Financial Economics - Assignment Example

ASB-3207 Financial Economics work assignment, 2006-07 session This assignment carries a 25% weighting in the overall module assessment. Submission deadline: Friday 15 December 2006 1. Consider a portfolio comprising two securities (A and B) with the following characteristics: Security E(Ri) (Ri) A 0.05 0.25 B 0.1 0.4 E(Ri) denotes the expected return for security i, and (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: (i) the expected portfolio return, E(RP), and (ii) the standard deviation of the portfolio return, (RP) for the following sets of portfolio weights: (xA, xB) = (1.5, –0.5), (1 0), (0.5, 0.5), (0, 1), (–0.5, 1.5) Assuming the correlation between the returns on securities A and B, denoted (RA,RB), is zero RP = XA RA + XB RB s2 = XA2s12 + XB2 s22 + 2 XA XB sAsB r Weights Returns(RP) Risk(s2) S1 S2 150.00% -50.00% 3% 18% 100.00% 0.00% 5% 6% 50.00% 50.00% 8% 6% 0.00% 100.00% 10% 16% -50.00% 150.00% 13% 38% (b) Using an analytical method, calculate the portfolio weights that are associated with the minimum variance portfolio. What is the minimum attainable variance? [HINT: - write down an equation for var(RP) in terms of xA and xB; - replace xB with (1–xA), to obtain an equation for var(RP) in terms of xA only; - differentiate the resulting expression with respect to xA, and set the derivative to zero; - solve the resulting equation for xA.] Min s2 = XA2sA2 + XB2 s22 + 2 XA XB sBs2 r Since r=0 and XB = (1-XA) Min s2 = XA2sA2 + (1-XA) 2 sB2 Min s2 = XA2sA2 + sB2 + XA2sB2 - 2 XAsB2 Differentiating with respect to XA and equating to 0, we get 0 = 2sA2 +2 XAsB2 - 2 sB2 0 = 2sA2 +2 (XAsB2 - sB2) 2s12 = 2 (XAsB2 - sB2) sA2 = XAsB2 - sB2 sA2 + sB2 = XAsB2 XA = sB2 / (sA2 + sB2) And XB = sA2 / (sB2 + sA2) Accordingly, Weights are 71.91% and 28.09% respectively for Security A and Security B, for minimum variance portfolio The return and risk are 6% and 4% respectively, for minimum variance portfolio (c) Repeat parts (a) and (b), assuming (RA,RB)=0.5. Weights Returns Risk (RA,RB)=0 Risk (RA,RB)=0.5 S1 S2 150.00% -50.00% 3% 18% 11% 100.00% 0.00% 5% 6% 6% 50.00% 50.00% 8% 6% 8% 0.00% 100.00% 10% 16% 16% -50.00% 150.00% 13% 38% 30% 71.91% 28.09% 6% 4% 7% (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 the results of parts (a) to (d), explain briefly how the correlation between the returns on individual securities affects the gain (in terms of reduction in risk) available to an investor who is willing to construct a diversified portfolio. When there is correlation between securities in portfolio then the risk is less compared to securities that are not related in portfolio. As, it can be seen in calculations above, the risk for portfolio of securities those are correlated with different weights varies from 6% to 30%. But the risk for portfolio of securities those are not correlated with different weights varies from 4% to 38%. The lower correlation value of securities in portfolio means lower list. Correlation can vary between -1 to +1. (20 marks) 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 (RM) = 0.5 RF = 0.04 (Ri, RM) denotes the correlation between the returns on security i and the returns on the market portfolio; (Ri) denotes the standard deviation of the returns on security i, E(RM) denotes the expected return on the market portfolio; (RM) denotes the standard deviation of the returns on the market portfolio; and RF denotes the risk-free rate of return. (a) Calculate i (beta) for each of the following: (i) Security A (ii) Security B BA = Covariance(RA, RM)/ (RA) Covariance(A,M)= (RA, RM) * (RA) * (RM) = 0.2 * 0.3 * 0.5 = 0.03 BA = 0.03 / 0.3 BA = 0.1 or 10% BB = Covariance(RB, RM)/ (RB) Covariance(B,M)= (RB, RM) * (RB) * (RM) = 0.4 * 0.7 * 0.5 = 0.14 BB = 0.14 / 0.7 BB = 0.2 or 20% (b) According to the Capital Asset Pricing Model (CAPM), what are the expected returns for securities A and B? RA = RF + BA(RM- RF) = 0.04 + 0.1 (0.1-0.04) = 0.046 or 4.6% RB = RF + BB(RM- RF) = 0.04 + 0.2 (0.1-0.04) = 0.052 or 5.2% (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. RA = RF + BA (RM- RF) The risk free rate of returns is 4%. Beyond that, the higher the risk, higher the returns. In the above Security Market Line, Security A gives 4.5% returns