## _071316 - INTRODUCTION ## Source details **Canonical URL:** [_071316 - INTRODUCTION](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/np/pp/eng/2016/_071316.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/np/pp/eng/2016/_071316.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/np/pp/eng/2016/_071316.pdf.json) --- ### Introduction: role and principles - Currency amounts are the number of units of each currency in the SDR basket and are used to determine the daily value of the SDR and the SDR interest rate (SDRi). - The value of the SDR (in U.S. dollars) is the sum of these currency amounts, valued at daily exchange rates of the currencies against the U.S. dollar, rounded to six significant digits. (See Rule O-1 and O-2(a), and Decision No. 15891-(15/109), adopted November 30, 2015.) - Currency amounts are determined on the last business day before the new SDR basket becomes effective (transition date) and remain fixed over the SDR valuation period; illustrative currency amounts are published in the lead up to the transition date. - Core principle preserved: the value of the SDR in U.S. dollars (and thus in all other currencies) must be the same on the transition day under the old and new baskets (equality condition). No change to this principle is proposed. - The existing methodology dates to staff papers in the 1970s and ‘80s and aims to ensure continuity and stability in the SDR value during transitions. ### Current practice of determining rounded currency amounts: process and issues - Key requirements: - Equality condition: the sum of the new currency amounts in U.S. dollars must produce the exact same SDR value (in U.S. dollars) as that prevailing on the transition day. - Weights correspondence: using average exchange rates for the three-month period ending on the transition date, the share of each currency in the SDR must correspond to its weight as determined by the IMF (Decision allows initial weights implied by currency amounts to deviate from Board-determined weights by up to half a percentage point (the tolerance limit)). - Uniform significant digits requirement: final amounts of all currencies should have the same number of significant digits. - Currency amounts should have between two and four significant digits, with a preference for lower significant digits. - If multiple qualifying sets exist, the set with the smallest average deviation of rounded currency amounts from unrounded ones should be selected (practically interpreted as the smallest root mean squared deviation). - Current methodology (summary of steps): - Step 1 – Calculating unrounded currency amounts Ci using the formula: - Ci = (Wi BEXi) / (Σ Wi BEXi n i=1) * TEXi / USD/SDR - where Wi is the Board-determined weight, BEXi is the three-month average exchange rate (base exchange rate) against the U.S. dollar, TEXi is the transition date exchange rate, and USD/SDR is the U.S. dollar per SDR on the transition date, expressed in six significant digits. - Step 2 – Rounding: truncate unrounded currency amounts at 2 significant digits, generate alternative amounts by adding/subtracting from the last digit (staff has iterated by +/-9 units since 2005), combine into potential baskets, test equality condition and tolerance limit; if no solution, repeat for 3 and then 4 significant digits. If multiple solutions exist, choose the set with minimum root mean squared deviation. - Practical drawbacks and numeric implications: - The iterative +/-9 approach across a five-currency basket can involve up to 7.5 million potential combinations (2.5 million combinations per significant digit), which is not feasible in one spreadsheet file. - The tolerance limit of half a percentage point can be relatively large given recent Board-adopted weights reported with 1 or 2 decimal places; the 2010 currency amounts implied deviations close to the full tolerance for the U.S. dollar and the euro. - Some rounding details (e.g., iteration range) are not explicitly specified in the Decision, reducing replicability. - Table 1 — Adopted and Implied Currency Weights in the 2010 SDR Review (selected lines): - U.S. Dollar: 2010 Review Weights (Percent) = 41.9; Adopted Currency Amounts (12/30/2010) = 0.660; Implied Weights (Percent) = 42.39; Implied Weights Deviation from Board-approved (pct point) = 0.49 - Euro: 2010 Review Weights (Percent) = 37.4; Adopted Currency Amounts (12/30/2010) = 0.423; Implied Weights (Percent) = 36.92; Implied Weights Deviation from Board-approved (pct point) = -0.48 - Japanese Yen: 2010 Review Weights (Percent) = 9.4; Adopted Currency Amounts (12/30/2010) = 12.1; Implied Weights (Percent) = 9.42; Implied Weights Deviation from Board-approved (pct point) = 0.02 - Pound