Fundamentals of Corporate Finance - Assignment Example

Question 1 (a) Starting from the theory developed by Markowitz (1952), it will be implemented a mean-variance analysis to choose the best combination of stocks, i.e. with the highest expected return for the level of risk taken. This theory is based on two assumptions: investors act rational and they are risk averse. From the data available for the period 2 October 2009 to 18 November 2011, it was computed the standard deviation for the stocks chosen - BARR and ABF, in order to find the covariance, using the formula: (1) The simulation of portfolios started from the fact that the sum of the weights of two stocks is equal to one and that short shelling is not permitted. Table no. 1. Simulation of Portfolios Formed with The BARR and ABF Stocks Weight BARR Weight ABF E(R) Standard deviation Coefficient of variation 1 0 0.49 % 3.74 % 7.62 0.9 0.1 0.47 % 3.48 % 7.40 0.8 0.2 0.45 % 3.24 % 7.21 0.7 0.3 0.43 % 3.02 % 7.05 0.6 0.4 0.41 % 2.84 % 6.96 0.5 0.5 0.39 % 2.69 % 6.95 0.4 0.6 0.37 % 2.59 % 7.06 0.3 0.7 0.35 % 2.53 % 7.33 0.2 0.8 0.32 % 2.53 % 7.79 0.1 0.9 0.30 % 2.58 % 8.49 0 1 0.28 % 2.68 % 9.46 The correlation coefficient can be calculated as follows: (2) The coefficient of correlation is positive and equal to 0.39, which shows a strong positive relation between the two stocks. Because this coefficient is less than 1, the standard deviation of any portfolio formed with the two securities is less than the weighted average of the standard deviations of BARR stock, respectively ABF (Ross et al., 2002). This is one of the benefits of diversification. The correlation coefficient among assets is the critical indicator when considering investing because for assets which are low or negatively correlated can reduce the risk of the portfolio, but unfortunately for the stocks chosen, this coefficient has a positive value (Elton and Martin, 1995). Figure no. 1 depicts the different weights of a portfolio held in the two stocks, which yield a curve of potential combinations. Figure no. 1. Mean-Standard Deviation Diagram for The Portfolios The standard deviation for the simulation of portfolios is a function not only of the standard deviations for the expected returns of individual assets, but also of the covariance between the expected returns for all the pairs of assets in the portfolio (Reilly and Brown, 2002). In order to choose the best combination of stocks, it was calculated the coefficient of variation as follows: v= (3) The portfolio with the lowest coefficient of variation is the equally weighted portfolio, i.e. wBARR= 50% and wABF=50 %. This portfolio has v=6.95, E(R)=0.39% and σ=2.69%. Although, in this case it was chosen this portfolio, in the real world this matter is related more with what type of investor is considering these stocks i.e. more or less risk averse. It was necessary to consider the minimum variance portfolio, which is the portfolio with the lower level of risk. Minimum variance portfolio is the portfolio of stocks that minimizes the total risk and is the portfolio with a return that has an equal covariance with every stock return (Hooda and Stehlík, 2011). Haugen and Hains (1975) documented the low-volatility anomaly, i.e. portfolios with lower risk, generated higher compounded returns. Even more recent studies (Clarke et al., 2006) arrived to the same conclusion. Table no. 2. Covariance Matrix of The Two Stocks (BARR and ABF) BARR ABF BARR 0.0014 0.0004 ABF 0.0004 0.0007 The weights for the minimum variance portfolio, after rescaling the values obtained from solving the two equations, are wBARR=23% and wABF= 77%. The expected return for this portfolio is 0.33%, whereas the standard deviation is 0.06%. One drawback of employing a mean-variance analysis to find efficient portfolios is that means and variances require estimation which is done with a degree of error. Although, in this case, there are only two stocks, for the real world where there are many stocks and other investments to choose from, the full use of this analysis as a tool for portfolio management is limited (Grinblatt and Titman, 2002). 