Capital asset pricing model
The CAPM is a model for pricing an individual security or portfolio. For individual securities, we make use of the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class. The SML enables us to calculate the reward-to-risk ratio for any security in relation to that of the overall market. Therefore, when the expected rate of return for any security is deflated by its beta coefficient, the reward-to-risk ratio for any individual  security in the market is equal to the market reward-to-risk ratio, thus: The market reward-to-risk ratio is effectively the market risk premium and by rearranging the above equation and solving for formula_2, we obtain the capital asset pricing model (CAPM). where: Restated, in terms of risk premium, we find that: which states that the "individual risk premium" equals the "market premium" times "β". Note 1: the expected market rate of return is usually estimated by measuring the arithmetic average of the historical returns on a market portfolio (e.g. S&P 500). Note 2: the risk free rate  of return used for determining the risk premium is usually the arithmetic average of historical risk free rates of return and not the current risk free rate of return. For the full derivation see Modern portfolio theory. Modified betas. There has also been research into a mean-reverting beta often referred to as the adjusted beta, as well as the consumption beta. However, in empirical tests the traditional CAPM has been found to do as well as or outperform the modified beta models. Security market line. The SML graphs the results from the capital asset pricing model (CAPM) formula. The  x-axis represents the risk (beta), and the "y"-axis represents the expected return. The market risk premium is determined from the slope of the SML. The relationship between β and required return is plotted on the "security market line" (SML), which shows expected return as a function of β. The intercept is the nominal risk-free rate available for the market, while the slope is the market premium, E("R")− "R". The security market line can be regarded as representing a single-factor model of the asset price, where β is the exposure to changes in the value of the Market. The  equation of the SML is thus: It is a useful tool for determining if an asset being considered for a portfolio offers a reasonable expected return for its risk. Individual securities are plotted on the SML graph. If the security's expected return versus risk is plotted above the SML, it is undervalued since the investor can expect a greater return for the inherent risk. And a security plotted below the SML is overvalued since the investor would be accepting less return for the amount of risk assumed. Asset pricing. Once the expected/required rate of return formula_2 is calculated  using CAPM, we can compare this required rate of return to the asset's estimated rate of return over a specific investment horizon to determine whether it would be an appropriate investment. To make this comparison, you need an independent estimate of the return outlook for the security based on either fundamental or technical analysis techniques, including P/E, M/B etc. Assuming that the CAPM is correct, an asset is correctly priced when its estimated price is the same as the present value of future cash flows of the asset, discounted at the rate suggested by CAPM. If the  estimated price is higher than the CAPM valuation, then the asset is overvalued (and undervalued when the estimated price is below the CAPM valuation). When the asset does not lie on the SML, this could also suggest mis-pricing. Since the expected return of the asset at time formula_21 is formula_22, a higher expected return than what CAPM suggests indicates that formula_23 is too low (the asset is currently undervalued), assuming that at time formula_24 the asset returns to the CAPM suggested price. The asset price formula_25 using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship  given by where formula_27 is the payoff of the asset or portfolio. Asset-specific required return. The CAPM returns the asset-appropriate required return or discount rate—i.e. the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. Betas exceeding one signify more than average "riskiness"; betas below one indicate lower than average. Thus, a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. Given the accepted concave utility  function, the CAPM is consistent with intuition—investors (should) require a higher return for holding a more risky asset. Since beta reflects asset-specific sensitivity to non-diversifiable, i.e. market risk, the market as a whole, by definition, has a beta of one. Stock market indices are frequently used as local proxies for the market—and in that case (by definition) have a beta of one. An investor in a large, diversified portfolio (such as a mutual fund), therefore, expects performance in line with the market. Risk and diversification. The risk of a portfolio comprises systematic risk, also known as  undiversifiable risk, and unsystematic risk which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all securities—i.e. market risk. Unsystematic risk is the risk associated with individual assets. Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio (specific risks "average out"). The same is not possible for systematic risk within one market. Depending on the market, a portfolio of approximately 30–40 securities in developed markets such as the UK or US will render the portfolio sufficiently diversified such that risk exposure  is limited to systematic risk only. In developing markets a larger number is required, due to the higher asset volatilities. A rational investor should not take on any diversifiable risk, as only non-diversifiable risks are rewarded within the scope of this model. Therefore, the required return on an asset, that is, the return that compensates for risk taken, must be linked to its riskiness in a portfolio context—i.e. its contribution to overall portfolio riskiness—as opposed to its "stand alone risk". In the CAPM context, portfolio risk is represented by higher variance i.e. less predictability. In other words , the beta of the portfolio is the defining factor in rewarding the systematic exposure taken by an investor. Efficient frontier. The CAPM assumes that the risk-return profile of a portfolio can be optimized—an optimal portfolio displays the lowest possible level of risk for its level of return. Additionally, since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset, (assuming no trading costs) with each asset value-weighted to achieve the above (assuming that any asset is infinitely divisible). All such optimal portfolios, i.e., one for each level of return , comprise the efficient frontier. Because the unsystematic risk is diversifiable, the total risk of a portfolio can be viewed as beta. Assumptions. All investors: Problems. In their 2004 review, economists Eugene Fama and Kenneth French argue that "the failure of the CAPM in empirical tests implies that most applications of the model are invalid". Roger Dayala goes a step further and claims the CAPM is fundamentally flawed even within its own narrow assumption set, illustrating the CAPM is either circular or irrational. The circularity refers to the price of total risk being a function of the price of covariance risk  only (and vice versa). The irrationality refers to the CAPM proclaimed ‘revision of prices’ resulting in identical discount rates for the (lower) amount of covariance risk only as for the (higher) amount of Total risk (i.e. identical discount rates for different amounts of risk. Roger’s findings have later been supported by Lai & Stohs.