Security Market Line

10 Key excerpts on "Security Market Line"

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  Risk, Uncertainty And The Agricultural Firm

  • Charles Britt Moss(Author)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  In the capital market model, this supply is typically in the form of equity capital investments (stocks). 9.3.1. Deriving the Security Market Line In capital market equilibrium, the expected value–standard deviation frontier depicted in Fig. 9.3 and Eq. (9.14) equals the Security Market Line (SML) depicted in Fig. 9.4, which is consistent with the standard CAPM relationship E [ r j ] = ¯ r + β [ E ( r m ) − ¯ r ] . (9.18) Starting with a two-asset portfolio, construct a portfolio using invest-ment i and asset m . E [ R p ] = aE [ R i ] + (1 − a ) E [ R m ] σ ( R p ) = [ a 2 σ 2 i + 2(1 − a ) a σ im + (1 − a ) 2 σ 2 m ] 1 2 . (9.19) [ ] i E R 1 i β Fig. 9.4. The Security Market Line. 192 Risk, Uncertainty and the Agricultural Firm Next, examine the risk–return relationship based on changes in the share of asset i . ∂ E [ R p ] ∂ a = E [ R i ] − E [ R m ] ∂σ ( R p ) ∂ a = 1 2 a 2 σ 2 i + 2(1 − a ) a σ im + (1 − a ) 2 σ 2 m − 1 2 (9.20) × 2 a σ 2 i − 2 σ 2 m + 2 a σ 2 m + 2 σ im − 4 a σ im . Consider what happens as the share held in asset i becomes small ∂σ ( R p ) ∂ a a → 0 = σ im − σ 2 m σ m = σ im − σ 2 m σ m σ m σ m = [ β − 1 ] σ m . (9.21) The risk–return relationship as the share in asset i becomes small, is then derived as ∴ ∂ E [ R p ] ∂ a ∂σ ( R p ) ∂ a a → 0 = E [ R i ] − E [ R m ] [ β − 1 ] σ m = 1 β − 1 E [ R i ] − E [ R m ] σ m . (9.22) This relationship then yields E [ R m ] − r f σ m ∴ 1 β − 1 E [ R i ] − E [ R m ] σ m = E [ R m ] − r f σ m E [ R i ] − E [ R m ] = ( β − 1) [ E [ R m ] − r f ] (9.23) = β [ E [ R m ] − r f ] − E [ R m ] + r f ⇒ E [ R i ] − r f = β [ E [ R m ] − r f ] . Market Models of Decision Making under Risk 193 The result in Eq. (9.23) implies that the premium (or increased expected return over the risk-free interest rate) is directly proportional to the premium on the market portfolio and the relative riskiness of the individual asset measured by its “beta” ( β ).

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  Quantitative Financial Analytics

  The Path to Investment Profits

  • Edward E Williams, John A Dobelman;;;(Authors)
  • 2017(Publication Date)
  • WSPC

    (Publisher)

  Sharpe contends that the price (and thus return) of a given security should not be determined in relation to its total risk, because the security will be combined with other securities in a portfolio and some of the individual risk will be eliminated by diversification (unless all the securities are perfectly positively correlated). Therefore, the return of the security should only contain a risk premium to the extent of the risk that will actually be borne (that is, that portion of the total risk which cannot be eliminated by diversification — which is variously called nondiversifiable risk or systematic risk). Figure 10.7. The SML If this logic is accepted, it is then possible to generate a Security Market Line (SML), as shown in Figure 10.7, where the return on individual securities is related to their covariance with the market. If capital markets are in equilibrium and all the other assumptions of this chapter hold, then the parameters of each security should lie on the SML. Furthermore, because the risk of a portfolio is the weighted sum of the nondiversifiable risk of its component securities, all portfolios should also fall on the SML in equilibrium. It should be noted that although all portfolios will fall on the SML, as a general rule no individual securities will fall on the CML (they will all lie to the right of the CML). Several additional points may be made regarding the SML. In the first place, several authors differentiate between defensive and aggressive securities on the basis of whether the covariance of the security with the market is less or greater than the variance of the market return itself. If C iM < σ 2 M, then the changes in the security’s return caused by market changes are less than the market changes themselves and the security is considered defensive

