The Capital Market Line

The Capital Market Line Individuals are not compelled to invest in risky assets. Earlier, we assumed the existence of a rate of interest at which funds could be borrowed or lent. This interest rate represents a risk free asset. An investor can lend at this rate in the knowledge that his return is certain, i.e. there will be no variability in the return. When a risk-free asset is introduced into the analysis, the options open to the investor are increased and improved, as he can now: 1. Invest all his funds in the risk-free asset. 2. Invest part of his funds in the risk-free asset and part in the risky assets. 3. Invest all his funds in risky assets. 4. Borrow at the risk-free rate in order to invest in risky assets (note: it is assumed that borrowing and lending rates are equal). For each of these alternatives, a rational investor will seek to minimize the risk associated with a given level of return. Consider the investor whose indifference curves are shown by the family IC. In Figure. A risk-free asset is introduced with return r and by definition, a standard deviation of zero. The indifference curve passing through C' also, by coincidence, passes through r, showing that the investor would get the same utility from an investment in the riskfree asset as from the investment in the risky portfolio represented by C'. The higher expected return on C' is just sufficient to compensate this investor for accepting its associated risks. However, the investor could reach a higher indifference curve by investing part of his funds in the risk-free asset and the remainder in risky assets. Suppose for the moment that the investor decides to retain the risky part of his portfolio as C'; he can then build a two-asset portfolio - the first asset being the riskfree asset, and the second being the set of risky assets represented by C'. E(r) is greater than r, and any desired return between E(r.) and r, can be obtained by varying the relative proportions of the total funds invested in the two assets. For example, if a return falling midway between E(r.) and r, was required, i.e. then this could be achieved by investing half the funds in the risk-free asset, and half in the risky portofolio. From equation, the expected return on this portofolio E(rp), is given by : But the introduction of risky asset C' into the portfolio brings uncertainty. The return becomes expected rather than certain. From equation 6.6, the variance of a portfolio consisting of the risk free asset and C' will be given by: By definition r, is zero, as there can be no variability of return on a risk-free asset. Furthermore, there can be no relation between the return on the risk-free asset and that on the risky asset, so Cov (r,r.) is also zero. Thus the standard deviation of the portfolio is directly proportional to the amount invested in the risky asset. The risk premium on the portfolio - the difference beween the expected return on the portfolio and the expected return on risk-free assets - will also vary in direct proportion to the amount invested in the risky asset. A straight line can therefore be drawn connecting r, and C', and by altering the proportion of his portfolio invested in the risky assets, C', an investor can construct a portfolio anywhere along this line. This enables the investor to reach a higher indifference curve, as can be seen in Figure However, the investor could reach a still higher indifference curve if he modified his portfolio of risky assets. The line r, C' is not unique: a straight line can be drawn connecting r, with any combination of risky assets. Portfolios can be constructed offering the risk and return shown by any line, simply by varying the proportion of risky assets. An infinite number of portfolios can therefore be constructed. However, the steeper the line connecting the risky portfolio with the riskfree asset, the better the risk/return trade-off. A straight line through r, tangential to the envelope curve represents the steepest of all these possible lines, and thus provides the most attractive investment opportunities. Inspection of Figure 6.6 shows that line r,M is superior to other possibilities such as r,C'. If an investor is prepared to accept the risk associated with C', his best option is to build a portfolio consisting of the risk-free asset and the portfolio of risky assets represented by M. This portfolio, P, vertically above C' on line r,M, would have the same standard deviation as C', but a higher expected return. The reader should note that the line r,M offers the optimal investment opportunities to all investors, whatever their individual tastes and preferences. For example, the preferred portfolio for the investor with indifference curves IC' now lies to the left of M along r,M; it is no longer C' as in Figure 6.4. It reflects a portfolio in which part of the investor's funds are invested in risk-free assets, the remainder being invested in the portfolio of risky assets represented by M. The optimal portfolio for the investor with indifference curves IC" lies to the right of M along r,M: it is no longer C" as in Figure. Portfolios along this section of the line reflect a situation in which the investor borrows funds at the rate r, in order to invest in the risky assets represented by M. It should be noted that, although the two preferred portfolios offer different expected returns and risk, the risky part of both portfolios is represented by the set of risky assets at M, known as the 'market portfolio'. The adjustment for individual taste is made by varying the proportion of the portfolio held in risky assets, and not by adjusting the risky assets themselves. For all individuals the investment decision is thus divided into two distinct steps: 1. Determining the optimal set of risky assets - given by M, and described as the 'market portfolio'. 2. Determining the preferred point along the line r, M - known as the capital market line' - by selecting the optimum mix between the risk-free asset r, and the risky asset M, according to tastes and preferences.