How many days to wait for a rebound after a fall in S&P 500 ?

I used different window sizes, from 10 to 99 days, to find the maximum in window.

For example, if a window of 20 days is used, the algorithm looks for the maximum in the past 20 days. After the maximum is found, there should be some local minima and local maxima occurred in these 20 days. The local maxima must be smaller than the maximum in the whole 20 days period. These maxima are actually the several rebounds of the fall from the maximum in the past 20 days.

As I tune the window size from 10 to 99 days, I can calculate the percentage of a rebound occurred in any one day of the whole window period. A valid rebound is the lowest price tomorrow not less than today's lowest price after the rebound. Obviously, the percentage should gradually drop as the window increases because there are more points in the window so the maxima should spread out. Nonetheless, there is still a high percentage of a rebound occurred in the next 7 days after a maximum is found for the window.

For example, the following table shows the cumulative frequency with a 20 days window. The maximum number of days may be 1 day more than the window size. It is because in the past 20 days, the trend is down and the maximum occurs 20 days ago, if tomorrow's price is higher than today's lowest price, it is counted as a rebound in the 21st day after the maximum. There is 11.65% chance of a rebound after a fall from a maximum found today. There is 81.6% of chance the rebound occurred in the next 7 days for a 20 days window. The percentage of a rebound in the next 7 days is more than 66% even for a 99 days window.

20 Days Window :

21 days	1348	Percentage	Cumulative Freq
0	157	11.65%	11.65%
1	341	25.30%	36.94%
2	217	16.10%	53.04%
3	127	9.42%	62.46%
4	89	6.60%	69.07%
5	65	4.82%	73.89%
6	45	3.34%	77.23%
7	59	4.38%	81.60%
8	32	2.37%	83.98%
9	28	2.08%	86.05%
10	30	2.23%	88.28%
11	21	1.56%	89.84%
12	18	1.34%	91.17%
13	14	1.04%	92.21%
14	22	1.63%	93.84%
15	19	1.41%	95.25%
16	12	0.89%	96.14%
17	15	1.11%	97.26%
18	13	0.96%	98.22%
19	16	1.19%	99.41%
20	7	0.52%	99.93%
21	1	0.07%	100.00%

The following tables list the cumulative frequency of a rebound in window of 10, 30, 40, 50, 90 and 99 days respectively.

10 Days Window :

11 days	1577	Percentage	Cumulative Freq
0	195	12.37%	12.37%
1	472	29.93%	42.30%
2	281	17.82%	60.11%
3	175	11.10%	71.21%
4	125	7.93%	79.14%
5	93	5.90%	85.03%
6	55	3.49%	88.52%
7	71	4.50%	93.02%
8	42	2.66%	95.69%
9	31	1.97%	97.65%
10	35	2.22%	99.87%
11	2	0.13%	100.00%

30 Days Window :

30 days	1216	Percentage	Cumulative Freq
0	136	11.18%	11.18%
1	286	23.52%	34.70%
2	167	13.73%	48.44%
3	110	9.05%	57.48%
4	72	5.92%	63.40%
5	57	4.69%	68.09%
6	34	2.80%	70.89%
7	54	4.44%	75.33%
8	31	2.55%	77.88%
9	20	1.64%	79.52%
10	28	2.30%	81.83%
11	16	1.32%	83.14%
12	19	1.56%	84.70%
13	15	1.23%	85.94%
14	20	1.64%	87.58%
15	19	1.56%	89.14%
16	13	1.07%	90.21%
17	16	1.32%	91.53%
18	13	1.07%	92.60%
19	13	1.07%	93.67%
20	7	0.58%	94.24%
21	9	0.74%	94.98%
22	14	1.15%	96.13%
23	6	0.49%	96.63%
24	7	0.58%	97.20%
25	4	0.33%	97.53%
26	3	0.25%	97.78%
27	9	0.74%	98.52%
28	2	0.16%	98.68%
29	9	0.74%	99.42%
30	7	0.58%	100.00%