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  Investments

  Analysis and Management

  • Gerald R. Jensen, Charles P. Jones(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  We also now know that under CMT all investors hold the same portfolio of risky assets, the market portfolio. Therefore, the risk that matters when we consider any security is its beta. • The major conclusion of the CAPM is as follows: The relevant risk of any secu-rity is the amount of risk that security contributes to a well-diversified portfo-lio, that is, its beta. The CAPM’s expected return–beta relationship The Security Market Line (SML) is the graphical depiction of the CAPM equation. The SML depicts the rela-tionship between risk and expected return for any asset, security, or portfolio. The CAPM posits a linear relationship between an asset’s risk and its expected return. The Security Market Line (SML) is shown in Figure 9.7. Required (expected) return is on the vertical axis and beta, the measure of risk, is on the horizontal axis. 6 The slope of the SML is the difference between the required return on the market index and RF. Security Market Line (SML) The graphical depiction of the CAPM Investments Intuition Investors first exposed to the concepts of beta and CAPM may hear someone say that high-beta stocks held last year produced a lower return than did low-beta stocks, and therefore something is wrong with this concept. This is a fallacy, however, because the CAPM relationship is an equilibrium relationship expected to prevail in the long run. High-beta stocks have greater risk than low-beta stocks and are expected to produce higher returns. However, they will not typ-ically produce higher returns every period and over all intervals of time. If they did, they would be less risky than low-beta stocks, not more risky. The correct statement is that over long periods of time, high-beta stocks should produce higher average returns. 6 In equilibrium the expected and required rates of return are the same for a security.

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  The Economics of Financial Markets

  • Roy E. Bailey(Author)
  • 2005(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Sometimes the SML is written in terms of the excess of the expected return over the risk-free rate, j − r 0 ; the SML then passes through the origin in the diagram. A practical application of the CAPM is to plot the observed average rates of return on individual assets, or portfolios of assets, against their estimated beta-coefficients. The CAPM predicts that the resulting graph will trace out the SML – that is, a line with intercept given by the risk-free rate and slope equal to the average return on the market portfolio minus the risk-free rate. It is always necessary to allow for some statistical error, so that the observations will not fall exactly on a line. Statistical analysis provides a systematic way of studying how well the data matches the prediction of the theory (see chapter 9). 3 Why would any investor ever wish to hold an asset with a negative beta-coefficient? It is, after all, a risky asset with an expected return lower than the risk-free rate. The answer is that such an asset might be included in a portfolio of assets because its covariance with the return on other assets enables the investor to control the risk on the portfolio as a whole. 152 The economics of financial markets ✲ ✻ ✟ j j r 0 1 M SML Fig. 6.3. The Security Market Line The SML views the CAPM prediction j = r 0 + M − r 0 j as a linear relationship between j and j . The model predicts that the average rates of return and beta-coefficients for all assets, and all portfolios of assets, will be located along the SML. 6.4.1 Disequilibrium How are divergences of average rates of return and beta-coefficients from the SML to be interpreted? One possibility is that the divergences are simply the result of statistical variation or mismeasurement; j , M and j are, after all, purely theoretical concepts – unknown parameters that reflect investors’ beliefs. As mentioned above, ways must be found for estimating these parameters in order to implement the theory.

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  The Capital Budgeting Decision

  Economic Analysis of Investment Projects

  • Harold Bierman, Jr., Seymour Smidt(Authors)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  is

  The relationship between expected return and beta is the Security Market Line; it is the major mathematical relationship of the capital asset pricing model. Figure 8.9 shows the Security Market Line. Note that β is measured on the X axis and expected return on the Y axis. M is the market portfolio with a beta of 1. The slope of the line is . For a security with beta of β

  i

  , the risk premium is

  If the risk is β = 1, then the expected return is and the investment has the same expected return as the market portfolio. The Security Market Line leads to a conclusion that if a security has more systematic risk, the market will require a higher return.

  Earlier we wrote that but since we can write the formulation for in a variety of ways by substituting for β. If we define then

  The equation for can be presented in a large number of different forms. All have ri as a function of and β or the components of β.

  Figure 8.9  The Security Market Line

  Table 8.1 Summary of graphed information

  Relationships	X axis	Y axis
  (Fig. 8.7 ) Capital market line	Standard deviation of rate of return	Expected rate of return on the market portfolio
  (Fig. 8.8 ) Security characteristic line	Excess market return	Excess return for security
  (Fig. 8.9 ) Security Market Line	Beta (risk)	Expected rate of return for a security

  Security A of Figure 8.9 is above the Security Market Line. It has a higher expected return than the market requires for its given risk. We can expect the price of A to increase so that its expected return decreases and A returns to the Security Market Line. Investors will buy A, driving its price up and its return down.

  Security B is below the Security Market Line. We can expect the price of B to decrease as investors sell it so that its expected return increases, and B will then sit on the Security Market Line. In equilibrium, all securities will be on the Security Market Line.