Sterling: 2010 Review Weights (Percent) = 11.3; Adopted Currency Amounts (12/30/2010) = 0.111; Implied Weights (Percent) = 11.27; Implied Weights Deviation from Board-approved (pct point) = -0.03 ### Staff proposal: simplify rounding to improve transparency and adherence to weights - Objective: simplify Step 2 and make final currency amounts easier to compute and closer to Board-adopted weights while preserving the equality principle. - Preferred approach: round all final currency amounts to five significant digits (uniform across currencies) using the sixth significant digit to determine rounding. - Step 1 (unrounded Ci calculation) remains unchanged and continues to meet Board-adopted weights and the equality condition. - In Step 2, Ci is rounded to five significant digits (based on the sixth significant digit) instead of the iterative truncation/adjustment at 2–4 significant digits. - Operational considerations and low-probability adjustment: - As long as the USD/SDR exchange rate is above 1, a solution would be found with five significant digits because the SDR value is specified in six significant digits (equivalent to five decimals while USD/SDR > 1). - If rounded currency amounts do not produce an SDR value that meets the equality condition, one rounded currency amount would be adjusted to meet the condition. - Staff proposes to adjust the currency amount for the currency with the largest weight in the SDR basket, i.e., the U.S. dollar, to ensure the least impact on relative weights and for operational simplicity. This adjustment would be conceptually similar to adjustments under paragraph 4(c) of the Decision. - If the USD/SDR rate were to fall below 1 (assessed as low likelihood), finding a solution would likely require rounding to six significant digits. - Staff back-testing indicates a low probability of needing an adjustment; the only historical case where the adjustment would have been needed is the 1995 Review (applied to December 31, 1995, rounded Ci would have given 1.48648 USD/SDR vs prevailing 1.48689 USD/SDR; USD currency amount adjustment example provided in staff back-test). - Proposed ancillary changes: - Eliminate the tolerance limit in the Decision because the single-solution rounding approach produces weights very close to Board-adopted weights, rendering the tolerance limit redundant. - Maintain uniformity of significant digits across currencies. - Advantages of the proposed methodology: - Simplicity: single-step rounding to five significant digits is easy to replicate in spreadsheets; staff consultations did not indicate operational issues with the expanded number of significant digits. - Closer adherence to Board-adopted weights: higher number of significant digits produces initial currency weights very close to Board-adopted weights. - Single solution: backing out a final currency amount to meet the equality condition yields a unique final set of currency amounts, eliminating extensive iterations and multiple-solution selection. - Continued continuity and stability: the equality condition is maintained so there is no discontinuity in SDR value across the transition. - Implementation timing and publication: - If approved by the Executive Board, paragraph 5 of the 2015 SDR Valuation Decision would be amended with immediate effect. - The same rounding method will be used to derive illustrative currency amounts to be published to assist users preparing for the changeover to the new SDR valuation on October 1, 2016. - Staff plans to publish a spreadsheet file describing the illustrative currency amount calculations in detail to facilitate user transition. ### Key rule: Determination of currency amounts (Paragraph 5) - The amounts of the currencies under paragraphs 3 and 4 above shall be determined in a manner that will ensure that the value of the special drawing right in terms of currencies on the last working day preceding the five-year period for which the determination is made will be the same under the valuation in effect before and after revision (“same value”), and shall be calculated in accordance with the following guidelines: - (a) The currency amounts calculated for the new basket will be rounded to five significant digits based on the sixth significant digit. If necessary to achieve the same value, an adjustment will be made to the amount of the currency against which the values of the other SDR basket currencies are determined in accordance with Rule O-2. - (b) If the calculations under (a) do not yield the same value in five significant digits, the calculations shall be made by applying the same guidelines but rounding currency amounts to six significant digits based on the seventh significant digit. ### Historical