Risk and Return of Equity and the Capital Asset Pricing Model

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 vice versa. 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 19.662%, 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 was −.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 such returns?

Appendix: The Three-Asset Case

Let us now examine a three-asset portfolio construction process using IBM, Dominion Resources, and BA securities:

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

(14)

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

(15)

$$ {\displaystyle \begin{array}{c}\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)\\ {}\kern2.25em \mathrm{let}\;{\mathrm{x}}_3=1-{\mathrm{x}}_1-{\mathrm{x}}_2\\ {}\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{\upsigma_1}^2+{{\mathrm{x}}_2}^2{\upsigma_2}^2+{\upsigma_3}^2{{\mathrm{x}}_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{12}+2{\mathrm{x}}_1{\mathrm{x}}_3{\upsigma}_{23}+2{\mathrm{x}}_2{\mathrm{x}}_3{\upsigma}_{23}\\ {}={{\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}\\ {}\kern1em +2{\mathrm{x}}_2\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right){\upsigma}_{23}\\ {}={{\mathrm{x}}_1}^2{\upsigma_1}^2+{{\mathrm{x}}_2}^2{\upsigma_2}^2+\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right)\left(1-{\mathrm{x}}_1-{\mathrm{x}}_2\right){\upsigma_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{12}+2{\mathrm{x}}_1{\upsigma}_{13}\\ {}\kern1em -2{{\mathrm{x}}_1}^2{\upsigma}_{13}-2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{13}+2{\mathrm{x}}_2{\upsigma}_{23}-2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{23}-2{{\mathrm{x}}_2}^2{\upsigma}_{23}\\ {}={{\mathrm{x}}_1}^2{\upsigma_1}^2+{{\mathrm{x}}_2}^2{\upsigma_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){\upsigma_3}^2+2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{12}\\ {}\kern1em +2{\mathrm{x}}_1{\upsigma}_{13}-2{{\mathrm{x}}_1}^2{\upsigma}_{13}-2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{13}+2{\mathrm{x}}_2{\upsigma}_{23}-2{\mathrm{x}}_1{\mathrm{x}}_2{\upsigma}_{23}-2{{\mathrm{x}}_2}^2{\upsigma}_{23}\end{array}} $$

(16)

$$ {\displaystyle \begin{array}{l}\frac{\partial {\sigma}_p^2}{\partial {x}_1}=2{\mathrm{x}}_1\left({\upsigma_1}^2+{\upsigma_3}^2-2{\upsigma}_{13}\right)+{\mathrm{x}}_2\left(2{\upsigma_3}^2+2{\upsigma}_{12}-2{\upsigma}_{13}-2{\upsigma}_{23}\right)-2{\upsigma_3}^2+2{\upsigma}_{13}=0\\ {}\frac{\partial {\sigma}_p^2}{\partial {x}_2}=2{\mathrm{x}}_2\left({\upsigma_2}^2+{\upsigma_3}^2-2{\upsigma}_{23}\right)+{\mathrm{x}}_1\left(2{\upsigma_3}^2+2{\upsigma}_{12}-2{\upsigma}_{13}-2{\upsigma}_{23}\right)-2{\upsigma_3}^2+2{\upsigma}_{23}=0\end{array}} $$

Let’s assume

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?

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

$$ \mathrm{E}\left({\mathrm{R}}_{\mathrm{p}}\right)=.333\left(.110+.095+.242\right)=.149 $$

$$ {\displaystyle \begin{array}{c}{\sigma}_p^2={(.333)}^2{(.0167)}^2+{(.333)}^2(.0407)\\ {}\kern1em +{(.333)}^2(.0799)+2(.333)(.333)\left(\hbox{--}.0103\right)\\ {}\kern1em +2(.333)(.333)\left(\hbox{--}.0041\right)+2(.333)(.333)(.0303)\end{array}} $$

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

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

The optimally weighted portfolio has an expected return of

$$ {\displaystyle \begin{array}{c}\mathrm{E}\left({\mathrm{R}}_{\mathrm{p}}\right)=.795(.110)+.2517(.095)+\left(\hbox{--}.0467\right)(.242)\\ {}=.0875+.0244+\left(\hbox{--}.0113\right)\\ {}=.1006\\ {}{\sigma}_p^2={(.795)}^2(.0167)+{(.2517)}^2(.0407)+{\left(\hbox{--}.0467\right)}^2\\ {}\kern1em (.0799)+2(.795)(.2517)\left(\hbox{--}.0103\right)\\ {}\kern1em +2(.795)\left(\hbox{--}.0467\right)\left(\hbox{--}.0041\right)+2(.2517)\left(\hbox{--}.0467\right)(.0303)\\ {}=.0106+.0026+.0002+\left(\hbox{--}.0041\right)+.0003+\left(\hbox{--}.0007\right)\\ {}{\sigma}_p^2=.0089\\ {}{\upsigma}_{\mathrm{p}}=.0943\end{array}} $$

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 (Table 16).

Table 16 Modeling TCBUER by OLS longer lags of LEI and WkUNCL time series and estimation of saturation (IIS and SIS) variables