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  Introduction to Corporate Finance

  • Laurence Booth, Ian Rakita(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  The new (or super) efficient frontier portfolios are composed of the risk-free rate and the tangent portfolio that offer the highest expected rate of return for any given level of risk. 9.2 Explain how modern portfolio theory is extended to develop the capital market line (CML). The “optimal” risky portfolio is the one that is tangent to the efficient frontier on a line that is drawn from RF, as shown in Figure 9.4. This portfolio will be the same for all investors. This optimal risky portfolio will be the market portfolio (M), which contains all risky securities. The value of this portfolio will equal the aggregate of the market val- ues of all the individual assets composing it. Therefore, the weights of these assets in the market portfolio will be represented by their pro- portionate weight in its total value. 9.3 Explain how the capital asset pricing model’s (CAPM) Security Market Line (SML) is developed from the capital market line (CML), and how the SML can be used to estimate the required return on individual securities and portfolios. Because of diversification, the individual risk can be diversified away, but the market risk remains. So the security price will finally depend on its co-movement with the market, which is measured by beta. Then CML is developed into SML. SML can be applied easily into asset pricing; see Equation 9.9. 9.4 List alternative risk-based pricing models and de- scribe how they differ from the CAPM. Alternative asset pricing models include the Fama-French model and the arbitrage pricing theory. The Fama-French model includes the market factor, value factor, and size factor, and is empirically suc- cessful. The arbitrage pricing theory introduces more factors, while CAPM has only one factor: the market portfolio. Questions and Practice Problems 9-25 insurance premium, p. 9-2 market portfolio, p. 9-7 market price of risk, p. 9-9 market risk premium, p. 9-16 new (or super) efficient frontier, p.

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  R for Programmers

  Quantitative Investment Applications

  • Dan Zhang(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  x value as M. M and B’ have the same risk, but M gets higher return, for it is above B’. That is, when the risks are the same, we will all choose the asset with the highest return.

  The slope of M is the biggest in the investible area, so we consider M the optimal portfolio of assets in the market.

  No matter what risk and return preferences the investors have, we can make the best investment plans for them as long as we find out the optimal portfolio and know the risk-free asset. For rational investors, they will only invest in the optimal portfolio, if found, regardless their return and risk preferences (Figure 2.5 ).

  Figure 2.4   The optimal portfolio.

  The portfolio of M is usually constituted with all the investible securities, and the proportion of each security asset is the relative market value of the security. The risk-free asset C is not included in M. People will always use CM connecting line for investment and for building the optimal portfolio.

  Figure 2.5   Capital market line.

  In a practical trading, the price of the financial assets may deviate for the impact of supply and demand in the market. When there’s price deviation, the market will automatically repair and restore the price to its equilibrium level.

  2.1.2.4 Capital Market Line (CML)

  The CM connection line fits the theory of portfolio selection which Markowitz put forward, where he considered the intersection point of the indifference curve of investors with risk aversion nature and the asset’s effective boundary line as the best portfolio. And the CM connection line is called the Capital Market Line (CML).

  The CML shows the simple linear relation between the expected return and the standard deviation of efficient portfolios.

  The CML determines the prices of securities, because the CML is the balance of risk and return in an efficient portfolio. If this balance is thrown off, another relation of risk and return will be formed beyond the CML.

  2.1.2.5 Portfolio Construction

  The CML is the optimal portfolio. If we find this portfolio, all of our money will be invested in it. We can get the risk and return we want by assigning different weights of investment to the risk-free asset C and the risk asset M; we can also use financial instruments to add leverage and magnify the risk and return, as shown in Figure 2.6

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  Operations Research Applications

  • A. Ravi Ravindran(Author)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  The CML begins at the risk-free rate on the vertical axis (which has risk 0) and is exactly tangent to the efficient frontier. The point where the CML touches the efficient frontier is the pair ( σ m , r m ), which is defined to be the “market” standard deviation and “market” expected return. By changing the relative proportions of riskless asset and market portfolio, an investor can obtain any combination of risk and return that lies on the CML. Because the CML is tangent to the efficient frontier at the market point, there are no other combinations of risky and riskless assets that can provide better expected returns for a given level of risk. Now, consider a particular investor portfolio i with expected return E ( r i ) and standard deviation σ i . For an investor to choose to hold this portfolio it must have returns comparable to the returns that lie on the CML. Thus, the following must hold: E ( r i ) = r f + σ i σ m ( r m − r f ) where r f is the risk-free rate. This equation is called the CAPM. The interpretation of the CAPM is that investors’ portfolios have an expected return that includes a reward for taking on risk. This reward, by the CAPM hypothesis, must be equal to the return that would be obtained from holding a portfolio on the CML that has an equivalent risk. Any remaining risk in the portfolio can be diversified away (by, for example, holding the market portfolio) so the investor does not gain any reward for the non-systematic, or diversifiable, risk of the portfolio. The CAPM argument applied to individual securities implies that the holders of individual securities will be compensated only for that part of the risk that is correlated with the market, or the so-called systematic risk. For an individual security j the CAPM relationship is E ( r j ) = r f + β j ( r m − r f ) where β j = ( σ j /σ m ) ρ mj and ρ mj is the correlation between asset j returns and the market returns. The Black–Litterman approach uses the CAPM in reverse.

