2: total differentiation yields Δd expression decomposed into quantity and valuation components.
  - Eqn 3 (price changes zero): Δdqty = (1/T)*{(1-d)*ds*ΔD – d*[es*ΔE + ps*ΔP + ys*ΔY + fs*ΔF]}
  - Eqn 4 (quantity changes zero): Δdval = (1/T)*{(1-d)*Δds*D – d*[E*Δes + P*Δps + Y*Δys + F*Δfs]}
- Rebalancing implications for data:
  - If dollar appreciates (Δds>0) and rebalancing implemented: Δdval>0 but Δdqty<0 (dollars sold), so data should show fall in dollar’s real share when dollar appreciates.
  - If dollar depreciates (Δds<0): Δdval<0 and under rebalancing Δdqty>0 (dollars bought), so data should show rise in dollar’s real share when dollar depreciates.
- Empirical regressions (changes in percentage points of real shares regressed on percentage changes in SDR exchange rates):
  - Δdqty = c1 + c2*Δds + c3*Δes + c4*Δps + c5*Δys + c6*Δfs + e1
  - Δeqty = c7 + c8*Δds + c9*Δes + c10*Δps + c11*Δys + c12*Δfs + e2
  - Δpqty = c13 + c14*Δds + c15*Δes + c16*Δps + c16*Δys + c17*Δfs + e3
  - Δyqty = c18 + c19*Δds + c20*Δes + c21*Δps + c22*Δys + c23*Δfs + e4
  - Δsqty = c24 + c25*Δds + c26*Δes + c27*Δps + c28*Δys + c29*Δfs + e5
- Expected signs under rebalancing:
  - Own-currency exchange rate coefficient negative: c2<0, c9<0, c16<0, c22<0, c29<0.
  - Other-currency exchange rate coefficients positive.
- Data:
  - COFER quarterly data for 1999Q1–2005Q4 (Panel 1 in Table 1 in millions of SDRs; Panel 2 excludes “Other Currencies”; Panel 3 real COFER data with exchange rates fixed at 1999Q1 levels; Panel 4 end-period SDR exchange rates).
  - Dollar share ~72–73 percent in 1999Q1–2002Q1; 70 percent in 2002Q2; 68 percent in 2005Q4.
  - Sub-period behavior:
    - 1999Q1–2002Q1: dollar appreciates by 9 percent and its real share falls from 72 percent to 69 percent (consistent with rebalancing).
    - 2002Q2–2005Q4: dollar depreciates by 7 percent and its real share rises from 69 percent to 70 percent (partial offset).
    - Whole period 1999Q1–2005Q4: dollar depreciates by 5 percent while its real share falls from 72 percent to 70 percent (rebalancing indications weaker over full period).
- Estimation method:
  - Seemingly Unrelated Regressions (SUR) used because Δdqty + Δeqty + Δpqty + Δyqty + Δsqty = 0 → contemporaneous correlations among errors.
  - Sum of coefficients of each exchange rate change across equations constrained to zero (e.g., c2 + c8 + c14 + c19 + c25 = 0).
  - Swiss franc equation exchange rate coefficients set as residuals (negative sum of others).
  - Dummy variable included for addition of new COFER reporters in 2003Q4.
  - Autoregressive (AR) terms added to certain equations to remove autocorrelation.

### IV. Econometric results (1999Q1–2005Q4)
- Regression specification notes:
  - Dependent variables: changes in real shares (percentage points).
  - Explanatory variables: percentage changes in SDR exchange rates; estimated coefficients measured in percentage points.
- Key statistically significant exchange rate impacts (probability value < 0.05):
  - SDR/dollar → dollar real share: -0.269 (standard error 0.088; probability value 0.003).
  - SDR/dollar → euro real share: 0.243 (standard error 0.079; probability value 0.003).
  - SDR/pound → Japanese yen real share: 0.054 (standard error 0.015; probability value 0.001).
  - SDR/pound → Swiss franc real share: 0.018 (standard error 0.066; probability value 0.003).
  - SDR/swiss franc → Japanese yen real share: 0.025 (standard error 0.012; probability value 0.048).
  - SDR/swiss franc → Swiss franc real share: 0.010 (standard error 0.056; probability value 0.004).
- Significance and interpretation:
  - Dollar appreciation leads to sales of dollars: decline of 0.27 percentage points in dollar’s real share; near-equal offset by purchases of euros: increase of 0.24 percentage points in euro’s real share → suggests rebalancing primarily via dollar–euro switches.
  - Yen purchases (0.04 points) help close gap between dollar sales and euro purchases but coefficient not statistically significant (probability value 0.179).
  - Pound and Swiss franc results: some significant positive impacts on minor currencies but lacking statistically significant offsetting sales; coefficients smaller reflecting lesser reserve importance.
  - One anomaly: franc appreciation associated with purchases of the franc (increase of 0.01 point) — contrary to rebalancing expectation.
  - Many other exchange rate coefficients insignificant, though many have expected signs.
- Other regression features:
  - Dummy variable for 2003Q4 new reporters is highly significant.
  - AR terms mostly significant where included.
  - Constants:
    - Euro equation constant: 0.30 percentage points (significant) → trend toward euro.
    - Yen equation constant: -0.17 percentage points (significant) → trend away from yen.
    - Dollar constant: -0.09 (large negative) but statistically insignificant (probability value 0.216).
- R-Squared by equation (Table 2):
  - Dollar: 0.615
  - Euro: 0.714
  - Japanese yen: 0.656
  - Pound sterling: 0.373
  - Swiss franc: 0.516
- Total observations: 27 (quarters) per equation.

