Means in money exchange operations
MM EANS IN MONEY EXCHANGE OPERATIONS
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Jacek Bojarski
Institute of MathematicsUniversity of Zielona GóraSzafrana 4A, PL 65-516 Zielona Góra, Poland [email protected]
Cinkciarz.plSienkiewicza 9, PL 65-001 Zielona Góra, Poland [email protected]
Janusz Matkowski
Institute of MathematicsUniversity of Zielona GóraSzafrana 4A, PL 65-516 Zielona Góra, Poland [email protected]
January 20, 2021 A BSTRACT
It is observed that in some money exchange operations, the applied n -variable mean M should beself reciprocally-conjugate, i.e. it should satisfy the equality M ( x , . . . , x n ) M (cid:18) x , . . . , x n (cid:19) = 1 , x , . . . , x n > . The main result says that the only weighted quasiarithmetic mean satisfying this condition is theweighet geometric mean.
Keywords
Mean · Money exchange · Weighted quasiarithmetic mean · Functional equation
Primary 91B26, 26E60 Secondary 39B22
Motivated by some money exchange operations, we consider the n -variable means M acting in the interval (0 , ∞ ) and satisfying the condition M ( x , . . . , x n ) M (cid:18) x , . . . , x n (cid:19) = 1 , x , . . . , x n > . (1)The role of this condition, on the model of work of two currency market analysts, acting in two different countries, isexplained in Section 2. In Section 3 we recall the basic notions concerning means ([1]). For a homeomorphic mapping ϕ and a mean M on (0 , ∞ ) , we define M [ ϕ ] , a ϕ -conjugate mean, and we remark that a mean M satisfies condition(1), iff M is self-reciprocally conjugate, which holds true, if and only if M [exp] is an odd mean on R (Remark 3). InSection 4 we recall some basic facts on weighted quasiarithmetic means. In Section 5 we determine the form of allodd weighted quasiarithmetic means in R (Proposition 1), and then, making use of Remark 3, we establish the formof all weighted quasiarithmetic means in (0 , ∞ ) satisfying condition (1) (Theorem 1). Assuming homogeneity of themean M , a "good" property of a mean, this result leads to Theorem 2, which says that the geometric weighted mean M = G p ,...,p n , G p ,...,p n ( x , . . . , x n ) := x p · . . . · x p n n , x , . . . , x n > is the only homogeneous n -variable weighted quasiarithmetic mean with weights p , . . . , p n > , p + . . . + p n = 1 ,satisfying condition (1) .Our considerations show that the geometric mean is the most proper tool in the money exchange operations. a r X i v : . [ m a t h . C A ] J a n eans in money exchange operations A P
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To explain the problem, consider, for example two currency market analysts in Great Britain and the United States,dealing with GBP and USD. For the sake of clarity, let us assume that the bid and ask rates are the same and norounding is applied.The UK analyst analyses the USD/GBP rate (denoted by x ), where USD is the base currency and GPB is the quotecurrency. The analyst in the United States analyses the rate of the same pair, but in a different ordering GBP/USD(denote it by y ), where GBP is the base currency and USD is the quote currency. Then, to avoid arbitration, for anygiven time t the condition x t · y t = 1 must be met. Suppose further that both analysts record the rates at the sametimes t , t , . . . , t n . Then, in Great Britain, the analyst will observe USD/GBP exchange rates ( x t , x t , . . . , x t n ) , and in the United States, GBP/USD exchange rates will be: ( y t , y t , . . . , y t n ) = (cid:18) x t , x t , . . . , x t n (cid:19) . Note that based on their observations, analysts know what the other analyst is observing. Analysts use many param-eters/indicators to facilitate inference. The basic one, frequently used, is the arithmetic mean of the course values(hourly, daily, weekly, etc.). But it is easy to see that the equality n n (cid:88) j =1 x t j = 1 n (cid:80) nj =1 1 x tj holds if and only if x t = . . . = x t n , which means that analysts knowing their own arithmetic mean do not know whatthe mean of the other analyst is.In this situation appears a natural problem of characterization of means M satisfying the relationship M ( x , x , . . . , x n ) = 1 M (cid:16) x , x , . . . , x n (cid:17) , n ≥ , x i > , i = 1 , , . . . , n. Let I ⊂ R be an interval and let n ∈ N , n ≥ be fixed.Recall that a function M : I n → I is called an n - variable mean in I , if min ( x , . . . , x n ) ≤ M ( x , . . . , x n ) ≤ max ( x , . . . , x n ) , ( x , . . . , x n ) ∈ I n , and the mean is called strict, if these inequalities are sharp for all nonconstant sequences ( x , . . . , x n ) ∈ I n .A mean M : I n → I is called symmetric , if M (cid:0) x σ (1) , . . . , x σ ( n ) (cid:1) = M ( x , . . . , x n ) for each permutation σ of { , . . . , n } .A mean M : (0 , ∞ ) n → (0 , ∞ ) is called homogeneous if M ( tx , . . . , tx n ) = tM ( x , . . . , x n ) , t, x , . . . , x n > . Remark 1.
