Capital asset pricing model

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An estimation of the CAPM and the security market line (purple) for the Dow Jones Industrial Average over 3 years for monthly data. CAPM-SML.svg
An estimation of the CAPM and the security market line (purple) for the Dow Jones Industrial Average over 3 years for monthly data.

In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio.

Contents

The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. CAPM assumes a particular form of utility functions (in which only first and second moments matter, that is risk is measured by variance, for example a quadratic utility) or alternatively asset returns whose probability distributions are completely described by the first two moments (for example, the normal distribution) and zero transaction costs (necessary for diversification to get rid of all idiosyncratic risk). Under these conditions, CAPM shows that the cost of equity capital is determined only by beta. [1] [2] Despite its failing numerous empirical tests, [3] and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton's portfolio problem), the CAPM still remains popular due to its simplicity and utility in a variety of situations.

Inventors

The CAPM was introduced by Jack Treynor (1961, 1962), [4] William F. Sharpe (1964), John Lintner (1965a,b) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. Sharpe, Markowitz and Merton Miller jointly received the 1990 Nobel Memorial Prize in Economics for this contribution to the field of financial economics. Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version was more robust against empirical testing and was influential in the widespread adoption of the CAPM.

Formula

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 , 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(Rm)− Rf. 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 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). [5] When the asset does not lie on the SML, this could also suggest mis-pricing. Since the expected return of the asset at time is , a higher expected return than what CAPM suggests indicates that is too low (the asset is currently undervalued), assuming that at time the asset returns to the CAPM suggested price. [6]

The asset price using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship given by

where is the future price of the asset or portfolio. [5]

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 (Markowitz) efficient frontier. CAL stands for the capital allocation line. Markowitz frontier.jpg
The (Markowitz) efficient frontier. CAL stands for the capital allocation line.

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: [7]

  1. Aim to maximize economic utilities (Asset quantities are given and fixed).
  2. Are rational and risk-averse.
  3. Are broadly diversified across a range of investments.
  4. Are price takers, i.e., they cannot influence prices.
  5. Can lend and borrow unlimited amounts under the risk free rate of interest.
  6. Trade without transaction or taxation costs.
  7. Deal with securities that are all highly divisible into small parcels (All assets are perfectly divisible and liquid).
  8. Have homogeneous expectations.
  9. Assume all information is available at the same time to 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". [3]

Roger Dayala [34] 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. [35]

See also

Related Research Articles

Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade". Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance.

<span class="mw-page-title-main">Eugene Fama</span> American economist and Nobel laureate in Economics

Eugene Francis "Gene" Fama is an American economist, best known for his empirical work on portfolio theory, asset pricing, and the efficient-market hypothesis.

<span class="mw-page-title-main">Risk premium</span> Measure of excess

A risk premium is a measure of excess return that is required by an individual to compensate being subjected to an increased level of risk. It is used widely in finance and economics, the general definition being the expected risky return less the risk-free return, as demonstrated by the formula below.

Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. The variance of return is used as a measure of risk, because it is tractable when assets are combined into portfolios. Often, the historical variance and covariance of returns is used as a proxy for the forward-looking versions of these quantities, but other, more sophisticated methods are available.

In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely believed to be an improved alternative to its predecessor, the capital asset pricing model (CAPM). APT is founded upon the law of one price, which suggests that within an equilibrium market, rational investors will implement arbitrage such that the equilibrium price is eventually realised. As such, APT argues that when opportunities for arbitrage are exhausted in a given period, then the expected return of an asset is a linear function of various factors or theoretical market indices, where sensitivities of each factor is represented by a factor-specific beta coefficient or factor loading. Consequently, it provides traders with an indication of ‘true’ asset value and enables exploitation of market discrepancies via arbitrage. The linear factor model structure of the APT is used as the basis for evaluating asset allocation, the performance of managed funds as well as the calculation of cost of capital. Furthermore, the newer APT model is more dynamic being utilised in more theoretical application than the preceding CAPM model. A 1986 article written by Gregory Connor and Robert Korajczyk, utilised the APT framework and applied it to portfolio performance measurement suggesting that the Jensen coefficient is an acceptable measurement of portfolio performance.

In finance, the beta is a statistic that measures the expected increase or decrease of an individual stock price in proportion to movements of the stock market as a whole. Beta can be used to indicate the contribution of an individual asset to the market risk of a portfolio when it is added in small quantity. It refers to an asset's non-diversifiable risk, systematic risk, or market risk. Beta is not a measure of idiosyncratic risk.

