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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.

**Finance** is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

In finance, **return** is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from the investment, such as interest payments or dividends. It may be measured either in absolute terms or as a percentage of the amount invested. The latter is also called the holding period return.

In financial accounting, an **asset** is any resource owned by the business. Anything tangible or intangible that can be owned or controlled to produce value and that is held by a company to produce positive economic value is an asset. Simply stated, assets represent value of ownership that can be converted into cash. The balance sheet of a firm records the monetary value of the assets owned by that firm. It covers money and other valuables belonging to an individual or to a business.

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 it 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.

In finance and economics, **systematic risk** is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggregate income. In many contexts, events like earthquakes and major weather catastrophes pose aggregate risks that affect not only the distribution but also the total amount of resources. If every possible outcome of a stochastic economic process is characterized by the same aggregate result, the process then has no aggregate risk.

**Market risk** is the risk of losses in positions arising from movements in market prices.:

In finance, the **beta** of an investment indicates whether the investment is more or less volatile than the market as a whole.

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.

**Jack Lawrence Treynor** was an American economist who served as the President of Treynor Capital Management in Palos Verdes Estates, California. He was a Senior Editor and Advisory Board member of the *Journal of Investment Management*, and was a Senior Fellow of the Institute for Quantitative Research in Finance. He served for many years as the editor of the CFA Institute's Financial Analysts Journal.

**William Forsyth Sharpe** is an American economist. He is the STANCO 25 Professor of Finance, Emeritus at Stanford University's Graduate School of Business, and the winner of the 1990 Nobel Memorial Prize in Economic Sciences.

**John Virgil Lintner, Jr.** was a professor at the Harvard Business School in the 1960s and one of the co-creators (1965a,b) of the 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:

**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 **risk–return spectrum** is the relationship between the amount of return gained on an investment and the amount of risk undertaken in that investment. The more return sought, the more risk that must be undertaken.

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).

For an individual, a **risk premium** is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset in order to induce an individual to hold the risky asset rather than the risk-free asset. It is positive if the person is risk averse. Thus it is the minimum willingness to accept compensation for the risk.

where:

- is the expected return on the capital asset
- is the risk-free rate of interest such as interest arising from government bonds
- (the
*beta*) is the sensitivity of the expected excess asset returns to the expected excess market returns, or also - is the expected return of the market
- is sometimes known as the
*market premium*(the difference between the expected market rate of return and the risk-free rate of return). - is also known as the
*risk premium* - denotes the correlation coefficient between the investment and the market
- is the standard deviation for the investment
- is the standard deviation for the market .

**Sensitivity** and **specificity** are statistical measures of the performance of a binary classification test, also known in statistics as a classification function, that are widely used in medicine:

A **correlation coefficient** is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.

In statistics, the **standard deviation** is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

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.

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 model has been found to do as well as or outperform the modified beta models.

The SML essentially 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 *securities 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*_{m})− *R*_{f}. The securities market line can be regarded as representing a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The equation of the SML is thus:

It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for 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.

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 undervalued (and overvalued 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 payoff of the asset or portfolio.^{ [5] }

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.

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.

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.

