11-Sigma? As reported on July 17, it has been claimed that US Financials recently experienced an eleven-standard-deviation price move; not just a black swan, but a black-hole swan!

Perhaps not surprisingly, US Financial 15 Split Corp has made a slight adjustment to their standard valuation page, namely a Fund Update dated July 18:

A myriad of issues have affected the financial markets and have had a dramatic impact on the Company’s portfolio. Overall financial markets continue to be adversely impacted by the confluence of record high commodity prices and the continuing credit related problems originating from the US sub prime lending market. These conditions have caused economic growth to slow considerably in both Canada and the United States while at the same time high commodity prices are beginning to lead to a marked increase in inflationary pressures. In particular, the dramatic increase in oil prices has become a large obstacle for economic recovery. The US Financial Services sector is down approx. 34% year to date and in the last month closed at its lowest level since 1997 (over 11 years).The combined effect of the market declines and the monthly distributions paid since inception has resulted in a decline in the net asset value of the Company to $9.25 as at July 15, 2008. The recent two day rally in the market has improved the net asset value of the Company by approximately 25% as at July 17, 2008.

One of their core holdings is Merrill Lynch, which got whacked today because of their writedowns, but let’s assume that the portfolio as a whole performed equally to the US S&P 500 Financials index, which is up another 3.05%

So, we’ll estimate the current net asset value of FTU units as 9.25 * 1.28 = $11.84.

Now, this asset coverage of slightly under 1.2:1 isn’t going to reverse the recent downgrade to Pfd-3. But just for fun, suppose we don’t need no stinking credit ratings. The prefs, FTU.PR.A, closed at 7.55-75, 15×10 today, after trading 800 – count ’em, 800 – shares in a range of 7.51-52.

So, say we can put on a huge position at $8.00. Our investment has asset coverage of just under 1.5:1 – not particularly good, but it’s not too long ago that issues were routinely given Pfd-2 credit ratings with this level of coverage – and it pays $0.525 annually until maturity 2012-12-1. That’s a yield of 6.56% on 4.5-year paper with asset coverage of 1.5-ish to 1. Which ain’t bad. And there’s the possibility of a bonus 25% being paid at the end of these 4.5-years if the units can avoid losing more than ~15% of their value over this time.

Which is kind of cool.

On the other hand, there’s some competition … the very ominously named “Mulvihill World Financial Split Corp” had asset coverage of just under 1.6:1 as of July 10, with no jiggery-pokery about market-value / par-value. It was downgraded recently to Pfd-2(low). It closed today at 8.80-85, 20×5, after trading 10,100 shares in a range of 8.77-87. At the closing bid, it yields 10.24%, way more than the Split Share Index … but remember, there is no bonus here – the yield calculation assumes full repayment of the $10 principal at maturity on 2011-6-30. Over 10% as a dividend on three-year paper is normally considered a good deal … but careful investors might wish to check the quarterly list of holdings to see if there have been any little accidents.

**Update, 2010-08-05**: See also Why Banks Failed the Stress Test.

11-Sigma my A**

First of all, I’m not going to pay to read the orignal story to get the exact details, because it sounds like BUNK. Somehow this 11-sigma is based on S&P Financial Select Sector SPDR (Symbol XLF) and its change over some period — a month, a year, a day I don’t know.

I did look up the price history of XLF back to its origin in 1998 and estimated the annualized standard deviation of the log change in XLF over a number of non-overlapping periods (stat-speak). We use log change (rather than percent change) for stock prices which have no probability of being less than zero.

For Daily changes: the standard deviation of ln(closing price / closing price the trading day before) since 1998 is 0.0170 (approximately 1.70%). We can annualize this by multiplying by Square Root (252 trading days in the year / Number of Trading days we looked at (i.e. 1 here)) to get an annualized log standard deviation of 0.27 (roughly 27%, though more up because Exp[0.27) = 1.31 or 31% up while Exp[-0.27] = 0.76 or 24% down).

Daily changes tend to be more volatile than weekly, which are a bit more volatile than monthly, quarterly or annual. By looking at non-overlapping periods since 1998, we find the annualized standard deviation of ln price change is:

2 weeks: 0.244

1 month: 0.203

3 months: 0.204

1 year: 0.171

But bear in mind we had 2407 trading days to get the 1-day standard deviation estimate and only 9 full years to use to get these standard deviation estimates. The accuracy of a standard deviation estimate increases with more data and decreases with higher standard deviation and shorter time frame, so, for low number of data is likely to underestimate the actual standard deviation (e.g. the 1 year estimate is the least reliable).

Anyway, XLF had (in the past, up to July 21, an annualized log price change standard deviation of somewhere in the range of 0.17 to 0.27).

