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Wednesday, December 24, 2014

MATH FOR BLOGGERS: highschool math revision for Gary Tanashian


Sigh.... I've left him alone for years but I just have to weigh in here.

NFTRH - math fail. Gary Tanashian needs some revision in highschool-level math.

This is going to take a while, but I enjoy math tutoring and I've got nothing else to do, so...


Quote:

What if I told you

Oh fuck he's still on his Morpheus shtick. Well, I guess referencing pop culture counts as valid economic commentary too.

there is a ‘no risk’ investment that will out perform the S&P 500, even assuming the S&P 500 will continue upward at the pace it has over the second half of 2014? Would you buy that investment?

Then consider owning cash, if you are a US resident denominated in USD. Very simply, SPX in USD units has been declining since the middle of the year, with the weekly trend by AROON having gone negative and what looks like a topping pattern forming.



Now, there are so many moving parts in currency exchange and differing costs all around the asset spectrum and cash, like gold, is a repository of liquidity. So there is some hyperbole in this post.

Actually, Gary, there's quite a lot of hyperbole in this post. Because you just posted a chart of the S&P 500 in US dollars squared. Your units are meaningless and thus your chart is meaningless. And the idea that cash trumps US equities is therefore nonsense. This is what happens when you randomly type things into Stockcharts without having any clue whatsoever what they are.

Let me explain exactly what you've done wrong in your chart, using math.

The US dollar index is

an index (or measure) of the value of the United States dollar relative to a basket of foreign currencies.

It is a weighted geometric mean of the dollar's value relative to other select currencies:

Euro (EUR), 57.6% weight
Japanese yen (JPY) 13.6% weight
Pound sterling (GBP), 11.9% weight
Canadian dollar (CAD), 9.1% weight
Swedish krona (SEK), 4.2% weight
Swiss franc (CHF) 3.6% weight

USDX goes up when the US dollar gains "strength" (value) when compared to other currencies.

In other words, if you combine the foreign currencies together into a hypothetical foreign currency that we'll call "Qs", where 1 "Q" is the weighted geometric mean of that stuff above, then the US dollar index is a measure of how many Qs one dollar buys.

When $USD goes up, it means one US dollar buys so many more Qs.

So the units for $USD are Q/$. One dollar used to buy you 80 Qs, but now it buys you 90 Qs because the value of Qs has gone down relative to the dollar. Following me still?

The units for S&P500 are $/S, where S is a constant unit of a basket of equities making up the index. It presently takes $2085 to buy one S unit of the S&P 500 index. Six years ago you could get the same basket of equities at an intraday low of $666. So the $SPX is dollars per S unit. Make sense? You got that?

The units for $SPX:$USD are therefore ($/S)/(Q/$), which is ($/S)*($/Q), which is dollars squared divided by a basket of equities and by Q units of Qs. In other words, Gary, you just charted a nonsense number. You charted dollars squared, divided by (the product of S units of equities and Q units of Qs).

Here, I'll do a sanity check for you that'll help you see that your chart above must be wrong.

First, here's a weekly chart of the $SPX:


This, by the way, is already a chart of "the S&P 500 [...] in the currency that denominates it". You must not understand what the S&P 500 index is: you don't need to do any fancy Stockcharts functions, it's already in dollars per S.

And it's been going up, right? I mean, look at the chart, right?

It certainly isn't "topping out in the currency that denominates it" if it's been hitting new highs in the currency that denominates it, nearly every month for the past 3 years, Gary, right?

Now let me show you something interesting, which you could have seen for yourself if you understood what it is you were charting instead of just randomly punching nonsense formulas into Stockcharts to produce charts showing meaningless squiggly lines rolling over.

The $USD is mainly derived from the Euro-USD exchange rate, so assuming the lower-weighted currencies aren't doing anything extremely different in magnitude to what the Euro has been doing, then the $EURUSD is an inexact but close analogue of the $USD.

So here's a chart of the $EURUSD:


What does the "1.22" reading today mean, Gary? Can you guess?

It means that one Euro buys 1.22 dollars. Does that sound right? Wanna look it up on Bloomie?

So the units for $EURUSD must be $/€: a $EURUSD reading of 1.22 today means you get 1.22 dollars per Euro. Thus $/€.