1 (b) In order to examine the performance of the weighted average portfolio (50% in BARR and 50% in ABF) over the period 25 November 2011 to 18 May 2012, it was computed the mean return of the two stocks. Further, it was analyzed the return of the portfolio, using the formula: (4) The return of the portfolio was equal to 0.08%, showing a lower performance than the one estimated (0.39%). The abnormal return computed as difference between effective and expected return is -0.31%, which shows an underperformance of the portfolio than the one expected. Although, the return of the portfolio was lower than the one expected, it still provided the investor with higher return than the one he would have obtained if he invested in only one of the two stocks. Moreover, a rational investor would be concerned not only with the profitability, but also with the probability of it arising, i.e. a risk-return tradeoff. Question 2 In the single-index model developed by Sharpe (1964), the return of a stock is the sum of three components: a constant, an indicator of the firm’s sensitivity to the market’s movements, and, finally, a random and unpredictable component, i.e. error. The equation for this model is shown below: Ri= αi + βiRM + ε (5) This model hypothesizes that correlation between stocks exists because each stock is dependent on an underlying systematic factor (Fabozzi and Markowitz, 2002). The reason for choosing as benchmark the market portfolio of risky assets is that rational investors (i.e. risk averse) will never hold a risky asset in isolation because they can eliminate a part of risk by diversification, and the market portfolio represents the extreme case of the most diversified portfolio (Howells and Bain, 2007). The estimates for this model applied for BARR and ABF stocks are shown in table no. 3. Table no. 3. Single-Index Model Estimates of The Returns for BARR and ABF Stocks BARR ABF Intercept (α) 0.0042 0.0022 Slope (β) 0.6332 0.5623 R squared (R2) 0.1911 0.2932 Steyx (ε) 0.0339 0.0227 The single-index models for the two stocks are: RBARR = 0.0042 + 0.6332 RM + 0.0339 RABF = 0.0022 + 0.5623 RM + 0.0227 The capacity of the index return to predict the move of the stocks is shown by the coefficient of determination (R squared), which has values of 19% for BARR stock and 29% for ABF, enough for the predictability of this model. Starting from these indicators, it was computed the expected return for the two stocks and for the period 25 November 2011 – 18 May 2012. Comparing these values with the effective values for that period, it can be said that the model is not completely accurate. The difference between the effective returns and the expected returns, i.e. abnormal returns registered negative values. Table no. 4. Abnormal Returns for BARR and ABF Stocks E(R)BARR E(R)ABF Ab(R)BARR Ab(R)ABF 8.48 % 6.75 % -3.48 % -3.89 % 3.46 % 2.17 % -0.48 % -4.95 % 2.15 % 0.99 % -0.50 % -0.80 % 5.25 % 3.81 % -6.22 % -2.24 % 4.52 % 3.14 % -5.09 % -2.87 % 4.69 % 3.30 % -3.87 % -2.22 % 3.89 % 2.57 % -6.26 % -1.76 % 5.02 % 3.60 % -0.84 % -1.30 % 3.91 % 2.59 % -5.04 % -0.94 % 5.72 % 4.24 % -3.28 % -3.47 % 3.35 % 2.08 % -3.91 % -1.23 % 4.42 % 3.05 % -8.65 % -0.20 % 4.27 % 2.92 % -2.19 % -3.49 % 3.63 % 2.33 % -7.30 % -4.05 % 3.60 % 2.31 % -5.55 % -1.89 % 4.73 % 3.34 % 2.87 % -3.92 % 2.67 % 1.46 % -4.68 % -0.29 % 3.02 % 1.77 % -6.21 % -0.95 % 3.28 % 2.01 % -6.16 % -3.24 % 3.10 % 1.85 % -1.01 % -3.18 % 5.07 % 3.64 % -4.30 % -0.61 % 3.89 % 2.57 % -4.83 % -2.74 % 2.44 % 1.24 % -8.08 % -0.59 % 2.95 % 1.72 % -0.96 % -1.96 % 0.36 % -0.65 % -0.80 % -3.83 % 8.48 % 6.75 % -3.48 % -3.89 % 3.46 % 2.17 % -0.48 % -4.95 % 2.15 % 0.99 % -0.50 % -0.80 % 5.25 % 3.81 % -6.22 % -2.24 % 4.52 % 3.14 % -5.09 % -2.87 % 4.69 % 3.30 % -3.87 % -2.22 % 3.89 % 2.57 % -6.26 % -1.76 % 5.02 % 3.60 % -0.84 % -1.30 % 3.91 % 2.59 % -5.04 % -0.94 % Question 3 For this question, it will be considered a government bond described in the following table by coupon rate, years to maturity, yield to maturity and par (face) value. Table no. 5. Bond Features Coupon rate: 10 % Years to maturity: 10 Yield to maturity: 13 % Par value: € 1000 The bond’s price changes as response to changes in interest rates, and the factors that affect the bond’s interest rate risk sensitivity are: the maturity of the bond (longer term bonds have greater interest rate risk), the coupon rate (a bond with a lower coupon rate