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2019/10/08 0:32:31
First edition 2018/06/04

The Siegel paradox and the two envelope paradox

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.

The Siegel paradox seems to resemble two envelope paradox. How is the similarity?

Original version of the Siegel paradox

On October 1, 2019, all the contents of the paragraph of “Siegel Paradox Original Version” were replaced.
Because I was misunderstanding the original version of the Siegel Paradox when I wrote that paragraph.

Paradox

If I rewrite the paradox presented in Siegel, J. J. (1972) as I have understood, it is as follows.
symbols meaning
c spot rate of foreign currency
ca anticipated spot rate of foreign currency
cf forward rate of foreign currency
rd rate of domestic yield
(total principal and interest
/ principal)
rf rate of foreign yield
(total principal and interest
/ principal)

If the investors are risk neutral, then:
  • indifference between the yields:
    (1) rd = E(crf/ca) =rfE(c/ca).
  • arbitrage condition:
    (2) rd =rf(c/cf).
  • Jensen's inequality:
    (3) E(c/ca) > c/E(ca).

Calculation example using imaginal data

(This paragraph was added on October 8, 2019)

I have tried the above calculation with the following imaginal data.

Anticipation by investers
Anticipated
ca

(imaginal)
Anticipated probability
(imaginal)
E(ca) 1/E(ca) Anticipated
1/ca
E(1/ca) 1/E(1/ca)
0.6556 1/10 0.7184 1.3918 1.525320 1.3996 0.7144
0.6692 2/5 1.494298
0.7605 2/5 1.314995
0.8104 1/10 1.234003

Interest rates and forward exhange rate
c
(imaginal)
rf/rd cf
=(rf/rd)c
0.6789 1.0524 0.7145
Thus, the above inequality (4) E(ca) > cf is hold.

Jensen's inequality
E(c/ca) c/E(ca)
0.9502 0.9450
Thus, the above inequality (3) E(c/ca) > c/E(ca) is hold.

My impression

This paradox is hard.
Is there a paradox in the first place?
I think that investors trade futures when the forward exchange rate is under their anticipation. (← Revised on October 8, 2019)
Therefore, inequality (4) E(ca) > cf seems not be a paradox.

After all
I got an impression that the above paradox is lesson of portfolio or hedging rather than paradox. (← Added on October 1, 2019)

Siegel paradoxes not relating forward exchange rate

(This header was added on October 1, 2019)

"Oranges and Apples" version of the Siegel paradox

Paradox

Reading Black, F. (1989), I learned the "Oranges and Apples" version of the Siegel paradox.
And I interpreted this paradox as follows, and I want this interpretation is not so wrong.
The exchange rate between apple and orange is now 1 to 1.
The apple man and the orange man made a trade as follows.
  • one apple from the apple man
  • one orange from the orange man
Next year, with equal probability the exchange rate will become 2 to 1 or 1 to 2.
Suppose that next year they will make a reverse trade as follows.
  • one orange from the apple man
  • one apple from the orange man
Then their expected earnings are as follows.
  • the apple man expects 1.25 apple
  • the orange man expects 1.25 orange
Therefore this trade will turn out to both men. Paradox !!!

My resolution

I think that the above "Oranges and Apples" version paradox has been caused by forgetting the third player.
So I tried to write the following tables which involves the third player.

case of 2 : 1
Stage Asset of
the apple man
Asset of
the orange man
Asset of
the third man
total
before exchange 1 apple 1 orange   1 apple
1 orange
after the first exchange 1 orange 1 apple   1 apple
1 orange
after the second exchange 2 apple 1/2 orange - 1 apple
1/2 orange
1 apple
1 orange

case of 1 : 2
Stage Asset of
the apple man
Asset of
the orange man
Asset of
the third man
total
before exchange 1 apple 1 orange   1 apple
1 orange
after the first exchange 1 orange 1 apple   1 apple
1 orange
after the second exchange 1/2 apple 2 orange 1/2 apple
-1 orange
1 apple
1 orange

mean of the above two cases
Stage Asset of
the apple man
Asset of
the orange man
Asset of
the third man
total
before exchange 1 apple 1 orange   1 apple
1 orange
after the first exchange 1 orange 1 apple   1 apple
1 orange
after the second exchange 1.25 apple 1.25 orange -0.25 apple
-0.25 orange
1 apple
1 orange

In this way, if the third player is involved, the total balance will be maintained.

Similarity to the two envelope paradox

I think that the above paradox resembles the "Paradox of the two envelopes which are greener than each other" but has different psychological mechanism.
Note :
About the later paradox, please see the paragraph "Paradox of the two envelopes which are greener than each other" of my page "An outline of the Two Envelopes Problem".

Similarity to the Missing dollar riddle

To my eyes, the above paradox is more similar to the missing dollar riddle than the two envelope paradox.
I think that the Siegel paradox and the missing dollar riddle are both related to overlooking items in accounting calculations.
Note :
About the missing dollar riddle, please see my page "The missing dollar riddle is as mysterious as the two envelope paradox".

