31331IFE_Topic 3_Interest Rate Parity

8/8/2019 31331IFE_Topic 3_Interest Rate Parity

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INTERNATIONAL PARITY CONDITIONS

The Forward Discount/ Premium

The forward premium / discount is the difference between the forwardexchange rate, F and the spot exchange rate, S, expressed as apercentage of the spot rate.

Thus, denote forward discount or premium between the home currency

and the foreign currency as fh , the forward currency as Ffh and thespot rate as S fh. Then:

100

=

fh

fhfh

fhS

SF or alternatively 100

=

hf

hfhf

hfS

SF

ExampleLet 1 year forward rate for in terms of $ be F$ = 1.46344 and S$ =1.4855.

%4.11004855.1

4855.146344.1100

$

$$

$ =

=

=

S

SF

Hence the is trading at a forward discount against the $. The forwardvalue for the in terms of $ is less than todays value of the in terms of$s

INTEREST RATE PARITY (IRP)

IRP establishes the linkages across- spot and forward currency markets- domestic and overseas security markets

IRP draws on the principle that, in equilibrium two investments exposed tothe same risks must yield identical returns.

IRP is ensured by arbitrage.

INTEREST RATE PARITY IN A PERFECTCAPITAL MARKET

In a perfect capital market, we assume there are:-

Many buyers and sellers all of whom are small relative to the size of themarket. This implies that all market participants are price takers.

no taxes and transactions costs

no uncertainty as to prices and market conditions

perfect capital mobility (i.e. there are no capital controls)


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We will relax these assumptions later to establish whether they alter theconclusions of our analysis.

We assume that the transactor is resident in the UK and that the UK is thehome country and the US is the foreign country. The UK based agent has 2

strategies for investing wealth

Home Run: This involves the UK resident investing in UK bonds

Suppose an agent can invest in a UK security that yields i UK per period. Ifthe interest is paid on maturity, the future value of this investment will be:

1(1+ iUK) (1)

Round Trip: This involves the UK based agent following a 3 stage approach

1. Take the 1 and convert it into US $ at the current spot St. In this analysis,

we define the spot rate St in terms of s per dollar. (e.g. if the spot rate is1= $2 in terms of $s per pound, the same rate expressed in terms of sper $ is = 0.5 s per dollar). To express a sterling sum in terms dollars ,

divided the sterling sum by St. This would yield 1x1/ St. If St = 0.5, 1 in

terms of $ = 1/St = 1/(0.5) = 2

2. Then invest this dollar sum in a US security identical to the UK instrumentabove to yield 1 x (1/ St) x (1+ iUS)

3. Next convert this dollar denominated terminal wealth back into sterling byexchanging it at the current forward rate, Ft. This yields:

1 x (1/ St) x (1+ iUS) x Ft(2)

In a perfect capital market, the terminal value implied by equation (1) shouldbe identical to that implied by equation (2). Thus,

1 x (1/ St) x (1+ iUS) x Ft = 1(1+ iUK)

We can rearrange this to give:

( )

( ))3(

1

11,

+

+

=

US

UK

t

t

i

i

S

F

( )

( ))4(

1

1,

+

=

US

USUK

t

tt

i

ii

S

SF

N.B. It is common to use an approximation for the Interest Rate Differential. Ifinterest rates are low, then the denominator in Equation 4 will be close to

subtracting 1 fromeach side gives:

ForwardPremium

Interest RateDifferential


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unity. In this case, the IRD is expressed as iuk- ius . This approximation onlyworks well when interest rates are low.

ARBITRAGE TO EXPLOIT DEVIATION FROM

INTEREST RATE PARITYDiagramatic Representation of IRP

Points A and B represent situations where interest rate parity holds i.e. wherethe forward premium = interest rate differential.

Points C1 , C2 & C3 represent situations in which the FP > IRD

Points D1 , D2 & D3 represent situations in which the FP < IRD

Market forces will ensure that deviations from interest rate parity areeliminated.

Consider FP > IRD

In this case: (F-S)/S > (iuk ius)/(1+ ius)

We can rewrite this as

F/S > (1+iuk)/(1+ius)

Re-arranging yields

(F/S)(1+ius) > (1+iuk) i.e. the value of the round trip is greater than the value

of the home run.

