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The purpose of this paper is to study on-line portfolio selection strategies for currency exchange markets and our focus is on the markets with presence of decrements. To this end, we first analyze the main factors arising in the decrements. Then we develop a cross rate scheme which enables us to establish an on-line portfolio selection strategy for the currency exchange markets with presence of decrements. Finally, we prove the universality of our on-line portfolio selections.

We are concerned with a currency exchange market in the presence of decrements. Our objective is to develop an on-line portfolio selection strategy for such a market. To this end, we utilise a cross rate method to establish a suitable algorithm scheme.

The problem of establishing on-line portfolio selection schemes for various financial markets has been well studied (see, e.g. the monograph [

The on-line portfolio selections can be identified as active portfolio strategies (see, e.g., [

Our consideration follows preliminarily the work of [

The paper is organised as follows. Section 2, the next section, starts with the mathe- matical framework setting for the currency exchange market with decrements. Then we introduce two kinds of exponential growth rates. We explicate the occurrence of decrements at the business day

The currency market we are concerned is described as follows. Let

・ We assume that the currency exchange market contains m different currencies, where

・ We suppose that currencies in the market can be perfectly divisible and that at the

・ For

・ We denote the vector of the return of m currencies at the

・ A portfolio vector

・ The return of the portfolio

・ The decrement factors

where

・ Let

・

An exchange rate at a given point in time represents the price of the involved currency with respect to a reference currency. It is clear that the currency exchange rates in the currency market depend on the both demand and supply for the currencies. There are two kinds of factors affecting the exchange rates which are: 1) Trade-related factor refers to the relative inflation rate, the income level and the government trade restriction; 2) Financial factor which refers to the relative interest rates and the capital flow restrictions (these two factors directly affect the demand and supply of the currencies). Thus the exchange rates should be varied with the two factors [

The following diagram takes the US dollars and the British pounds as an example, showing the relationships of the four elements between the currency demand and the supply, respectively.

The vertical axis represents the price of US dollars in Sterling (i.e., changing of the exchange rate) and horizontal axis represents the quantity of US dollars demand by the Sterlings.

・ Solid line 1 represents the demand of the US dollars in the UK , when the price of the US dollars goes to expensive the demand of the US dollars will be lower, otherwise when the US dollars depreciates with the Sterlings the demand of the US dollars will be higher.

・ Solid line 2 represents the supply of the US dollars in the UK , when the price of the US dollars becomes too expensive the supply of the US dollars will get higher due to fewer US dollars would then buy more Sterlings, the UK commodities are cheaper therefore more supply of the US dollars to be purchased in the UK. On the contrary, when the US dollars depreciate with the Sterlings the supply of the US dollars should be lower. The crossing point of the two solid lines represents the equilibrium exchange rate.

・ Changes in the relative inflation rate will affect international trade activities, which influence the demand and the supply for currencies and therefore affect the exchange rates. The impact of rising the UK inflation rate while the US inflation remains the same are represented by the dotted lines 3 and 4, the increasing of the UK inflation rate would cause an increasing demand for US goods in the UK, leading to an increase in the demand for the US dollars in the UK. At the same time the inflation rises in the UK will reduce the export that indicates the supply of the US dollars will be reduced. Therefore, the crossing point of the lines 3 and 4 is the new exchange rate, which means the appreciation of the US dollars.

・ Changes in the relative interest rates also affect the investment in foreign securities, which influences the demand and the supply for currencies and therefore influences the exchange rates. The impact of rising the UK interest rates while the US interest rates remain the same are represented by the dotted lines 5 and 6. When the interest rates increase in the UK, the investors will reduce the hold of the US dollars and invest Sterling to earn the high interests in the UK, thus the demand of the US dollars is reduced and the supply is increased. If the price of the US dollars is reduced to be lower, then the investors would establish more bank deposits in the UK.The crossing point of the lines 5 and 6 is the new exchange rate, which means the depreciation of the US dollars.