40 Days Window :

40 days	1123	Percentage	Cumulative Freq
0	131	11.67%	11.67%
1	239	21.28%	32.95%
2	152	13.54%	46.48%
3	97	8.64%	55.12%
4	67	5.97%	61.09%
5	52	4.63%	65.72%
6	27	2.40%	68.12%
7	46	4.10%	72.22%
8	28	2.49%	74.71%
9	21	1.87%	76.58%
10	26	2.32%	78.90%
11	15	1.34%	80.23%
12	16	1.42%	81.66%
13	13	1.16%	82.81%
14	15	1.34%	84.15%
15	17	1.51%	85.66%
16	11	0.98%	86.64%
17	13	1.16%	87.80%
18	11	0.98%	88.78%
19	13	1.16%	89.94%
20	5	0.45%	90.38%
21	10	0.89%	91.27%
22	13	1.16%	92.43%
23	6	0.53%	92.97%
24	7	0.62%	93.59%
25	4	0.36%	93.94%
26	3	0.27%	94.21%
27	8	0.71%	94.92%
28	2	0.18%	95.10%
29	8	0.71%	95.81%
30	8	0.71%	96.53%
31	8	0.71%	97.24%
32	4	0.36%	97.60%
33	5	0.45%	98.04%
34	4	0.36%	98.40%
35	2	0.18%	98.58%
36	3	0.27%	98.84%
37	4	0.36%	99.20%
38	2	0.18%	99.38%
39	4	0.36%	99.73%
40	3	0.27%	100.00%

50 Days Window :

50 days	1042	Percentage	Cumulative Freq
0	117	11.23%	11.23%
1	230	22.07%	33.30%
2	129	12.38%	45.68%
3	90	8.64%	54.32%
4	57	5.47%	59.79%
5	42	4.03%	63.82%
6	23	2.21%	66.03%
7	42	4.03%	70.06%
8	21	2.02%	72.07%
9	19	1.82%	73.90%
10	24	2.30%	76.20%
11	14	1.34%	77.54%
12	16	1.54%	79.08%
13	11	1.06%	80.13%
14	14	1.34%	81.48%
15	18	1.73%	83.21%
16	9	0.86%	84.07%
17	14	1.34%	85.41%
18	9	0.86%	86.28%
19	12	1.15%	87.43%
20	7	0.67%	88.10%
21	8	0.77%	88.87%
22	11	1.06%	89.92%
23	7	0.67%	90.60%
24	8	0.77%	91.36%
25	4	0.38%	91.75%
26	4	0.38%	92.13%
27	6	0.58%	92.71%
28	1	0.10%	92.80%
29	7	0.67%	93.47%
30	6	0.58%	94.05%
31	9	0.86%	94.91%
32	3	0.29%	95.20%
33	6	0.58%	95.78%
34	3	0.29%	96.07%
35	1	0.10%	96.16%
36	2	0.19%	96.35%
37	4	0.38%	96.74%
38	2	0.19%	96.93%
39	4	0.38%	97.31%
40	3	0.29%	97.60%
41	2	0.19%	97.79%
42	0	0.00%	97.79%
43	6	0.58%	98.37%
44	0	0.00%	98.37%
45	4	0.38%	98.75%
46	0	0.00%	98.75%
47	3	0.29%	99.04%
48	3	0.29%	99.33%
49	3	0.29%	99.62%
50	4	0.38%	100.00%

90 Days Window :

87 days	889	Percentage	Cumulative Freq
0	95	10.69%	10.69%
1	188	21.15%	31.83%
2	105	11.81%	43.64%
3	77	8.66%	52.31%
4	48	5.40%	57.71%
5	31	3.49%	61.19%
6	18	2.02%	63.22%
7	34	3.82%	67.04%
8	18	2.02%	69.07%
9	17	1.91%	70.98%
10	20	2.25%	73.23%
11	11	1.24%	74.47%
12	15	1.69%	76.15%
13	10	1.12%	77.28%
14	14	1.57%	78.85%
15	15	1.69%	80.54%
16	8	0.90%	81.44%
17	10	1.12%	82.56%
18	12	1.35%	83.91%
19	10	1.12%	85.04%
20	6	0.67%	85.71%
21	6	0.67%	86.39%
22	7	0.79%	87.18%
23	6	0.67%	87.85%
24	8	0.90%	88.75%
25	1	0.11%	88.86%
26	4	0.45%	89.31%
27	4	0.45%	89.76%
28	1	0.11%	89.88%
29	6	0.67%	90.55%
30	5	0.56%	91.11%
31	8	0.90%	92.01%
32	1	0.11%	92.13%
33	5	0.56%	92.69%
34	1	0.11%	92.80%
35	1	0.11%	92.91%
36	6	0.67%	93.59%
37	2	0.22%	93.81%
38	3	0.34%	94.15%
39	4	0.45%	94.60%
40	3	0.34%	94.94%
41	1	0.11%	95.05%
42	1	0.11%	95.16%
43	4	0.45%	95.61%
44	0	0.00%	95.61%
45	6	0.67%	96.29%
46	1	0.11%	96.40%
47	1	0.11%	96.51%
48	4	0.45%	96.96%
49	3	0.34%	97.30%
50	3	0.34%	97.64%
51	1	0.11%	97.75%
52	0	0.00%	97.75%
53	1	0.11%	97.86%
54	0	0.00%	97.86%
55	0	0.00%	97.86%
56	2	0.22%	98.09%
57	1	0.11%	98.20%
58	1	0.11%	98.31%
59	2	0.22%	98.54%
60	0	0.00%	98.54%
61	1	0.11%	98.65%
62	2	0.22%	98.88%
63	0	0.00%	98.88%
64	1	0.11%	98.99%
65	0	0.00%	98.99%
66	0	0.00%	98.99%
67	0	0.00%	98.99%
68	0	0.00%	98.99%
69	0	0.00%	98.99%
70	2	0.22%	99.21%
71	1	0.11%	99.33%
72	0	0.00%	99.33%
73	0	0.00%	99.33%
74	1	0.11%	99.44%
75	0	0.00%	99.44%
76	0	0.00%	99.44%
77	1	0.11%	99.55%
78	0	0.00%	99.55%
79	1	0.11%	99.66%
80	0	0.00%	99.66%
81	0	0.00%	99.66%
82	0	0.00%	99.66%
83	0	0.00%	99.66%
84	0	0.00%	99.66%
85	2	0.22%	99.89%
86	0	0.00%	99.89%
87	1	0.11%	100.00%

