Covered Interest Parity and Transaction Costs: A synthesis of theory and practice
Wolfgang Schultze* University of Augsburg May 1994
Abstract: This paper analyses departures from covered interest parity (CIP) from the perspective of financial institutions’ procedures for conducting forward transactions. If it is in fact true that brokers actually compute the prices for forward transactions based on interest rate-differentials and on formulae that are consitent with CIP, then there should be no discrepancies from covered interest parity at all. The paper finds that devitations from CIP – often referred to as the neutral band – should be attributed to problems in measurement, not to fundamental factors. Keywords: covered interest parity, currency valuation, forward transactions
*Wolfgang Schultze, School of Accounting & Control, University of Augsburg, Universitaetsstr. 16, 86159 Augsburg, Germany. Tel: +49-821-598-4130, E-mail:
[email protected].
2
Covered Interest Parity and Transaction Costs: A synthesis of theory and practice
1. Introduction Covered interest parity (CIP) is “one of the best-documented relationships in international finance”.1 However it is still not clear whether it holds or not. Most papers in international finance conclude that there are generally smaller or larger deviations from CIP – due to transaction costs, political risks and liquidity premiums.2 These deviations are generally refered to as the “neutral band” around the parity-function in which arbitrage is not profitable and consequently the international capital markets do not diminish these deviations.3 On the other hand it is widely acknowledged that brokers actually compute the prices for forward transactions based on interest rate-differentials.4 If that is true, then there should be no discrepancies from covered interest parity at all.5 Therefore the neutral band should be attributed to problems in measurement, not to other factors. The first part of this paper reviews the existing literature about CIP and the empirical evidence on the extent to which it holds. In the second part the law of one price is applied consistently to the spot/forward exchange and the securities markets, deriving no-arbitrage conditions for these markets under recognition of "real-world"-transaction cost. These conditions are then used to develop a set of measures for parity and the width of the Neutral Band. The final part concludes that the standard CIP-relationship in combination with estimated transaction-costs is not appropriate for the measurement of covered interest arbitrage in a world with bid-ask-spreads. 2. Studies on covered interest parity There is extensive literature about the question whether CIP holds and to which extent. The existing literatur can be divided into two groups: The first is mainly concerned with market efficiency and is therefore interested in the occurrence of deviations from CIP and arising 1 2 3 4 5
Shapiro (1992), pg. 53. See e.g. Giddy, I. H. (1994), pg. 63; Levi, M. D. (1990), pg. 171. See Levi (1990), pp. 171. See Levi (1990), pg. 157; Shapiro (1992), pg. 53. In fact, a hand-book for management of currency-risk exposure of the German Sparkassen-group, a major commercial banking organization, "Die Absicherung von Fremdwährungsrisiken" by von Hiering/Jacobs, states a formula to compute the Swap-premium (discount)in the forward market, which is nothing but an operationalized version of the no-arbitrage-condition of CIP, as is shown in the Appendix.
3 possibilities for arbitrage and excess returns. The second is interested in its implications for macroeconomics and confines itself to the question whether CIP is an accurate concept for the description of the foreign exchange markets on average and in the long run. THORNTON (1993) concludes that this holds true and "the CIP-assumption used in many theoretical models is appropriate, so long as it is not required to hold at every point in time."6 This article is interested in revealing the reasons for deviations in the short run. The first group has mainly tried to estimate the percentage transactions costs involved in the arbitrage process to see if observed deviations from CIP could be explained by them. These transaction cost cause a "Neutral Band" around the parity-line. Inside this band arbitrage is not profitable and market forces fail to drive deviations from the line back to parity. Older studies have estimated these costs to be between 0.18% (BRANSON 1969), 0.15% (FRENKEL/LEVICH 1975) up to more than 1% (FRENKEL/LEVICH 1977). These estimates, however, cannot be taken too serious as their data had some serious flaws regarding their simultaneity. Using data that were taken without a time difference from a Reuters screen, MCCORMICK (1979) found the transactions cost in forward and spot market to be only 0.09% compared to 0.13%, 0.26% and 1.03% in FRENKEL/LEVICH (1977). Both studies used data of Eurocurrency-interest rates in addition to the usual method of estimating the cost of lending and borrowing by US/UK-Treasury-Bills. This was suggested by ALIBER (1973), who found that securities issued in different financial centers were different in political risk and thus less comparable. More important, however, seems the notion that Eurocurrencies are what is actually used by banks when computing their forward quotations7. And as MAASOUMI/PIPPENGER (1989) have pointed out, "in order to resolve this issue, we need to analyze the data that traders and arbitrageurs actually use in making their decisions." Using these different data, MCCORMICK like FRENKEL/LEVICH find transaction costs to explain 100% of the observed deviations from CIP − that is, no unexploited opportunities for profit to exist. Up to this point, all studies used round-trip arbitrage as the relevant criterion for estimating transaction cost. Starting with one currency, they always ended up with the same currency after having gone through the covered interest arbitrage (CIA)- process of, for example, starting with $, lending them, selling them forward and borrowing £, selling those spot, "thus effectively
6 7
See pg. 177. See Frenkel/Levich (1977), pg. 1216 fn.
