Questions and Answers of Investments Analysis And Management

5.3 In Exercises 5.1 and 5.2, find the density of V11.2, and express it as an explicit function of p (recall that V11.2 V11–V12V22–1V.21), and the elements of V.
5.2 In Exercise 5.1, find the marginal density of V11, the upper left 2 × 2 submatrix of V. Express the density as an explicit function of ρ and the elements of V11.
5.1 Let pj(t) denote the price of security j at time t, and let xj(t) ≡ log pj(t)–log pj(t–1). Suppose there are q securities in the same industry so that their price changes are correlated in
4.10 y = (y¡): p × 1 follows a multinomial distribution with parameters θ = (θi): p × 1, and . We want to estimate γ = θ’θ. Show that , the MLE of γ is , where , ; Show that: Hint: Expand
4.9 Let (X, Y) denote scalar random variables following the law L(X, Y) = N(0,Σ),Let (X1 , Y1 ), ... , (XN , YN) denote independent bivariate observations. Evaluate:
4.8 Suppose V : p × p and L(V) = W(Σ,p,n), p ≤ n, > 0. Find the asymptotic distribution of | V | .
4.7 he asymptotic posterior distribution for θ given the sample data.Suppose X: p × 1 is random and all third order moments of the components of X exist. Let f(X) denote a scalar function of X. The
4.6 Suppose W : p × p and W = W’. Show that for all α ≠ 0, if the latent roots of W lie within the unit circle. Moreover, if the latent roots are sufficiently small, |I + W|α ≅ exp (α tr
sample.
4.5 Let X: p × 1 follow the law N(θ,I). X1, ... ,XN are independent observations of X. Suppose the prior density of θ ≡ (θi) is given by p(θ) = 1, for 0 ≤ θi ≤ 1 for every i = 1, ... ,p,
4.4 Suppose X1,...,XN are independent observations all following the law N(θ,1). Adopt a natural conjugate prior distribution for θ and find t
4.3 Referring to Example (4.6.1), find the prior distribution of θ6; P{θ6 ≤ 4}; the asymptotic marginal posterior distribution of θ6 given the n observations.
4.2 Let X : p × 1 denote the scores of unemployed males 18-60 years of age, in a ghetto area, on a battery of p tests of their job skills and abilities. Let Xj, j = 1, ... ,N denote the vector of
4.1 Let X: p × 1 denote the numbers of sales for each of p items of a door-todoor salesman, and let X1, ... ,XN denote the independent sales vectors of N salesman (selling the same items) on a
3.14 Explain how you would test a set of data for multivariate normality. (Hint: See Andrews et al., 1973].
3.13 Find the distribution functions corresponding to the characteristic functions:[Use the inversion formula.]
3.12 Show that for x > 0,
3.11 In Dempster, 1963, an invariance argument was used to cast doubt upon sampling theory methods of making inferences about the covariance matrix in a large class of distributions. Discuss the
3.10 Show that if X and Y are independent random variables, X and Y are also uncorrelated. Show that if X and Y follow a bivariate Normal distribution and X and Y are uncorrelated, then X and Y are
3.9 In an experiment, r0 successes were obtained out of n0 binomial trials, with probability of success p on a single trial. Suppose, a priori, that p follows a beta distribution [see (6.3.1); this
3.8 Suppose the joint density of X and Y is given by Find the means, variances, and correlation for X and Y.
3.7 It is of interest to study sentiment of people toward racially integrated neighborhoods. Let X1 denote time an individual has lived in a given neighborhood; X2 denotes his feelings toward an
3.6 Suppose L(X|θ) = N(θ, 0), where x’ = (x1,x2), x1 denotes change in personal income, x2 denotes change in personal consumption, and Suppose there is one observation x = (5,3)’. Test the
3.5 Let X : p × 1 denote a vector of random variables representing various observable quantities related to learning rate, and suppose for > 0, L(X|θ, ) = N(θ, ). It is found that the first q
3.4 The scores on a battery of three tests are given by z’ = (z1,z2,z3). Suppose 100 people are tested and the results are summarized from observations z1, . . . ,z100 by where z and V are the
3.3 Let X denote the change in the number of international confrontations into which a nation entered from the quarter century preceding World War II to the quarter century following World War II;
3.2 Let X : 5 × 1 denote the vector of demand for 5 separate items in a retail business, and L(X|θ, ) = N(θ, ). Suppose that for fixed, θ follows the natural conjugate prior distribution L(θ| )
3.1 Let x1 denote the change in the number of users of the hallucinogenic drug LSD in Chicago from 1968 to 1969; x2 denotes the change in the number of users of heroin; x3 denotes the change in the
2.17 Write each of the following quadratic forms in terms of a symmetric matrix; 2x1² + 4 x1X2 + x2², 3x1²–2 x1x2 + 4x2². Are they positive definite?
