Risk and Return of Equity, the Capital Asset Pricing Model, and Stock Selection for Efficient Portfolio Construction

Abstract

Individual investors must be compensated for bearing risk. It seems intuitive to the reader that there should be a direct linkage between the risk of a security and its rate of return. We are interested in securing the maximum return for a given level of risk, or the minimum risk for a given level of return. The concept of such risk-return analysis is the efficient frontier of Harry Markowitz (1952, 1959). If an investor can invest in a government security, which is backed by the taxing power of the federal government, then that government security is relatively risk-free. The 90-day Treasury bill rate is used as the basic risk-free rate. Supposedly, the taxing power of the federal government eliminates default risk of government debt issues. A liquidity premium is paid for longer-term maturities, due to the increasing level of interest rate risk. Investors are paid interest payments, as determined by the bond’s coupon rate, and may earn market price appreciation on longer bonds if market rates fall or losses if market rates rise. During the period from 1928 to 2017, Treasury bills returned 3.44%, longer-term (10-year Treasury) government bonds earned 5.15%, and corporate stocks, as measured by the stock of the S&P 500 index, earned 11.53% annually, as measured by the mean annual return. The annualized standard deviations are 3.06%, 7.72%, and 19.66%, respectively, for Treasury bills, Treasury bonds, and S&P stocks. The risk-return trade-off has been relevant for the 1928–2017 period. The correlation coefficient between annual returns for Treasury bills and the S&P 500 stock returns were −0.030 for the 1928–2017 time period. This was essentially no correlation between Treasury bills and large stocks, as measured by the S&P 500 stock. The correlation coefficient between annual returns for Treasury bonds and the S&P 500 stock returns was 0.30 for the 1928–2017 time period. Why do corporate stocks offer investors higher returns for stocks than bonds?

Appendices

Appendices

14.1.1 Appendix A: The Three-Asset Case

Let us now examine a three-asset portfolio construction process using IBM, Dominion Resources, and BA securities for the 2012–2016 time period.

$$ \mathrm{E}\left(\mathrm{Rp}\right)=\sum \limits^{\underset{N}{i=1}}{\mathrm{x}}_{\mathrm{i}}\ \mathrm{E}\left({\mathrm{R}}_{\mathrm{i}}\right) $$

(14.17)

$$ {\sigma}_p^2\sum \limits_{i=1}^N\sum \limits^{\overset{j=1}{N}}{\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}{\upsigma}_{\mathrm{i}\mathrm{j}} $$

(14.18)

$$ \mathrm{E}\left(\mathrm{Rp}\right)={\mathrm{x}}_1\mathrm{E}\left({\mathrm{R}}_1\right)+{\mathrm{x}}_2\mathrm{E}\left({\mathrm{R}}_2\right)+{\mathrm{x}}_3\mathrm{E}\left({\mathrm{R}}_3\right) $$

let x₃ = 1 − x₁ − x₂

$$ {\displaystyle \begin{array}{l}\mathrm{E}\left(\mathrm{Rp}\right)={\mathrm{x}}_1\mathrm{E}\left({\mathrm{R}}_1\right)+{\mathrm{x}}_2\mathrm{E}\left({\mathrm{R}}_2\right)+\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right)\mathrm{E}\left({\mathrm{R}}_3\right)\\ {}{\sigma}_p^2={{\mathrm{x}}_1}^2{\sigma_1}^2+{{\mathrm{x}}_2}^2{\sigma_2}^2+{\sigma_3}^2{{\mathrm{x}}_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{12}+2{\mathrm{x}}_1{\mathrm{x}}_3{\sigma}_{23}+2{\mathrm{x}}_2{\mathrm{x}}_3{\sigma}_{23}\\ {}\kern1.5em ={{\mathrm{x}}_1}^2{\upsigma_1}^2+{{\mathrm{x}}_2}^2{\upsigma_2}^2+{\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right)}^2{\upsigma_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{12}+2{\mathrm{x}}_1\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right){\upsigma}_{13}\\ {}\kern2em +2{\mathrm{x}}_2\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right){\sigma}_{23}\\ {}\kern1.5em ={{\mathrm{x}}_1}^2{\sigma_1}^2+{{\mathrm{x}}_2}^2{\sigma_2}^2+\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right)\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right){\sigma_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{12}+2{\mathrm{x}}_1{\sigma}_{13}\\ {}\kern2em -2{{\mathrm{x}}_1}^2{\sigma}_{13}-2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{13}+2{\mathrm{x}}_2{\sigma}_{23}-2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{23}-2{{\mathrm{x}}_2}^2{\sigma}_{23}\\ {}\kern1.5em ={{\mathrm{x}}_1}^2{\sigma_1}^2+{{\mathrm{x}}_2}^2{\sigma_2}^2+\left(1-2{\mathrm{x}}_1-2{\mathrm{x}}_2+2{\mathrm{x}}_1{\mathrm{x}}_2+{{\mathrm{x}}_1}^2+{{\mathrm{x}}_2}^2\right){\sigma_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{12}\\ {}\kern2em +2{\mathrm{x}}_1{\sigma}_{13}-2{{\mathrm{x}}_1}^2{\sigma}_{13}-2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{13}+2{\mathrm{x}}_2{\sigma}_{23}-2{\mathrm{x}}_1{\mathrm{x}}_2{\sigma}_{23}-2{{\mathrm{x}}_2}^2{\sigma}_{23}\end{array}} $$

(14.19)

$$ \frac{\partial {\sigma}_p^2}{\partial {x}_1}=2{\mathrm{x}}_1\left({\sigma_1}^2+{\sigma_3}^2-2{\sigma}_{13}\right)+{\mathrm{x}}_2\left(2{\sigma_3}^2+2{\sigma}_{12}-2{\sigma}_{13}-2{\sigma}_{23}\right)-2{\sigma_3}^2+2{\sigma}_{13}=0 $$

$$ \frac{\partial {\sigma}_p^2}{\partial {x}_2}=2{\mathrm{x}}_2\left({\sigma_2}^2+{\sigma_3}^2-2{\sigma}_{23}\right)+{\mathrm{x}}_1\left(2{\sigma_3}^2+2{\sigma}_{12}-2{\sigma}_{13}-2{\sigma}_{23}\right)-2{\sigma_3}^2+2{\sigma}_{23}=0 $$

Let’s assume

Asset	Security
1	IBM
2 3	D BA

The optimal portfolio weights involve selling IBM short and going long on DuPont and Dominion. How does the portfolio using optimal weights compare to an equally weighted portfolio of the three assets? (Table 14.9)

Table 14.9 Appendix. CRP Stock data

The equally weighted portfolio has an expected return of 14.9% and a 13.42% standard deviation.

