A Heuristic for Fat-Tailed Stock Market Returns

Figures & data

Table 1. Daily Value-Weighted Market Rates of Return, Post 1962/07/03, 15,479 Days

Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic (Z′)
−0.1713	−17.05	0.0000646	61.2	0.00000000	−65	−3.83	−4.09
−0.0900	−8.98	0.0003230	12.2	0.00000000	−19	−3.41	−3.44
−0.0709	−7.08	0.0006459	6.1	0.00000000	−12	−3.22	−3.21
−0.0645	−6.44	0.0009689	4.1	0.00000000	−10	−3.10	−3.11
−0.0557	−5.57	0.0012919	3.1	0.00000001	−8	−3.01	−2.97
−0.0502	−5.03	0.0018087	2.2	0.00000024	−7	−2.91	−2.87
−0.0397	−3.98	0.0031006	1.3	0.00003443	−4	−2.74	−2.63

Note. Explanations: These are summary statistics for a dataset of value-weighted CRSP returns including dividends from 1962/07/03 to 2023/12/31. Columns 1 and 2 show some extreme-left tail rates of return (here, a sampling from the worst to the 48th worst) and their equivalent Z scores (net of mean, divided by standard deviation). Columns 3 and 4 describe the empirically observed cumulative distribution. Columns 5 and 6 show the incorrect inference that would be obtained by applying the Gaussian normal distribution table to the Z values in the second column. Column 7 inverts the Gaussian cumulative distribution table to a Gaussian normal Z value that would match the empirical CDF in column 3. Column 8 uses the adjustment heuristic, based on the formula $Z\prime =-1.25-\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}(-Z),$ where Z is in the second column. Interpretation: Researchers are intuitively familiar with Z statistics and T statistics. The adjustment-heuristic fits the left tail well. It is even sufficient not to be statistically distinguishable from the empirical frequency inference, because of the limited sample data.

Source: Data/empirical.R: 2024/04/16 and CRSP.

Figure 1. Z Scores for Daily Market Rates of Return

Note. Explanations: The blue line is the Z score that best matches the empirical probability for the value-weighted market rate of return when using the normal distribution tables, beginning in 1962. It is the same case as that in Table 1. The red line is the same for the equal-weighted market rate of return, beginning in 1973. The thick black line is the quick-and-dirty adjustment-heuristic of $Z\prime =-1.25-\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}(-Z).$ The diagonal straight line on the right is the unadjusted Z score. Interpretation: The heuristic formula can be recommended for all negative Z scores, even though it is less conservative between −0.2 and −1.9. (For these Z scores, the difference between the adjusted and unadjusted Z score is modest, and the adjusted Z’ fits the empirical return distribution better than Z.) For Z scores lower than, say, −13, an extra further adjustment could be wise—although such events are so rare, and thus difficult to assess in the empirical stock return distribution, that this may not even be the case.) Source: data/plotusethis.R: 2024/04/16 and CRSP.

Table 2. Daily Value-Weighted Market Rate of Return, Later Starts

Panel A: Post 1973/01/01, 12,861 days
Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic (Z′)
−0.1713	−16.08	0.00007774	50.8	0.00000000	−58	−3.78	−4.03
−0.0900	−8.46	0.00038871	10.2	0.00000000	−17	−3.36	−3.39
−0.0709	−6.68	0.00077742	5.1	0.00000000	−11	−3.16	−3.15
−0.0645	−6.07	0.00116614	3.4	0.00000000	−9	−3.04	−3.05
−0.0557	−5.25	0.00155485	2.5	0.00000007	−7	−2.96	−2.91
−0.0534	−5.04	0.00178807	2.2	0.00000023	−7	−2.91	−2.87
−0.0423	−4.00	0.00334292	1.2	0.00003119	−5	−2.71	−2.64
Panel B: Post 1998/01/01, 6,541 days
Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic (Z′)
−0.1201	−9.73	0.0001528	25.9	0.00000000	−22	−3.61	−3.53
−0.0829	−6.73	0.0007642	5.2	0.00000000	−11	−3.17	−3.16
−0.0662	−5.38	0.0015283	2.6	0.00000004	−7	−2.96	−2.93
−0.0589	−4.79	0.0022925	1.7	0.00000084	−6	−2.83	−2.82
−0.0507	−4.13	0.0030567	1.3	0.00001806	−5	−2.74	−2.67
−0.0493	−4.01	0.0036680	1.1	0.00002993	−5	−2.68	−2.64

Note. Explanations: See Table 1. Interpretation: For these two commonly used samples for the stock market statistical distribution, the heuristic fits well. Source: Data/empirical.R: 2024/04/16 and CRSP.

