Applying Percentage Weights in a Geometric Dollar Index

# Calculate Federal Reserve Dollar Indexes


# Calculate Federal Reserve Dollar Indexes












I'm attempting to validate the formula used to calculate the Federal Reserve Dollar Indexes, as published on January 15, 2019, in this document: https://www.federalreserve.gov/econres/notes/feds-notes/revisions-to-the-federal-reserve-dollar-indexes-20190115.html

The formula states that the index value "I" at time "t" should be computed as follows:

```
I = I(t-1) * Product(j=1..26, [e(j,t)/e(j,t-1)]^w(j,t))
```

Here, I(t-1) represents the previous index value (from the previous day), j=1..26 denotes the 26 currency exchange rates that constitute the Nominal Broad USD index, e(j,t) represents the current exchange rate of currency "j" (the price of 1 Dollar in currency "j"), e(j,t-1) represents the previous exchange rate, and w(j,t) corresponds to the weight of currency "j" in the index on the current day.

The list of currencies comprising the index, along with their respective weights (updated annually), can be found in this publication: https://www.federalreserve.gov/releases/h10/weights/.

To validate the formula, I collected the closing prices of all 26 currencies for June 13, 2023 (t), and June 12, 2023 (t-1), as well as the index values for those two dates. Using Excel, I performed the calculation as described above, expecting to reproduce the index value for time "t". However, the obtained result significantly deviates from the expected value.

I obtained the official exchange rates and index values from this page: https://www.federalreserve.gov/releases/h10/20230620/. However, this source lacks data for 9 exchange rates. To retrieve their daily closing values, I referred to:

https://www.exchangerates.org.uk/USD-ILS-exchange-rate-history.html https://www.exchangerates.org.uk/USD-ARS-exchange-rate-history.html https://www.exchangerates.org.uk/USD-CLP-exchange-rate-history.html https://www.exchangerates.org.uk/USD-COP-exchange-rate-history.html https://www.exchangerates.org.uk/USD-IDR-exchange-rate-history.html https://www.exchangerates.org.uk/USD-PHP-exchange-rate-history.html https://www.exchangerates.org.uk/USD-RUB-exchange-rate-history.html https://www.exchangerates.org.uk/USD-SAR-exchange-rate-history.html https://www.investing.com/currencies/usd-vnd-historical-data

Here are my data and calculations in csv format:

Country;Weight;12.06.2023;13.06.2023;e(t)/e(t-1);[e(t)/e(t-1)]^weight Argentina;0,353;246,71;247,12;1,00166187;1,000586325 Australia;1,211;0,6747;0,6776;1,004298207;1,005207486 Brazil;1,714;4,8773;4,8567;0,995776352;0,992771587 Canada;13,104;1,3377;1,3298;0,994094341;0,925318597 Chile;0,723;788,6;803,7;1,019147857;1,013807483 China;14,567;7,1437;7,1585;1,002071756;1,030607098 Colombia;0,64;4184;4176;0,998087954;0,998775869 Hong Kong;1,177;7,8335;7,8336;1,000012766;1,000015025 India;2,954;82,4300;82,3800;0,999393425;0,998209238 Indonesia;0,744;14871;14870;0,999932755;0,999949969 Israel;0,877;3,6001;3,5598;0,988805867;0,990175958 Japan;5,49;139,7200;140,0000;1,002004008;1,011051618 Korea;3,594;1288,6100;1271,4800;0,986706606;0,953041476 Malaysia;1,566;4,6180;4,6140;0,999133824;0,998643901 Mexico;13,784;17,3080;17,2140;0,994568985;0,927683065 Philippines;0,618;56,075;55,912;0,997093179;0,998202586 Russia;0,501;83,792;83,949;1,001873687;1,000938279 Saudi Arabia;0,466;3,7494;3,7501;1,000186697;1,000086996 Singapore;2,008;1,3443;1,3416;0,99799152;0,995971054 Sweden;0,588;10,8244;10,7179;0,990161117;0,994202957 Switzerland;3,046;0,9099;0,9053;0,994944499;0,984680445 Taiwan;2,652;30,7600;30,6700;0,997074122;0,992259313 Thailand;1,275;34,6800;34,6500;0,999134948;0,99889719 United Kingdom;4,654;1,2489;1,2611;1,009768596;1,046281477 Vietnam;2,37;23,48;23,495;1,000638842;1,001514717 Euro Area;19,325;1,0747;1,0792;1,004187215;1,084098745 INDEX;100,001;120,0069;119,5948;;0,933218371

[Last line: INDEX; sum of weigth; index value 12.6; index value 13.6.; ; Product of all values of last column]

As observed, the published index value for June 13th is stated as 119.5948. However, based on my calculations using the provided formula (120.0069 * 0.933218371), I obtained a result of 111.9926437.

I am unable to identify the specific error in my calculations that led to this discrepancy. I have carefully followed the formula and utilized the closing prices of the respective currencies on June 12th and June 13th, as well as the corresponding weights.

If possible, I would appreciate any guidance or assistance in pinpointing the source of the discrepancy and correcting my calculations to obtain the accurate index value for June 13th.

## Answer by jamacoe (score 2)

https://quant.stackexchange.com/a/75957

Found the error myself: I had to devide each weight by 100, such that the sum of weights is 1, not 100. Now the final result is 119.9239846, which is just 0,33 off the expected result.