How to convert an historic value denoted in foreign currency to a modern value in local currency?

Question

I want to convert GBP in 1968 to USD in 2018, in order to "translate" an amount paid by an individual in Britain fifty years ago. My intention is that an American reader today will be able to better understand the value of the money if it is given in inflation-adjusted USD.

In 1968, £1 exchanged to \$2.38. \$1 in 1968 adjusts to \$7.22 in 2018. £1 in 1968 adjusts to £17.02 in 2018. Finally, £1 exchanges to \$1.34 in 2018.^[1]

If we do the exchange first, then £1 (1968) is equivalent to \$17.18 (2018). If we adjust for inflation first, then £1 (1968) is equivalent to \$22.81 (2018). Which is the correct way to do this? Why?

Addendum:

An answer to a similar question suggests that the exchange rate, in some form, should be commutative with inflation adjustment:

Also, it does not matter which transformation you do first, as its purely multiplicative.

I am not sure whether the author is referring to calculations using the formula that he defines as the "real exchange rate" as "purely multiplicative", or whether he/she meant that the exchange rate and inflation are ideally always going to be commutative (and therefore this case deviates from the ideal by a degree). The answer does not go into very much detail and it is beyond my understanding of economics to determine what the "respective price level" between Britain and the US is in 1968 or 2018.

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^{[1] I am getting my inflation data from officialdata.org. While I have found on other websites slightly different conversion factors for inflation since 1968 in both currencies, they are only a few cents different from the figures I used here. The nominal exchange rate is, by contrast, an historical fact.}

Answer 1

For what purpose?

Money over time acts as inflation rate, wage bundle, production good bundle, %gdp, %gdp/capita. Each represents a different “meaning” of money over time.

The US and UK economies have different consumption bundles, wage structures, skill structures, industrial structures, government policy, market agents and social meanings. 1968UK and 2018UK also vary and 1968UK and 2018UK vary in these domains.

Your question is unanswerable without the commodity or expense, the volume or market (wholesale, retail, government) and the purpose.

See measuringworth.com for essays.

Answer 2

The difference you get is due to the different methods that are followed (in the two countries over the period of time) to compute inflation rates. If those were exact, then you would witness the putative commutativity.

Have you tried to make your calculations backward? Tracing year after year any potential spread? Hopefully, approaches have been standardized and you may notice when by doing so.

BTW, I would go for using only the country's trajectory of inflation rates that would (subjectively admittedly) look the most exact, the most exhaustive or the most consensual. In the latter case, USA's would be the one to consider.