4 exchanging current dollars for current dollars."8 So their Neutral Band of deviations from CIP was given by9: F − F0 ≤ t + t * + tS + t F F0 DEARDORFF (1979) noticed that arbitrageurs didn't necessarily have to end up with the same currency, if they were in need for a different one. This fact would save them t S or t F in the spot F − F0 or forward market. This reduces the Neutral Band to: ≤ t + t * − tS − t F F0 Using the FRENKEL/LEVICH (1975)-data, DEARDORFF estimated the width of the band to amount to at most 0.044% compared to 0.145% at round-trip10. In 1981 CALLIER developed this idea further, applying it to the securities markets, where either t or t* can be saved and thus reduces the band to: F − F0 ≤ tS + t F − t − t * F0 Taking up these conditions BAHMANI-OSKOEE/DAS (1985) found that "out of 79 monthly estimates none of them satisfied inequality (2)" (that is Deardorff's) "and only 13 of them satisfied inequality (7)", (that is CALLIERS contribution) "indicating that there were other factors than transactions cost causing deviations outside the Neutral Band."11 But as MAASOUMI/PIPPENGER (1989) noticed, their data again had some serious flaws regarding the simultaneity-criterion. Using corrected data (but still using T-Bills) they estimate transaction costs to explain about 20% instead of 0.73% of the deviations from CIP. CLINTON (1988) used data of the Euromarket from a Reuters information service screen and "careful attention has been given to accurate timing."12 He introduced the idea of using a swap instead of two separate spot and forward transactions13 into the debate, which reduces the cost to those of a swap. As t F = t S + tW , he was able to replace t F − t S by tW in the band for spot and
8 9
10
11 12 13
See Deardorff (1979), pg. 355. I follow Deardorff's notation with t S = transaction cost in the spot-market, t F = transaction cost in the forward market, t = transaction cost in the domestic securities market, t * = transaction cost in the foreign securities market, F =actual forward quotation and F0 =CIP price of a forward. It is interesting to note that he computed this figure from his inequality for equilibrium in spot and forward markets, not from the one stated above for simultaneous non-trivial equilibrium in spot/forward markets, that is, with transactions taking place, as then the result would have been .019-.025=-.006, which doesn't make sense! See pg. 798. See pg. 359. He stated very correctly that there is no such thing as an outright price of a forward quoted in the trading room, but that it is added from the two separate quotes for spots and swaps, thus showing the nature of a forward transaction as nothing but a combination of the two.
5 forward-market-equilibrium w − wo ≤ t + t * −tw14 and in the band for the securities-markets w − wo ≤ tw − t * −t . These two inequalities determine the amount to which deviations from CIP can be explained by transaction costs by their relevant minimum values, defining the Neutral band as w − wo ≤ min(t + t * −tw , tw − t * −t ) . Making his data even more accurate to real life by using observed bid-ask-spreads as measures for the transaction costs,15 he finds that there are deviations from CIP beyond the neutral band, but that these "profit opportunities appear to be both small and short-lived, even though they are not rare."16 So all these contributions have narrowed the neutral band further and further, reducing the explanatory value of transaction costs substantially. All of them measured deviations from CIP by taking some midpoint-value of an observed forward-price and comparing it to a theoretical value given by the standard CIP S (1 + id ) condition F0 = , then comparing their difference with the estimated amount of (1 + i f ) transaction cost. In the next section a different approach is introduced and CIP is redefined in the face of transaction costs by thoroughly applying the law of one price. 3. Covered interest parity, transaction cost and the Law of One Price Covered interest parity is a straightforward application of the law of one price.17 The basic condition states that investing (borrowing) in one currency should not be more or less profitable than exchanging the funds into a different currency, then investing (borrowing), and covering the exchange risk by a forward sale (purchase), thus making the transaction as riskless as, and 1 basically equivalent to the domestic transaction: (1 + id ) = (1 + i f ) F .18 S By using this basic relationship, each of the four variables can be determined by the other three. As interest rates and the spot rate are determined by other economic factors, domestic as well as foreign, such as monetary policy, international trade etc., it is mainly the forward rate that should S (1 + id ) make the condition hold. This determines the forward rate to be: F0 = , which says that a (1 + i f ) forward transaction should not cost more than a combined transaction of simultaneous borrowing, lending and a spot. The simultaneity is crucial because time lags would cause risk due to price 14
