Influencia de las variables de primer curso para Ingeniería Industrial

We now extend Markowitz’s analysis with two additional elements: a financial (money) market and market equilibrium. In a portfolio context, the introduction of a financial mar-ket adds a new investment to the investment opportunity set: the risk-free asset. As usual we will assume a perfect financial market, where unlimited amounts of money can be bor-rowed or lent at the same rate and without transaction costs. As before, the introduction of a financial market may look trivial, but it has profound effects. It drastically changes the shape of the efficient frontier, so that all investors hold combinations of the risk-free asset and the so-called market portfolio, as we shall see.

Market equilibrium requires a set of market clearing prices, which ensure that investors’ collective demand equals the supply for every asset. The process in which those

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7 When short selling is allowed, there is no maximum return portfolio: return can always be increased by shorting more stocks with a low return and buying those with a high return.

69 3.2 Selecting and pricing portfolios

prices are established can be thought of as being directed by a market manager (the ‘Wal-rasian auctioneer’ from classical economics). This manager announces a set of prices and each investor informs the market manager how much of each asset they are willing to hold at those prices. If collective demand exceeds supply, the market manager raises the prices and if supply exceeds demand they are lowered. The new prices are announced and the process continues until demand equals supply. The result is that all assets are held; in equilibrium there is no excess demand (i.e. some investors want to invest more at current prices) or excess supply (i.e. some assets are not held at current prices). This includes the risk-free asset; the risk-free interest rate is set such that borrowing equals lending.

As Figure 3.9 illustrates, introducing the risk-free asset changes the efficient fron-tier from the curve BC to the straight line that runs from the risk-free interest rate rf

through the tangency point M with the risky investment opportunity set. This tangency point gives the line its steepest possible slope, which means the highest possible expected return per additional unit of risk. In all points except M, the new frontier offers a higher expected return for given levels of risk than the old frontier BC. In point M both offer the same expected return. The straight line is called the capital market line (CML) and all investors will choose their optimal positions along it. This means that all investors only hold combinations of two assets (or funds), the risk-free asset and the tangency portfolio M, regardless of the risk preferences that are expressed in their indifference curves. This remarkable result is called two fund separation.

A B

C

Ind.2

Ind.1

r_f

M

Risk (σ_p) Return

(E[r_p])

Figure 3.9 The capital market line

Note that portfolio M is chosen as the tangency point that gives the CML its maximum slope and M is therefore determined by the expected returns, variances and covariances of the risky investment universe plus the risk-free interest rate. It is not determined by the risk preferences of investors. But as it is the tangency point, two fund separation obtains

and all investors will want to hold portfolio M. If all investors⁸ want to hold portfolio M, market equilibrium requires that it contains all assets in the risky universe. Otherwise, there would be excess supply of the assets not in it. Hence, portfolio M is called the market portfolio. Equilibrium further requires that the prices are such that there is no excess demand or supply of the assets in the market portfolio. In equilibrium, with the equilibrium prices established, the price of each risky asset determines its weight in M.

So each risky asset is held in the proportion of its market value relative to the total market value for all risky assets (i.e. each asset is held according to its market value weight). By consequence, all investors who hold risky assets, hold them in the same proportions, i.e.

they hold a fraction of the market portfolio M.

The risk-return preferences of individual investors are expressed in the way they com-bine the riskless asset with the market portfolio M. Individual 2 in Figure 3.9, who is more risk averse than individual 1, lends some of her money risk-free and invests the rest in the risky market portfolio. Individual 1, who is less risk averse, borrows money risk-free and invests this together with his own money in the market portfolio. This increases his expected return, but also his risk, beyond those of the market portfolio. Notice that both are better off by the introduction of a financial market: they jump to higher indif-ference curves. No investor is worse off and only investors who (happened to) hold the market portfolio are not better off.

An expression for the CML can be derived as follows. If we invest a fraction x in the risk-free asset and a fraction (1 − x) in the market portfolio, the expected return on this portfolio E(rp)is the weighted average of the risk-free interest rate rf and the expected return on the market portfolio E(rm), so E(rp) = x × rf + (1 − x) × E(rm). Its risk is σp = (1 − x) × σm since the variance of and the covariance with a constant are both zero. We can write the weight x in terms of the variances: σp/σ_m = (1 − x) ⇒ x = 1 − (σp/σ_m). Substituting this back into the return relation eliminates x and we get:

E(r_p)= risk or the market price of risk and σp is the volume of risk. The market price of risk is the same for all market participants, it is independent of the individual degrees of risk aversion. This, in turn, means that managers of firms or investment funds do not have to know the risk preferences of their investors, they can simply use the market price of risk.

This facilitates the separation of ownership and management.

Finally, it will be clear from the above that the capital market line is an equilibrium risk-return relation (or pricing relation) for efficient portfolios. In efficient portfolios, all risk comes from the fraction of the market portfolio M. The capital market line cannot be used for individual assets or other inefficient portfolios. To price these investments we need a different pricing relation, such as the Capital Asset Pricing Model.

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8 All but the most risk averse, who only invest in the risk-free asset.

71 3.3 The Capital Asset Pricing Model