# capital asset pricing model

### 1. Introduction

The CAPM is a mathematical model of portfolio theory (Markowitz) and equilibrium in financial markets, published for the first time in 1964 by William Sharpe (Nobel prize for this contribution in 1990 with Markowitz and Miller) and then taken up by several scholars ( Lintner and Mossini especially).

The basis of CAPM is the relationship between the yield of a security and its risk (that the model is measured through a single risk factor known as "Beta").

### 2. CAPM

CAPM is a theoretical model for calculating the equilibrium price of a financial asset. It states that the expected return of an asset is a linear function of risk-free return and the systematic risk of the activity, multiplied by the risk premium of the market. According to the CAPM, the only relevant risk to the investor is systematic (market risk), that is the risk that cannot be eliminated through diversification.

Two activities in balance with the same expected return must also have the same systematic risk (measured by beta), although the overall risk (as measured by standard deviation) of the two activities may be different.

The reason for the possible difference between the overall risk (standard deviation) and systematic risk (beta) resides in the fact that a portion of the overall risk can be eliminated through diversification.

The part of risk that can be eliminated through diversification is called diversifiable risk (or risk-specific) and represents a risk unique to a specific company. Investors do not need to be compensated for bearing that risk because it can be avoided through diversification, and thus the risk diversifiable not affect expected returns.

The contrary, there is a portion of risk that cannot be eliminated, regardless of how investors diversify their portfolios. This risk is systematic risk (or market risk or non-diversifiable), which can be thought of as the risk of the market as a whole. The exposure of an asset to this risk is summarized by measurement of beta activity.

Market equilibrium requires that the demand for risky assets is equal to the supply of risky assets. The demand for risky assets is represented by the optimal portfolio chosen by investors, while the offer is represented by the market portfolio. The main conclusion of the CAPM actually lies in identifying the market portfolio as the portfolio of tangency between demand and supply in balance. Therefore, under the assumptions of CAPM, the balance between risk and return of an asset can be expressed as:

E (Ri) = Rf + Bi [E (RM) - Rf]

where E (Ri) = expected return of the i-th;

Rf = risk-free return that is risk-free return of an asset (such as short-term government securities),

Bi = beta coefficient of i-th,

and (RM) = expected return on the market.

Note that [E (RM) - Rf] is the risk premium of the market (market risk premium), which represents the extent to which the performance of the market portfolio exceeds the return on risk-free. It 'best to make clear that the market portfolio is the portfolio that covers all market activities, where the weights of each activity are represented by market capitalization (also called market value) of each activity divided by the total capitalization of all risky assets (such weights are those weights capitalization). The market portfolio should include all equities and bonds traded on organized markets and over-the-counter and also non-financial assets (such as real estate or consumer durable), and human capital. Therefore, the exact composition of the market portfolio cannot be observed. Consequently, the use of the CAPM requires the use of proxy for the market portfolio. A proxy often used consists of the stock indices such as S & P / Mib for the Italian market or S & P500 for the U.S. market. The equilibrium relation between risk and return according to the theory of CAPM is a very simple graphical representation. The formula of the CAPM can be seen as a linear relationship between expected return and beta, known as the Securities Market Line (SML).

The assumptions of the CAPM (I)

1. There are no transaction costs.
2. The assets are infinitely divisible.
3. There are no income tax.
4. No one can affect the price of securities with their operations.
5. The operators make their choices only on the basis of returns expected and the risk.
6. You can take "short" positions are securities.
7. You can borrow and you can invest in risk-free license no limit.
8. All investors set the time horizon in the same way.
9. All investors have the same set of inputs (yields, volatilities and correlations).
10. Tutte assets are traded on the market and are priced.

### 3. Drawbacks of CAPM

Some of the criticisms of the CAPM are:

The CAPM is based on very restrictive assumptions, such as those that investors concerned only with expected return and risk.

The CAPM cannot be tested empirically.

The CAPM assumes that there is only one factor

The CAPM has received support from empirical tests.

Although these criticisms are well founded, it should be recognized that the CAPM has the advantage of be simple to understand and use. Furthermore, it was demonstrated that the low success of empirical tests of the CAPM may be partly due to problems in data and non-stationary of beta.

- The degree of riskiness of a security cannot be entirely (if at all) captured in an indicator which simply indicates the volatility of a security relative to the market.

- The model in my opinion are many, too many assumptions (from the efficient market without informational asymmetries, transaction costs, "not always" well received by the market and there are some issues too much general as the quadratic utility of wealth and the calculation of risk premium, but in fact are fundamental to the use of CAPM).

- Exactly the fair value of any financial asset, it decides the market in every moment and that we must do, otherwise if it were possible to reach a single universal value through the calculations, however, fairly trivial, there would be no market.

Alternative theories to the CAPM

APT: the return of any action depends in part on macroeconomic phenomena ( "factors") and partly by phenomena disturbance, specific events company.

Zero-Beta model (Black):

### 4. CAPM VS dividend valuation model

The basic model for evaluating the intrinsic value of an action - even from the chronological point of view, as first developed as a model for valuing shares - is that based on discounted dividends. According to this model, the value of a share is given by the present value of dividends that it expected. In an infinite horizon of evaluation, the value of the stock is a perpetuity of dividends where the discount rate is the cost of equity (e.g. determined using the CAPM).

The Dividend Discount Model (DDM), which is based on the same logic evaluation of bonds, it works properly for investments with a defined time horizon. But one of the key features of the actions is the vagueness of their maturity. Therefore in order to calculate the value of an action on a timescale defined by the DDM, it is necessary to estimate the expected price of the end of period itself. And in this regard emerges with immediacy the main theoretical limit of DDM as the intrinsic value of current is based on an estimate of the forward price, which is actually the value you are trying to estimate.

Although the value of action is based on dividends, the estimate on the horizon defined dividends not only provides an adequate intrinsic value. Companies not able to generate value may in fact distribute dividends in the medium term resorting to debt, while firms with good profitability may decide not to distribute any dividend.

A solution adopted to solve this problem consists in assuming that the dividend paid last year of explicit forecast period will remain forever. In this case, the price at maturity will be equal to the capitalization of a perpetuity with constant rate (obtained by relating the dividend over the cost of equity).

An alternative is to assume that the dividend received at the end of explicit forecast period will continue to grow indefinitely at a constant growth rate (known as g). In this case the value of the forward price is obtained by capitalizing on a perpetuity growing at rates = DIV / (Cost of equity - g). This model is known Gordon Growth Method.

The main limitation of Dividend Discount Method is that dividends are not a measure of value creation over a defined time horizon, despite the value of an action is determined by the same dividends. Company that creates value because it can borrow in order to distribute dividends, and highly profitable firms may decide not to distribute any dividend. A method to overcome the limitations of DDM, however, by adopting the same focus on cash flows, is the Discounted Cash Flow Method (DCF).

### 5. Conclusion

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