Factor Replication with Industry Stratification

Figures & data

Table 1. Comparing Costless and Costed Factor Benchmarks for QR1000 IS 100%

	MCWI	FI	EW	DW-1	DW-2	DW-3	DRPW	DMV	MOM	QLTY
Annualized Geometric Ret. (%):	10.89	10.84	10.77	10.92	10.98	10.96	10.81	9.42	8.92	10.74
Annualized Volatility (%):	14.72	15.30	16.84	16.33	15.75	15.16	14.69	13.55	15.74	16.37
Value of $1:	26.14	25.77	25.26	26.41	26.87	26.70	25.60	17.18	14.85	25.07
Ratio to Benchmark:	0.97	0.88	0.81	0.87	0.91	0.95	0.73	0.52	0.41	0.62
Annualized Turnover (%):	12.84	43.01	72.42	54.09	37.24	22.77	93.94	153.31	226.11	120.28
Trade of $1:	26.09	156.37	225.18	167.71	110.35	59.60	322.00	315.81	647.99	412.60
Cost of $1:	0.13	0.91	1.43	0.98	0.59	0.30	2.16	4.40	4.63	3.13
% of Costless:	97.29	87.70	81.36	86.75	91.45	95.14	73.40	52.16	40.87	62.44
Eroded Value:	0.73	3.61	5.79	4.03	2.51	1.36	9.27	15.76	21.48	15.08
Tracking Error Diff Monthly Benchmark (%):	0.01	0.04	0.05	0.04	0.02	0.01	0.08	0.17	0.24	0.12
Information Ratio:	−0.20	−0.92	−1.55	−1.18	−0.74	−0.42	−1.86	−1.82	−4.13	−0.82

Notes: Reducing factor replication costs while maintaining high-fidelity tracking is crucial to attaining viable factors. This table compares four dimensions of the relative performance between each factor’s costless and costly benchmarks (i.e., all stocks or 100% stratification) including performance, turnover, transaction cost, and replication quality metrics. In contrast to Market Capitalized Weighted Indexation, factors do not enjoy automatic rebalancing and require dynamic rebalancing to replicate factors accurately, generating substantial portfolio turnover and corresponding large transaction costs that erode factor performance. After including transaction costs, each benchmark factor suffers severe performance degradation relative to its costless benchmark. The key driver of this performance degradation is turnover. Factor returns are affected in proportion to each factor’s turnover. These results demonstrate, consistent with the extant literature that, after costs, efficient factor replication is crucial for viable factors.

Table 2. Percentage Coverage by Industry for QR1000 Using NS (Panel A) and IS (Panel B) by GICS

GICS/% Strat.	10	20	30	40	50	60	70	80	90	100
A: QR1000 percentage coverage using NS, by GICS
Energy	33%	52%	52%	52%	67%	67%	67%	67%	67%	100%
Materials	80%	100%	100%	100%	100%	100%	100%	100%	100%	100%
Industrials	49%	69%	79%	79%	79%	79%	79%	79%	100%	100%
Consumer Discretionary	57%	81%	100%	100%	100%	100%	100%	100%	100%	100%
Consumer Staples	35%	44%	44%	60%	84%	84%	100%	100%	100%	100%
Health	28%	42%	63%	63%	73%	100%	100%	100%	100%	100%
Financials	47%	61%	66%	79%	100%	100%	100%	100%	100%	100%
Information Technology	27%	30%	34%	41%	62%	62%	62%	78%	78%	100%
Communication Services	30%	46%	67%	82%	100%	100%	100%	100%	100%	100%
Utilities	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
Real Estate	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
B: QR1000 percentage coverage using IS, by GICS
Energy	59%	75%	85%	92%	95%	97%	99%	99%	100%	100%
Materials	44%	59%	70%	78%	85%	90%	94%	97%	99%	100%
Industrials	60%	76%	84%	90%	93%	96%	97%	99%	99%	100%
Consumer Discretionary	57%	73%	82%	88%	92%	95%	97%	98%	99%	100%
Consumer Staples	62%	77%	85%	90%	94%	97%	98%	99%	100%	100%
Health	53%	74%	85%	91%	94%	96%	98%	99%	100%	100%
Financials	61%	76%	85%	90%	93%	96%	98%	99%	100%	100%
Information Technology	70%	84%	90%	94%	96%	98%	99%	99%	100%	100%
Communication Services	54%	76%	85%	92%	95%	97%	98%	99%	100%	100%
Utilities	42%	62%	73%	80%	86%	91%	94%	97%	99%	100%
Real Estate	0%	71%	71%	86%	86%	94%	94%	97%	97%	100%

Notes: Using Global Industry Classification Standard (GICS) industry classification, we see the cumulative market capitalization for each percentile stratification level for naïve (NS) and industry stratification (IS). In bold is the stratification level where a threshold coverage of greater than 75% of industry market capitalization is attained. IS results in higher industry coverage at lower stratification. In contrast, NS induces a strong industry selection bias toward industries with large stocks, resulting in unbalanced industry exposure. Since IS provides better factor benchmark industry coverage than NS, it should provide higher fidelity returns to benchmark factors.

Figure 1. TE for EW Factor Using IS, NS, and LP with QR1000

Notes: Due to onerous replication costs, investors must accurately reproduce factors efficiently. To reduce replication costs, portfolio stratification includes only a portion of the entire factor portfolio. Here, we compare factor replication quality for industry stratification (IS), naïve stratification (NS), and linear programming optimization (LP) using tracking error (TE) by stratification level. NS incrementally removes the smallest, costliest stock in the entire factor replication portfolio. IS removes the most costly stock by industry. LP optimizes stock weights in the stratified replication portfolio by minimizing the tracking error (TE). IS, NS, and LP have differing goals. The primary purpose of NS is to reduce the costs of factor replication by strictly removing the smallest, costliest stocks in the entire replicating portfolio. The primary goal of IS is to both retain similar industry exposures to the factor benchmark and eliminate each industry’s smallest, costliest stocks. LP’s explicit goal is to minimize the TE. For the equal weight factor (EW), IS provides the lowest TE for each stratification level, demonstrating the importance of industry coverage for accurate stratified factor replication.

Table 3. Monthly Index Tracking Error (TE, ΔTE) between IS and NS Portfolios for the QR1000 Market Capitalized Ranking

	FI		EW		DW-1		DW-2		DW-3
%	IS	NS	IS	NS	IS	NS	IS	NS	IS	NS
Stratification	ΔTE		ΔTE		ΔTE		ΔTE		ΔTE
10%	0.69	0.77	1.29	1.4	1.17	1.26	1	1.08	0.81	0.87
	−8**		−11**		−10**		−8**		−5**
40%	0.27	0.28	0.64	0.72	0.55	0.61	0.45	0.49	0.33	0.36
	−1*		−8**		−6**		−4**		−3**
70%	0.11	0.12	0.33	0.35	0.27	0.29	0.21	0.22	0.14	0.15
	−1*		−3**		−2**		−1**		−1**

Table

	DRPW		DMV		MOM		QLTY
%	IS	NS	IS	NS	IS	NS	IS	NS
Stratification	ΔTE		ΔTE		ΔTE		ΔTE
10%	1.11	1.22	1.08	1.19	1.27	1.41	1.23	1.35
	−12**		−10*		−13**		−12**
40%	0.56	0.62	0.62	0.66	0.63	0.69	0.63	0.68
	−6**		−4**		−6**		−5**
70%	0.29	0.31	0.4	0.41	0.35	0.38	0.33	0.35
	−2**		−1		−3**		−2**

Notes: These results validate the core intuition of our industry stratification (IS) methodology and demonstrate the general applicability and benefits of IS over naïve stratification (NS) for factor replication. This table presents the tracking error (TE) and TE difference (ΔTE) for 10%, 40%, and 70% stratification levels, and includes two-sided p values using a Wilcoxon matched pairs test (with *indicating significance at the 5% level and **significance at the 1% level). For almost all factors and stratifications, IS leads to a highly statistically significant lower TE than NS. The TE difference between IS and NS, ΔTE, is nearly always negative and significant indicating that IS has a lower TE than NS.

Figure 2. ΔTE for All Factors with QR1000

Notes: Reducing factor replication costs by simply removing the most costly, small stocks leads to inaccurate factor replication. Instead, it is important to remove the most costly, small stock by industry, not simply the overall smallest stock. The differences in absolute tracking error, $\Delta \mathrm{TE},$ for the industry (IS) and naïve (NS) stratified portfolios, $\Delta \mathrm{TE}=\left|{\mathrm{TE}}_{\mathrm{IS}}\right|\text{-}\mathrm{ }\left|{\mathrm{TE}}_{\mathrm{NS}}\right|,$ by stratification level is shown. The value of $\Delta \mathrm{TE}$ is almost always less than zero for all factors, denoting enhanced TE for IS versus NS. The prevalent negative difference between $\Delta \mathrm{TE}$ means that NS has a larger TE than IS for all factors for almost all stratification levels.

Figure 3. ΔICI for All Factors with QR1000

Notes: Naïve stratification (NS) retains the largest stock in a factor replication portfolio, irrespective of industry, inducing a strong industry selection bias toward industries with large stocks. This industry selection bias is responsible for the poor performance of the NS methodology. Here, we see the differences in industry coverage (ΔICI; see Kacperczyk et al. (2005)) for the industry stratification (IS) and NS portfolios, ${\Delta \mathrm{ICI}\equiv \mathrm{ }\mathrm{ICI}}^{\mathrm{IS}}-{\mathrm{ICI}}^{\mathrm{NS}}\mathrm{ }$ by stratification for each factor are always negative. This demonstrates that IS provides closer industry coverage to the factor benchmark than NS.

Table 4. QR1000 EW and LP Factor DMRS Regression

DMRS factor/
% Strat.	10%	40%	70%
A: QR1000 EW Factor DMRS Regression: ΔTE versus αi, βi, hi, si, ri, ci, and ΔICI by stratification %
αi	0.14	−0.03	0.01
βi	−0.01	0.01	0.01
si	0.00	0.01	0.00
hi	0.02	−0.01	−0.01
ri	−0.04	0.00	0.00
ci	−0.02	0.01	0.01
ΔICIi	12.66	18.34	65.80
t(αi)	1.79*	−0.71	0.77
t(βi)	−0.65	2.81**	2.9**
t(si)	−0.09	1.09	−0.41
t(hi)	0.77	−1.20	−1.51
t(ri)	−1.77*	−0.14	0.86
t(ci)	−0.49	0.88	0.90
t(ΔICIi)	3.5**	1.64	3.18**
B: QR1000 LP Factor DMRS Regression: ΔTE versus αi, βi, hi, si, ri, ci, and ΔICI by stratification %
αi	0.05	0.00	0.02
βi	−0.02	−0.02	−0.02
si	−0.01	−0.01	−0.02
hi	0.02	−0.01	−0.01
ri	0.01	−0.05	−0.03
ci	0.08	0.01	0.01
ΔICIi	4.40	146.06	219.57
t(αi)	1.15	−0.09	0.64
t(βi)	−1.26	−1.59	−1.84*
t(si)	−0.26	−0.38	−1.16
t(hi)	0.8	−0.53	−0.69
t(ri)	0.5	−1.98**	−1.7*
t(ci)	1.9*	0.3	0.54
t(ΔICIi)	0.51	16.77**	20.82**

Notes: The importance of industry coverage for high-fidelity factor replication is highlighted. Stratification may lead to inadvertent risk tilts, responsible for the superior tracking error (TE) of IS observed in Table 3. Here, we show that changes in industry coverage, not standard risk factors, explain differences between differences in TE (ΔTE). The dependent variable is ΔTE. The independent variables are the DMRS risk factors and the difference in industry coverage between industry (IS) and naive (NS) stratification, ΔICI. Panel A applies to EW and Panel B to LP. ΔICI is often significant suggesting that differences in industry coverage explain differences in ΔTE (for all factors see Table B.1, Online Supplemental Material Appendix B). LP generates inferior TE due to its poor industry coverage. Its ΔICI coefficient in Panel B is on between 6 and 10 times larger for LP than EW in Panel A.

Figure 4. ΔAnnual Turnover and ΔEroded Value for All Factors

Notes: Of primary concern when replicating factors with high fidelity is industry coverage not cost reduction. For a given stratification level and factor, we see the difference between industry (IS) and naïve stratification (NS) Annual Turnover (ΔAnnual Turnover, Panel A) and Eroded Value (ΔEroded Value, Panel B). Positive across all stratification levels, IS always suffers both a higher annualized turnover and higher eroded value than NS. Since NS removes the smallest (and highest-cost) stock, ipso facto, it generates lower annual dollar turnover (and smaller eroded value). Table 3 shows that the cost mitigation benefits of NS do not translate to TE enhancements. Industry coverage is more important.

Table 5. QR1000 Dollar Ending Value ($) for EW Factor IS, NS, and LP by Stratification Percentage for Costless and Costed Portfolios

%	IS		NS		LP
Strat.	Costless EW ($)	Costed EW ($)	Costless EW ($)	Costed EW ($)	Costless EW ($)	Costed EW ($)
10%	$27.14	$17.98	$24.29	$17.25	$26.54	$18.54
40%	$31.84	$23.85	$30.20	$23.55	$29.79	$22.68
70%	$33.01	$24.60	$31.22	$23.90	$30.35	$22.22
100%	$31.05	$25.26	$31.05	$25.26	$34.95	$25.94

Notes: By design, linear programming (LP) should select investment weights for each stratified portfolio to reduce both tracking error (TE) and the costs of factor replication. Here, the costless ending value assumes no trading costs, whereas the costed results include turnover costs in forming the LP portfolio. LP delivers competitive dollar-ending value. Even though the LP 100% stratified portfolio, has a slightly larger TE (see Figure 1), it enjoys both the best costless and costed dollar ending value. No approach dominates across all stratifications.

Table 6. QR1000 Economic Significance ($millions) between IS and NS Portfolios

% Strat. /	FI	EW	DW-1	DW-2	DW-3	DRPW	DMV	MOM	QLTY
Factor
10%	$774	$712	$795	$839	$784	$608	$463	$713	$184
40%	−$121	$317	$252	$188	$108	$136	−$39	$474	−$954
70%	$197	$711	$532	$379	$187	$67	−$444	$227	$911

Notes: As our economic measure, we take the difference in dollar-ending value between industry (IS) and naively stratified (NS) portfolios, representing the value of a factor investment after 30 years given the initial $1 billion investment. We report the difference (in $millions of dollars) between the ending value of the IS and NS factor portfolios after costs at different stratification levels. We see that the magnitude of the economic benefits of industry stratification (IS) is large. Given the initial $1 bn investment, IS enhancements are often worth hundreds of millions of dollars. As these differences are often positive, IS tends to be closer to the benchmark ending values than NS.

Kacperczyk, M., C. Sialm, and L. Zheng. 2005. “On the Industry Concentration of Actively Managed Equity Mutual Funds.” The Journal of Finance 60 (4): 1983–2011. doi:10.1111/j.1540-6261.2005.00785.x.

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