EXAMPLE 73 The Effect of Commercial Paper Issuance on Stock Prices
Commercial paper CP is unsecured short-term corporate debt that, like U.S. Treasury bills, is characterized by a single payment at maturity. When a company enters the CP market for the first time, how do stock market participants react to the announcement of the CP ratings Nayar and Rozeff 1994 addressed this question using data for the period October 1981 to December 1985. During this period, 132 CP issues 96 industrial and 36 non-industrial received an initial rating in Standard amp Poor's...
Recovery Rates by Seniority
Industry Group Ex-Utilities Sample Group Number of Average Standard Number of Average Standard Seniority Observations Price Deviation Observations Price Deviation Senior Unsecured 32 77.74 18.06 189 42.56 24.89 Source Altman and Kishore 1996, Table 5 . This is the average price at default and is a measure of recovery rate. Assume that the populations recovery rates of utilities, recovery rates of non-utilities are normally distributed and that the samples are independent. The population...
EXAMPLE 49 BankCorps Earnings per Share 4
In Example 4-8, you calculated the expected value of BankCorp's EPS as 2.34, which is your forecast. Now you want to measure the dispersion around your forecast. Table 4-4 shows your view of the probability distribution of EPS for the current fiscal year. 8 For simplicity, we model all random variables in this chapter as discrete random variables, which have a countable set of outcomes. For continuous random variables, which are discussed along with discrete random variables in the chapter on...
EXAMPLE 34 Median and Arithmetic Mean The Case of the PriceEarnings Ratio
Suppose a client asks you for a valuation analysis on the seven-stock U.S. common stock portfolio given in Table 3-10. The stocks are equally weighted in the portfolio. One valuation measure that you use is P E, the ratio of share price to earnings per share EPS . Many variations exist for the denominator in the P E, but you are examining P E defined as current price divided by the current mean of all analysts' EPS estimates for the company for the current fiscal year Consensus Current EPS in...
Performance in Forecasting Quarterly Earnings per Share
Investment analysts often use earnings per share EPS forecasts. One test of forecasting quality is the zero-mean test, which states that optimal forecasts should have a mean forecasting error of 0. Forecasting error Predicted value of variable -Actual value of variable. You have collected data shown in the table above for two analysts who cover two different industries Analyst A covers the telecom industry Analyst B covers automotive parts and suppliers. A. With x as the population mean...
TABLE 324 Arithmetic Mean Annual Return and Standard Deviation of Returns US
S amp P500 U.S. small stock U.S. long-term corporate U.S. long-term government U.S. 30-day T-bill Using the information in Table 3-24, address the following 1. Calculate the coefficient of variation for each asset class given. 2. Rank the asset classes from most risky to least risky using CV as a measure of relative dispersion. 3. Determine whether there is more difference between the absolute or the relative riskiness of the S amp P 500 and U.S. small stocks. Use the standard deviation as a...
rw1 taW InlJW55
For our example, r0 gt 1 In Si S0 ln l R0 lt 1 ln 34.50 30 ln 1.15 0.139762. Thus, 13.98 percent is the continuously compounded return from t 0 to t 1. The continuously compounded return is smaller than the associated holding period 33 Continuous compounding treats time as essentially continuous or unbroken, in contrast to discrete compounding, which treats time as advancing in discrete finite intervals. Continuously compounded returns are the model for returns in so-called continuous time...
The Future Value Of A Series Of Cash Flows
In this section, we consider series of cash flows, both even and uneven. We begin with a list of terms commonly used when valuing cash flows that are distributed over many time periods. An annuity is a finite set of level sequential cash flows. An ordinary annuity has a first cash flow that occurs one period from now indexed at t 1 . An annuity due has a first cash flow that occurs immediately indexed at t 0 . A perpetuity is a perpetual annuity, or a set of level never-ending sequential cash...