$$ {\displaystyle \begin{array}{l}\mathrm{E}\left({\mathrm{R}}_{\mathrm{p}}\right)=.333\left(.110+.095+.242\right)=.149\\ {}{\sigma}_p^2={(.333)}^2{(.0167)}^2+{(.333)}^2(.0407)+{(.333)}^2(.0799)+2(.333)(.333)\left(\hbox{--} .0103\right)+2\\ {}(.333)(.333)\left(\hbox{--} .0041\right)+2(.333)(.333)(.0303)\end{array}} $$

$$ {\sigma}_p^2=.0019+.0045+.0081+\left(\hbox{--} .0023\right)+\left(\hbox{--} .0009\right)+.0067=.0180 $$

$$ {\sigma}_{\mathrm{p}}=.1342 $$

The optimally weighted portfolio has an expected return of

$$ {\displaystyle \begin{array}{l}\mathrm{E}\left({\mathrm{R}}_{\mathrm{p}}\right)=.795(.110)+.2517(.095)+\left(\hbox{--} .0467\right)(.242)\\ {}=.0875+.0244+\left(\hbox{--} .0113\right)\\ {}=.1006\end{array}} $$

$$ {\displaystyle \begin{array}{l}{\sigma}_p^2={(.795)}^2(.0167)+{(.2517)}^2(.0407)+{\left(\hbox{--} .0467\right)}^2(.0799)+2\ (.795)(.2517)\left(\hbox{--} .0103\right)+\\ {}2(.795)\left(\hbox{--} .0467\right)\left(\hbox{--} .0041\right)+2(.2517)\left(\hbox{--} .0467\right)(.0303)\\ {}=.0106+.0026+.0002+\left(\hbox{--} .0041\right)+.0003+\left(\hbox{--} .0007\right)\\ {}{\sigma}_p^2=.0089\\ {}{\upsigma}_{\mathrm{p}}=.0943\end{array}} $$

The optimally weighted portfolio has an expected return of 10.06% and a standard deviation of 9.43%. Portfolio risk can be significantly reduced through diversification.

