LOS h: Explain interest rate parity, and illustrate covered interest arbitrage.

Interest rate parity is given by:

Forward FCC = Spot FCC × [(1 + r_domestic) / (1 + r_foreign)], or alternatively
Spot FCC = Forward FCC × [(1 + r_foreign) / (1 + r_domestic)] = 0.89348 × (1.0190 / 1.0338) = 0.88069

Note that in this question, the dollar is the foreign currency and the Euro is the domestic currency.

F/S= (1 + r_D) / (1 + r_F) where the currency is quoted as FCC
F = (1.04/1.05)(0.400) = 0.396

If (r_D ? r_F) is approximately equal to the forward premium, which is (Forward D/F) ? Spot(D/F) / Spot(D/F), then no arbitrage opportunities exist.

0.08 ? 0.10 ? (5 ? 5.102) / 5.102.

-0.02 ? -0.01999.

According to interest rate parity the currency with the lower interest rate is expected to appreciate so the Argentina rate of 7% is approximately 4% less than the Brazilian rate of 7 + 4 = 11%.

Forward rate (DC/FC) = Spot Rate (DC/FC) × [(1 + domestic rate) / (1 + foreign rate)],
Forward rate = 1 / 1.65 (KPW/$) × (1.09 / 1.10) = 0.60055 $/KPW, or 1.665 KPW/$.
Alternatively, forward rate = 1.65 (KPW/$) × (1.10 / 1.09) = 1.665 (KPW/$).

Using the following interest rate parity equation:

Forward_DC/FC=Spot_DC/FC × [(1 + r_domestic) / (1 + r_foreign )] 

Solving for the forward rate:  Forward_DC/FC = 4 × [(1 + 0.08) / (1 + 0.06)]

= 4(1.08) / (1.06)

= 4(1.01887)

= 4.07547

Using the following IRP equation: Forward_FCC = Spot_FCC × [(1 + r_domestic) / (1 + r_foreign )] 

Solving for the spot rate: Spot_FCC = Forward_FCC × [(1 + r_foreign) / (1 + r_domestic)] 

                                    = [(1 + 0.09) / (1 + 0.07)](5)

                                    = (1.09 / 1.07)(5)

                                    = 5.09

Forward_FCC / Spot_FCC = (1 + r_domestic) / (1 + r_foreign).

Spot_FCC = Forward_FCC (1 + r_foreign) / (1 + r_domestic) = (5.00)(1.07) / (1.09) = 4.908

Interest rate parity is given by:

Forward_FCC = 0.5500 × (1.075/1.06) = NZD:USD 0.55778

F/S = (1 + r_domestic) / (1 + r_foreign). Note in this equation exchange rates are quoted as Domestic/Foreign.

S = F (1 + r_F) / (1 + r_D) = (4.00)(1.06) / (1.08) = 3.93

Forward(DC/FC) = Spot (DC/FC)[(1 + r domestic) / (1 + r foreign)]

(2.0010)(1.04/1.07)

(2.0010)(0.972)

= 1.9450

Forward (DC/FC) = Spot (DC/FC)[(1 + r_domestic) / (1 + r_foreign)]

(80 CY/EGP)[(1 + 0.055) / (1 + 0.06)]

(80)(0.99528)

= 79.6226

The approximate forward premium/discount is given by the interest rate differential. This differential is: 6.25% ? 5.15% = 1.10%. Since Tunisia has higher interest rates, its currency will be at a discount in the forward market. This discount equals: 0.011 × 0.8105 = 0.0089. Since the exchange rate is quoted in CHF:TND, as a depreciating currency, it will take more TND to buy one CHF. The forward rate is thus: 0.8105 + 0.0089 = CHF:TND 0.8194. In other words, the CHF is stronger in the forward market.

If (r_D ? r_F) > Forward premium, which is (Forward D/F) ? Spot(D/F) / Spot(D/F), then you would borrow foreign currency and lend out local currency. If the domestic rate is high relative to the hedged foreign rate, you would borrow foreign currency units and then sell them for domestic currency units at the spot rate, lend these domestic currency units at the domestic interest rate and simultaneously sell just enough domestic currency forward so that you can repay your foreign loan.

Interest rate parity requires that:
(Forward/Spot) = [(1+r_D)/(1+r_F)]
(1.2545/1.1975) = [1.13665/1.085]
So, interest rate parity holds and no arbitrage opportunity exists.