OK, a quick glance at the scale on the right shows there hasn't really been a lot of variation in these three years, but you can still see that the Euro has dropped constantly against USD in the past 6 months.

What would that say about the value of the S&P 500 in Euros, Gary?

It would mean that over the past 6 months the S&P 500, denominated in Euros, must have been going up faster than the S&P 500 denominated in dollars, right?

So the chart of the S&P 500 in Euros would look like this:


And thus, over the last six months, the S&P 500 has been appreciating more in Euros than it is in US dollars. Which makes sense because the Euro has been going down, so it takes more and more Euros to buy the same S unit of equities in the S&P 500.

Look at the units of the $SPX:$EURUSD function. It's ($/S)/($/€), or ($/S)*(€/$). Dollars cancel out and thus the 1713 reading means "for 1713 euros you get one S unit of equities in the S&P 500".

So if the $USD is (usually, assuming no extreme moves in non-Euro currencies) a close analogue of $EURUSD, then your $SPX:USD chart should look the same, no?

NO.

The reason it doesn't is because the units for $USD are inverted relative to $EURUSD.

See above: $USD is Q/$. $EURUSD is $/€.
 
Here's why:

Here is the formula to calculating USDX:

USDX = 50.14348112 × EUR/USD^(-0.576) × USD/JPY^(0.136) × GBP/USD^(-0.119) × USD/CAD^(0.091) × USD/SEK^(0.042) × USD/CHF^(0.036)

See that, above? The negative exponent used for EUR/USD is telling you that USDX is calculated using (among other things) the inverse of EUR/USD.
 
So, and this is my whole point, YOU CAN NOT USE THE STOCKCHARTS DIVISION FUNCTION (:) with $USD.

Gary, if what you wanted to do in your post was show the value of the S&P 500 ex-US dollar, in an attempt to expose the ludicrous fiction of the US stock market as asset prices are inflated into the stratosphere by the evil money-printing of Bernanke Yellen, or whatever you're reading from the Soviet KGB disinformation stooges running Zerohedge today, then you have to multiply $SPX by $USD.

If you do that, it will give you a chart that looks nearly exactly the same as the $SPX:$EURUSD chart above as long as the Yen, GBP and CAD haven't done anything particularly different than the Euro recently. (I keep adding that caveat because it's vital to always remember what's under the hood with every index and ETF you ever look at.)

In other words, the S&P 500 has been appreciating stronger in foreign currencies than in the US currency over the past 6 months.

How do you chart it? I don't know if there is a multiplier function on Stockcharts, and Jojo couldn't think of one either when I asked him a couple years ago.

But UDN is an ETF of the inverse of $USD. It has crappy volume, so it generates large error bars, but it's the best I've found if I'm going to use a divisor function to look at the value of things ex-USD (like gold). Dividing by UDN is, with some illiquidity-generated error, the same as multiplying by $USD.

Here's $SPX:UDN:


Oh lookie dat! It looks nearly the same as the $SPX:$EURUSD chart above, and it should since the Euro is such a big part of the $USD index.

And so it affirms that the S&P 500, over the past 6 months, has accelerated in appreciation ex-USD. Which means it's an even better investment for foreigners than it is for Americans.

Which, to me, is definitive proof that the US has entered a secular bull market where it attracts all the world's capital to power a 10- or 20- year thrust in the US economy.

I don't know how else you could spin it, Gary, but I'm sure you'll try. Maybe the above chart is instead proof of an evil plot to artificially inflate US equity values by evilly manipulating forex to cause the US dollar to evilly appreciate, thus evilly causing the mother of all asset bubbles (the US dollar) that can only terminate in (ahem) a "massive repudiation of debt as a dishonest system comes apart at the seams"? Evilly?

But it certainly doesn't say that
the S&P 500 may be topping out in the currency that denominates it and over time holding that currency as opposed to the broad US stock market could become a favorable thing to be doing.
Now, at least in the reality the rest of the world inhabits, assuming you wrote that sentence in English and not some strange constructed language, it looks to me like you just suggested to your subscribers to "over time" leave US equities to go to cash. Which is senseless at the start of a secular bull market. And, as I've shown, your suggestion was based on a nonsense chart.

Then again, if a person doesn't believe in a US secular bull market they'll grasp at anything to disprove it. Even a nonsense chart.


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