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*Julian D. A. Wiseman*

**Abstract:** The currencies of the first wave of EMU entrants entered EMU at inconveniently non-round conversion rates.
This need not be the case for subsequent entrants, who should prefer conversion rates which are round, and which have round inverses.

**Contents: **
Table of conversion rates with round inverses,
Introduction,
The desiderata,
The case of the UK,
Conclusion.

**Publication history:** only here. Usual disclaimer and copyright terms apply.

The conversion factors for the first wave of EMU entrants were determined at the end of 1998. This determination was done by formula, into which were input several variables including the £/$ exchange rate, and because of this formula it was not possible for round numbers to be chosen. Thus the conversion rates are as unround as EUR 1 = DEM 1.95583, which causes even worse inconveniences such as DEM 1,000,000 = 511,291.8812 (with infinitely more decimal places to follow).

For any subsequent EMU entrants not already part of version two of the Exchange Rate Mechanism (so, for any entrants other than Denmark), a conversion rate would be chosen by agreement of the Council of Ministers. This council could choose less inconvenient numbers.

What properties would be convenient in a conversion rate between the € and a National Currency Unit (‘NCU’)?

As with the first wave, the €/NCU conversion rate should be exactly expressible to six significant figures.

And its inverse, the NCU/€ rate, should also be exactly expressible to six significant figures.

These conditions greatly narrow the field.
Mathematically, any such conversion rate must expressible in the form 2^{n} × 5^{m}, with integer *n* and *m*.
And if we restrict ourselves to possible €/NCU conversion rates with the above properties that are at least 0.5 and less than 1000, then there are only fifty-seven, and they are shown in the
table on the right
along with their inverses and values of *n* and *m*.

As has been argued previously (such as in a letter to my aunt dated March 1997), UK entry into EMU would be foolish. Let us hope that it doesn’t happen, and hope even more that it doesn’t happen through improper politicisation of the taxpayers’ money. But if it were to happen, what conversion rates would be convenient?

Well, it is unlikely that the other ministers would agree to a € as strong as 0.78125 or 0.8; and hopefully unlikely that the British would agree to one as weak as 0.5. That leaves two choices: 0.625 (implying £5 = €8); and 0.64 (implying £16 = €25). Both of these are close to the recent trading range of the euro.

Hence the probability distribution of €/£ conversion rates, assuming entry, contains (as of May 2002) substantial spikes at 0.625 and 0.64. This reasoning might be of interest to those who trade foreign-exchange options.

*Julian D. A. Wiseman, May 2002*

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