#### CAPITAL MARKET LINE (CML) (Encyclopedia)

According to the Markowitz theory, where all assets are risky, the optimal portfolio is the one located in the efficient part of the frontier and it depends on the investor’s attitude toward risk. Investors have a different risk aversion so they will choose differently; each of them will choose a portfolio made up of distinct combinations of securities.

By introducing a risk-free asset that could be bought and sold in a short period of time, (for instance, it is possible to grant and obtain loans at a certain rate), Tobin has shown that the efficient frontier is linear and the choice of portfolio (composed exclusively of risky assets) is independent from the individual’s attitude toward risk.

This portfolio is along the concave frontier and has the higher ratio : it can be easily found by plotting a line that intersects the y-axis in R_{f} and that is tangent to the concave set. All investors have the same opportunities and the same tangency-portfolio. So the CML derives from the efficient part of the minimum variance frontier connected with the possibility to buy/sell a risk-free asset. CML is the line that represents the entire market.

When expectations are all the same and there is a single interest rate, the concave frontier and the linear one coincide among investors; if they want to maximise the expected utility of their portfolio, they will choose a unique mix P to be combined, depending on their attitude toward risk, with the risk-free asset in order to obtain the optimum portfolio.

When each investor wants to hold P (even in combination with R_{f}), the aggregate P will include all the existing securities; the exclusion of some securities from P (for instance, no one holds those assets) implies a supply excess, therefore an imbalance in the market. Those securities will not be part of the portfolio because, given the risk, they presuppose an inadequate return, so their price will necessarily decrease in order to eliminate the existing imbalance.

If investors choose the same risky mix P, and R_{f} is unique, the linear frontier resulting from the combination between P and R_{f} is unique. This separation line, made by perfectly broad-based and efficient portfolios, is called the *Capital Market Line* (CML). Investors who want to invest a portion of their wealth in a risk-free asset, place their mix in the part [R_{f} … P] of the line. The second part of the frontier starts from P and includes the combinations ("borrowing portfolios") chosen by those who have a "limited" risk aversion. These investors take out loans at the rate R_{f} in order to buy higher amounts of the portfolio P. So the CML starts from the separation theorem and extends its practical implications: if there is a risk-free asset, investors choose a risky portfolio P independently from their attitude toward risk, which determines the portion of wealth to be invested in the mix P and in the risk-free asset. Basically, the choice of portfolio P is separated from individual preferences and, if the expectations are homogeneous, each investor will choose a unique risky portfolio (portfolio P) that has to be combined with the risk-free asset R_{f}. The risky mix of the "cautious" investor is the same as the one that belongs to the "aggressive" investor: the difference lies in the decision to grant or obtain loans (wealth repartition between P and risk-free asset) at the rate R_{f}. Therefore, the portfolio strategy suggested is a "passive" one: buy the mix P and carefully consider the investor’s attitude toward risk.

The CML can be easily plotted in the risk-return space because the intercept R_{f }is known and the inclination comes out from the ratio between the vertical distance from P and R_{f }and their horizontal distance ().

The fundamental equation is the following:

Figure. Capital Market Line (CML)

Therefore, the CML draws the linear frontier composed of portfolios/combinations between the risk-free asset and the market portfolio P, and describes the risk/return relationship for efficient and perfectly broad-based mixes. From this point of view, the question is whether the relationship will still be linear for single stocks or for broad-based mixes, and what correct measure of risk will be associated with each category previously mentioned. When creating the portfolio, the investor is interested in knowing the contribution of each security to the mix variance and to the mix return.

According to the market model CAPM, the investor, in any case, will choose the market portfolio P, by considering its risk and its return. With regard to the return, it is easy to see the proportional contribution given to E(R_{m}) by the single asset with respect to its own E(R_{i}).

Ignoring the analytic derivation, we can see how, in equilibrium, the asset contribution to should be compensated in terms of E(R_{p}). Therefore, if two securities have the same , they should produce the same return, otherwise it would be more convenient to reduce the amount of the less lucrative asset in order to increase the amount of the more profitable one. Similarly, if two assets have the same expected return they should have the same .

Bibliography:

SALTARI, E., 1997, Introduzione all’Economia Finanziaria, NIS (La Nuova Italia Scientifica), Roma;

BECCHETTI, L., CICIRETTI, R., TRENTA, U., 2007, *Modelli di Asset Pricing I: Titoli Azionari*, in Il Sistema Finanziario Internazionale, Michele Bagella, a cura di, Giappichelli, Torino.

Editor: Rocco CICIRETTI

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