Attributing change to the correct letter

[terjeja]

I have an equation like this:

A / B = (A / C) / (B / C)

A / B = 1 / 1.10
A / C = 1 / 10
B / C = 1 / 9

However, these values changes to:

A / B = 1 / 1
A / C = 1 / 9
B / C = 1 / 9

It is obviously easy to see that the change has happened to A while B and C stays the same.

But how can I prove this with math, and at the same time calculate the value of each letter A, B and C before and after, and how large the change has been in each?

And how would this work if I took it one step further, as if all values had changed at the same time, and I had to find the change attributed to each letter after the change. Could it still be solved?

 

[Steven G]

Only 1 divided by 1=1, ie 1/1 = 1. This guarantees us that 1/1.10 is NOT 1. That is A/B is NOT 1. So to go on to say that A/B = 1/1 = 1 is NOT correct.

Can we see the whole problem so we might know why A/B had its value changed? Bases solely on your post, it is not correct that A/B = 1.

 

[terjeja]

Ok. It is actually my attempt at rewriting the real problem into a more easily explainable form. I guess I made an error.

The real problem is actually with currencies:

Friday (approximately):
EUR / USD = 1.1716
EUR / NOK = 10.93
USD / NOK = 9.356

Today:
EUR / USD = 1.17237
EUR / NOK = 10.89947
USD / NOK = 9.29639

The rates fluctuate constantly, so they might be a tiny bit off from when I wrote the first until I wrote the last currency-pair.

However, my question is regards how I can identify the change in value from one day until the next. I am not looking for arbitrage, but just to see which of the currencies that have moved, and by how much. Is it the NOK that has strengthened? Is it the USD? Or maybe both the USD and the EUR while the NOK has become weaker? Can this be calculated with the above information?

 

[Cubist]

You should have posted this under the finance section to get the best chance of an answer. Anyway... (moved to Finance forum now)

I am not looking for arbitrage, but just to see which of the currencies that have moved, and by how much. Is it the NOK that has strengthened? Is it the USD? Or maybe both the USD and the EUR while the NOK has become weaker? Can this be calculated with the above information?

Yes - I think this can be calculated - kind of (read below). This is my first look into FOREX, so I'm not an expert. You must seek other sources (and feel free to challenge my thinking) but I'll share my thoughts...

On any ONE particular day (say Friday):- then any two of the three exchange rates should imply the other one. If this isn't the case then arbitrage should be possible. For the purposes of your question, and since you're not interested in arbitrage, let's assume a very fluid market (with no arbitrage), so we would only have two figures per day that describe the relative strength of the 3 currencies. We may as well use the two exchange rates with the common denominator...

Friday:
Let f1=EUR / NOK
f2=USD / NOK

Today:
t1=EUR / NOK
t2=USD / NOK

(It may be possible to get better figures for the above rates by using the third figure of each day, that we're currently ignoring, to guess what the rates would be after arbitrage has taken place. I won't consider this idea for now!)

From these figures there is no way of knowing if there was a common strengthening or weakening (by the same factor) across all three currencies. We can only tell their relative growth compared to each other.

Let "strength" of NOK on Friday be 1 therefore strength of EUR on Friday is f1 and strength of USD on Friday is f2

Let "strength" of NOK today be "n" therefore strength of EUR today is t1*n and strength of USD today is t2*n

Growth of NOK is n
Growth of EUR is t1*n/f1
Growth of USD is t2*n/f2

Let's assume the common growth factor is 1 (the amount they commonly move by) therefore
(Growth of NOK) * (Growth of EUR) * (Growth of USD) = 1
n * (t1*n/f1) * (t2*n/f2) = 1
can you rearrange the above equation to "n=", and then find what n would be for your particular figures above? Can you use this to determine the relative growth between the three currencies? Please post back with your work and any questions you have.

 

[terjeja]

Thanks for your help.

To rearrange the equation, I believe I this is the correct way:

n * (t1*n/f1) * (t2*n/f2) = 1

n * (t1/1*n/f1) * (t2/1*n/f2) = 1

n * ((t1*n)/f1) * ((t2*n)/f2) = 1

n/1 * ((t1*n)/f1) * ((t2*n)/f2) = 1

n³*t1*t2 / f1*f2 = 1

n³ = (f1*f2) / (t1*t2)

Here are the exact values:

Date	USD_NOK	EUR_USD	EUR_NOK
2020-08-09	9.03267	1.17732	10.60526
2020-08-10	8.97862	1.17749	10.58188

If I plug these numbers in, I get the following:

(10.60526*10.58188 )/ (9.03267*8.97862) = 1.3837524954018579

Then I take the 1/3 power of this and get:

1.1143444960716027

Does that mean that the NOK increased in value by about 1.1143 % that day?

