Future exchange rates and Siegel's paradox
aa r X i v : . [ q -f i n . M F ] M a y FUTURE EXCHANGE RATES AND SIEGEL’S PARADOX
KEIVAN MALLAHI-KARAI AND PEDRAM SAFARI
Abstract.
Siegel’s paradox is a fundamental question in international trade aboutexchange rates for futures contracts and has puzzled many scholars for over fortyyears. The unorthodox approach presented in this article leads to an arbitrage-free solution which is invariant under currency re-denominations and is symmetric, asexplained. We will also give a complete classification of all such aggregators inthe general case. The formula obtained in this setting therefore describes all thenegotiated no-arbitrage forward exchange rates in terms of a reciprocity function.Keywords: international trade, forward exchange rates, futures contract, discountbias, Siegel’s paradox Introduction
Siegel’s paradox, discussed by Siegel (1972), is a discount bias in future exchangerates and is often discussed in connection with Nalebuff (1989) puzzle in the contextof expected values. It can briefly be described in simple terms as follows. Let usassume that there are two possible states of the world ω and ω at a specific time T in the future, both having an equal chance of 50% of occurring, where the exchangerate of Euros to US dollars is expected to be e and e , respectively. It appears thatan investor who wants to exchange Euros to US dollars at time T would consider theexpected value ( e + e ) as the spot price for a futures contract at time 0. Meanwhile,for their trading counterpart, who wants to perform the reciprocal exchange of USdollars to Euros, this price would be ( e − + e − ) in their own currency, which isan obvious disagreement. This is the content of Siegel’s paradox — that these (risk-neutral) investors cannot both opt for the arithmetic mean as the spot price for afutures contract. In other words, the source of the paradox lies in the fact that thearithmetic and harmonic means do not coincide. To appear in Global Finance Journal. https://doi.org/10.1016/j.gfj.2018.04.007
Authors’ affiliations and E-mails:
Keivan Mallahi-Karai, Jacobs University, Campus Ring I, 28759 Bremen, Germany. [email protected] .Pedram Safari (corresponding author), Department of Mathematics and Institute for QuantitativeSocial Science, Harvard University, Cambridge, MA 02138, USA. [email protected] .c (cid:13) http://creativecommons.org/licenses/by-nc-nd/4.0/ . It should not be surprising that this problem is of great significance in internationaltrade. The Bank for International Settlements estimates that the daily turnover inforeign exchange markets far exceeds $1 trillion (Obstfeld & Rogoff (1996)). Therehave been numerous theoretical attempts at understanding this paradox, as well asenormous amount of empirical testing. To begin with, Siegel (1972) seems to haveviewed it not as a paradox, but as a relationship between foreign exchange prices andinterest rates under risk neutrality. However, in the absence of interest rates, theparadox will not fade away. He suggested that the investors in the ”small country”choose a biased estimator, as their own forward rate, which lies between the geometricand harmonic means. Roper (1975) suggested that the paradox may be resolved if theinvestors take their forward profits in the foreign currency. However, if the investorswish to take their profits in their own domestic currencies, they have to be risk-averse(Beenstock (1985)). A nice overview of Siegel’s paradox could be found in Edlin(2002). None of the proposed solutions, to our knowledge, suggests a specific forwardexchange rate that is acceptable to risk-neutral investors in both currencies.In their classic text on international macroeconomics, Obstfeld & Rogoff (1996)dedicate an entire section to Siegel’s paradox and its ramifications. They analyze theempirical tests for prediction bias in forward rates in detail and develop a stochasticmonetary model to understand forward exchange pricing. They explain that in theirmodel, if (relative) purchasing power parity holds, the expected real returns will bezero for one party if it is zero for the other, but the argument fails when the PPP failsfor any reason. They also suggest that the forward exchange rate in the equilibriumends up to be a negotiated rate between the expected value of the