Currency forward arbitrage

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jaychou

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Q:The spot rate for British pounds is $1.76. The U.S. risk-free rate is 5.1 percent, and the U.K. risk-free rate is 6.2 percent; both are compounded annually. One- year forward contracts are currently quoted at a rate of $1.75.
im good with the logic of getting into position to arbitrage: buy GBP and enter forward contract to convert my GBP at higher price. Now according to curriculum i should take 1.76/1.062 and buy GBP 0.9416?
What? Spot rate divide by GBP risk free? Now what ?
 
What if its undervalue instead? Spot rate/ US risk free?
 
If the forward rate were correct it’d be $1.76/£ × (1.051/1.062) = $1.7418/£. At $1.75/£, pounds are too expensive in the forward market. Thus, you want to sell pounds in the forward market, and buy them in the spot market.
So:
Borrow $100 @ 5.1% for one year. (You can borrow more if you’re daring.)
Convert that $100 to $100 ÷ $1.76/£ = £56.82.
Invest £56.82 @ 6.2% for one year; it grows to £60.34.
Convert £60.34 to £60.34 × $1.75/£ = $105.60.
Pay off your loan of $105.10, and go wild with $0.50!
 
S2000magician wrote:
If the forward rate were correct it’d be $1.76/£ × (1.051/1.062) = $1.7418/£. At $1.75/£, pounds are too expensive in the forward market. Thus, you want to sell pounds in the forward market, and buy them in the spot market.
So:
Borrow $100 @ 5.1% for one year. (You can borrow more if you’re daring.)
Convert that $100 to $100 ÷ $1.76/£ = £56.82.
Invest £56.82 @ 6.2% for one year; it grows to £60.34.
Convert £60.34 to £60.34 × $1.75/£ = $105.60.
Pay off your loan of $105.10, and go wild with $0.50!
Click to expand...
can you explain to me why in the curriculum they showed 1.76/1.062 as i showed above?
 
jaychou wrote:
S2000magician wrote:If the forward rate were correct it’d be $1.76/£ × (1.051/1.062) = $1.7418/£. At $1.75/£, pounds are too expensive in the forward market. Thus, you want to sell pounds in the forward market, and buy them in the spot market.
So:
Borrow $100 @ 5.1% for one year. (You can borrow more if you’re daring.)
Convert that $100 to $100 ÷ $1.76/£ = £56.82.
Invest £56.82 @ 6.2% for one year; it grows to £60.34.
Convert £60.34 to £60.34 × $1.75/£ = $105.60.
Pay off your loan of $105.10, and go wild with $0.50!
Click to expand...
can you explain to me why in the curriculum they showed 1.76/1.062 as i showed above?
Click to expand...
Without reading the curriculum I cannot be sure, but it appears that they didn’t mean 1.76 ÷ 1.062 (which makes no sense), but the combination of 1.76 & 1.062; i.e., buy pounds & $1.76/£ and invest them at 6.2%; you’re then buying pound at 0.9416 (= 1/1.062) of their future value.
It’s a confusing way of writing it.
 
no they really meant it:
this is what they did: take $1.76/1.062 = $1.6573, use this amount to buy 1/1.062 = £0.9416. Hold this amount for 1 year and get £1.
 
ok i think i get what they meant. What they are trying to say is, i want to hold this amount in £1/1.062 so that after 1 year, i will have £1. now what is the equivalent in $ @ current spot rate ? it is £0.9416 * $1.76 = $1.6573. I know i have to borrow this amount in US dollar
 
I just wanna know the genius’ name who wrote that section in the curriculum … he must be walking with his back to front every day !
 
This is from my own notes:

How to arbitrage:
  1. Determine which interest rate to borrow. Borrow the lower interest rate (A)
  2. Other currency (B) will be reinvested at a higher interest rate
  3. Determine how much you’ll have to repay A back at the end of the period (X)
  4. Contract will involve exchanging (X) for (B) at expected forward rate
  5. Profit.
 
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