t w for percentage cost of a swap, where w is the actual percentage swap-premium and w0 is the CIP-value for
the premium in the proxy-form i-i*. Earlier studies usually use values estimated by triangular arbitrage. 16 See pg. 367. 17 See Levi (1990), pp. 122. 18 I am using direct price quotations, that is, e.g. $/£, where S is the spot and F the forward quotation outright, i the d domestic interest rate and i f the foreign interest rate. 15
6 changes and thus would make the combined action not equivalent to the single. Higher risk would be rewarded by the market and the law of one price need not hold. If the same basic condition that two equal things should be priced the same is applied to forwards with real-world transaction cost, two different prices for purchases and sales arise. The abovementioned formula is no longer able to determine the two different prices. The law of one price needs to be applied separately for each price. A forward purchase should be the same as buying spot at S(ask), investing the acquired currency at the foreign lending rate i lf , and borrowing the needed funds at the domestic borrowing rate idb , altering the above condition to: 1 + idb , F4 ( ask ) = S ( ask ) 1 + i lf A forward sale is equivalent to selling spot at S(bid), borrowing the sold currency at foreign borrowing rate i bf , and investing the funds received from the transaction at the domestic lending rate idl :
F4 (bid ) = S (bid )
1 + idl . 1 + i bf
An arbitrageur thus has the choice between going through the forward market or achieving the same thing by the combined actions stated above. He can make a profit by an offsetting transaction, which would be a forward sale at F (bid ) of the funds acquired at F4 ( ask ) , if F (bid ) > F4 ( ask ) or in the case of an indirect sale at F4 (bid ) ,he can buy funds at F ( ask ) and make a cash-profit if F4 (bid ) > F ( ask ) .This is what has been called "round-trip"-arbitrage in the literature. This can be illustrated by the following graph which shows the course of action at clockwise round-trip arbitrage:19 (1 + idl )
$0
$1
1 F (ask )
S (bid )
£0
19
See Deardorff (1979).
1 (1 + i bf )
£1
7 As outlined above it reflects a combined forward sale and an actual forward purchase of pounds. The next graph illustrates counterclockwise round-trip arbitrage, which is an actual forward sale of pounds acquired by combined action: 1 (1 + idb )
$0
$1
1 S (ask )
F (bid )
£0
£1
(1 + i ) l f
In the following this process shall be termed 4-corner arbitrage, because it uses all 4 markets20. Other proposed forms of arbitrage do not use all four markets and do not end up where they started, and are therefore called one-way-arbitrage21. These forms of arbitrage thus require a need for the particular currency, arising from ordinary activities such as exports etc. If this is not the case, existing opportunities are of no value. Only if one is in need for future pounds to pay for, say, a delivery of 1 ton of Worcestershire Sauce, one will be able to exploit an opportunity given by a lower future pound price through combined transactions than by an actual forward purchase. This is illustrated in the following graph.
20 21
That is also the reason why I called the implied values of the price for the combined transaction As suggested by Deardorff (1979).
F4 .
8 ($1 → £1 ): 1 (1 + idb )
$0
$1
1 S (ask )
1 F ( ask )
£0
£1
(1 + i ) l f
From this example one can see that there are three different basic forms of one-way-arbitrage, depending on how far one goes in the square. I want to call these 3-corner, 2-corner and 1-cornerarbitrage. 3-corner-arbitrage reflects alternatives for the forward market, when funds are received/needed in the future, or the spot market, when funds are received/needed at present. 2corner-arbitrage reflects different strategies when funds are received today but needed in a different currency in the future ($ 0 → £1 , £ 0 → $1 ), or when funds are needed at present, but will be available only in a different currency in the future ($1 → £ 0 , £1 → $ 0 ). 1-corner-arbitrage is an alternative for the securities markets22, when funds are available today, but needed in the future in the same currency ($ 0 → $1 , £ 0 → £1 ) or when funds are needed today, but will only be available in the future in the same currency ($1 → $ 0 , £1 → £ 0 ). If we follow the same line of thought as at the beginning of this section for modeling a forward transaction with the help of the three other markets, we can see that a spot transaction can be 1 modeled in a similar way: A spot purchase at will be equivalent to a forward purchase at S ( ask ) 1 1 and borrowing the needed currency in the meantime at , being able to invest the F ( ask ) (1 + i bf ) not yet needed dollars at (1+ idl ) . Noting this the law of one price gives the following result for (1 + i bf ) the price of a spot-purchase ($ 0 → £ 0 ): S ( ask ) = F ( ask ) . (1 + idl )
22
This is the alternative considered by Callier (1981), that gave rise to his 4 conditions in addition to Deardorff's.
