1.Investment Management Equations1. Rate of re turn on an asset or portfolioReturn = (End-of-period wealth - beginning-of-period weal th )/(begin n ing-of~pe ri od wealth)2. Actual margin in a stock purchase(nXmp) -[(I-im) X /,p X ,,]am =>I X mp3. Market price at which a margin purchaserwill receive a margin call_ (I - ;1>1) X ppmp- 1 - mm4. Actual margin in a short sale[(spXn) X ( I +im)] -(mpX>l)am = Xmp n5. Market price at which a short seller willreceive a margin call.Ip X (1 + ;1>1)mp = 1+ mm6. Expe cted return on a portfolio,rp = LX;ri; =17. Covariance between two securitiesau = PjUjUj8. Standard deviation of a portfolio[ .v ] 1/2Up = ~ ~ XiXi0;,9. Standard deviation of a two-asset portfolio[ j 1 ) 2 ~ I /~U p = XjUI + X,1O2 + 2XIX"PI2IT ,U 2j10. The market model"j = Uj + {3;/ + e,11. Beta from the market model13.= Uil, .,Uj12. Securitv variance from the market modelu; = l3~u7 + 0;,.13. Security cov,~riance from the market modelU,j = f3if3jUj14. Variance of a portfolio (by market and unique risk)IT~ = f3~ u; + u;p15. Market risk ofa portfolio.ve,= L x;f3;; =116. Unique risk of a portfoliosuZ/J = L XTu;,j =117. Capital Market Line[( .11 - Ii)]r" = j + - - - - u p0.1118. Variance of the market portfolio. .a~, = L L X,.,Xj.fU,j, =1 j =119. Security Market Line (covariance and beta versions)r; = j + [( r.1I ~ r/)] Ui.1Ifr .1r; = i + (1"11 - Tj)f3,20. Beta from the CAP:.I13= 17, .11t "U ,121. Two-factor modelri = a, + bil!j + b2F..! + e,22. Variance of a-secu rity (1.>) factor andnon-factor risk. two-factor model)u7 = 1r"ull + til~(T~2 + 2hdbi2 COV(Jj. f;) + lr;;23. Covariance between two securities (two-factor model)U,j = bilbj,u;"1 + hiA2U;"2+ (b" hJ2 + [J2bjl) Coo(fj f ;)24. Arbitrage Pricing Theory (two-factor model)1-;=Ao+A,II" +A21121-i = i+ (,) denotes the expected return over the forthcoming time period onsecurities having the same amount of risk as security j, given the information avail-able at the beginning of the period. Last, Pi. denotes the price ofsecurityj at the be-ginning of the period. Thus, Equation (4.1) states that the expected price for anysecurity at the end of the period (t + 1) is based on the securitys expected normal(or equilibrium) rate of return during that period. In turn, the expected rate of re-turn is determined by (or is conditional on) the information set available at the startof the period, eJ>/. Investors are assumed to make full and accurate use of this avail-able information set. That is, they do not ignore or misinterpret this information insetting their return projections for the coming period.What is contained in the available information set eJ>,? The answer depends onthe particular form of market efficiency being considered. In the case of weak-form94 CHAPTER 4 123. efficiency, the information set includes past price data. In the case of semistrong-form efficiency, the information set includes all publicly available information rele-vant to establishing security values, including the weak-form information set. Thisinformation would include published financial data about companies, governmentdata about the state of the economy, earnings estimates disseminated by companiesand securities analysts, and so on. Finally, in the case of strong-form efficiency, theinformation set includes all relevant valuation data, including information known onlyto corporate insiders, such as imminent corporate takeover plans and extraordinarypositive or negative future earnings announcements.If markets are efficient, then investors cannot earn abnormal profits trading onthe available information set cI>/ other than by chance. Again employing Farnas no-tation, the level of over- or undervaluation ofa security is defined as:Xj. t+l =!JJ,I+I - E(Pj, 1+1 1J (4.2)where Xj, 1+1 indicates the extent to which the actual price for security j at the end ofthe period differs from the price expected by investors based on the informationavailable at the beginning of the period. As a result, in an efficient market it mustbe true that:(4.3)That is, there will be no expected under- or overvaluation of securities based on theavailable information set. That information is always impounded in security prices.4.2.3 Security Price Changes As a Random WalkWhat happens when new information arrives that changes the information set 1such as an announcement that a companys earnings havejust experienced a signif-icant and unexpected decline? In an efficient market, investors will incorporate anynew information immediately and fully in security prices. New information is justthat: new, meaning a surprise. (Anything that is not a surprise is predictable andshould have been anticipated before the fact .) Because happy surprises are about aslikely as unhappy ones, price changes in an efficient market are about as likely to bepositive as negative. A securitys price can be expected to move upward by an amountthat provides a reasonable return on capital (when considered in conjunction withdividend payments), but anything above or below such an amount would, in an ef-ficient market, be unpredictable.In a perfectly efficient market, price changes are random." This does not