Here in figure 1b, all the points along the ‘efficient curve’ provide investors with best return given certain risk level. It is a matter of different preferences which are up to individuals’ different utility function (Frank, 2003) and determine which to choose for different investors. However, if an alternative investment opportunity is introduced where investors can freely invest and borrow at risk free rate, we simply conjured more efficient market opportunities where a line starts from risk free point and also tangents the ‘efficient curve’ (illustrated in graph below).
Written in Principles of Corporate Finance (Brealey and Myers, 2003), ‘since borrowing is merely negative lending, you can extend the range of possibilities to the right of ‘T’ by borrowing funds at an interest rate of rf and investing them as well as your own money in portfolio’. The efficient portfolio at the tangency is better than all the others, as it offers the highest ratio of risk premium to standard deviation. However, so far, what we have done is just the essential basis for deriving the CAPM; the graph above is not the CAPM we are pursuing, as the main goal of measuring risky assets in the market has not been fulfilled. According to Financial theory and corporate policy (Copeland and Weston, 1946), now, let us suppose a portfolio consisting of a % invested in risky asset and (1 – a %) in the market portfolio, which will have the following mean and standard deviation:
E(Rp) = aE(Ri) + (1 – a )E(Rm)
σ(Rp) = [a2σi2 + (1-a)2 σm2 + 2a (1-a) σim]1 ∕2
(Where σi2=the variance of the risky asset; σm2= the variance of the market portfolio; σim = the covariance between the risky asset and the market portfolio)
And also, the change in the mean and standard deviation with respect to the percentage of the portfolio, ‘a’, invested in asset is determined as follows (using simple differentiation):
∂E(Rp) / ∂a = E(Ri) – E(Rm) ;
∂σ(Rp)/ ∂a = 0.5 [a2 σi2 + (1-a)2 σm2 + 2a (1-a) σim] 1 ∕2 . [2aσi2 - 2σm2 + 2aσm2 + 2σim – 4aσim]
Yet, here comes the problem: this actually violates our pre-assumptions, which clearly states that when market is in equilibrium, there should be no access of demand and supply (demand = supply). Consequently, in equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights. As a result, the market portfolio already contains the risky asset held according to its market value weight (Sharpe and Treynor). Therefore, the percentage ‘a’ in the above equations is the excess demand for an individual risky asset, where it should be equal to zero. Simplifying previous equations, by substituting ‘a = 0’, we could get:
∂E(Rp) / ∂a = E(Ri) – E(Rm) -------1
∂σ(Rp)/ ∂a = (σim - σm2) / σm --------2
Then, by dividing equation ‘1’ by ‘2’, we could arrive at the slope of the risk-return trade-off evaluated at point M, in the market equilibrium, is
By the same token, as the capital market line is also an equilibrium relationship, given market efficiency, the tangency portfolio must be the market portfolio where all assets are held according to their market value weights. We can use the same method to derive the slope of the capital market line shown as follow:
E(Rm) – Rf
σm
(Where σm is the standard deviation of the market portfolio) In the tangency point (pint ‘T’ in the graph), the slope of market efficient portfolio curve should be equal to the slope of capital market line. Therefore, by equating the slopes together and rearranging, the final relationship can be obtained:
E(Ri) = Rf + [R(Rm) – Rf] σim / σm2
This is known as the Capital Asset Pricing Model (CAPM). The equation is also shown graphically below, where it is also called the security market line. The ‘BETA’ here is simply the covariance between returns on risky asset and market portfolio.
BETA = β = σim / σm2 = COV (Ri, Rm) / VAR (Rm)
Brealey and Myers also argued in their book Principles of Corporate Finance (Brealey and Myers, 2003): ‘the horizontal axis is BETA not Standard Deviation, because the market only rewards systematic risk bearers, and the non-systematic risk bearers are idiots’. In other words, risky assets should never be isolated from portfolio context, the contribution of which to portfolio is our best interest. From the result (CAPM) we have arrived at, we actually developed a relationship showing that the equilibrium rates of return on all risky assets are a function of their covariance with the market portfolio. In the terminology of CAPM, the price of risk is the slope of the line, the difference between the expected rates of return on the market portfolio and the risk-free rate of return; whereas the quantity of risk is measured by ‘BETA’ (Copeland and Weston, 1946).