perspective on calculation of Currency Amounts (Box 1) - Stability principle: - The standard basket method of valuation of the SDR was adopted in 1974 to ensure the stability of the SDR in terms of the major currencies under floating exchange rates. - Two stability-preserving features adopted in 1974: - Use of an average exchange rate of each currency relative to the US dollar during a three-month base period ending the last business day before a new basket comes into effect (the transition date) to convert Board-adopted currency weights (agreed weights) into currency amounts (CA). - Requirement that the value of the SDR under a new basket equal the prevailing value on the transition date (equality condition). - Rounding and significant-digits evolution: - 1970s: CA uniformly rounded to two significant digits; met equality principle with strict adherence to agreed weights. - 1980: Reduction in number of basket currencies from 16 to five required changes: - Allowed a deviation within a tolerance limit of half a percentage point from agreed weights. - Increased maximum number of significant digits in CA from two to four, with no uniformity requirement across currencies. - 1985 Decision: introduced uniformity of significant digits across currencies; tolerance limit and rounding guidelines as of 1985 have since remained in effect. - Practical consequence: - Changes in the 1980s increased complexity, introduced a search space and permutation steps above and below truncated CA digits, producing a procedure viewed as complex and final CA non-transparent and difficult to replicate by SDR users. ### Significant digits and their use in SDR valuation (Box 2) - Definition and interpretation: - “The number of significant digits (or significant figures) of a number ‘gives some idea of the precision or reliability of that number.’” - Significant digits differ from number of decimals; counted from the first non-zero digit and include nonzero digits, zeros sandwiched between non-zero digits, and trailing zeros after a decimal; leading zeros are not significant. - Examples from the source: - “42”, “4.2”, and “0.0042” all have two significant digits. - 0.0402 has four decimals but only three significant digits. - The number “3200” can have two or four significant digits depending on measurement precision. - Historical usage in SDR valuation: - Initial SDR value set equal to 0.888671 grams of fine gold, equivalent to one U.S. dollar, expressed in six significant digits. - Since inception of the basket method, currency amounts have been expressed in two to four significant digits. - The value of the USD/SDR under Rule O-2 is rounded to six significant digits. ### Proposed amendments (Annex I: Redlined Version of Proposed Decision) - Proposed amendment text (preserved verbatim in redlined form) to Paragraph 5 of Decision No. 15891-(15/109), adopted November 30, 2015, includes: - (a) The currency amounts calculated for the new basket will be expressed in two rounded to five significant digits based on the sixth significant digit. provided that the deviation of the percentage share of each currency in the value of the special drawing right, resulting from the application of the average exchange rates for July- September, from the percentage weight as determined under paragraphs 3 and 4(c) above is the minimum on average and will not exceed one half percentage point for any currency. If necessary to achieve the same value, an adjustment will be made to the amount of the currency against which the values of the other SDR basket currencies are determined in accordance with Rule O-2. - (b) If the calculations under a solution cannot be obtained by the application of the guidelines set forth in (a) do not yield the same value in five significant digits, above, the calculations shall be made by applying the same guidelines but expressing rounding the amount of each currency amounts in three to six significant digits based on the seventh significant digit, and if no solution is found with three significant digits then the calculation shall be made applying the same guidelines but expressing the amount of each currency in four significant digits. - (c) If more than one solution is found in the calculation at the level of two, three, or four significant digits, the solution that has the smallest average deviation will be employed. *Source: _071316 - INTRODUCTION (July 13, 2016), IMF staff paper on Currency Amounts in the SDR Basket.