Question

the interest rate in the united states are 2% per year while the interest rate in the united kingdom is 1% per year the spot rate and the forward rate between the us dollars ad the british pounds are as follows.

S(USD/GBP)   bid price is 1.3290 and the ask price is 1.3300

F6(USD/GBP) bid price is 1.3195 and the ask price is 1.3200

you can borrow usd 100,000 or gbp 100,000 in the united kingdom and your investment horizon is half a year

are you able to make a gauranteed profit through a covered interest arbitrage? if so explain clearly how you are able to take advantage of it and calculate the amount of profit you will be able to make show your calculations

Answer 1

Pound = p (notation)

Currency quote is as $/p. Here assume, p as base currency which is also assumed as the domestic currency. Then covered interest rate parity condition for 6-months is:

[1+r(p) * (180/360) ] = S($/p) * [ 1 + r($) * (180/360) ]

Here:

interest rate in united states, r($) = 2% p.a

interest rate in the united kingdom, r(p) = 1%

Spot price ($/p) = 1.3290/1.3300

6m forward ($/p) = 1.3195/1.3200

According to covered interest rate parity, no-arbitrage 6month forward rate should be:

F($/p) = 1.329 * [ 1 + 0.02 * (180/360) ] / [ 1 + 0.01 * (180/360) ]

• F($/p) = 1.329 * [ 1 + 0.02 * 0.5 ] / [ 1 + 0.01 * 0.5]
• F($/p) = 1.329 * [ 1 + 0.01]/[1+0.005]
• F($/p) = 1.329 * [ 1.01/1.005]
• F($/p) = 1.329 * 1.00495
• F($/p) = 1.3356
• To have no arbitrage profits, 6month forward rate to buy 1 pound must be: 1 p = $1.3356

However, the quoted 6 month forward rate given is to buy one pound, 1p = $1.3200. So pound is cheaper in the quoted forward market. So long the 6 month forward contract to buy 1 p = $1.3200

There is a risk-less profit opportunity.

Steps to profit from arbitrage is:

• Borrow p100,000 at 1% for 6 months and convert to $ at spot rate. Sell 1 p = $1.329 (Sell base currency at Bid rate) and get 100,000 * $1.329 = $132,900
• Invest $132,900 at 2% for 6 months
• Enter into 6 months Forward contract to buy 1 p = $1.3200 (6m quoted forward ask rate)

At T =0.5 (after 6 months)

• Receive investment proceeds in $ to get = $132,900 * [ 1 + {0.02 * (180/360) } ] = $132900 * [ 1 + { 0.02 * 0.5} ] = $132900 * [ 1.01] = $134,229
• Oblige the forward contract and buy 1 p = $1.32 and receive in pounds the amount $134,229 / $1.32 per pound = p 134,229 / 1.32 = p 101,688.60
• Repay the borrowed funds in pound with interest at 1% for 6 months for total amount 100,000 * [ 1 + 0.01 * (180/360)] = 100,000 * [ 1 + 0.01 * 0.5] = 100,000 * 1.005 = 100500
• Remaining amount in pounds = 101,688.60 - 100,500 = p 1188.636

Risk-less arbitrage profit in pounds = 1,188.64 (rounded to 2 decimals)