2.16 Let Find the latent roots and latent vectors of: 4A, A + 5I, A–1, A², A½
2.15 Find the ranks of the following matrices:
2.14 Given the vectors a = (3,1,2)’, and b= (–1,7,4)’, find: b’b, a’b, aa’, 5a + 2b, a’Hb, where H = diag(2,–1,7).
2.13 Write the following equations in matrix form:
2.12 Suppose V: p × p is positive definite and V is partitioned as where V11: q × q and V11 is positive definite. Define ƒ ≡ ∫ V (n–p–1)/2dV12, where n > p + 1, and the integration extends
2.11 Define ƒ(Z) = E[tr ( –Z)²], for : p × p, Z: p × p, = ’, Z = Z’, and the expectation is taken with respect to the distribution of . Show that ƒ(Z) is minimized for Z = E( ). It will be
2.10 In Exercise 2.1, find the latent roots of . In Chapter 9 it will be seen that these latent roots will correspond to the variances of the principal components under an intraclass covariance
2.9 Suppose A and B are defined as in Exercise 2.1. Evaluate or determine the following: A ⊗ B, A ⊗ B ,tr (A ⊗ B), (A ⊗ B)–1, the latent roots of A ⊗ B, Is A ⊗ Ba covariance matrix?
2.8 Let x : p × 1 denote a vector of positive control variables associated with the administration of a hospital. That is, x can be set as desired to achieve a particular effect.Let ƒ(x) denote an
2.7 Let xj denote the time required to perform job j, j = 1,...,5. Suppose the cost of a project is given by C(x) = x’Ax, where x’ = (x1,...,x5), and Assuming the units are defined so that = 1,
2.6 Find the pseudoinverse of Is positive definite? What is the advantage of computing the pseudoinverse instead of any other generalized inverse?
2.5 Let X1, X2, X3 denote changes in the incidence of crime from 1967 to 1968, for three types of crime. Suppose there is so much data available that the matrix of variances and covariances of
2.4 Define Find the latent roots and all sets of normalized latent vectors of A. Is A a covariance matrix? Is A singular? What are the latent roots of A²?
2.3 Suppose y : n × 1 is a vector of response times of n individuals in a learning experiment, u : n × 1 and v : n × 1 are vectors of scores of the n individuals on each of two types of tests, and
2.2 Let xt denote sales of a company during period t, and suppose the year is divided into four periods. Let the matrix of variances and covariances of xi and xj have the structure Thus, sales during
2.1 Let Uj : k × 1 denote a vector of k scores on a battery of tests given to a group of children in city j. It is believed that and the matrix of covariances between Ui and Uj, i ≠ j, is Define
To illustrate the leverage impact, assume that the stock in Example 5‐3 goes up 20 percent from $90 to $108, for a gain of $18 × 100 shares, or $1,800. The investor has a $1,800/$4,500 = 40
Assume in the previous example that the maintenance margin is 30 percent. If the price of the stock drops to $75, the actual margin percentage will be 40.0 percent [($7,500 − $4,500)/$7,500].
Confirm the expected portfolio returns in column 1. The correlation coefficient, ρ, is 0.15. Return (%) Standard deviation (%) Covariance EG&G 25 30 112.5 GF 23 25
Based on your analysis, you think that next year’s return for General Foods will range from 1 to 15 percent as described earlier. The expected value of the probability distribution for General
The variance and standard deviation for General Foods, using the information reported above, is calculated in Table 7-1.Table 7-1 TABLE 7-1 Calculating the Standard Deviation Using Expected
With equal dollar amounts invested in three securities, the portfolio weights are 0.333, 0.333, and 0.333. For an equal‐weighted portfolio of five securities, each security would have a portfolio
Consider a three‐stock portfolio consisting of stocks G, H, and I with expected returns of 12, 20, and 17 percent, respectively. Assume that 50 percent of investable funds are invested in security
The risk of a portfolio declines quickly as more securities are added. Using Equation 7-8 and assuming that each security’s standard deviation is 20 percent, the risk of a 100‐security portfolio
Consider the returns between Southeast Utilities and Precision Instruments for the period 2005–2014. The summary statistics for these two stocks are as follows:Assume, for expositional purposes, we
Extending Example 7-6, the correlation between Southeast Utilities and Precision Instruments returns is +0.29. In order to focus on the effects of a changing correlation coefficient, we continue to
Using the same data as Example 7-7, let’s consider portfolio risk. Recall that the correlation coefficient between Southeast Utilities and Precision Instruments is +0.29. For illustration purposes,
An analyst considering 100 securities must estimate [100(99)]/2 = 4,950 unique covariances. For 250 securities, the number is [250(249)]/2 = 31,125 unique covariances.
Consider the 25‐month bear market that occurred during 2000–2002. A 100 percent stock portfolio (Wilshire 5000 Index) would have lost about 44 percent of its value, while an investor who chose a
Consider the period around the 2008 financial crisis. The stock market hit a record high on October 9, 2007, and officially entered a declining phase by June 30, 2008. The typical U.S. stock index
Assume that the expected return on portfolio M is 13 percent, with a standard deviation of 20 percent, and that RF is 5 percent. The slope of the CML isIn our example, a risk premium of 0.40
In Figure 9‐5, Security A’s beta of 1.5 indicates that, on average, Security A’s returns exhibit 1.5 times the average sensitivity to market return changes, both up and down. A security whose
To illustrate the estimation of the market model, we use return data for the Coca‐Cola company (ticker symbol “KO”). Fitting a regression equation to 60 months of return data along with
In Figure 9‐8, Investor G plots two securities relative to the SML. Based on Investor G’s analysis, Security X has a relatively high estimated return derived from fundamental analysis and plots
Assume that the beta for IBM is 1.15. Also assume that RF is 5 percent and that the expected return on the market is 12 percent. The required return for IBM can be calculated as KIBM = 0.05 +
An investor holds a portfolio of stocks that she thinks is influenced by only two basic economic factors—inflation and the economy’s output. Diversification once again plays a role, because the
Assume Magna Corporation annual‐pay preferred stock has a 7 percent dividend rate and a $100 par value. If an investor’s required return on Magna preferred is 8.5 percent, the investor’s
Assume that Summa Corporation is currently paying $1 per share in dividends and investors expect dividends to grow at the rate of 7 percent a year for the foreseeable future. For investments at this
For Summa, the estimated value today, V0 , is $13.38 , and for the end of Period 1, using D2 in the numerator of Equation 10-5, it isThis estimated value at the end of Period 1 is 7 percent higher
For Summa, assume that the discount rate used, k, is 16 percent instead of 15 percent, with other variables held constant: In this example, a 1‐percentage‐point increase in k results in an 11.14
Assume that for Summa the growth rate, g, is 8 percent instead of 7 percent, with other variables held constant: In this example, a 1‐percentage‐point increase in g results in a 15.3 percent
Assume that for Summa the discount rate increases to 16 percent, and the growth rate declines to 4 percent: Vo $1(1.04) 0.16 0.04 = $8.67
In early 2012, Standard & Poor’s estimated 2012 earnings for Cliff’s Natural Resources (CLF) of $12.18. S&P estimated an appropriate P/E for CLF to be 7.8. Multiplying these two numbers together
In November 2014, J.P. Morgan Chase had a P/B ratio of 1.1, whereas Coca‐Cola’s P/B was 5.8, and the average P/B for the S&P 500 was 2.4. Differences in the level of fixed assets and capital
In November 2014, Amazon had an enterprise value to EBITDA (EV/EBITDA) ratio of 38.4, whereas Barnes & Noble’s EV/EBITDA was 4.7. This dramatic difference in ratios reflects, among other things,