Table 3. Empirically Estimated Kurtosis and Student T Degrees of Freedom

			Empirically estimated
Frequency	Description	Sample	Kurtosis	df of T dist
Daily	Value-weighted stock market	–1962	19.2	2.09
		1962–	19.0	3.05	(Table 1)
		1973–	17.9	2.78	(Table 2A)
		1998–	12.0	3.27	(Table 2B)
	Log-value-weighted stock market	−1962	18.6	2.08
		1962–	21.6	3.03
		1973–	20.4	2.77
		1998–	12.3	3.32
	Equal-weighted stock market	–1962	26.8	1.91
		1962–	18.6	2.68
		1973–	18.6	7.41
		1998–	13.5	3.21
	Small decile	1973–	98.0	7.30	(Table 5A)
	Large decile		18.5	5.01	(Table 5B)
Monthly	Value-weighted stock market	–1962	10.8	2.89
		1962–	4.8	6.25
		1973–	4.9	6.11
		1998–	4.1	7.03	(Table 4)
	Log-value-weighted stock market	−1962	9.5	2.76
		1962–	5.6	5.59
		1973–	5.7	5.44
		1998–	4.6	5.74
	Equal-weighted stock market	–1962	16.5	2.28
		1962–	5.8	4.75
		1973–	6.2	4.48
		1998–	5.0	4.66
	Small decile	1973–	9.7	3.16
	Large decile		4.7	6.24

Note. Explanations: The empirical kurtosis for daily returns ranges from a low of 12 to a high of 100. For monthly returns, it ranges from a low of 4 to a high of 17. The empirically best-fitting T distributions have degrees of freedom ranging from 1.91 to 7.41. The two do not necessarily align, because the best fitting T distribution is also fit to higher moments, not just the kurtosis. Moreover, if the true degrees of freedom were as estimated, the true kurtosis would be infinite for all degrees of freedom less than 4.0. Interpretation: The overall stock market and even extreme size deciles have distributions that are similarly fat-tailed, regardless of how and when they are measured. Thus, the heuristic is likely to fit well. Source: data/empirical.R: 2024/04/16 (grep for ’Best df_T is’) and CRSP.

Table 4. Monthly Value-Weighted Market Rate of Return, 1998, 312 Months

Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic (Z′)
−0.1849	−4.08	0.003185	26.2	0.00002229	−5	−2.73	−2.66
−0.1011	−2.31	0.015924	5.2	0.01056976	−2	−2.15	−2.09

Note. Explanations: See Table 1. These data are monthly and begin on 1998/01/01. Interpretation: Even for the distribution of monthly rates of return, with their higher estimated numbers of degrees of freedom (i.e., more resemblance to a Gaussian normal distribution than the dailies), the heuristic fits well. Source: Data/empirical.R: 2024/04/16 and CRSP.

Table 5. Extreme CRSP Size Decile Rates of Return, Post 1973, 12,861 Days

Panel A: 01 Smallest capitalization decile
Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z (Z′)
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic
−0.1375	−14.37	0.00007774	50.8	0.00000000	−46	−3.78	−3.91
−0.0805	−8.44	0.00038871	10.2	0.00000000	−17	−3.36	−3.38
−0.0687	−7.21	0.00077742	5.1	0.00000000	−13	−3.16	−3.22
−0.0561	−5.90	0.00116614	3.4	0.00000000	−9	−3.04	−3.02
−0.0485	−5.11	0.00155485	2.5	0.00000016	−7	−2.96	−2.88
−0.0476	−5.01	0.00171033	2.3	0.00000027	−7	−2.93	−2.86
−0.0378	−3.99	0.00357615	1.1	0.00003254	−4	−2.69	−2.63
Panel B: 10 largest capitalization decile
Obsrvd		Empirical obsrvd		Bad inference		Corrected normal Z
RoR	Z	Emp CDF	Per Yr	Normal CDF	$\approx \mathrm{10\hspace{0.17em}}̂$	Optimal	Heuristic (Z′)
−0.1828	−16.84	0.00007774	50.8	0.00000000	−63	−3.78	−4.07
−0.0879	−8.12	0.00038871	10.2	0.00000000	−16	−3.36	−3.34
−0.0709	−6.56	0.00077742	5.1	0.00000000	−11	−3.16	−3.13
−0.0630	−5.83	0.00116614	3.4	0.00000000	−9	−3.04	−3.01
−0.0571	−5.28	0.00155485	2.5	0.00000006	−7	−2.96	−2.91
−0.0540	−5.00	0.00163259	2.4	0.00000029	−7	−2.94	−2.86
−0.0429	−3.98	0.00318744	1.2	0.00003423	−4	−2.73	−2.63

Note. Explanations: See Table 1. These data begin on 1973/01/01. Interpretation: Even for the daily distributions of portfolios with small and big firms, the heuristic fits well. Source: Data/empirical.R: 2024/04/16 and CRSP.

