1. International Finance
Lecture 10:
Covered interest rate parity
Read: Chapter 6 ( )
Aaron Smallwood Ph.D.

2. Interest Rates and Exchange Rates
One of the most important relationships in international finance is the relationship between interest rates and the exchange rate.
The setup:
Suppose a trader has the ability to borrow or lend in both the domestic market and a foreign market.
Denote the domestic annualized interest rate as it and denote the foreign annualized interest rate as it*.
Denote the spot and forward domestic currency price of the foreign currency as St and Ft. Suppose the forward contract matures in M days.

3. Interest Rate Adjustment
The forward contract matures in M days.
Interest rates are quoted in annualized terms. We need to adjust interest rates to allow a comparison:

4. Borrowing in the domestic currency; lending in the foreign
If I borrow one unit of the domestic currency, in M days, I will repay:
To lend in the foreign currency, I must convert domestic currency into foreign currency. For each unit of domestic currency I have, I receive 1/St units of the foreign currency.

5. Lending
Now I lend the proceeds in the foreign country…I have 1/St units of the foreign currency…I will receive:
Problem…these proceeds are in foreign currency units…I want the proceeds in domestic currency. I could have acquired a forward contract, to sell (forward) foreign currency proceeds in M periods. The result:

6. The result: Suppose What if,
Then, to profit, I could borrow in the domestic currency, convert the proceeds into foreign currency, lend in the foreign market, and convert proceeds back into domestic currency using a forward contract.
What if,
I can still profit…Start by borrowing in the foreign currency.

7. Implications The no arbitrage condition implies:
The equation, known as the no arbitrage condition, has important implications.
To illustrate, suppose the equation did not hold.
Example, suppose:
it: 6.00% (annualized interest rate in the US for an asset maturing in one month).
it*: 5.25% (annualized interest rate in Germany for a similar asset maturing in 1 month).
St: $ (dollar price of the euro on the spot market).
Ft: $1.30 (assume asset matures in 30 days time).

8. An arbitrage opportunity exists:
First, interest rates are adjusted:
We have:
As thus:
PROFIT TIME!

9. How do we profit
Start by borrowing in the foreign country. Let’s do it big! Let’s borrow €10,000,000.
We will have to repay:
€10,000,000* = €10,043,750
Note, as a result of our actions, demand for loanable funds in Germany increases. Foreign interest rates increase.
Convert euros and lend in the US.
€10,000,000*$ = $13,653,700.
Lend at .5% yielding:
13,653,700*(1.005) = $13,721,
Note, two things happen here. On the spot market, supply of euros increases, driving down St.
Supply of loanable funds increases in the US, driving down it.

10. Last step…
Finally, you use the pre-existing forward contract to sell the dollar proceeds for euros. The result:
$13,721,968.50/1.30 = 10,555,
Profit: €10,555, €10,043,750 =
€511,
Note, in the final step, you sell forward dollars. You are buying forward euros. This likely causes, Ft to rise.

11. No arbitrage opportunities?
NOT ONCE YOU HAVE LEFT THE MARKET!
Recall, our arbitrage opportunity existed because:
However, as a result of your actions:
1. Foreign interest rates rise.
2. The spot rate falls.
3. Domestic interest rates fall.
4. The forward rate rises.

12. No arbitrage
Thus, we can expect smart traders will eliminate profitable arbitrage opportunities quickly when they exist. Thus, as a rule:
Implications: Suppose domestic interest rates fall as a result of, say, monetary policy.
To ensure equilibrium:
1. Foreign interest rates could also fall…
2. and/or The forward rate could fall.
3. and/or…The spot rate COULD rise. An increase in the spot rate implies a DOMESTIC CURRENCY DEPRECIATION.

13. Covered Interest Rate Parity
The no arbitrage condition is frequently re-arranged in a more convenient way:

14. (Fb/Sa)(1 + i¥l)  (1 + i$ b)  0
Deviations from CIRP?
Transactions Costs: Suppose we thought we could profit from borrowing in the US and lending in Japan.
The interest rate available to an arbitrageur for borrowing, ib,may exceed the rate she 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.
Taxation differences on capital gains.

