Friday, September 30, 2011

Warren Buffett didn't make 20% on August 26th...

...he potentially made over 63%!

And his investment is asymmetrically depended on the performance of the Bank of America shares. He can gain when shares go up, but doesn't lose when they go down. Unless the bank fails.

How does it works?

Chart: Bank of America, 3M, 2011-09-30;

Bank of America is down again today. So far -2.6% at $6.18. That's 11.6% below the price on the day when Warren Buffett invested $5 billion in the company ($6.99), and 16.6% below the strike price of warrants granted to him ($7.14).

Nevertheless, one cannot say that Mr. Buffett is down on his investment.

The Buffett's investment consists of two elements:

  • preferred shares paying 6% dividend
  • 10 years warrants for 700 million BofA shares with the exercise price of $7.14

Preferred shares are more like a subordinate debt rather than ordinary shares. Their performance is not connected with market fluctuations. As long as BofA don't go bust, they pay a nice dividend and retain their value.

Warrants are equivalent of call options on BofA shares. Based on the historic 10 years BofA volatility of 55% and using 10 year T-bonds yield of 1.83% we can estimate the value of the warrants using Black-Scholes formula. The result is $4.5 per warrant.

Since Bank of America granted 700 million warrants to Mr. Buffett, the potential value of them is about $3.15 billion. $3.15 billion on top of $5 billion investment makes a 63% return.

There are two critical assumptions in this investment, though:
  1. Bank of America does not go bankrupt
  2. Bank of America does not require a substantial government bail-out that wipes out existing shareholders and decimates share price
Even if you assume a high discount rate (say 50%) for calculating the potential value of the preferred shares purchased by Buffett, the total ROI on investment is above 10%.

Waiting for the break out

Chart: Risk-on / risk-off investor behavior; Bloomberg BRIEF

Thursday's Bloomberg BRIEF describes the recent bipolar investors behavior dubbed "risk-on / risk-off". 

Such behavior seems connected with the investors' uncertainty about the prospects of the economy. While many still believe in the recovery, similar number worry about a possible "double-dip".

And the prolonging crisis in Europe does not help...

As a result the distribution of the daily changes of the S&P500 index for the previous 20 sessions seems to resemble the investors assessment of the economy:

Chart: distribution of daily S&P500 changes, 20 days through 2011-09-29

Pretty unusual distribution (BTW: kurtosis of -1.2587 is characteristic to Wigner semicircle distribution) in connection with high volatility makes the market difficult for trend-oriented investors.

But the market won't stay range-bound forever, will it?

Monday, September 26, 2011

Probability of Lotto

There is currently a noticable accumulation in the Polish Lotto - 50 million PLN or $15.2 mln / 11.3 mln EUR.

Since there are 6 numbers to select from 49, according to the basic probability calculations, the chance that some single ticket will win is 1 to 13,983,816.

Hence, the expected value for a single ticket would be 50 million PLN * 1/13983816 or 3.57 PLN. Seems, someone buying some 13.98 million tickets (each at 3 PLN, which requires an investment of 42 mln PLN) cannot lose, right?

Not exactly.

It may happen there will be more than one winner. Let's suppose, some 10 million tickets will be bought in total by various people. Then, the probabilities of 2, 3, 4 or 5 distinctive winners are as following:

> dbinom(2:5,10e6,1/choose(49,6))

[1] 0.1250688763 0.0298127636 0.0053298679 0.0007622907

In other words, there is a pretty large chance (16.1%), there will be 2 or more wining tickets.

OK, so what is the expected value given the number of tickets bought?

Assuming that between 1 and 50 million tickets will be bought, the expected value decreases from 3.512 to 1.319:

Chart: expected value [PLN] vs number of tickets sold [mln]

For 10 million tickets, the EV is equal to 2.9659, which is below the price of the ticket (3 PLN):

> Nwinners <- 100 # max. simultaneous winners in history - 80

> Nbets <- 10e6

> Vwin <- 50e6

> Wwin <- sum(dbinom(1:Nwinners,Nbets,1/choose(49,6))/sum(dbinom(1:Nwinners,Nbets,1/choose(49,6)))*(Vwin/(1:Nwinners)))

> Wwin
[1] 41475080

> Wwin/Vwin
[1] 0.8295016

> Wwin/choose(49,6)
[1] 2.965934

And there is also a 10% tax on winnings...