future spot ratefrom one investor’s perspective and that of their counterpart — in other words, if E T is the future spot rate at time T in the future, the negotiated rate will be between E ( E T ) and 1 /E (1 / E T ) , which is consistent with our findings.In this paper, we take a non-traditional approach and seek an aggregator thatis arbitrage-free, symmetric and invariant under redenominations. These conditionswill be defined precisely in the next section. We will show that under these naturalaxioms, the only possible aggregator for Siegel’s paradox will be the geometric mean.Note that for any pair of positive numbers, the geometric mean always lies betweenthe arithmetic and harmonic means. We will go even further and give a completeclassification of the aggregators satisfying the generalized axioms in any dimension.The main result (Theorem 4.2) gives a formula for these aggregators in terms of thegeometric mean and a reciprocity function.Our approach not only provides an unbiased common ground to the exchangeproblem in Siegel’s paradox, but might eventually even shed a new light in under-standing other questions in future discount rates, such as in the Weitzman-Gollierpuzzle (Gollier & Weitzman (2010)). UTURE EXCHANGE RATES AND SIEGEL’S PARADOX 3 The Model
In this model, we will assume that there are two possible states of the world ω and ω at a fixed time T in the future. We consider the possible values of foreignexchange rate (we use European and American currency) EUR/USD at time T andassume that it can attain values R EUR / USD ( ω ) = e , R EUR / USD ( ω ) = e . The reciprocal exchange rates for USD in terms of Euros will be 1 /e , /e , that is, R USD / EUR ( ω ) = e − , R USD / EUR ( ω ) = e − . Siegel’s paradox shows that investors who want to exchange Euros to US dollarsat time T and the ones who want to exchange US dollars to Euros at time T cannotboth use the arithmetic mean as the spot price for the futures contract at time 0.Our point of view is that the arithmetic mean (or the expected value with respectto probability distribution (1 / , / “aggregator” to be used in thiscontext. We will consider three natural axioms that any reasonable aggregator mustsatisfy. Underlying each axiom is what may be viewed as an invariance principle stating the aggregate price must be preserved under certain transformations of thegiven data. Surprisingly, these three axioms are strong enough to determine a uniqueaggregator. Let us denote the aggregator by A ( e , e ). Our axioms are as follows. • Symmetry . We will assume that there is no particular order on the twopossible states of the world, hence the aggregator must be indifferent to theorder in which the rates are listed. This implies that A ( e , e ) = A ( e , e ) . • Re-denomination . Upon redenomination of currencies, say, replacing a Euroby 100 Euro cents, the exchange rates have to be adjusted accordingly, hereby a factor of 100. More generally, if we denote the redenomination factor by λ , the following identity must hold: A ( λe , λe ) = λ · A ( e , e ) . • Reciprocity . This axiom explicitly precludes what happens in Siegel’s para-dox. It can also naturally be viewed as a no-arbitrage constraint for the ex-change rates, stipulating that the aggregator prevents the possibility of makingrisk-less money by exchanging Euros to Dollars and then back into Euros (as-suming, obviously, that there are no transaction costs). This condition can beexpressed as A ( e − , e − ) = A ( e , e ) − . FUTURE EXCHANGE RATES AND SIEGEL’S PARADOX The Geometric Mean
We will identify the set E of, say, EUR/USD (and USD/EUR) exchange rates withthe set of positive real numbers. An aggregator is a function A : E × E → E , whichaggregates the two possible exchange rates e , e into one deterministic rate A ( e , e ).The following theorem gives a complete characterization of the aggregators A thatsatisfy the Symmetry, Redenomination and Reciprocity axioms. Theorem 3.1.
Let A : E × E → E be an aggregator, which satisfies the followingaxioms: A1.
Symmetry: For all e , e ∈ E , we have A ( e , e ) = A ( e , e ) . A2.
Redenomination: For all e , e ∈ E and every λ > , we have A ( λe , λe ) = λ · A ( e .e ) . A3.
Reciprocity: For all e , e ∈ E , we have A ( e − , e − ) = A ( e , e ) − . Then, A ( e , e ) = √ e e for all e , e ∈ E. Proof.