14.1.2 Appendix B: ICs

• Time Period: 1/31/01 to 5/31/2021

• Various U.S., non-US, and global universes

Universe	Factor	Mean	Std	T-stat
ACW	CTEF	0.040	0.080	7.67
ACW	FCF_YLD	0.024	0.058	6.44
ACW	BR2	0.029	0.080	5.63
ACW	BR1	0.026	0.074	5.28
ACW	FY2RV3	0.033	0.100	5.00
ACW	FY1RV3	0.025	0.091	4.28
ACW	ROIC	0.022	0.083	4.13
ACW	ROE	0.022	0.080	4.11
ACW	AX_Profitability	0.019	0.071	3.94
ACW	ROA	0.020	0.090	3.32
ACW	IBES_EPS_5Y_GTOS	0.015	0.069	3.25
ACW	CP	0.018	0.087	3.14
ACW	DP	0.019	0.094	3.07
ACW	FEP1	0.023	0.114	3.01
ACW	IBES_EXP_DY	0.019	0.099	3.00
ACW	OCFYLD	0.016	0.081	2.85
ACW	GROSS_MARGIN	0.015	0.078	2.84
ACW	AX_MidTermMomentum	0.029	0.155	2.84
ACW	EP	0.017	0.093	2.82
ACW	RCP	0.013	0.071	2.73
ACW	FEP2	0.024	0.133	2.69
ACW	IBES_FY1_EPS_G	0.018	0.103	2.68
ACW	IBES_EPS_5Y_GRO	0.013	0.076	2.66
ACW	IBES_FY1_DPS_G	0.015	0.089	2.63
ACW	EBITDA_EV	0.016	0.095	2.59
ACW	Sales_Assets	0.010	0.067	2.31
ACW	PM121	0.023	0.164	2.10
ACW	YOY_EPS_G	0.008	0.069	1.78
ACW	PM91	0.017	0.158	1.66
ACW	REP	0.007	0.066	1.59
ACW	SALES_EV	0.008	0.080	1.50
ACW	AX_Growth	0.007	0.075	1.48
ACW	ES	0.008	0.080	1.45
ACW	STDEV	0.015	0.186	1.25
ACW	ALTMANZ	0.007	0.088	1.22
ACW	PM61	0.011	0.145	1.15
ACW	PM31	0.009	0.119	1.11
ACW	CURRENT-RATIO	0.003	0.064	0.78
ACW	AX_Value	0.005	0.114	0.74
ACW	SP	0.005	0.105	0.72
ACW	IBES_PTG_RET	0.005	0.136	0.61
ACW	AX_Size	0.001	0.089	0.12
ACW	AX_Liquidity	0.000	0.118	−0.05
ACW	YOY_SALES_G	−0.001	0.086	−0.19
ACW	IBES_EPS_LTG	−0.002	0.079	−0.35
ACW	AX_DividendYield	−0.006	0.103	−0.36
ACW	AX_EarningsYield	−0.009	0.137	−0.39
ACW	BP	−0.004	0.118	−0.48
ACW	RSP	−0.008	0.118	−1.01
ACW	Debt_Equity	−0.005	0.065	−1.16
ACW	AX_ShortTermMomentum	−0.010	0.110	−1.33
ACW	RBP	−0.013	0.120	−1.66
ACW	AX_ExRateSensitivity	−0.006	0.051	−1.79
ACW	AX_Volatility	−0.022	0.173	−1.91
ACW	IBES_FY1_EPS_DISP	−0.018	0.136	−2.04
ACW	Percent_Accrual	−0.007	0.051	−2.16
ACW	CHG_DEBT	−0.006	0.044	−2.21
ACW	AX_Leverage	−0.008	0.048	−2.42
ACW	CAPEX_DEP	−0.014	0.082	−2.57
ACW	CHG_SHARES	−0.015	0.073	−3.25
ACW	IBES_REC_MEAN	−0.016	0.071	−3.41
ACW	IBES_REC_MEAN_3M	−0.015	0.043	−5.13
ACWXUS	CTEF	0.043	0.080	8.14
ACWXUS	FCF_YLD	0.024	0.052	6.92
ACWXUS	BR2	0.032	0.078	6.33
ACWXUS	BR1	0.028	0.073	5.94
ACWXUS	FY2RV3	0.033	0.100	5.09
ACWXUS	IBES_EXP_DY	0.030	0.095	4.78
ACWXUS	DP	0.027	0.088	4.59
ACWXUS	FY1RV3	0.027	0.091	4.48
ACWXUS	ROIC	0.024	0.089	4.17
ACWXUS	ROE	0.023	0.087	4.05
ACWXUS	AX_Profitability	0.021	0.077	3.98
ACWXUS	CP	0.020	0.081	3.72
ACWXUS	EP	0.020	0.093	3.36
ACWXUS	FEP1	0.025	0.117	3.33
ACWXUS	EBITDA_EV	0.019	0.089	3.30
ACWXUS	ROA	0.020	0.094	3.26
ACWXUS	AX_MidTermMomentum	0.031	0.151	3.10
ACWXUS	RCP	0.014	0.067	3.08
ACWXUS	FEP2	0.027	0.137	2.97
ACWXUS	IBES_EPS_5Y_GTOS	0.013	0.069	2.95
ACWXUS	GROSS_MARGIN	0.013	0.076	2.65
ACWXUS	IBES_FY1_EPS_G	0.017	0.102	2.60
ACWXUS	IBES_EPS_5Y_GRO	0.012	0.077	2.47
ACWXUS	PM121	0.024	0.159	2.31
ACWXUS	IBES_FY1_DPS_G	0.014	0.090	2.30
ACWXUS	YOY_EPS_G	0.010	0.072	2.14
ACWXUS	REP	0.009	0.069	1.99
ACWXUS	AX_Growth	0.010	0.077	1.90
ACWXUS	Sales_Assets	0.008	0.069	1.82
ACWXUS	PM91	0.018	0.155	1.74
ACWXUS	SALES_EV	0.008	0.078	1.58
ACWXUS	STDEV	0.017	0.180	1.42
ACWXUS	ES	0.008	0.083	1.40
ACWXUS	ALTMANZ	0.007	0.088	1.26
ACWXUS	PM61	0.011	0.146	1.14
ACWXUS	PM31	0.008	0.120	0.96
ACWXUS	AX_Value	0.007	0.118	0.94
ACWXUS	SP	0.005	0.103	0.71
ACWXUS	IBES_PTG_RET	0.005	0.130	0.63
ACWXUS	CURRENT_RATIO	0.002	0.060	0.47
ACWXUS	AX_Size	0.002	0.081	0.31
ACWXUS	YOY_SALES_G	0.000	0.089	0.07
ACWXUS	BP	0.000	0.115	−0.05
ACWXUS	IBES_EPS_LTG	−0.001	0.074	−0.12
ACWXUS	AX_DividendYield	−0.005	0.093	−0.28
ACWXUS	AX_EarningsYield	−0.016	0.134	−0.68
ACWXUS	AX_ExRateSensitivity	−0.003	0.055	−0.74
ACWXUS	AX_ShortTermMomentum	−0.008	0.112	−1.06
ACWXUS	RSP	−0.009	0.117	−1.13
ACWXUS	AX_Liquidity	−0.010	0.107	−1.36
ACWXUS	Debt_Equity	−0.007	0.073	−1.39
ACWXUS	RBP	−0.012	0.123	−1.44
ACWXUS	AX_Volatility	−0.017	0.162	−1.61
ACWXUS	CAPEX_DEP	−0.010	0.075	−2.00
ACWXUS	Percent_Accrual	−0.007	0.056	−2.03
ACWXUS	IBES_FY1_EPS_DISP	−0.017	0.113	−2.26
ACWXUS	CHG_SHARES	−0.010	0.070	−2.28
ACWXUS	AX_Leverage	−0.009	0.050	−2.64
ACWXUS	CHG_DEBT	−0.010	0.052	−2.89
ACWXUS	IBES_REC_MEAN	−0.016	0.073	−3.40
ACWXUS	IBES_REC_MEAN_3M	−0.018	0.045	−6.09
EAFE	CTEF	0.035	0.110	4.82
EAFE	FCF_YLD	0.017	0.061	4.17
EAFE	BR2	0.025	0.102	3.75
EAFE	FY2RV3	0.031	0.142	3.36
EAFE	RCP	0.016	0.080	3.04
EAFE	AX_Profitability	0.021	0.106	2.90
EAFE	BR1	0.018	0.097	2.82
EAFE	IBES_EXP_DY	0.027	0.147	2.79
EAFE	DP	0.024	0.133	2.73
EAFE	ROIC	0.021	0.116	2.73
EAFE	FY1RV3	0.021	0.126	2.60
EAFE	ROE	0.019	0.121	2.44
EAFE	ROA	0.019	0.124	2.34
EAFE	CP	0.014	0.097	2.22
EAFE	FEP1	0.021	0.150	2.16
EAFE	IBES_EPS_5Y_GTOS	0.013	0.090	2.12