Annual Change: from $36.68 July 13, 2007 (a Friday, there is no Jul 15 trading day in 2007) to $17.17 July 15, 2008. This is a log change of ln(17.17/36.68) = -0.76. Compared with an annualized standard deviation of 0.27, this represents -2.8 sigma. Compared with a less reliable annualized standard deviation estimate of 0.17, this represents -4.4 sigma. Neither is anywhere near -11 sigma!!!!!!!!

I also looked at 6-month, 3-month, 1-month, 2 week and one day changes in XLF to July 15, 2008, with no change representing worse than -5.3 sigma. For example:

On Friday June 13, 2008 XLF closed at $23.38, so the log price change is ln(17.17/23.38) = -0.31 over one month (Times Sqrt(12) = -1.07 annualized change). -1.07 is -4.0 sigma compared with 0.27 annually; or -5.3 compared with 0.203 annually (the standard deviation for 114 monthly changes since 1998).

I have no idea where the 11-sgima came from: annual change? Usage of percent changes rather than the more rigorous log changes? data mining for an unusal period with an unusually high number of sigma? or some math error (like multiplying a month standard deviation by 12 instead of sqrt(12) to annualize???

IF stock and index prices were actually log normally distributed, even a 4 or 5-sigma event would be somewhat rare (-4 sigma = probability of 0.0032% or one in 32,000 months, days, years, or whatever). However, stock and index prices have stochastic but mean-reverting volatility, so tend to have fatter tails and a sharper peak than pure log normal distributions.

From my reading of the investment literature, it is not unusual to have a 4-5 sigma move (on a daily basis) at least once a year in one or more markets. Thus, I don’t consider the behaviour of XLF, nor of financial stocks in general over the past month, quarter or year, out of the realm of reasonable possibility GIVEN their historical volatility. But, it sure ain’t no 11-sigma.

Why this is important to Prefblog relates to split shares where a 50% fall in an individual stock, index, or conceivably even a market, is not that uncommon and will WIPE out capital unit investors (cf FTU) and cause split share pref owners considerable grief. So many of these structured products have too much potential to blow up — I would guess that several to 10 percent of Pfd-2 split shares have eventually gone into or near default — which is vastly more than the rating agencies’ probability models for ordinary companies would suggest.

If you really want to know a decent (though perhaps high side) market estimate of the probability of something, look at publicly traded put options. XLF closed today at $20.75 and a 17-month Jan 2010 put option at $10.00 strike price is worth about 0.30 (mid bid-ask): it costs 1.5% of the present value to insure against a drop larger than 50% in 17 months (about -2.3 sigma using 0.27 annualized; 1.15% probability).

If I get a chance, I will look at James’ idea that the FTU.PR.A might be an intriguing investment. I worry about MER drag, especially of fixed costs now amortized over a tiny base. The FTU common stub is a call option on the underlying. The Pref has some characteristics of a short put, so should trade below intrinsic value because of the short put time premium. (after all, with put call parity, if the capital unit has call time premium value, the pref share has to be short put time premium value of a comparable magnitude — assuming a yield — because the sum of the parts is not more valuable than NAV). To top it off, FTU and FTU.PR.A have stinky liquidity…..

Anyway, before I digress too far, my main message is DON’T believe every 11-sigma story you hear! The internet (and newspaper) is often full of baloney and we need reasoned judgement to cut through it.

Sorry, FTU option logic not quite right here:

Capital unit is a call option, therefore

Pref share is short the call option (Covered write)

Capital Unit + Pref Share = NAV (minus option plus option cancel out).

Shouldn’t stay up this late 😉

In the comments to the

Naked Capitalismpost, it turned out that nobody actually read the original claim!The consensus was that to the extent that standard deviation means anything, it was probably around a 4-5 sigma move. To the extent that SD doesn’t mean anything … holy smokes, that was a big jump!

Because of the fat tails, I believe 4-5 sigma measured in this way has the probability characteristics closer to 2-2.5 sigma — a few percent chance; once every 20-40 years (if using annual data).

However 4-5 sigma on a MONTHLY basis with 10 years of past data means a few percent adjusted probability (still), but there were 120 chances for such a move, so it becomes a near certainty. The odds of NOT having a 2-sigma negative monthly move at least once in 10 years are (1-2.275%) ** 120 = 6.3% (where ** is a power to avoid crashing the reply window with a caret) and NOT having a 2.5 sigma move are less than 50/50.

Maybe because I am a few years older than you and invested through the 1980s boom and 1987 crash, a couple of mini-crashes in the 1990s, 9/11 etc, I am not at all surprised as seeing things happen despite having only a few percent probability (on both the upside and downside).

For financials, the preceding 10-year history was very quiet, with only a near-recession in 2001, so using 10 years of data is bound to underestimate inherent annual volatility. For technology stocks, the 10-year history includes the tech boom and bust, so is likelier to be a fair estimator of annual volatility.

For Pref shares, I would argue the environment has been basically positive since about 1982 — when long term interest rates began a 25-year slide. I’m sure lots of bad stuff happened to prefs in the 1970s as real interest rates and inflation soared. Unfortunately, we don’t have indices or records of that period to help us understand the current period, but I am not too surprised by a 20% decline for discount prefs in 18 months.

