Solution Assignment 3 Essay

Solution Assignment 3 Essay Solution Assignment 3 Essay Solutions to Assignment 3 1. a. using excel Stock A Stock B i. alpha -.609 2.964 ii. beta 1.183 1.021 iii. standard deviation of Residuals 4.676 4.983 iv. correlation with Market .757 .684 v. Average of the Market = 3.005 vi. Variance of the Market = 20.908 vii. First, we need Rf from SML 2.964 = RF + 1.183 [ 3.005 – RF] ïÆ'ž solving for RF = 3.352 Therefore for next year E(R A) = 3.352 + 1.183 (5 – 3.352) = 5.3 b . (i ) From single‑index model use: Rj = ÃŽ ±i - ÃŽ ²i Rm RA = ‑.609 + 1.183(3.005) = 2.946 RB = 6.032 RC = 3. 556 From the single‑index model the variance is: ÏÆ'2 i = ÃŽ ²i 2 ÏÆ'2 m + ÏÆ'ei2 ÏÆ'2 A = (1.183)2(20.908) + (4.677)2 = 51.14 ÏÆ'2 B = 46.62 ÏÆ'2 C = 265. 0 *The answers should be identical whichever way means and variances are computed. Any slight differences are due to rounding errors in the calculations. (ii) RA = 2.946 ÏÆ'2 A = 51.15 RB = 6.031 ÏÆ'2 B = 46.61 RC = 3. 554 ÏÆ'2 C = 265.0 c. (i ) Under the single‑index model covariance: cov(i j) = ÃŽ ²i ÃŽ ²j ÏÆ'm2 CovAB = (1.183)(1.021)(20.908) = 25.254 CovAC = 57.433 CovBC = 49.568 (ii) From the historic data itself: cov(i j) = ÃŽ £ (1/T-1)(Ri ‑ Ri)(Rj ‑ Rj) CovAB = 18.462 CovAC = 61.618 CovBC = 54.085 The calculations of covariances are different because the single‑index model computes covariances as if the correlation between residuals from the equation Ri = ÃŽ ±i + ÃŽ ²i Rm + ei are zero [cov(ei ej) = 0]. While computing covariance from historic data is equivalent to incorporating the historic level of cov(ei ej) into the measurement of covariance. d. For a portfolio made up of one‑half stocks A and B: (i) Expected return and standard deviation under the single‑index model: Rp = 1/2(2.946) + 1/2(6.032) = ÏÆ'p = [(1/2)2(51.14) + (1/2)2(46.62) + 2(1/3)2(25.25)+2(1/3)2(57.43) = (ii) Expected return and standard deviation using historical data: Rp = 1/2(2.946) + 1/2(6.031) = ÏÆ'p = [(1/2)2(51.15) + (1/2)2(46.61) + +2(1/3)2(18.46) = 2. a)We know by the CAPM:.18 = .04 + (.11 - .06) ï  ¢j which gives ï  ¢j = 2 The CAPM assumes that the market is in equilibrium and that investors hold efficient portfolios, i.e., that all portfolios lie on the security market line. b) Let â€Å"y† be the percent invested in the risk-free asset. Portfolio return is the point on the market line where 18% = y (4%) + (1 - y) (11%) and y = -1. Therefore, (1-y) = 2, i.e., the individual should put 200% of his portfolio into the market portfolio. 3. Assuming that the company pays no dividends, the one period expected rate of return, E(Rj) = [E(P1) - P0 ] / P0 where E(P1) = $179. Using the CAPM, we have E(Rj) = Rf + [E(Rm) - Rf] ï  ¢j = [E(P1) - P0 ] / P0 Substituting in the appropriate numbers and solving for P0, we have .08 + [.18 - .08]2.0 = [$100 - P0]/ P0 and solving for P0 = $154.3 4. Using the definition of the correlation coefficient, we have .8 = and cov (K, M) = .8(.25) (.2) = .04 Using the definition of Beta, we can calculate the systematic risk of MF: ï  ¢k = .04/(.2)2 = 1.0 The systematic risk of a portfolio is a weighted average of asset’s ï  ¢Ã¢â‚¬Ëœs. If â€Å"y† is the percent of MF, ï  ¢P = (1 - y) ï  ¢F+ y ï  ¢K or .8 = (1 - y ) 0 + y 1.0 or y =80% In this case the investor would invest an amount equal to 80 percent of his wealth in MF in order to obtain a portfolio with a ï  ¢ of .8 5. a) Using E(RP) = Rf + [E(Rm) - Rf] ï  ¢P to solve for ï  ¢P=2.2 b)We know that efficient portfolios have no unsystematic risk. The total risk is ï  ³2P= ï  ¢2P ï  ³2m + ï  ³2ï  ¥ and since the unsystematic risk of an efficient portfolio, ï  ³2ï  ¥ is zero, ï  ³P = ï  ¢P ï  ³m = 2.2 (.18) = .396 or 39.6% c)The definition of correlation is CorrJ m = cov (RJ,Rm) ï  ³J ï  ³m To find cov(Rj,Rm), use the definition of ï  ¢j = cov(Rj,Rm) ï  ³2 m Solving, we get Corr J m = 1.0, which indicates that the efficient portfolios are perfectly correlated with the market (and with each other). 6. We know from the CAPM : .13 = .04 + (.08)ï  ¢ J , solving which gives ï  ¢J= .1.125 If the rate of return covariance with the market