at 10% risk. Security B gives 5.2% returns at 20% risk. The slope of the security market line is the risk(B-Beta). (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. (10 marks) 3. This question refers to Figure 1. (a) In the Capital Asset Pricing Model (CAPM), what are the values of i (beta) for securities A, B and C? From the Picture E(RA)=0.06 (RA)=0.7 E(RB)=0.08 (RB)=0.6 E(RC)=0.11 (RC)=0.8 E(RM)=0.1 (RM)=0.5 RF = 0.05 BA = (RA - RF) / (RM - RF) BA = 0.2 or 20% BB = (RB - RF) / (RM - RF) BB = 0.6 or 60% BC = (RC - RF) / (RM - RF) BC = 1.2 or 120% (b) What are the residual variances for securities A, B and C? Total Risk for security = Market Related Risk + residual variance for security 2(RA) = BA2 2 (RM) + residual variance for security Residual variance for A = 2(RA) - BA2 2 (RM) = (0.7)2 – (0.2)2 (0.5)2 = 0.48 or 48% 2(RB) = BB2 2 (RM) + residual variance for security Residual variance for B = 2(RB) - BB2 2 (RM) = (0.6)2 – (0.6)2 (0.5)2 = 0.27 or 27% 2(RC) = BC2 2 (RM) + residual variance for security Residual variance for C = 2(RC) – BC2 2 (RM) = (0.8)2 – (1.2)2 (0.5)2 = 0.28 or 28% (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 (RC)>(RA). The systematic risk is common for all the securities in the market. So, the systematic risk is common for the securities A and C too. The unsystematic risk is unique to each and every security. The risk of Securities A and C are 6% and 11% respectively. The systematic risk has to be less than 6%. Assuming that systematic risk is x %, the unsystematic risk for security A would be 6%-x% and for Security C would be 11%-x%. Since 11%-x% would be greater than 6%-x%, the Security C has higher risk compared to Security A. (d) Consider portfolio P, in which 30% is invested in security A, 40% in security B and 30% in security C. What is the value of E(RP), the expected return for portfolio P? Security Returns Risk Weight alpha A 6% 70% 30% 1.80% B 8% 60% 40% 3.20% C 11% 80% 30% 3.30% Weighted Returns (RP) 8.30% (e) In the Capital Asset Pricing Model (CAPM), what is the value of P (beta) for portfolio P? RP=RF+BP (RM-RF) So BP= (RP - RF) / (RM-RF) BP= (0.083 - 0.05) / (0.1-0.05) BP= 0.66 or 66% (f) Verify that the portfolio beta is the weighted average of the betas of the securities that comprise the portfolio, i.e. P=0.3A+0.4B+0.3C. Security Weight Beta A 30% 6% B 40% 24% C 30% 36% Weighted Beta (BP) 66.00% (10 marks) 4. (a) Discuss the main theoretical limitations of the CAPM. (20 marks) The Capital Asset Pricing Model (CAPM) is an equilibrium model which specifies the relationship between risk and required rates of return on assets when they are held in well-diversified portfolios. The CAPM requires an extensive set of assumptions: All investors are single-period expected utility of terminal wealth maximizers, who choose among alternative portfolios on the basis of each portfolios expected return and standard deviation. All investors can borrow or lend an unlimited amount at a given risk-free rate of interest. Investors have homogeneous expectations (that is, investors have identical estimates of the expected values, variances, and covariances of returns among all assets). All assets are perfectly divisible and perfectly marketable at the going price, and there are no transactions costs. There are no taxes. All investors are price takers (that is, all investors assume that their own buying and selling activity will not affect stock prices). The quantities of all assets are given and fixed. Apart from those assumptions, the following are other limitations of CAPM The market portfolio is a theoretical concept. There is no consensus on which proxy for the market portfolio is best. It is hard to estimate the risk free rate of return on projects under different economic environment. The CAPM is really just a single period model. It is not possible to use the CAPM for projects which last for more than one year. Complications in decision-making cannot be modeled easily. Markets are really inefficient and imperfect All investors are not well diversified Disagreement over time period to estimate Beta Only focuses on single factor Beta may shift over time Beta calculation sensitive to historical time period used The model makes unrealistic assumptions The