will have greater interest rate risk), the yield to maturity (a lower yield to maturity will result in greater interest rate risk). A measure of interest rate risk is Macaulay duration, which is an indicator of the elasticity of bond price changes to changes in the interest rate (Ross et al., 2009). Table no. 6. Macaulay and Modified Duration Settlement date: 1/1/2012 Maturity date: 1/1/2021 Annual coupon rate: 10 % Yield to maturity: 13 % Face value (% of par): 100 Coupons per year: 1 Macaulay duration: 6.09 Modified duration 5.38 In this case, the Macaulay duration is equal to 6.09 years. This indicator is more effective at predicting the bond price moves for small changes in interest rates, but for larger interest rate changes, Macaulay duration is a less accurate indicator (Fabozzi, 2000). Another indicator, which gives more accurate prediction, is modified duration, computed as follows: m= (6) For the chosen bond, the modified duration is equal to 5.38 years. Further, this indicator is used to determine the change in price, by considering the percentage change in yield. Change in price = Current price (-Modified duration)(Change in interest rates) (7) The computations for the change in price of the analyzed bond (starting from the value obtained for modified duration) can be seen in the following table. This change could have been computed also by starting with Macaulay duration, and dividing it by one plus the yield to maturity. Table no. 7. Change in Bond’s Price Change in YTM: -1 % Current bond price: € 846.05 Change in bond price: € 45.59 New bond price: € 891.64 If the required yield to maturity decreases from 13% to 12%, the price of the bond will increase to € 891.64 from € 846.05. The duration measure and its limitations can be improved by using a different measure namely convexity (Fabozzi, 1999). Convexity is interpreted as the rate of change of the slope of the price-yield curve, expressed as a fraction of the bond price (Bodie et al., 2009). The formula for computing convexity is: (8) The computations for this indicator are depicted in table no. 7. Table no. 8. Convexity Measure Time (t) Cash flow PV(CF)       1 100 88.49 2 100 78.31 3 100 69.30 4 100 61.33 5 100 54.27 6 100 48.03 7 100 42.50 8 100 37.61 9 100 33.28 10 1100 324.04 Convexity:     46.84 This bond exhibits positive convexity, which means that the price appreciation will be greater than the price depreciation for a large change in yield (Fabozzi and Drake, 2009). Reference List Bodie, Z., Kane, A. and Marcus, A. (2009) Investments, 8th edn. New York: McGraw-Hill/Irwin. Clarke, R., De Silva, H. and Thorley, S. (2006) Minimum-Variance Portfolios in the U.S. Equity Market, The Journal of Portfolio Management, 33(1): 10–24. Elton, E.J. and Martin, J. (1995) Modern Portfolio Theory and Investment Analysis, 5th edn. New York: John Wiley & Sons. Fabozzi, F.J. (1999) Duration, Convexity, and Other Bond Risk Measures. New Jersey: John Wiley & Sons. Fabozzi, F.J. (2000) Bond Markets, Analysis and Strategies. New Jerse: John Wiley & Sons. Fabozzi, F.J. and Drake, P. (2009) Capital Markets, Financial Management, and Investment Management. New Jersey: John Wiley & Sons. Fabozzi, F.J. and Markowitz, H. (2002) The Theory and Practice of Investment Management. New Jersey: John Wiley & Sons. Grinbalt, M. and Titman, S. (2002) Financial Markets and Corporate Strategy, 2nd edn. New York: McGraw-Hill/Irwin. Haugen, F. and Hains, R. (1975) Risk and The Rate of Return on Financial Assets: Some Old Wine in Financial Bottles, Journal of Financial and Quantitative Analysis, 10(5): 775–784. Hooda, D. and Stehlík, M. (2011) Portfolio Analysis of Investments in Risk Management, The Open Statistics and Probability Journal, 3(1): 21-26. Howells, P. and Bain, K. (2007) Financial Markets and Institutions. Harllow: Pearson Education Limited. Markowitz, H. (1952) Portfolio Selection, Journal of Finance, 7(1): 77–99. Reilly, F. and Brown, K. (2002) Investment Analysis and Portfolio Management, 7th edn. Boston: South-Western College Pub. Ross, S., Westerfield, R. and Jordan, B. (2009) Fundamentals of Corporate Finance, 9th edn. New York: McGraw-Hill/Irwin. Sharpe, W.F. (1964) Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance, 19(1): 425-42. Read More