Alternately:

(1 + 0.13665) = [(1 + 0.085)(1.2545) / 1.1975]

1.13665 = [(1.085)(1.2545) / 1.1975]

1.13665 = 1.36113 / 1.1975

1.13665 = 1.13665, therefore no arbitrage profit can be made.

If 1 + r_D < (1 + r_F)(forward rate) / spot rate, then borrow domestic, lend out foreign, and convert back to domestic.

Borrow foreign if 1 + r_D> [(1 + r_F)(forward rate)] / spot rate

1 + 0.0802 > [(1 + 0.1102)(1.8)] / 1.9

1.0802 > 1.99836 / 1.9

1.0802 > 1.0518 therefore borrow foreign (dollars) and lend domestic (francs).

Alternatively, U.S. rate is 11.02 ? 8.02 = 3% higher and USD is at (1.8 ? 1.9) / 1.9 = 5.3% discount since USD will fall more than the extra 3% interest, better to lend francs.

Let us first check if an arbitrage opportunity exists. Applying the interest rate parity theorem, we have:

Forward rate = 0.1432 × 1.072/1.066 = 0.1440 $/ADF > 0.1430 $/ADF (quoted forward rate)

This implies that an arbitrage opportunity exists. The inequality implies that ADF is mispriced (weak) in the forward market or is underpriced relative to the dollar. We should buy ADF in the forward market and sell the dollar in the spot market. This requires that we borrow in Andorra and convert the francs into dollars at the spot rate. Invest the proceeds in U.S. securities @ 7.2%, and simultaneously enter into a forward transaction where we sell the dollars for ADF @ 0.1430 $/ADF. Assuming that we borrow 1 ADF today and convert it into dollars, we will have 0.1432 dollars to invest at 7.2% for one year. After one year we will have 0.1432 × 1.072 = 0.1535 dollars. At that point, we will owe an Andorran bank 1 × 1.066 or 1.066 ADF, including interest. We will need to convert enough dollars at the forward rate to pay off this loan. At the forward contract rate, we will need to convert 1.066 × 0.1430 = 0.1524 dollars into ADF to pay off our obligation. This will leave us with an arbitrage profit of 0.1535 ? 0.1524 = 0.0011 dollars.

This equation is Interest Rate Parity rearranged! If the term on the left (1 + r_DC), is less than the term on the right, it means that the domestic rate is low relative to the hedged foreign rate. Therefore, there is a profitable arbitrage from borrowing the domestic currency and lending at the foreign interest rate.

Because we lend in the foreign market, we say that the funds flow out of the domestic economy.

Use the following formula to determine if an arbitrage opportunity exists and which currency to borrow.

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate then borrow foreign.

1.0723 > [(1.0694)(1.70)] / 1.73

1.0723 > 1.81798 / 1.73

1.0723 > 1.0509, therefore borrow foreign (pounds).

Alternatively, the dollar is appreciating. [(1.73 ? 1.70) / 1.70] = 1.76% and the $U.S. interest rate is higher. Clearly, investing in $U.S. (and borrowing pounds) is the way to go.

Note that peso is at a forward premium (less pesos per CAD in the future) and that peso interest rate is higher. Therefore it is clear there are arbitrage profits from lending in pesos and borrowing CAD.

First convert to Canadian dollars to determine the amount of interest due at the end of the year. (15,000,000 MXN) × (CAD/569.87 MXN) = 26,321.79 CAD.

26,321.79 CAD × 0.0745 = 1,960.97 CAD interest due at the end of the year.

Lend out pesos 15,000,000 pesos × 1.0802 = 16,203,000 pesos received at the end of the year.

Convert to Canadian dollars (16,203,000 MXN) × (CAD/526.78 MXN) = 30,758.57 CAD.

Subtract the original loan amount and interest: 30,758.57 ? 26,321.79 (original loan) ? 1,960.97 (interest) = 2,475.81 CAD profit.

Convert the remainder back to pesos: (2,475.81 CAD) × (526.78 MXN/CAD) = 1,304,207.19 peso profit.

Borrow 10,000 GBP at 6.54% = 654 GBP interest due at the end of the year.
Convert to NZD: (10,000 GBP) × (1 NZD/0.41 GBP) = 24,390 NZD.
Lend out NZD at 7.03% interest: (24,390 NZD) × (1.0703) = 26,104.88 NZD.
Convert back to GBP: (26,104.88 NZD) × (0.45 GBP/NZD) = 11,747.20 GBP.
11,747.20 GBP ? 10,000 GBP (original amount borrowed) ? 654 GBP interest = 1,093.20 GBP profit.