Never mind the standard deviation stuff, it would be most interesting to know how many times the daily change in any major sector exceeded the 13% one-day and 25% two-day change in financials.

Presumably October 19, 1987, with a 20% drop in the entire market would dominate the list!

Unfortunately, my reply of last night was wiped out by a thunderstorm, so I will try to recall:

Speaking only for XLF (Select Sector SPDR ETF) going back to 1998, the top 10 positive and bottom 10 negative 1-day changes are in the range 6 to 8.5% (except for the 12.3% recently). This is out of 2407 data. These big swings occurred in 1999, 2000, 2002, 2004 and 2007-08, so are somewhat spread out and not all recent. This period does not cover 1970-75, 1980-84 nor 1990-93 where I expect financials also had huge one day ups and downs. Thus, I guess that if you can observe 6-9%, then 12% in one day is not unreasonable as an extreme event.

For other sectors, I don’t have the data. Since 1971, the Nasdaq Composite has had 8 single day changes in the 8-13% range (in almost 10,000 days). This is one every 5 years — but they are not evenly distributed and tend to cluster! For example, 3 of 8 ocurred in October 1987.

Since 1950 the S&P-500 has had 15 single day changes in the 5-9% range EXcluding the Oct 19, 1987 crash (almost 15,000 days; one every 4 years). Four of these other 15 days were in October 1987 — two up and two down to roughly cancel out (leaving the main effect of the actual crash day).

Obviously, with sectors more concentrated than broad indices, even wider daily sectoral swings are possible.

The good news for long term investors is that large drops tend to be followed (eventually) by large rises. Usually the individual drops are larger (though the 12.3% one-day rise in financials last week was 50% larger than the previous record one-day drop). And, of course, prefs took a long time to recover from surging inflation in the 1970s. If we don’t have a repeat of much higher interest rates, prefs could still come back. In the meantime, we just have to enjoy our regular income, the entertainment from the occasional arbitrage trade, and try to steer clear of the defaults and delistings.

Wow! Cool! Thanks!

So we could say, for instance, that the chance of XLF experiencing a change in excess of 6% in a single day is (10 + 10) / 2407 = 0.83%. Fitting this number into a confidence interval as defined by Wikipedia, this then implies that a change in excess of 6% is about 2.6 sigma. Which then implies that a change in excess of 12%, like the one we’re discussing, is about 5.2 sigma.

Which is really hokey math, for sure, but estimating sigma by counting outliers is, to me, better than estimating sigma by counting normal days … given that you know already that the distribution is non-gaussian and standard deviation is meaningless.

my reply of last night was wiped out by a thunderstormMy extremely expensive Bell Canada Hosting is giving me an increasing number of problems … who needs thunderstorms when you’ve got Bell? I will be changing hosts in the near future – until then, I suggest that long posts be copied to notepad (or even just the clipboard, if you’re using Windows) and only then should one push the button.

And – oh yeah – 5.2 sigma as estimated in my previous post is 1-million + to one, or about one trading day in 4,000 years. Although we have to go back less than 21 years to the crash of ’87, when I feel quite confident that financials experienced a bigger down day. So, frankly, I don’t take this sigma stuff all that seriously.

The biggest problem with estimating 5.2 sigma this way is that the sample with the 20 extreme values was too short — only 10 years, and a pretty quiet ten years at that.

Here’s another way to estimate the odds of a 12.3% change in financials without the assumption of a normal distribution needed for confidence intervals. We just compare XLF with S&P-500 daily changes over the same period (1998 to present):

S&P 500 since 1998: Top 20 moves (+ or -) are in the 3.7 to 6.0% range, only 4 of which overlap with the Top 16 since 1950 (which were 5-9% plus 22.3% for Black Monday).

Therefore, for extreme moves, I would say that:

(a) Recall XLF in 10 years saw the Top 20 in the 6 to 8.5% plus 12.3% for the recent bounce.

(b) Comparing XLF with S&P-500 for 10 years, we could say XLF is about 1.6X more volatile than the S&P-500 (e.g. 6%/3.7% or 8.5%/6% or 12.3%/6% taking ratios of some of these extreme values as XLF/S&P-500).

(c) Returning to the 58-year history, we would then apply the 1.6X factor to S&P-500 extreme values to get an estimate for extreme XLF over 58 years as:

1.6X 5-9% (or Black Monday 22.3%) = 8 to 14% (BM 36%).

Now we see that using a longer database, the Top 20 of about 15,000 days (say 1 in 1000 days) of XLF could involve a change in the range of 8-14%. Thus, 12.3% is not 1 in one million, but probably more like one in 5 to 10,000 or once every 20-40 years — with the chances of a real extreme Black Monday-type move IN ADDITION.

QED

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