* ### INTRODUCTION ______________________________________________________________________ 2 ### _071316 - INTRODUCTION ______________________________________________________________________ 2 ### Introduction: role and principles - Currency amounts are the number of units of each currency in the SDR basket and are used to determine the daily value of the SDR and the SDR interest rate (SDRi). - The value of the SDR (in U.S. dollars) is the sum of these currency amounts, valued at daily exchange rates of the currencies against the U.S. dollar, rounded to six significant digits. (See Rule O-1 and O-2(a), and Decision No. 15891-(15/109), adopted November 30, 2015.) - Currency amounts are determined on the last business day before the new SDR basket becomes effective (transition date) and remain fixed over the SDR valuation period; illustrative currency amounts are published in the lead up to the transition date. - Core principle preserved: the value of the SDR in U.S. dollars (and thus in all other currencies) must be the same on the transition day under the old and new baskets (equality condition). No change to this principle is proposed. - The existing methodology dates to staff papers in the 1970s and ‘80s (see Box 1) and aims to ensure continuity and stability in the SDR value during transitions. ### Current practice of determining rounded currency amounts: process and issues - Key requirements: - Equality condition: the sum of the new currency amounts in U.S. dollars must produce the exact same SDR value (in U.S. dollars) as that prevailing on the transition day. - Weights correspondence: using average exchange rates for the three-month period ending on the transition date, the share of each currency in the SDR must correspond to its weight as determined by the IMF (Decision allows initial weights implied by currency amounts to deviate from Board-determined weights by up to half a percentage point (the tolerance limit)). - Uniform significant digits requirement: final amounts of all currencies should have the same number of significant digits. - Currency amounts should have between two and four significant digits, with a preference for lower significant digits. - If multiple qualifying sets exist, the set with the smallest average deviation of rounded currency amounts from unrounded ones should be selected (practically interpreted as the smallest root mean squared deviation). - Current methodology (summary of steps): - Step 1 – Calculating unrounded currency amounts Ci using the formula: - Ci = (Wi BEXi) / (Σ Wi BEXi n i=1) * TEXi / USD/SDR - where Wi is the Board-determined weight, BEXi is the three-month average exchange rate (base exchange rate) against the U.S. dollar, TEXi is the transition date exchange rate, and USD/SDR is the U.S. dollar per SDR on the transition date, expressed in six significant digits. - Step 2 – Rounding: truncate unrounded currency amounts at 2 significant digits, generate alternative amounts by adding/subtracting from the last digit (staff has iterated by +/-9 units since 2005), combine into potential baskets, test equality condition and tolerance limit; if no solution, repeat for 3 and then 4 significant digits. If multiple solutions exist, choose the set with minimum root mean squared deviation. - Practical drawbacks and numeric implications: - The iterative +/-9 approach across a five-currency basket can involve up to 7.5 million potential combinations (2.5 million combinations per significant digit), which is not feasible in one spreadsheet file. - The tolerance limit of half a percentage point can be relatively large given recent Board-adopted weights reported with 1 or 2 decimal places; the 2010 currency amounts implied deviations close to the full tolerance for the U.S. dollar and the euro. - Some rounding details (e.g., iteration range) are not explicitly specified in the Decision, reducing replicability. - Table 1 — Adopted and Implied Currency Weights in the 2010 SDR Review: - Currency / 2010 Review Weights (Percent) / Adopted Currency Amounts (12/30/2010) / Implied Weights (Percent) / Implied Weights Deviation from Board-approved (pct point) - U.S. Dollar / 41.9 / 0.660 / 42.39 / 0.49 - Euro / 37.4 / 0.423 / 36.92 / -0.48 - Japanese Yen / 9.4 / 12.1 / 9.42 / 0.02 - Pound Sterling / 11.3 / 0.111 / 11.27 / -0.03 ### Staff proposal: simplify rounding to improve transparency and adherence to weights - Objective: simplify Step 2 and make final currency amounts easier to compute and closer to Board-adopted weights while preserving the equality principle. - Preferred approach: round all final currency amounts to five significant digits (uniform across currencies) using the sixth significant digit to determine rounding. - Step 1 (unrounded Ci calculation) remains unchanged and continues to meet Board-adopted weights and the equality condition. - In Step 2, Ci is rounded to five significant digits (based on the sixth significant digit) instead of the iterative