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[Spun off from the 7702 thread.]

Whenever the topic of annual floors and caps comes up, the question arises of: What do the returns look like? How many times is the annual return of the S&P500 below the XX% floor, and how many times is it above the YY% cap?

I computed the rolling 12-month returns of the S&P500 index (excluding dividends) beginning Jan 1975 and ending Dec 2012. That's 37 years, and 456 rolling annual periods.

The worst 12-month loss was -45%.
The best 12-month gain was +53%.

Here's the table of the distribution of the returns. (Explanations following.)
Gain	Cnt	Pct	Cum pct	Weighted	Floor/Cap
-45% 7 1.5% 1.5% -0.6140
-35% 1 0.2% 1.8% -0.0713
-30% 5 1.1% 2.9% -0.3015
-25% 11 2.4% 5.3% -0.5428
-20% 14 3.1% 8.3% -0.5373
-15% 14 3.1% 11.4% -0.4145
-12% 10 2.2% 13.6% -0.2412
-10% 7 1.5% 15.1% -0.1458
-9% 4 0.9% 16.0% -0.0746 ** These are eliminated **
-8% 6 1.3% 17.3% -0.0987
-7% 5 1.1% 18.4% -0.0713
-6% 4 0.9% 19.3% -0.0482
-5% 6 1.3% 20.6% -0.0592
-4% 3 0.7% 21.3% -0.0230
-3% 5 1.1% 22.4% -0.0274
-2% 8 1.8% 24.1% -0.0263
-1% 3 0.7% 24.8% -0.0033
0% 7 1.5% 26.3% 0.0077 0.0077
1% 9 2.0% 28.3% 0.0296 0.0296
2% 9 2.0% 30.3% 0.0493 0.0493
3% 8 1.8% 32.0% 0.0614 0.0614
4% 12 2.6% 34.6% 0.1184 0.1184
5% 9 2.0% 36.6% 0.1086 0.1086
6% 19 4.2% 40.8% 0.2708 0.2708
7% 15 3.3% 44.1% 0.2467 0.2467
8% 8 1.8% 45.8% 0.1491 0.1491
9% 16 3.5% 49.3% 0.3333 0.3333
10% 13 2.9% 52.2% 0.2993 0.2993
11% 12 2.6% 54.8% 0.3026 0.3026
12% 47 10.3% 65.1% 1.3914 5.4211
15% 42 9.2% 74.3% 1.6118
20% 39 8.6% 82.9% 1.9243
25% 34 7.5% 90.4% 2.0504 ** These all become 12% **
30% 25 5.5% 95.8% 1.7818
35% 11 2.4% 98.2% 0.9046
40% 8 1.8% 100.0% 0.8158
53% 100.0%
Total 456 9.8421 7.3980

# periods with returns...
< 0% 113
>= 0% 343
>= 12% 206


Each row represents one bucket of periodic returns.
"Gain" is the annual gain.
"Cnt" is the number of periods with a return in that bucket. For example, 7 periods had a gain of 0% to 1%. That's the 0% bucket.
"Pct" and "Cum pct" is the percentage of that cnt of periods.

The next question is: How much does each bucket contribute to your overall gain/loss?
"Weighted" is the average gain of the bucket multiplied by the Pct.
For example, 7 periods were in the 0% bucket, which has average gain of 0.5%, and that happens 1.5% of the time. That bucket help you only a little bit.
Another bucket had 7 periods, that's the -10% bucket, with an average loss of -9.5%.
The -10% bucket hurts a lot more than the 0% bucket helps.
Basically, a big Weighted value is a big contribution to the overall gain, and a negative weighted value is harmful to the overall gain.

The total row is the sum of all those individual weights, which is 9.8421.

"Floor/cap" column:
Obviously, if you get rid of negative values, the total will be higher, which means that the overall gain will be higher. Setting a floor of 0% does just that.

But they also impose a cap -- in this example 12% cap. Everything above 12% is capped to 12%. The floor/cap column is the weighted values is all the buckets below 0% discarded and buckets 12% and above treated as exactly 12%.

The total for this column is the total with the floor & cap. That is, all the returns below 0% are discarded, and all the gains above 12% are treated as 12%.
That total is 7.3980.

This combination of 0% floor & 12% cap is *worst* overall return. Overall, the loss buckets that got discarded helped you less than having the large gain capped buckets hurt you.
In order to get the same total, the cap would have to be 28%.

However, it is true that the floor completely eliminates the 113 times with a loss.
There is something to be said for avoiding the 7 times that had a -45% loss.
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This is interesting in the statistical abstract. It could apply if the markets were actually "a random walk" and had no trends, or if outlier rallies (or drops) were just as likely as previous similar rallies (or drops.)

That's not the case, though. The majority of the major annual rallies occur after significant down trends... and this has a major reduction effect on the benefits of capturing "all the rally upside" above caps.

To give a very simple example,
If Trader A's position in a market drops 20%, we all know it requires a rally of 25% to reach the prior water level.

When Trader B's posiiton in the same market is traded with a 0 floor and 12 cap, Trader B lost nothing in the down year, and gained 12% in the following year (though "missing out" on the 13% above her cap.)

