23.1. Assumptions of Capital Market Theory

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1 NPTEL Course Course Title: Security Analysis and Portfolio anagement Course Coordinator: Dr. Jitendra ahakud odule-12 Session-23 Capital arket Theory-I Capital market theory extends portfolio theory and develops a model for pricing all risky assets. In this regard Capital asset pricing model (CAP) 1 allows us to determine the required rate of return for any risky asset Assumptions of Capital arket Theory Capital market theory which builds on the arkowitz portfolio theory, is having the following assumptions: All investors are arkowitz efficient investors who want to target points on the efficient frontier. Investors can borrow or lend any amount of money at the risk-free rate of return (R). All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return. All investors have the same one-period time horizon such as one-month, six months, or one year. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. There are no taxes or transaction costs involved in buying or selling assets. There is no inflation or any change in interest rates, or inflation is fully anticipated. Capital markets are in equilibrium. This means that we begin with all investments properly priced in line with their risk levels. 1 1

2 23.2. The Capital Asset Pricing odel: Expected Return and Risk CAP indicates what should be the expected or required rates of return on risky assets. This helps to compare an estimated rate of return to the required rate of return implied by CAP i.e., to know a certain stock is over/under valued. With the explanation of the existence of a risk-free asset resulted in deriving a capital market line (CL) that became the relevant frontier it explains that an asset s covariance with the market portfolio is the relevant risk measure. This can also be used to determine an appropriate expected rate of return on a risky asset by using the approach of CAP and to value an asset by providing an appropriate discount rate to use in dividend valuation models The Concept of Risk-Free Asset Risk-free asset which provides the risk-free rate of return (R) is an asset with zero variance and have zero correlation with all other risky assets. Following the approach for the determination of covariance between assets i and risk free () asset: Because the returns for the risk free asset are certain, i.e., the, and thus Thus Ri = 0 E(Ri), and Ri - E(Ri) = 0. Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero. Similarly the correlation between any risky asset i, and the risk-free asset (), would be zero because the r Covi Combining a Risk-Free Asset with a Risky Portfolio 2 i Cov i = 0. The expected rate of return for a portfolio that includes a risk-free asset is the weighted average of the two returns: E(R ) W (R) (1-W )E(R ) Where, w = the proportion of the portfolio invested in the risk-free asset, E(Ri) = the expected rate of return on risky Portfolio i n Cov [R -E(R )][R -E(R )]/n i i i i1 port i 2 ost of the explanations in this section is derived from the Reilly, Frank. and Brown, Keith, Investment Analysis & Portfolio anagement, 7th Edition (pp ), Thomson Soth-Western. 2

3 Thus the expected variance for a two-asset portfolio is: E( ) w (1 w ) 2w (1-w )r port i,i i Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula E( ) (1 w ) i port Given the variance formula the standard deviation is: E( ) (1 w ) i port (1w ) i 2 2 Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio. Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets. The efficient frontier which gives higher return is the point where the line is tangent to the frontier. The set of portfolio possibilities along Line R- dominates all portfolios below Point. For example, an investor could attain a risk and return combination between the R and Point (Point C) by investing one-half of the portfolio in the risk-free asset (that is, lending money at the R) and the other half in the risky portfolio at Point. 3

4 23.5. Risk-Return Possibilities with Leverage From the above discussion it is clear that to attain a higher expected return than is available at point (in exchange for accepting higher risk). Either invest along the efficient frontier beyond point, such as point D or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point. If an investor borrow an amount equal to 40 percent of the original wealth at the risk-free rate, w will not be a positive fraction but, rather, a negative 60 percent (w = 0.60). The effect on the expected return for your portfolio is: E( R ) w ( R) (1 w ) E( R ) Port =.60( R) (1 (.40) E( R ) =.60R 1.40 E( R ) If for example the R=.04 and ER ( ER ( ) (.60)(.04) 1.40(.18) 0.23 Port )=.18, then Thus return will increase in a linear fashion along the Line R- because the gross return is increases with the high level of risk. Therefore, both return and risk increase in a linear fashion along the original Line R-, and this extension dominates everything below the line on the original efficient frontier. Thus, we have a new efficient frontier: the straight line from the R tangent to Point. This line is referred to as the capital market line (CL). Because the CL is a straight line, it implies that all the portfolios 4