It will be expedient to make a logarithmic change of coordinates. In order todo this, we set α ( x, y ) = log( A ( e x , e y )), where log denotes the natural logarithm.It is easy to verify that A1 – A3 correspond to the following properties of the newfunction α , defined for all real numbers. ˆA1. For all x, y ∈ R , we have α ( x, y ) = α ( y, x ) . ˆA2. For all x, y ∈ R , and all λ > , we have α ( x + λ, y + λ ) = α ( x, y ) + λ. ˆA3. For all x, y ∈ R , we have α ( − x, − y ) = − α ( x, y ) . Now, we consider the difference h ( x, y ) = α ( x, y ) − x + y . It is clear that ˆA1 and ˆA3 imply(1) h ( x, y ) = h ( y, x ) , h ( − x, − y ) = − h ( x, y ) . From ˆA2 it follows that for all real values of λ, we have(2) h ( x + λ, y + λ ) = h ( x, y ) . Applying (2) to λ = − ( x + y ) , we get h ( x, y ) = h ( − y, − x ) = − h ( y, x ) = − h ( x, y ) , UTURE EXCHANGE RATES AND SIEGEL’S PARADOX 5 which shows that h ( x, y ) = 0 . Hence α ( x, y ) = x + y . Now, by translating this backto the original variables, we obtain A ( e , e ) = exp( α (log e , log e )) = exp (cid:18) log e + log e (cid:19) = √ e e . (cid:3) Remark 3.2.
Using the geometric mean as an aggregator has the following curiousinterpretation. It is clear that from the point of view of an investor who is interestedin a Euro to USD exchange in the future, the value of the geometric mean aggregator A ( e , e ) = √ e e can be seen as the expected value with respect to a new probabilitydistribution P [ ω ] = p, P [ ω ] = 1 − p, hence √ e e = pe + (1 − p ) e . The new probability measure always gives a larger weight to the state of the worldin which e i is smaller. Similarly, the investor who wants to exchange US dollars toEuros in the future can also view the value of the aggregator as the expected valuewith respect to some probability measure. What is interesting is that this probabilitymeasure assigns p to ω and 1 − p to ω . The reason for this is:(1 − p ) 1 e + p e = pe + (1 − p ) e e e = √ e e e e = 1 √ e e . Remark 3.3.
The proof of theorem 3.1 could be re-interpreted in the language ofgroup actions. To see this more clearly, let us consider the more general case of ag-gregators with n variables A ( e , . . . , e n ) . One can define analogously an n -variablefunction h, which now should be invariant under all the permutations P σ of the coor-dinates, as well as the translation group T λ along the vector e λ = ( λ, λ, . . . , λ ) . More-over, h has to be odd with respect to the reflection R across the origin. It is easy to seethat the group Γ generated by the transformations P σ , T λ and R is a finite extension ofa group isomorphic to the additive group of real numbers consisting of T λ . In fact, anyelement of this group can be represented by γ ( x , . . . , x n ) = ǫ ( γ )( x σ (1) , . . . , x σ ( n ) )+ e λ , where ǫ ( γ ) = ± , σ is a permutation, and λ ∈ R . Alternatively, one can representthis group by matrices of the form (cid:18) ± P σ λ j (cid:19) , where P σ is the permutation matrix associated to σ and j is the column vector whoseentries are all equal to 1 . One can show that ǫ : Γ → {± } is a character of Γ andthe axioms are in fact equivalent to the following condition on h acting on Γ-orbits: h ( γ x ) = ǫ ( γ ) h ( x ) . When n = 2 , the Γ-orbits are never injective, and this is indeedthe key to the uniqueness part of the theorem. For n ≥ , however, one can see thatthe Γ-action is generically free, that is the map γ γ · x is a bijection. FUTURE EXCHANGE RATES AND SIEGEL’S PARADOX The General Case
Our approach gives a deterministic answer to Siegel’s paradox when there are twoequally likely exchange rates in the future, namely that the geometric mean is the only possible aggregator subject to Symmetry, Redenomination and Reciprocity axioms.One might think of other variations of this problem as well, for example when thereare more than two such possible future exchange rates e , . . . , e n . An aggregator thenis a function A : E n → E, where E n denotes the n -fold Cartesian product of E withitself. We formulate the following axioms: A1.
Symmetry:
For all e , . . . , e n ∈ E , we have A ( e , . . . , e n ) = A ( e σ (1) , . . . , e σ ( n ) )for any permutation σ of the set { , , . . . , n } . A2.
Scaling:
For all e , . . . , e n ∈ E and every λ >
0, we have A ( λe , . . . , λe n ) = λ · A ( e , . . . , e n ) . A3.
