How to Resolve Forex Expectation Paradox

[Metronome]

In this video (37:27 - 42:21), it is proved that buying a Euro with a Dollar and buying a Dollar with a Euro can both have positive expected value without varying the probabilistic structure of the problem statement.

My initial reaction was that this might be structurally equivalent to the necktie paradox. However, I cannot figure out how to map one paradox to the other.

Is there something else going on here? If a gambler came across a wager with this probabilistic structure (more likely than a forex trader, since the probabilities of future trading success are never pinned down precisely), how could they identify that their positive expected value is somehow misleading?

 

[Cubist]

An interesting video. Think about the first calculation of expected value (before the chalk colour is changed).

Consider a wild exchange rate prediction of 100$per£. The exact opposite of this would be 1/100 = 0.01$per£, right?
If both of these rates are equally likely, and they are truly the opposite of each other, then shouldn't the average of these two exchange rates be 1.00$per£ rather than (100 + 0.01)/2 = 50.005$per£ ?

 

[Metronome]

An interesting video. Think about the first calculation of expected value (before the chalk colour is changed).

Consider a wild exchange rate prediction of 100$per£. The exact opposite of this would be 1/100 = 0.01$per£, right?
If both of these rates are equally likely, and they are truly the opposite of each other, then shouldn't the average of these two exchange rates be 1.00$per£ rather than (100 + 0.01)/2 = 50.005$per£ ?

I mean, the geometric average of $100 and $.01 is $1, so I can see where you get the figure, but I'm not sure I see why the trader/investor/gambler should use that average rather than $50.005 when evaluating the trade; I will take a coin flip to multiply or divide my capital by 100 any time.

 

[Cubist]

I mean, the geometric average of $100 and $.01 is $1, so I can see where you get the figure, but I'm not sure I see why the trader/investor/gambler should use that average rather than $50.005 when evaluating the trade; I will take a coin flip to multiply or divide my capital by 100 any time.

There is no trade. The lecturer is talking about exchange rates.

Now let's consider some exchange trades, one now and one in the predicted future. Bob had $2 and converted one of his dollars into a pound. Now Bob has $1 and £1. The prediction is that he will end up with either 1 + 100*1 dollars OR 1 + 1/100 dollars after converting the pound back to dollars in the future. The expected amount is (1 + 100*1)*.5 + (1+1/100)*.5 = $51.005. However, it shouldn't be surprising that Dave who had £2 and converted one of them to a dollar also has the same expected amount of pounds by going through a similar process (but the other way around). After all, both parties currently have $1 and £1.

Of course, in the future, one person will have gained currency and the other will have lost (they won't both gain at the same time). Whoever gained - did they really gain a lot or is it just that their currency is now worth far less on the global market? (and their country must have had hyper-inflation during this time interval so everything in the shops would cost a lot more).

I haven't had time to double check this but I think (hope) that I'm correct!

 

[Metronome]

Now let's consider some exchange trades, one now and one in the predicted future. Bob had $2 and converted one of his dollars into a pound. Now Bob has $1 and £1. The prediction is that he will end up with either 1 + 100*1 dollars OR 1 + 1/100 dollars after converting the pound back to dollars in the future. The expected amount is (1 + 100*1)*.5 + (1+1/100)*.5 = $51.005. However, it shouldn't be surprising that Dave who had £2 and converted one of them to a dollar also has the same expected amount of pounds by going through a similar process (but the other way around). After all, both parties currently have $1 and £1.

I agree that since Bob and Dave have identical portfolios, those portfolios should each have equal expected value regardless of how they acquired them, but that doesn't seem to explain the weirdness.

Suppose the quoted transaction occurred by Bob directly trading $1 to Dave for £1. Who got the better deal, in expectation? The symmetry of the problem suggests that the transaction increased the expectation of both portfolios, but then the linearity of the expected value operator implies that they should repeat the trade to increase each portfolio's expectation again by an equal amount, but after that Bob has Dave's original portfolio and Dave has Bob's, so they should use the reasoning behind the first trade to justify a trade back to increase each portfolio's expectation even more, and continue on indefinitely, which seems absurd.

 

[Cubist]

Please try to think about what EXACT set of trades you'd do to secure the unlimited profit (and then let me know ?). Who are the two parties in each trade, what exactly changes hands and when does each trade happen? (now or in the future). The date/time of a trade is very important. Remember that you can't use a time machine to travel forward, receive a profit from a trade, and then travel back to now with that profit to perform another trade in the present - and perhaps try to enter an infinite loop.

