cross currency arb

how would you do the math on this?

bid ASK EURO:$ 0.8 .8012

$:YEN 12.80 12.818

EURO:YEN 10.10 10.102


A. ,13,661 YEN

B 16,810 YEN


Isn’t there something missing in this question?

thanks. he has access to 1,000,000 yen

answer is A

can somebody provide the working out plz

start with 1 million yen.

Convert to euro by dividing by ask: 1,000,000 / 10.102 = 98,990

Convert to USD by multiplying by bid: 98,990 x 0.8 = 79,192

Convert back into yen by multiplying by bid: 79,192 x 12.8 = 1,013,661

i think? anyone else care to comment?

Sorry for the terse email. I had to write so much, so couldn’t really care about the format etc. Please point out if my approach is not correct or I have made any errors in the calculation. Here is what I do:

Here are market rates given to you. EUR:USD 0.8-0.8012 USD:YEN 12.80-12.818 EUR:YEN 10.10-10.102 Now the very first thing you have to figure out, where you are starting from. In this case you are starting from 1,000,000 YEN. Now you can go to either path either you can buy dollar for YEN or EUR for YEN. To determine which path to follow, you have to find out which currency is overvalued/undervalued between USD and EUR, since you are starting from YEN. To do this let’s note down the market rate for EUR:USD = 0.8-0.8012 Now calculate the rate for EUR:USD from the other two rates given, which are: USD:YEN YEN/USD 12.80-12.818 EUR:YEN YEN/EUR 10.10-10.102 since you know you are going to calculate EUR:USD, which is nothing but USD/EUR, so you can restate this as USD/YEN*YEN/EUR. Out of these two, what is given to you, you know you already have EUR:YEN(YEN/EUR), but you don’t have (YEN:USD)USD/YEN. To get this simply inverse the USD:YEN rate. Remember now bid will become ask and ask will become bid because logically bid can’t be greater than ask. so the new rate after taking inverse will be YEN:USD = 1/12.818-1/12.80 = 0.07802-0.07813 So here are the rates you have so far: EUR:YEN 10.10-10.102 YEN:USD 0.07802-0.07813 EUR:USD is simply going to be cross multiplication of these bid and ask rate. Why we do the crossing is because we want to find out the lowest bid and highest ask. so EUR:USD = 10.102*0.07802-10.10*0.07813 = 0.78816-0.78911. You would have again noticed that I have flipped the bid and ask again. This is done to make sure my ask is higher than the bid. Here is what we have so far: Market Rate: EUR:USD 0.8-0.8012 Caclulated Rate: EUR:USD 0.78816-0.78911 Since It is evident from these two exchange rate that EUR is Overvalued in the market, dollar must be undervalued. **This is the key step** Now to determine which path to choose from here, first find out the possible paths. You could either go to YEN–>(USD–>EUR)–YEN or YEN–>(EUR–>USD)–>YEN. Since we know that we have a comarison between USD(undervalued) and EUR(Overvalued), we have to choose the path in which we sell the overvalued security and buy undervalued security. This can be determined only by looking at the paths to see what gets exchanged for what. In the first path, we are exchanging EUR for USD, means you must have to sell USD in order to buy EUR, which is going against the wind. You are trying to buy overvalued security and sell undervalued security, this path will result in the loss. Now in the second one you are buying USD(undervalued) and selling EUR(Overvalued). So in the seocnd path you should realize the profit. Keep the market rates and path Handy with you: Market Rate: EUR:USD USD/EUR 0.8-0.8012 USD:YEN YEN/USD 12.80-12.818 EUR:YEN YEN/EUR 10.10-10.102 Path: YEN–>(EUR–>USD)–>YEN With 1,000,000 YEN you want to buy EUR. Look for the exchange rate of EUR:YEN(YEN/EUR). To determine whether to use the bid or ask, ask yourself looking at the exchange rate(YEN/EUR), whether you are going up or down. In this case you are going from YEN–>EUR, means you are going down in the exchange rate YEN/EUR. Apply “Up-the-bid and down-the-ask rule”. In this case since you are going down you are going to use ask rate. YEN 1,000,000/10.102(YEN/EUR) = EUR 98,990.29895 Now we are going EUR–>USD in the cycle. Now for EUR–>USD, in the exchange rate of USD/EUR, you are going up, so up-the-bid means use bid rate. EUR 98,990.29895*0.80(USD/EUR) = USD 79,192.23916 Now for USD–>YEN in the exchange rate YEN/USD, you are going up so use bid. USD 79,192.23916*12.80(YEN/USD)=YEN 1,013,660.661 Remember you had started with 1,000,000 YEN so the profit made is YEN 1,013,660.661 - 1,000,000 YEN = 13,660.66125

lol not email, post*…ughh

*not security, currency

The shortcut to this approach is, try both paths and see which one yields profit for you. But if one path leads to the loss, doesn’t mean the seond one will be profit automatically. You have to make sure you are calculating both.

@sgupta , amazing explanantion . I had trouble with this concept , now its sorted . for the purpose of exam I will not check the overvalued or undervalued rates to determine which path to choose , rather i would pick any path and see whether any arb is present .

thanks a ton

Awesome write up sgupta. I haven’t spent much time on this stuff, I have a few questions. So the only reason to calculate the cross rate is to decide which path to take, not to actually use the rates in your calculation for Arb? When calculating the bid ask for Euro:$ should you multiply the bids together to get the new bid and the asks to get the new ask?


$:Yen = .07802 - .07813

Yen:Euro = 10.10 - 10.102

Thus: $:Euro = (.07802*10.10) . 788002 - (.07813*10.102) . 7892692

Your conclusion is still the same.


@Factor Hedge I had trouble with this concept too. I think this is one of the hardest one. I am yet to figure out swap calculations.

@CFAWorldwide: You can multiply or divide depending upon what rates are given to you. Division is in fact easier as you don’t have to flip the bid and ask so many times, in fact not even once. But for people who have problems with exchange rate conventions, I wanted to make it easier. Yes, the cross rate is only for the path selection. I think randomly also you should be fine but you are increasing the probability of getting the questions wrong as the concept is bit harder and it takes time to realize where goes what.