Sunday, 13 May 2012

Risk mispricing

Derivative positions are accumulating profits… while sweeping on to a catastrophe

"JP Morgan May Be a Trading Accident Waiting To Happen" is what you may like to read on "Jesse’s cross road café blog". JPM’s two billion dollar loss on credit derivatives is making it more and more obvious that the quadrillion derivative positions outstanding are really the accident waiting to happen.
But how can risk models be so flawed that financial collapse is looming behind the corner? Well, just listen to John Butler, author of “The Golden Revolution” as interviewed by Jim Puplava on ‘Financial Sense’, especially the section from 21’30” to 28’30” dealing with risk taking.

An economy Nobel prize leading to perdition

Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term Black–Scholes options pricing model. Merton and Scholes received the 1997 Nobel Prize in Economics (The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for their work.
A key assumption in the Black-Scholes model is that the stock price follows a geometric Brownian motion with constant drift and volatility. This assumption implies that a Gaussian distribution (the bell shaped curve) can be used in the model, allowing its complete mathematical derivation. This assumption is built on rather shaky ground and it is not too difficult to verify that the day-to-day evolution of a market index is not following the expected geometric Brownian motion.
You may assume that none of the above key assumptions is valid, however the constant drift (average) and volatility (standard deviation) show up as parameters in the Gaussian distribution. Hence their variability can be taken into account.
The misconception on a Gaussian distribution of stock price movements in fact goes back to the dawn of the 20th century, when a young French mathematician Louis Bachelier published his thesis: 'Théorie de la spéculation'... at a time when there was no computing power to check the validity of the hypothesis. The work gained notoriety only well after the author died in 1946. Eugene Fama took up the idea in a 1965 popular article: "Random Walks In Stock Market Prices".


Distribution analysis of Daily variations of the DOW JONES INDUSTRIAL

The longest time series publicly available is that of the Dow Jones Industrial, for which you can go back to October 1, 1928 or nearly 84 years. This comes to 20996 day-to-day variations observed till May 9, 2012. The day-to-day variations are calculated in percentage, the statistics are summarized in Table 1.
A mean of 0.0189542% is the daily gain that brought the DJ up from 240 on Oct 1 of 1928 to 12,835 last Wednesday May 9, 2012. The average gain calculated from the distribution is slightly higher, but the true value is within the confidence limits of the estimated one. The mean is way lower than the standard deviation of 1.161%. Using the Black-Scholes hypothesis of a normal distribution, 68% of the daily fluctuations would then be within 1.161% of that tiny mean. The normal distribution is perfectly symmetric, however we notice that the real distribution is skew (-0.168) to the left. This implies that catastrophic swoons are more probable than manic rallies.
At the right side you find Kurtosis, which is an indication for the “peakedness” of the statistics. A normal distribution has a kurtosis of 3 or an excess kurtosis of 0 (which is the usual way to express it). The distribution of the daily variations of the DJ has an excess kurtosis over 20, making the distribution very much “peaked” (leptokurtic if you prefer the math terminology). Such distribution has a slender top and seems to drop more rapidly than the equivalent normal distribution, however it has fatter tails: making extreme variations more likely.

Table 1: Distribution analysis of Daily variations of the DOW JONES INDUSTRIAL since Oct 1928

Moments
N
20996
Sum Weights
20996
Mean
0.02571175
Sum Observations
539.8439
Std Deviation
1.1610099
Variance
1.347944
Skewness
-0.168028
Kurtosis
20.691268
Uncorrected SS
28313.9657
Corrected SS
28300.085
Coeff Variation
4515.48378
Std Error Mean
0.00801249

Basic Statistical Measures
Location
Variability
Mean
0.025712
Std Deviation
1.16101
Median
0.041909
Variance
1.34794
Mode
0.000000
Range
37.95195


Interquartile Range
0.99759

Basic Confidence Limits Assuming Normality
Parameter
Estimate
95% Confidence Limits
Mean
0.02571
0.01001
0.04142
Std Deviation
1.16101
1.15001
1.17222
Variance
1.34794
1.32253
1.37411

Now if we leave out variations in excess of + 8%, it may be illustrative to plot the distribution observed:
Daily variations of the DJ Industrial in frequency intervals 0.4% wide. The distribution is truncated to losses and gains between -8% and 8%, leaving out 25 extreme observations

Does this limitation affect the calculated values for average, standard deviation and other parameters ? You bet: leaving out only 25 extreme observations reduces the standard deviation significantly, as can be checked in Table 2.
The truncated distribution is (even more) skew to the left but the kurtosis has dropped to a more modest 6.56. The above graph however still proves the essential: at the very center, you find more observations than for the corresponding normal distribution (the red curve). The frequency drops off rapidly at either side. Yet between 3 and 4 (or over 3 times the standard deviation) you find extreme variations to be more frequent than what is predicted by a normal distribution.

Table 2: Statistical key-characteristics of the truncated distribution of daily variations of the DJ

Moments
N
20971
Sum Weights
20971
Mean
0.02158836
Sum Observations
452.72952
Std Deviation
1.09369169
Variance
1.19616151
Skewness
-0.2921284
Kurtosis
6.55813798
Uncorrected SS
25093.2806
Corrected SS
25083.5069
Coeff Variation
5066.11728
Std Error Mean
0.00755241

The distribution illustrates very well how the usual bull markets progress: you typically have small gains outnumbering small losses. Close to break-even, the difference is quite pronounced. On the losing edge, the frequency of losses ranging from ‑0.8% to -1.2% still is lower than the gains ranging from 0.8% to 1.2%, but further away from break-even, frequencies seem to balance. Ultimately swoons are not compensated by rallies anymore.

"6 sigma" events

The largest percentage gain on the DJ ever was a manic 15.34% surge on March 15 of 1933. The most jaw dropping swoon is still the 22.61% panic loss on Oct 19, 1987. A similar analysis has been posted before: See “Volatility persists”  (using the data series for the Euro-stoxx 50 and the S&P500)

The 25 extreme observations left out in the above graph are not 'anomalies' isolated in time. Larger fluctuations typically tend to occur in avalanche. The typical market crisis is a succession of exasperating swoons intermingled by few manic rallies. This makes volatility surge to epic levels.

The mere fact that these 'black swan' events have been so extremely rare and that almost all the time potential gains are so large, leads traders to taking more risk. The surging benefit for their company translates into ever larger bonuses.  Reassured by past success, traders start adopting an ever bolder and risk tolerant attitude; they ultimately end piling up unhedged derivative positions.  They feel backed by the Black & Scholes theory for derivative pricing.  Eventually traders could be taken down by the main flaw in this theory: the non-gaussian nature of the day-to-day average price changes.
Their unhedged derivative positions may blow up the enterprise, possibly taking down the whole financial sector in a series of cascading defaults.

This may sound exagerated at first: yet remember the Long Term Capital management hedge fund (LTCM, one of the first of its kind).  The LTCM went belly up in 1998 following heavy losses on their bond portfolio after the Russian default (under the late Boris Jeltsin.)  Severe liquidity problems rising among the LTCM creditors urged former FED chairman Alan Greenspan to abruptly cut the FED interest rate and provide $3.6 Billion in emergency loans to stave off a cascade of defaults. And one to remember: Members of LTCM's board of directors included Myron S. Scholes and Robert C. Merton, who shared the above mentioned 1997 Nobel Memorial Prize in Economic Sciences.

Bond Pricing

As compared to equity, the price of bonds tends to vary only very little. Key-factors contributing to price moves are:
  1. the 'risk-free' short term interest rate set by the Central Bank (the FED in the US);
  2. the steepness of the duration curve for different bond maturities, hence the interest rate of the sovereign debt for various maturities;
  3. the specific credit default risk, whereby credit agencies issue a rating for major debtors and monitor that credit rating over time.
Foreign currency exchange risk is not considered here: bond investors need to consider the risk-reward balance of such operation. Nor do I consider bonds with specific clauses, such as calls (by the issuer) or puts (by the holder) on certain dates or convertible bonds.

As bonds come closer to maturity, the duration dependency fades and the yield on these bonds tends to approach the short-term interest during the last few months. Credit default swaps aim to eliminate only the third risk factor. By design, a bond holder may take a credit default swap with a bond insurer against payment of a premium. That premium typically would be less than the difference between the bond yield and the 'risk-free' interest rate on sovereign debt.

Not only do bond prices vary little over time, the credit default risk is of an even more static nature, always being positive but close to zero. If there is one derivative that cannot be modelled by a 'Gaussian distribution', the credit default swap is a textbook example. And yet ...


Credit risk and moral hazard

In the world of credit default swaps, it is "The International Swaps and Derivatives Association" - essentially the issuers of credit default swaps - that determines whether default swaps are triggered.  For example the Greek debt restructuring (whereby creditors lost half their principal and the remainder is paying a low rate on a longer term) is NOT recognized a 'credit event'. The International Swaps and Derivatives Association is both judge and party.  Unfortunately this is hardly an exception: moral hazard is ubiquitous because issuers of default swaps have such large portfolio of those derivatives that they rarely have the capital necessary to assume pay-out of any major credit event. At the other end, there are the fraudulent bets, taken by parties who don't own the bonds insured: who are NOT creditors and are merely placing a bet on a company or public authority declaring bankruptcy.
Why am I thinking about the Greek myth when Augias was asked to clean that stable?

2 comments:

  1. 1) Excellent writing. Please indicate more clearly with vertical lines on the distribution graph where the mean is and why it is skew left as it is not immediately obvious.

    2) Could you conduct the same analysis on the gold market? I would gladly attempt (and probably fail) to do so myself if I could get a table of the daily closing price of gold.

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  2. Thanks for your interest. Deriving the mean and skewness from the graph is less than evident. As you read from the table, the mean is really very small. As such it would show up in the middle (considering the scale). Skewness and Kurtosis are calculated from the series of observations and are in the first table. As for the gold price, there exists a series of London AM and PM fixes. The publicly available information goes back to the beginning of the 21st century (if I recall well). Better than nothing and probably that series is long enough to check for a very similar behaviour.

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