A

Interest Rate Differential

ForwardPremium


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In this case, rational agents will

1. Sell UK securities, driving down UK bond prices and driving up UK interest

rates (i.e. iuk rises)2. Sell s and buy $s. Recall that we defined both S and F in terms of s

per dollar. If there is selling pressure on the pound then the pound willdepreciate, (e.g. if the 1 = $2 , this implies that $1 = 0.5, if the pounddepreciates to 1 = $1.5, this implies that $1= 0.667) In terms of ourdefinition of exchange rate S increases. Note that situations such aspoints C1-C3 will result in a capital outflow from the UK.

3. Buy US bonds. Increased demand for the fixed supply of US bonds will

drive up US bond prices (i.e. ius falls)4. Buy pounds forward in order to cover the proceeds of the investment in US

securities. Increased buying pressure on s in the forward exchange ratemarket will cause the forward to appreciate. (e.g. if F is 1 = $1.70 then

$1 = 0.5882: If F appreciates to 1 = $1.75 then $1 = $0.5714) Thus,increased buying pressure on forward s means that F falls

Taking these actions and their effects on market prices we can establish that:-

If F falls and S rises, then (F-S)/S will decrease

If iuk rises and ius falls then (iuk ius)/(1+iuk) will rise

Thus, the result of these market transactions is to drive down the forwardpremium and simultaneously drive up the interest rate differential. This

process is termed ARBITRAGE. Arbitrage results when 2 assets of similarrisk trade at different prices in different locations. Agents will buy one assetand sell the other. In this case, the home run is sold and the round trip isbought. This results in the profitable opportunity being eliminated by risk freetrading. Eventually situations such as those described by points C1 to C3above disappear and market forces work to restore interest rate parity.

Consider FP < IRD

In this case: (F-S)/S < (iuk ius)/(1+ ius)

We can rewrite this as

F/S < (1+iuk)/(1+ius)

Re-arranging yields

(F/S)(1+ius) < (1+iuk) i.e. the value of the home run is greater than the valueof the round trip.

In this case, rational agents will

1. Sell US securities -> ius rises


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2. Buy s/ sell dollars in the spot market -> S decreases

3. Buy UK bonds -> iuk falls4. Sell s forward/ purchase $s forward -> F increases

Taking these actions and their effects on market prices we can establish that:-

If F rises and S falls, then (F-S)/S will increase

If iuk falls and ius rises then (iuk ius)/(1+iuk) will fall

In this case, arbitrage erodes pricing anomalies by causing the FP toincrease and the IRD to reduce. This will halt when interest rate parity isrestored.

IRP AND IMPERFECT CAPITAL MARKETS

A perfect capital market assumes, inter alia

- no transaction costs- no taxes- certainty and- no capital controls

We next examine how our understanding changes if we relax theseassumptions

Transaction Costs

In the above account, IRP is ensured because profit maximising agentsengage in arbitrage, which involves undertaking transactions in both theforeign exchange and domestic and foreign capital markets. In the realworld, all of these trades invite transaction costs.

This implies that deviations from IRP can exist because of the transactionscosts involved in eliminating them are greater than the profits / returns tobe earned. For any given interest rate differential, transactions costs will

create an upper and lower limit on the forward discount / premium withinwhich arbitrage will not be profitable. In other words, transactions costengender a neutral band within which covered interest arbitrage will NOToccur.


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Points A, B, C and D lie within the neutral band and are equilibrium, giventransaction costs.

Points W and X lie out with the neutral band and will result in a capitalinflow and consequent exchange rate appreciation (If the exchange rate isdefined in terms of $s per ).

Point Y and Z lie out with the neutral band and will result in capital outflowand consequent exchange rate depreciation (If the exchange rate isdefined in terms of $s per ).

Note that, there is no reason to expect that deviations from IRP within theneutral band will be either systematically positive or negative. Transactioncosts engender deviations from IRP but NOT in any systematic direction.

Taxes

In most financial markets, transactions attract taxes such as stamp dutiesand transfer taxes. Since these are directly related to trading activity, they

can be conceptualised in a way similar to transaction costs and would tendto widen the neutral band.