UPDATE: On March 30, 1994 there was 80 winning tickets. The probability of such an event is 4.985477*10^-86, even if ALL the Poles would have bought one ticket.

> dbinom(80,38*10^6,1/choose(49,6))
[1] 4.985477e-86

The probability drops to 1.529368*10^-131 with 10 million tickets bought.

Just as a reference, the number of the atoms in the observable universe is estimated at 10^80.

Sunday, September 25, 2011

A bigger bubble?

Do you think gold is a bubble? If yes, what would you say about Apple?

Chart: Apple vs gold, 10 years;

Friday, September 23, 2011


Over the last nearly three years researches at CERN have been observing some interesting discrepancies in time it takes neutrinos to travel between laboratories in Switzerland and Italy.

The difference is very tiny - just 60.7 ns (equivalent to 17,77 meters at c) on a distance of 731,278 meters which translates into a 0.0000248 difference in the speed of light. In other words, the measured speed of neutrinos is 292,799.3 km/s instead of 292,792 km/s, or 7 kilometers per second larger.

Even though the physicists at CERN spent significant time on verification of their surprising measurements, its too early to declare the result definitive. It definitely requires verification at other laboratories.

But if it is confirmed, this tiny rift can transform into huge hole in the contemporary physics.

BTW: In 2008, researches from Fermilab published results of their MINOS experiment, in which the speed of neutrinos was measured at +0.000051, or 299,807.3 - 15 kilometers per second larger than c. However, the possible error was much higher than in the case of OPERA in CERN.

Kweku Adoboli vs Leo Apotheker ;)

Kweku Adoboli is a UBS trader accused of losing $2.3 billion in a series of rogue transactions. Since September 15th, when the loss was announced, share price of UBS felt down by 14.62%, decreasing UBS market capitalization by some $6.8 billion.

Foto: Kweku Adoboli; Mirror

Leo Apotheker was the CEO of HP till yesterday. Since September 30th, 2010 when he was appointed, the capitalization of HP felt by 47.17% or by $40.44 billion.

Foto: Leo Apotheker; CBS News

Thursday, September 22, 2011

Banks squeezed by regulations

Moody's downgraded debt of Bank of America, Citibank and Wells Fargo yesterday, saying the banks may not be too big to fail anymore.

In response Bank of America shares plummeted 7.54% to $6.36, below the $6.99 price before Warren Buffett invested $5 billion in the company in August. The price of bank's shares is also nearly 11% below the exercise level of  warrants granted to Buffett. Fortunately for Mr. Buffett, part of the short-term impact should be reduced by 6% dividend from preferred shares he owns, and the warrants can be exercised anywhere over the 10 year period.

Nevertheless, Moody's action reflects increasing pressure banks both in the U.S. and Europe are facing from new regulations introduced after the crisis of 2008 - mainly Dodd-Frank and Basel III.

The former was conceived as a way to allow safe unwinding of financial institutions without the need of government bail outs. The later increases capital requirements for banks.

According to Basel III, banks should increase their Tier I capital (i.e. common shares + retained earnings) / Risk Weighted Assets ratio to 4% and additionally create a "capital conservation buffer" of 2.5%.

However, there is a significant problem with with the way of calculating Risk Weighted Assets (RWA) under Basel III. The regulations assign no risk to sovereign debt. As a result, European banks that already at least partially implemented Basel III rules, face a difficult situation connected with their holdings of peripheral eurozone bonds.

