INTEREST-RATE PARITY

INTEREST-RATE PARITY

Now we will look at the connections between interest rates across countries. Assume that there is a different nominal interest rate in each country. We again think of the united states as the home country and the united kingdom as the foreign country. The nominal interest rate in the united states, denoted by i, is the expressed as dollar paid per year per dollar held as U.S bonds. The nominal rate i^f in the united kingdom is expressed as pounds paid per year per pound held as U.K bonds.

Consider a household that has one U.S dollar in year t and is choosing between holding a U.S. bond or a U.K bond. (the household could reside in either country). Option 1, holding a U.S bond, is a straightforward. The amount of dillars received in year t + 1 depends only on the U.S interest rate : 

(18.6)                                              Option 1 : hold U.S bond

Dollar received in year t + 1 = 1 + i

The second option to use the exchange market to obtain U.K pounds, buy a U.K bond, and use the exchange market a year later to convert back to dollars. In this scenario it will be important that the nominal exchange rate between pounds and dollars might change over time. Therefore, we use the time subscript to  denote the nominal exchange rate, Ɛ_t for year t. In year t, the household can exchange its S1.00 for Ɛ_t  pounds. By holding a U.K bond with the nominal interest rate i ^f , the household receives Ɛ_t ( 1+ i ^f ) of pounds in year t +1. In year t +1, each pound can be exchanged for dollars at the rate of 1/ Ɛ_t+1 dollars per pound. Therefore, if the household convert its pounds back to dollars in year t + 1, it recceives Ɛ_t ( 1+ i ^f ) dollars per pound. Therefore, if the household converts its pounds back to dollars in year t + 1, it receives  Ɛ_t ( 1+ i ^f ) / Ɛ_t+1  o option second dollars. Thus, the second option yields:

(18.7)                         Option 2: use exchange market and hold U.K bond

Dollars received in year t + 1 = Ɛ_t ( 1+ i ^f ) / Ɛ_t+1

If there are no cost for using the exchange rate market and holding U.K bonds, the two options must, in equilibrium, yield the same number of dollars in year t + 1. Otherwise, all households would hold bonds only in the country with the highest yield and borrow only in the country with the lowest yield. There for,equations (18.6) and (18.7) give the equilibrium condition as

1+i = Ɛ_t ( 1+ i ^f ) / Ɛ_t+1

return on holding US bond ₌ return on using exchange market and holding U.K bond

before we interpret equation (18.8), we will find it helpful to simplify the result by using some algebra. Start by rearranging the terms to get

1+ i ^f = (1+i ). (Ɛ_t-1/ Ɛ_t )

The growth rate of the nominal exchange rate is

Δ Ɛ_t/ Ɛ_t = (Ɛ_t-1/ Ɛ_t )/ Ɛ_t

Δ Ɛ_t/ Ɛ_t = Ɛ_t+1/ Ɛ_t – 1

We can therefore substitute (1+ Δ Ɛ_t/ Ɛ_t ) for Ɛ_t+ 1/Ɛ_t in the equation above to get

1+ i ^f = (1+i ). (1+ Δ Ɛ_t/ Ɛ_t )

If we multiply out the two term on the rigth-hand side, we get

1+ i ^f = 1+i + 1+ Δ Ɛ_t/ Ɛ_t + i . Δ Ɛ_t/ Ɛ_t

The last term,  i . Δ Ɛ_t/ Ɛ_t  tends to be be small-infact, this term become negligible if we consider very short periods rather than years. Therefore, we neglect this term. If we cancel out the 1s on each side of the equation and move i from the right-hand side to the left-hand side, we get the result we are looking for :

i ^f- i = Δ Ɛ_t/ Ɛ_t

interest-rate differential = growth rate of nominal exchange rate

To understand equation ( 18.9), imagine that the pound-dollar exchange rate, Ɛ_t  is risiving over time-that is, the dollar is becoming more valuable compared to the pound at the rate Δ Ɛ_t/ Ɛ_t . in order for U.s and U.K bonds to yield the same return in dollars, the U.K nominal rate i ^f  has to exceed the U.S nominal rate, i by the growth rate of the nominal exchange rate, Δ Ɛ_t/ Ɛ_t.

