Interest_rate_parity

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The interest rate parity is the basic identity that relates interest rates and exchange rates. The identity is theoretical, and usually follows from assumptions imposed in economics models. There is evidence that supports as well as rejects interest rate parity.

Interest rate parity is an arbitrage condition which says that the returns from borrowing in one currency, exchanging that currency for another currency and investing in interest-bearing instruments of the second currency, while simultaneously purchasing futures contracts to convert the currency back at the end of the investment period, should be equal to the returns from purchasing and holding similar interest-bearing instruments of the first currency. If the returns are different, investors could theoretically arbitrage and make risk-free returns.

Looked at differently, interest rate parity says that the spot and future prices for currency trades incorporate any interest rate differentials between the two currencies.

Two versions of the identity are commonly presented in academic literature: covered interest rate parity and uncovered interest rate parity.

Contents 1 Covered interest rate parity 1.1 An example 2 Uncovered interest rate parity 2.1 Uncovered interest parity example 2.2 Uncovered vs. covered interest parity example 3 Cost of carry model

Covered interest rate parity

Covered interest parity (also called interest parity condition) means that the following equation holds:

$(1\; +\; i\_\backslash \$)\; =\; (F/S)\; (1\; +\; i\_c)\backslash ;$

where:

• $i\_\backslash \$$ is the domestic interest rate implied by debt of a given maturity;
• $i\_c$ is the interest rate in the foreign country for debt of the same maturity;
• $S$ is the spot exchange rate, expressed as the price in domestic currency ($) of one unit of the foreign currency c, i.e. $/c;
• $F$ is the forward exchange rate implied by a forward contract maturing at the same time as the domestic and foreign debt underlying $i\_\backslash \$$ and $i\_c$. F is expressed in the same units as S, namely $/c.

Taking natural logs of both sides of the interest parity condition yields:

$i\_\backslash \$\; =\; i\_c\; +\; ln(F/S)\backslash ;$

where all interest rates are now the continuously compounded equivalents. ln(F/S) is the forward premium, the percentage difference between the forward rate and the spot rate. Covered interest parity states that the difference between domestic and foreign interest rates equals the forward premium. When , the forward price of the foreign currency will be below the spot price. Conversely, if $i\_\backslash \$\; >\; i\_c$, the forward price of the foreign currency will exceed the spot price.

Covered interest parity assumes that debt instruments denominated in domestic and foreign currency are freely traded internationally (absence of capital controls), and have similar risk. If the parity condition does not hold, there exists an arbitrage opportunity. (see covered interest arbitrage and an example below).

The interest parity condition may also be expressed as:

$i\_\backslash \$\; =\; i\_c\; +\; \backslash frac\; \{F\; -\; S\}\; \{S\}\; (1\; +\; i\_c)$

The following common approximation is valid when S is not too volatile:

$i\_\backslash \$\; =\; i\_c\; +\; \backslash frac\; \{F\; -\; S\}\; \{S\}.$

An example

In short, assume that

.

This would imply that one dollar invested in the US < one dollar converted into a foreign currency and invested abroad. Such an imbalance would give rise to an arbitrage opportunity, where in one could borrow at the lower effective interest rate in US, convert to the foreign currency and invest abroad.

The following rudimentary example demonstrates covered interest rate arbitrage (CIA). Consider the interest rate parity (IRP) equation,

$(1\; +\; i\_\backslash \$)\; =\; (F/S)(1\; +\; i\_c)\backslash ;$

Assume:

• the 12-month interest rate in US is 5%, per annum
• the 12-month interest rate in UK is 8%, per annum
• the current spot exchange rate is 1.5 $/£
• the forward exchange rate implied by a forward contract maturing 12 months in the future is 1.5 $/£.

Clearly, the UK has a higher interest rate than the US. Thus the basic idea of covered interest arbitrage is to borrow in the country with lower interest rate and invest in the country with higher interest rate. All else being equal this would help you make money riskless. Thus,

• Per the LHS of the interest rate parity equation above, a dollar invested in the US at the end of the 12-month period will be,

$1 · (1 + 5%) = $1.05

• Per the RHS of the interest rate parity equation above, a dollar invested in the UK (after conversion into £ and back into $ at the end of 12-months) at the end of the 12-month period will be,

$1 · (1.5/1.5)(1 + 8%) = $1.08

Thus one could carry out a covered interest rate (CIA) arbitrage as follows,

1. Borrow $1 from the US bank at 5% interest rate.
2. Convert $ into £ at current spot rate of 1.5$/£ giving 0.67£
3. Invest the 0.67£ in the UK for the 12 month period
4. Purchase a forward contract on the 1.5$/£ (i.e. cover your position against exchange rate fluctuations)

At the end of 12-months

1. 0.67£ becomes 0.67£(1 + 8%) = 0.72£
2. Convert the 0.72£ back to $ at 1.5$/£, giving $1.08
3. Pay off the initially borrowed amount of $1 to the US bank with 5% interest, i.e $1.05

The resulting arbitrage profit is $1.08 − $1.05 = $0.03 or 3 cents per dollar.

Obviously, arbitrage opportunities of this magnitude would vanish very quickly.