Table 6. Fama–French Daily Factors, Post 1973

		Empirically estimated
Region	Factor	Kurtosis	df of T dist
USA	SMB	19.3	4.60
	HML	12.9	2.47
	RMW	11.3	2.70
	CMA	12.4	3.28
	MOM	21.1	2.20
	REV	9.5	3.07
	STREV	33.6	2.12
EUR	SMB	9.6	3.32
	HML	9.9	2.65
	RMW	11.5	4.16
	CMA	8.2	2.90
JPN	SMB	11.1	3.80
	HML	7.9	2.67
	RMW	10.2	3.45
	CMA	9.6	2.90

Note. Explanations: See Table 3. Factors are as named by Ken French. Interpretation: The factor portfolios have distributions that are similarly fat-tailed, regardless of how and when they are measured. The heuristic would thus fit well. Source: Data/empirical.R: 2024/04/16 and Ken French’s website.

Table 7. Individual Stocks

	Empirically estimated
	Kurtosis	df of T
Mean	49.1	3.96
Sd	192.2	19.28
Smallest	3.1	0.48
1%	5.6	1.12
5%	7.3	1.51
25%	11.4	2.21
Median	16.9	2.67
75%	32.2	3.22
95%	140.0	4.74
99%	624.7	8.32
Biggest	8,509.x	859.8x
N = 10,585 Stocks

Note. Explanations: See Table 3. Estimated statistics are as in earlier tables, but here they are for individual stocks with at least 2,500 rates of returns from 1973/01/01. Interpretation: There are about 5% of stocks for which even the heuristic would have underestimated the left tail (though far less so than the unadjusted Z score), and there are fewer than 1% of stocks for which the heuristic would have been worse than the assumption of a plain normal distribution. Source: Data/fattailed-individual-stocks.R: 2024/04/16 and CRSP.

Table A1. Well-Fitting Formula on Z Scores Below −3

Used	dfT	Kurtosis		a	b	Heuristic is more conservative below Z of
		Simul	(Theory)
37,381	2.0		(∞)	−2.07	−0.57	−2.62
49,997	2.2		(∞)	−1.90	−0.64	−2.47
59,266	2.4		(∞)	−1.77	−0.70	−2.37
65,178	2.6		(∞)	−1.66	−0.76	−2.30
68,497	2.8		(∞)	−1.58	−0.82	−2.25
70,013	3.0		(∞)	−1.51	−0.88	−2.21
70,332	3.2		(∞)	−1.46	−0.93	−2.18
69,877	3.4		(∞)	−1.40	−0.98	−2.15
68,933	3.6		(∞)	−1.35	−1.02	−2.13
67,688	3.8		(∞)	−1.31	−1.07	−2.11
66,267	4.0		(∞)	−1.27	−1.11	−2.09
58,648	5.0	8.2	(9.0)	−1.09	−1.31	−1.99
52,018	6.0	5.9	(6.0)	−0.94	−1.48	−1.86
46,746	7.0	5.0	(5.0)	−0.81	−1.63
36,574	10.0	4.0	(4.0)	−0.53	−1.96
15,542	100.0	3.1	(3.1)	0.61	−3.23
13,699	1000.0	3.0	(3.0)	0.80	−3.44
	Recommended Heuristic:			−1.25	−1.00	−1.88

Note. Explanations: Based on a simulated perfect interval grid of 10 million draws from a Student T distribution with the degrees of freedom as described in the second column, this table shows the coefficients on the appropriate inference from a raw Z score to the best-as-if-Gaussian-normal Z score in the formula $Z\prime =a+b·\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}(-Z ),$ fitting only drawn values with scaled values (Z scores) of less than −3 (the number of which is indicated in the first column). (The parenthetical value in the kurtosis is the theoretically correct one [$3+6/({\text{df}}_T-4)$ for ${\text{df}}_T>4$].). Interpretation: The adjustment-heuristic formula of $-1.25-\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}(-Z )$ used in the paper is geared roughly toward a Student T distribution with 3 to 3.5 degrees of freedom. It is more conservative for Z values less than −1.88. (Note: There are actually two roots. The second root, in the latter case, is about −0.2, visible also in Figure 1. With lesser kurtosis, the $g(·)$ no longer intersects at all.). Source: Data/find-good-formula.R: 2024/04/16, 2024/06/19. No data.