Taxes on income and capital gain are likely to have a different impact.IRP as set out above balances pre-tax returns from two alternativeinvestment options. Clearly, investors ought to be interested in post-taxreturns. When the tax rates applied to the two options are identical thentaxes do not influence the choice of the investor in favour of one or theother and the original parity condition remains valid.

If income and capital gains are taxed differently at Ty and Tk respectively,

then Tk will apply to currency gains and Ty to investment income. Thus,


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( )

( )( )

y

US

USUK

k

t

tmtT

i

iiT

S

SF

+

=

11

)1(3,

or

( )

( ) )1(

1

1

3,

k

y

US

USUK

t

tmt

T

T

i

ii

S

SF

+

=

If at Ty > Tk then a new interest parity line will emerge with a shallowerslope than the pre-tax case.

Different tax regimes in different countries with different treatment ofincome and capital gains can lead to different arbitrage opportunities fordifferent sets of foreign nationals. This, in theory could cast doubts overwhether IRP will always hold. It is possible, however, that suchtransactions tend to be made by international investors with low / zero taxrates. This would imply that IRP would hold.

Uncertainty

Although, all prices involved in covered interest rate arbitrage areobserved there may be some uncertainty, ex ante, about the actual pricesat which deals can be struck. In addition, the forward contract is subjectto default / credit risk. In addition, there are risks in operating offshorefrom the possible existence of introduction ofcapital controls.

For these reasons, arbitrage may be inhibited resulting in a widening of theneutral band around the traditional parity line.

Empirical Evidence

There are 2 main ways of assessing whether IRP holds in real world financialmarkets

Neutral band analysis

Regression analysis

Neutral band analysis involves plotting out the forward premium versus theinterest rate differential and establishing whether the majority of points liewithin the neutral band. This analysis was pioneered by Frenkel and Levich inpapers published in the 1970s.

In Frenkel and Levich (1975), 3 month Treasury Bill rates for the US and UKand for the US and Canada and appropriate spot and forward rates wereexamined for the period January 1962 to November 1967. It was establishedthat circa 80% of the observations lie within the neutral band generated withreference to the appropriate transactions costs. It was suggested that the

deviations from IRP were caused political risk of capital controls which causedarbitrageurs to hesitate in eliminating pricing anomalies. Indeed, Frenkel and


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Levich conduct the exercise with reference to the interest rate differentialsobserved on offshore or Euro bills. These are bills denominated in $s, s andCanadian $s but placed and traded outside the country in whose currency theassets are denominated. (e.g. $dollar securities, CAN $ securities and sterlingsecurities traded in one offshore centre such as Paris or Luxembourg). Thus,

there is no political risk of capital controls. In this case, circa 100% of theobservations lie within the transaction cost generated neutral band.

In Frenkel and Levich (1977), the authors extend the analysis to 3 differenttime periods

Period 1:- 1962-67, a period of pegged/fixed exchange rates in which theforeign exchange markets were calm. This is termed the tranquil peggedexchange rate period

Period 2:- 1968-69 turbulent pegged exchange rate period

Period 3:- 1973-75 the managed float period

In Periods 1 and 3, circa 80% of observations for Treasury Bills and circa100% for Euro Bills lie within the neutral band. However, in the turbulentperiod 2, a significantly greater proportion of observations lie out with thetransactions cost neutral band suggesting that uncertainty is the culprit.

It should be noted that these results and their interpretation can be questionedon the basis that the trades which violate IRP could not have actually beenmade (i.e. were the observed IRDs and FPs actually contemporary). Toobviate this, later studies concentrate on real time data which can isolate

times at which prices and thus IRDs and FPs present real arbitrageopportunities. When this is undertaken, there are very few violations of IRP.

Regression Analysis. This involves estimating the following relationship:-

FPt = + IRDt + ut

If IRP holds = 0 and = 0

In general, regression evidence tend to support IRP but has been criticised byTaylor who argues that whilst the relationship holds on average, there may be

large residuals which represent arbitrage opportunities. As MacDonaldargues, it is possible that these tests indicate IRP may hold on average over aperiod but not at any point in time within that period. He concludes thatregression analysis can establish the stylised fact of IRP but says virtuallynothing about market efficiency

References

Levich: International Financial Markets (2nd Ed) Chapter 5Hallwood & MacDonald: International Money & Finance (3rd Edition): Ch 3

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31331IFE_Topic 3_Interest Rate Parity