meanthat prices are irrational. On the contrary, prices are quite rational. Because infor-mation arrives randomly, changes in prices that occur as a consequence of that in-formation will appear to be random, sometimes being positive and sometimes beingnegative. However, these price changes are simply the consequence of investors re-assessing a securityS prospects and adjusting their buying and selling appropriately.Hence the price changes are random but rational.As mentioned earlier, in an efficient market a securitys price will be a good es-timate of its investment value, where investment value is the present value of the se-curitys future prospects as estimated by well-informed and capable analysts. In awell-developed and free market, major disparities between price and investmentvalue will be noted by alert analysts who will seek to take advantage of their discov-eries. Securities priced below investment value (known as underpriced or underval-ued securities) will be purchased, creating pressure for price increases due to theincreased demand to buy. Securities priced above investment value (known as over-priced or overvalued securities) will be sold, creating pressure for price decreases dueto the increased supply to sell. As investors seek to take advantage of opportunitiesEfficient Markets, Investment Value, and Market Price 95 124. created by temporary inefficiencies, they will cause the inefficiencies to be reduced,denying the less alert and the less informed a chance to obtain large abnormal prof-its. As a consequence of the efforts of such highly alert investors, at any time a secu-ritys price can be assumed to equal the securitys investment value, implying thatsecurity mispricing will not exist.4.2.4 Observations about PerfectlyEfficient MarketsSome interesting observations can be made about perfectly efficient markets.1. Investors should expect to make a fair return on their investment but no more.This statement means that looking for mispriced securities with either technical analy-sis (whereby analysts examine the past price behavior of securities) or fundamentalanalysis (whereby analysts examine earnings and dividend forecasts) will not prove tobe fruitful. Investing money on the basis of either type of analysis will not generateabnormal returns (other than [or those few investors who turn out to be lucky) .2. Markets will be efficient only if enough investors believe that they are not ef-ficient. The reason for this seeming paradox is straightforward; it is the actions of in-vestors who carefully analyze securities that make prices reflect investment values.However, if everyone believed that markets are perfectly efficient, then everyonewould realize that nothing is to be gained by searching for undervalued securities,and hence nobody would bother to analyze securities. Consequently, security priceswould not react instantaneously to the release of information but instead would re-spond more slowly. Thus, in a classic "Catch-22" situation, markets would becomeinefficient if investors believed that they are efficient, yet they are efficient becauseinvestors believe them to be inefficient.3. Publicly known investment strategies cannot be expected to generate abnor-mal returns. If somebody has a strategy that generated abnormal returns in the pastand subsequently divulges it to the public (for example, by publishing a book or ar-ticle about it), then the usefulness of the strategy will be destroyed. The strategy,whatever it is based on, must provide some means to identify mispriced securities. Thereason that it will not work after it is made public is that the investors who know thestrategy will try to capitalize on it, and in doing so will force prices to equal invest-ment values the moment the strategy indicates a security is mispriced. The action ofinvestors following the strategy will eliminate its effectiveness at identifying mispricedsecurities.4. Some investors will display impressive performance records. However, theirperformance is merely due to chance (despite their assertions to the contrary). Thinkof a simple model in which half of the time the stock market has an annual returngreater than Treasury bills (an "up" market) and the other half of the time its re-turn is less than T-bills (a "down" market). With many investors attempting to fore-cast whether the stock market will be up or down each year and acting accordingly,in an efficient market about half the investors will be right in any given year and halfwill be wrong. The next year, half of those who were righ t the first year will be rightthe second year too. Thus, 1/4 (= 1/2 X 1/2) of all the investors will have been rightboth years. About half of the surviving investors will be right in the third year, so intotal 1/8 (= 1/2 X 1/2 X 1/2) of all the investors can show that they were right allthree years. Thus, it can be seen that (1/2) T investors will be correct every year overa span of Tyears. Hence if T = 6, then 1/64 of the investors will have been correctall five years, but only because they were lucky, not because they were skillful. Con-sequently, anecdotal evidence about the success of certain investors is misleading. In-deed, any fees paid for advice from such investors would be wasted.96 CHAPTER 4 125. 