Solution to Smart Beta Factors

Solution to Smart Beta Factors#

# Output from previous execution

	MKT	SMB	HML	RMW	CMA	UMD
date
2025-06-30	0.0486	-0.0002	-0.0160	-0.0319	0.0145	-0.0264
2025-07-31	0.0198	-0.0015	-0.0127	-0.0029	-0.0207	-0.0096
2025-08-31	0.0185	0.0488	0.0441	-0.0069	0.0207	-0.0354

# Output from previous execution

	Name	Unit	Construction	Description
MKT	Market	Excess Return	Market-cap-weighted	US Equities
SMB	Size	Excess Return	Small Minus Big	Long small stocks and short big stocks
HML	Value	Excess Return	High Minus Low	Long value (high book-to-market) stocks and sh...
RMW	Profitability	Excess Return	Robust Minus Weak	Long profitability (income statement) and shor...
CMA	Investment	Excess Return	Conservative Minus Agressive	Long stocks with low (conservative) investment...
UMD	Momentum	Excess Return	Up Minus Down	Long stocks that have high recent returns and ...
RF	Risk-free rate	Total Return	Tbills	NaN

# Output from previous execution

	Annualized Mean	Annualized Vol	Annualized Sharpe
MKT	0.0876	0.1561	0.5607
RMW	0.0440	0.0829	0.5311
UMD	0.0603	0.1534	0.3933
CMA	0.0283	0.0725	0.3903
HML	0.0260	0.1088	0.2392
SMB	0.0061	0.1013	0.0604

# Output from previous execution

	Annualized Mean	Annualized Vol	Annualized Sharpe
MKT 1980-2001	0.0835	0.1592	0.5242
SMB 1980-2001	0.0028	0.1092	0.0257
HML 1980-2001	0.0528	0.1110	0.4751
RMW 1980-2001	0.0487	0.0912	0.5341
CMA 1980-2001	0.0460	0.0772	0.5958
UMD 1980-2001	0.1088	0.1497	0.7272
MKT 2002-2025	0.0914	0.1535	0.5952
SMB 2002-2025	0.0092	0.0935	0.0984
HML 2002-2025	0.0012	0.1064	0.0113
RMW 2002-2025	0.0397	0.0745	0.5326
CMA 2002-2025	0.0118	0.0676	0.1749
UMD 2002-2025	0.0152	0.1558	0.0976

# Output from previous execution

../_images/047cb49bb861e2da37aeed802808b7a69c96439af15b5e6aaa05e38e3b38e050.png

# Output from previous execution

	Annualized Mean	Annualized Vol	Annualized Sharpe
MKT 2015-Present	0.1179	0.1574	0.7491
RMW 2015-Present	0.0400	0.0726	0.5509
UMD 2015-Present	0.0201	0.1374	0.1464
CMA 2015-Present	-0.0091	0.0821	-0.1114
HML 2015-Present	-0.0163	0.1299	-0.1255
SMB 2015-Present	-0.0238	0.1032	-0.2305

# Output from previous execution

../_images/cd24369b74f56f2ef61cefa720f824144d3acc16e73040f30d235057966a5ef2.png

# Output from previous execution

../_images/9aff409e8d90be9d218bc42defaea1bdb055eff5b38ed966364f56c8eeed4978.png

# Output from previous execution

	Tangency Weights	Annualized Mean	Annualized Sharpe
CMA	0.3214	0.0283	0.3903
RMW	0.3018	0.0440	0.5311
MKT	0.2186	0.0876	0.5607
UMD	0.1125	0.0603	0.3933
SMB	0.0668	0.0061	0.0604
HML	-0.0212	0.0260	0.2392

# Output from previous execution

../_images/ca45fc7a3d522c40cf5163ffbbaaee73b9d491bb1dae79b3557523b6f72af8f6.png

# Output from previous execution

../_images/a01ae1ce965cb9e93c0fd237d325330ea67f4edf559a7fdebedf1c8ccccf45d0.png

# Output from previous execution

	Tangency Weights	Annualized Mean	Annualized Sharpe	Avg Correlation	MKT Correlation	SMB Correlation	HML Correlation	RMW Correlation	CMA Correlation	UMD Correlation
CMA	32.1%	2.8%	39.0%	23.6%	-34.7%	-5.1%	67.7%	13.9%	100.0%	0.0%
RMW	30.2%	4.4%	53.1%	12.9%	-25.1%	-41.2%	21.9%	100.0%	13.9%	7.7%
MKT	21.9%	8.8%	56.1%	4.0%	100.0%	22.7%	-20.8%	-25.1%	-34.7%	-17.9%
UMD	11.2%	6.0%	39.3%	10.3%	-17.9%	-6.1%	-21.6%	7.7%	0.0%	100.0%
SMB	6.7%	0.6%	6.0%	11.4%	22.7%	100.0%	-2.2%	-41.2%	-5.1%	-6.1%
HML	-2.1%	2.6%	23.9%	24.2%	-20.8%	-2.2%	100.0%	21.9%	67.7%	-21.6%