EAFE	EP	0.016	0.118	2.01
EAFE	FEP2	0.022	0.176	1.92
EAFE	GROSS_MARGIN	0.011	0.097	1.73
EAFE	IBES_EPS_5Y_GRO	0.010	0.090	1.65
EAFE	AX_MidTermMomentum	0.020	0.181	1.65
EAFE	EBITDA_EV	0.011	0.111	1.56
EAFE	AX_Growth	0.009	0.095	1.42
EAFE	ES	0.010	0.108	1.41
EAFE	IBES_FY1_DPS_G	0.009	0.114	1.14
EAFE	PM121	0.013	0.189	1.07
EAFE	AX_Value	0.010	0.140	1.06
EAFE	PM31	0.011	0.157	1.03
EAFE	IBES_FY1_EPS_G	0.008	0.122	1.00
EAFE	PM91	0.012	0.181	0.99
EAFE	Sales_Assets	0.006	0.092	0.96
EAFE	STDEV	0.012	0.204	0.93
EAFE	REP	0.005	0.083	0.89
EAFE	YOY_EPS_G	0.004	0.087	0.77
EAFE	SALES_EV	0.005	0.106	0.77
EAFE	PM61	0.007	0.175	0.63
EAFE	IBES_PTG_RET	0.005	0.151	0.48
EAFE	SP	0.002	0.130	0.23
EAFE	CURRENT_R	0.001	0.083	0.20
EAFE	BP	0.002	0.142	0.17
EAFE	ALTMANZ	0.001	0.099	0.12
EAFE	RSP	0.000	0.130	0.03
EAFE	Percent_Accrual	0.000	0.070	−0.10
EAFE	RBP	−0.002	0.134	−0.23
EAFE	AX_EarningsYield	−0.008	0.168	−0.28
EAFE	YOY_SALES_G	−0.002	0.095	−0.36
EAFE	AX_DividendYield	−0.012	0.140	−0.47
EAFE	AX_ExRateSensitivity	−0.002	0.072	−0.47
EAFE	IBES_FY1_EPS_DISP	−0.005	0.123	−0.64
EAFE	CHG_SHARES	−0.005	0.092	−0.81
EAFE	AX_Size	−0.005	0.101	−0.83
EAFE	IBES_EPS_LTG	−0.005	0.081	−0.92
EAFE	Debt_Equity	−0.007	0.094	−1.14
EAFE	AX_Liquidity	−0.010	0.126	−1.19
EAFE	AX_ShortTermMomentum	−0.012	0.136	−1.22
EAFE	CHG_DEBT	−0.004	0.054	−1.23
EAFE	AX_Volatility	−0.017	0.192	−1.36
EAFE	AX_Leverage	−0.006	0.064	−1.41
EAFE	CAPEX_DEP	−0.010	0.070	−2.09
EAFE	IBES_REC_MEAN	−0.011	0.078	−2.19
EAFE	IBES_REC_MEAN_3M	−0.011	0.058	−3.00
Europe	CTEF	0.037	0.123	4.60
Europe	BR2	0.035	0.120	4.38
Europe	BR1	0.029	0.112	3.94
Europe	FY2RV3	0.034	0.151	3.43
Europe	FY1RV3	0.028	0.133	3.27
Europe	ROIC	0.028	0.151	2.85
Europe	AX_Profitability	0.027	0.148	2.61
Europe	ROE	0.022	0.143	2.36
Europe	ROA	0.026	0.167	2.36
Europe	AX_MidTermMomentum	0.033	0.215	2.34
Europe	PM121	0.033	0.215	2.33
Europe	ES	0.024	0.149	2.30
Europe	FCF_YLD	0.011	0.073	2.28
Europe	IBES_EPS_5Y_GRO	0.018	0.121	2.23
Europe	IBES_FY1_DPS_G	0.021	0.145	2.21
Europe	PM91	0.027	0.207	1.99
Europe	IBES_EPS_5Y_GTOS	0.016	0.127	1.87
Europe	Sales_Assets	0.016	0.134	1.83
Europe	AX_Growth	0.014	0.119	1.83
Europe	SALES_EV	0.013	0.111	1.81
Europe	IBES_FY1_EPS_G	0.016	0.156	1.57
Europe	YOY_EPS_G	0.010	0.102	1.57
Europe	Percent_Accrual	0.006	0.063	1.41
Europe	EP	0.011	0.123	1.36
Europe	CURRENT_R	0.009	0.106	1.36
Europe	GROSS_MARGIN	0.010	0.115	1.29
Europe	PM31	0.013	0.166	1.22
Europe	PM61	0.014	0.185	1.18
Europe	STDEV	0.017	0.234	1.13
Europe	ALTMANZ	0.008	0.129	0.97
Europe	EBITDA_EV	0.008	0.135	0.94
Europe	FEP1	0.010	0.167	0.94
Europe	REP	0.006	0.097	0.87
Europe	CP	0.005	0.124	0.64
Europe	IBES_EXP_DY	0.004	0.136	0.50
Europe	DP	0.004	0.132	0.46
Europe	FEP2	0.005	0.194	0.42
Europe	RCP	0.001	0.096	0.17
Europe	IBES_PTG_RET	0.002	0.206	0.14
Europe	AX_Value	0.001	0.188	0.05
Europe	SP	−0.001	0.161	−0.07
Europe	YOY_SALES_G	−0.002	0.104	−0.24
Europe	AX_EarningsYield	−0.017	0.208	−0.46
Europe	BP	−0.007	0.194	−0.58
Europe	IBES_EPS_LTG	−0.004	0.098	−0.69
Europe	AX_ExRateSensitivity	−0.006	0.117	−0.81
Europe	AX_ShortTermMomentum	−0.008	0.138	−0.82
Europe	AX_Size	−0.008	0.104	−1.22
Europe	AX_Liquidity	−0.011	0.130	−1.28
Europe	AX_Volatility	−0.019	0.216	−1.36
Europe	AX_DividendYield	−0.044	0.170	−1.47
Europe	IBES_FY1_EPS_DISP	−0.017	0.171	−1.54
Europe	RBP	−0.020	0.156	−1.94
Europe	RSP	−0.021	0.159	−2.02
Europe	IBES_REC_MEAN_3M	−0.010	0.071	−2.14
Europe	IBES_REC_MEAN	−0.013	0.091	−2.18
Europe	AX_Leverage	−0.011	0.078	−2.19
Europe	CHG_DEBT	−0.010	0.066	−2.32
Europe	CAPEX_DEP	−0.013	0.078	−2.60
Europe	Debt_Equity	−0.019	0.113	−2.62
Europe	CHG_SHARES	−0.014	0.066	−3.18
EM	CTEF	0.050	0.081	9.36
EM	BR1	0.039	0.075	7.90
EM	MQ	0.058	0.121	7.29
EM	FCF_YLD	0.030	0.064	7.14
EM	BR2	0.037	0.080	7.10
EM	FY2RV3	0.036	0.090	6.16
EM	FY1RV3	0.033	0.085	5.85
EM	ROE	0.030	0.092	4.95
EM	ROIC	0.029	0.093	4.71
EM	IBES_EXP_DY	0.030	0.101	4.59
EM	DP	0.029	0.097	4.57
EM	EP	0.027	0.093	4.44
EM	CP	0.025	0.085	4.38
EM	RCP	0.018	0.068	4.05
EM	FEP1	0.030	0.114	4.03
EM	IBES_FY1_EPS_G	0.026	0.101	3.98
EM	FEP2	0.033	0.128	3.97
EM	AX_MidTermMomentum	0.035	0.143	3.73
EM	AX_Profitability	0.022	0.085	3.68
EM	YOY_EPS_G	0.018	0.075	3.60
EM	REP	0.018	0.079	3.52
EM	IBES_EPS_5Y_GTOS	0.017	0.076	3.43
EM	ROA	0.020	0.093	3.37
EM	IBES_FY1_DPS_G	0.018	0.084	3.20
EM	EBITDA_EV	0.021	0.100	3.14
EM	PM121	0.030	0.154	2.96
EM	IBES_EPS_5Y_GRO	0.014	0.076	2.85
EM	GROSS_MARGIN	0.015	0.095	2.46
EM	ALTMANZ	0.017	0.113	2.33
EM	PM91	0.022	0.152	2.20
EM	AX_Growth	0.011	0.089	1.90
EM	Sales_Assets	0.009	0.071	1.85
EM	AX_Size	0.010	0.086	1.84
EM	STDEV	0.017	0.170	1.53
EM	PM61	0.013	0.139	1.48
EM	SALES_EV	0.007	0.085	1.32
EM	IBES_PTG_RET	0.011	0.130	1.25