Forecast XNumber of Analysts nj
Because the sample is a small fraction of the number of analysts who follow this stock, assume that we can ignore the finite population correction factor. Assume that the analyst forecasts are normally distributed. A. What are the mean forecast and standard deviation of forecasts B. Provide a 95 percent confidence interval for the population mean of the forecasts. 8. Thirteen analysts have given the following fiscal-year earnings forecasts for a stock
Forecast XNumber of Analysts n
Because the sample is a small fraction of the number of analysts who follow this stock, assume that we can ignore the finite population correction factor. A. What are the mean forecast and standard deviation of forecasts B. What aspect of the data makes us uncomfortable about using Rabies to construct confidence intervals for the population mean forecast 9. Explain the differences between constructing a confidence interval when sampling from a normal population with a known population variance...
EXAMPLE 310 The Range and the Mean Absolute Deviation
Having calculated mean returns for the two mutual funds in Example 3-7, the analyst is now concerned with evaluating risk. TABLE 3-15 repeated Total Returns for Two Mutual Funds, 1998-2002 TABLE 3-15 repeated Total Returns for Two Mutual Funds, 1998-2002 Based on the data in Table 3-15 repeated above, answer the following 1. Calculate the range of annual returns for A SLASX and B PRFDX, and state which mutual fund appears to be riskier based on these ranges. 2. Calculate the mean absolute...
EXAMPLE 67 Biases in Investment Research
An analyst is reviewing the empirical evidence on historical U.S. equity returns. She finds that value stocks i.e., those with low P Bs outperformed growth stocks i.e., those with high P Bs in some recent time periods. After reviewing the U.S. market, the analyst wonders whether value stocks might be attractive in the United Kingdom. She investigates the performance of value and growth stocks in the U.K. market from January 1990 to December 2003. To conduct this research, the analyst does the...
Portfolios Deviations from Benchmark Return 19922003
3. A. Calculate the frequency, cumulative frequency, relative frequency, and cumulative relative frequency for the portfolio's deviations from benchmark return, given the set of intervals in the table below. Cumulative Cumulative Relative Relative Return Interval Frequency Frequency Frequency Frequency -9.19 lt A lt -4.55 -4.55 lt B lt 0.09 0.09 lt C lt 4.73 4.73 lt D lt 9.37 B. Construct a histogram using the data. C. Identify the modal interval of the grouped data. 4. Tracking risk also...
EXAMPLE 26 The Bank Discount Yield
Suppose a T-bill with a face value or par value of 100,000 and 150 days until maturity is selling for 98,000. What is its bank discount yield Solution. For this example, the dollar discount, D, is 2,000. The yield on a bank discount basis is 4.8 percent, as computed with Equation 2-3 The bank discount formula takes the T-bill's dollar discount from face or par as a fraction of face value, 2 percent, and then annualizes by the factor 360 150 2.4. The price of discount instruments such as T-bills...
EXAMPLE 56 The Expected Number of Defaults in a Bond Portfolio
Suppose as a bond analyst you are asked to estimate the number of bond issues expected to default over the next year in an unmanaged high-yield bond portfolio with 25 U.S. issues from distinct issuers. The credit ratings of the bonds in the portfolio are tightly clustered around Moody's B2 Standard amp Poor's B, meaning that the bonds are speculative with respect to the capacity to pay interest and repay principal. The estimated annual default rate for B2 B rated bonds is 10.7 percent. 1. Over...
EXAMPLE 41 Profiting from Inconsistent Probabilities
You are examining the common stock of two companies in the same industry in which an important antitrust decision will be announced next week. The first company, SmithCo Corporation, will benefit from a governmental decision that there is no antitrust obstacle related to a merger in which it is involved. You believe that SmithCo's share price reflects a 0.85 probability of such a decision. A second company, Selbert Corporation, will equally benefit from a go ahead ruling. Surprisingly, you...
Fraction of the 500 Companies Meeting the Criterion
Company will increase combined sales growth rate Balance sheet impact manageable If the criteria are independent, how many companies will pass the screen 8. You apply both valuation criteria and financial strength criteria in choosing stocks. The probability that a randomly selected stock from your investment universe meets your valuation criteria is 0.25. Given that a stock meets your valuation criteria, the probability that the stock meets your financial strength criteria is 0.40. What is the...
US Dollar Daily Returns in Percent
U.S. Bonds German Bonds Italian Bonds Expected Standard U.S. Bonds German Bonds Italian Bonds Source Kool 2000 , Table 1 excerpted and adapted . A. Using the data given above, construct a covariance matrix for the daily returns on U.S., German, and Italian bonds. B. State the expected return and variance of return on a portfolio 70 percent invested in U.S. bonds, 20 percent in German bonds, and 10 percent in Italian bonds. C. Calculate the standard deviation of return for the portfolio in Part...
FIGURE 35 Center of Gravity Analogy for the Arithmetic Mean
When the fulcrum is placed at 8, the bar is perfectly balanced. Equation 3-3, multiplying both sides of the equation by n nX X The sum of the deviations from the mean can thus be calculated as follows 1 1 Deviations from the arithmetic mean are important information because they indicate risk. The concept of deviations around the mean forms the foundation for the more complex concepts of variance, skewness, and kurtosis, which we will discuss later in this chapter. An advantage of the...
Probability Function of Bedolf and Zedock Returns
Note Entries are joint probabilities. Note Entries are joint probabilities. 15. You have developed a set of criteria for evaluating distressed credits. Companies that do not receive a passing score are classed as likely to go bankrupt within 12 months. You gathered the following information when validating the criteria Forty percent of the companies to which the test is administered will go bankrupt within 12 months P nonsurvivor 0.40. Fifty-five percent of the companies to which the test is...
TABLE 21 Computing IRR
Notation Used on Numerical Value Most Calculators for This Problem Because the IRR of 14.50 percent is greater than the opportunity cost of the project, the project should benefit Kageyama's shareholders. Whether it uses the IRR rule or the NPV rule, Kageyama makes the same decision Build the factory. In the previous example, value creation is evident For a single 1,000 million payment, Kageyama creates a project worth 1,089.55 million, a value increase of 89.55 million. Another perspective on...
The Future Value Of A Single Cash Flow
In this section, we introduce time value associated with a single cash flow or lump-sum investment. We describe the relationship between an initial investment or present value PV , which earns a rate of return the interest rate per period denoted as r, and its future value FV , which will be received N years or periods from today. The following example illustrates this concept. Suppose you invest 100 PV 100 in an interest-bearing bank account paying 5 percent annually. At the end of the first...
EXAMPLE 313 Applying Chebyshevs Inequality
According to Table 3-22, the arithmetic mean monthly return and standard deviation of monthly returns on the S amp P 500 were 0.97 percent and 5.65 percent, respectively, during the 1926-2002 period, totaling 924 monthly observations. Using this information, address the following 1. Calculate the endpoints of the interval that must contain at least 75 percent of monthly returns according to Chebyshev's inequality. 2. What are the minimum and maximum number of observations that must lie in the...
EXAMPLE 57 Probability That a Lending Facility Covenant Is Breached
You are evaluating the bonds of a below-investment-grade borrower at a low point in its business cycle. You have many factors to consider, including the terms of the company's bank lending facilities. The contract creating a bank lending facility such as an unsecured line of credit typically has clauses known as covenants. These covenants place restrictions on what the borrower can do. The company will be in breach of a covenant in the lending facility if the interest coverage ratio, EBITDA...
