Financial Management Principles and Applications Titman Keown Martin Twelfth Edition

Financial Management Principles and Applications Titman Keown Martin Twelfth Edition

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 10: 1-292-02306-6 ISBN 13: 978-1-292-02306-9 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

Regardless of Your Major Statistics permeate almost all areas of business. Because financial markets provide rich Using Statistics sources of data, it is no surprise that the tools used by statisticians are so widely used in finance. In this chapter, we use the basic tools of descriptive statistics, such as the mean and measures of dispersion, to analyze the riskiness of potential investments. These tools, which are essential for the study of finance, are widely used in all business disciplines as well as in the social sciences. A good understanding of statistics is extremely useful, regardless of your major. Your Turn: See Study Question 1. 1 Realized and s of Return and Risk We begin our discussion of risk and return by defining some key terms that are critical to developing an understanding of the risk and return inherent in risky investments. We will focus our examples on the risk and return encountered when investing in various types of securities in the financial markets but the methods we use to measure risk and return are equally applicable to any type of risky investment, such as the introduction of a new product line. Specifically, we provide a detailed definition of both realized and expected rates of return. In addition, we begin our analysis of risk by showing how to calculate the variance and the standard deviation of historical, or realized, rates of return. Calculating the Realized Return from an Investment If you bought a share of stock and sold it one year later, the return you would earn on your stock investment would equal the ending price of the share (plus any cash distributions such as dividends) minus the beginning price of the share. This gain or loss on an investment is called a cash return, which is summarized in Equation (1) as follows: Return = Ending Distribution + - Beginning (Dividend) Consider what you would have earned by investing in one share of Dick s Sporting Goods (DKS) stock at the end of May 2008 and then selling that share one year later at the beginning of June 2009. Substituting into Equation (1), you would calculate the cash return as follows: Return = Ending Distribution + - Beginning = +17.80 + 0.00-23.15 = - +5.35 1Dividend2 In this instance, you would have realized a loss of $5.35 on your investment, because the firm s stock price dropped over the year from $23.15 down to $17.80 and the firm did not make any cash distributions to its stockholders. The method we have just used to compute the return on Dick s Sporting Goods stock provides the gain or loss we experienced during a period. We call this the cash return for the period. In addition to calculating a cash return, we can calculate the rate of return as a percentage. As a general rule, we summarize the return on an investment in terms of a percentage return, because we can compare these percentage rates of return across different investments. The rate of return (sometimes referred to as a holding period return) is simply the cash return divided by the beginning stock price, as defined in Equation (2): (1)

Return = Return Beginning = Ending Distribution + - Beginning 1Dividend2 Beginning Table 1 contains beginning prices, dividends (cash distributions), and ending prices spanning a one-year holding period for five public firms. We use this data to compute the realized rates of return for a one-year period of time beginning on October 8, 2008, and ending with October 9, 2009. To illustrate, we calculate the rate of return earned from the investment in Dick s Sporting Goods stock as the ratio of the cash return (found in Column D of Table 1) to your investment in the stock at the beginning of the period (found in Column A). For this investment, your rate of return is a whopping 45% $7.37/15.32. Even though Dick s paid no cash dividends, its stock price rose from $15.32 at the beginning of the period to $22.69, or by $7.37 over the year you would have earned a 45 percent rate of return on the stock if you had bought and sold on these dates. Notice that all the realized rates of return found in Table 1 are positive except for Walmart (WMT), which experienced a negative rate of return. Does this mean that if we purchase shares of Walmart stock today we should expect to realize a negative rate of return over the next year? The answer is an emphatic no. The fact that Walmart s stock earned a negative rate of return in the past is evidence that investing in stock is risky. So, the fact that we realized a negative rate of return does not mean we should expect negative rates of return in the future. Future returns are risky and they may be negative or they may be positive; however, P Principle 2: There Is a Risk Return Tradeoff tells us that we will expect to receive higher returns for assuming more risk (even though there is no guarantee we will get what we expect). Calculating the Expected Return from an Investment We call the gain or loss we actually experienced on a stock during a period the realized rate of return for that period. However, the risk return tradeoff that investors face is not based on realized rates of return; it is instead based on what the investor expects to earn on an investment (2) Table 1 Measuring an Investor s Realized Return from Investing in Common Stock Stock s Return Beginning (Oct. 8, 2008) Ending (Oct. 9, 2009) Distribution (Dividend) Rate Company A B C D C B A E D/A Dick s Sporting Goods (DKS) $15.32 $22.69 $0.00 $ 7.37 45.0% Duke Energy (DUK) 16.38 15.82 1.16 $ 0.60 1.8% Emerson Electric (EMR) 32.73 37.75 1.32 $ 6.34 19.4% Sears Holdings (SHLD) 57.74 67.86 0.00 $10.12 17.5% Walmart (WMT) 55.81 49.68 1.06 (5.07) 9.1% Legend: We formalize the return calculations found in Columns D and E using Equations (1) and (2): Column D ( or Dollar Return) Return = Ending Distribution + - Beginning = P End + Div - P Beginning (1) 1Dividend2 Column E ( Return) Return, r = Return Beginning = P End + Div - P Beginning P Beginning (2)