Between January 1998 and March 2000, the Amex Internet Index rose from 87 to 689. Therefore, in just over two years, this index showed a gain of almost 700 percent. Such was the euphoria that
At the peak of the NASDAQ market rise, which occurred on March 10, 2000, Cisco had a P/E ratio of about 150, Yahoo about 650, and JDS Uniphase about 640. One year later, the same companies had P/E
Consider the performance of the T. Rowe Price Growth and Income Fund for a recent multi-year period, as shown in Figure 11-1. Notice how this fund’s total returns and the total returns for the S&P
Google went public in August 2004 at $85, a price regarded by many at the time as ridiculously high. By May of 2005, the price had tripled. By late 2007, the price was over $600 a share but fell
Perhaps the best foreign example of the impact of the overall market on investors is Japan. In the 1980s, Japan seemed invincible in its economic performance, and the Nikkei stock index reflected
Think of one investor allocating her portfolio as 90 percent NASDAQ stocks and 10 percent cash equivalents and another investor doing the opposite, 90 percent cash equivalents and 10 percent NASDAQ
Vanguard offers a large selection of index funds, which allows investors to duplicate various market segments at a very low cost. Some examples are as follows: 1. The 500 Index Portfolio consists of
Technical analysts look for patterns in stock prices that can be exploited. Figure 12‐3 shows 30 days of price changes for Botox Company. Notice that in every case but one, whenever the price
Although an extreme example, the Boston Company International Small‐Cap fund had a 2007 distribution equal to $23.17 per share. If, earlier that year, you had invested $100,000 in the fund, your
Suppose an analyst wants to study the relationship between earnings surprises and stock prices. An event study would quantify the relationship between quarterly earnings announcements (which contain
The Motley Fool is a popular and well‐known source of investment information and ideas for individual investors. Operating a website and also producing some books and other publications, this
In March 2001, following the stock market plunge and weakening of the economy, it was reported that the index of leading economic indicators declined in February for the fourth time in five months.
The business cycle–stock‐price relationship is illustrated by what happened in 2000–2001. Following a strong run‐up in the late 1990s, the U.S. stock market peaked in March 2000, and the
The S&P 500 increased about 150 percent between the end of 1994 and December 1998. Stock prices are a function of both corporate earnings and the P/E ratio. At the end of 1994, the P/E based on
Consider the following headlines from financial media: “Markets Fall on Absence of Rate Cut” “Recent Rise in Long‐Term Interest Rates May Mean Trouble for the Stock Market” “As
Consider the medical appliances and equipment industry. Intuitive Surgical, Inc. (ISRG) pioneered a robotic surgery machine that revolutionized certain surgical procedures by making possible only
Consider General Electric, a classic industrial company that has been in business for more than 100 years. Today it is well known for making CT scanners, jet engines, locomotives, gas turbines,
Using NAICS codes, the plastics product manufacturing industry is coded 3261. Within this code are several breakdowns, including, among others, plastic pipe and pipe fitting manufacturing (326122),
Would it surprise you to learn that in early 2012 the home building industry was ranked next to last out of all industries ranked by The Value Line Investment Survey? Probably not, given what
In certifying Coca‐Cola’s financial statements, the accounting firm preparing the statements makes a statement such as, “In our opinion, the reported financial statements present fairly, in all
FAS 133, which is concerned with financial derivatives and hedging, was issued in June 1998. The standard and its supporting documents now total more than 800 pages. There have been over 200
For a take on earnings quality, consider J.P. Morgan Chase’s 2011 third‐quarter earnings report. The company raised its quarterly earnings by almost $2 billion by using a debit valuation
Using the data for Coca‐Cola from Exhibits 15‐1 and 15‐2 (all $ values in billions):Substitute the following for the equations: Exhibits 15‐1Exhibits 15‐2 Net income/sales =
Using data from Exhibits 15‐2 and 15‐3, the FCFF and FCFE for Coca‐Cola for 2014 are as follows (all $ values in billions):FCFF is generally greater than FCFE; however, in this case, the low
Assume that the current one‐year bond rate (tR1) is 7 percent and the two forward rates are 7.5 percent (t+1r1) and 8.2 percent (t+2r1). The rate for a three‐year bond, (1+tR3), would be R3 =
To calculate the RCY an investor would earn from a 12 percent reinvestment rate, add the total return from coupons shown in Table 17‐1, $7,738, to the maturity value of the bond, $1,000, to obtain

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