5. Professional investors should fare no better in picking securities than ordinaryinvestors. Prices always reflect investment value, and hence the search for mispricedsecurities is futile. Consequently, professional investors do not have an edge on or-dinary investors when it comes to identifying mispriced securities and generatingabnormally high returns.6. Past performance is not an indicator of future performance. Investors whohave done well in the past are no more likely to do better in the future than are in-vestors who have done poorly in the past. Those who did well in the past were mere-ly lucky, and those who did poorly merely had a streak of misfortune. Because pastluck and misfortune do not have a tendency to repeat themselves, historical perfor-mance records are useless in predicting future performance records. (Of course, ifthe poor performance was due to incurring high operating expenses, then poor per-formers are likely to remain poor performers.)4.2.5 Observations about Perfectly EfficientMarkets with Transactions CostsThe efficient market model discussed earlier assumed that investors had free accessto all information relevant to setting security prices. In reality, it is expensive to col-lect and process such information. Furthermore, investors cannot adjust their port-folios in response to new information without incurring transaction costs. How doesthe existence of these costs affect the efficient market model? Sanford Grossmanand joseph Stiglitz considered this issue and arrived at two important conclusions.l1. In a world where it costs money to analyze securities, analysts will be able toidentify mispriced securities. However, their gain from doing so will be exactly off-set by the increased costs (perhaps associated with the money needed to procuredata and analytical software) that they incur. Hence, their gross returns will indicatethat they have made abnormal returns, but their net returns will show that they haveearned a fair return and nothing more. Of course, they can earn less than a fair re-turn in such an environment if they fail to properly use the data. For example, byrapidly buying and selling securities, they may generate large transactions costs thatwould more than offset the value of their superior security analysis.2. Investors will do just as well using a passive investment strategy where they sim-ply buy the securities in a particular index and hold onto that investment. Such astrategy will minimize transaction costs and can be expected to do as well as any pro-fessionally managed portfolio that actively seeks out mispriced securities and incurscosts in doing so. Indeed, such a strategy can be expected to outperform any pro-fessionally managed portfolio that incurs unnecessary transaction costs (by, for ex-ample, trading too often). Note that the gross returns of professionally managedportfolios will exceed those of passively managed portfolios having similar invest-ment objectives but that the two kinds of portfolios can be expected to have similarnet returns._ _ TESTING FOR MARKET EFFICIENCYNow that the method for determining prices in a perfectly efficient market has beendescribed, it is useful to take a moment to consider how to conduct tests to determineif markets actually are perfectly efficient, reasonably efficient, or not efficient at all.There are a multitude of methodologies, but three stand out. They involve con-ducting event studies, looking for patterns in security prices, and examining the in-vestment performance of professional money managers.Efficient Markets. Investment Value. and Market Price 97 126. INSTITUTIONAL ISSUESThe Active versus Passive DebateThe issue of market efficiency has clear and impor-tant ramifications for the investment management in-dustry, At stake are billions of dollars in investmentmanagement fees, professional reputations, and,some would argue, even the effective functioning ofour capital markets.Greater market efficiency implies a lower prob-ability that an investor willbe able to consistently iden-tify mispriced securities. Consequently, the moreefficient is a security market, the lower is the expect-ed payoff to investors who buy and sell securities insearch of abnormally high returns. In fact, when re-search and transaction costs are factored into theanalysis, greater market efficiency increases thechances that investors "actively" managing their port-folios will underperform a simple technique of hold-ing a portfolio that resembles the composition of theentire market. Institutional investors have respond-ed to evidence of significant efficiency in the U.S.stock and bond markets by increasingly pursing aninvestment approach referred to as passive manage-ment. Passive management involves a long-term, buy-and-hold approach to investing. The investor selectsan appropriate target and buys a portfolio designedto closely track the performance of that target. Oncethe portfolio is purchased, little additional tradingoccurs, beyond reinvesting income