In finance, Jensen's alpha is used to determine the abnormal return of a security or portfolio of securities over the theoretical expected return. It is a version of the standard alpha based on a theoretical performance instead of a market index.

Alpha is a measure of the active return on an investment, the performance of that investment compared with a suitable market index. An alpha of 1% means the investment's return on investment over a selected period of time was 1% better than the market during that same period; a negative alpha means the investment underperformed the market. Alpha, along with beta, is one of two key coefficients in the capital asset pricing model used in modern portfolio theory and is closely related to other important quantities such as standard deviation, R-squared and the Sharpe ratio.

A market anomaly in a financial market is predictability that seems to be inconsistent with theories of asset prices. Standard theories include the capital asset pricing model and the Fama-French Three Factor Model, but a lack of agreement among academics about the proper theory leads many to refer to anomalies without a reference to a benchmark theory. Indeed, many academics simply refer to anomalies as "return predictors", avoiding the problem of defining a benchmark theory.

<span class="mw-page-title-main">Security market line</span>

Security market line (SML) is the representation of the capital asset pricing model. It displays the expected rate of return of an individual security as a function of systematic, non-diversifiable risk. The risk of an individual risky security reflects the volatility of the return from security rather than the return of the market portfolio. The risk in these individual risky securities reflects the systematic risk.

The consumption-based capital asset pricing model (CCAPM) is a model of the determination of expected return on an investment. The foundations of this concept were laid by the research of Robert Lucas (1978) and Douglas Breeden (1979).

Roll's critique is a famous analysis of the validity of empirical tests of the capital asset pricing model (CAPM) by Richard Roll. It concerns methods to formally test the statement of the CAPM, the equation

In asset pricing and portfolio management the Fama–French three-factor model is a statistical model designed in 1992 by Eugene Fama and Kenneth French to describe stock returns. Fama and French were colleagues at the University of Chicago Booth School of Business, where Fama still works. In 2013, Fama shared the Nobel Memorial Prize in Economic Sciences for his empirical analysis of asset prices. The three factors are (1) market excess return, (2) the outperformance of small versus big companies, and (3) the outperformance of high book/market versus low book/market companies. There is academic debate about the last two factors.

A portfolio manager (PM) is a professional responsible for making investment decisions and carrying out investment activities on behalf of vested individuals or institutions. Clients invest their money into the PM's investment policy for future growth, such as a retirement fund, endowment fund, or education fund. PMs work with a team of analysts and researchers and are responsible for establishing an investment strategy, selecting appropriate investments, and allocating each investment properly towards an investment fund or asset management vehicle.

Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.

<span class="mw-page-title-main">Low-volatility anomaly</span>

In investing and finance, the low-volatility anomaly is the observation that low-volatility stocks have higher returns than high-volatility stocks in most markets studied. This is an example of a stock market anomaly since it contradicts the central prediction of many financial theories that taking higher risk must be compensated with higher returns.

In finance, active return refers the returns produced by an investment portfolio due to active management decisions made by the portfolio manager that cannot be explained by the portfolio's exposure to returns or to risks in the portfolio's investment benchmark; active return is usually the objective of active management and subject of performance attribution. In contrast, passive returns refers to returns produced by an investment portfolio due to its exposure to returns of its benchmark. Passive returns can be obtained deliberately through passive tracking of the portfolio benchmark or obtained inadvertently through an investment process unrelated to tracking the index.

Returns-based style analysis (RBSA) is a statistical technique used in finance to deconstruct the returns of investment strategies using a variety of explanatory variables. The model results in a strategy's exposures to asset classes or other factors, interpreted as a measure of a fund or portfolio manager's investment style. While the model is most frequently used to show an equity mutual fund’s style with reference to common style axes, recent applications have extended the model’s utility to model more complex strategies, such as those employed by hedge funds.

In portfolio management, the Carhart four-factor model is an extra factor addition in the Fama–French three-factor model, proposed by Mark Carhart. The Fama-French model, developed in the 1990, argued most stock market returns are explained by three factors: risk, price and company size. Carhart added a momentum factor for asset pricing of stocks. The Four Factor Model is also known in the industry as the Monthly Momentum Factor (MOM). Momentum is the speed or velocity of price changes in a stock, security, or tradable instrument.

Nontraded assets are assets that are not traded on the market. Human capital is the most important nontraded assets. Other important nontraded asset classes are private businesses, claims to government transfer payments and claims on trust income.

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