9 By the same means selling spot ( £ 0 → $ 0 ) must be equivalent to selling forward, investing the currency
in
the meantime (1 + i lf ) . S (bid ) = F (bid ) (1 + idb )
and
borrowing
the
needed
dollars.
This
yields:
Again an arbitrageur has the choice between the left and the right-hand-side of the equations. If the equality does not hold, there will be a chance for one-way-arbitrage. On the other hand roundtrip-arbitrage will be profitable if the currency that was acquired by combined transactions can be (1 + i bf ) sold at a higher price, that is, when S (bid ) > F ( ask ) , or when the regularly bought (1 + idl ) (1 + i lf ) currency can be sold by combined action at a higher price: . S ( ask ) < F (bid ) (1 + idb ) In the same way as 3-corner-arbitrage yielded the conditions for the forward and spot-market, 1corner-arbitrage provides us with the conditions for equilibrium in the securities-markets by modeling borrowing (or lending) by a combination of spot- and forward-purchase and sale together with lending (or borrowing) in the other security-market. A $-investment would thus be equivalent to buying pounds spot, investing those, and selling the proceeds forward, so not experiencing any risk. ($ 0 → $1 ): (1 + idl )
$0
$1
1 S (ask )
F (bid )
£0
£1
(1 + i ) l f
Following this approach all possible forms of arbitrage can be determined and the conditions under which arbitrage is not profitable can be derived. This is summarized in the following:
10 4-corner-arbitrage:
(1 + i bf )
($ 0 → $ 0 ):
S (bid ) < F ( ask )
( £ 0 → £ 0 ):
S ( ask ) > F (bid )
($1 → $1):
F (bid ) < F4 ( ask) = S ( ask)
( £1 → £1 ):
(1 + idl ) (1 + i lf ) (1 + idb )
(1 + idb ) (1 + i lf ) (1 + idl ) F ( ask) > F4 (bid ) = S (bid ) (1 + i bf )
3-corner-arbitrage: ($ 0 → £ 0 ):
S ( ask ) = F ( ask )
( £ 0 → $ 0 ):
S (bid ) = F (bid )
($1 → £1 ): ( £1 → $1 ):
(1 + i bf ) (1 + idl ) (1 + i lf )
(1 + idb ) (1 + idb ) F ( ask ) = S ( ask ) (1 + i lf ) (1 + idl ) F (bid ) = S (bid ) (1 + i bf )
2-corner-arbitrage: ($ 0 → £1): ( £ 0 → $1): ($1 → £ 0 ): ( £1 → $ 0 ):
l (1 + idl ) (1 + i f ) = F ( ask ) S ( ask ) S (bid )(1 + idl ) = F (bid )(1 + i lf ) S ( ask )(1 + idb ) = F ( ask )(1 + i bf ) S (bid ) F (bid ) = (1 + i bf ) (1 + idb )
1-corner-arbitrage: ($ 0 → $1 ): ( £ 0 → £1 ): ($1 → $ 0 ):
F (bid ) (1 + i lf ) = W (bid )(1 + i lf ) S ( ask ) S (bid ) (1 + i lf ) = (1 + idl ) = W ( ask )(1 + idl ) F ( ask ) F ( ask ) 1 (1 + idb ) = (1 + i bf ) = (1 + i bf ) S (bid ) W ( ask ) (1 + idl ) =
11 ( £1 → £ 0 ):
(1 + i bf ) =
S ( ask ) 1 (1 + idb ) = (1 + idb ) F (bid ) W (bid )
These last conditions consider the fact that a swap W can be used instead of separate spot and forward transactions, where a swap-purchase W ( ask ) is a combination of spot sale and a forward repurchase
on the basis of the S (bid ) S (bid ) 1 W ( ask ) = = = F ( ask ) S (bid ) + P ( ask ) (1 + pask )
same
spot-price
bid23:
and a swap-sale W (bid ) is a combination of a spot purchase and a forward-resale on the basis of F (bid ) S ( ask ) + P (bid ) the same spot-price ask: W (bid ) = = = (1 + pbid ) S ( ask ) S ( ask ) We can also apply this to 4-corner-arbitrage, as there a joint use of the spot and forward-markets is possible. Note, however, that for the two other forms of arbitrage a swap is not possible, as in those cases spot and forward-transactions are alternatives, and not being used together! ($ 0 → $ 0 ):
1
($1 → $1):
1
b b F (ask ) (1 + i f ) 1 (1 + i f ) = S (bid ) (1 + idl ) W ( ask) (1 + idl ) l (1 + i lf ) F (bid ) (1 + i f ) = W ( bid ) S (ask) (1 + idb ) (1 + idb ) S ( ask) (1 + idb ) 1 (1 + idb ) = F (bid ) (1 + i lf ) W (bid ) (1 + i lf ) S (bid ) (1 + idl ) (1 + idl ) = W ( ask) F (ask) (1 + i bf ) (1 + i bf )
These results boil down to only 2 basic conditions for no round-trip-arbitrage. 3-corner and 1corner-arbitrage determine the equilibrium values for F , S , id , i f . And 2-corner-arbitrage determines their interrelationship. 4-corner-arbitrage: ($ 0 → $ 0 , £1 → £1 ): ($1 → $1 , £ 0 → £ 0 ):
23
(1 + idl ) = F4 (bid ) (1 + i bf ) (1 + idb ) F (bid ) < S ( ask) = F4 ( ask) (1 + i lf ) F ( ask) > S (bid )
I use P as the outright swap-premium, while p denotes the percentage swap-premium, which is the same is Clinton's w , only that I do not use the approximation form i-i*.
12 in swap-form: ($ 0 → $ 0 , £1 → £1 ): ($1 → $1 , £ 0 → £ 0 ):
(1 + idl ) (1 + idl ) ⇒ ( 1 + p ) > ask (1 + i bf ) (1 + i bf ) (1 + i lf ) (1 + idb ) 1 > W (bid ) ⇒ ( 1 + p ) < bid (1 + idb ) (1 + i lf ) 1 > W ( ask)
4. Measurement of Deviations If we account for costs other than the ones included in the spreads, such as costs for searching an arbitrage-opportunity or fees24 etc., the above equations have to be relaxed to inequalities in a way that it must be less costly to use the standard market than using a combination of the others. This is not true for 2-corner-arbitrage, as both sides of the equations are combinations. If we consider this, we can now solve the no-arbitrage conditions for the F -values, that are required to make the conditions hold: 3-corner-arbitrage: ($ 0 → £ 0 ):
S ( ask ) ≤ F ( ask )
( £ 0 → $ 0 ):
S (bid ) ≥ F (bid )
($1 → £1 ): ( £1 → $1 ):
(1 + i bf ) (1 + idl ) (1 + i lf )
⇒ F ( ask ) ≥ S ( ask )
(1 + idl ) = F3 ( ask ) (1 + i bf )
⇒ F (bid ) ≤ S (bid )
(1 + idb ) = F3 (bid ) (1 + i lf )
(1 + idb ) (1 + idb ) F ( ask) ≤ S ( ask) = F4 ( ask) (1 + i lf ) (1 + idl ) F (bid ) ≥ S (bid ) = F4 (bid ) (1 + i bf )
2-corner-arbitrage: ($ 0 → £1): ( £ 0 → $1):
24
l (1 + idl ) (1 + i f ) (1 + idl ) = ⇒ F ( ask ) = S ( ask ) = F2 ( ask ) F ( ask ) S ( ask ) (1 + i lf ) (1 + idl ) l l S (bid )(1 + id ) = F (bid )(1 + i f ) ⇒ F (bid ) = S (bid ) = F1 (bid ) (1 + i lf )
At this point it should be mentioned that the spreads we are talking about are interbank-spreads. Customers usually experience far higher spreads and thus costs. It is obvious that the kind of arbitrage that is discussed here will mainly be an opportunity for professional currency-dealers. Brokerage fees will thus rarely occur. The cost meant here will mainly be unmeasurable cost like effort, searching, etc.