・ The income level affects the demand amount of imports. Hence, it influences the exchange rate. Rising the income level in the UK while the US income level keeps the same will lead to increasing of the demand for US goods, but the supply of US dollars is not changed. Then the crossing point of the lines 7 and 2 is the new exchange rate, which shows an appreciation of the US dollars.

・ The taxation represents a kind of market frictions. It affects both the demand and the supply. The more tax the government has the less inflation occurs. Hence the taxation and the inflation rates have a negative correlation which means when taxation goes up the inflation rate goes down. As we known, the interest rates and the inflation rates have a negative correlation, we let taxation in the digram follow the change of interest rates as the dotted lines 5 and 6.

The above four elements are inter-dependent on one another. For example, when the income level goes up and the taxation goes up simultaneously, then that the inflation goes down leads to the interest rate goes up.

We let

while when four elements goes down, we let

The on-line portfolio is an active portfolio strategy, the investor using all the historical data of the

・ Let the portfolio for the day

where f is a function of the returns of the currency and the previous portfolios.

・ Without the decrements, the whole investment can be increased by the following factor

・ Let

here we assume

・ With the decrements, the whole investment can be increased by the following factor

・ Let

・ Let

For self-financing portfolios, for

At the beginning of the

The decrement of the currency i is

We assume that the decrements remain the same for both sale and buy, then the total decrement at the beginning of the

In the day

We define

On the other hand, we have that

If

At the end of the day k, we set a portfolio contains m currencies in the currency exchange market as

where the

Suppose that

As the funds at the end of the trading day k are equal to the beginning of the trading day

From above, we known that at the beginning of the day

In the currency exchange market, investors usually prefer to exchange their currencies in portfolios to earn more profit, but simultaneously, investors need to pay more attention to those decrements linking to any costs incurred in the trading which might cause a gain reduction of the portfolios.

At the trading day

where

The first form of that

Next, suppose

which is just the first order approximation in the Taylor expansion of the function

For

which is clearly a positive continuous convex function of

With these

1) Combining (24), (25) and (27) we get

2) Combining (24), (26) and (27) we get

It is clear that the function

Applying the Lagrange’s method (see, e.g., [

where

Another portfolio vector

We call (31) the update rule of the Exponential Growth the of Fund with Decrements (EGRFD) for the

One can see in the update rules (30) and (31) that there are two selective elements

Remark 3.1. At the day

Remark 3.2. In the real world currency exchange market, the investors prefer to remove the unprofitable currencies and to the add profitable currencies meanwhile to avoid the decrements, therefore, update rules (30) and (31) could be utilised.

In the currency exchange market, an important issue is to predicate the return of the investment. In this section, we would like to establish a method to select good currencies to replacing those bad ones in the portfolio. Our new prediction method is the so-called cross rate method, which concerns the order of the currencies returns rather than the returns of currencies themselves.

Here we only consider two currencies to discuss the order of the currency return for the aim of maximising the gain of the portfolio. To discuss the order of the currency, we introduce the cross rate step by step. The idea follows [

Given a sequence of daily returns

as the order of the return vector

We say that the sequence

for all

The segment is defined as

where J and K are integers such that

where

Let the cross rate of

We divide the sequence

Let

and define

We outline three steps to get the prediction of a

・ To estimate

For

where

・ One can predict the order of the return of the currencies, here we predicate the order as following

and

where

・ Follow the step 2 we have

and for MPO2

From above three steps we get the

In the last subsection, we develop the CR method for the strictly unequal sequence

We define segment as

where

Clearly,

where

Same as before, we divide the sequence

Similar to the subsection 4.1, the prediction of

・ The prediction of the

and

・ In terms of

and

・ For MPO1', we have

and for MPO2', we have

For the segment

and if

then CR(MPCR,MPO,

The investors can rearrange their portfolio in the profitable direction by using the two update rules FFGD (30) or the EGRFD (31) with effective CR(MPCR, MPO,