More conservative way of calculating capital adequacy is to simply divide bank equity by assets. When you use this formula to compare U.S. banks to their European peers, you get quite a scary picture:

Chart: Equity / Assets ratio for selected banks; data as of 2011-06-30 based on Google Finance

Wednesday, September 21, 2011

Problems in Europe overshadow Operation Twist

Chart: S&P500 futures 1 day 2011-09-21;

As anticipated, Fed has announced the "Operation Twist" today. The declared scale of the operation is a little larger than anticipated. And as a result markets have sunk by 3%... (still, -2.94% is better than

Early comments point to the wording of the FOMC statement about the "significant downside risks to the economic outlook, including strains in global financial markets" (previously: "downside risks to the economic outlook have increased").

Is it really so strange that FOMC confirms there is a culminating problem in Europe, or is there something more?

Since reading even a pretty short FOMC statement is not so easy when markets are reacting aggressively in the real time, many people probably haven't noticed yet a number of slightly positive changes in the wording:


  • indicators suggest a deterioration in overall market conditions > indicators point to continuing weakness in overall market conditions
  • household spending has flattened out > household spending has been increasing at only modest pace
  • inflation picked up earlier in the year > inflation appears to have moderated since earlier this year
  • The Committee now expects a somewhat slower pace of recovery over coming quarters > The Committee to expect some pickup in the pace of recovery over coming quarters
It seems, Fed has observed some weak signals of the improvements in the economic conditions. The economy is still very fragile, especially the labor market. There is also a significant risk that - when realized - can have a tremendous impact of financial markets.

Probably what has been missing in  the FOMC statement is some kind of declaration that Fed is monitoring the situation and is willing and able to act in case of negative developments. Unless there is nothing Fed can do to mitigate this particular risk and market reaction is correct?

Meanwhile, USDJPY has managed to bounce back from the previously mentioned 76.00 defense level. Will see how long it will last...

Chart: USDJPY & USDEUR 1 day 2011-09-21;

Comparing two recent forex interventions

Chart: USDCHF vs USDJPY 3 months, 2011-09-21

Back in August, Bank of Japan intervened in USDJPY. As a result, the Japanese yen felt by nearly 3%. For a single day...

A few days later, the Swiss National Bank's intervention followed. EURCHF shot up by nearly 6%. Supported by gossip about further actions, the franc continued to weaken for a number of days,  till the beginning of September, when the fear of another SNB interventions waned.

Then SNB stroke again and pushed EURCHF by another 9% in one day. Just as declared, the Swiss bank has remained determined to defend the 1.2000 level by all means.

Most recently a new wave of rumors hit the market. Some say, SNB may move the target rate to 1.2500 from 1.2000. It has been enough to push market above 1.2200.

So far, SNB's actions demonstrate that central bank is able to successfully intervene in the forex market if only it plays on the right sight of the market (selling UNLIMITED locally currency as opposite to selling limited foreign reserves) and is determined enough to stick to its pledge.

In the meantime, the Japanese play a strange game. They declare intervention, spend quite substantial amount of money on a single action and... back away.

As a result, USDJPY is more than 8% down since a year ago, when BoJ started its recent wave of the interventions.

Most recently, the decreased volatility in USDJPY suggested that BoJ may be intervening in the market in more subtle and effective way in order to defend the 76.00 line.

As USDJPY has just resumed its march down, and currently is less than 0.5% above the imaginary Maginot Line, the day of the test of the BoJ's resolve is coming...

Monday, September 19, 2011

Stabilizing financial markets through forex

Have you looked recently on the performance of the EURCHF and USDJPY crosses?

They are basically flat:

Chart: EURCHF vs USDJPY 10 days, 2011-09-19;

Both the Swiss and Japanese currency are starting to resemble the tightly controlled Chinese yuan (renminbi):

Chart: EURCHF, USDJPY and USDCNY, 3 months, 2011-09-19;

As suggested earlier, forex volatility of some crosses may disappear for some time.

Volatility still remains present in the emerging market crosses and... in gold, which has become a hard currency proxy of some kind over the recent years.

However, the latest increase in the activity of central banks, who provide virtually unlimited funds through currency swaps and asymmetric market operations, may lead to the further reduction in overall market volatility. There is not a central bank able to "print' gold though, so nobody can put a floor under the price of gold (see also "Gold - correction or trend change?").