parameters of the model cannot be estimated precisely Firm may have changed during the estimation period The relationship between betas and returns is weak (b) Describe Roll’s critique of the early empirical tests of the CAPM. Roll questioned whether it is even conceptually possible to test the CAPM. Roll showed that the linear relationship which prior researchers had observed in graphs resulted from the mathematical properties of the models being tested, hence that a finding of linearity proved nothing about the validity of the CAPM. Rolls work did not disprove the CAPM theory, but he did show that it is virtually impossible to prove that investors behave in accordance with the theory. In general, evidence seems to support the CAPM model when it is applied to portfolios, but the evidence is less convincing when the CAPM is applied to individual stocks. Nevertheless, the CAPM provides a rational way to think about risk and return as long as one recognizes the limitations of the CAPM when using it in practice. (20 marks) (c) How successfully does the Arbitrage Pricing Theory (APT) address the weaknesses of the CAPM that you have identified in parts (a) and (b)? The CAPM is a single-factor model, while the Arbitrage Pricing Theory (APT) can include any number of risk factors. It is likely that the required return is dependent on many fundamental factors such as the GNP growth, expected inflation, and changes in tax laws, and that different groups of stocks are affected differently by these factors. Thus, the APT seems to have a stronger theoretical footing than does the CAPM. Sensitivities relate individual asset return to multiple economic factors such as Inflation, Interest rates, Economic growth, Industry effects, Market risk premium and Exchange Rates The CAPM was developed as a model for investment appraisal and share valuation. The CAPM is a one year model and not suitable for projects longer than one year. The arbitrage pricing model (APM) is a model that was developed out of the CAPM and considers various numbers of independent factors which may affect the share price. CAPM consider the risk in form of the beta factor. The expected rate of return using APM considers factors such as unanticipated inflation, changes in the expected level of industrial production; changes in the risk premium on bonds, and unexpected changes in term structure of interest rates. APM betas Measures of unknown economic factors driving asset returns Standardized covariances between the individual security return and the unknown factor values Measures the sensitivity of a change in the return on a single security to the changes in the set of factors included in the model. More general pricing model Advantages of APT over CAPM Intercept term not specified Does not require utility assumption Relies on arbitrage for derivation Does not require efficient and perfect markets as long as some arbitragers exist CAPM - Capital Asset Pricing Model APM - Arbitrage Pricing Model utility based valuation model that posits a linear relationship between the returns of an asset and the market risk premium arbitrage based valuation model that posits a multi-linear relationship between the returns of an asset and the values of a set of multiple unknown economic factors Utility based derivation Arbitrage based derivation Single factor = market risk premium Multiple factors = not defined Intercept term = risk free rate Intercept term = not defined Requires perfect markets for derivation Requires only arbitrage for derivation Requires well diversified investors for Derivation Requires only arbitrage for derivation Theoretical model specifies CAPM Beta as single systematic risk measure Theoretical model specifies unknown set of “sensitivities” to unknown economic factors However, the APT faces several major hurdles in implementation, the most severe being that the apt does not identify the relevant factors--a complex mathematical procedure called factor analysis must be used to identify the factors. To date, it appears that only three or four factors are required in the APT, but much more research is required before the APT is fully understood and presents a true challenge to the CAPM. (20 marks) (Word length for Q4  1,200) Figure 1 Read More