The CML and Quantitative Analyses
We have observed diagrammatically that if capital markets are efficient, all rational investors would ideally hold the market portfolio (M) irrespective of their risk attitudes. By finding the point of tangency between the efficiency frontier (F-F1) and the capital market line (CML) then borrowing or lending at the risk-free rate (rf) it is also possible for individual investors to achieve a desired balance between risk and return elsewhere on the CML.
Obviously, portfolios whose risk-return characteristics place it below the CML are inefficient and could be improved by altering their composition. It is also possible that an investor might "beat the market" (if only by luck rather than judgment) so that the portfolio's risk-return profile would lie above the CML, making it "super-efficient". However, if markets are efficient without access to insider information (as portfolio theorists assume) then this will be a temporary phenomenon.
Like the work of Fisher and Markowitz before him, Tobin's theorem is another landmark in the development of financial theory, which you ought to read at source. At the very least you need to be able to manipulate the following statistical equations, which we shall apply to an Exercise in Chapter Four of our companion text (PTFAE).
So, let us begin by redefining our general portfolio risk formula based on Equation (7) for the standard deviation (which you first encountered in Chapter Two). Combining the market variance of returns (a2 ) with the variance of risk-free investments (a2c):
The first point to note is that because the variability of risk-free returns is obviously zero, their variance (a2f) and standard deviation (af ) equals zero. The second and third terms of Equation (27), which define the variance of the risk-free investment and the correlation coefficient, disappear completely. Thus, Equation (27) for the portfolio's standard deviation simplifies to:
Rearranging the terms of Equation (28) with only one unknown and simplifying, we can also determine the proportion of funds (x) invested in the market portfolio. Given any investor's preferred portfolio and the market standard deviation of returns (a and a ):
In Chapter Two we defined the expected return for a two-asset portfolio R(P) as the weighted average of expected returns from two investments or projects, R(A) and R(B), where the weights are the proportional funds invested in each. Mathematically, this is given by:
The equation can be adapted to calculate the expected return (rp) for any portfolio that includes a combination of risky and risk-free investments, whose returns are (rm) and (rf) respectively.
The Market Price of Risk or Risk Premium
Because the CML is a simple linear regression line, its slope (am) is a constant, measured by:
The expected return for any portfolio on the CML (rp ) can also be expressed as:
Given rf (the risk-free rate of return) which is the intercept illustrated in Figure 4.2 (where ap equals zero) r is still the market portfolio return and a and a define market risk and the risk of the particular m f m p I portfolio, respectively.
The constant slope of the CML (am) defined by Equation (30) is called the market price of risk. It represents the incremental return (rm - rf) obtained by investing in the market portfolio (M) divided by market risk (am). In effect it is the risk premium added to the risk-free rate (sketched in Figure 4.2) to establish the total return for any particular portfolio's risk-return trade off.
For example, with a risk premium am defined by Equation (30), the incremental return from a portfolio bearing risk (ap) in relation to market risk (am ) is given by:
This can be confirmed if we were to compare a particular portfolio return with that for the market portfolio. The difference between the two (rp - rm) equals the market price of risk (am) times the spread (a - a ). pm
To summarise, the expected return for any efficient portfolio lying on the CML comprising the market portfolio, plus either borrowing or lending at the risk free rate can be expressed by simplifying Equations (30) and (31), so that:
In other words, the expected return of an efficient portfolio (rp) equals the risk-free rate of return (rf) plus a risk premium (am.ap).This premium reflects the market's risk-return trade-off (am) combined with the portfolio's own risk (ap).