Central banks can directly operate on the forex and - partially - on the fixed income markets. Equities and commodities (with the exception of precious metals) are officially off-limits. Nevertheless, since all the financial markets are interconnected, actions on some markets affect the others.

Currently I still have no clear picture of what may be the long term consequences of the above mentioned central banks activities. Most probably many financial markets - not only forex - will stabilize or start raising for reduced risk. Many correlations will disappear or reverse. Various investment strategies may stop working. But what then?

Friday, September 16, 2011

Gold - correction or trend change?

While the five central banks decide to cooperate in providing dollar liquidity, gold is falling:

Fig. Gold futures vs USDJPY and USDCHF

Even though historically Q4 was on average positive for gold, September tends to be quite volatile, with significant downside potential.

Gold has been recently positively correlated to safe heaven currencies such as Swiss franc and Japanese yen (both in relation to U.S. dollar), and at the same time negatively correlated to equities.

Since mid-August, the raise of the yen against dollar has been stopped. In September SNB semi-fixated franc against the euro, simultaneously causing USDCHF to raise.

Basically price of gold has been recently driven by three fundamental factors: fear about global economy, possibility of high inflation and low interest rates. While economy is improving extremely slowly, inflation is haunting emerging markets, and interest rates should remain on low level through 2013, all these factors are probably already discounted in the price of gold. Unless a new fear factor emerges, gold may stagnate. And slow correction may easily turn into disorderly liquidation...

So far, the proposed gold-crude pair trade would be in the positive territory.

Thursday, September 15, 2011

Bounce from the exteme oversold conditions

Deutsche Bank is raising more than 10% today:

Source: Reuters

Seems the relief rally from the extreme oversold conditions has materialized.

The question remains how far it can go...

BTW: I wonder whether this is in any way connected with the flop at UBS ;)


EOD results:

The coordinated action of the five central banks definitely helped the European banks.

Since the end of last week the cumulative performance is:

  • BNP Paribas +2,35%
  • Deutsche Bank +5,17%
  • Societe Generale +5,07%

Tuesday, September 13, 2011

What distribution does the stock market follow?

Fig WIG20 and daily changes

Przemek Biecek's "Na przełaj przez Data Mining z pakietem R" (Across Data Mining with the R package)  is a fascinating tutorial for people who would like to quickly implement selected data mining concepts in R.

Today I've browsed though the chapter about analyzing the distributions of daily changes of equity prices and applied it to the WIG20 index of the Warsaw Stock Exchange.

No surprises here :)

> summary(daily_changes)
      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-0.1416000 -0.0100500  0.0002639  0.0006617  0.0113000  0.1479000 

> skewness(daily_changes)
[1] -0.04857761

> kurtosis(daily_changes)
[1] 4.414738

Daily changes do not follow normal distribution:

> mec <- mean(daily_changes)
> sdc <- sd(daily_changes)
> ad.test(daily_changes,pnorm,mec,sdc)

        Anderson-Darling GoF Test

data:  daily_changes  and  pnorm 
AD = 61.0893, p-value = 1.288e-07
alternative hypothesis: NA 

Fig. observed distribution of the daily changes of WIG20 vs fitted normal, Cauchy, Laplace and Stable distributions

Among the tested distributions, the best results can be obtained with stable distribution with the following parameters:

 Stable Distribution

Estimated Parameter(s):
       alpha         beta        gamma        delta 
1.480000e+00 6.700000e-02 1.101051e-02 6.523883e-05 

Fig. Fitting of the stable distribution

> ad.test(daily_changes,pstable,sf_a,sf_b,sf_g,sf_d)

        Anderson-Darling GoF Test

data:  daily_changes  and  pstable 
AD = 1.361, p-value = 0.2134
alternative hypothesis: NA 

Hence, even that we can expect daily changes to remain in the (-5.5%, +5.5) range for more than 95% of the time, sometimes we can get a nasty surprise...

