I couldn't find this by googling - if A is an n by n matrix, can you get A^k strictly faster than you can get a product of k arbitrary n by n matrices?
Just spending 30 seconds looking at it, think of it like the bit shifting solution to multiplication, do a related faster operation then correct.
You can reduce A^k to be a lot faster if you calculate A^2 then A^4 then A^8 .. A^N where N is 2^(Log(k)-1) (log base 2) then multiply A^N by A^(k-N) (which you can speedup using the same method if it is larger then 4)
That was a bit rough and quick, but the basic idea, is that doing (nxn)^k would likely be reduced to O(log(k)n^2.373) versus the more naive O(kn^2.373) which is a speedup of O(k/log(k)) (or it's inverse, I am not sure how it is best to represent the ratio) which is decent, I am sure there is a better solution out there.
Multiplying k arbitrary matrices can be done in less time than O(kn^omega). But certainly for k > 3 you wouldn't multiply naively k times as you've pointed out.
Ah yes, one can compute P(x) a polynomial such that P(A) = 0. If you wish to compute A^n for n much bigger than the degree of P you can compute S(x) = x^n mod P(x) and then just compute S(A).
However, algorithms that do actually improve the complexity of powering when this technique isn't applicable do exist. Here is a paper on a recent one:
I think this tendency to mythologize certain financial derivatives is weird. The concept of a macro is way more complex than the concept of a credit default swap. To demonstrate, here is (in my opinion) an explanation of credit default swaps that a person with no financial background should be able to understand.
A bond is a contract created and sold by an "issuer". The issuer can be a government or a company. Ownership of the bond entitles you to payments from the issuer. The specific number and size of payments varies from bond to bond. Once the issuer sells the bond to someone, that person can sell the bond to anyone else they want.
A credit default swap is a contract between two parties (neither of which is necessarily the aforementioned issuer) that are usually called the protection buyer and the protection seller. This contract is made in reference to someone called the reference entity. The protection buyer agrees to make a series of payments to the protection seller in exchange for the protection seller's promise that, in the case of a "credit event", they will give the protection buyer either some specified amount of money or some specified amount of bonds. The nature of the payments and the meaning of "credit event" vary from contract to contract, but generally a "credit event" is understood to have occurred if the reference entity (which is always an issuer) fails to make payments on some of the bonds it has sold. Either party in the contract is free to find someone else to take up their side of the trade at whatever price they can negotiate.
The only thing you needed to know to understand that was what a contract was, but if you want to understand macros, you really need to know what an interpreter is.
It's also complicated by the legal framework. My (limited) understanding of the problems with CDOs in 2008 was that multiple buyers could purchase insurance on the same bond. It's like if I could buy fire insurance on your house. If a lot of us did that and your house burned down, the insurer would have been on the hook for many times the cost of the underlying asset. This is why we had to bail out AIG.
When it's spelled out like that, the idea seems preposterous. But the legal framework allowed it -- in certain contexts. AIG et al call them "CDOs" instead of just "insurance" so they could avoid the more-restrictive legal framework governing the insurance industry.
And given that a typical CDO is the size of a Manhattan phone book (remember those?), there obviously is a lot of nuance that a lot of people did not understand.
You're thinking of CDS, not CDO. CDS stands for credit default swap(s) and CDO stands for collateralized debt obligation. You're right that a CDO would be really big if you printed out a formal specification, but that's because a CDO is special trust where the trustee buys and sells different securitized products and tranches out the payments to shareholders in the trust.
Also - AIG could have made the same mistake with conventional home insurance. Say they only keep 100 dollars of cash around and they decide to insure a million houses, each with a value of one dollar. If one ten thousandth of the houses burn down, AIG goes bankrupt. So it's not true that selling a dollar notional of home insurance is less risky than selling a dollar notional of CDS, because it could easily be the case that the expected payout on the CDS is higher. CDS are just harder to price. There are very robust statistics about houses burning down - the statistics on whether homeowners would default were a lot trickier to deal with.
I think you're right about the basics of what a CDS is, but this doesn't really cover it. You also need to understand how the price of a CDS moves and why. Which can lead to much more complicated systems.
You're right, of course, but my point (and maybe I didn't make it very effectively) was that to acquire a similarly thorough understanding of macros would require way more information - I wasn't trying to make the point that a complete description of CDS could be given in a couple of paragraphs.
I think that in this case the reporter doesn't understand what's going on behind the abstraction.
One thing that might happen is that some country's equivalent of our treasury dept will realize that they're not going to be able to make the next set of payments on their bonds.
Because USD aren't legal tender in the Eurozone, so you couldn't actually buy anything with them. You could change your euros into pounds and move to Britain, except your job, family, friends, and so forth are still back home (and once everyone gets that idea, it's not going to work for a million reasons).