or minor rebal-ancings necessary to accurately track the target. Be-cause the selected target is usually (although notnecessarily) a hroad, diversified market index (for ex-ample, the S&P 500 for domestic common stocks),passive management (see Chapter 23) is commonlyreferred to as "indexation," and the passive portfoliosare called "index funds."Active management, on the other hand, involvesa systematic effort to exceed the performance of a se-lected target. A wide array of active management in-vestment approaches exists-far too many to sum-marize in this space. Nevertheless, all active manage-ment entails the search for mispriced securities or formispriced groups ofsecurities. Accurately identifyingand adroitly purchasing or selling these mispriced se-curities provides the active investor with the poten-tial to outperform the passive investor.Passive management is a relative newcomer tothe investment industry. Before the mid-I 960s, it wasaxiomatic that investors should search for mispricedstocks. Some investment strategies had passive over-tones, such as buying "solid, blue-chip" companies forthe "long term." Nevertheless, even these strategiesimplied an attempt to outperform some nebulouslyspecified market target. The concepts of broad di-versification and passive management were, for prac-tical purposes, nonexistent.Attitudes changed in the 1960s with the popu-larization of Markowitzs portfolio selection concepL~(see Chapter 6), the introduction ofthe efficient mar-ket hypothesis (see this chapter), the emphasis on"the market portfolio" derived from the Capital AssetPricing Model (see Chapter 9), and various academ-ic studies proclaiming the futility of active manage-ment. Many investors, especially large institutionalinvestors, began to question the wisdom of activelymanaging all of their assets. The first domestic com-mon stock index fund was introduced in 1971. Bytheend ofthe decade, roughly $100 million was investedin index funds.Today, hundreds of billions ofdollarsare invested in domestic and international stock andbond index funds. Even individual investors have be-come enamored of index funds. Passively managedportfolios are some of the fastest growing productsoffered by many large mutual fund organizations.Proponents of active management argue thatcapital markets are inefficient enough to justify the4.3.1 Event StudiesEvent studies can be carried out to see just how fast security prices actually react tothe release of information. Do they react rapidly or slowly?Are the returns after theannouncement date abnormally high or low, or are they simply normal? Note thatthe answer to the second question requires a definition of a "normal return" for agiven security. Typically "normal" is defined by the use ofsome equilibrium-based assetpricing model, two of which will be discussed in Chapters 9 and 11. An improperlyspecified asset pricing model can invalidate a test of market efficiency. Thus, eventstudies are really joint tests, as they simultaneously involve tests of the asset pricing98 CHAPTER 4 127. search for mispriced securities. They may disagree onthe degree of the markets inefficiencies. Technicalanalysts (see Chapter 16), for example, tend to viewmarkets as dominated by emotionally driven and pre-dictable investors, thereby creating numerous profitopportunities for the creative and disciplined investor.Conversely, managers who use highly quantitative in-vestment tools often view profit opportunities as small-er and less persistent. Nevertheless, all activemanagers possess a fundamental belief in the exis-tence of consistently exploitable security mispricings.As evidence they frequently poin t to the stellar trackrecords of certain successful managers and variousstudies identifying market inefficiencies (see Appen-dix A, Chapter 16, which discusses empirical regu-larities).Some active management proponents also in-troduce into the active-passive debate what amountsto a moralistic appeal. They contend that investorshave virtually an obligation to seek out mispriced se-curities, because their actions help remove these mis-pricings and thereby lead to a more efficientallocation ofcapital. Moreover, some proponents de-risively contend that passive management implies set-tling for mediocre, or average, performance.Proponents of passive management do not denythat exploitable profit opportunities exist or thatsome managers have established impressive perfor-mance results. Rather, they contend that the capitalmarkets are efficient enough to prevent all but a fewwith inside information from consistently being ableto earn abnormal profits. They claim that examples ofpast successes are more likely the result of luck ratherthan skill. If 1,000 people flip a coin ten times, theodds are that one ofthem will flip all heads. In the in-vestment industry, this person is crowned a brilliantmoney manager.Passive management proponents also argue thatthe expected returns from active management are ac-tually less than those for passive management. Thefees charged by active managers are typically muchhigher than those levied by passive managers. (Thedifference averages