Theoretical limitations of the CAPM - Essay Example

? Financial Economics By Business of Consider a portfolio comprising two securities (A and B) with the following characteristics Security E (Ri) o (Ri) A 0.05 0.25 B 0.1 0.4 E (Ri) denotes the expected return for security i and the o (Ri) denotes the standard deviation of the returns on security i. a) Assuming the correlation between the returns on securities A and B, denoted ? (RA, RB), is zero, calculate the returns on security. The expected portfolio return E [Rp] = w1 E[A] + w2E[B] Let w1 be x and w2 be y such that E [Rp] = w1 (0.05) + w2 (0.1), since, w1 and w2 are not given we find the expected portfolio return as E [Rp] = 0.05w1 +0.1w2 Assume the ratio is 1:1 E [Rp] = 0.05 + 0.1 =1.05 The standards deviation of the portfolio return o (RP) We use the following formula, and then we find a positive square root to get the standard deviation of the portfolio s2p = (wA)2s2A + (wB)2s2B + 2wAwB SA, B s2p = wA (0.25)2 + wB (0.4)2 +2 wAwB (1) = 0.252 + 0.42 + (2* 0.25) = 0.7225 For the following sets of portfolio (XA, XB) = (1.5, -0.5), (1, 0), (0.5, 0.5), (0, 1), (-0.5, 1.5) b) Using analytical method, calculate the portfolio weights that are associated with the minimum variance portfolio. What is the minimum attainable variance? s2p = (XA)2s2XA + (1-XA)2s2 (1-XA) + 2XA(1-XA) sXA, (1-XA) Divide both sides by s2 p = (XA)2XA + (1-XA)2(1-XA) + 2XA(1-XA) XA(1-XA)s-1 Differentiate to obtain p = (XA)2XA + (1-XA)2(1-XA) c) Repeat parts (a) and (b), assuming the ? (RA, RB) = 0.5 d) Using your results from parts a to c, sketch the combination lines for the two cases of ?(RA, RB)=0 and ?(RA, RB) = 0.5 e) With reference to results of parts (a) to (d), explain briefly how the correlation between the returns on individuals’ securities affects the gain (in terms of reduction in risk) available to an investor who is willing to construct a diversified portfolio. 2. Consider the following information about the characteristics of two securities A and B; the market portfolio, M and the risk-free rate of return: Security (Ri, Rm) (Ri) A 0.2 0.3 B 0.4 0.7 E (RM) =0.1 o (RM) =0.5 RF= 0.04 ?(Ri, RM) denotes the correlation between the returns on security I and the returns on market portfolio; ?(RI) denotes the standard deviation of the returns on security I, E(RM) denotes the expected return on the market portfolio; o (RM) denotes the standard deviation of the returns on the market portfolio; and RF denotes the risk-free rate of return. A) Calculate (beta) for each of the following: i) Security A Beta = 0.1-0.04 0.2- 0.04 = 0.06/0.16 =0.375 ii) Security B Beta = 0.1-0.04 0.7-0.04 =0.0457 b. According to the Capital Asset Pricing model (CAPM), what are the expected returns for securities of A and B? We use the CAPM equation Ki = Krf + bi(Km - Krf) Where: Ki = the required return for the individual security Krf = the risk-free rate of return bi = the beta of the individual security Km = the expected return on the market portfolio (Km - Krf) is called the market risk premium KA = 0.04 + 0.375 (0.3) = 0.1525 For B KB =0.04 + 0.0457 (0.7) = 0.07199 c) Write down an expression for the security market line. Draw a sketch of the security market line, and indicate the positions of securities A and B on this line. Explain briefly how you would interpret the security market line Ki = Krf + bi(Km - Krf) Ki = 0.004 +0.375 (0.3) Ki = Krf + bi(Km - Krf) Ki = 0.004 +0.375 (0.7) E(Ri) B Undervalued Overvalued RM 0.5A 0 1 The assets above are undervalued because they reflect a high return. d. Write down expressions for the characteristic lines for securities A and B. Draw sketches of the characteristic lines for securities A and B. Explain briefly how you would interpret the characteristic lines. E(Ri) B Undervalued Overvalued RM 0.5A 0 1 3. This question refers to Figure 1. a) In the Capital Asset Pricing Model (CAPM), what are the values of ((betai) for securities A, B and C? A= 0.06 B= 0.08 C= 0.10 b) What are residual variances for securities A, B and C A=0.7 B= 0.6 C= 0.8 c) With reference to the notions of systematic risk and unsystematic risk, evaluate the claim that security C should be viewed as more risky than security A because o(RC> o(RA). C as a relatively higher o(RC) so it becomes susceptible to the volatile market and on the other hand, A remains less volatile and hence viewed as less risky. d) Consider portfolio P, in which 30% is, invested in security, 40% in security B and 30% in security C. What is the value OF E (RP), the expected return for portfolio P? A=30% =0.3, E (A) = 0.06 B=40% =0.4, E (A) = 0.08 C=30%=0.3, E (A) = 0.11 We use this formula E [Rp] = w1 E[A] + w2E[B] 0.3 *0.06 +0.4 *0.08* 0.3*0.11 =0.083 e) In the Capital Asset Pricing Model CAPM, what is the value of (betap) for portfolio P? We use this formula Ki = Krf + bi(Km - Krf) f) Verify that the portfolio beta is the weighted average of the securities that comprise the portfolio, i.e. ?p=0.3 ?A+0.4?B+0.3?C. 1. Discuss the main theoretical limitations of the CAPM The CAPM model indicates the association amid the asset risk and the anticipated return. It is used in pricing risky securities. Consider the CPM equation below E (Ri) =RF +?i [E (RM) - RF] The deviations experienced during the application of this model are attributed to the numerous assumptions. It is assumed that, all investors are risk vulnerable. Due to this, they select the most proficient portfolio based on projected return, variation and standard deviation. The assumption that, the deviation of returns is a sufficient extent of risk is not correct, because there are numerous factors attributed to returns (PHILLIPS, 2007). This may include inflation rate, which the model does not take into consideration. According to PHILLIPS, (2007) the CAPM needs asset returns to be generally distributed, with mean of zero. However, in the real world, the assets