99 Days Window :

94 days	868	Percentage	Cumulative Freq
0	91	10.48%	10.48%
1	192	22.12%	32.60%
2	98	11.29%	43.89%
3	74	8.53%	52.42%
4	53	6.11%	58.53%
5	33	3.80%	62.33%
6	19	2.19%	64.52%
7	35	4.03%	68.55%
8	17	1.96%	70.51%
9	19	2.19%	72.70%
10	17	1.96%	74.65%
11	14	1.61%	76.27%
12	13	1.50%	77.76%
13	6	0.69%	78.46%
14	8	0.92%	79.38%
15	17	1.96%	81.34%
16	5	0.58%	81.91%
17	11	1.27%	83.18%
18	9	1.04%	84.22%
19	11	1.27%	85.48%
20	4	0.46%	85.94%
21	6	0.69%	86.64%
22	10	1.15%	87.79%
23	5	0.58%	88.36%
24	7	0.81%	89.17%
25	0	0.00%	89.17%
26	2	0.23%	89.40%
27	7	0.81%	90.21%
28	0	0.00%	90.21%
29	6	0.69%	90.90%
30	5	0.58%	91.47%
31	6	0.69%	92.17%
32	1	0.12%	92.28%
33	6	0.69%	92.97%
34	1	0.12%	93.09%
35	1	0.12%	93.20%
36	5	0.58%	93.78%
37	3	0.35%	94.12%
38	2	0.23%	94.35%
39	3	0.35%	94.70%
40	4	0.46%	95.16%
41	1	0.12%	95.28%
42	0	0.00%	95.28%
43	4	0.46%	95.74%
44	0	0.00%	95.74%
45	3	0.35%	96.08%
46	0	0.00%	96.08%
47	2	0.23%	96.31%
48	3	0.35%	96.66%
49	2	0.23%	96.89%
50	3	0.35%	97.24%
51	1	0.12%	97.35%
52	0	0.00%	97.35%
53	2	0.23%	97.58%
54	0	0.00%	97.58%
55	0	0.00%	97.58%
56	2	0.23%	97.81%
57	1	0.12%	97.93%
58	1	0.12%	98.04%
59	2	0.23%	98.27%
60	0	0.00%	98.27%
61	1	0.12%	98.39%
62	2	0.23%	98.62%
63	0	0.00%	98.62%
64	1	0.12%	98.73%
65	1	0.12%	98.85%
66	0	0.00%	98.85%
67	0	0.00%	98.85%
68	0	0.00%	98.85%
69	0	0.00%	98.85%
70	3	0.35%	99.19%
71	1	0.12%	99.31%
72	0	0.00%	99.31%
73	0	0.00%	99.31%
74	0	0.00%	99.31%
75	0	0.00%	99.31%
76	0	0.00%	99.31%
77	1	0.12%	99.42%
78	0	0.00%	99.42%
79	1	0.12%	99.54%
80	0	0.00%	99.54%
81	0	0.00%	99.54%
82	0	0.00%	99.54%
83	0	0.00%	99.54%
84	0	0.00%	99.54%
85	1	0.12%	99.65%
86	1	0.12%	99.77%
87	1	0.12%	99.88%
88	0	0.00%	99.88%
89	0	0.00%	99.88%
90	0	0.00%	99.88%
91	0	0.00%	99.88%
92	0	0.00%	99.88%
93	0	0.00%	99.88%
94	1	0.12%	100.00%