In the above example, some combination of the following would occur to reestablish Covered Interest Parity and extinguish the arbitrage opportunity:

• US interest rates will go up
• Forward exchange rates will go down
• Spot exchange rates will go up
• UK interest rates will go down

Uncovered interest rate parity

The uncovered interest rate parity postulates that

$(1\; +\; i^\backslash \$\_\{t,t+1\})\; =\; \backslash frac\; \{E\_t[\; S\_\{t+1\}\; ]\}\; \{S\_t\}\; (1\; +\; i^c\_\{t,t+1\}).\backslash ;$

The equality assumes that the risk premium is zero, which is the case if investors are risk-neutral. If investors are not risk-neutral then the forward rate ($F\_\{t,t+1\}$) can differ from the expected future spot rate ($E\_t[\; S\_\{t+1\}\; ]$), and covered and uncovered interest rate parities cannot both hold.

The uncovered parity is not directly testable in the absence of market expectations of future exchange rates. Moreover, the above rather simple demonstration assumes no transaction cost, equal default risk over foreign and domestic currency denominated assets, perfect capital flow and no simultaneity induced by monetary authorities. Note also that it is possible to construct the UIP condition in real terms, which is more plausible.

Uncovered interest parity example

An example for the uncovered interest parity condition: Consider an initial situation, where interest rates in the US (home country) and a foreign country (e.g. Japan) are equal. Except for exchange rate risk, investing in the US or Japan would yield the same return. If the dollar depreciates against the yen, an investment in Japan would become more profitable than a US-investment - in other words, for the same amount of yen, more dollars can be purchased. By investing in Japan and converting back to the dollar at the favorable exchange rate, the return from the investment in Japan, in the dollar term, is higher than the return from the direct investment in the US. In order to persuade an Investor to invest in the US nonetheless, the dollar interest rate would have to be higher than the yen interest rate by an amount equal to the devaluation (a 20% depreciation of the dollar implies a 20% rise in the dollar interest rate).

Note: Technically, a 20% depreciation in the dollar only results in an approximate rise of 20% in U.S. interest rates. The exact form is as follows: Change in spot rate (Yen/Dollar) equals the dollar interest rate minus the yen interest rate, with this expression being divided by one plus the yen interest rate.

Uncovered vs. covered interest parity example

Let\'s assume you wanted to pay for something in Yen in a month\'s time. There are two ways to do this.

• (a) Buy Yen forward 30 days to lock in the exchange rate. Then you may invest in dollars for 30 days until you must convert dollars to Yen in a month. This is called covering because you now have covered yourself and have no exchange rate risk.

• (b) Convert spot to Yen today. Invest in a Japanese bond (in Yen) for 30 days (or otherwise loan Yen for 30 days) then pay your Yen obligation. Under this model, you are sure of the interest you will earn, so you may convert fewer dollars to Yen today, since the Yen will grow via interest. Notice how you have still covered your exchange risk, because you have simply converted to Yen immediately.

• (c) You could also invest the money in dollars and change it for Yen in a month.

According to the interest rate parity, you should get the same number of Yen in all methods. Methods (a) and (b) are covered while (c) is uncovered.

• In method (a) the higher (lower) interest rate in the US is offset by the forward discount (premium).
• In method (b) The higher (lower) interest rate in Japan is offset by the loss (gain) from converting spot instead of using a forward.
• Method (c) is uncovered, however, according to interest rate parity, the spot exchange rate in 30 days should become the same as the 30 day forward rate. Obviously there is exchange risk because you must see if this actually happens.

General Rules: If the forward rate is lower than what the interest rate parity indicates, the appropriate strategy would be: borrow Yen, convert to dollars at the spot rate, and lend dollars.

If the forward rate is higher than what interest rate parity indicates, the appropriate strategy would be: borrow dollars, convert to Yen at the spot rate, and lend the Yen.

Cost of carry model

A slightly more general model, used to find the forward price of any commodity, is called the cost of carry model. Using continuously compounded interest rates, the model is:

$\backslash \; F\; =\; S\; e^\{(r+s-c)t\}$

where $F$ is the forward price, $S$ is the spot price, $e$ is the base of the natural logarithms, $r$ is the risk free interest rate, $s$ is the storage cost, $c$ is the convenience yield, and $t$ is the time to delivery of the forward contract (expressed as a fraction of 1 year).

For currencies there is no storage cost, and c is interpreted as the foreign interest rate. The currency prices should be quoted as domestic units per foreign units.

If the currencies are freely tradeable and there are minimal transaction costs, then a profitable arbitrage is possible if the equation doesn\'t hold. If the forward price is too high, the arbitrageur sells the forward currency, buys the spot currency and lends it for time period t, and then uses the loan proceeds to deliver on the forward contract. To complete the arbitrage, the home currency is borrowed in the amount needed to buy the spot foreign currency, and paid off with the home currency proceeds of forward contract.

Similarly, if the forward price is too low, the arbitrageur buys the forward currency, borrows the foreign currency for time period t and sells the foreign currency spot. The proceeds of the forward contract are used to pay off the loan. To complete the arbitrage, the home currency from the spot transaction is lent and the proceeds used to pay for the forward contract.

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