INTERNATIONAL FINANCE

Unit 4

Interest Rate Parity

• "The parity conditions are manifestations of the law of one price, the idea that two equivalent things must sell for the same price across different locations or markets. These parity conditions provide key insights into how foreign exchange rates are determined and how to forecast foreign exchange rates.”

• Arbitrage: "the act of simultaneously buying and selling the same or equivalent assets or commodities for the purpose of making certain, guaranteed profits”

• If there are arbitrage opportunities, the market cannot be in equilibrium.

Interest Rate Parity Relationship

S$/£ ×F$/£ = (1 + i£) (1 + i$)

Consider alternative one-year investments for $100,000: 1. Invest in the U.S. at i$. Future value = $100,000 × (1 + i$) 2. Trade your $ for £ at the spot rate, invest $100,000/S$/£ in Britain at i£

while eliminating any exchange rate risk by selling the future value of the British investment forward.

Interest Rate Parity Carefully Defined

S$/£ F$/£Future value = $100,000(1 + i£)×

S$/£ F$/£(1 + i£) × = (1 + i$)

Since these investments have the same risk, they must have the same future value (otherwise an arbitrage would exist)

Formally,

Interest Rate Parity Defined

IRP is sometimes approximated as

i$ – i¥ ≈ S

F – S

1 + i$ 1 + i¥ S$/¥

F$/¥=

• Depending upon how you quote the exchange rate (as $ per ¥ or ¥ per $) we have:

Interest Rate Parity Carefully Defined

1 + i$ 1 + i¥

S¥/$ F¥/$=

1 + i$ 1 + i¥ S$/¥

F$/¥=or

…so be a bit careful about that.

• No matter how you quote the exchange rate ($ per ¥ or ¥ per $) to find a forward rate, increase the dollars by the dollar rate and the foreign currency by the foreign currency rate:

Interest Rate Parity Carefully Defined

…be careful—it’s easy to get this wrong.

1 + i$ 1 + i¥

F$/¥ = S$/¥ ×or1 + i$ 1 + i¥F¥/$ = S¥/$ ×

• People´s expectations thus become self-fulfilling. – Expectation plays a key role in exchange rate determination. When

people “expect” the exchange rate to go up in the future, it goes up now. • Exchange rate behavior will be driven by news events

– Exchange rate will tend to exhibit a dynamic and volatile short term behavior, responding to various news events.

• Based on those considerations, we could adjust this equation:

• Where E(e) is the expected rate of change in the exchange rate. This is known as UNCOVERED INTEREST RATE PARITY

Interest Rate Parity

i$ – i¥ ≈ S

F – S i$ – i¥ ≈ E (e)

• If every trader borrow in US as much as possible (1), lend in the UK (2), buy the GBP spot (3) and at the same time sell the GBP forward (4), the following adjustaments will occur: 1. Interest rate will rise in US (i$ increase) 2. Interest rate will fall in UK (iGBP decrease) 3. GBP will appreciate in the spot market (Spot increase) 4. Pound will depreciate in the Forward market (Forward

decrease)

IRP adjustment mechanics

• If, annual interest rate is 5% in US and 8% in UK • Uncovered IRP suggests that the GBP is expected to

depreciate against the USD by about 3%, that is E(e) ≈ - 3%

• Though unlike IRP, Uncovered Interest Rate Parity often doesn´t hold giving rise to uncovered interest arbitrage opportunities

Uncovered Interest Rate Parity

If IRP failed to hold, an arbitrage (Covered Interest Arbitrage) would exist. It’s easiest to see this in the form of an example.

Consider the following set of foreign and domestic interest rates and spot and forward exchange rates.

IRP and Covered Interest Arbitrage

Spot exchange rate S($/£) = $2.0000/£ 360-day forward rate F360($/£) = $2.0100/£ U.S. discount rate i$ = 3.00% British discount rate i£ = 2.49%

A trader with $1,000 could invest in the U.S. at 3.00%, in one year his investment will be worth

$1,030 = $1,000 ´ (1+ i$) = $1,000 ´ (1.03) Alternatively, this trader could 1. Exchange $1,000 for £500 at the prevailing spot rate, 2. Invest £500 for one year at i£ = 2.49%; earn £512.45 3. Translate £512.45 back into dollars at the forward

rate F360($/£) = $2.01/£, the £512.45 will be worth $1,030.

IRP and Covered Interest Arbitrage

According to IRP only one 360-day forward rate, F360($/£), can exist. It must be the case that

F360($/£) = $2.01/£

Why?