Let J ⊂ R be an interval, ϕ : J → I be a homeomorphic mapping of J onto I . If M : I n → I is a mean,then the function M [ ϕ ] : J n → J defined by M [ ϕ ] ( x , . . . , x n ) := ϕ − ( M ( ϕ ( x ) , . . . , ϕ ( x n ))) , ( x , . . . , x n ) ∈ J n , is a mean in J. The mean M [ ϕ ] is called ϕ -conjugate of M .Moreover, if J = I and M [ ϕ ] = M , then M is said to be ϕ -self conjugate. Taking for ϕ the reciprocal function, i.e. ϕ ( t ) = t for t > , we get the following Remark 2.
A mean M : (0 , ∞ ) n → (0 , ∞ ) satisfies condition (1) if and only if, it is reciprocally self-conjugate. Let us note the following obvious 2eans in money exchange operations
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Remark 3.
Let n ∈ N , n ≥ . A mean M : (0 , ∞ ) n → (0 , ∞ ) satisfies (1) if and only if, the exponentially conjugatemean M [exp] : R n → R , M [exp] ( t , . . . , t n ) := ln M (cid:0) e t , . . . , e t n (cid:1) , t , . . . , t n ∈ R , it is odd, that is M [exp] ( − t , . . . , − t n ) = − M [exp] ( t , . . . , t n ) , t , . . . , t n ∈ R . By the way let us note that there is no even mean.
Recall a description of the quasiarithmetic means, one of the most important classes of means. These means are closelyrelated to the conjugacy notion, as they are conjugate to the weighted arithmetic means.
Remark 4 (see for instance Chapter III in [3]) . Let ϕ : I → R be a continuous and strictly monotonic function, andlet p , . . . , p n > be such that p + . . . + p n = 1 . Then the function A [ ϕ ] p ,...,p n : I n → I, given by A [ ϕ ] p ,...,p n ( t , . . . , t n ) := ϕ − n (cid:88) j =1 p j ϕ ( t j ) , is a strict mean in I, and it is called a weighted quasiarithmetic mean; the function ϕ is referred to as its generator,and p , . . . , p n as its weights. A [ ϕ ] p ,...,p n is symmetric iff p = . . . = p n = n ; and then this mean, denoted by A [ ϕ ] , is called quasiarithmetic.Moreover, if ϕ, ψ : I → R , then A [ ψ ] p ,...,p n = A [ ϕ ] p ,...,p n if and only if ψ = aϕ + b for some real a, b , a (cid:54) = 0 . Remark 5 ([3], p.68) . Let I = (0 , ∞ ) . The following conditions are equivalent: ( i ) the mean A [ ϕ ] p ,...,p n is homogeneous; ( ii ) for some a, b, r ∈ R , a (cid:54) = 0 , ϕ ( t ) = (cid:26) a t r + b if r (cid:54) = 0 a log t + b if r = 0 , t ∈ (0 , ∞ ) ;( iii ) for some r ∈ R , A [ ϕ ] p ,...,p n = (cid:16)(cid:80) nj =1 p j x rj (cid:17) /r if r (cid:54) = 0 n (cid:81) j =1 x p j j if r = 0 , x , . . . , x n ∈ (0 , ∞ ) . Applying the last result of Remark 1 we obtain the following
Proposition 1.