The entire sequence

Lemma 5.1. Assume either

or

then CR(MPCRi, MPOj,

Proof. We only show that CR(MPCR1, MPO2,

close interval from

than half of the correct order of the segment

□

By the above lemma, one can see that if both

same intervals

Next, we recall from [

Lemma 5.2. If a sequence

Based on this lemma, the profitability of the FFGD and the EGRFD can be obtained. We state the following

Theorem 5.1. Let the sequence of cross rate

(1) if

then the FFGD or the EGRFD with CR(MPCRi, MPOj,

(2) if

then the FFGD or the EGRFD with CR(MPCRi, MPOj,

Proof. We only prove case (1) and the proof of case (2) is similar. We let

for

due to that if either (61) or (62) holds, then CR(MPCR1, MPO2,

Next, applying Lemma 5.2 to (60), we have

This completes the proof. □

This theorem combined with Subsection 4.1 gives the reasons for the selection of the MPCR1 and the MPCR2.

Remark 5.1. In practice, aiming to get the high profit, one suggest investor to choose two currencies with one of

greater than

then the portfolio strategy actually has

The feature of universality has been discussed in many papers. Here we follow [

Let us define the exponential growth rate of investment of the

then the corresponding exponential growth rate of investment with decrements of the FFGD or the EGRFD algorithm can be defined as follows

where for all k and i,

Theorem 5.2. For any

(a) for the FFGD algorithm (30),

(b) for the EGRFD algorithm (31),

(c) If (30) is the selection strategy, we have

(d) If (31) is the selection strategy, we have

where

The best currency in the buy-and-hold portfolio will be selected by the two update rules FFGD and EGRFD step by step. In practice, we divide any fixed integer

where

With the above division,

Let us finally consider the sequence of the return

where

With the similar approach, we have for the FFGD

Hence, we get following corollary

Corollary 1. A universal portfolio strategy w.r.t.the set of all buy-and-hold portfolios can be represented as the FFGD or the EGRFD algorithm with the linear prediction (containing the prediction under EMH). In a long-term investment, the exponential growth rate of funds with decrements in terms of the FFGD or the EGRFD algorithm is larger or equal to the exponential growth rage of funds achieved by the single best currency.

Remark 5.2. When

We have shown that the investors follow two update rules with the cross rate method to obtain more than half probability to reschedule the portfolio vector profitability and these update rules can also be considered as universality for the investors to measure on-line portfolios, by following the main ideas of [

This paper introduces a universal prediction method for on-line portfolio selection. We introduce the decrements first, which formed by four elements: the inflation rates, the interest rates, the income level and the taxation. These four elements strongly influence the volatility of the currency price during the transaction. In the present paper, we treat the decrements as any costs during the trading or we can say that any reduction of the profit during the transaction. To optimise the portfolio, we introduce two update rules with the decrements for any sequence of return vectors which maximise the increment of the investment and minimising the decrements.

We divide the sequence of the relative prices into the equal segments, and then we

predict the order of the currencies returns by the cross rate whether belongs to

is useful for the active portfolio strategy. More transactions in a trading day means that the currency has high price volatility, which implies greater decrement amount.

In our consideration, we focus on the on-line portfolio selection with the prediction method in the currency exchange markets. This strategy has showed success in [

We would like to mention that we have not yet tested our scheme developed in this paper with existing data from the currency exchange markets. Also, in the present paper, we focus only on the decrements. It would be interesting to extend our consideration in combining other factors, for instance, the transaction costs, the information costs, etc., just mention a few. We plan to do these topics in our future work.

We thank the referee for constructive comments.

Ren, P.P. and Wu, J.L. (2016) On-Line Portfolio Selection for a Currency Exchange Market. Journal of Mathe- matical Finance, 6, 471-488. http://dx.doi.org/10.4236/jmf.2016.64038