Now if you're talking about people with investments in EUR who could switch that over to USD, well yeah, the exchange rate doesn't seem to have crashed. Hm, can you short a currency somehow?
Now if you're talking about people with investments in EUR who could switch that over to USD, well yeah, the exchange rate doesn't seem to have crashed.
With all the doom and gloom about the Euro, why hasn't it crashed against other currencies?
What effect would a breakup of the Euro zone actually have on the Euro?
Of course you will need Euros for day to day expenses. The idea would be to move your savings onto USD or GBP before the exchange rates spike and convert them back into $local_currency after things in europe settle down.
...can you short a currency somehow?
If you have the right brokerage account you can trade currency futures and options but be careful. Such trading is generally very leveraged and the possibility of losing all of your principle is very real(1) so be sure you understand the risks.
It's a lot easier to keep your money in a foreign currency these days. You could, say, put all your euros in a dollar denominated US bank account and then just use an ATM to convert only what you're about to spend back into euros.
Just a note: Short and leveraged ETFs are designed to match intraday movement and are poorly suited for long-term holding. It says as much on the website.
To wit: YTD EUR/USD is down .2% and that ETF is down 5%.
Yes, exactly what I'm getting at. That's the point of my parent...people should do it now BEFORE mass-inflation so that they can convert it to a more stable currency (USD/GBP/CAD).
I agree that once everyone starts doing it there isn't a point, but now...geezz..I don't see what you have to lose.
If you believe that currency market are somewhat efficient, the chances of a catastrophe happening are already priced in. For example, the Eur vs. Swiss Francs historically hovered around the 1,50 mark for a long time; now it's at 1,23. The swiss franc is considered clearly overvalued and there is speculation that their national bank wants to push it back up to 1,30.
Anyway if the Euro problems get solved, you would probably lose at least 10-15% of your assets. Of course maybe such an "insurance" is worth it, but it is by no means an easy decision.
Not that I have a solution to this problem, but who does the filtering once you remove the job offer as a prerequisite? I can't conceive of a situation where the people that get put in charge of that process are actually qualified.
In every other country there is a central government who have a department of immigration who accept applications from outsiders and judge them on their experience, qualifications, education etc.
You should try having a functional and effective national government - just a thought?
In every other country there is a central government who have a department of immigration who accept applications from outsiders and judge them on their experience, qualifications, education etc.
How many of those countries have a net INFLOW of immigrants from the United States? How many have a net outflow of immigrants to the United States, under the current United States system?
Edit: To clarify (since someone downvoted me), you are making a true statement, but I don't see how it is in conflict with anything I'm saying, or what other point it supports.
What Taleb is saying is still true. If I have a 400k market salary and I give up 200k of it for the right to 1% of the profits I generate, then I still maximize the expected value of my compensation that year by maximizing the size of my bets. I could bet a billion dollars on a coin flip, get 9.8 million (after recouping foregone salary) on heads and lose 200k on tails. Moreover, there is a well established history of traders that lost large amounts of money finding gainful employment regardless. See Boaz Weinstein, for example.
That would work if we wanted to stop banks from transferring losses to shareholders - the real problem is that they transfer losses to taxpayers. Pre-IPO Goldman Sachs was probably considered "systemically important" enough for their losses to have been covered in the event of a large trading loss.
Yeah, there needs to be a clear distinction in advance between entities which can be bailed out (and thus must be highly regulated) and entities which can experiment/innovate freely (but which can't socialize their losses).
LTCM would have clearly been "not bailed out", except it was. Admittedly not directly by the USG/FR, but at the direction of the Federal Reserve.
And even then it's tricky, because a non-regulated entity can make a bet with a regulated one and end up getting bail-out money by virtue of that bet. That seems undesirable, but you also can't really stiff the non-regulated entity, because then nobody will ever want to bet with the regulated one again, thereby destroying its ability to hedge.
I wonder what the costs to the economy would be if we had Canadian levels of regulation of the financial industry, or even greater (something like how utilities themselves are regulated).
Canada believes in allowing only a few, highly regulated banks. Since there are only a handful of banks, it makes it very easy to regulate them.
But this model does hurt the little guys. Since Canadian banks form an oligopoly, they charge you for lending them money; account maintenance fees are standard. This hurts the smallest savers the most. Similarly, there is minimal competition in setting interest rates. Finally, financial innovation, done right, does help people---think Vanguard, index funds, ETFs, etc. All much more limited and much more expensive in Canada (I think).
In contrast to Canada, I think we would be better off with many small institutions, lightly regulated. Then they can compete for your business and do not need taxpayer support when they fail. When they become too big, break them up.
The banking system in Canada is not significantly different than the United States, except that it is just a banking industry. The U.S. has a banking/gambling industry. Actually it's more of a gambling industry with a small banking sideline.
IMHO they would also have been considered "politically important" enough to have their losses covered, given their long history of private => public => private employment.
