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Author: Rayvt Big gold star, 5000 posts Top Favorite Fools Add to my Favorite Fools Ignore this person (you won't see their posts anymore) Number: of 76398  
Subject: Distributions of S&P500 returns Date: 3/31/2013 5:34 PM
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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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