Security Analysis and Portfolio Management - Math Problem Example

Security Analysis and Portfolio Management Customer inserts his/her name University’s name Date Outline Problem 16, page 185 Problem 29 page 214 Problem 20 page 255 Problem 20 page 305 Proble11, page 343 References Appendix – excel files Problem 16, page 185 a). The probability on put option of the HPR can be calculated using the following table At this point we should not that the purchase is 100 and the put option is 12. If dividends of the asset are paid during the life of the option the underlying value of the asset is expected to decrease. Put option increases with the payment of dividends while call option decreases with the payments of dividends. As the time to expiration increase both put option become more valuable. This happens because the longer the time to expiration the more time is available for the underlying asset to move which increases the value of the both put option (Poon and Granger, 2003). b). cost of the index fund and put option The probability distribution of the HPR consisting of one share of the index fund under put option is shown below; The cost of the index fund plus the cost of the put option is 112. c). When one buys a put option is guaranteed his money regardless the state of the market. Therefore it acts as an insurance of recovering your investment. It is a form of hedging for the investor. It is used to protect a downside of the stock market. The option usually gives the buyer rights to be able to sell underlying asset at a price which is fixed at a time prior to its expiration date which the buyers pays the price for the right. If the strike price is less than underlying asset the option is not exercised and expires worthless; while, the strike price is more than the underlying assets the put option owner exercise the option and sells the stock at the strike price and claims the difference between the asset market value . Problem 29 page 214 The client has a degree of risk aversion of A=3.5. If the investor chooses to invest in a passive portfolio, he will select a proportion that matches his risk perception. It is calculated as follows; Proportion = Proportion == =0.429 This means that optimal proportion for the client is: 42.9% invested in the risk portfolio and 57.1% invested in T- bills. b) The fee that one expects to pay or charge a customer for this is equal to the original fees regardless the asset allotted. Asset mix is irrelevant in the amount of fee charged to client as it depends on amount spend not on the mix. A fee or sales charge is often assessed when a fund is purchased. When this occurs, the fund is referred to as a front-end local fund. When the fee is assessed upon redemption of the shares in a hedge fund, the fund is referred to as a back-end local fund. Often the exit fee is reduced as the investor’s holding period increases. This encourages the investor to leave the funds under the control of the management company for a longer time. These include investment advisory fees and administrative expenses. Some funds also charge services fees. These fees cover the commission paid to brokers called trail commissions. These fees are assessed annually whereas the black-end and front-end loads are onetime charges. The investors should carefully consider the nature and extent of fees that the various hedge funds and can significantly impact performance over time. The fees are disclosed in the fund’s prospectus as are many others facts about the funds its investment philosophy and management style (Santa-Clara and Yan, 2006). Problem 20 page 255 The correlation is a simple way of summarizing the correlation between independent variables (predictors) and also between independent variables (predictors) and dependent variable (Steinberg, 2010). In the case under review, the correlation matrix is provided in the following: . The correlation matrix identifies the correlation between dependent variable and independent variables as positive relationship between independent variables and one dependent variable. Also, there is a positive correlation between all three independent variables identified for this study. From the correlation table above imported from excel it shows that only decades 1930s to decades 1920s, 1940s 1950s 1960s and 1990s have a negative correlation. Other decades have a positive correlation. The table shows that there is a strong positive correlation between 1940s and other decades after 1940s except 1970s. It also shows that decades 1980s and 1990s have a stronger correlation coefficient. From the table above it shows that there is negative correlation between returns of small companies stocks and long term government , intermediate –term government and treasury bills. However, it should be noted that the negative correlation is weak as it is less than -0.2. Large company has positive correlation with other class of assets except correlation. However the positive correlation is weak as none of them is more than 0.4. Long term government assets have positive correction with intermediate term government and treasury bills assets which is strong. The degree of correlation that both of them have is as thus, the correlated the degree of correlation of Large company stocks is far greater than that of Small company stocks to the extent that is only correlated -1.69% whilst Large company stock is correlated nearly 3.8% positive. Large co. stocks - This segment posse the largest risk when compared with other figures within the table, the correlations values are well above the zero margins. The changes in this value are determined with the risk factors associated with bond market (Santa-Clara and 437). The period dedicated for this analysis is justifiable in the sense that most companies and firms with large market capitalization recorded a wonderful trend in their stock market gains, with those gaining recording values well above those recorded from the small stocks companies. The time span shows a positive trend that was largely beneficial to most investors who had send their money in this section of the market. What comes out clearly is the desire by most of the investors to assess market capitalization risks to enable them predicts the possible outcome in the future. This period is justifiable because it not only provides appositive outcome throughout the entire period but can also be used to analyze the possible strengths that ensured the market remain in this upward trend throughout the entire period, by analyzing the betas in this case, the investments with lots of risks posed higher values for betas than those which are safer Small co. stocks - This section shows a much lower risk value when compared with other figures within the table, most of the correlations values are well below the zero margins. The changes in this value are due to less stressing factors in the market segment. This segment of the analysis poses a more or less predictive outcome that would be least expected. The risks of investing in this section are thus high because the betas recorded are higher. Although it is still important to note that these are figures which do not give a possible outcome in the future, this long duration analysis puts those interested in investing at a cross road of making decisions. Long term government - This segment shows a steady correlation value across all the other aspects with minimal variations within and across. Probably this is the most appropriate section that posses a more predictable outcome in the investment segment. The entire period shows a positive response although at a much lower rate of turnover, this is associated with fewer betas and thus the risks likely are minimal (Jordan and Miller, 2008). However, the returns from such investments are not as significant or as attractive as most investors would prefer. Intermediate-term government: - This section posses a relatively significant correlation of values that are also steady and significant. The changes in this value are determined with the risk factors associated with bond market as well. This section shows a most appropriate and a more predictable outcome in the investment segment. The entire period shows a positive response although at a much lower rate of turnover, this is associated with fewer betas and thus the risks likely are minimal. The period dedicated for this analysis is justifiable because most companies and firms with small market capitalization recorded a wonderful trend in their stock market gains, with those gaining recording values well above those recorded from the large caps. Problem 20 page 305 Sharpe optimal is closely related to the procedure of obtaining a Markowitz efficient frontier. This is no surprise, as both methods are used to achieve an optimal balance of risk and return for investment portfolios. In the Spreadsheet Analysis, we in put formulas for the portfolio return, standard deviation, and Sharpe ratio, we can solve for the portfolio weights that give us the highest possible Sharpe ratio. We do not have to worry about the weight in bonds because the weight in bonds is simply equal to one minus the weight in stocks. The Sharpe ratio is computed as a portfolio’s risk premium dividend by the standard deviation of the portfolio’s return: Sharpe ratio = Rp – Rt σp in this case, the portfolio risk premium is the raw portfolio return less a risk-free return, that is,Rp – Rf, which we know is the basic reward for bearing risk. The return standard deviation, σp, is a measure of risk. Return standard deviation is a measure of the total risk for a security or a portfolio. Thus, the Sharpe ratio is a reward-to-risk ratio that focuses on total risk. Because total risk is used to make the adjustment, the Sharpe ratio is probably most appropriate for evaluating relatively portfolios (Sharpe, 1970). In our case the alpha is being double current figures. Alpha is computed as the raw portfolio return less the expected portfolio return predicted by the capital asset pricing model (CAPM). To compute alpha, we compare the actual return, Rp, to the predicted return. The difference is the alpha, denoted p: P = RP – E(Rp) = Rp – {Rf + [E(Rm) – Rf] x Bp } Thus alpha is the excess return above or below the security market line, and, in this sense; it can be interpreted