The different predicted interest rates of $, £, Euro etc are strongly linked to the future exchange rate predictions between these currencies.

NPV (net present value) is a calculation to work out the value of a future dollar in terms of today's dollars.

--

If $1 in the future is predicted to have the same worth as $1 now
and £1 in the future is predicted to have the same worth as £100 now
and the future exchange rate is predicted to be $100 = £1

then

the NPV of the dollar side of the future exchange rate ( $100) is $100
the NPV of the pound side of the future exchange rate ( £1) is £100
this implies the current exchange rate should be $100 = £100 or in other words $1 = £1. If this was any different to the actual current exchange rate then arbitrage could take place (and profit made) and pretty soon the market would move and everything would match up.

--

The lecturer is calculating an expected exchange rate (not an expected profit) and therefore ought to be using a geometric average IMO

If we are evaluating the profit from a future trade - with two separate predictions for intervening interest rates and future exchange rates - you'd have to calculate the profit of each scenario in terms of NPV (otherwise it's like comparing number of cars vs number of tanks, or number of rowing boats vs number of jet skis, etc. because future currency won't have the same value in each of the different scenarios, but NPVs are OK to compare across different scenarios)

Hope this helps. It can be very confusing.

 

[Metronome]

Thinking about this problem is leaving me more convinced of it being equivalent to a necktie paradox or two envelope paradox, of which I know some solutions but struggle to find them satisfying.

I believe time can be more or less eliminated from the model. Bob (tracks expectation in USD and starts with $2) and Dave (tracks expectation in GBP and starts with £2) make as many trades as they wish at the £1/$1 exchange rate (treated as simultaneous and without market impact), then a fair coin would be flipped to simulate one currency appreciating to 100x the other, but since we care about expectation, the flip does not actually have to be executed.

Bob's portfolio's initial expected value is E[$2] = $2. He trades $1 for £1 and it becomes E[$1] + E[£1] = $51.005. He trades $1 for £1 again and it becomes E[£2] = $100.01. This pattern could have been extended indefinitely if Bob's portfolio had started with more USD, thus E[trading $1 for £1] = $49.005.

Restating the same sequence of actions from the counterparty's perspective, Dave's portfolio's initial expected value is E[£2] = £2. He trades £1 for $1 and it becomes E[£1] + E[$1] = £51.005. He trades £1 for $1 again and it becomes E[$2] = £100.01. This pattern could have been extended indefinitely if Dave's portfolio had started with more GBP, thus E[trading £1 for $1] = £49.005.

Because E[trading $1 for £1] > $0, Bob wants to perform the action "trade $1 for £1" as many times as possible, and because E[trading £1 for $1] > £0, Dave wants to perform the action "trade £1 for $1" as many times as possible. Unfortunately, Bob is out of USD so cannot trade for more GBP, and Dave is out of GBP so cannot trade for more USD. However, Bob can observe that E[trading £1 for $1] > £0 and Dave can observe that E[trading $1 for £1] > $0. Since USD is never worth a nonpositive quantity of GBP and vice-versa, Bob sees that E[trading £1 for $1] > $0 and Dave sees that E[trading $1 for £1] > £0, so Bob will trade £1 for $1 and end up with a portfolio worth more than $100.01. By a symmetric argument, Dave ends up a portfolio worth more than £100.01.

The above argument could be repeated to suggest that the expectation of each portfolio grows indefinitely as more trading occurs, regardless of sequence, but the absurdity is already present, since we have previously computed that E[$1] + E[£1] = $51.005 = £51.005.

 

[JeffM]

There is no paradox.

We start out at [imath]\dfrac{1 \text { dollar}}{1 \text { ounce of hummus}}.[/imath]

If in a year, the price of hummus is [imath]\dfrac{1.25 \text { dollars}}{1 \text { ounce of hummus}}[/imath], then we say that the price of hummus (in terms of dollars) has gone up by 25%.

If in a year, the price of hummus is [imath]\dfrac{0.80 \text { dollars}}{1 \text { ounce of hummus}}[/imath], then we say that the price of hummus (in terms of dollars) has gone down by 20%.