> stabledist::pstable(0.055,sf_a,sf_b,sf_g,sf_d)-stabledist::pstable(-0.055,sf_a,sf_b,sf_g,sf_d) # prob. of (-5.5%, +5.5%)

> stabledist::pstable(-0.1,sf_a,sf_b,sf_g,sf_d)  # probability of -10% or less

It is still worth keeping in mind that short term distribution can be highly distorted (see here and here) and affected by the recent volatility.

You can see the complete source code here.

Monday, September 12, 2011

Short term relief?

BERLIN (Dow Jones) 20:54 CET -- The European Union's current treaty doesn't foresee the possibility of an exit by Greece from the euro zone, German Finance Minister Wolfgang Schaeuble said in an interview with the ZDF public broadcaster Monday.

--German Finance Minister Schaeuble says Greek euro exit not possible due to EU treaty
--Schaeuble says contingency plans for a Greek insolvency are not an actual government plan
--Schaeuble comments come as others in government think loudly about Greek insolvency
--Schaeuble criticized such comments as unsettling markets 

How will the financial markets react to that?

Definitely these declarations do not end "the Greek tragedy".

But given the extreme oversold conditions of the European banks, they can initiate at least a short relief rally.

Fig. BNP, DB and SG 1YR performance

This may also help for some time:

FT 2011-09-12 20:20 UK Italy turns to China for help in debt crisis

Italy’s centre-right government is turning to cash-rich China in the hope that Beijing will help rescue it from financial crisis by making “significant” purchases of Italian bonds and investments in strategic companies.

Volatility clustering and characteristic of distribution

Benoit Mandelbrot observed that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes", what was named volatility clustering.

This behavior is not clearly visible in relation between single day changes, where randomness keeps the picture fuzzy:

Fig. Absolute one day change vs next day absolute change (S&P500)

It becomes more visible when one analyzes relations between cumulative changes over n days (say: 5-10-20-30) and the identical proceeding period:

Fig. Absolute cumulative change over n days against previous n days period (S&P500)

Hence we can assume that the distribution of expected price changes for a given period is not a constant, but depends on the previous volatility.

Volatility clustering is also the basis of the family of ARCH volatility models, used with varying successes in VaR calculation. 

So what can be done to improve the performance of volatility modeling?

Sunday, September 11, 2011

The end of floating exchange rates is coming?

During the weekend's G7 finance ministers' meeting, Japan was probing participants about possible new forex intervention and didn't meet with any significant objections.

Japans was trying to stop the strengthening of its currency without success for some time now.

However, the recent action by the Swiss National Bank has suggested a new, potentially more fruitful approach.

With inflation at 0.2% (identical to the Swiss'), Japan shouldn't be worrying about dangers connected with "printing" empty money.

Should we expect then another declaration of providing UNLIMTED funds by one more central bank, soon?

Even more interesting question is, how the other major central banks - such as Fed and ECB - will react to another ultimate intervention.

Many studies in the past concluded that interventions are ineffective.

But take a look on the performance of Chinese yuan (renminbi) versus US dollar over the last five years (blue line in the center of the chart):

It's true that yuan is a controlled currency, which value is basically set by the Chinese central bank, and its availability is limited to foreigners.

The famous Black Wednesday of September 16th, 1992 demonstrated that single central bank is not able to defend the value of currency, especially when operating under certain exchange regime, for limited size of its foreign reserves.

Nevertheless, there are at least two situations when central bank cannot lose:

  1. when it intends to weaken "its" currency - it can simply print ANY amount of money to flood the market
  2. when two central banks agree to cooperate in order to regulate the exchange rate for their currencies - they can provide each other with UNLIMTED funds using swap agreements

In the previous years, we were witnesses to various central banks' decisive actions:

In the coming days, we can see ultimate intervention in USDJPY by BoJ.

All these moves give rise to fears about the possibility of  "currency wars". Paradoxically, what we can see instead is closer cooperation between central banks and more regulation of the forex market.

Germany and France keep pushing for introduction of the financial transaction tax (Tobin tax), which should cover forex. Most probably it will limit the volume of the currency transactions and possibly reduce volatility.