All investors:^{ [7] }

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

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] }

- The traditional CAPM using historical data as the inputs to solve for a future return of asset i. However, the history may not be sufficient to use for predicting the future and modern CAPM approaches have used betas that rely on future risk estimates.
^{ [8] } - Most practitioners and academics agree that risk is of a varying nature (non-constant). A critique of the traditional CAPM is that the risk measure used remains constant (non-varying beta). Recent research has empirically tested time-varying betas to improve the forecast accuracy of the CAPM.
^{ [9] } - The model assumes that the variance of returns is an adequate measurement of risk. This would be implied by the assumption that returns are normally distributed, or indeed are distributed in any two-parameter way, but for general return distributions other risk measures (like coherent risk measures) will reflect the active and potential shareholders' preferences more adequately. Indeed, risk in financial investments is not variance in itself, rather it is the probability of losing: it is asymmetric in nature. Barclays Wealth have published some research on asset allocation with non-normal returns which shows that investors with very low risk tolerances should hold more cash than CAPM suggests.
^{ [10] } - The model assumes that all active and potential shareholders have access to the same information and agree about the risk and expected return of all assets (homogeneous expectations assumption).
^{[ citation needed ]} - The model assumes that the probability beliefs of active and potential shareholders match the true distribution of returns. A different possibility is that active and potential shareholders' expectations are biased, causing market prices to be informationally inefficient. This possibility is studied in the field of behavioral finance, which uses psychological assumptions to provide alternatives to the CAPM such as the overconfidence-based asset pricing model of Kent Daniel, David Hirshleifer, and Avanidhar Subrahmanyam (2001).
^{ [11] } - The model does not appear to adequately explain the variation in stock returns. Empirical studies show that low beta stocks may offer higher returns than the model would predict. Some data to this effect was presented as early as a 1969 conference in Buffalo, New York in a paper by Fischer Black, Michael Jensen, and Myron Scholes. Either that fact is itself rational (which saves the efficient-market hypothesis but makes CAPM wrong), or it is irrational (which saves CAPM, but makes the EMH wrong – indeed, this possibility makes volatility arbitrage a strategy for reliably beating the market).
^{ [12] }^{ [13] } - The model assumes that given a certain expected return, active and potential shareholders will prefer lower risk (lower variance) to higher risk and conversely given a certain level of risk will prefer higher returns to lower ones. It does not allow for active and potential shareholders who will accept lower returns for higher risk. Casino gamblers pay to take on more risk, and it is possible that some stock traders will pay for risk as well.
^{[ citation needed ]} - The model assumes that there are no taxes or transaction costs, although this assumption may be relaxed with more complicated versions of the model.
^{ [14] } - The market portfolio consists of all assets in all markets, where each asset is weighted by its market capitalization. This assumes no preference between markets and assets for individual active and potential shareholders, and that active and potential shareholders choose assets solely as a function of their risk-return profile. It also assumes that all assets are infinitely divisible as to the amount which may be held or transacted.
^{[ citation needed ]} - The market portfolio should in theory include all types of assets that are held by anyone as an investment (including works of art, real estate, human capital...) In practice, such a market portfolio is unobservable and people usually substitute a stock index as a proxy for the true market portfolio. Unfortunately, it has been shown that this substitution is not innocuous and can lead to false inferences as to the validity of the CAPM, and it has been said that due to the inobservability of the true market portfolio, the CAPM might not be empirically testable. This was presented in greater depth in a paper by Richard Roll in 1977, and is generally referred to as Roll's critique.
^{ [15] } - The model assumes economic agents optimise over a short-term horizon, and in fact investors with longer-term outlooks would optimally choose long-term inflation-linked bonds instead of short-term rates as this would be more risk-free asset to such an agent.
^{ [16] }^{ [17] } - The model assumes just two dates, so that there is no opportunity to consume and rebalance portfolios repeatedly over time. The basic insights of the model are extended and generalized in the intertemporal CAPM (ICAPM) of Robert Merton,
^{ [18] }and the consumption CAPM (CCAPM) of Douglas Breeden and Mark Rubinstein.^{ [19] } - CAPM assumes that all active and potential shareholders will consider all of their assets and optimize one portfolio. This is in sharp contradiction with portfolios that are held by individual shareholders: humans tend to have fragmented portfolios or, rather, multiple portfolios: for each goal one portfolio — see behavioral portfolio theory
^{ [20] }and Maslowian portfolio theory.^{ [21] } - Empirical tests show market anomalies like the size and value effect that cannot be explained by the CAPM.
^{ [22] }For details see the Fama–French three-factor model.^{ [23] } - Roger Dayala
^{ [24] }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 tota 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^{ [25] }.

**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 prices, interest rates and shares, 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.

**Rational pricing** is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

**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. It uses the variance of asset prices as a proxy for risk.

In finance, **arbitrage pricing theory** (**APT**) is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. The model-derived rate of return will then be used to price the asset correctly—the asset price should equal the expected end of period price discounted at the rate implied by the model. If the price diverges, arbitrage should bring it back into line.

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 index instead of a market index.

The **Treynor reward to volatility model**, named after Jack L. Treynor, is a measurement of the returns earned in excess of that which could have been earned on an investment that has no diversifiable risk, per each unit of market risk assumed.

**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.

**Business valuation** is a process and a set of procedures used to estimate the economic value of an owner's interest in a business. Valuation is used by financial market participants to determine the price they are willing to pay or receive to effect a sale of a business. In addition to estimating the selling price of a business, the same valuation tools are often used by business appraisers to resolve disputes related to estate and gift taxation, divorce litigation, allocate business purchase price among business assets, establish a formula for estimating the value of partners' ownership interest for buy-sell agreements, and many other business and legal purposes such as in shareholders deadlock, divorce litigation and estate contest. In some cases, the court would appoint a forensic accountant as the joint expert doing the business valuation.

In finance, **diversification** is the process of allocating capital in a way that reduces the exposure to any one particular asset or risk. A common path towards diversification is to reduce risk or volatility by investing in a variety of assets. If asset prices do not change in perfect synchrony, a diversified portfolio will have less variance than the weighted average variance of its constituent assets, and often less volatility than the least volatile of its constituents.

The **single-index model (SIM)** is a simple asset pricing model to measure both the risk and the return of a stock. The model has been developed by William Sharpe in 1963 and is commonly used in the finance industry. Mathematically the SIM is expressed as:

**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

**Security characteristic line** (SCL) is a regression line, plotting performance of a particular security or portfolio against that of the market portfolio at every point in time. The SCL is plotted on a graph where the Y-axis is the excess return on a security over the risk-free return and the X-axis is the excess return of the market in general. The slope of the SCL is the security's beta, and the intercept is its alpha.

**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.

**Returns-based style analysis** 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 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. Returns based strategies that use factors such as momentum signals have been popular to the extent that industry analysts incorporate their use in their Buy/Sell recommendations.

In investing, **downside beta** is the element of beta that investors associate with risk in the sense of the uncertain potential for loss. It is defined to be the scaled amount by which an asset tends to move compared to a benchmark, calculated only on days when the benchmark’s return is negative.

In investing, **dual-beta** is a concept that states that a regular, market beta can be divided into downside beta and upside beta. Thus, dual stands for two betas, upside and downside. The fundamental principle behind dual-beta is that upside and downside betas are not the same. This is in contrast to what the Capital Asset Pricing Model assumes, which is that upside and downside betas are identical. Moreover, Fama and French (1992) demonstrated that beta is an imperfect measure of investment risk.

In investing, **upside beta** is the element of traditional beta that investors do not typically associate with the true meaning of risk. It is defined to be the scaled amount by which an asset tends to move compared to a benchmark, calculated only on days when the benchmark’s return is positive.

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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