13
(1 + idb ) = F1 (ask) (1 + i bf )
($1 → £ 0 ):
S ( ask)(1 + idb ) = F (ask)(1 + i bf ) ⇒ F (ask) = S (ask)
( £1 → $ 0 ):
S (bid ) F (bid ) (1 + idb ) = ⇒ F ( bid ) = S ( bid ) = F2 (bid ) (1 + i bf ) (1 + idb ) (1 + i bf )
1-corner-arbitrage: ($ 0 → $1 ): ( £ 0 → £1 ): ($1 → $ 0 ): ( £1 → £ 0 ):
F (bid ) (1 + idl ) (1 + i lf ) ⇒ F (bid ) ≤ S ( ask ) = F2 ( ask ) S ( ask ) (1 + i lf ) S (bid ) (1 + idl ) (1 + i lf ) ≥ (1 + idl ) ⇒ F ( ask) ≥ S (bid ) = F1 (bid ) F ( ask ) (1 + i lf ) F ( ask ) (1 + idb ) (1 + idb ) ≤ (1 + i bf ) ⇒ F ( ask ) ≥ S (bid ) = F2 (bid ) S (bid ) (1 + i bf ) S ( ask) (1 + idb ) b b (1 + i f ) ≤ (1 + id ) ⇒ F (bid ) ≤ S (ask) = F1 ( ask) F (bid ) (1 + i bf ) (1 + idl ) ≥
or in swap-form: ($ 0 → $1 ): ( £ 0 → £1 ): ($1 → $ 0 ): ( £1 → £ 0 ):
S ( ask) + P(bid ) (1 + idl ) (1 + i lf ) ⇒ (1 + pbid ) ≤ S (ask) (1 + i lf ) S (bid ) (1 + idl ) l l (1 + i f ) ≥ (1 + id ) ⇒ (1 + pask ) ≥ S (bid ) + P(ask) (1 + i lf ) S (bid ) + P(ask ) (1 + idb ) (1 + idb ) ≤ (1 + i bf ) ⇒ (1 + pask ) ≥ S (bid ) (1 + i bf ) S (ask) (1 + idb ) (1 + i bf ) ≤ (1 + idb ) ⇒ (1 + pbid ) ≤ S (ask) + P(bid ) (1 + i bf ) (1 + idl ) ≥
So all the conditions only yield 8 different values for F . In fact there are only 4 basic Fi 's, each with a bid and an ask side. These have to be compared to the actual realizations of the forward prices. Doing so reveals a possible opportunity for the particular kind of arbitrage. The following summary reviews the basic no-arbitrage-conditions. It also shows again how stringent the different conditions are. 4-corner-arbitrage: F (bid ) < F4 ( ask )
or
(1 + idb ) (1 + pbid ) < (1 + i lf )
14
F ( ask ) > F4 (bid )
or
(1 + pask ) >
(1 + idl ) (1 + i bf )
3-corner-arbitrage: F ( ask ) ≤ F4 ( ask )
F ( ask ) ≥ F3 ( ask )
F (bid ) ≥ F4 (bid )
F (bid ) ≤ F3 (bid )
2-corner-arbitrage:25 F ( ask ) = F2 ( ask ) = F ( ask ) = F1 ( ask ) F (bid ) = F2 (bid ) = F (bid ) = F1 (bid )
(1 + idl ) (1 + idb ) ⇒ = (1 + i lf ) (1 + i bf )
1-corner-arbitrage: F ( ask ) ≥ F2 (bid )
F ( ask ) ≥ F1 (bid )
F (bid ) ≤ F2 ( ask )
F (bid ) ≤ F1 ( ask )
These results imply some general conclusions: It is actually 2-corner-arbitrage that will determine the exact value of the forward-price. 3- and 4-corner-arbitrage form the Neutral Band: the amount of allowed deviations are given by F3 and F4 in the relevant direction and the conditions under 1corner-arbitrage are automatically met if those under 2-corner are satisfied and F ( ask ) > F (bid ) is true26. The parity-values implied by 2-corner-arbitrage become: (1 + idm ) Fp ( ask ) = S ( ask ) (1 + i mf ) (1 + idm ) , Fp (bid ) = S (bid ) (1 + i mf ) (1 + idl ) (1 + idb ) (1 + idl ) (1 + idb ) (1 + idm ) when noting that , = ⇒ = = (1 + i lf ) (1 + i bf ) (1 + i lf ) (1 + i bf ) (1 + i mf ) where i m is the interest-rate at midpoint . 25
2-corner-arbitrage yields:
(1 + idl ) (1 + idb ) = = F ( ask ) F ( ask ) = S ( ask ) = F1 ( ask) 2 (1 + i lf ) (1 + i bf ) (1 + idl ) (1 + idb ) = F (bid ) = S (bid ) = F ( bid ) F ( bid ) = S ( bid ) = F2 (bid ). 1 (1 + i lf ) (1 + i bf )
F ( ask) = S ( ask)
26
And it will generally be the case that the price of a purchase is higher than the price of a sale, so that F1 ( ask ) = F2 ( ask ) = F ( ask ) > F (bid ) and F1 (bid ) = F2 (bid ) = F (bid ) < F ( ask ) .