That is, the interest rate differential has to compensate for increase in the nominal exchange rates, which favors holding U.S bonds.

In practice, changes in nominal exchange rates are not known precisely in advance. Therefore, the growth rate Δ Ɛ_t/ Ɛ_t  in equation (18.9) must be replaced by expected growth rate, (Δ Ɛ_t/ Ɛ_t )^e. When we make this change, we get an important result called interest-rate parity:

Key equation (interest-rate parity :

i ^f- i =( Δ Ɛ_t/ Ɛ_t )^e

interest-rate differential = expected growth rate of nominal exchange rate

The idea behind equation (18.10) is that for the two interest rate to offer the same deal-to have parity in the returns offered on bonds in the two countries-the difference in nominal interest rate,     i ^f- i, has to compansate for the expected growth rate of the nominal exchange rate,               ( Δ Ɛ_t/ Ɛ_t )^e.

            A number of real-world consideration prevent interest-rate parity from holding excatly.these considerations include uncertainties about asset return and exchange-rate movement, the tax treatment of interest income in different countries, and governmental main developed countries, departures from interest-rate parity are sometimes substantial in the short run but tend to be small in the long run.

            We can use the result on interest-rate parity to compare real interest rate across countries. To make this comprasion, we need our earlier result for purchasing-power parity in relative form:

Δ Ɛ_t/ Ɛ = π^f- π

We can revise this condition to express the term as expected rates of change:

( Δ Ɛ_t/ Ɛ_t )^e = (π^f )- π^e

If we substitute this result for ( Δ Ɛ_t/ Ɛ_t )^e into the interest-rate parity condition from equation (18.10), we get

i ^f- i = (π^f )- π^e

interest-rate differential = difference in expected inflation rates

we can rearrange the term to get an important result about real interest rates :

key equation (equality of expected real interest rates across countries :

i ^f - (π^f )^e = i - π^e

foreign expected real interest rate = home expected real interest rate

Thus, the combination of the interest-rate parity condition (equetion 18.10) with the PPP condition in relative form (18.11) implies that expected real interest rates are the same in foreign country ( the united kingdom ) and the home country (the united states).

For advanced countries,expected real interest rates on government securities are not equal in practice, but the differences are usualy not too large. One reason for the discrepancies is that, as discussed before, the PPP condition in relative form (equation 18.11) does not always hold, that is, real exchange rates are not always expected to be constant. We mentioned that, for advanced countries, real exchange rates tended to adjust over time toward values close to 1.0. Consider, for example,switzerland and sweden, which were expensive compared to the united states in 2004-the real exchange rate in table 18.2 were 0,69 and 0,77,respectively. Our prediction was that these real exchange rates would increase toward 1.0 in the long run.

Recall that the formula for the real exchange rate is

Real exchange rate =           Ɛ          

                                                            P^f / P

Our prediction of a rising real exchange rate for switzerland and sweden means that the expected growth rate of the nominal exchange rate,  ( Δ Ɛ_t/ Ɛ_t )^e , must be greater than the expected growth of P^f/P, which equal the difference between the expected inflation rates, (π^f )^e - π^e. Therefore, instead of the equality in equation 18.11, we have the inequality :

( Δ Ɛ_t/ Ɛ_t )^e   >  (π^f )- π^e

If we substitute this inequality into the interest rate parity condition in equalition 18.10 we get

i ^f- i   >  (π^f )- π^e

rearranging terms, we have

i ^f - (π^f )^e   >  i – π

foreign espected real interest rate > home expected real interest rate

            Therefore, our prediction for a country that initially more expensive than the unites states-such as switzerland and sweden with their low real wxchange rates in 2004-is that the expected real interest rates in the countries will be higher than in the united states. One way to think about this result is that we predict that relatively more expensive countries will become cheaper iver time. To make this adjustment, the countries must have relatively low inflation rates, which correspond to relatively high real interest rates.