EM	ES	0.007	0.084	1.20
EM	SP	0.009	0.112	1.19
EM	IBES_EPS_LTG	0.005	0.085	0.87
EM	YOY_SALES_G	0.004	0.086	0.78
EM	AX_Value	0.005	0.124	0.61
EM	CURRENT_RATIO	0.002	0.070	0.51
EM	AX_DividendYield	0.000	0.108	−0.01
EM	AX_ExRateSensitivity	−0.001	0.068	−0.13
EM	PM31	−0.002	0.120	−0.24
EM	BP	−0.003	0.128	−0.32
EM	RSP	−0.003	0.114	−0.38
EM	RBP	−0.007	0.119	−0.88
EM	AX_EarningsYield	−0.021	0.140	−0.88
EM	Debt_Equity	−0.005	0.077	−1.00
EM	AX_Liquidity	−0.010	0.145	−1.02
EM	AX_ShortTermMomentum	−0.010	0.110	−1.33
EM	AX_Volatility	−0.015	0.154	−1.50
EM	CAPEX_DEP	−0.007	0.072	−1.55
EM	AX_Leverage	−0.012	0.063	−2.84
EM	CHG_DEBT	−0.012	0.057	−3.15
EM	CHG_SHARES	−0.014	0.062	−3.37
EM	IBES_FY1_EPS_DISP	−0.026	0.111	−3.63
EM	Percent_Accrual	−0.013	0.056	−3.64
EM	IBES_REC_MEAN	−0.028	0.080	−5.30
EM	IBES_REC_MEAN_3M	−0.027	0.055	−7.58
Japan	RCP	0.028	0.106	4.05
Japan	FCF_YLD	0.021	0.088	3.71
Japan	IBES_EXP_DY	0.034	0.143	3.63
Japan	MQ	0.035	0.155	3.47
Japan	DP	0.030	0.146	3.13
Japan	CP	0.021	0.123	2.62
Japan	SALES_EV	0.022	0.129	2.56
Japan	BP	0.029	0.179	2.45
Japan	IBES_PTG_RET	0.027	0.179	2.27
Japan	SP	0.023	0.159	2.17
Japan	AX_Value	0.025	0.188	2.02
Japan	CTEF	0.019	0.142	2.01
Japan	BR2	0.017	0.131	1.95
Japan	RBP	0.022	0.177	1.89
Japan	RSP	0.021	0.172	1.83
Japan	FEP2	0.023	0.196	1.83
Japan	EBITDA_EV	0.018	0.153	1.77
Japan	AX_Profitability	0.015	0.136	1.63
Japan	FEP1	0.017	0.177	1.45
Japan	FY2RV3	0.015	0.161	1.38
Japan	CURRENT_R	0.010	0.120	1.30
Japan	Sales_Assets	0.010	0.120	1.29
Japan	ES	0.011	0.140	1.13
Japan	BR1	0.009	0.129	1.03
Japan	EP	0.008	0.164	0.78
Japan	ROA	0.008	0.162	0.75
Japan	AX_ExRateSensitivity	0.005	0.121	0.67
Japan	IBES_EPS_5Y_GRO	0.003	0.108	0.46
Japan	IBES_EPS_5Y_GTOS	0.003	0.108	0.40
Japan	IBES_FY1_EPS_DISP	0.004	0.148	0.37
Japan	ALTMANZ	0.003	0.148	0.35
Japan	FY1RV3	0.003	0.155	0.33
Japan	GROSS_MARGIN	0.003	0.144	0.33
Japan	STDEV	0.005	0.235	0.33
Japan	IBES_FY1_EPS_G	0.003	0.153	0.30
Japan	ROIC	0.003	0.151	0.28
Japan	REP	0.002	0.135	0.28
Japan	IBES_EPS_LTG	0.002	0.113	0.20
Japan	AX_MidTermMomentum	0.001	0.212	0.08
Japan	AX_DividendYield	0.002	0.207	0.05
Japan	AX_Growth	−0.001	0.130	−0.08
Japan	PM31	−0.003	0.167	−0.25
Japan	AX_EarningsYield	−0.011	0.196	−0.30
Japan	PM121	−0.005	0.211	−0.35
Japan	ROE	−0.003	0.137	−0.36
Japan	CHG_DEBT	−0.002	0.075	−0.36
Japan	AX_Volatility	−0.007	0.231	−0.44
Japan	AX_Liquidity	−0.006	0.200	−0.44
Japan	PM91	−0.008	0.200	−0.60
Japan	PM61	−0.008	0.188	−0.62
Japan	IBES_FY1_DPS_G	−0.006	0.138	−0.67
Japan	IBES_REC_MEAN	−0.005	0.115	−0.69
Japan	IBES_REC_MEAN_3M	−0.005	0.083	−0.83
Japan	CHG_SHARES	−0.005	0.084	−0.98
Japan	YOY_EPS_G	−0.008	0.118	−0.99
Japan	Debt_Equity	−0.013	0.143	−1.34
Japan	AX_Leverage	−0.013	0.125	−1.55
Japan	AX_Size	−0.011	0.112	−1.55
Japan	YOY_SALES_G	−0.015	0.129	−1.78
Japan	Percent_Accrual	−0.011	0.095	−1.79
Japan	CAPEX_DEP	−0.010	0.077	−2.02
Japan	AX_ShortTermMomentum	−0.033	0.175	−2.63
R1000	CTEF	0.030	0.117	3.88
R1000	FY2RV3	0.025	0.125	2.99
R1000	FCF_YLD	0.018	0.096	2.83
R1000	BR2	0.018	0.105	2.67
R1000	FY1RV3	0.019	0.107	2.64
R1000	Sales_Assets	0.018	0.106	2.53
R1000	FEP1	0.021	0.136	2.33
R1000	FEP2	0.022	0.146	2.28
R1000	BR1	0.014	0.097	2.25
R1000	DJ_SENT	0.011	0.082	2.00
R1000	ROIC_QTR	0.014	0.108	1.94
R1000	SALES_EV	0.015	0.120	1.91
R1000	IBES_EPS_5Y_GRO	0.013	0.108	1.90
R1000	ROE	0.012	0.105	1.79
R1000	IBES_EPS_5Y_GTOS	0.012	0.106	1.78
R1000	ROIC	0.013	0.110	1.76
R1000	AX_MidTermMomentum	0.021	0.192	1.67
R1000	PR_SENT	0.008	0.070	1.65
R1000	IBES_FY1_EPS_G	0.013	0.124	1.65
R1000	ROA	0.012	0.114	1.60
R1000	IBES_FY1_DPS_G	0.010	0.101	1.43
R1000	CP	0.012	0.131	1.40
R1000	SP	0.012	0.140	1.31
R1000	GROSS_MARGIN	0.009	0.104	1.26
R1000	PM91	0.013	0.184	1.11
R1000	RCP	0.008	0.106	1.09
R1000	OCFYLD	0.008	0.113	1.08
R1000	EP	0.008	0.122	1.06
R1000	EBITDA_EV	0.009	0.141	1.02
R1000	PM121	0.013	0.191	1.00
R1000	AX_Growth	0.007	0.104	0.99
R1000	ES	0.007	0.101	0.97
R1000	CHG_DEBT	0.003	0.051	0.81
R1000	REP	0.004	0.090	0.74
R1000	PM61	0.008	0.168	0.74
R1000	IBES_PTG_RET	0.008	0.177	0.69
R1000	PM31	0.007	0.146	0.69
R1000	ALTMANZ	0.005	0.115	0.67
R1000	STDEV	0.009	0.225	0.58
R1000	PMUS	0.008	0.188	0.58
R1000	DP	0.004	0.150	0.38
R1000	IBES_EXP_DY	0.001	0.149	0.12
R1000	Debt_Equity	0.001	0.091	0.10
R1000	AX_Size	0.001	0.112	0.08
R1000	AX_Leverage	0.000	0.093	0.06
R1000	IBES_REC_MEAN_3M	0.000	0.056	−0.01
R1000	CURRENT_RATIO	−0.001	0.110	−0.08
R1000	IBES_EPS_LTG	−0.002	0.135	−0.19
R1000	YOY_EPS_G	−0.001	0.080	−0.20
R1000	AX_DividendYield	−0.009	0.179	−0.28
R1000	AX_EarningsYield	−0.013	0.185	−0.39
R1000	RSP	−0.005	0.156	−0.49
R1000	AX_Value	−0.005	0.138	−0.58
R1000	BP	−0.005	0.133	−0.60
R1000	YOY_SALES_G	−0.005	0.107	−0.67