If F360($/£) ¹ $2.01/£, an astute trader could make money with one of the following strategies:

Interest Rate Parity & Exchange Rate Determination

If F360($/£) > $2.01/£ i. Borrow $1,000 at t = 0 at i$ = 3%. ii. Exchange $1,000 for £500 at the prevailing spot rate, (note that £500 = $1,000 ÷ $2/£) invest £500 at 2.49% (i£) for one year to achieve £512.45 iii. Translate £512.45 back into dollars, if F360($/£) > $2.01/£, then £512.45 will be more than enough to repay

your debt of $1,030.

Arbitrage Strategy one way

If F360($/£) < $2.01/£

i. Borrow £500 at t = 0 at i£= 2.49% . ii. Exchange £500 for $1,000 at the prevailing spot rate, invest $1,000 at 3% for one year to achieve $1,030. iii. Translate $1,030 back into pounds, if F360($/£) < $2.01/£, then $1,030 will be more than enough to repay

your debt of £512.45.

Arbitrage Strategy or another

How long will this arbitrage opportunity last? – Only for a short while, as as soon as

deviations from Interest Rate Parity are detected, informed trades will inmediately carry out COVERED INTEREST ARBITRAGE (CIA) transactions and IRP will be restored quite quickly.

Arbitrage Strategies

You are a U.S. importer of British woolens and have just ordered next year’s inventory. Payment of £100M is due in one year.

IRP and Hedging Currency Risk

IRP implies that there are two ways that you fix the cash outflow to a certain U.S. dollar amount:

a) Put yourself in a position that delivers £100M in one year—a long forward contract on the pound. You will pay (£100M)($2.01/£) = $201M in one year.

b) Form a money market hedge as shown below.

Spot exchange rate S($/£) = $2.00/£ 360-day forward rate F360($/£) = $2.01/£ U.S. discount rate i$ = 3.00% British discount rate i£ = 2.49%

To form a money market hedge: 1. Borrow $195,140,989.36 in the U.S. (in one year you will owe

$200,995,219.05). 2. Translate $195,140,989.36 into pounds at the spot rate S($/£) =

$2/£ to receive £97,570,494.68 3. Invest £97,570,494.68 in the UK at i£ = 2.49% for one year. 4. In one year your investment will be worth £100 million—exactly

enough to pay your supplier.

IRP and a Money Market Hedge

Where do the numbers come from? We owe our supplier £100 million in one year—so we know that we need to have an investment with a future value of £100 million. Since i£ = 2.49% we need to invest £97,570,494.68 at the start of the year.

Money Market Hedge

How many dollars will it take to acquire £97,570,494.68 at the start of the year if S($/£) = $2/£?

£97,570,494.68 = £100,000,000

1.0249

$195,140,989.36 = £97,570,494.68 × $2.00 £1.00

• This is the same idea as covered interest arbitrage.

• To hedge a foreign currency payable, buy a bunch of that foreign currency today and sit on it. – Buy the present value of the foreign currency payable

today. – Invest that amount at the foreign rate. – At maturity your investment will have grown enough

to cover your foreign currency payable.

Money Market Hedge

• Transactions Costs – The interest rate available to an arbitrageur for borrowing, ib may

exceed the rate he can lend at, il. – There may be bid-ask spreads to overcome, Fb/Sa < F/S – Thus

(Fb/Sa)(1 + i¥l) - (1 + i¥ b) £ 0 • Capital Controls

– Governments sometimes restrict import and export of money through taxes or outright bans.

Reasons for Deviations from IRP

• If PPP holds and thus the differential inflation rates between countries are exactly offset by exchange rates changes, countries ´ competitive positions in world export markets will not be systematically affected by exchange rates.

• If there are deviations from PPP, changes in nominal exchange rates cause changes in the real exchange rates, affecting the international competitive positions of countries,, which in turn, would affect countries ´ trade balances.

Purchasing power parity

• The Economist each year compiles local prices of Big Macs around the world and computes the so-called “Big Mac PPP” – The exchange rate that would equalize the hamburger prices between

America and elsewhere – Comparing this PPP and the eactual exchange rate, a currency may be

judged to be either undervalued or overvalued • Big Mac $3.57 in US and 12.50 yuan in China • Big Mac PPP 3.50 yuan per $ • Actual exchange rate is 6.83 yuan per $, implying that the yuan is vastly

undervalued.

