A truly riskless system?

[laserwolf]

Quote from sambian:
...But why don't we calculate the balance in euros and see what happens? The account started from 1000 euros and finished with 421.875*8 = 3375. So the system actually made profit ...[/B]

1. Start at eur/usd =1 and we have $1000 (or 1000 euro), we buy 500 euros and 500 dollars.
2. eur/usd = 0.5, we have 1500 euro. profit of 500 euro!
or if
2. eur/usd = 2.0, we have $1500. profit of $500!

a truly riskless system!

 

[sambian]

Quote from laserwolf:

1. Start at eur/usd =1 and we have $1000 (or 1000 euro), we buy 500 euros and 500 dollars.
2. eur/usd = 0.5, we have 1500 euro. profit of 500 euro!
or if
2. eur/usd = 2.0, we have $1500. profit of $500!

a truly riskless system!

You are starting to get it

 

[Martinghoul]

Quote from sambian:
IMHO, your newly created formulas for calculating expected values should be in the schoolbooks.
You calculate expected values for trading dollars by multiplying logarithms of prices, instead of the prices themselves. Can you answer why? When you buy eur/usd and usd/eur, what do you expect to get, dollars/euros or logarithms?

Who said my formulas are newly created? I just used the simple formula for logarithmic returns, which is, in fact, in all the schoolbooks.

If you believe in your calculations, I can offer you a deal. We will throw a coin and every time it comes up heads, you pay me $2, while evey time it comes up tails I pay you $0,5. You can multiply the probabilities of heads and tails (0,5) by the logarithms of the dollars which we bet and you will get expected value of zero.
If using logarithms of the outcomes is the right way to calculate expected values, prove it by accepting my offer

But now you're contradicting your own assumptions by offering me a game that violates your own random walk definitions. Remember that the B&S paper says "...distribution of stock prices at the end of any finite interval is log-normal". If you want to play a game, let's design it in a consistent manner, with two outcomes, where I either double my money or halve it.

Essentially, you're refuting my point by engaging in the same exact inconsistency that is your problem in the first place.

 

[sambian]

Quote from Martinghoul:

Who said my formulas are newly created? I just used the simple formula for logarithmic returns, which is, in fact, in all the schoolbooks.

You use logarithms to calculate expected value. I have never seen somebody using logarithms for calculating expected value of trades. And when I trade I don't expect to get logarithms, but dollars or euros.

Quote from Martinghoul:

But now you're contradicting your own assumptions by offering me a game that violates your own random walk definitions. Remember that the B&S paper says "...distribution of stock prices at the end of any finite interval is log-normal". If you want to play a game, let's design it in a consistent manner, with two outcomes, where I either double my money or halve it.

I was not contradicting my assumptions, I was just offering you a game where the logarithms of our payoffs are equal. For some strange reason you decline it, while you still claim that expected values should be calculated using logarithms, not prices.
But I'm ready to accept your proposal. Let's design a game in a consistent manner, with two outcomes, where I either double my money or halve it. Let's say that in the first flip I give you $10 and if a coin comes up heads, you give me $20, while if it comes up tails you give me $5. What is the expected value of my bet in dollars? It's 0.5*5+0.5*20 = 12.5. This is the expected value as everybody knows it.
But you came up with the brilliant idea to calculate expected value of logarithms, not of dollars. Let's see where your logic can lead us:
I bet $10, ln(10)=2.302585093
I have 0.5 probability to get ln(5)=1.609437912
and 0.5 probability to get ln(20)=2.995732274
0.5*ln(5)+0.5*ln(20)=2.302585093
So far so good. In this game the expected values of the natural logarithms of the outcomes will equal the natural logarithms of my bets. According to you, this means that I can not beat you at this game, except through luck. In the long run my expected value is 0, following your logic. But I still want to try this game. Actually I want to play it for as many coin flips as possible. I'm telling you in advance that I will bet half of my bankroll on each coin flip. Are you prepared to play this game with me?