truncation/adjustment at 2–4 significant digits. - Operational considerations and low-probability adjustment: - As long as the USD/SDR exchange rate is above 1, a solution would be found with five significant digits because the SDR value is specified in six significant digits (equivalent to five decimals while USD/SDR > 1). - If rounded currency amounts do not produce an SDR value that meets the equality condition, one rounded currency amount would be adjusted to meet the condition. - Staff proposes to adjust the currency amount for the currency with the largest weight in the SDR basket, i.e., the U.S. dollar, to ensure the least impact on relative weights and for operational simplicity. This adjustment would be conceptually similar to adjustments under paragraph 4(c) of the Decision. - If the USD/SDR rate were to fall below 1 (assessed as low likelihood), finding a solution would likely require rounding to six significant digits. - Staff back-testing indicates a low probability of needing an adjustment; the only historical case where the adjustment would have been needed is the 1995 Review (applied to December 31, 1995, rounded Ci would have given 1.48648 USD/SDR vs prevailing 1.48689 USD/SDR; USD currency amount adjustment example provided in staff back-test). - Proposed ancillary changes: - Eliminate the tolerance limit in the Decision because the single-solution rounding approach produces weights very close to Board-adopted weights, rendering the tolerance limit redundant. - Maintain uniformity of significant digits across currencies. - Advantages of the proposed methodology: - Simplicity: single-step rounding to five significant digits is easy to replicate in spreadsheets; staff consultations did not indicate operational issues with the expanded number of significant digits. - Closer adherence to Board-adopted weights: higher number of significant digits produces initial currency weights very close to Board-adopted weights (see Table 2 comparisons). - Single solution: backing out a final currency amount to meet the equality condition yields a unique final set of currency amounts, eliminating extensive iterations and multiple-solution selection. - Continued continuity and stability: the equality condition is maintained so there is no discontinuity in SDR value across the transition. - Implementation timing and publication: - If approved by the Executive Board, paragraph 5 of the 2015 SDR Valuation Decision would be amended with immediate effect. - The same rounding method will be used to derive illustrative currency amounts to be published to assist users preparing for the changeover to the new SDR valuation on October 1, 2016. - Staff plans to publish a spreadsheet file describing the illustrative currency amount calculations in detail to facilitate user transition. ### Table 2 (summary note) - Table 2 in the source compares weights implied by currency amounts under the current and proposed methodologies across past SDR Valuation Reviews (1980, 1985, 1990, 1995, 2000 excluded, 2005, 2010, 2015), showing that deviations from Board-adopted weights under the proposed methodology are on the order of magnitude of 0.0000xx (very small) for historical examples and materially closer than under the current methodology. (Detailed numeric lines are in the source table.) *Source: _071316 - INTRODUCTION (July 13, 2016), IMF staff paper on Currency Amounts in the SDR Basket.* ### 5. The amounts of the currencies under paragraphs 3 and 4 above shall be determined in a ### _071316 - 5. The amounts of the currencies under paragraphs 3 and 4 above shall be determined in a ### Key rule: Determination of currency amounts (Paragraph 5) - The amounts of the currencies under paragraphs 3 and 4 above shall be determined in a manner that will ensure that the value of the special drawing right in terms of currencies on the last working day preceding the five-year period for which the determination is made will be the same under the valuation in effect before and after revision (“same value”), and shall be calculated in accordance with the following guidelines: - (a) The currency amounts calculated for the new basket will be rounded to five significant digits based on the sixth significant digit. If necessary to achieve the same value, an adjustment will be made to the amount of the currency against which the values of the other SDR basket currencies are determined in accordance with Rule O-2. - (b) If the calculations under (a) do not yield the same value in five significant digits, the calculations shall be made by applying the same guidelines but rounding currency amounts to six significant digits based on the seventh significant digit. ### Historical perspective on calculation