Being the most liquid of the financial markets, forex is not an easy place for arbitrageurs. Introduction of the new market mechanisms can disturb this balance and create new interesting opportunities.

Extreme oversold conditions nearly reached by European banks

Fig. Extremes for BNP Paribas

Fig. Extremes for Deutsche Bank

Fig. Extremes for Societe Generale

The most stable condition indicating extreme oversold level for the three banks mentioned before is price deviation from long moving average. Based on the historic data, extreme oversold level is reached at -0.3 for BNP Paribas, -0.4 for Deutsche Bank, and -0.5 for Societe Generale.

The second most indicative signal is the price change over the previous year. The threshold levels are -0.5 for BNP, -0.7 for DB and -1.0 for SG (weak signal).

The current levels of the indicators are:

        BNP    DB     SG
 DEV MA -0.383 -0.410 -0.544
 REL SD  0.242  0.241  0.364
 YOY %  -0.618 -0.627 -0.931

It seems that based on the deviation from long MA, all three banks have already reached the extreme oversold level, and their annual price change is close to confirming that.

It doesn't mean, the prices cannot go even lower, especially in a shock situation. The extreme oversold conditions have been calculated using daily data for the last 18-24 years (depending on availability). The present may be totally different from the past, though... (so at least in the case of the French banks it may be worth waiting till Moody's downgrades their debt this week)

High probability trading opportunity in European banking stocks?

On Friday gossip about the potential Greek default has pushed the stock price of Bank of America to the level they had been before Warren Buffett invested $5B in the company.

The same fear is driving down prices of the European banks. From the beginning of the year, Societe Generale dropped 56.29%, Deutsche Bank 40.17% and BNP Paribas 37.25% (based on the ADRs).

While Bank of America is definitely in the red this year (EPS= -1.65), the fall of the stock prices of the still profitable European banks has actually made them pretty attractive from the P/E perspective (4.05 for SG, 4.62 for BNP and 10.10 for DB).

There is however one critical risk factor that weights heavily on the value of the European banks - the stability of the eurozone. Even if the banks direct exposure to the sovereign debt of the unstable countries may be limited, there still is a significant risk of domino and secondary effects caused by the default of even the smallest eurozone member country.

On the other hand it is hard to imagine German or French government allowing their largest banks to fall...

The previous analysis of the extreme oversold levels for S&P500, may suggest that the prices of the European banks may be close to the levels offering high probability long trading opportunity. The model requires re-calibration for higher volatility of individual stocks. And one needs to assume that the banks will survive the possible shock.

Stay tuned :)

Tuesday, September 6, 2011

Sterilizing SNB intervention with ECB help?

Swiss National Bank has announced today it will defend EURCHF at the 1.2000 level at any cost.

It has declared it is prepared to sell UNLIMTED amounts of CHF. This means buying adequate amounts of EUR.

In theory, SNB can print unlimted amount of Swiss francs.

By buying euro, SNB will increase its already substantial foreign reserves. This creates a risk of loss if the value of these reserves decreases. However, by providing the floor for EURCHF exchange rate, SNB can defend the CHF-denominated value of its holdings.

Hence, if EUR strengthens, SNB makes - at least on paper - a profit on its position.

And if EUR weakens, SNB can defend it by buying even more EUR with unlimited funds.

There is a problem with unwinding the position though. The larger the EUR position held by the SNB, the harder it is to sell it without affecting the markets and pushing EURCHF down.

One possibility would be to park accumulated EUR at the ECB, using a swap construction similar to the Dollar Liquity Swap Lines previously provided by Fed.

SNB could swap its EUR with ECB for CHF at the fixed rate of 1.2000, securing the CHF-denominated value of its foreign holdings.

However, banks would need to somehow square the interest differential between CHF and EUR. For example, while CHF LIBOR O/N rate is currently (minus) -0,00333%, EUR LIBOR O/N stands at 0,83438%, meaning that on each one billion of EURCHF, ECB would have to pay roughly 8,4 mln EUR of interest daily.

Unless both banks set the IR for the swap at 0%.

BTW: Can BOJ perform similar intervention?