 

[MasterAtWork]

Quote from sambian:

You focus on only one possible outcome - eur/usd only falls and stays at some level. You draw the conclusion that the system would lose. But you calculate the losses in dollars. And you have errors in the calculations. This is how they should be:
1. Start at eur/usd =1, we buy 500 euros and 500 dollars.
2. eur/usd = 0.5, we have $750. We buy euros with half of our dollars, and keep the other half, i.e. we buy 750 euros and keep 375 dollars.
3. eur/usd = 0.25, we have $562.5. We buy euros with half of our dollars, and keep the other half, i.e. we buy 1125 euros and keep 281,25 dollars.
4. eur/usd = 0.125, we have $421.875. You came up with $140.625 because of erroneous calculations.
But why don't we calculate the balance in euros and see what happens? The account started from 1000 euros and finished with 421.875*8 = 3375. So the system actually made profit
Another thing - let's consider what other possibilities did you have at the starting point, when the price was 1. You could have chosen to buy only euros or only dollars. If the markets are efficient and the price follows a random walk, you have no reason to prefer holding dollars instead of holding euros. So let's say you decided to hold euros. Then eur/usd falls to 0,125. You have only 125 dollars, which is significantly worse than if you followed my system. The loss of my system is lower than if you bought and held, this is why I use the term "riskless".

Hi Sambian,


So you started with $500 and €500

1-If €/$ = 1 then your portfolio is worth $1000 or €1000

2-If €/$ = 0.5 your portfolio is always $500 + €500
But is worth $500+$250=$750 or €1000+€500=€1500

Now “We buy euros with half of our dollars, and keep the other half,”

That means : portfolio= $250+€1000 ($250 is now €500) (its worth keeps being €1500 or $750)

3- if €/$ = 0.25 (4€ equal $1)

Portfolio is $250+€1000
It’s now worth $250+ $(1000/4)=$250+$250=$500
Or €(250*4)+€1000=€1000+€1000=€2000
…
May I miss something obvious and I would be sorry for that but what’s new ?

Masteratwork

 

[Martinghoul]

sambian, we shall play the game, let's first make sure we have an explicit understanding...

Firstly, can you make sure you understand the notion of "logarithmic return"? Just like arithmetic return, t's a quantity that's expressed in percentage terms...

Secondly, can you confirm that, in order for your strategy to work, you require eur/usd to be a random walk as defined in the B&S paper?

 

[sambian]

Quote from MasterAtWork:

2-If €/$ = 0.5 your portfolio is always $500 + €500
But is worth $500+$250=$750 or €1000+€500=€1500

Now “We buy euros with half of our dollars, and keep the other half,”

That means : portfolio= $250+€1000 ($250 is now €500) (its worth keeps being €1500 or $750)

Our portfolio is worth $750, we buy euros with $375, not with $250. This means that we buy €750 and our portfolio consists of $375 and €750.

Quote from MasterAtWork:

3- if €/$ = 0.25 (4€ equal $1)

Portfolio is $250+€1000
It’s now worth $250+ $(1000/4)=$250+$250=$500
Or €(250*4)+€1000=€1000+€1000=€2000

Our portfolio is $375+€750.
It is now worth $375+750/4=$375+$187.5=$562.5
Or 375*4+€750=€1500+€750=€2250

 

[sambian]

Quote from Martinghoul:

Firstly, can you make sure you understand the notion of "logarithmic return"? Just like arithmetic return, t's a quantity that's expressed in percentage terms...

I understand this notion. The returns in our game in percentage terms are +100% of my bet when I win and -50% of my bet when I lose.

Quote from Martinghoul:

Secondly, can you confirm that, in order for your strategy to work, you require eur/usd to be a random walk as defined in the B&S paper?

Yes.

 

[MasterAtWork]

Quote from sambian:

Our portfolio is worth $750, we buy euros with $375, not with $250. This means that we buy €750 and our portfolio consists of $375 and €750.


Our portfolio is $375+€750.
It is now worth $375+750/4=$375+$187.5=$562.5
Or 375*4+€750=€1500+€750=€2250

I agree your datas If you want, but I still don't get your point.
You got a portfolio that was worth $1000 ans is now $562.5. Where on earth did you make some profits ? I'm really sorry but don't get it.

I hope that it's more subtle than "I make profit in euros, but a loss in $".
So please, what'is your point ?

Masteratwork

 

[sambian]

Quote from MasterAtWork:

I agree your datas If you want, but I still don't get your point.
You got a portfolio that was worth $1000 ans is now $562.5. Where on earth did you make some profits ? I'm really sorry but don't get it.

I hope that it's more subtle than "I make profit in euros, but a loss in $".
So please, what'is your point ?

Masteratwork

My portfolio is now $562.5, but if I just bought and held euros, it would be $250. And the idea of the system is that it's profitable in both currencies in the long run, when there are up and down movements. You take an example of only two movements, both of them are down movements, and then you ask "Where on earth did you make some profits". I have written very clearly in the title and many times in the article that the system is profitable in the long run, if the price follows a random walk. It is also useful if markets are efficient and you can not predict future prices. If you can predict when eur/usd will be 0.25, of course the best is to bet all your money on it.