of Currency Amounts (Box 1) - Stability principle: - The standard basket method of valuation of the SDR was adopted in 1974 to ensure the stability of the SDR in terms of the major currencies under floating exchange rates. - Two stability-preserving features adopted in 1974: - Use of an average exchange rate of each currency relative to the US dollar during a three-month base period ending the last business day before a new basket comes into effect (the transition date) to convert Board-adopted currency weights (agreed weights) into currency amounts (CA). - Requirement that the value of the SDR under a new basket equal the prevailing value on the transition date (equality condition). - Rounding and significant-digits evolution: - 1970s: CA uniformly rounded to two significant digits; met equality principle with strict adherence to agreed weights. - 1980: Reduction in number of basket currencies from 16 to five required changes: - Allowed a deviation within a tolerance limit of half a percentage point from agreed weights. - Increased maximum number of significant digits in CA from two to four, with no uniformity requirement across currencies. - 1985 Decision: introduced uniformity of significant digits across currencies; tolerance limit and rounding guidelines as of 1985 have since remained in effect. - Practical consequence: - Changes in the 1980s increased complexity, introduced a search space and permutation steps above and below truncated CA digits, producing a procedure viewed as complex and final CA non-transparent and difficult to replicate by SDR users. ### Significant digits and their use in SDR valuation (Box 2) - Definition and interpretation: - “The number of significant digits (or significant figures) of a number ‘gives some idea of the precision or reliability of that number.’” - Significant digits differ from number of decimals; counted from the first non-zero digit and include nonzero digits, zeros sandwiched between non-zero digits, and trailing zeros after a decimal; leading zeros are not significant. - Examples from the source: - “42”, “4.2”, and “0.0042” all have two significant digits. - 0.0402 has four decimals but only three significant digits. - The number “3200” can have two or four significant digits depending on measurement precision. - Historical usage in SDR valuation: - Initial SDR value set equal to 0.888671 grams of fine gold, equivalent to one U.S. dollar, expressed in six significant digits. - Since inception of the basket method, currency amounts have been expressed in two to four significant digits. - The value of the USD/SDR under Rule O-2 is rounded to six significant digits. ### Proposed amendments (Annex I: Redlined Version of Proposed Decision) - Text proposed to amend Paragraph 5 of Decision No. 15891-(15/109), adopted November 30, 2015: - The determination language is amended to preserve the “same value” requirement and to set detailed rounding and expression rules as follows (text preserved verbatim from the redlined version): - (a) The currency amounts calculated for the new basket will be expressed in two rounded to five significant digits based on the sixth significant digit. provided that the deviation of the percentage share of each currency in the value of the special drawing right, resulting from the application of the average exchange rates for July- September, from the percentage weight as determined under paragraphs 3 and 4(c) above is the minimum on average and will not exceed one half percentage point for any currency. If necessary to achieve the same value, an adjustment will be made to the amount of the currency against which the values of the other SDR basket currencies are determined in accordance with Rule O-2. - (b) If the calculations under a solution cannot be obtained by the application of the guidelines set forth in (a) do not yield the same value in five significant digits, above, the calculations shall be made by applying the same guidelines but expressing rounding the amount of each currency amounts in three to six significant digits based on the seventh significant digit, and if no solution is found with three significant digits then the calculation shall be made applying the same guidelines but expressing the amount of each currency in four significant digits. - (c) If more than one solution is found in the calculation at the level of two, three, or four significant digits, the solution that has the smallest average deviation will be employed. *Source: _071316 - 5. The amounts of the currencies under paragraphs 3 and 4 above shall be determined in a; International Monetary Fund PDF content as provided.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/np/pp/eng/2016/_071316.pdf_