in the future. We can think of the rate of return that will ultimately be realized from making a risky investment in terms of a range of possible return outcomes, much like the distribution of grades for a class at the end of the term. The expected rate of return is the weighted average of the possible returns, where the weights are determined by the probability that it occurs. To illustrate the calculation of an expected rate of return, consider an investment of $10,000 in shares of common stock that you plan to sell at the end of one year. To simplify the computations we will assume that the stock will not pay any dividends during the year, so that your total cash return comes from the difference between the beginning-of-year and endof-year prices of the shares of stock, which will depend on the state of the overall economy. In Table 2 we see that there is a 20 percent probability that the economy will be in recession at year s end and that the value of your $10,000 investment will be worth only $9,000, providing you with a loss on your investment of $1,000 (a 10 percent rate of return). Similarly, there is a 30 percent probability the economy will experience moderate growth, in which case you will realize a $1,200 gain and a 12 percent rate of return on your investment by year s end. Finally, there is a 50 percent chance that the economy will experience strong growth, in which case your investment will realize a 22 percent gain. Column G of Table 2 contains the products of the probability of each state of the economy (recession, moderate growth, or strong growth) found in Column B and the rate of return earned if that state occurs (Column F). By adding up these probability-weighted rates of return for the three states of the economy, we calculate an expected rate of return for the investment of 12.6 percent. Equation (3) summarizes the calculation in Column G of Table 2, where there are n possible outcomes. of Return 3E1r24 = Return 1 (r 1 ) of Return 1 + Return 2 (Pb 1 ) 1r 2 2 of Return 2 + g + Return n 1Pb 2 2 1r n 2 of Return n (3) 1Pb n 2 We can use Equation (3) to calculate the expected rate of return for the investment in Table 2, where there are three possible outcomes, as follows: E1r2 = 1-10,.22 + 112,.32 + 122,.52 = 12.6, Measuring Risk In the example we just examined, we expect to realize a 12.6 percent return on our investment; however, the return could be as little as 10 percent or as high as 22 percent. There are two methods financial analysts can use to quantify the variability of an investment s returns. The Table 2 State of the Economy Calculating the Expected Return for an Investment in Common Stock of the State of the Economy a (Pb i ) End-of-Year Selling for the Stock Beginning of the Stock Return from Your Investment Percentage Return Return/Beginning of the Stock Product Rate of Return of State of the Economy Column A Column B Column C Column D Column E C D Column F E D Column G B F Recession 20% $ 9,000 $10,000 $(1,000) 10% $1,000 $10,000 2.0% Moderate growth 30% 11,200 10,000 1,200 12% $1,200 $10,000 3.6% Strong growth 50% 12,200 10,000 2,200 22% $2,200 $10,000 11% Sum 100% 12.6% a The probabilities assigned to the three possible economic conditions have to be determined subjectively, which requires management to have a thorough understanding of both the investment cash flows and the general economy.

first is the variance of the investment returns and the second is the standard deviation, which is the square root of the variance. Recall that the variance is the average squared difference between the individual realized returns and the expected return. To better understand this we will examine both the variance and the standard deviation of an investment s rate of return. Calculating the Variance and Standard Deviation of the Return on an Investment Let s compare two possible investment alternatives: 1. U.S. Treasury Bill. A short-term (maturity of one year or less) debt obligation of the U.S. government. The particular Treasury bill that we consider matures in one year and promises to pay an annual return of 5 percent. This security has a risk-free rate of return, which means that if we purchase and hold this security for one year, we can be confident of receiving no more and no less than a 5 percent return. The term risk-free security specifically refers to a security for which there is no risk of default on the promised payments. 2. Common Stock of the Ace Publishing Company. A risky investment in the common stock of a company we will call Ace Publishing Company. The probability distribution of an investment s returns contains all the possible rates of return from the investment that might occur, along with the associated probabilities for each outcome. Figure 1 contains a probability distribution of the possible rates of return that we might realize on these two investments. The probability distribution for a risk-free investment in Treasury bills is illustrated as a single spike at a 5 percent rate of return. This spike indicates that if you purchase a Treasury bill, there is a 100 percent chance that you will earn a 5 percent annual rate of return. The probability distribution for the common stock investment, however, includes returns as low as 10 percent and as high as 40 percent. Thus, the common stock investment is risky, whereas the Treasury bill is not. Figure 1 Distribution of Returns for a Treasury Bill and the Common Stock of the Ace Publishing Company A probability distribution provides a tool for describing the possible outcomes or rates of return from an investment and the associated probabilities for each possible outcome. Technically, the following probability distribution is a discrete distribution because there are only five possible returns that the Ace Publishing Company stock can earn. The Treasury bill investment offers only one possible rate of return (5%) because this investment is risk-free. of occurrence 1.0 0.4 0.35 0.3 0.25 0.2 0.15 0.1 Treasury bill Publishing Co. Chance or of Occurrence Return on Investment 1 chance in 10 (10%) 10% 2 chances in 10 (20%) 5% 4 chances in 10 (40%) 15% 2 chances in 10 (20%) 25% 1 chance in 10 (10%) 40% 0.05 0 10% 5% 15% 25% Possible returns 40% >> END FIGURE 1

Using Equation (3), we calculate the expected rate of return for the stock investment as follows: E1r2 = 1.1021-10,2 + 1.20215,2 + 1.402115,2 + 1.202125,2 + 1.102140,2 = 15, Thus, the common stock investment in Ace Publishing Company gives us an expected rate of return of 15 percent. As we saw earlier, the Treasury bill investment offers an expected rate of return of only 5 percent. Does this mean that the common stock is a better investment than the Treasury bill because it offers a higher expected rate of return? The answer is no, because the two investments have very different risks. The common stock might earn a negative 10 percent rate of return or a positive 40 percent, whereas the Treasury bill offers only one positive rate of 5 percent. One way to measure the risk of an investment is to calculate the variance of the possible rates of return, which is the average of the squared deviations from the expected rate of return. Specifically, the formula for the return variance of an investment with n possible future returns can be calculated using Equation (4) as follows: Variance in Rates of Return 1s 2 2 = Return 1 1r 1 2-2 of Return E1r2 of Return 1 1Pb 1 2 + Return 2 1r 2 2-2 of Return E1r2 of Return 2 1Pb 2 2 + g + Return 3 1r n 2-2 of Return E1r2 of Return n (4) 1Pb n 2 Note that the variance is measured using squared deviations of each possible return from the mean or expected return. Thus, the variance is a measure of the average squared deviation around the mean. For this reason it is customary to measure risk as the square root of the variance which, as we learned in our statistics class, is called the standard deviation. For Ace Publishing Company s common stock, we calculate the variance and standard deviation using the following five-step procedure: Step 1. Calculate the expected rate of return using Equation (3). This was calculated previously to be 15 percent. Step 2. Subtract the expected rate of return of 15 percent from each of the possible rates of return and square the difference. Step 3. Multiply the squared differences calculated in Step 2 by the probability that those outcomes will occur. Step 4. Sum all the values calculated in Step 3 together. The sum is the variance of the distribution of possible rates of return. Note that the variance is actually the average squared difference between the possible rates of return and the expected rate of return. Step 5. Take the square root of the variance calculated in Step 4 to calculate the standard deviation of the distribution of possible rates of return. Note that the standard deviation (unlike the variance) is measured in rates of return. Table 3 illustrates the application of this procedure, which results in an estimated standard deviation for the common stock investment of 12.85 percent. This standard deviation compares to the 0 percent standard deviation of a risk-free Treasury bill investment. The investment in Ace Publishing Company carries higher risk than investing in the Treasury bill because it can potentially result in a return of 40 percent or possibly a loss of 10 percent. The standard deviation measure captures this difference in the risks of the two investments.