# Output from previous execution

../_images/24ae333352437465343a989da133cd7ff7e2eff9bcf4daf031a4729f80570d33.png

# Output from previous execution

	Tangency Weights with 4 Factors	Tangency Weights with 6 Factors
MKT	0.3765	0.2186
SMB	-0.0512	0.0668
HML	0.3653	-0.0212
UMD	0.3094	0.1125
RMW	NaN	0.3018
CMA	NaN	0.3214

# Output from previous execution

	Annualized Mean	Annualized Vol	Annualized Sharpe
Tangency Portfolio with 4 Factors Portfolio	0.0608	0.0666	0.9138
Tangency Portfolio with 6 Factors Portfolio	0.0482	0.0401	1.2013
Tangency Portfolio without HML Portfolio	0.0482	0.0402	1.2003

# Output from previous execution

	Agric	Food	Soda	Beer	Smoke	Toys	Fun	Books	Hshld	Clths	...	Boxes	Trans	Whlsl	Rtail	Meals	Banks	Insur	RlEst	Fin	Other
date
2025-04-30	-0.0103	-0.0214	0.0116	-0.0616	0.0483	-0.0671	0.1377	0.0164	-0.0383	-0.0801	...	-0.0179	-0.0308	0.0163	0.0097	-0.0299	-0.0245	-0.0851	-0.0792	-0.0054	-0.0030
2025-05-31	0.1264	-0.0326	-0.0009	-0.0382	0.0424	0.0517	0.0615	0.0380	0.0372	0.0989	...	0.0211	0.0689	0.0200	0.0609	0.0330	0.0706	-0.0499	0.0089	0.0752	0.0009
2025-06-30	0.0500	-0.0170	-0.0117	-0.0201	0.0041	0.0627	0.0992	0.0327	-0.0425	0.0167	...	0.0143	0.0441	0.0121	0.0241	0.0162	0.0616	-0.0028	0.1014	0.0819	-0.0192
2025-07-31	-0.0228	-0.0072	-0.0446	0.0304	-0.0618	0.0147	-0.0908	-0.0367	-0.0344	0.0216	...	-0.0076	-0.0239	-0.0020	0.0278	-0.0285	0.0048	-0.1008	0.1033	0.0415	-0.0266
2025-08-31	0.0243	-0.0025	0.0305	0.0311	0.0349	0.0651	0.0462	0.0531	0.0289	0.0387	...	0.0161	0.0383	0.0116	0.0047	0.0050	0.0473	0.0711	0.0541	-0.0058	0.0407

5 rows × 49 columns

# Output from previous execution

	Alpha	Annualized Alpha	R-Squared
Agric	0.0010	0.0117	0.3421
Food	0.0001	0.0015	0.4551
Soda	0.0013	0.0154	0.3025
Beer	0.0008	0.0099	0.4148
Smoke	0.0034	0.0411	0.2654
Toys	-0.0028	-0.0337	0.5102
Fun	0.0033	0.0391	0.6072
Books	-0.0031	-0.0367	0.6889
Hshld	-0.0011	-0.0127	0.5547
Clths	-0.0019	-0.0227	0.6190
Hlth	-0.0034	-0.0407	0.4409
MedEq	0.0014	0.0164	0.5958
Drugs	0.0021	0.0256	0.4894
Chems	-0.0029	-0.0351	0.7452
Rubbr	-0.0004	-0.0045	0.6453
Txtls	-0.0031	-0.0367	0.5470
BldMt	-0.0024	-0.0286	0.7525
Cnstr	-0.0029	-0.0353	0.6315
Steel	-0.0027	-0.0324	0.6283
FabPr	-0.0024	-0.0287	0.4245
Mach	-0.0004	-0.0053	0.7540
ElcEq	-0.0004	-0.0048	0.7372
Autos	0.0011	0.0127	0.5278
Aero	-0.0006	-0.0070	0.5994
Ships	-0.0032	-0.0389	0.5046
Guns	0.0003	0.0041	0.3245
Gold	0.0013	0.0151	0.0495
Mines	-0.0016	-0.0196	0.4589
Coal	-0.0033	-0.0396	0.2126
Oil	-0.0024	-0.0288	0.4556
Util	0.0003	0.0040	0.3614
Telcm	0.0005	0.0062	0.5804
PerSv	-0.0041	-0.0496	0.5848
BusSv	-0.0007	-0.0081	0.8463
Hardw	0.0035	0.0420	0.6669
Softw	0.0057	0.0684	0.7450
Chips	0.0053	0.0636	0.7498
LabEq	0.0017	0.0201	0.7279
Paper	-0.0036	-0.0430	0.6730
Boxes	-0.0003	-0.0041	0.5829
Trans	-0.0019	-0.0226	0.7088
Whlsl	-0.0014	-0.0166	0.7523
Rtail	0.0018	0.0211	0.6866
Meals	-0.0002	-0.0028	0.6371
Banks	-0.0018	-0.0211	0.7740
Insur	-0.0013	-0.0154	0.6612
RlEst	-0.0043	-0.0511	0.6054
Fin	0.0016	0.0197	0.8129
Other	-0.0035	-0.0422	0.5837

# Output from previous execution

Lambda represents the premium calculated by the cross-section regression and the historical premium is the average of the factor excess returns

	TS MAE	TS Annualized MAE
MKT + HML + RMW + UMD Cross-Section Regression	0.0021	0.0246

# Output from previous execution

AQR Time-Series Average Absolute Alpha: 0.0021
AQR Time-Series Average Absolute Annualized Alpha: 0.0246

# Output from previous execution

Lambda represents the premium calculated by the cross-section regression and the historical premium is the average of the factor excess returns
Lambda represents the premium calculated by the cross-section regression and the historical premium is the average of the factor excess returns
Lambda represents the premium calculated by the cross-section regression and the historical premium is the average of the factor excess returns
Lambda represents the premium calculated by the cross-section regression and the historical premium is the average of the factor excess returns

	Eta	Annualized Eta	R-Squared	TS MAE	TS Annualized MAE	CS MAE	CS Annualized MAE
FF5 Cross-Section Regression	0.0050	0.0599	0.3765	0.0026	0.0314	0.0010	0.0120
FF3 Cross-Section Regression	0.0052	0.0627	0.3504	0.0020	0.0244	0.0010	0.0120
AQR Cross-Section Regression	0.0063	0.0755	0.2066	0.0021	0.0246	0.0011	0.0136
CAPM Cross-Section Regression	0.0069	0.0832	0.0093	0.0017	0.0210	0.0013	0.0152

# Output from previous execution

	CAPM Weights	AQR Weights	FF3 Weights	FF5 Weights
MKT	1.0000	0.2765	0.6142	0.2273
HML	NaN	0.1836	0.5019	-0.0989
RMW	NaN	0.3546	NaN	0.3687
UMD	NaN	0.1853	NaN	NaN
SMB	NaN	NaN	-0.1161	0.0806
CMA	NaN	NaN	NaN	0.4223

# Output from previous execution

	FF5 + Momentum Weights	FF3 + Momentum Weights
MKT	0.2186	0.3765
HML	-0.0212	0.3653
SMB	0.0668	-0.0512
RMW	0.3018	NaN
CMA	0.3214	NaN
UMD	0.1125	0.3094

# Output from previous execution

	CAPM R-Squared	AQR R-Squared	FF3 R-Squared	FF5 R-Squared
Aero	0.5431	0.5994	0.5776	0.5971
Agric	0.3333	0.3421	0.3573	0.3619
Autos	0.4828	0.5278	0.5014	0.5030
Banks	0.6126	0.7740	0.7666	0.7819
Beer	0.3244	0.4148	0.3518	0.4336
BldMt	0.6949	0.7525	0.7492	0.7800
Books	0.6551	0.6889	0.6911	0.7022
Boxes	0.5549	0.5829	0.5683	0.5787
BusSv	0.8439	0.8463	0.8687	0.8736
Chems	0.6845	0.7452	0.7284	0.7461
Chips	0.6714	0.7498	0.7323	0.7461
Clths	0.5607	0.6190	0.5739	0.6291
Cnstr	0.6107	0.6315	0.6552	0.6728
Coal	0.1926	0.2126	0.2315	0.2342
Drugs	0.4666	0.4894	0.4995	0.5190
ElcEq	0.7351	0.7372	0.7394	0.7420
FabPr	0.4010	0.4245	0.5041	0.5174
Fin	0.7713	0.8129	0.7947	0.8209
Food	0.3541	0.4551	0.4041	0.4781
Fun	0.5861	0.6072	0.5952	0.5995
Gold	0.0488	0.0495	0.0556	0.0691
Guns	0.2483	0.3245	0.2727	0.3351
Hardw	0.5946	0.6669	0.6354	0.6541
Hlth	0.4058	0.4409	0.4351	0.4905
Hshld	0.4862	0.5547	0.5043	0.5819
Insur	0.5518	0.6612	0.6573	0.6653
LabEq	0.6806	0.7279	0.7477	0.7534
Mach	0.7400	0.7540	0.7728	0.7731
Meals	0.5816	0.6371	0.5919	0.6470
MedEq	0.5855	0.5958	0.5955	0.6045
Mines	0.4269	0.4589	0.4667	0.4726
Oil	0.3584	0.4556	0.4508	0.4619
Other	0.5770	0.5837	0.5810	0.5844
Paper	0.5991	0.6730	0.6381	0.6858
PerSv	0.5593	0.5848	0.5863	0.6197
RlEst	0.5268	0.6054	0.6735	0.6908
Rtail	0.6643	0.6866	0.6666	0.6889
Rubbr	0.6343	0.6453	0.6837	0.7044
Ships	0.4482	0.5046	0.4999	0.5369
Smoke	0.1821	0.2654	0.2312	0.2944
Soda	0.2449	0.3025	0.2734	0.3064
Softw	0.6270	0.7450	0.7454	0.7547
Steel	0.5857	0.6283	0.6405	0.6450
Telcm	0.5661	0.5804	0.5829	0.5917
Toys	0.4963	0.5102	0.5305	0.5509
Trans	0.6672	0.7088	0.6955	0.7210
Txtls	0.4311	0.5470	0.5863	0.6183
Util	0.2723	0.3614	0.3609	0.3805
Whlsl	0.7393	0.7523	0.7741	0.7974

# Output from previous execution

	Avg R-Squared
FF5	0.5918
AQR	0.5719
FF3	0.5679
CAPM	0.5226

# Output from previous execution

	CAPM		AQR		FF3		FF5
Premium	Cross-Sectional	Historical	Cross-Sectional	Historical	Cross-Sectional	Historical	Cross-Sectional	Historical
Factor
CMA	NaN	NaN	NaN	NaN	NaN	NaN	-0.0221	0.0283
HML	NaN	NaN	-0.0323	0.0260	-0.0210	0.0260	-0.0259	0.0260
MKT	0.0079	0.0876	0.0172	0.0876	0.0388	0.0876	0.0403	0.0876
RMW	NaN	NaN	0.0175	0.0440	NaN	NaN	0.0187	0.0440
SMB	NaN	NaN	NaN	NaN	-0.0396	0.0061	-0.0414	0.0061
UMD	NaN	NaN	0.0003	0.0603	NaN	NaN	NaN	NaN

# Output from previous execution

	Eta	Annualized Eta	R-Squared	MKT Lambda	Treynor Ratio	Annualized Treynor Ratio	MKT Annualized Lambda	MKT Historical Premium	MKT Annualized Historical Premium	TS MAE	...	RMW Annualized Historical Premium	UMD Annualized Historical Premium	SMB Lambda	SMB Annualized Lambda	SMB Historical Premium	SMB Annualized Historical Premium	CMA Lambda	CMA Annualized Lambda	CMA Historical Premium	CMA Annualized Historical Premium
CAPM Cross-Section Regression	0.0069	0.0832	0.0093	0.0007	11.5196	138.2352	0.0079	0.0073	0.0876	0.0017	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
AQR Cross-Section Regression	0.0063	0.0755	0.2066	0.0014	NaN	NaN	0.0172	0.0073	0.0876	0.0021	...	0.0440	0.0603	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
FF3 Cross-Section Regression	0.0052	0.0627	0.3504	0.0032	NaN	NaN	0.0388	0.0073	0.0876	0.0020	...	NaN	NaN	-0.0033	-0.0396	0.0005	0.0061	NaN	NaN	NaN	NaN
FF5 Cross-Section Regression	0.0050	0.0599	0.3765	0.0034	NaN	NaN	0.0403	0.0073	0.0876	0.0026	...	0.0440	NaN	-0.0035	-0.0414	0.0005	0.0061	-0.0018	-0.0221	0.0024	0.0283

4 rows × 33 columns

# Output from previous execution

	TS Annualized MAE	CS Annualized MAE
CAPM	0.0210	0.0152
AQR	0.0246	0.0136
FF3	0.0244	0.0120
FF5	0.0314	0.0120