Let n ∈ N , n ≥ . Suppose that ϕ : (0 , ∞ ) → R is a continuous strictly monotonic function, and p , . . . , p n > are such that p + . . . + p n = 1 . Then the following two conditions are equivalent: ( i ) the weighted quasiarithmetic mean A [ ϕ ] p ,...,p n : (0 , ∞ ) n → (0 , ∞ ) satisfies (1) , i.e. A [ ϕ ] p ,...,p n ( x , . . . , x n ) A [ ϕ ] p ,...,p n (cid:18) x , . . . , x n (cid:19) = 1 , x , . . . , x n > ii ) there are a, b ∈ R , a (cid:54) = 0 such that ϕ satisfies the functional equation ϕ (cid:18) x (cid:19) = aϕ ( x ) + b, x ∈ (0 , ∞ ) . Proof.
Note that (1) holds if and only if A [ ϕ ] p ,...,p n ( x , . . . , x n ) = 1 A [ ϕ ] p ,...,p n (cid:16) x , . . . , x n (cid:17) , x , . . . , x n > , A P
REPRINT that is, if and only if A [ ϕ ] p ,...,p n = A [ ϕ ◦ β ] p ,...,p n , where β : (0 , ∞ ) → (0 , ∞ ) denotes the reciprocal function β ( x ) = x . In view of Remark 1 (see also [3], p.66), thisequality holds if and only if there are a, b ∈ R , a (cid:54) = 0 such that ( ϕ ◦ β ) ( x ) = aϕ ( x ) + b for all x ∈ (0 , ∞ ) , that isif, and only if (2) holds. By Remark 3, a mean M on (0 , ∞ ) satisfies condition (1) iff the mean M [exp] is odd on R . Stimulated by this fact weprove
Proposition 2.
Let n ∈ N , n ≥ and let the positive numbers p , . . . , p n such that p + . . . + p n = 1 be fixed.Suppose that ϕ : R → R is a continuous strictly monotonic function . Then the following conditions are equivalent: ( i ) the quasiarithmetic weighted mean A [ ϕ ] p ,...,p n : R n → R is odd; ( ii ) the function ϕ − ϕ (0) is odd or, equivalently, ϕ ( − t ) + ϕ ( t ) = 2 ϕ (0) , t ∈ R . Proof.
To prove the implication (i) = ⇒ (ii), assume that A [ ϕ ] p ,...,p n is an odd function. By the definition of A [ ϕ ] p ,...,p n itmeans that ϕ − n (cid:88) j =1 p j ϕ ( − t j ) = − ϕ − n (cid:88) j =1 p j ϕ ( t j ) , t , . . . , t n ∈ R . Choosing arbitrarily j ∈ { , . . . , n − } and setting p := p + . . . + p j ,t = . . . = t j = s, t j +1 = . . . = t n = t, we hence get ϕ − ( pϕ ( − s ) + (1 − p ) ϕ ( − t )) = − ϕ − ( pϕ ( s ) + (1 − p ) ϕ ( t )) , s, t ∈ R . For s = ϕ − ( u ) , t = ϕ − ( v ) , we have ϕ − (cid:0) pϕ (cid:0) − ϕ − ( u ) (cid:1) + (1 − p ) ϕ (cid:0) − ϕ − ( v ) (cid:1)(cid:1) = − ϕ − ( pu + (1 − p ) v ) , u, v ∈ ϕ ( R ) . and, taking ϕ of both sides pϕ (cid:0) − ϕ − ( u ) (cid:1) + (1 − p ) ϕ (cid:0) − ϕ − ( v ) (cid:1) = ϕ (cid:0) − ϕ − ( pu + (1 − p ) v ) (cid:1) , u, v ∈ ϕ ( R ) . Hence, setting ψ := ϕ ◦ (cid:0) − ϕ − (cid:1) , (2)we get ψ ( pu + (1 − p ) v ) = pψ ( u ) + (1 − p ) ψ ( v ) , u, v ∈ ϕ ( R ) . This equation and the Daróczy-Páles identity (see [2]) p (cid:18) p u + v − p ) u (cid:19) + (1 − p ) (cid:18) pv + (1 − p ) u + v (cid:19) = u + v , x, y ∈ R easily imply that ψ satisfies the Jensen functional equation ψ (cid:18) u + v (cid:19) = ψ ( u ) + ψ ( v )2 , u, v ∈ ϕ ( R ) . Since ψ is continuous in the interval ϕ ( R ) , in view of Theorem 1, p. 315 in M. Kuczma [4], there are some constant a, b ∈ R , a (cid:54) = 0 such that ψ ( u ) = au + b, u ∈ ϕ ( R ) . Hence, by (2), ϕ ◦ (cid:0) − ϕ − (cid:1) ( u ) = au + b, u ∈ ϕ ( R ) , A P