Reciprocity:
For all e , . . . , e n ∈ E , we have A ( e − , . . . , e − n ) = A ( e , . . . , e n ) − . Having known the result for two rates, it would be natural to assume that the n -aggregator in this case should again be the geometric mean A ( e , . . . , e n ) = n √ e . . . e n . This function definitely satisfies the generalized axioms we have proposed above andwould be the aggregator of choice. However, as the following proposition shows, thisis not the only possibility. Let us call a collection of functions log-convex if their logarithms constitute a convex set. Recall that a set is convex if for any two points A and A in the set and any α ∈ [0 , , the convex combination (1 − α ) A + αA isalso in the set. Proposition 4.1.
For n > , there are infinitely many aggregators A : E n → E satisfying axioms A1 – A3 . The collection of all such aggregators is log -convex.Proof. Let e (1) ≤ · · · ≤ e ( n ) denote the order statistics of the sequence e , . . . , e n . In other words, e (1) is the minimum of e , . . . , e n , e (2) is the second smallest term,etc. For n ≥
3, let A ( e , . . . , e n ) denote the median of e , . . . , e n . When n is odd,this is defined to be e ( n +12 ) , and when n is even, it is defined to be √ e ( n ) e ( n +1) . Weare going to show that the median is an aggregator satisfying A1 – A3 . Notice thatthe order statistics are invariant under permutations and scale with λ > A3 . Note that the order statistics for1 /e , . . . , /e n are simply 1 /e ( n ) ≤ · · · ≤ /e (1) . This proves the claim.One can verify that if A and A are two aggregators satisfying A1 – A3 , and α ∈ [0 , , then A α = A − α A α is also an aggregator satisfying A1 – A3 . One canalso check that these aggregators are all different if A and A are different (in thiscase, the geometric mean and the median). This proves that there are infinitely manyaggregators. Moreover, since log A α is a convex combination of log A and log A , the collection of aggregators satisfying A1 – A3 is log-convex. (cid:3) UTURE EXCHANGE RATES AND SIEGEL’S PARADOX 7
We can in fact go further and characterize all aggregators satisfying A1 – A3 inany dimension. As before, consider the order statistics e (1) ≤ · · · ≤ e ( n ) and formthe consecutive ratios e ( n ) /e ( n − , e ( n − /e ( n − , . . . , e (2) /e (1) . We have the followinggeneral characterization theorem.
Theorem 4.2.
Any aggregator A : E n → E satisfying A1 – A3 is of the form (3) A ( e , . . . , e n ) = ( e . . . e n ) /n β ( e ( n ) /e ( n − , . . . , e (2) /e (1) ) , where β : ( R ≥ ) n − → R + is a function satisfying β ( u , . . . , u n − ) β ( u n − , . . . , u ) = 1 . Conversely, for any such function β, the aggregator A defined by (3) satisfies A1 – A3 . For simplicity, let us call a function β satisfying β ( u , . . . , u n − ) β ( u n − , . . . , u ) = 1a reciprocity function. Note that the constant function β = 1 is always a reciprocityfunction, and using it in formula (3) produces the familiar geometric mean aggregator.For a non-trivial example, let n = 3 and take β ( u , u ) = ( u /u ) / to obtain themedian as another aggregator, as we have verified before. Proof.
