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The Siegel paradox seems to resemble two envelope paradox. How is the similarity?
Because I was misunderstanding the original version of the Siegel Paradox when I wrote that paragraph.
If the investors are risk neutral, then:
I have tried the above calculation with the following imaginal data.
Anticipation by investers
Interest rates and forward exhange rate
Thus, the above inequality (4) E(ca) > cf is hold.
Jensen's inequality
Thus, the above inequality (3) E(c/ca) > c/E(ca) is hold.
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)
And I interpreted this paradox as follows, and I want this interpretation is not so wrong.
So I tried to write the following tables which involves the third player.
case of 2 : 1
case of 1 : 2
mean of the above two cases
In this way, if the third player is involved, the total balance will be maintained.
I think that the Siegel paradox and the missing dollar riddle are both related to overlooking items in accounting calculations.
I understand this paradox as follows.
I wrote the following table to verify it.
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.
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.
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)
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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.
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
ca (imaginal) |
Anticipated probability (imaginal) |
|
|
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) |
|
cf = |
---|---|---|
0.6789 | 1.0524 | 0.7145 |
Jensen's inequality
|
|
---|---|
0.9502 | 0.9450 |
My impression
This paradox is hard.-
In the first place, I have little knowledge about the forward market.
I read the English language Wikipedia article "Forward exchange rate" (revision at 12:42, 5 September 201) and accepted the equation (2) above.
However I cannot understand equation (1).
(↑ Revised on October 8, 2019) - I don't know about the relationship between equations (1) and (2) above. Therefore, I cannot find a paradox in the results of combination of them. (← Aevised on October 8, 2019)
- Beenstock, M. (1985) is a paper about the paradox, but it is too difficult for me.
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
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.
Suppose that next year they will make a reverse trade as follows.
The apple man and the orange man made a trade as follows.
- one apple from the apple man
- one orange from the orange man
Suppose that next year they will make a reverse trade as follows.
- one orange from the apple man
- one apple from the orange man
- the apple man expects 1.25 apple
- the orange man expects 1.25 orange
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 |
|
after the first exchange |
|
1 apple |
1 apple 1 orange |
|
after the second exchange | 2 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 |
|
after the first exchange |
|
1 apple |
1 apple 1 orange |
|
after the second exchange |
|
|
|
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 |
|
after the first exchange |
|
1 apple |
1 apple 1 orange |
|
after the second exchange |
|
|
|
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".
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".
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.
("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 !!!
year (real) |
|
|
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) .
Thereforethe 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.
Then for each period, RA/B + RB/A =
On the other hand,
Therefore
(↑ Revised on Jun 8, 2018.)
I wrote the following table to verify it.
year (real) |
|
|
RUSD/JPY α |
RJPY/USD | RUSD/JPY + RJPY/USD |
|
---|---|---|---|---|---|---|
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.
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).
year (real) |
|
|
rate of change of the exchange rate |
rate of change of the exchange rate |
---|---|---|---|---|
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 !!!
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) |
|
|
rate of change of the exchange rate |
rate of change of the exchange rate |
---|---|---|---|---|
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
-
Beenstock, M. (1985)
Forward Exchange Rates and" Siegel's Paradox".
Oxford Economic Papers, 37(2), 298-303.
-
Black, F. (1989)
Universal hedging: Optimizing currency risk and reward in international equity portfolios.
Financial Analysts Journal, 45(4), 16-22.
-
Siegel, J. J. (1972)
Risk, interest rates and the forward exchange.
The Quarterly Journal of Economics, 303-309.
Terms
-
Jensen's inequality
Please see the English language Wikipedia article "Jensen's inequality".
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