15
If it is actually customary in banks to calculate the forward premium in accordance with (see the (1 + idm ) Appendix) F0 = S , then the resulting value will be between the values calculated above: (1 + i mf ) Fp ( ask ) > F0 > Fp (bid ). So banks will be able to pick their forward-prices simply by plugging into the above formula the relevant purchase/sale-price of the spot S , effectively using the Fp formula. Deviations should not exist. Consistency with the round-trip and three-corner-arbitrage conditions will also be achieved, as we can generally note that: (1 + idl ) (1 + idm ) (1 + idb ) (1 + idm ) and , as in general i b > i m > i l . < > b m l m (1 + i f ) (1 + i f ) (1 + i f ) (1 + i f ) Therefore we can conclude that F3 ( ask ) < Fp ( ask ) . F4 ( ask ) > Fp ( ask ).
F3 (bid ) > Fp (bid ). F4 (bid ) < Fp (bid ) .
Comparing this with the no-arbitrage conditions derived earlier: F4 ( ask ) ≥ F ( ask ) ≥ F3 ( ask ) F3 (bid ) ≥ F (bid ) ≥ F4 (bid ) we see that Fp meets the required conditions. These results show that in general the banks' conventional calculation of their forward-prices should make arbitrage impossible. The advantage of this approach is that any deviations could be exactly located and the kind of arbitrage-opportunity be uncovered. It is not enough to realize that there are deviations, but it is essential to know what kind of opportunity it is. It could only be exploited if someone is in the particular situation to make use of a uncovered opportunity. If one is not about to receive £ next period, he cannot sell them forward to make use of an opportunity F (bid ) < F4 (bid ) . Only someone who has future pounds can do that, otherwise F ( ask ) < F4 (bid ) must hold to make a profit. This is a reason why recorded deviations outside the band might exist. Studies often use end-of-day-data. At the end of the day banks will usually have completed their regular orders and will mainly close their open positions. Chances are high that few people will be in need for a particular currency with a particular maturity at that time, if this opportunity should arise. So it is better to use data that were recorded during regular trading hours. This approach also saves from estimating transaction cost, as the Fi -values automatically reflect them. An important property is that they are variable here, and so the results will be subject to less error, as the spreads fluctuate with the volatility of the market. So a fixed band of estimated transaction cost will be too narrow in times of busy trading and heavy market fluctuations-exactly when it is the most interesting to know if markets react efficiently. To express its width in
16 percentage terms is difficult and doesn't make much sense as the spreads vary with marketactivities and not as a percentage of the relevant price F , S , i . For example the bid/ask spread of the Deutsche Mark/US-$ quotations ranges from 1-2 basis-points in quiet trading to 5-10 points and more at highly volatile times. In percentage terms at a rate of 1.7 t S would vary from 0.006% to 0.06%. So it is of no use to fix it. The Neutral Band involved in the above conditions still includes t w + t * + t at round-trip and t + t * − t w at one-way-arbitrage, but they are not estimated but defined by the conditions. The width can graphically be illustrated in the following way. One way arbitrage is not possible in between the F4 / F3-values: F(ask)
F4(ask)
F(bid)
F3(bid)
FP(ask)
FP(bid)
F3(ask)
F4(bid)
S ( ask )
(1+idm ) (1+i m f )
S ( bid )
m (1 + id ) m (1 + i f )
Round-trip-arbitrage is not profitable within the band between the F4 -values: F
F4(ask) Fp(ask) Fp(bid) F4(bid)
S ( mid )
m (1 + id ) m (1 + i f )
Combination of the bands yields: F
F4(ask) F3(bid) Fp(ask) Fp(bid) F3(ask) F4(bid)
17
S ( mid )
m (1 + id ) m (1 + i f )
So the width of the band cannot be estimated by adding up estimated transaction-cost-figures, but it is determined by the relative size of all the involved variables. It is possible to calculate an average value by taking the average difference between the F4 / F3 -values and the parity-prices. 5. Conclusion Unfortunately I did not have data available in the high quality necessary to test this approach. One would have to use highly simultaneous price-quotations, at best randomly picked as to avoid any biases arising from scheduled trading-activities over the course of the day, such as the fixing or market-closing. They have to be of the same kind as the ones used by arbitrageurs/traders. However I have shown that the existence of transaction costs in the form of bid-ask-spreads changes the basic CIP-identity to two separate ones for either a purchase or a sale and creates a Neutral Band that cannot be described by static estimated transaction-costs, but is determined dynamically by the spot- and forward-market-conditions for no arbitrage. Also it was shown that forward-prices computed by the banker's formula will usually meet the conditions for zero arbitrage-profits. Therefore CIP should hold very closely. The only way in which an arising deviation might be explained is that it was computed wrongly or not adjusted to price-changes.