Let us prove the converse first. For any aggregator A defined as above, wecan see that it satisfies A1 – A3 by verifying how the two factors on the right-handside of the formula transform under symmetry, scaling and reciprocals. We know how( e . . . e n ) /n transforms, because it already satisfies A1 – A3 . It is also pretty clearthat the value of β ( e ( n ) /e ( n − , . . . , e (2) /e (1) ) is invariant under a scaling of e , . . . , e n , as well as their permutations, so all we need to check is how it transforms if e , . . . , e n are replaced by their reciprocals e − , . . . , e n − ; this will just reverse the order of thevariables e ( n ) /e ( n − , . . . , e (2) /e (1) , so we get the reciprocal value for β , since it is areciprocity function.This argument meanwhile implies that if formula (3) holds, then the aggregator A automatically satisfies A1 and A2 , and satisfies A3 exactly when the function β is a reciprocity function, since the n -tuple ( e ( n ) /e ( n − , . . . , e (2) /e (1) ) can assume anyarbitrary value in ( R ≥ ) n − by an appropriate choice of e , . . . , e n . Now, let us prove the other direction of the theorem. Since A is invariant underpermutations, we can assume without loss of generality that e ≤ · · · ≤ e n . We nowneed to establish the existence of a reciprocity function β : ( R + ) n − → R + such that A ( e , . . . , e n ) = ( e . . . e n ) /n β ( e n /e n − , . . . , e /e ) . As in the case of n = 2 , we set h ( x , . . . , x n ) = log( A ( e x , . . . , e x n )) − x + · · · + x n n . One can again see that as a result of A1 – A3 , h is symmetric, translation-invariant,and satisfies h ( − x , . . . , − x n ) = − h ( x , . . . , x n ) . For u , . . . , u n − ∈ R + , set β ( u n − , u n − , . . . , u ) = exp( h (0 , log u , log( u u ) , . . . , log( u . . . u n − )) . FUTURE EXCHANGE RATES AND SIEGEL’S PARADOX
The claim now follows by combining these formulas for e = e x , . . . , e n = e x n . Wejust need to keep in mind that, by translation invariance, h ( x , . . . , x n ) = h (0 , x − x , . . . , x n − x ) . As we noted earlier, since formula (3) holds and A satisfies the reciprocity axiom A3 , β has to be a reciprocity function. However, to see this explicitly, write h (0 , log u , log( u u ) , . . . , log( u . . . u n − ) = − h (0 , − log u , . . . , − log( u . . . u n − ))= − h (log( u . . . u n − ) , log(( u . . . u n − )) , . . . , − h (0 , log u n − , . . . , log( u . . . u n − )) , where we have translated the variables by log( u . . . u n − ) in the second step andused symmetry in the last. Exponentiating the first and the last terms here produce β ( u n − , u n − , . . . , u ) and β ( u , u , . . . , u n − ) − . (cid:3) Discussion
We have been able to classify all aggregators satisfying A1 – A3 in terms of thegeometric mean and a reciprocity function β, as in theorem 4.2. For n = 2 , thisfunction β is identical to 1, so we obtain a unique aggregator in this case, but there areplenty of options in other cases. It would be interesting to find out an interpretationof the reciprocity function β in economic terms. This could help our understandingof aggregators and may even point to some natural axioms or constraints that couldnarrow down the possibilities for an aggregator or even determine it uniquely. In allcases, however, the geometric mean stands out as the trivial aggregator of choice.Another direction in which the problem could be generalized would be when thefuture exchange rates e , . . . , e n are not necessarily equally likely, but occur withprobabilities p , . . . , p n , respectively. It would be fair to assume that the followingweighted geometric mean would be the natural aggregator in this case.(4) A ( e , . . . , e n ) = e p e p . . . e p n n . One could in fact argue for this aggregator, at least when p , . . . , p n are all rationalnumbers, but should take extra care in interpreting the Symmetry axiom A1 . Todemonstrate this case, let us assume that there are only two possible future exchangerates e and e , occurring with probabilities m/ ( m + n ) and n/ ( m + n ) , respectively.We can recast this situation as the case of m + n equally likely future exchange rates,where m of those possibilities are e and the rest are e . Then we can apply theorem 4.2to find all the aggregators in this case, in particular our distinguished geometric mean,which would be a weighted one, as suggested in equation (4). This argument could beadapted to extend to any number of future exchange rates with rational probabilities,but runs into difficulty when the probabilities p , . . . , p n , contain irrational numbers.Fixing this issue may require an appropriate modification of the Symmetry axiom orformulation of an additional Continuity axiom to extend the results to the generalcase and could open up new avenues for investigation.
UTURE EXCHANGE RATES AND SIEGEL’S PARADOX 9 Acknowledgements
We wish to thank Hamed Ghoddusi for introducing Siegel’s paradox to us andcommenting on an earlier draft of this paper. We would also like to thank HazhirRahmandad and Hassan Tehranian for their helpful comments on the paper whichled to improvements in the text, as well as Ken Rogoff for his interest in our work.
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