Now that is how we normally think about prices in the US because our medioum of exchange, standard of value, and legal tender is the dollar. We are thinking in terms of the dollar. It is, to use the technical term, the numeraire.

Now let's think about foreign exchange.

We start out at [imath]\dfrac{1 \text { dollar}}{1 \text { euro}}[/imath] a year ago.

If the price of euros is [imath]\dfrac{1.25 \text { dollars}}{1 \text { euro}}[/imath], then we say that the value of euros (in terms of dollars) has gone up by 25% (just like the hummus example). But that means the value of a dollar in terms of euroes has gone down. How much?
Well, obviously,

[math]\dfrac{1 \text { euro}}{1.25 \text { dollar}} = \dfrac{0.8 \text { euros}{1 \text { dollar}}.[/math]
The value of the dollar has gone down by 20% in terms of euroes. It depends on our frame of reference whether euroes appreciated by 25% in terms of dollars or whether dollars depreciated by 20% in terms of euros. If you are an American who bought euros a year ago for dollars, you will enjoy your vacation in France. If you are French and bought dollars a year ago for euros, you will not be happy that you did not.

When we are talking about the price of money, we need to realize that we can choose either currency as the numeraire (the numerator), and that in general the reciprocal of a number equals itself only if the number is 1.

The guy said it was a brain teaser, not a paradox. He carefully started with a 1 to 1 ratio and chose a pair of numbers whose reciprocals equaled each other. If you play around with foreigh exchange, you quickly get used to playing with reciprocals and frames of reference. There is no esoteric anything going on. It is just that we Americans are not used to thinking about anything but dollars in the numerator of a price. It's a Brit laughing at the stupid colonists across the Atlantic.

 

[Cubist]

Thinking about this problem is leaving me more convinced of it being equivalent to a necktie paradox or two envelope paradox, of which I know some solutions but struggle to find them satisfying....

It seems to me that you're deliberately attempting to construct a paradox rather than trying to understand how forex actually works. Is this correct? (This might be an interesting challenge)

Let me repeat the scenario in your post, as I understand it, in steps...

1. Initially:- Bob starts with $2 and Dave starts with £2
2. Now Bob and Dave have the opportunity to swap their dollars and pounds between each other at the exchange rate of £1 for $1
3. They both write down the currency that they hold just before the coin flip
4. There's a coin flip...
- heads implies $1 is henceforth exchanged for £100
- tails implies £1 is henceforth exchanged for $100
5. Now Bob and Dave still have EXACTLY the same currency that they had in step 3
6. Bob wants $ since he's American. He uses the new exchange rate to convert any £ that he is holding into $
7. Dave wants £ since he's English. He uses the new exchange rate to convert any $ that he is holding into £
8. They both write down the currency that they hold

Steps 6 & 7 weren't very clear in your post, but I think that they are necessary for the whole scenario to make sense.

Here are the only possible outcomes...

 BEFORE       COIN=HEADS     COIN=TAILS    EXPECTED AMOUNT
 FLIP
(step 3)      (step 8)       (step 8)       (step 8)

Bob  Dave      Bob   Dave     Bob   Dave    Bob     Dave
£ $  £ $       $     £        $     £       $       £
=========      ==========     =========     ==========
0 2  2 0       2     2        2    2        2       2
1 1  1 1       1.01  101      101  1.01     51.005  51.005
2 0  0 2       0.02  200      200  0.02     100.01  100.01

Bob and Dave both have maximum expected value when they exchange all their starting currency at step 2. But after they do this, you can clearly see that there's no incentive for them to swap anything back to their original currencies (at step 2) since this would reduce their expected amounts.

 

[JeffM]

@Cubist

There is no transaction specified. It is simply playing with numeraires. It’s a magic trick, a brain teaser. The expected values are just distractions from a fact about reciprocals of fractions and confusion about frames of reference.

What’s a 25% appreciation of the Euro in terms of dollars is a 20% depreciation of the dollar in terms of Euros. What is a 20% depreciation of the Euro in terms of dollars is a 25% appreciation of the dollar in terms of Euros. Then he uses a probability of 1/2 to calculate an expected values. Everything is symmetric numerically, but in one case the expected value is an expected percentage appreciation of the value of the Euro in terms of dollars, which, because of the numbers chosen, equals the expected depreciation of the dollar in terms of Euros.

Do not play liar’s poker with this guy.