A Introduction to Sociology Statistics

A Introduction to Sociology Statistics Sociological research can have three distinct goals: description, explanation, and prediction. The description is always an important part of the research, but most sociologists attempt to explain and predict what they observe. The three research methods most commonly used by sociologists are observational techniques, surveys, and experiments. In each case, measurement is involved that yields a set of numbers, which are the findings, or data, produced by the research study. Sociologists and other scientists summarize data, find relationships between sets of data, and determine whether experimental manipulations have affected some variable of interest. The word statistics has two meanings: The field that applies mathematical techniques to the organizing, summarizing, and interpreting of data. The actual mathematical techniques themselves. Knowledge of statistics has many practical benefits. Even a rudimentary knowledge of statistics will make you better able to evaluate statistical claims made by reporters, weather forecasters, television advertisers, political candidates, government officials, and other persons who may use statistics in the information or arguments they present. Representation of Data Data are often represented in frequency distributions, which indicate the frequency of each score in a set of scores. Sociologists also use graphs to represent data. These include pie graphs, frequency histograms, and line graphs. Line graphs are important in representing the results of experiments because they are used to illustrate the relationship between independent and dependent variables. Descriptive Statistics Descriptive statistics summarize and organize research data. Measures of central tendency represent the typical score in a set of scores. The mode is the most frequently occurring score, the median is the middle score, and the mean is the arithmetic average of the set of scores. Measures of variability represent the degree of dispersion of scores. The range is the difference between the highest and lowest scores. The variance is the average of the squared deviations from the mean of the set of scores, and the standard deviation is the square root of the variance. Many kinds of measurements fall on a normal, or bell-shaped, curve. A certain percentage of scores fall below each point on the abscissa of the normal curve. Percentiles identify the percentage of scores that fall below a particular score. Correlational Statistics Correlational statistics assess the relationship between two or more sets of scores. A correlation may be positive or negative and vary from 0.00 to plus or minus 1.00. The existence of a correlation does not necessarily mean that one of the correlated variables causes changes in the other. Nor does the existence of a correlation preclude that possibility. Correlations are commonly graphed on scatter plots. Perhaps the most common correlational technique is Pearsons product-moment correlation. You square the Pearsons product-moment correlation to get the coefficient of determination, which will indicate the amount of variance in one variable accounted for by another variable. Inferential Statistics Inferential statistics permit social researchers to determine whether their findings can be generalized from their samples to the populations they represent. Consider a simple investigation in which an experimental group that is exposed to a condition is compared with a control group that is not. For the difference between the means of the two groups to be statistically significant, the difference must have a low probability (usually less than 5 percent) of occurring by normal random variation. Sources: McGraw Hill. (2001). Statistics Primer for Sociology. mhhe.com/socscience/sociology/statistics/stat_intro.htm