REPRINT whence, setting u = ϕ ( t ) , we get ϕ ( − t ) = aϕ ( t ) + b, t ∈ R . (3)Replacing here first t by − t , and then applying this equation again, we get ϕ ( t ) = aϕ ( − t ) + b = a [ aϕ ( t ) + b ] + b = a ϕ ( t ) + ab + b, whence (cid:0) − a (cid:1) ϕ ( t ) = ab + b, t ∈ R . (4)Since the right side is constant, it follows that a = 1 . Consequently, either a = 1 or a = − . If a = 1 then by (4) the number b must be and, by (3) we get ϕ ( − t ) = ϕ ( t ) , t ∈ R , that is a contradiction, as the function ϕ is strictly monotonic.If a = − then b in (4) can be arbitrary and (3) becomes ϕ ( − t ) = − ϕ ( t ) + b, t ∈ R . Setting here t = 0 gives ϕ (0) = − ϕ (0) + b , whence b = 2 ϕ (0) . Thus, by (3), ϕ ( − t ) − ϕ (0) = − [ ϕ ( t ) − ϕ (0)] , t ∈ R , so the function ϕ − ϕ (0) is odd, which proves the implication (i) = ⇒ (ii).To prove the reversed implication assume that ϕ − ϕ (0) is odd. Then, the inverse function ϕ − ( y + ϕ (0)) is alsoodd. Applying these facts in turn, by the definition of A [ ϕ ] p ,...,p n , we have A [ ϕ ] p ,...,p n ( − t , . . . , − t n ) = ϕ − n (cid:88) j =1 p j ϕ ( − t j ) = ϕ − n (cid:88) j =1 p j [ ϕ ( − t j ) − ϕ (0)] + ϕ (0) = ϕ − − n (cid:88) j =1 p j [ ϕ ( t j ) − ϕ (0)] + ϕ (0) = − ϕ − n (cid:88) j =1 p j [ ϕ ( − t j ) − ϕ (0)] + ϕ (0) = − ϕ − n (cid:88) j =1 p j ϕ ( t j ) = −A [ ϕ ] p ,...,p n ( t , . . . , t n ) for all t , . . . , t n ∈ R , which completes the proof. We begin this section with the following
Remark 6.
Let n ∈ N , n ≥ , and J ⊂ (0 , ∞ ) be a fixed interval. If γ : J → (0 , ∞ ) is a continuous strictlymonotonic function, and the real numbers p , . . . , p n > are such that p + . . . + p n = 1 , then ( i ) the function M [ γ ] p ,...,p n : I n → I given by M [ γ ] p ,...,p n ( x , . . . , x n ) := γ − n (cid:89) j =1 [ γ ( x j )] p j is a strict mean in I ; A P
REPRINT ( ii ) M [ γ ] p ,...,p n = A [log ◦ γ ] p ,...,p n . Proof.
Indeed, for arbitrary x , . . . ., x n > , M [ γ ] p ,...,p n ( x , . . . , x n ) = γ − n (cid:89) j =1 [ γ ( x j )] p j = γ − exp log n (cid:89) j =1 [ γ ( x j )] p j = (log ◦ γ ) − n (cid:88) j =1 p j (log ◦ γ ) ( x j ) = A [log ◦ γ ] p ,...,p n ( x , . . . , x n ) , so (ii) holds true. Since A [log ◦ γ ] p ,...,p n is a quasiarithmetic mean, condition (ii) implies (i).By analogy to the quasiarithmetic means, the mean M [ γ ] p ,...,p n could be called a weighted quasigeometric mean, thefunction γ its generator, and p , . . . , p n , the weights.This remark, Remark 3 and Proposition 1 give the following characterization of weighted quasiarithmetic (or weightedquasigeometric) means on (0 , ∞ ) and satisfying condition (1). Theorem 1.