5 on the CL are perfectly positively correlated. The more conservative the investor the more is placed in risk-free lending and the less borrowing. The more aggressive the investor the less is placed in risk-free lending and the more borrowing. ost aggressive investors would use leverage to invest more in portfolio The Concept of arket Portfolio In the above example because Portfolio lies at the point of tangency, it has the highest portfolio possibility line, and everybody will want to invest in Portfolio and borrow or lend to be somewhere on the CL. This portfolio must, therefore, include all risky assets. If a risky asset were not in this portfolio in which everyone wants to invest, there would be no demand for it and therefore it would have no value. This portfolio that includes all risky assets is referred to as the market portfolio. Because the market portfolio contains all risky assets, it is a completely diversified portfolio, which means that all the risk unique to individual assets in the portfolio is diversified away. This unique (diversifiable) risk is also referred to as unsystematic risk. 3 The optimal portfolio is at the highest point of tangency between Capital arket Line (CL) and the efficient frontier. The CL leads all investors to invest in the same risky asset portfolio, the portfolio. Individual investors should only differ regarding their position on the CL, which depends on their risk preferences. The risk preference of individual investors can be regarded as their choice or preferable position along the CL line with respect to their financing decision. In other words when an investor decides to invest in the market Portfolio nad wants to be on the CL is because of the investment decision. But the decision either to borrow or to lend to attain the preferred risk position on the CL is known as the financing decision. This division of the investment decision from the financing decision is known as the separation theorem. The major points of separation theorem are as follows: The CL leads all investors to invest in the portfolio Individual investors should differ in position on the CL depending on risk preferences How an investor gets to a point on the CL is based on financing decisions 3 ost of the explanations in this section is derived from the Reilly, Frank. and Brown, Keith, Investment Analysis & Portfolio anagement, 7th Edition (pp ), Thomson Soth-Western. 5

6 Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio Investors preferring more risk might borrow funds at the R and invest everything in the market portfolio The decision of both investors is to invest in portfolio along the CL (the investment decision) The decision to borrow or lend to obtain a point on the CL is a separate decision based on risk preferences (financing decision) Tobin refers to this separation of the investment decision from the financing decision, the separation theorem Capital arket Line The concept of capital arket Line (CL) states that the relevant risk measure for risky assets is their covariance with the arket portfolio (), which is referred to as their systematic risk. Since all individual risky assets are a part of the portfolio, the rates of return in relation to the returns for the portfolio using the following linear model: R R it i i t Where, Ri,t is return for asset i during period t, ai is constant term for asset i, i is the slope coefficient for asset i, Rt is the return for the portfolio during period t, is the random error term. 6

7 Slope of the CL is the market price of risk for efficient portfolios, or the equilibrium price of risk in the market The Security arket Line The security market line (SL) visually represents the relationship between risk and the expected or the required rate of return on an asset. The equation of SL, together with estimates for the return on a risk-free asset and on the market portfolio, can generate expected or required rates of return for any asset based on its systematic risk. Given this standardized measure of systematic risk, the SL graph can be expressed as follows: The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m). The equation for the risk-return line is: R -R E(R i) R (Cov 2 i,) Cov R i, 2 We then define i as beta ( ). (R -R) Cov i, 2 Thus the equation becomes: E(R i) R i (R -R) systematic risk. Here the Beta can be viewed as a standardized measure of Determining the Expected Rate of Return for a Risky Asset The expected rate of return of a risk asset is determined by the R plus a risk premium for the individual asset. The risk premium for a security i is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (R-R). E(R i) R i (R -R) Following points are important to remember with respect to the determination of expected rate of return for risky assets: In equilibrium, all assets and all portfolios of assets should plot on the SL Any security with an estimated return that plots above the SL is under priced 7