### V. Conclusions — substantive findings and implications
- Main finding: econometric results suggest portfolio rebalancing is likely the dominant dynamic allocation strategy in reserve management for 1999–2005.
  - Reserve managers appear to purchase depreciating currencies and sell appreciating currencies, consistent with stabilizing diversification.
  - Rebalancing transactions would tend to offset market trend movements in currency exchange rates, reducing the likelihood that reserve diversifications create pressure and disrupt exchange markets.
- Rebalancing helps explain relative stability of currency shares over long periods:
  - Largest one-quarter shift observed in study period: two percentage point decline in dollar’s share in 2002Q2; only one such shift for dollar in period.
  - Reserve portfolio revisions appear to be implemented gradually or determinants of optimal portfolios did not change abruptly during study period.
- Specific empirical insights:
  - Rebalancing dominated by switches between the dollar and the euro (coefficients for dollar/euro switches in response to dollar exchange rate changes considerably larger than those among minor currencies).
  - Rebalancing decisions appear demarcated:
    - Valuation changes in major currencies → switches primarily between major currencies.
    - Valuation changes in minor currencies → switches primarily among minor currencies.
  - Trend shifts independent of exchange rate effects:
    - Trend into euros at expense of yen (positive euro constant, negative yen constant); dollar constant negative but statistically insignificant.
- Caveat: longer-term analysis may be affected by changes in desired optimal allocations; rebalancing over short intervals may be obscured by gradual changes in optimal shares over longer intervals.

*Source: IMF COFER Database; International Financial Statistics; content extracted from _wp07293 - Bibliography (COFER data and regressions for 1999Q1–2005Q4).*

### APPENDIX I

### APPENDIX I

### Empirical model (three-currency simplification)
- The empirical model assumes only 3 currencies: dollar, euro, and the yen.
- The model consists of three equations (dollar, euro, yen):
  - Δdqty = c1 + c2*Δds + c3*Δes + c5*Δys + e1 (dollar equation)
  - Δeqty = c7 + c8*Δds + c9*Δes + c11*Δys + e2 (euro equation)
  - Δyqty = c18 + c19*Δds + c20*Δes + c22*Δys + e4 (yen equation)

### Interpretation of estimated exchange-rate coefficients
- Example: estimated coefficients for the dollar exchange rate in the three equations:
  - c2 = -0.27
  - c8 = 0.24
  - c19 = 0.03
- Interpretation: a one percentage point appreciation in the dollar’s exchange rate implies:
  - a 0.27 percentage point decline in the dollar’s real share,
  - a 0.24 percentage point increase in the euro’s real share,
  - a 0.03 percentage point increase in the yen’s real share.
- These coefficient signs suggest dollars are sold in exchange for euros and yen.

### Changes in real shares (analytical structure)
- Definitions and identities:
  - d = share of the dollar in total reserves
  - e = share of the euro in total reserves
  - y = share of the yen in total reserves
  - T = total reserves in SDRs
  - d + e + y = 1 (three-currency case)
- Equations for changes in real shares (Eqns 6–8):
  - Δdqty = (1/T)*{(1-d)*ds*ΔD – d*[es*ΔE + ys*ΔY ]} (change in dollar’s real share) Eqn 6
  - Δeqty = (1/T)*{(1-e)*es*ΔE – e*[ds*ΔD + ys*ΔY ]} (change in euro’s real share) Eqn 7
  - Δyqty = (1/T)*{(1-y)*ys*ΔY – y*[ds*ΔD + es*ΔE ]} (change in yen’s real share) Eqn 8

### Constraint linking currency exchanges
- The exchange constraint implied by selling dollars in exchange for euros and yen:
  - - ds*ΔD = es*ΔE + ys*ΔY    Eqn 9

### Numerical substitution and solution (example)
- Substitute empirical coefficients (as proportions of total reserves) into Eqns 6–8, bringing T to the other side:
  - - 0.0027*T = (1-d)*ds*ΔD – d*[es*ΔE + ys*ΔY ]  (change in dollar’s real share)
  - 0.0024*T = (1-e)*es*ΔE – e*[ds*ΔD + ys*ΔY ]   (change in euro’s real share)
  - 0.0003*T = (1-y)*ys*ΔY – y*[ds*ΔD + es*ΔE ]   (change in yen’s real share)
- One solution (by substitution using Eqn 9) is:
  - ds*ΔD = - 0.0027*T
  - es*ΔE = 0.0024*T
  - ys*ΔY = 0.0003*T
- Verification steps shown in text:
  - Substituting ys*ΔY = - ds*ΔD – es*ΔE into dollar and euro equations yields ds*ΔD = - 0.0027*T and es*ΔE = 0.0024*T.
  - Substituting these into the constraint yields ys*ΔY = 0.0027*T – 0.0024*T = 0.0003*T.
- Moving T over gives the coefficients as fractions of total reserves:
  - - 0.0027 = c1 = (ds*ΔD)/T
  - 0.0024 = c8 = (es*ΔE)/T
  - 0.0003 = c19 = (ys*ΔY)/T

### Interpretation in amounts and conversion to own-currency units
- The exchange-rate coefficients for a given exchange rate (e.g., the dollar) tell the amounts of dollars, euros, and yen, respectively, in percent of the total SDR reserves, that are bought or sold given a one percentage point change in that exchange rate.
- Conversion to own units (divide by the SDR exchange rates):
  - ΔD = -0.0027*T/ds
  - ΔE = 0.0024*T/es
  - ΔY = 0.0003*T/ys

*Source: APPENDIX I*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07293.pdf_