Another Siegel paradox not relating forward exchange rate

(This section was added on Jun 7, 2018. The title was revised on October 1, 2019)

Paradox

Black, F. (1989) presents another Siegel's paradox about rate of change of exchange rate.
I understand this paradox as follows.
Let A and B be currencies, and let A/B and B/A be exchange rates.
("X/Y" means how much Y is needed to buy one X.)
And let RA/B be the rate of change of A/B and let RB/A be the rate of change of B/A.
Then for each period, sum of RA/B and RB/A is always positive.
And sum of the average of RA/B and the average of RB/A is always positive.
But this fact contradicts the expectation that exchange is a zero sum game. Paradox !!!
And I wrote the following table to explain this paradox.

year
(real)
USD/JPY JPY/USD RUSD/JPY RJPY/USD
2005 110.2182 0.00907    
2006 116.2993 0.0086 0.05517 -0.05229
2007 117.7535 0.00849 0.0125 -0.01235
2008 103.3595 0.00967 -0.12224 0.13926
2009 93.5701 0.01069 -0.09471 0.10462
2010 87.7799 0.01139 -0.06188 0.06596
2011 79.807 0.01253 -0.09083 0.0999
Average     -0.05033 0.05752

My calculation

Let RA/B be α.
Then for each period, RA/B + RB/A = α2 / (1 + α) ≥ 0 because α > -1.
On the other hand,
the average of RA/B + the average of RB/A = the average of (RA/B + RB/A).
Therefore the average of RA/B + the average of RB/A ≥ 0.
(↑ Revised on Jun 8, 2018.)
In this way, what seemed like a paradox is a mathematically correct calculation.

I wrote the following table to verify it.

year
(real)
USD/JPY JPY/USD RUSD/JPY
α
RJPY/USD RUSD/JPY + RJPY/USD α2 / (1 + α)
2005 110.2182 0.00907        
2006 116.2993 0.0086 0.05517 -0.05229 0.00288 0.00288
2007 117.7535 0.00849 0.0125 -0.01235 0.00015 0.00015
2008 103.3595 0.00967 -0.12224 0.13926 0.01702 0.01702
2009 93.5701 0.01069 -0.09471 0.10462 0.00991 0.00991
2010 87.7799 0.01139 -0.06188 0.06596 0.00408 0.00408
2011 79.807 0.01253 -0.09083 0.0999 0.00907 0.00907

One more Siegel paradox not relating forward exchange rate

(This section was added on Jun 7, 2018. The title was revised on October 1, 2019)

Paradox

I found a web page which presented another kind of the Siegel pardox as follows.
From 2008 to March, 2014, among JPY and USD, the rates of change of the exchange rates were as follows.

year
(real)
USD/JPY JPY/USD rate of change of the exchange rate
USD/JPY
rate of change of the exchange rate
JPY/USD
2008 103.3595 0.00967    
2009 93.5701 0.01069 -0.0947 0.1046
2010 87.7799 0.01139 -0.0619 0.0660
2011 79.807 0.01253 -0.0908 0.0999
2012 79.7905 0.01253 -0.0002 0.0002
2013 97.5957 0.01025 0.2231 -0.1824
2014 105.9448 0.00944 0.0855 -0.0788
Average     0.0102 0.0016

This table says that both averages are positive.
But this fact contradicts the expectation that exchange is a zero sum game. Paradox !!!
This paradox seems have been made with reference to Black, F. (1989).

My resolution

I think that this paradox is explained by the following fact.
The absolute value of the rate of change of upward change tends to be larger than the absolute value of the rate of change of downward change.
The following table clearly shows this rule.

year
(fictitious)
USD/JPY JPY/USD rate of change of the exchange rate
USD/JPY
rate of change of the exchange rate
JPY/USD
3018 100 0.01    
3019 110 0.00909 0.1 -0.0909
3020 100 0.01 -0.0909 0.1
Average     0.0045 0.0045

Furthermore, the table which is presented in the previous paragraph "Another Siegel paradox not relating forward exchange rate" shows that the average of the rate of change of the exchange rate USD/JPY is negative on the course to the yen appreciation.
So this paradox may be a fake imitation of the Siegel paradox.

Source of the paradoxical feeling derived by the above paradoxes which are not relating forward exchange rate

(Added on Jun 8, 2018. The title was revised on October 1, 2019)

I think that the paradoxical feeling of the Siegel paradox is caused by a confusion of the mean value of amounts and the mean value of rates.

My imression

I got an impression that the above paradoxes are parodies of the two envelope paradox. (← Revised on October 1, 2019)
Actually the calculations of the above versions of the Siegel paradox are correct.
About the above paradoxes not relating forward exchange rate, I think that the cause of the paradoxes are wrong interpretations of these calculations. (← Revised on October 1, 2019, October 8,2019)

Reference

Terms


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