Big Mac

• The exchange rate between two currencies should equal the ratio of the countries’ price levels:

Purchasing Power Parity and Exchange Rate Determination

S($/£) = P£ P$

S($/£) = P£ P$

£150 $300

= = $2/£

l For example, if an ounce of gold costs $300 in the U.S. and £150 in the U.K., then the price of one pound in terms of dollars should be:

• PPP probably doesn’t hold precisely in the real world for a variety of reasons. – Haircuts cost 10 times as much in the developed world as in the

developing world. – Film, on the other hand, is a highly standardized commodity that is

actively traded across borders. – Shipping costs, as well as tariffs and quotas can lead to deviations

from PPP. • PPP-determined exchange rates still provide a valuable benchmark.

Evidence on PPP

• Suppose the spot exchange rate is $1.25 = €1.00 • If the inflation rate in the U.S. is expected to be 3% in the next year

and 5% in the euro zone, • Then the expected exchange rate in one year should be

$1.25×(1.03) = €1.00×(1.05)

Purchasing Power Parity and Exchange Rate Determination

F($/€) = $1.25×(1.03) €1.00×(1.05)

$1.23 €1.00

=

• The euro will trade at a 1.90% discount in the forward market:

Purchasing Power Parity and Exchange Rate Determination

$1.25 €1.00

= F($/€) S($/€)

$1.25×(1.03) €1.00×(1.05) 1.03

1.05 1 + p$ 1 + p€

= =

Relative PPP states that the rate of change in the exchange rate is equal to differences in the rates of inflation—roughly 2%

• Notice that our two big equations today equal each other:

Purchasing Power Parity and Interest Rate Parity

= = F($/€) S($/€)

1 + p$ 1 + p€

PPP

1 + i€ 1 + i$ =

F($/€) S($/€)

IRP

• We could also reformulate our equations as inflation or interest rate differentials:

Expected Rate of Change in Exchange Rate as Inflation Differential

= F($/€) – S($/€)

S($/€) 1 + p$ 1 + p€

– 1 = 1 + p$ 1 + p€

– 1 + p€ 1 + p€

= F($/€) S($/€)

1 + p$ 1 + p€

= F($/€) – S($/€)

S($/€) p$ – p€ 1 + p€

E(e) = ≈ p$ – p€

Expected Rate of Change in Exchange Rate as Interest Rate Differential

= F($/€) – S($/€)

S($/€) i$ – i€ 1 + i€

E(e) = ≈ i$ – i€

• An increase (decrease) in the expected rate of inflation will cause a proportionate increase (decrease) in the interest rate in the country.

• For the U.S., the Fisher effect is written as: 1 + i$ = (1 + r$ ) × E(1 + p$)

Where r$ is the equilibrium expected “real” U.S. interest rate E(p$) is the expected rate of U.S. inflation i$ is the equilibrium expected nominal U.S. interest rate

The Fisher Effect

If the Fisher effect holds in the U.S. 1 + i$ = (1 + r$ ) × E(1 + p$)

and the Fisher effect holds in Japan, 1 + i¥ = (1 + r¥ ) × E(1 + p¥)

and if the real rates are the same in each country r$ = r¥

then we get the International Fisher Effect:

International Fisher Effect

E(1 + p¥) E(1 + p$)1 + i$

1 + i¥ =

If the International Fisher Effect holds,

International Fisher Effect

then forward rate PPP holds:

E(1 + p¥) E(1 + p$)1 + i$

1 + i¥ =

and if IRP also holds

1 + i$ 1 + i¥

S¥/$ F¥/$

= E(1 + p¥) E(1 + p$)

= S¥/$ F¥/$

• Efficient Markets Approach • Fundamental Approach • Technical Approach

Forecasting Exchange Rates

• Financial Markets are efficient if prices reflect all available and relevant information.

• As less arbitrage available a better sign of efficiency • If this is so, exchange rates will only change when new information

arrives, thus: St = E[St+1]

and Ft = E[St+1| It]

• Predicting exchange rates using the efficient markets approach is affordable and is hard to beat.

Efficient Markets Approach

• Involves econometrics to develop models that use a variety of explanatory variables. This involves three steps: – step 1: Estimate the structural model. – step 2: Estimate future parameter values. – step 3: Use the model to develop forecasts.

• The downside is that fundamental models do not work any better than the forward rate model or the random walk model.

• Sentiment, catastrophes…

Fundamental Approach

• Technical analysis looks for patterns in the past behavior of exchange rates.

• Chart analysis • Clearly it is based upon the premise that history repeats itself.

Technical Approach