15. FRUH and UIP Ft = E(St+1) if investors are risk neutral.
Since investors are assumed to be rational, E(St+1) = St+1 + εt+1 where εt+1 is a random (unforecastable) forecast error.
Then Ft = St+1 + εt+1 and the forward rate is an unbiased predictor of the future spot rate.
From CIP =
(1 + i*t)
(it – i*t) St
Ft – St

16. FRUH and UIP = (1 + i*t) (it – i*t) St E(St+1) – St
Then it must be the case that
This is the Uncovered Interest Parity condition.
It will only hold if investors are risk neutral or equivalently they do not care about the currency denomination of the assets they hold =
(1 + i*t)
(it – i*t) St
E(St+1) – St
E(st+1) – st=it-it*

17. FRUH and UIP
If uncovered interest parity (UIP) holds then FRUH is true and investors are risk neutral.
Risk neutrality implies that investors have no currency preference in which their investments are denominated.
Assets with identical risk characteristics but denominated in different currencies will be viewed as perfect substitutes.

18. Purchasing Power Parity
Purchasing Power Parity and Exchange Rate Determination
PPP Deviations and the Real Exchange Rate

19. Purchasing Power Parity in a Perfect Capital Market
Purchasing power parity (PPP) is built on the notion of arbitrage across goods markets and the Law of One Price.
The Law of One Price is the principle that the same goods will sell for the same price in two markets, taking into account the exchange rate.

20. Purchasing Power Parity
Let PUS and PCHINA represent the weighted average price level for goods in the U.S. and Chinese market baskets respectively.
Absolute PPP predicts that these two price measures will be equal after adjusting for the exchange rate: PUS = S$/RMB  PCHINA
Absolute PPP requires that the consumption baskets are identical across the two countries.

21. Purchasing Power Parity and Exchange Rate Determination
The exchange rate between two currencies should equal the ratio of the countries’ price levels:
S($/RMB) =
PCHINA P$
For example, an ounce of gold on March 12 cost $ in the U.S. and RMB8, in China. Then the price of one dollar in terms of RMB should be:
S(RMB/$) = P$
PRMB
$1,370.70
RMB8,413.36 =
= ¥6.138

22. Relative Purchasing Power Parity
Note, that %Δp = π, the rate of inflation
Relative PPP states that the rate of change in the exchange rate is equal to the differences in the rates of inflation:
%Δs =
(1 + ¥ )
($ – ¥)
≈ $ – ¥
If U.S. inflation is 5% and Chinese inflation is 8%, the yuan should ordinarily depreciate by 2.78% or approximately 3%.

23. Ex-Ante PPP
Ex-Ante PPP says that relative PPP will hold in an expected value sense, i.e.
Where E is the expectations operator signifying that E(·) is an expected value.

24. PPP Deviations and the Real Exchange Rate
The real exchange rate is
If absolute PPP holds then
Absolute PPP implies that price indices in all countries are computed with the same weights and the same basket of goods. Since this is never true, relative PPP is the more appropriate form. Under relative PPP, RERt will be constant but differ from unity.

25. Purchasing Power Parity and Overvalued or Undervalued Currencies
Example
Base period nominal exchange rate = $1.50/£
Prices of U.S. goods had risen by 8%
Prices of U.K. goods had risen by 4%
PPP spot rate = $1.50/£  1.08/1.04 = $1.5577/£
A nominal exchange rate of $1.5577/£ would reestablish PPP in comparison to the base period.
Nominal exchange rates greater than $1.5577/£ represent £ “overvaluation” ($ undervaluation), while rates less than $1.5577/£ represent $ “overvaluation” (£ undervaluation).

26. Purchasing Power Parity and Overvalued or Undervalued Currencies
Nominal exchange rates greater than the PPP implied exchange rate represent foreign currency overvaluation, while nominal exchange rates less than the PPP implied exchange rate represent domestic overvaluation (or foreign undervaluation).