R1000	AX_Volatility	−0.015	0.215	−1.03
R1000	AX_MidCap	−0.018	0.080	−1.09
R1000	RBP	−0.011	0.135	−1.20
R1000	AX_Liquidity	−0.012	0.150	−1.20
R1000	CAPEX_DEP	−0.007	0.082	−1.26
R1000	AX_ShortTermMomentum	−0.013	0.142	−1.33
R1000	Percent_Accrual	−0.007	0.072	−1.46
R1000	IBES_REC_MEAN	−0.009	0.096	−1.48
R1000	IBES_FY1_EPS_DISP	−0.017	0.134	−1.90
R1000	AX_ExRateSensitivity	−0.016	0.116	−2.06
R1000	CHG_SHARES	−0.016	0.090	−2.78
R2000	CTEF	0.042	0.086	7.51
R2000	OCFYLD	0.033	0.072	7.02
R2000	FCF_YLD	0.033	0.080	6.37
R2000	FEP2	0.045	0.117	5.84
R2000	FEP1	0.046	0.121	5.82
R2000	DJ_SENT	0.022	0.056	5.57
R2000	AX_Size	0.031	0.091	5.21
R2000	ROA	0.035	0.104	5.12
R2000	BR2	0.022	0.066	5.04
R2000	ROIC_QTR	0.034	0.098	4.99
R2000	RCP	0.028	0.087	4.96
R2000	FY2RV3	0.025	0.077	4.94
R2000	BR1	0.021	0.065	4.86
R2000	ROIC	0.034	0.109	4.82
R2000	REP	0.032	0.104	4.77
R2000	ROE	0.034	0.111	4.66
R2000	Sales_Assets	0.025	0.083	4.58
R2000	EP	0.034	0.117	4.50
R2000	AX_Growth	0.021	0.074	4.38
R2000	CP	0.031	0.112	4.20
R2000	FY1RV3	0.019	0.068	4.19
R2000	STDEV	0.046	0.169	4.12
R2000	IBES_FY1_EPS_G	0.020	0.074	4.07
R2000	SALES_EV	0.024	0.096	3.87
R2000	EBITDA_EV	0.032	0.127	3.83
R2000	PR_SENT	0.011	0.042	3.75
R2000	IBES_EPS_5Y_GRO	0.017	0.068	3.73
R2000	IBES_EPS_5Y_GTOS	0.018	0.074	3.67
R2000	AX_MidTermMomentum	0.027	0.133	3.05
R2000	SP	0.024	0.120	3.00
R2000	IBES_FY1_DPS_G	0.020	0.104	2.81
R2000	PM91	0.021	0.124	2.56
R2000	PM61	0.018	0.113	2.41
R2000	GROSS_MARGIN	0.012	0.076	2.38
R2000	DP	0.019	0.122	2.31
R2000	PM121	0.018	0.132	2.10
R2000	PM31	0.011	0.097	1.72
R2000	ALTMANZ	0.008	0.076	1.70
R2000	PMUS	0.014	0.124	1.67
R2000	AX_MidCap	0.023	0.068	1.65
R2000	YOY_EPS_G	0.005	0.048	1.56
R2000	AX_EarningsYield	0.037	0.134	1.47
R2000	Debt_Equity	0.008	0.090	1.41
R2000	BP	0.009	0.105	1.31
R2000	ES	0.005	0.058	1.24
R2000	AX_Leverage	0.006	0.080	1.21
R2000	IBES_EXP_DY	0.008	0.140	0.86
R2000	AX_DividendYield	0.021	0.136	0.84
R2000	AX_Value	0.006	0.106	0.84
R2000	RBP	0.002	0.086	0.38
R2000	CHG_DEBT	0.001	0.040	0.23
R2000	YOY_SALES_G	0.000	0.069	0.04
R2000	RSP	−0.003	0.102	−0.52
R2000	IBES_PTG_RET	−0.006	0.133	−0.74
R2000	AX_ExRateSensitivity	−0.004	0.068	−0.89
R2000	IBES_EPS_LTG	−0.007	0.113	−0.95
R2000	AX_Liquidity	−0.010	0.112	−1.39
R2000	CAPEX_DEP	−0.006	0.052	−1.63
R2000	AX_ShortTermMomentum	−0.012	0.094	−1.77
R2000	IBES_REC_MEAN	−0.010	0.070	−2.26
R2000	CURRENT_RATIO	−0.014	0.088	−2.48
R2000	AX_Volatility	−0.038	0.169	−3.39
R2000	Percent_Accrual	−0.017	0.072	−3.61
R2000	LowPrice	−0.035	0.126	−4.26
R2000	CHG_SHARES	−0.023	0.079	−4.42
R2000	IBES_REC_MEAN_3M	−0.011	0.038	−4.58
R2000	IBES_FY1_EPS_DISP	−0.032	0.096	−5.14
WORLD	CTEF	0.031	0.094	4.76
WORLD	FCF_YLD	0.020	0.072	4.00
WORLD	BR2	0.025	0.094	3.77
WORLD	BR1	0.021	0.087	3.43
WORLD	ROIC	0.022	0.097	3.35
WORLD	FY2RV3	0.029	0.125	3.33
WORLD	IBES_EPS_5Y_GTOS	0.021	0.091	3.29
WORLD	ROE	0.021	0.095	3.25
WORLD	AX_Profitability	0.018	0.084	3.06
WORLD	ROA	0.020	0.106	2.75
WORLD	IBES_EPS_5Y_GRO	0.017	0.092	2.71
WORLD	FY1RV3	0.020	0.111	2.57
WORLD	GROSS_MARGIN	0.015	0.089	2.39
WORLD	Sales_Assets	0.012	0.083	1.99
WORLD	AX_MidTermMomentum	0.023	0.170	1.96
WORLD	IBES_FY1_DPS_G	0.013	0.101	1.86
WORLD	RCP	0.010	0.086	1.75
WORLD	ES	0.012	0.096	1.74
WORLD	OCFYLD	0.011	0.090	1.73
WORLD	FEP1	0.015	0.127	1.69
WORLD	IBES_FY1_EPS_G	0.013	0.120	1.62
WORLD	AX_Volatility	0.020	0.187	1.56
WORLD	AX_Liquidity	0.012	0.122	1.37
WORLD	FEP2	0.014	0.150	1.37
WORLD	ALTMANZ	0.009	0.098	1.36
WORLD	PM31	0.013	0.134	1.35
WORLD	AX_EarningsYield	0.010	0.107	1.32
WORLD	PM91	0.014	0.168	1.22
WORLD	EP	0.008	0.103	1.17
WORLD	PM121	0.013	0.175	1.09
WORLD	CURRENT_RATIO	0.006	0.076	1.06
WORLD	AX_DividendYield	0.007	0.107	0.98
WORLD	PM61	0.010	0.154	0.96
WORLD	IBES_EXP_DY	0.008	0.121	0.92
WORLD	DP	0.007	0.120	0.91
WORLD	CP	0.006	0.101	0.83
WORLD	STDEV	0.009	0.196	0.68
WORLD	AX_Growth	0.004	0.097	0.65
WORLD	SALES_EV	0.004	0.098	0.62
WORLD	AX_Size	0.003	0.086	0.57
WORLD	REP	0.003	0.073	0.56
WORLD	YOY_EPS_G	0.003	0.076	0.55
WORLD	EBITDA_EV	0.004	0.108	0.52
WORLD	IBES_PTG_RET	0.005	0.150	0.49
WORLD	CHG_DEBT	0.001	0.045	0.31
WORLD	IBES_EPS_LTG	0.001	0.094	0.14
WORLD	Percent_Accrual	−0.001	0.056	−0.14
WORLD	SP	−0.002	0.123	−0.23
WORLD	YOY_SALES_G	−0.003	0.091	−0.46
WORLD	RSP	−0.005	0.133	−0.60
WORLD	Debt_Equity	−0.004	0.074	−0.86
WORLD	AX_Value	−0.008	0.124	−0.90
WORLD	AX_ExRateSensitivity	−0.004	0.057	−0.92
WORLD	RBP	−0.010	0.128	−1.10
WORLD	BP	−0.013	0.134	−1.43
WORLD	CAPEX_DEP	−0.008	0.077	−1.51
WORLD	IBES_FY1_EPS_DISP	−0.018	0.142	−1.86
WORLD	IBES_REC_MEAN_3M	−0.006	0.048	−1.88
WORLD	AX_Leverage	−0.008	0.061	−1.91
WORLD	IBES_REC_MEAN	−0.014	0.079	−2.50
WORLD	CHG_SHARES	−0.016	0.081	−2.77

