Investments involve predictions about the future, and those predictions can be uncertain. The formal representation of that uncertainty is accomplished through the use of probability theory, such that every uncertain number X is now represented as the probability distribution function P(X).
Given two investments which give uncertain returns X and Y respectively, how does one choose between them? Under decision theory, a utility function U is used to compare <U(X)> and <U(Y)>, and the investment with the higher utility is chosen. For the sake of simplicity, modern portfolio theory models U using a quadratic function, where U(X) = X – r*X^2, with a greater r meaning more risk adversity.
Given a quadratic utility function, it is possible to evaluate the expected utility of all linear combinations of X and Y by just knowing <X>, <Y>, <XY>, <X^2>, <Y^2>. This result generalizes to any number of investments. Using this information, for any given return, one can find the linear combination which minimizes the variance, forming the optimal portfolio for that return. The set of all optimal portfolios forms the efficient frontier in the diagram below.
Now examine the effect of adding a risk-free asset. Taking a linear combination of the risk-free asset and any given portfolio, you can achieve any new portfolio with the same Sharpe ratio (ratio of the difference between return and risk-free rate to the standard deviation) as the given portfolio. The capital market line represents the best portfolios that can be formed this way – they consist of combinations of the risk-free asset and the market portfolio, which is the efficient frontier portfolio with the largest Sharpe ratio.
Under the CAPM pricing model, it is assumed that stocks function as components for building market portfolios, and stockholders are only compensated for risk which cannot be diversified away by the other components. Approximating the market portfolio with the SP500, we approximate this non-diversify-able risk using the covariance between the stock and the SP500. That covariance is scaled by the variance of the market portfolio and called beta.
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This is the standard classroom treatment of the concept of risk, and I am uncomfortable with it in ways that I will describe in the next post. For additional reading, you have wikipedia: