# how to calculate covered interest arbritrage

Please some1 can guide me the steps to calculate covered interest arbitrage given the following information.

Spot rate \$0.85 / SF

Three month forward for SF \$0.80 / SF

Three month Interest rate for SF annualized 12%

Three month Interest rate for USD annualized 18%

Which currency would you borrow and how much is the arbitrage profit ?

find the arbitrage free forward rate first (by the way are these rates in LIBOR?)

F0 = 0.85 x (1.18/1.12)90/365 = 0.861

since the quoted forward rate is underpriced (lower than the arbitrage free forward), you would borrow SF, use the proceed to buy USD at spot, invest USD, enter into the forward at 0.8. 3 months later, collect USD investment and convert it at the agreed forward rate at 0.8 and pay off the loan on SF

assuming a SF1,000,000 loan.

1. borrow SF1,000,000 convert to USD -> USD850,000 (at spot rate of 0.85)

2. USD850,000 invested for 3 months -> USD885,408 (at 18% interest rate, ending value rounded up, assuming 30-365 day convention and annual compounding)

3. convert USD885,408 back to SF (at the agreed forward rate of 0.8) -> SF1,106,850

4. loan amount need to be repaid in SF (again, assuming 30-365 day convention and annual compounding) -> SF1,028,338

5. collect a net difference of SF78,551 (1,106,850 - 1,028,338)

note that in step 3, if the agreed forward rate is 0.861 (the arbitrage free forward), there would be no profit (there might still be negative or positive arbitrage profit due to rounding errors)

if quoted forward rate is underpriced, you buy and invest the price currency (USD) at spot

if quoted forward rate is overpriced, you buy and invest the base currency (SF) at spot

if the question asks you to start with a currency, you just have to mix it around.

for example, if in my explanation you have to start with USD. you enter into a forward agreement at 0.8 first, borrow SF at SF rate, convert it into USD and use the proceed to invest in USD. 3 months later, convert the necessary amount of USD back to SF to repay SF loan and keep the leftovers.

same thing really but the ending profits will be in USD instead of at SF.

Can you explain how do we understand when do we buy/sell in forward and sell/buy in spot.

Here the implied forward rate was greater than the market forward rate. so we bought usd at spot and sold in forward.

What if implied forward rate was less, then what would have been the case?

if forward rate is overpriced, you borrow USD, buy and invest SF and sell in forward. let’s say the forward rate is 0.9

1. borrow USD1,000,000 convert to SF -> SF1,176,470

2, investment in SF yields SF1,209,810

1. SF converted into USD at 0.9 -> USD1,088,828

2. USD loan payment -> USD1,041,656. net difference of USD47,172

Thank you so much

If you will end up with too many USD, start by borrowing USD.

If you will end up with too many SF (San Francisco?), start by borrowing SF.

For this part, you would use the forward rate to convert back to SF, but how will the ending profits be in USD? Are you saying you would convert the one-year-ahead borrowed SF amount to USD using the forward rate?

steps:

1. borrow SF1000,000, convert to USD -> USD850,000

2. USD850,000 invested for 3 months -> USD885,408 (at 18% interest rate, value rounded up, assuming 30-365 day convention and annual compounding)

3. using forward rate of 0.8, calculate the amount of USD to repay SF loan (SF1,028,338 is the amount you need to pay -> equivalent USD822,670).

4. you get USD885,408 but only pays USD822,670 at the end, a profit of USD62,738

if you notice, USD62,738 is approximately equal to SF78,551 at a forward rate of 0.8

Awesome, thanks. I was trying this conversion (bolded), but used the expected spot (instead of the forward) to go back and forth between the profit in SF and the profit in USD.

I was told you just borrow the lower yielding rate…is this not true?

Not necessarily.

You have to compare the quoted forward rate with the rate implied by interest rate parity.

If I well understand this example:

Since the market forward rate of \$ 0.8/SF is lower than that implied by interest rate parity, \$0.861/SF, then SF is undervalued vs dollar (we are supposed to buy more dollars in the future than now with one unit of SF currency), then we should borrow SF and invest in USD because it is relatively cheaper to do that taking into account both the forward price and the forward rates of both currencies that will prevail in the near future.

Is that reasoning right?