Variable	Definition
FY1RV3	Three-month revisions of 1-year-ahead I/B/E/S earnings per share revisions
FY2RV3	Three-month revisions of 2-year-ahead I/B/E/S earnings per share revisions
BR2	Two-tear-ahead I/B/E/S forecast breadth
BR1	One-tear-ahead I/B/E/S forecast breadth
FCF_YLD	Forecasted free cash flow yield
CTEFOCFROIC	Proprietary forecasted free cash flow
IBES_EPS_5Y_GTOS	I/B/E/S 3–5 year forecasted growth rate
MQ	McKinley capital management prioritary quant score
CURRENT_R	Current ratio
AX_Profitability	Axioma profitability factor return
ROIC	Return on invested capital
IBES_EPS_5Y_GRO	Five-year-ahead I/B/E/S forecast EPS growth rate
IBES_FY1_DPS_G	One-year-ahead I/B/E/S forecast dividend growth rate
STDEV	One year annualized standard deviation
ALTMANZ	Altman Z-score
AX_Size	Axioma size factor return
ES	Corporate exports
IBES_FY1_EPS_G	One-year-ahead I/B/E/S forecast EPS growth rate
REP	Relative earnings to price ratio
AX_MidTermMomentum	Axioma medium-term momentum factor return
ROA	Return on assets ratio
ROE	Return on equity ratio
Sales_Assets	Sales-to-assets ratio
GROSS_MARGIN	Profits-sales ratio
YOY_EPS_G	Year-to-year EPS growth
PM61	Price momentum 6-month momentum
AX_Growth	Axioma growth factor return
IBES_EPS_LTG	I/B/E/S 3–5 year forecasted growth rate
PM91	Price momentum 9-month (net reversion) momentum
PM121	Price momentum 12-month (net reversion) momentum
AX_Liquidity	Axioma liquidity factor return
SALES_EV	Sales-to-enterprise value ratio
DP	Dividends-to-price ratio
IBES_EXP_DY	I/B/E/S forecasted dividend yield to price ratio
PM31	Price momentum 3-month (net reversion) momentum
AX_Volatility	Axioma volatility factor return
AX_DividendYield	Axioma dividend yield factor return
AX_ShortTermMomentum	Axioma short-term momentum factor return
RCP	Relative earnings to price ratio
IBES_REC_MEAN	I/B/E/S recommendation mean ratio
Percent_Accrual	(Net income − Operating cash flow)/Net income ratio
SP	Sales to price ratio
FEP1	I/B/E/S 1-year-ahead forecasted earnings to price ratio
EP	Earnings to price ratio
CP	Cash flow to price ratio
FEP2	I/B/E/S 2-year-ahead forecasted earnings to price ratio
IBES_PTG_RET	I/B/E/S targeted price to current price ratio
EBITDA_EV	EBITDA-to-enterprise value
BP	Book value to price ratio
AX_Value	Axioma value factor return
AX_EarningsYield	Axioma earnings yield factor return
RSP	Relative sales to price ratio
RBP	Relative book value to price ratio
YOY_SALES_G	Year-to-year sales growth
AX_ExRateSensitivity	Axioma exchange ratio factor return
Debt_Equity	Debt-to-equity ratio
CHG_SHARES	Percentage change in stock shares
CHG_DEBT	Percentage change in debt
IBES_REC_MEAN_3M	I/B/E/S 1-year-ahead forecasted earnings to price ratio
IBES_FY1_EPS_DISP	I/B/E/S 1-year-ahead forecasted standard deviation earnings to price ratio
AX_Leverage	Axioma leverage factor return
CAPEX_DEP	Capital expenditures-to-depreciation ratio