Properties and Uses
According to Principles of Corporate Finance (Brealey and Myers, 2003), there are several properties of the CAPM that are important: first of all, assuming market perfection, in equilibrium, every asset must be priced as that is risk-adjusted required rate of return falls exactly on the straight line. In other words, there are no excess demand and supply, all investors are satisfied. Secondly, the measure of risk for individual assets is linearly additive, when the assets are combined into portfolio (Archer and Ambrosio, 1970). For example, if we put a % of our wealth into asset X, which systematic risk of βx, and b % of our wealth into asset Y, with systematic risk of βy, then the beta of the resulting portfolio, βp, is simply the weighted average of the betas of the individual securities (βp = a βx + b βy). This is extremely useful for us, because all we need to measure the systematic risk of portfolios is the betas of the individual assets.
The wide applications of the theory of CAPM could be almost seen everywhere in Finance. The useful model does not only give investors benchmarks (Treasury bills and the market portfolio) for security performance, but also does provide quantitative relationship between ‘risk’ and ‘return’ in the market. Because it provides a quantifiable measure of risk for individual assets, the CAPM is an extremely useful tool for valuing risky assets. In corporate context, it facilitates shareholders with reasonable and accurate ruler of ‘expectations of return’ so that financial managers’ performances could be measured and monitored in mutually-agreed basis. Furthermore, it provided strong theoretical basis for the development of latter financial theories (dividend policy theory, capital structure theory and so on). It is also used to develop decision-making rules for the selection of investment projects by the firm, for measurement of the firm’s cost of capital, and for capital structure (optimal debt / equity ratio) decisions.
Limitations of CAPM
Unfortunately we are living in an imperfect world, where nothing is perfect ----neither is the CAPM. As explained previously, the CAPM is a hypothetical model and lies on some perfect-market pre-assumptions which are not necessarily true in reality. In literature, Roll’s critique (1977) on the CAPM’s measurement of performance points out the invalidity of the CAPM, even given perfect market conditions (Archer and Ambrosio, 1970). Therefore, people have tried to develop different new models (Arbitrage Pricing Theory, Consumption CAPM and so forth) instead of CAPM, but no one gives perfect solution to explaining the reality. And, as stated in Principles of Corporate Finance (Brealey and Myers, 2003), ‘defenders of the CAPM emphasize that it is concerned with expected returns, whereas we can observe only actually returns’. In addition, empirical testing of the CAPM and the birth of post form, which introduce errors in the end of the equation, suggested that although the CAPM does not conform to the reality, it does give accurate long-term predictions and trends of what is going on in the real financial market. For short, the Capital Asset Pricing Model is the best-known model of risk and return, which is plausible and widely used but far from perfect.
Conclusion
All in all, although the CAPM is imperfect, it does give us significant implications and ideas of ‘how risk is measured and priced in the market’. It also helps us with investment decision-makings and corporate policies. Therefore, having recognized its problems, we should accept it as the ‘one of the best but imperfect solutions’.
Bibliographies
Archer, S. and Ambrosio, C. (1970). The theory of business finance. The McMillan company. New York.
Brealey, R. and Myers, S. (2003). Principles of Corporate Finance. McGraw – Hill.
Brian, B. and Butler, D. (1993). A dictionary of Finance and Banking. Oxford University Press
Ben, S. and Robert, H. (2001). Principles of Economics. McGraw-Hill. New York.
Copeland, T and Weston, J. (1946). Financial theory and corporate policy. Addison-Wesley publishing company. USA.
Frank, R. (2003). Microeconomics and behaviour. McGraw – Hill.
Markowitz, H. (1952). ‘Portfolio Selection’. Journal of Finance. 7:77 – 91 March.
Horne, V. (1983). Financial management and policy. Prentice-Hall International.
Sharpe, W. (1964). ‘Capital Asset Prices: a theory of market equilibrium under conditions of risk.’ Journal of Finance. 19: 425 – 442 (September).
References
Archer, S. and Ambrosio, C. (1970). The theory of business finance. The McMillan Company. New York.
Ben, S. and Robert, H. (2001). Principles of Economics. McGraw-Hill. New York.
Brealey, R. and Myers, S. (2003). Principles of Corporate Finance. McGraw – Hill.
Copeland, T and Weston, J. (1946). Financial theory and corporate policy. Addison-Wesley publishing company. USA.
Frank, R. (2003). Microeconomics and behaviour. McGraw – Hill.