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4.1. The Market Portfolio and Tobin's Theorem

We have already explained that if an individual or company objective is to minimize the standard deviation of an investment's expected return, this could be determined by reference to indifference curves, which calibrate attitudes toward risk and return. In Chapter Three (PTFA) and the summary of Chapter Two (PTFAE) we graphed an equation of indifference between portfolio risk and portfolio return for any rational investor relative to their optimum portfolio.

the Determination of an Optimum Portfolio: The Multi-Asset Case

Figure 4.1: the Determination of an Optimum Portfolio: The Multi-Asset Case

Diagrammatically, you will recall that the optimum portfolio is determined at the point where one of the investor's indifference curves (risk-return profile) is tangential to the frontier of efficient portfolios. This portfolio (point E on the curve F-F1 in Figure 4.1) is optimal because it provides the best combination of risk and return to suit their preferences.

However, apart from the computational difficulty of deriving optimum portfolios using variance-covariance matrix calculations (think 1950's theory without twenty-first century computer technology-software) this policy prescription only concerns wholly risky portfolios.

But what if risk-free investments (such as government stocks) are included in portfolios? Presumably, investors who are totally risk-averse would opt for a riskless selection of financial and government securities, including cash. Those who require an element of liquidity would construct a mixed portfolio that combines risk and risk-free investments to satisfy their needs.

Thus, what we require is a more sophisticated model than that initially offered by Markowitz, whereby the returns on new investments (risk-free or otherwise) can be compared with the risk of the market portfolio.

Fortunately, John Tobin (1958) developed such a model, built on Markowitz efficiency and the perfect capital market assumptions that underpin the Separation Theorem of Irving Fisher (1930) (with which you should be familiar).

Tobin demonstrates that in a perfect market where risky financial securities are traded with the option to lend or borrow at a risk-free rate, using risk-free assets, such as government securities.

Investors and companies need not calculate a multiplicity of covariance terms. All they require is the covariance of a new investment's return with the overall return on the efficient market portfolio.

To understand what is now termed Tobin's Separation Theorem, suppose every stock market participant invests in all the market's risky securities, with their expenditure in each proportionate to the market's total capitalization. Every investor's risky portfolio would now correspond to the market portfolio with a market return and market standard deviation, which we shall denote as M, rm and sm, respectively.

Tobin maintains that in perfect capital markets that are efficient, such an investment strategy is completely rational. In equilibrium, security prices will reflect their "true" intrinsic value. In other words, they provide a return commensurate with a degree of risk that justifies their inclusion in the market portfolio. Obviously, if a security's return does not compensate for risk; rational investors will want to sell their holding. But with no takers, price must fall and the yield will rise until the risk-return trade-off once again merits the security's inclusion in the market portfolio. Conversely, excess returns will lead to buying pressure that raises price and depresses yield as the security moves back into equilibrium.

This phenomenon is portrayed in Figure 4.2, where M represents the 100 per cent risky market portfolio, which lies along the efficiency frontier of all risky investment opportunities given by the curve F-F1.

The Capital Market Line

Figure 4.2: The Capital Market Line

Now, assume that all market participants can not only choose risky investments with the return rm in the market portfolio M. They also have the option of investing in risk-free securities (such as short-term government stocks) at a risk-free rate, rf. According to their aversion to risk and their desire for liquidity, we can now separate their preferences, (hence the term Separation Theorem). Investors may now opt for a riskless portfolio, or a mixed portfolio, which comprises any preferred combination of risk and risk-free securities.

Diagrammatically, investors can combine the market portfolio with risk-free investments to create a portfolio between rf and M in Figure 4.2. If a line is drawn from the risk-free return rf on the vertical axis of our diagram to the point of tangency with the efficiency frontier at point M, it is obvious that part of the original frontier (F-F1) is now inefficient.

Below M, a higher return can be achieved for the same level of risk by combining the market portfolio with risk-free assets. Since rf denotes a riskless portfolio, the line rf -M represents increasing proportions of portfolio M combined with a reducing balance of investment at the risk-free rate.

Of course, as Fisher first explained way back in 1930, if capital markets are perfect (where borrowing and lending rates are equal) there is nothing to prevent individuals from borrowing at the risk-free rate to build up their investment portfolios. Tobin therefore adapted this concept to show that if investors could borrow at a risk-free rate and invest more in portfolio M using borrowed funds, they could construct a portfolio beyond M in Figure 4.2.

To show this, the line rf to M has been extended to point P and beyond to CML. The effect eliminates the remainder of our original efficiency frontier. Any initial efficient portfolios lying along the curve M-F1 are no longer desirable. With borrowing (leverage) there are always better portfolios with higher returns for the same risk. The line (rf -M-CML) in Figure 4.2 is a new portfolio "efficiency frontier" for all investors, termed the Capital Market Line (CML).

Activity 1

To illustrate the purpose of the CML, let us assume that historically an investment company has passively held a market portfolio (M) of risky assets. This fund tracks the London FT-SE 100 (Footsie) on behalf of its clients.

However, with increasing global uncertainty the company now wishes to manage their portfolio actively, introducing risk-free investments into the mix and even borrowing funds if necessary.

Using Figure 4.2 for reference, briefly explain how the company's new strategy would redefine its optimum portfolio (or portfolios) if it is willing to borrow up to point P?

The portfolio lending-borrowing line (rf -M-P) in Figure 4.2 is the new efficiency frontier (CML) for all the company's portfolio constituents. Portfolios lying along the CML between rf and M are constructed by placing a proportion of their available funds in the market portfolio and the residual in risk-free assets. To establish a portfolio lying halfway up the line rf -M, the company should divide funds equally between the two.

Portfolios lying along the CML beyond M (for example, P in the diagram) are constructed by placing all their funds in M, plus an amount borrowed at the risk-free rate (rf). The amount borrowed would equal the ratio of the line rf - M: M-P.

 
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