ERP 070 X 0029 020 X 0021 010 X 0073 00318 or 32 percent jRP w2 x2Ri w v2R2 wi
0.70 2 0.167281 0.20 2 0.367236 0.10 2 0.403225 2 0.70 0.20 0.022307 2 0.70 0.10 0.025972 2 0.20 0.10 0.269367 0.081968 0.014689 0.004032 0.006246 0.003636 C. The standard deviation of this portfolio is Vct2 Rp 0.121346 i 2 0.348348, or 34.8 percent. 13. A covariance matrix for five assets has 5 X 5 25 entries. Subtracting the five diagonal variance terms, we have 25 - 5 20 off-diagonal entries. Because the covariance matrix is symmetric, only 10 entries are unique 10 20 2 . Hence, you must...
Other Measures Of Location Quantiles
Having discussed measures of central tendency, we now examine an approach to describing the location of data that involves identifying values at or below which specified proportions of the data lie. For example, establishing that 25,50, and 75 percent of the annual returns on a portfolio are at or below the values -0.05,0.16, and 0.25, respectively, provides concise information about the distribution of portfolio returns. Statisticians use the word quantile or fractile as the most general term...
EXAMPLE 512 Potential Gains from Market Timing A Monte Carlo Simulation 1
All active investors want to achieve superior performance. One possible source of superior performance is market timing ability. How accurate does an investor need to be as a bull- and bear-market forecaster for market timing to be profitable What size gains compared with a buy-and-hold strategy accrue to a given level of accuracy Because of the variability in asset returns, a huge amount of return data is needed to find statistically reliable answers to these questions. Chua, Woodward, and To...
FIGURE 32 Histogram of SP 500 Monthly Total Returns January 1926 to December
o o o o o o o o o o o o o o o o o o o 05i5 gt 55C gt 55 55C 0CiC gt 5555550C gt C gt 5553 o o o o o o o o o o o o o o o o o o o 05i5 gt 55C gt 55 55C 0CiC gt 5555550C gt C gt 5553 amp gt ' , gt ' W V V V V lt V 'O' iO iP' S gt ' o 0 o 0 o lt 5 gt 55 O O s- w o o o o o o o o Another form of line graph is the cumulative frequency distribution. Such a graph can plot either the cumulative absolute or cumulative relative frequency against the upper interval limit. The cumulative frequency...
EXAMPLE 413 Inferring Whether DriveMeds EPS Met Consensus EPS
You are still an investor in DriveMed stock. To review the givens, your prior probabilities are P EPS exceeded consensus 0.45, P EPS met consensus 0.30, and P EPS fell short of consensus 0.25. You also have the following conditional probabilities P DriveMed expands I EPS exceeded consensus 0.75 P DriveMed expands I EPS met consensus 0.20 P DriveMed expands I EPS fell short of consensus 0.05 Recall that you updated your probability that last quarter's EPS exceeded the consensus estimate from 45...
EXAMPLE 315 Calculating the Sharpe Ratio
In earlier examples, we computed the various statistics for two mutual funds, Selected American Shares SLASX and T. Rowe Price Equity Income PRFDX , for a five-year period ending in December 2002. Table 3-25 summarizes selected statistics for these two mutual funds for a longer period, the 10-year period ending in 2002. TABLE 3-25 Mutual Fund Mean Return and Standard Deviation of Return, 1993-2002 TABLE 3-25 Mutual Fund Mean Return and Standard Deviation of Return, 1993-2002 The U.S. 30-day...
EXAMPLE 52 OnePeriod Stock Price Movement as a Bernoulli Random Variable
Suppose we describe stock price movement in the following way. Stock price today is S. Next period stock price can move up or down. The probability of an up move is p, and the probability of a down move is 1 - p. Thus, stock price is a Bernoulli random variable with probability of success an up move equal to p. When the stock moves up, ending price is uS, with u equal to 1 plus the rate of return if the stock moves up. For example, if the stock earns 0.01 or 1 percent on an up move, u 1.01....