14.1.3 Appendix C: Matrix Algebra

In finance matrix, algebra is very useful. Many programing languages, including Excel, have built in functions to do calculations easily with matrix operations. Matrices are tables of numbers with finite number of rows and columns. A matrix is described by giving the number of rows first followed by the number of columns, its dimensions. We define scalars (single numbers) as simply 1x1 matrices, vectors as 1xN, (a row vector with N elements), or as Mx1 (a column vector with M elements).

Let A denote an M x N and B denote an N x L matrix:

$$ {A}_{M\times N}=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1N}\\ {}{a}_{21}& {a}_{22}& \cdots & {a}_{2N}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{a}_{M1}& {a}_{M2}& \cdots & {a}_{MN}\end{array}\right),\kern0.75em {B}_{N\times L}=\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \cdots & {b}_{1L}\\ {}{b}_{21}& {b}_{22}& \cdots & {b}_{2L}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{b}_{N1}& {b}_{N2}& \cdots & {b}_{NL}\end{array}\right) $$

Let F denote the product of A and B, then F is an M x L matrix:

$$ {F}_{M\times L}={A}_{M\times N}\times {B}_{N\times L} $$

Note that first matrix’s number of columns and the second matrix’s number of rows must be the same in the multiplication operation! If necessary, transpose one of the matrices to obtain this property. In Excel, multiplication operation is done by using the function MMULT(range 1, range 2).

Transpose of matrix A_MxN denoted by A^T, or A’, is an N x M matrix:

$$ {A}_{N\times M}^T=\left(\begin{array}{cccc}{a}_{11}& {a}_{21}& \cdots & {a}_{M1}\\ {}{a}_{12}& {a}_{22}& \cdots & {a}_{M2}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{a}_{1N}& {a}_{2N}& \cdots & {a}_{MN}\end{array}\right) $$

In Excel, the function is TRANSPOSE(Range).

Let X and Y denote two N x 1 vectors, then the difference X – Y is given by an N x 1 vector D:

$$ {D}_{N\times 1}={X}_{N\times 1}-{Y}_{N\times 1}=\left(\begin{array}{c}{x}_1\\ {}{x}_2\\ {}\vdots \\ {}{x}_N\end{array}\right)-\left(\begin{array}{c}{y}_1\\ {}{y}_2\\ {}\vdots \\ {}{y}_N\end{array}\right)=\left(\begin{array}{c}{x}_1-{y}_1\\ {}{x}_2-{y}_2\\ {}\vdots \\ {}{x}_N-{y}_N\end{array}\right) $$

14.1.3.1 Matrix algebra in Excel

Let numbers in cells A1:C1 be the matrix A (1x3) and numbers in E1:G3 be the matrix B (3x3). Product of A and B will be matrix with 1 row and 3 columns. Select 3 columns, say A3:C3, and enter formula =MMULT(A1:C1,E1:G3). You must use Ctrl+Shift+Enter! It will show as { =MMULT(A1:C1,E1:G3)}. The {} indicates that it is an array function.

To transpose matrix A, select 3 rows, say A5:A7, and enter the formula =TRANSPOSE(A1:C1). Then Ctrl+Shift+Enter. It will show as {=TRANSPOSE(A1:C1)}.

Here is a simple example:

$$ {A}_{1\times 3}\times {B}_{3\times 3}={Z}_{1\times 3}=\left[2\kern0.5em 3\kern0.5em 4\right]\ \left[\begin{array}{ccc}13& -8& -3\\ {}-8& 10& -1\\ {}-3& -1& 11\end{array}\right]=\left[-10\kern0.5em 10\kern0.5em 35\right] $$

$$ {A}_{3\times 1}^T=\left[\begin{array}{c}2\\ {}3\\ {}4\end{array}\right] $$

14.1.3.2 Portfolio Statistics with Matrix Algebra

Expected return and variance of a portfolio of N securities is calculated using the following two equations:

$$ E\left[r{}_P\right]=\sum \limits_{i=1}^N{x}_iE\left[r{}_i\right] $$

$$ \kern2.7em {\sigma}_P^2=\sum \limits_{i=1}^N\sum \limits_{j=1}^N{x}_i{x}_j{\sigma}_{ij} $$

where x_i is the weight of security i in the portfolio, σ_ij is the covariance of security i with the security j. Sum of weights is always 1.

Let X be Nx1 matrix, column vector, representing the weights of the securities in the portfolio, R be Nx1 matrix representing expected returns of securities, and Ω be NxN covariance matrix of securities. Then, expected return and the variance of portfolio in matrix algebra will be:

$$ E\left[R{}_P\right]={X}_{1 xN}^T\times {R}_{Nx1}=\left({x}_1\kern0.5em {x}_2\kern0.5em \cdots \kern0.5em {x}_N\right)\left(\begin{array}{c}{r}_1\\ {}{r}_2\\ {}\vdots \\ {}{r}_N\end{array}\right) $$

$$ {\sigma}_P^2={X}_{1 xN}^T\times {\Omega}_{Nx N}\times {X}_{Nx1}=\left({x}_1\kern0.5em {x}_2\kern0.5em \cdots \kern0.5em {x}_N\right)\left(\begin{array}{cccc}{\sigma}_{11}& {\sigma}_{12}& \cdots & {\sigma}_{1N}\\ {}{\sigma}_{21}& {\sigma}_{22}& \cdots & {\sigma}_{2N}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{\sigma}_{N1}& {\sigma}_{N2}& \cdots & {\sigma}_{NN}\end{array}\right)\left(\begin{array}{c}{x}_1\\ {}{x}_2\\ {}\vdots \\ {}{x}_N\end{array}\right) $$

and,

$$ {X}_{1 xN}^T\times {\underline{1}}_{Nx1}=\left({x}_1\kern0.5em {x}_2\kern0.5em \cdots \kern0.5em {x}_N\right)\left(\begin{array}{c}1\\ {}1\\ {}\vdots \\ {}1\end{array}\right)=1 $$

Unit vector, 1, is used to make sure that sum of weights is 1!

To demonstrate it in Excel, we will use the example of two security portfolio of IBM and D presented in chapter 14. An equally weighted portfolio of IBM and D has the following statistics:

$$ E\left[{r}_P\right]=\left(0.5\kern0.5em 0.5\right)\left(\begin{array}{c}0.0024\\ {}0.0054\end{array}\right)=0.0039 $$

$$ {\sigma}_P^2=\left(0.5\kern0.5em 0.5\right)\left(\begin{array}{cc}0.004212& 0.000211\\ {}0.000211& 0.001529\end{array}\right)\left(\begin{array}{c}0.5\\ {}0.5\end{array}\right)=0.001541 $$

$$ {\sigma}_P=\sqrt[2]{0.001541}=0.039251 $$

Formulas in cells B6, C6, and H6 are:

• B6: =MMULT(TRANSPOSE(H3:H4),B3:B4)

• C6: =SQRT(MMULT(MMULT(TRANSPOSE(H3:H4),F3:G4),H3:H4))

• H6: =MMULT(TRANSPOSE(H3:H4),I3:I4)

If we change the weights in cells H3 and H4 to 0.245 and 0.755, respectively, portfolio stats will be: