Categories:
Statistical theory and methods
In an otherwise good article in today’s Wall Street Journal, “Just Say Nein to Eurobonds,” Matthew Will writes:
From 1980 to the launch of the euro in January 1999, the Italian lira and Portuguese escudo lost 108% and 244% of their value against the U.S. dollar, respectively. Greece devalued the drachma 583% against the dollar from 1981 until its euro entry in January 2001.
Impossible. An item, whether a good, a service, or a currency, cannot lose more than 100% of its value unless it becomes garbage that you need to pay someone to haul away.
READER COMMENTS
Ken B
Jul 2 2012 at 10:59am
You’re not just right. You’re 110% right.
david
Jul 2 2012 at 12:04pm
It can’t lose more than 100% of the value against itself, but it can lose more than 100% of the value against something else, surely?
By simply changing in absolute value that is more than 100% of something else?
David O. Cushman
Jul 2 2012 at 12:37pm
The quoted percentage figures turn out to be the percentage appreciation of the U.S. dollar against given European currency. The confusion highlights the importance of which number is the base in ordinary percent calculations. It makes a lot of difference here. Taking differences of logs gives a unique answer but might be harder to explain to typical WSJ readers. (In two of the cases at hand the result is bigger than 1, sort of like a percent bigger than 100 for the depreciation.) Anyway, I certainly agree that Will wrote badly.
Sierra
Jul 2 2012 at 12:44pm
I agree with david. The value relative to something else can be expressed as a multiplier/percentage relative to the current value of that item. If one euro is worth two dollars and those positions switch, then the value of the dollar is now improved by its own full value, or 100%. The euro’s value will have decreased by its full value, or 100%. However if the euro were to lose even more value (i.e. its value goes below its full value relative to the dollar it was originally compared to) it can be expressed as 100% plus whatever additional percentage (110% for example).
MG
Jul 2 2012 at 12:53pm
Professor Henderson is right as usual. The writer could have said that the dollar appreciated all of these 100%+ figures against the club med currencies; or that the club med currencies depreciated between 50% to 80% against the dollar. I am sure that the writer simply erred in mixing the metrics, but has he stumbled into shedding some light into the pain of currency asset liability mismatches? (A bit of what afflicts many club med insitutions.) The “large numbers” bring home (impressionalistically) the pain of owning a club med asset (which admittedly can only lose 100% of its value) funded by borrowing in usd, when your only means of repayment are in club med currency. A long position can only lose 100% of its value; the loss on a short position is unlimited.
David O. Cushman
Jul 2 2012 at 2:06pm
Sierra: Your example is not quite right. You start with 1 euro = 2 dollars and say that the positions switch. I assume you mean that now 2 euros = 1 dollar. So, we went from 2 dollars per euro to 0.5 dollars per euro, or from 0.5 euros per dollar to 2 euros per dollar. In ordinary percentage terms the appreciation of the dollar in terms of the euro is 100(2/0.5 – 1) = 300%. (It is 400% of its original value.) The appreciation of the euro is 100(0.5/2 – 1) = -75%. Negative appreciation is depreciation, so depreciation is 75%. (The euro is 25% of its original value.) The depreciation of the euro would be 100% only if it became worthless. The mistake equivalent to that of the WSJ writer Will would be in the present hypothetical case to say that the euro depreciated by 300%. Now, if we take the (natural) log difference, we get: ln(2) – ln(0.5) = 1.39. This means that the dollar’s appreciation rate is equivalent to 139% compounded continuously. The euro depreciates at a 139% rate compounded continuously. So we get the same numerical answer (except for sign) from either currency’s point of view (and in this case a depreciation value greater than 100%).
Sierra
Jul 2 2012 at 3:26pm
David Cushman – I stand corrected, thank you. 🙂
Ken B
Jul 2 2012 at 3:29pm
Sierra et al are missing a trick. The reporter did not say they lost 244% of some other item’s value. That’s easy. I could lose 10,000% of a penny’s value by dropping a single. But no-one talks like that and it’s not what he said. He said they lost 244% of *their own* value. And that means they became a liability, worse than useless. Like smelly garbage.
A raccoon
Jul 2 2012 at 6:24pm
You’re not making any sense.
Matt Will
Jul 2 2012 at 9:27pm
As the author, I will explain.
Currency values are relative, not absolute. They cannot be thought of as having 100% value and only being able to lose 100% of their value. Price levels can rise and infinitely lose money. The short sale of a derivative is an ideal example of something that can lose much more than 100%. Think of the Weimer Republic where it took 2 marks to buy a loaf of bread and a few years later it took 2 million marks to buy the same loaf of bread. Would you not say it lost its value by much more than 100%? Foreign Exchange currencies trade at forward premiums and discounts. These figures can lead to loss of purchasing value far in excess of 100%. The math below is the source of the article. Unfortunately, the English language in a brief space does not allow for a complete explanation.
Sorry for the formatting
Currency per $1.00
Start dates Jan 1980
Drachma start 4/13/81
German Mark 1980-2000…………. 1.72 to 1.69 …….. +1.74 %
Ended 1/15/1999
Italian Lira 1980-2000 ………………. 800 to 1670 …….. -108 %
Ended 1/15/1999
Greek Drachma 1981-2001……….. 53 to 362…… -583 %
Ended 12/29/2000
Portuguese Escudo 1980-2000 …… 50 to 172 ……. -244 %
Ended 1/15/1999
David R. Henderson
Jul 3 2012 at 1:32am
@Matt Will,
Thanks for responding. Rather than going through all your numerical examples, I’ll do it with your German hyperinflation one. If it took 2 marks to buy a loaf of bread and then later it took 2 million marks, then the mark lost 1,999,998/2,000,000 of its value. That’s still just a little less than 100%. As long as a positive number falls to a much lower positive number, it can’t fall by 100%. If it can’t fall to less than zero, then it can’t fall by more than 100%. Do you see?
Ken B
Jul 3 2012 at 8:17am
@A raccoon:
If it became merely worthless, it would have lost 100% of its value. To lose more than 100% of its value it must have become a liability, something that will cost you to be rid of.
A raccoon
Jul 3 2012 at 3:23pm
@Ken B:
Yes, but why would you compare a liability to smelly trash? Smelly trash is delicious and I dine upon it every night. Your math is fine, but your metaphor is bad (unlike smelly trash, which is good).
/s/ A racoon.
A Racoon
Jul 3 2012 at 3:43pm
Professor Henderson-
Mr. Will could also say the mark depreciated by 1382% (vis a vis bread) and he thinks most of the traders would intuitively understand. Your rendering, Mr. Will might argue, troubles his audience to do quick math to translate into their language. In that case, Mr. Will would have done his job (i.e., communicate) poorly.
I have no idea, but that seems most important.
Mike Rulle
Jul 4 2012 at 5:45pm
You are both half right and this is more complex than you think. It depends which currency you are measuring gains and losses in. The “home guy” can lose an infinite amount in his home currency. The “foreign guy” can only lose 100% in his home currency. What is being ignored is opportunity cost, which is why this is such a great illusion. Lets use stocks as a more intuitive example. One can pretend a stock is a foreign currency.
What is my return on the following investment?
It is 1997 or so. I short 1 share of Apple at $16 in my diversified portfolio. I receive $16 which is now invested in being short Apple. I am still short Apple today (pretend it is $600) What is the percent I lost on that investment?
David, do you still believe it is 100%? Of course not. But your Weimar example is also true. What if I bought $16 dollars with one share of Apple stock in 1997? In Apple Stock terms my 16 dollars would be worth 50 cents. So the person living in Apple country only lost 98% in Apple stock terms.. How does one square this circle or apparent paradox? .
This circle gets squared by realizing that while these 2 Apple trades are identical trades, one is measured in real dollars lost, while the other is an opportunity loss and is not measured. It is always possible to lose more than 100% given the right poorly chosen trade. Yes, if I bought 1 dollars worth of DM in 1920, by 1928 I could only lose a max of 100%, as David says. But the German who shorted the dollar in 1920 lost 1000s of percent in his home currency, like the Apple stock shorter. The guy who bought DM lost 99% or more in his home currency, but lost thousands of percents in DM terms in opportunity loss.
Obviously, we are not addressing PPP or anything like that. Just showing one can lose more than 100%, and as Apple example shows, we ignore opportunity loss at our own peril.
It makes me feel better that its not just me who finds these ideas complicated.
David O. Cushman
Jul 5 2012 at 11:26am
David Henderson first wrote, “An item, whether a good, a service, or a currency, cannot lose more than 100% of its value unless it becomes garbage that you need to pay someone to haul away.”
The example of the loss on a short sale exceeding 100% is, in fact, an example of something becoming garbage. To get out if his obligation to eventually close out the short sale, to have it “hauled away,” the investor has to pay more than his original investment.
The currency assets that WSJ writer Will was writing about had not become garbage, and almost surely never will, unless they become not only worthless but the holders are obliged to eliminate the currency entries in their financial accounts and incur a cost to do so.
By the way, the “investment” in the short is not really the $16, but the fees and interest involved in borrowing the share from the broker.
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