may be needed to be irregular. In addition, the assumption that all investors have similar expectations on security returns and less competitive market is incorrect. It is correct to say that, the investors choose a market portfolio that is in line with the purpose of risk-return outline. The imagination of friction- free market, which has no transaction cost, taxes trade precincts, is not practical and does not exist (PHILLIPS, 2007). Diverse investors have diverse investment abilities; hence, their investment outlay is an aspect to be taken into consideration in determining their resourceful portfolio. Otherwise, a wrong decision may lead to inappropriate selection of a portfolio, which might not be resourceful. It should be noted that in any decision making, the aspect of preference takes the center stage. This is because; every investor is bound to experience his own preference. Equally, due to the market portfolio’s nature of diverse investments opportunities, which in some cases are not observable, investors, use stock indexes as a proxy. This leads to inaccurate inferences about the soundness of CAPM. According to (PHILLIPS, 2007), CAPM assumes that, only two dates exists for an investor to undertake transactions such that, no extra time left to re-balance and use portfolios repeatedly. Moreover, it is normal that, diversification cannot remove systematic risk proceeds on all securities attributed to the market portfolio proceeds. This is a pure contrast to the CAPM’s assumption that investors should consider systematic risks because they can be eliminated through diversification. In conclusion, the extent of CAPM model’s limitations are bound on three primary aspects, which include the investors’ preferences on investment portfolios and investment aptitudes in relation to the investment outlay, which influences decision making on the type of investment to be considered. Secondly, the aspect of diversification: Diversification does not remove systematic risk returns on all securities attributed to market portfolio return (PHILLIPS, 2007). This is in contrast with the CAPM’s assumption that investors should consider systematic risks because diversification can purge them. b) Describe Roll’s Critique of the early empirical tests of the CAPM. This critical is one of the most common concerning, the validity of the CAPM equation. E (Ri) = Rf + ?i [E (Rm) – Rf ] The equation above assumes that, criticism that one of these properties indicates this risk –free proceeds. It indicates a risk – free souk. The investment portfolios make CAPM. Roll criticized the model in two primary ways. The mathematical fact ensures that, there are no properties of traverse sectional association between the graphs (BODURTHA, & MARK, 2001). It is realizable that, the associations between systematic and anticipated rate of return is 11%. The intercept created is divalent to the equation RZ=O, implying that, the return on the least variance is zero beta portfolio (BODURTHA, & MARK, 2001). Equally, the mean-variance proficient portfolio satisfies the CAPM equation wholly (BODURTHA, & MARK, 2001). This implies that, the index portfolio applied in estimating the betas may be attributed to the edge securities, and it does not represent the correct true market portfolio is efficient. This model proves that the portfolio proceeds and betas have nearly a linear association, even if an inefficient sample leads to the non-linear association amid the betas and particular securities and their expected returns (BODURTHA, & MARK, 2001). CAPM’s prediction on the efficiency of the global markets may not be correct in the real world because of its use of proxy for true market portfolio. Therefore, the assumptions attributed to the CAPM are superfluous. It becomes hard to observe the expected returns on the investment opportunities and hence hard to test the CAPM. In conclusion, Roll analyzes the CAPM on two primary aspects. The aspect of the properties of the CAPM equation, which indicates a risk –free profits, which is not true in the real world. Another contention involves the use of proxy in the true market in predicting, and this does not give a true reflection. How successfully does the Arbitrage Pricing Theory (ATP) address the weakness of the CPM that you identified in parts (a) and (b)? This is a universal theory, which implies that the price of a benefit has an influence on the stock price. The expected return on financial assets is seen as a linear equation intended to express diverse micro economic factors. Their sensitivity is represented by beta (CHAPPELL, 2007). The ATP applies the factor model, which assumes that security returns are always driven by a substantial factor, which is more of the market portfolio. Indeed, the ATP can perceive as a simplified CAPM that eliminates and improves the purpose of theory to be more bendable (CHAPPELL, 2007). Consider the following ATP model affected by few factors. Rit =RF +?1, iF1t +?2, iF2t +…+?k,i Fkt +?it Take the common factors affecting the return on assets be multiple and observable such as GDP inflation and interest rates (CHAPPELL, 2007). When considering the ATP, there is no need to assume that the utility functions of investors rely on only the expected returns and return on variance. In addition, the assumption of standard distribution of returns is redundant. It is essential to note that, the ATP is fulfilled if the arbitrators in the market construct prices approach stability, while the CAPM is fulfilled if all the investors choose appropriate portfolios only. List of references BODURTHA, J. N., & MARK, N. C. (2001). Testing the CAPM with Time-Varying Risks and Returns. The Journal of Finance. 46, 1485-1505. CHAPPELL, J. B. (2007). ATP. Burlington, N.C., Scientific Publications Division, Carolina Biological Supply Co PHILLIPS, J. (2007). CAPM. New York, McGraw-Hill. Read More