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  Quantitative Financial Economics

  Stocks, Bonds and Foreign Exchange

  • Keith Cuthbertson, Dirk Nitzsche(Authors)
  • 2005(Publication Date)
  • Wiley

    (Publisher)

  According to the CAPM/SML, the average excess monthly return (say over 60 months) on each asset ER i − r should be proportional to that asset’s beta – a security with a high beta has a high risk and therefore should earn a high average return: Excess return on security i Excess return on security j = (ER i − r) (ER j − r) = β i β j (39) S E C T I O N 5 . 4 / B E TA A N D S Y S T E M AT I C R I S K 135 Required return = SML and actual return = SML Asset’s beta, b i Q (buy) S (sell) P Actual return Expected return 0.5 1 1.2 T (sell) M Securities that lie above (below) the SML have a positive (negative) ‘alpha’, indicating a positive (negative) ‘abnormal return’, after correcting for ‘beta risk’. r a Q > 0 a S < 0 Figure 8 Security Market Line, SML The SML can be used to try to pick underpriced and overpriced stocks. To see this, consider a security S (Figure 8) with a beta of 0.5. You could replicate the beta of security S by buying a portfolio with 50% in the safe asset (β = 0) and 50% in a security with a beta of unity (i.e. β p = 0.5(0) + 0.5(1) = 0.5). But this synthetic portfolio would lie on the SML (at P) and have a higher expected return than S. Hence, S would be sold, since its actual return is less than its equilibrium return given by its beta. If S were sold, its current price would fall and this would raise its expected return, so that S moved towards P. (Similarly, security T with β i = 1.2 could be duplicated by borrowing 20% of your wealth at the safe rate and using your own funds plus borrowed funds to invest in a security with a β i = 1.) Alternatively, consider a security like Q also with β i = 0.5 (Figure 8) but which currently has a higher average return than indicated by the SML. An investor should purchase Q. Securities like Q and S are currently mispriced (i.e. they are not on the SML), and a speculator might short sell S and use the funds to purchase Q.

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  Modern Portfolio Theory and Investment Analysis

  • Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, William N. Goetzmann(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  This generalized asset pricing model was first derived by Bawa and Lindenberg (1977) who showed that it corresponded with an equilibrium model where agents have utility functions for wealth displaying diminishing absolute risk aversion. They further show that if returns are multi- variate normal or Student t, the risk measure collapses to  MV , and the standard CAPM result follows. QUESTIONS AND PROBLEMS 1. Assume that the following assets are correctly priced according to the Security Market Line. Derive the Security Market Line. What is the expected return on an asset with a beta of 2? 2. Assume the Security Market Line given below. Assume that analysts have estimated the beta on two stocks as follows:  x  0.5 and  y  2. What must the expected return on the two securities be in order for them to be a good purchase? 3. Assume that over some period, a CAPM was estimated. The results are shown below. Assume that over the same period, two mutual funds had the following results: Fund A Actual return  10% Beta  0.8 Fund B Actual return  15% Beta  1.2 What can be said about the fund performance? 4. Consider the CAPM line shown below. What is the excess return of the market over the risk-free rate? What is the risk-free rate? 308 PART 3 MODELS OF EQUILIBRIUM IN THE CAPITAL MARKETS 5. Write the CAPM shown in Problem 4 in price form. 6. Show that the standard CAPM should hold even if short sales are not allowed. 7. Assume that an asset exists with R – 3  15% and  3  1.2. Further assume the Security Market Line discussed in Problem 1. Design the arbitrage opportunity. 8. If the following assets are correctly priced on the Security Market Line, what is the return of the market portfolio? What is the risk-free rate? 9. Given the Security Market Line What must be the returns for two stocks, assuming their s are 1.2 and 0.9? BIBLIOGRAPHY 1. Aivazian, Varouj. “The Demand for Assets under Conditions of Risk: Comment,” Journal of Finance, XXXII, No.

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