Trader A has to actually get a 37% return in order to merely 'catch up' to Trader B. Since the current rally gave them both 25%, Trader A is *STILL* down, despite capturing "ALL" the 25% windfall rally.

Of course, if Trader A gets several consecutive years of rallies greater than Trader B's caps, then Trader A may indeed supercede.

This is why a floor/cap hedge outperforms in volatile markets, and particularly excels when there are significant bearish periods. When there is a strong certainty of straight rally years greater than a particular cap, then going naked is better.

The simple question to ask is;
What is the rough probability of consecutive S&P rally years greater than 12%, without intermediary years below zero to wipe out the above-cap gains?

Everybody can make their own guess...
Mine is that the probability is low... or extremely low.

Dave Donhoff
Leverage Planner
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...That's not the case, though. The majority of the major annual rallies occur after significant down trends... and this has a major reduction effect on the benefits of capturing "all the rally upside" above caps.
Absolutely. You have grasped a point which eludes many many authors and writers. See: http://ssrn.com/abstract=1908469

Losses tend to cluster, and gains also tend to cluster. So the simple statistics that people toss about don't tell the whole story.

This is why a floor/cap hedge outperforms in volatile markets, and particularly excels when there are significant bearish periods. When there is a strong certainty of straight rally years greater than a particular cap, then going naked is better.
Yup.
The challenge is designing a strategy that avoids the bad outcomes but keeps the good outcomes. This is extremely difficult.

The market risk is there. It is what it is. It cannot be eliminated, only shifted around. If A wants to get rid of risk he can only do so by sloughing it off to B. B will only take on this risk if he gets paid for doing so. Insurance companies are pretty good at pricing risk and rarely undercharge for it. They change the risk profile that A is exposed to, and they charge A for doing so.

And this whole discussion is more appropriate for the M.I. board than this board. ;-)

What is the rough probability of consecutive S&P rally years greater than 12%, without intermediary years below zero to wipe out the above-cap gains?
That's a good question.

The table I posted did not address sequence of returns, nor was it meant to.
The table merely answers the question, "So, how often did the historical returns fall into these various buckets, anyway? Just how many times were capped off?"
And secondarily, "What is the density of the returns in the buckets -- how much did each bucket contribute to the overall return?"

I put this in a different thread because it's really a separate issue from the IUL srategy, where the sequence of returns *does* matter.
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And this whole discussion is more appropriate for the M.I. board than this board. ;-)
I'll have to go revisit that board when I have time... but last I was there it seemed it was all stock picking mechanics, rather than trade structuring.

Maybe we'll have to open yet *another* board for technical & risk trade structuring for "unfair advantages" ;~)
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Hi RayVT

And this whole discussion is more appropriate for the M.I. board than this board. ;-)

I disagree because 'learning' is appropriate for ALL the boards. You will reach people on this board who do not visit the M.I. board.

Your and Dave's thread, the back and forth of basic concepts... is why the Fool is so valuable.

thanks :-)
ralph
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I disagree because 'learning' is appropriate for ALL the boards. You will reach people on this board who do not visit the M.I. board.

This :
And this whole discussion is more appropriate for the M.I. board than this board. ;-)

wouldn't cue you to look there ? "do not" = "choose not"

All boards becoming all topics is but one of the things that makes TMF far less useful than it once was.
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Well, since there is at least 3 of us foraging of into the weeds, let me take the risk of focusing a spotlight on what might already be obvious;

The table merely answers the question, "So, how often did the historical returns fall into these various buckets, anyway? Just how many times were capped off?"

'How often' is irrelevant from a P&L perspective. The old trader's mantra applies; A system works when the average of all winning trades is greater than the average of all losers, minus the ability to survive the worst statistical drawdown.

In my prior 'Traders A/B' example, trader B lost on 100% of the rallies, yet still blew away trader A. Of course, a mere 2-trade market history is hardly sufficient to illustrate anything near reality... but as we proceed with our comparison illustrations, the clouds will begin to clear and I predict you'll see that this is the functional result in the overwhelming majority of market periods.

The IUL industry has coined the phrase "Zero is your Hero" (and I am sure that rankles some of the political hardcores... LOL!) but from a strategic trading perspective, it really does pan out.

Of course, you are absolutely right in that blocking or repositioning risk comes at costs... and in order to 'buy' the elimination of annual downside risks you have to give up some measure of upside risks.

The trader's/investor's question then, again, boils down to;
'At what trade structure will my cost (average trade losses) be less than my average gains?'

A 12% S&P hurdle (as I pointed out previously) is somewhat daunting... but more complex index blends (which take advantage of lower option spreading costs) push that "average gains" hurdle (via higher caps) to 17%-21%

At that level the trade is simply a no brainer... at least for the trade structure, on its mathematical face. Of course, then all the other factors have to be double checked (short term versus long term costs, liquidity, systemic risk exposures, etc.)

Dave Donhoff
Leverage Planner
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Here is a graph that shows all the rolling 12-month returns, both IUL-type capped and raw uncapped.
However, the graph caps the raw gains at +/- 25% to keep the scale viewable.

This visibly shows how returns & losses tend to cluster.

http://i1131.photobucket.com/albums/m543/rayvt/chart-7_zpsdb...
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