18 Appendix:
Calculation of the forward premium symbols used:
i = interest rate p.a. p = forward premium p.a. P = forward premium outright = F - S t = number of days
Formulas used in Germany according to von Werner Hiering/Rolf Jacobs: Die Absicherung von Fremdwährungsrisiken" (Managing currency risk exposure)
P+S = F simple approximation: Pt =
exact form:
Pt =
S (id − i f ) t 360
P+S−S P F −S t = = = (id − i f ) S S S 360
S (id − i f ) t 360 + i f t
The identity with the interest rate parity condition can be shown in the following way: 1 1 (1 + id ) = (1 + i f ) F = (1 + i f ){S (1 + p)} S S (1 + id ) = (1 + i f )(1 + p)
(1 + p) =
(1 + id ) (1 + i f )
Considering the fact that i and p are given rates per annum we can calculate them for shorter periods as t t it = i ⋅ and− > p t = p ⋅ 360 360
19 Considering this we can write the parity condition: t t t (1 + id ) = (1 + i f )(1 + p ) 360 360 360
t t 360 (1 + id ) ( + id ) t 360 = 360 t (1 + p )= t t 360 360 (1 + i f ) ( + if ) 360 360 t 360 360 360 + id + id − − if t t p = t −1 = t 360 360 360 + i + if f t t p
i −i (i − i )t t = d f = d f 360 360 + i 360 + i f ⋅ t f t
Pt = p
S ⋅ (id − i f ) ⋅ t t ⋅S = 360 360 + i f ⋅ t
So the formula used to calculate the forward premium is consistent with the interest rate parity condition. Therefore IRP should hold.
20 References Aliber, Robert Z. (1973): "The Interest Rate Parity Theorem: A Reinterpretation." Journal of Political Economy, December 1973, pp. 1451-1459. Bahmani-Oskooee, Mohsen, and Satya P. Das (1985): "Transaction Costs and the Interest Parity Theorem", Journal of Political Economy, August 1985, pp. 793-799. Branson, William H. (1969): "The Minimum Covered Interest Differential Needed for International Arbitrage Activity," Journal of Political Economy, December 1969, pp. 1029-1034. Chrystal, K. Alec (1984): "A guide to Foreign Exchange Markets." Fed. Reserve Bank of St. Louis Review, March 1984, pp. 5-18 Clinton, Kevin (1988): "Transaction Costs and Covered Interest Arbitrage: Theory and Evidence." Journal of Political Economy, April 1988, pp. 358-370 Deardorff, Alan V. (1979): "One-Way Arbitrage and Its Implications for Foreign Exchange Markets," Journal of Political Economy, April 1979, pp. 351-364. Frenkel, Jacob A., and Richard M. Levich (1975): "Covered Interest Arbitrage: Unexploited Profits?" Journal of Political Economy, April 1975, pp. 325-338. ____________(1977):" Transaction Costs and Interest Arbitrage: Tranquil Vs. Turbulent Periods." Journal of Political Economy, December 1977, pp. 1209-1226. ____________(1979):" Covered Interest Arbitrage and Unexploited Profits? Reply." Journal of Political Economy, April 1979, pp. 418-422. Giddy, Ian H. (1976): "An Integrated Theory of Exchange Rate Equilibrium." Journal of Financial and Quantitative Analysis, December 1976, pp. 883-92 ____________(1977):" Exchange Risk: Whose View?" Financial Management, Summer 1977, pp. 23-33. ____________ (1994): Global Financial Markets, Lexington et al. 1994. Robert W. Kolb (Ed.) (1993): The International Finance Reader,Kolb Publishing Co., Miami, 1993 Levi, Maurice D. (1990): "International Finance: The Markets and Financial Management of Multinational Business", McGraw Hill, 2nd edition, 1990. Maasoumi, Esfandiar and Pippenger, John (1989): "Transaction Costs and the Interest-rate Parity Theorem: Comment." Journal of Political Economy, 1989, vol. 97. Shapiro, Alan C. (1992): "Multinational Financial Management", Allyn and Bacon, 4th. edition, 1992. Solnik, Bruno (1990): "International Parity Conditions and Exchange Risk: A Review." Journal of Banking and Finance, August 1978, pp. 281-293. Thornton, Daniel L. (1993): "Tests of Covered Interest Rate Parity" in Robert W. Kolb (RWK) The International Finance Reader, Miami, 1993, pp. 166-177.