as a measure of by how much the portfolio “beat the market”. Portfolio A plots above the Security Market Line (SML) and has a positive alpha. Portfolio B has a zero alpha. Portfolio C plots below the SML and has a negative alpha return (E (Rp)) RA A RB Rf C RC Portfolio beta (Bp) In calculating Sharpe optimal portfolio, we use the following E (Rp) = xS E (RS) + xB E (RB)………. Where xS and xB are the percentages invested in the funds and the variance on a portfolio of these two assets is: σp2 = xS2σS2 + xB2σB2 + 2xSxBσSσB Corr(RS, RB) Our task is to find the value of xs and xB that make this ratio as large as possible. This looks like a tough job, but, as our nearby Spreadsheet Analysis box shows, it can be done relatively easily. The sharpe ratio of 0.4994 is shown in the table below. Sharpe –optimal portfolio, the key to the problem is to first determine the expected return and the standard deviation for the portfolio. Thus the problem of calculating VaR statistics for a sharpe-optimal portfolio is the same for any portfolio once the appropriate portfolio weights are determined(Poon and Granger, 2003). The Sharpe ratio of 0.4942 strongly indicates that portfolio could have done better with investing in stock rather than their market. That ratio indicates in a way the amount of interest lost by a potential investor when they invested in the portfolio instead of the market. Proble11, page 343 a). In this case McKay, should increase his portfolio by investing in high beta assets. This will increase the variability of the portfolio thus increasing returns. Desired risk levels are then achieved through combining portfolio M with lending and borrowing. The straight line depicted in graph below is referred to as the capital market line (CML). All investors will end up with portfolios somewhere along the CML. however, not all securities or portfolios would lie along the CML from the deprivation of the efficient frontier we know that all portfolios, except those that are efficient, lie below the CML (Graham, 2004). Observing the CML tells us something about the market price of risk. The equation of CML (connecting the riskless asset with a risky portfolio) is: Re=𝝈e Where the subscript e denotes efficient portfolio, the term (rm-rf) can be thought of as the extra return that can be gained by increasing the level of risk (standard deviation) on an efficient portfolio by one unit. The entire second term on the right of the equation is thus the market price of risk times the amount of risk in the portfolio. The expression Rf is the price of time; that is, it is the price paid for delaying consumption for one period (Norton, 2006). The expected return on an efficient portfolio is: (Price of time) + (price of risk) (amount of risk) Although this equation sets the return of an efficient portfolio, we need to go beyond to deal with returns on non efficient portfolios or individual securities. For well-diversified portfolios, non-systematic risk tends to go to zero, and the only relevant risk is systematic risk measured by beta. Because we assume that investors are concerned only with expected return and risk the only dimensions of a security that need be of concern are expected return and beta. b). to reduce risk exposure If he wants to reduce risk exposure without borrowing or lending McKay should reduce the systematic risk by substituting low beta investments for high beta stocks. This will reduce the volatility of the portfolio thus reducing risk exposure. Using the security market line we will not that there will be a loss of portfolio efficiency if original diversification is not maintained. It describes the expected return for all assets and portfolios of assets efficient or not the difference between return on any two assets can be related simply to their difference in beta. The higher beta is for any security the higher must be its expected return. The relationship between beta and expected return is linear (Fisher and Jordan, 2006). As we said that the risk of any stock could be divided into systematic and unsystematic risk. Beta is an index of systematic risk. This equation suggests that systematic risk is of no consequence. It is not total variance of return that affects returns, only that part of the variance in returns that cannot be eliminated by diversification. References Fisher, D. E. & Jordan, R. J. (2006). Security analysis and portfolio management. Prentice hall of India private limited Graham, B. (2004). Securities Analysis. New York: McGraw Hill. Jordan, B. & Miller, T. (2008). Fundamentals of investments: Valuation and management. Boston: McGraw-Hill Irwin Norton, R. (2006). Investments. New York: Thomson Higher Education. Poon, S. & Granger, C., (2003). Forecasting financial market volatility: a review, Journal of Economic Literature, 41, pp.478–539. Santa-Clara, P. & Yan, S. (2006). “Crashes, Volatility, and the Equity Premium: Lessons from S&P 500 Options.” Review of Economics and Statistics 92:4 (2006): 435-451. Print. Sharpe, W. (1970). Portfolio Theory and Capital Markets. New York: McGraw-hill. Steinberg, W. J. (2010). Statistics Alive! London: SAGE Publications. Read More