Let n ∈ N , n ≥ , and let the positive numbers p , . . . , p n such that p + . . . + p n = 1 be fixed. Supposethat γ : (0 , ∞ ) → (0 , ∞ ) is continuous and strictly monotonic. Then the following conditions are equivalent: ( i ) the mean M [ γ ] p ,...,p n : (0 , ∞ ) → (0 , ∞ ) satisfies condition M [ γ ] p ,...,p n ( x , . . . , x n ) M [ γ ] p ,...,p n (cid:18) x , . . . , x n (cid:19) = 1 , x , . . . , x n > ii ) the generator γ satisfies the functional equation γ ( x ) γ (cid:18) x (cid:19) = [ γ (1)] , x > . The main result of this paper reads as follows:
Theorem 2.
Let n ∈ N , n ≥ and let the positive numbers p , . . . , p n be such that p + . . . + p n = 1 . Suppose that γ : (0 , ∞ ) → (0 , ∞ ) is continuous and strictly monotonic. Then the following conditions are equivalent: ( i ) the mean M [ γ ] p ,...,p n : (0 , ∞ ) n → (0 , ∞ ) is homogeneous in (0 , ∞ ) and M [ γ ] p ,...,p n ( x , . . . , x n ) M [ γ ] p ,...,p n (cid:18) x , . . . , x n (cid:19) = 1 , x , . . . , x n > ii ) there are positive a, b ∈ R , a (cid:54) = 0 such that γ ( t ) = e b t a , t > iii ) M [ γ ] p ,...,p n = G p ,...,p n , where G p ,...,p n ( x , . . . , x n ) := x p · . . . · x p n n , x , . . . , x n > , is the weighted geometric mean.Proof. Assume (i). By Remark 6(ii) we have M [ γ ] p ,...,p n = A [log ◦ γ ] p ,...,p n . The homogeneity of the weighted quasiarith-metic mean A [log ◦ γ ] p ,...,p n implies (see [3] p. 68) that there are a, b, r ∈ R , a (cid:54) = 0 such that log ◦ γ ( t ) = (cid:26) at r + b if r (cid:54) = 0 a log t + b if r = 0 , t > . If r (cid:54) = 0 then γ ( t ) = b exp ( at r ) , t > , A P
REPRINT whence γ ( t ) γ (cid:18) t (cid:19) = b exp ( at r ) b exp (cid:18) a (cid:18) t (cid:19) r (cid:19) = b exp (cid:0) a (cid:0) t r + t − r (cid:1)(cid:1) , so the function (0 , ∞ ) (cid:51) t → γ ( t ) γ (cid:0) t (cid:1) is not constant.If r = 0 then γ ( t ) = e b t a , t > , whence γ ( t ) γ (cid:18) t (cid:19) = (cid:0) e b t a (cid:1) (cid:0) e b t − a (cid:1) = (cid:0) e b (cid:1) = [ γ (1)] , t > . Now (ii) follows from Theorem 1.Assume (ii). Then γ − ( u ) = e − ba u a , u ∈ γ (0 , ∞ ) , and, taking into account that p + . . . + p n = 1 , we have, for all x , . . . , x n > , M [ γ ] p ,...,p n ( x , . . . , x n ) = γ − n (cid:89) j =1 [ γ ( x j )] p j = e − ba n (cid:89) j =1 (cid:0) e b x aj (cid:1) p j /a = n (cid:89) j =1 x p j j = G p ,...,p n ( x , . . . , x n ) , so (iii) holds true.Since the implication ( iii ) = ⇒ ( i ) is obvious, the proof is complete. Remark 7.
Let us note that the main results of this paper can be also extended to the class generalized weightedquasiarithmetic means of the form M [ g ,...,g n ] ( x , . . . , x n ) := n (cid:89) j =1 g j − n (cid:89) j =1 g j ( x j ) where g , . . . , g n : (0 , ∞ ) → (0 , ∞ ) are continuous strictly increasing (or strictly decreasing) functions and n (cid:81) j =1 g j isa single variable function being the product of g . . . g n (see [5]). Also in this case, if M [ g ,...,g n ] is homogeneous andsatisfies (1), then M [ g ,...,g n ] = G p ,...,p n for some positive p , . . . , p n such that p + . . . + p n = 1 . We conclude this paper with the following
Remark 8.
In the family of all homogeneous weighted quasiarithmetic means, only the geometric mean is self recip-rocally conjugate, i.e. satisfies condition (1).
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An Introduction to the Theory of Functional Equations, Cauchy’s Equation and Jensen’s Inequality .Polish Scientific Editors and Silesian University, 1985.[5] J. Matkowski. Generalized weighted quasi-arithmetic means.