FIGURE 37 Properties of a Skewed Distribution
Distribution Skewed to the Right Positively Skewed Distribution Skewed to the Left Negatively Skewed Reprinted with permission from Fixed Income Readings for the Chartered Financial Analysf Program. Copyright 2000, Frank J. Fabozzi Associates, New Hope, PA. the median, which is less than the mode.40 Investors should be attracted by a positive skew because the mean return falls above the median. Relative to the mean return, positive skew amounts to a limited, though frequent, downside compared...
TABLE 28 ShortTerm Money Market Yields
Which yield or yields are appropriate for finding the present value of the 1,000 to be received in 150 days Solution. The holding period yield is appropriate, and we can also use the money market yield and effective annual yield after converting them to a holding period yield. Holding period yield 2.0408 percent . This yield is exactly what we want. Because it applies to a 150-day period, we can use it in a straightforward fashion to find the present value of the 1,000 to be received in 150...
EX PXX PX2X2 PXnXn X PXXi47
where X is one of n possible outcomes of the random variable X.8 The expected value is our forecast. Because we are discussing random quantities, we cannot count on an individual forecast being realized although we hope that, on average, forecasts will be accurate . It is important, as a result, to measure the risk we face. Variance and standard deviation measure the dispersion of outcomes around the expected value or forecast. Definition of Variance. The variance of a random variable is the...
n
The variance is the average of the squared deviations around the mean, and the standard deviation is the positive square root of variance. In computing sample variance s2 and sample standard deviation, the average squared deviation is computed using a divisor equal to the sample size minus 1. The semivariance is the average squared deviation below the mean semideviation is the positive square root of semivariance. Target semivariance is the average squared deviation below a target level target...
EXAMPLE 24 TimeWeighted Rate of Return
Strubeck Corporation sponsors a pension plan for its employees. It manages part of the equity portfolio in-house and delegates management of the balance to Super Trust Company. As chief investment officer of Strubeck, you want to review the performance of the in-house and Super Trust portfolios over the last four quarters. You have arranged for outflows and inflows to the portfolios to be made at the very beginning of the quarter. Table 2-5 summarizes the inflows and outflows as well as the two...
Symmetry And Skewness In Return Distributions
Mean and variance may not adequately describe an investment's distribution of returns. In calculations of variance, for example, the deviations around the mean are squared, so we do not know whether large deviations are likely to be positive or negative. We need to go beyond measures of central tendency and dispersion to reveal other important characteristics of the distribution. One important characteristic of interest to analysts is the degree of symmetry in return distributions. If a return...
Client Portfolio
Source www.multexinvestor.com 51 In particular, diluted EPS is for continuing operations and before extraordinary items and accounting changes. Source www.multexinvestor.com 51 In particular, diluted EPS is for continuing operations and before extraordinary items and accounting changes. Based only on the information in the above table, calculate the following for the portfolio A. i. Arithmetic mean P E ii. Median P E B. i. Arithmetic mean P S ii. Median P S C. i. Arithmetic mean P B ii. Median...
EXAMPLE 43 Probability of a Limit Order Executing
You have two buy limit orders outstanding on the same stock. A limit order to buy stock at a stated price is an order to buy at that price or lower. A number of vendors, including an Internet service that you use, supply the estimated probability that a limit order will be filled within a stated time horizon, given the current stock price and the price limit. One buy order Order 1 was placed at a price limit of 10. The probability that it will execute within one hour is 0.35. The second buy...
EXAMPLE 46 Conditional Probabilities and Predictability of Mutual Fund
The purpose of the Kahn and Rudd 1995 study, introduced in Example 4-2, was to address the question of repeat mutual fund winners and losers. If the status of a fund as a winner or a loser in one period is independent of whether it is a winner in the next period, the practical value of performance ranking is questionable. Using the four events defined in Example 4-2 as building blocks, we can define the following events to address the issue of predictability of mutual fund performance Fund is a...
MSCI Germany Index Total Returns 19932002
5. To describe the distribution of observations, perform the following A. Create a frequency distribution with five equally spaced classes round up at the second decimal place in computing the width of class intervals . B. Calculate the cumulative frequency of the data. C. Calculate the relative frequency and cumulative relative frequency of the data. D. State whether the frequency distribution is symmetric or asymmetric. If the distribution is asymmetric, characterize the nature of the...
The Graphic Presentation Of Data
A graphical display of data allows us to visualize important characteristics quickly. For example, we may see that the distribution is symmetrically shaped, and this finding may influence which probability distribution we use to describe the data. In this section, we discuss the histogram, the frequency polygon, and the cumulative frequency distribution as methods for displaying data graphically. We construct all of these graphic presentations with the information contained in the frequency...
Kurtosis In Return Distributions
In the previous section, we discussed how to determine whether a return distribution deviates from a normal distribution because of skewness. One other way in which a return distribution might differ from a normal distribution is by having more returns clustered closely around the mean being more peaked and more returns with large deviations from the mean having fatter tails . Relative to a normal distribution, such a distribution has a greater percentage of small deviations from the mean...
Summary
In this chapter, we applied the concepts of present value, net present value, and internal rate of return to the fundamental problem of valuing investments. We applied these concepts first to corporate investment, the well-known capital budgeting problem. We then examined the fundamental problem of calculating the return on a portfolio subject to cash inflows and outflows. Finally we discussed money market yields and basic bond market terminology. The following summarizes the chapter's key...
Population Standard Deviation Formula The population standard deviation
defined as the positive square root of the population variance, is where x is the population mean and N is the size of the population. Using the example of the P Es for BJ, COST, and WMT, according to Equation 3-12 we would calculate the variance, 26.5542, then take the square root V26.5542 5.1531 or approximately 5.2. Both the population variance and standard deviation are examples of parameters of a distribution. In later chapters, we will introduce the notion of variance and standard...
Solving For Rates Number Of Periods Or Size Of Annuity Payments
In the previous examples, certain pieces of information have been made available. For instance, all problems have given the rate of interest, r, the number of time periods, N, the annuity amount, A, and either the present value, PV, or future value, FV. In real-world applications, however, although the present and future values may be given, you may have to solve for either the interest rate, the number of periods, or the annuity amount. In the subsections that follow, we show these types of...
Net Present Value And Internal Rate Of Return
In applying discounted cash flow analysis in all fields of finance, we repeatedly encounter two concepts, net present value and internal rate of return. In the following sections we present these keystone concepts. We could explore the concepts of net present value and internal rate of return in many contexts, because their scope of application covers all areas of finance. Capital budgeting, however, can serve as a representative starting point. Capital budgeting is important not only in...
Introduction 1
As investment analysts, much of our work includes evaluating transactions involving present and future cash flows. In the chapter on the time value of money TVM , we presented the mathematics needed to solve those problems and illustrated the techniques for the major problem types. In this chapter we turn to applications. Analysts must master the numerous applications of TVM or discounted cash flow analysis in equity, fixed income, and derivatives analysis as they study each of those topics...
F Discounted Cash Flow J Applications
After completing this chapter, you will be able to do the following Calculate and interpret the net present value NPV and the internal rate of return IRR of an investment. Contrast the NPV rule to the IRR rule. Distinguish between money-weighted and time-weighted rates of return. Calculate the money-weighted and time-weighted rates of return of a portfolio. Calculate bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill. Convert among...
Info Isz
iv. Total 66,242.54 7,350.30 73,592.84 You should be willing to pay 73,592.84 for this instrument. ii. Recognize the problem as a delayed annuity. Delaying the payments requires two calculations. iii. Use the formula for the present value of an annuity Equation 1-11 . 1 to bring the four payments of 10,000 back to a single equivalent lump sum of 33,121.27 at t 2. Note that we use t 2 because the first annuity payment is then one period away, giving an ordinary annuity. Notation Used on...