8 Any security with an estimated return that plots below the SL is overpriced A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor Compare the required rate of return to the expected rate of return for a specific risky asset using the SL over a specific investment horizon to determine if it is an appropriate investment Additional Readings: Alexander, Gordon, J., Sharpe, William, F. and Bailey, Jeffery, V., Fundamentals of Investment, 3 rd Edition, Pearson Education. Bodie, Z., Kane, A, arcus,a.j., and ohanty, P. Investments, 6 th Edition, Tata cgraw-hill. Bhole, L.., and ahakud, J. (2009), Financial institutions and markets.5th Edition, Tata cgraw Hill (India). Fisher D.E. and Jordan R.J., Security Analysis and Portfolio anagement, 4th Edition., Prentice-Hall. Jones, Charles, P., Investment Analysis and anagement, 9 th Edition, John Wiley and Sons. Prasanna, C., Investment Analysis and Portfolio anagement, 3rd Edition, Tata cgraw-hill. Reilly, Frank. and Brown, Keith, Investment Analysis & Portfolio anagement, 7th Edition, Thomson Soth-Western. Additional Questions with Answers Session-23: Capital arket Theory-I 1. What is the meaning of Capital Asset Pricing odel? Ans. Asset pricing theory helps us to understand the cross-sectional differences in expected returns for a given set of assets. It offers a framework to identify and measure risk as well as assign rewards for risk bearing. The Capital Asset Pricing odel (CAP) indicates what should be the expected or required rates of return on risky assets. It helps to value an asset by providing an appropriate discount rate to use in dividend valuation models. It can be used to compare an estimated rate of return to the required rate of return implied by CAP for over/under valued securities CAP will allow you to determine the required rate of return for any risky asset. This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models ajor Assumptions: 8

9 i. All investors are arkowitz efficient investors who want to target points on the efficient frontier. ii. Investors can borrow or lend any amount of money at the risk-free rate of return (R). iii. All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return. iv. All investors have the same one-period time horizon such as one-month, six months, or one year. v. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. vi. There are no taxes or transaction costs involved in buying or selling assets. vii. There is no inflation or any change in interest rates, or inflation is fully anticipated. viii. Capital markets are in equilibrium. This means that we begin with all investments properly priced in line with their risk levels. 2. How Risk-Free Asset can influence portfolio expected risk and return prospective? Ans. Risk-Free Asset: An asset with zero standard deviation Zero correlation with all other risky assets Provides the risk-free rate of return (R) Will lie on the vertical axis of a portfolio graph The existence of a risk-free asset resulted in deriving a capital market line (CL) that became the relevant frontier The covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero. Combining a Risk-Free Asset with a Risky Portfolio Expected return: the weighted average of the two returns is a linear relationship. E(R port ) W(R) (1- W)E(R i) Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio. 3. What is the difference between Capital arket Line and Securities arket Line? Capital arket Line: Slope of the CL is the market price of risk for efficient portfolios, or the equilibrium price of risk in the market. Relationship between risk and expected return for portfolio P (Equation for CL): E(R ) E(Rp ) σ p σ The CL leads all investors to invest in the portfolio. Individual investors should differ in position on the CL depending on risk preferences. Risk averse 9

10 investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio Security arket Line (SL): SL estimates the return of a single security relative to its exposure to systematic risk. It helps to find out that whether the expected return on the securities is better than the risk which he is taking. The Security arket Line (SL): R E(R ) R i - R (Cov 2 i, Implication of SL for determining the Expected Rate of Return for a Risky Asset ) In equilibrium, all assets and all portfolios of assets should plot on the SL Any security with an estimated return that plots above the SL is underpriced Any security with an estimated return that plots below the SL is overpriced A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor Compare the required rate of return to the expected rate of return for a specific risky asset using the SL over a specific investment horizon to determine if it is an appropriate investment CL a special case of SL. While CL represents Return potential and risk involved in all financial asset across the Capital market, SL is the linear relationship between the expected return of security and its systematic risk, the expected return comparing a riskfree return plus a risk premium. 4. What are the Characteristics of the arket Portfolio? Ans. Characteristics arket Portfolio: All risky assets must be in portfolio, so it is completely diversified: Includes only systematic risk All securities included in proportion to their market value Contains worldwide assets: Financial and real assets ost important implication of the CAP All investors hold the same optimal portfolio of risky assets The optimal portfolio is at the highest point of tangency between Capital arket Line (CL) and the efficient frontier The portfolio of all risky assets is the optimal risky portfolio Called the market portfolio 10