Does it also mean I can switch which one is n? So for example, I say that I want to find the strengthening of the others:

So by using the same formula, and switching USD_NOK with EUR_USD like this:

pow(((1.17732*1.17749)/(9.03267*8.97862)),1/3) = 0.2576 I have found that EUR strengthened by 0.2576 % that day, and lastly

pow(((1.17732*1.17749)/(10.60526*10.58188)),1/3) = 0.2312 which means that the USD strengthened with 0.2312 ?

In other words, NOK increased in value a lot more than the other two this day? Is this correct?

Thanks in advance

 

[Cubist]

Thanks for your help.

You're welcome

n³ = (f1*f2) / (t1*t2)

I agree with the above. Well done!

If I plug these numbers in, I get the following:
(10.60526*10.58188 )/ (9.03267*8.97862) = 1.3837524954018579
Then I take the 1/3 power of this and get:
1.1143444960716027

I get something different. I'm pretty sure that you plugged the wrong numbers in, here's the definitions of the variables (from post#4) and the numbers that I used...

Past:
Let f1=EUR / NOK=10.60526
f2=USD / NOK=9.03267

Today:
t1=EUR / NOK=10.58188
t2=USD / NOK=8.97862

Using these values in the above formula I get n=1.00274, do you agree?

Then the relative growth between the three currencies will be:-
NOK n = 1.0027400
EUR t1*n/f1 = ?
USD t2*n/f2 = ?

And what conclusions can you draw from the numbers you obtain?

 

[terjeja]

Ahh... Yes. I get the same n=1.00274 as you do. I did plug in the same currency within the same parenthesis instead of as it should be.

So that means I get this:
NOK n = 1.0027400
EUR t1*n/f1 = 1.0005294
USD t2*n/f2 = 0.9967398

That means that the NOK strengthened by 0.274 %, EUR strengthened by 0.05 % and USD weakened by 0.33 % that particular day?

However, if I exchange SEK for NOK in the calculations above, with these numbers for SEK:

Date	USD_SEK	EUR_SEK	EUR_USD
2020-08-09	8.76218	10.28461	1.17732
2020-08-10	8.75321	10.29956	1.17749

I would expect the strength for EUR and USD to stay the same. However, they do not.

I get:
SEK: 99.9857233328263
EUR: 100.1310654083961
USD: 102.45553221122834

(I have added it to a function that create correct results for the above calculations. So my math-skills should be removed as a source of discrepency)

def CurrencyStrength(table):
Cur1 = pow(((table.iloc[0][1]*table.iloc[0][0])/(table.iloc[1][1]*table.iloc[1][0])),1/3)*100
Cur2 = table.iloc[1][1]*((Cur1/100)/table.iloc[0][1])*100
Cur3 = 8.97862*((Cur1/100)/table.iloc[0][0])*100
return Cur1, Cur2, Cur3

CurrencyStrength(table[['USD_NOK', 'EUR_NOK','EUR_USD']][-2:])
-> 100.27400175027617 100.05294105389328 99.67397851313254
vs
CurrencyStrength(table[['USD_SEK', 'EUR_SEK','EUR_USD']][-2:])
-> 99.9857233328263 100.1310654083961 102.45553221122834

Here USD have strengthened more than the EUR, which is the opposite result from above. Is there something wrong with the math from the beginning? Shouldn't we expect the calculations for USD and EUR to stay the same?

 

[Cubist]

Ahh... Yes. I get the same n=1.00274 as you do. I did plug in the same currency within the same parenthesis instead of as it should be.

So that means I get this:
NOK n = 1.0027400
EUR t1*n/f1 = 1.0005294
USD t2*n/f2 = 0.9967398

Cool, that's what I got.

That means that the NOK strengthened by 0.274 %, EUR strengthened by 0.05 % and USD weakened by 0.33 % that particular day?

Almost, but you missed out some key words (which might explain the rest of your post). Better to say, "That means that the NOK strengthened by 0.274 %, EUR strengthened by 0.05 % and USD weakened by 0.33 % relative to each other during that period."

Remember:-

From these figures there is no way of knowing if there was a common strengthening or weakening (by the same factor) across all three currencies. We can only tell their relative growth compared to each other.

So we don't have enough info to say how these three currencies are performing when compared to others. But, on the positive side, you don't need to limit your analysis to three currencies. You could consider 4,5,6,..,or all currencies using the above technique (obviously the equation will need extending, or perhaps you can generalise it to work with "n" currencies). Then you'd get a better idea of how each is performing on a global stage.

But you still won't know whether or not there has been a common strengthening (or weakening) globally (if this even makes sense! Maybe we could purchase more goods from an off-planet alien ). But perhaps a common strengthening would indicate that the whole world is prospering at that time, living standards are improving for everyone, and the only way to tell this would be via some other means/ metrics.

NOTE: I've never looked at forex in any depth before so please double check my logic.

 

[Cubist]

I would expect the strength for EUR and USD to stay the same. However, they do not.

OK, that's a very interesting observation. I agree with your logic. Perhaps there's something wrong about my suggestions for analyzing these rates. I'll have a think. If any other helper knows about the forex market then please join in!

EDIT: I've entered the figures for SEK into a spreadsheet and I get results that are more consistent between EUR and USD:-


@terjeja please double check how your code performs the rate lookup. There also seems to be something funny going on in your calculation because your relative growth figures multiplied together 0.9998572 * 1.0013107 * 1.0245553 = 1.0257517 but it ought to equal 1.000000

 

[terjeja]

Would this really be a forex-problem more than a general math problem? Isn't it really relatively simple algebra if we really look at only the isolated problem?

But what if we try to expand this a little. Here is a table with every currency mentioned three times (actual close of day prices):

Date	EUR_GBP	GBP_AUD	GBP_USD	AUD_USD	EUR_AUD	EUR_USD
2020-08-09	0.89794	1.82871	1.30682	0.71534	1.64414	1.17732
2020-08-10	0.89967	1.82710	1.30804	0.71849	1.64364	1.17749

If we use the same logic;

GBP = n

Past:
f1 = GBP_AUD = 1.82871
f2 = GBP_USD = 1.30682
f3 = AUD_USD = 0.71534
f4 = EUR_AUD = 1.64414
f5 = EUR_USD = 1.17732

Today:
t1 = GBP_AUD = 1.82710
t2 = GBP_USD = 1.30804
t3 = AUD_USD = 0.71849
t4 = EUR_AUD = 1.64364
t5 = EUR_USD = 1.17749

Will the formula for n as in change in GBP be: nˆ6 = (f1*f2)*(f3*f4)*(f5*f6) / (t1*t2)*(t3*t4)*(t5*t6) ?
What for the others? Or will this be all mixed up since GBP is first in a few of the currencypairs and last in others?

(My end goal is really to create a function that can calculate the relative strengt of every currency-pair of the available 71 for each day. But I need to get the math correct first).

 

[Cubist]

Would this really be a forex-problem more than a general math problem? Isn't it really relatively simple algebra if we really look at only the isolated problem?

Often the hardest part of a math problem is distilling a need down to set of equations/ expressions. There's a lot of scope for making mistakes here, and it can involve a lot of research. Yes, the algebra after that stage can sometimes be the simplest part of the question.

I'm keen for you to know that my posts may be incorrect and they certainly come with no guarantee of any kind. I don't know what you intend to use this information for, and if you intend to trade money then I strongly urge that you don't. Trading isn't something you can pick up in 5 mins otherwise everyone would be making money.

But what if we try to expand this a little. If we use the same logic;

Sorry, the formula you quote won't work mainly because the inputs aren't in the required form.

One problem is that many of the currency pairs that you wrote are reversed (see * below). But an even bigger problem is that GBP isn't even present in many of the pairs. Also you only have a total of 4 currencies {GBP, AUD, USD, EUR} therefore you shouldn't have f4,f5,t4 or t5 when using the above method. This all indicates that you didn't really understand the method in post#4.

If that set of rates must be used then there's much more involved in the calculation. Sorry, I'm not willing to go that much farther into this subject myself.

*- I don't think that it's fully accurate to reverse the rate symbols by simply taking the reciprocal (1/r) of the rate because there may be a buy-sell spread involved.