First determine if arbitrage opportunities exist by using the following equation:

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate, then borrow foreign (dollars).

1.0589 > [(1.0575)(556.75)] / 581.23

1.0589 > 588.763 / 581.23

1.0589 > 1.01296, therefore, borrow foreign (dollars).

Borrow $10,000 at 5.75%, interest = $575 due at the end of the year. Convert to pesos using the spot rate: ($10,000) × (581.23 MXN/USD) = 5,812,300 pesos.

Lend out at 5.89%: (5,812,300 pesos) × (1.0589) = 6,154,644.47 pesos. Convert to dollars: (6,154,644.47 MXN) × (USD/556.75 MXN) = $11,054.59. $11,054.59 ? $10,000 (original amount borrowed) ? $575 (interest) = $479.59 profit.

Use the following formula to determine if an arbitrage opportunity exists and which currency to borrow.

if 1 + r_D > [(1 + r_F)(Forward rate)] / Spot rate, then borrow foreign.

1.0613 > [(1.0601)(1.193)] / 1.265

1.0613 > 1.265 / 1.265

1.0613 > 1 therefore borrow foreign (British pound) and lend domestic (Canadian dollar).

If the forward rate is lower than what the interest rate parity indicates, the appropriate strategy would be: borrow pounds, convert to dollars at the spot rate, and lend dollars. (Study Session 4, LOS 18.h)

If the forward rate is higher than what interest rate parity indicates, the appropriate strategy would be: borrow dollars, convert to pounds at the spot rate, and lend the pounds. (Study Session 4, LOS 18.h)

Given the above relationship, interest rate parity does not hold.

(If interest parity held, 1.70 = 1.85 × (1.08 / 1.10), but 1.85 × (1.08 / 1.10) = 1.82).

Therefore, an arbitrage opportunity exists.

To determine whether to borrow dollars or pounds, express the foreign rate in hedged US$ terms (by manipulating the equation for IRP). We get:

(1.70 / 1.85) × 1.10 = 1.0108, which is less than 1.08 (U.S. rate), so we should start by borrowing British pounds and lending U.S. dollars.

Arbitrage Example:

1. Today:
   1. borrow 5,000 GBP @ 10%
   2. buy $9,250 with the proceeds of the loan (5,000 GBP × 1.85).
   3. lend $9,250 @ 8%
   4. buy 5,500 GBP one year in the future @ 1.70 $/￡. This guarantees your loan repayment of 5,000 GBP × 1.1 = 5,500 GBP.
2. One year later, close out your position:
   1. collect the proceeds of your loan: $9,990 = $9,250 × 1.08
   2. buy 5,500 GBP with your forward contract → cost = 5,500 GBP × 1.70 = $9,350
   3. pay off your loan of 5,500 GBP
   4. reap your profits: $9,990 ? $9,350 = $640
   5. Alternately, you could say that the arbitrage profit is 376.47 GBP. Bob Bowman is a US investor so we left his profits in USD. 640 USD = 1.70 × 376.47 GBP.

(Study Session 4, LOS 18.h)

Interest Rate Parity

Interest rates between countries and their exchange rates (spot and futures) must be in equilibrium at all times or else there will be arbitrage opportunities. Interest rate parity says that:

F_0,t = S₀ [(1+r_US) / (1+r_FX)]

Where:

S0	=	the current exchange rate in the spot market
F0,t	=	the current exchange rate in the forward of futures market
rUS	=	the risk-free interest rate in the U.S.
rFX	=	the risk-free interest rate in the foreign market

Note: the above currency quotes are expressed in $/FX. (Study Session 4, LOS 18.h)

The PPP assumption is that the future spot exchange rate will change exactly as the inflation rates affect the values of each currency. For the computation, raise the U.S. inflation rate to the 5th power (because of 5 years) and divide it by the Indian inflation rate raised to the 5th power. Then multiply the result by the spot exchange rate. ((1.03)⁵ / (1.05)⁵) × 0.02046 = $0.01858. (Study Session 4, LOS 19.h)

The exact methodology of the covered interest rate parity (IRP) is:

expected spot rate in one period (FC/DC) = spot rate today (FC/DC) × [(1 + R_FC) / (1 + R_DC)].

Setting up this equation gives us E(S₁) = 189.76 Yen/USD × (1.08 / 1.05) = 195.18 Yen/USD. (Study Session 4, LOS 18.h)