anywhere from .30% to 1.00% ofthe managers assets under managernent.) Further,passively managed portfolios usually experience verysmall transaction costs, whereas, depending on theamount oftrading involved, active management trans-action costs can be quite high. Thus it is argued thatpassive managers will outperform active managers be-cause of cost differences. Passive management thusentails settling for superior, as opposed to mediocre,results.The active-passive debate will never be totally re-solved. The random "noise" inherent in investmentperformance tends to drown out any systematic evi-dence of investment management skill on the part ofactive managers. Subjective issues therefore dominatethe discussion, and as a result neither side can con-vince the other of the correctness of its Viewpoint.Despite the rapid growth in passively managedassets, most domestic and international stock andbond portfolios remain actively managed. Many largeinstitutional investors, such as pension funds, havetaken a middle ground on the matter, hiring bothpassive and active managers. In a crude way,this strat-egy may be a reasonable response to the unresolvedactive-passive debate. Assets cannot all be passivelymanaged-who would be left to maintain securityprices at "fair" value levels? On the other hand, it maynot take many skillful active managers to ensure anadequate level of market efficiency. Further, the evi-dence is overwhelming that managers with above-av-erage investment skills are in the minority of thegroup currently offering their services to investors.models validity and tests of market efficiency. A finding that prices react slowly to in-formation might be due to markets being inefficient, or it might be due to the useof an improper asset pricing model, or it might be due to both.As an example of an event study, consider what happens in a perfectly efficientmarket when information is released. When the information arrives in the market-place, prices will react instantaneously and, in doing so, will immediately move totheir new investment values. Figure 4.6(a) shows what happens in such a marketwhen good news arrives; Figure 4.6(b) shows what happens when bad news arrives.Note that in both cases the horizontal axis is a time line and the vertical axis is thesecuritys price, so the figure reflects the securitys price over time. At time 0 (t = 0),Efficient Markets, Investment Value, and Market Price 99 128. pA ~------------------~ P B f--------------------C~1=0(a) Arrival of Good NewsP B f--------------------,Time*~ pA L- __C~1=0(b) Arrival of Bad NewsTime*Figure 4.6The Effect of Information on a Firms Stock Price in a Perfectly Efficient Market.*Information is assumed to arrive at time t = O.information is released about the firm. Observe that, shortly before the news arrived,the securitys price was at a price P ~ With the arrival of the information the price im-mediately moves to its new equilibrium level of Pwhere it stays until some additionalpiece of information arrives. (To be more precise, the vertical axis would representthe securitys abnormal return, which in an efficient market would be zero untilt = 0, when it would briefly be either significantly positive, in the case of good news,100 CHAPTER 4 129. t= 0 1= I Time*(a) Slow Reaction of Stock Price to Informationpll - - - - - - - - - - -~ P B f-------.1:Q.,1=0 1= I Time*(b) Overreaction of Stock Price to InformationFigure 4.7Possible Effectsof Information on a Firms Stock Price in an Inefficient Market. Information is assumed to arrive at time I = O.or significantly negative, in the case of bad news. Afterward, it would return to zero.For simplicity, prices are used instead ofreturns, because over a short time period sur-rounding t = 0 the results are essentially the same.)Figure 4.7 displays two situations that cannot occur with regularity in an efficientmarket when good news arrives. (A similar situation, although not illustrated here,cannot occur when bad news arrives.) In Figure 4.7(a) the security reacts slowly toEfficient Markets, Investment Value, and Market Price 101 130. the information, so it does not reach its equilibrium price pA until t = 1. This situ-ation cannot occur regularly in an efficient market, because analysts will realize aftert = 0 but before t = 1 that the security is not priced fairly and will rush to buy it. Con-sequently, they will force the security to reach its equilibrium price before t = 1. In-deed, they will force the security to reach its equilibrium price moments after theinformation is released at t = O.In Figure 4.7(b) the securitys price overreacts to the information and then slow-ly settles down to its equilibrium value . Again, this situation cannot occur with reg-ularity in an efficient market. Analysts will realize that the security is selling above itsfair value and will proceed to sell it if they own it or to short sell it if they do not ownit. As a consequence of their actions, they will prevent its price from rising above itsequilibrium value, pA, after the information is released.Many event studies have been made about the reaction of security prices, par-ticularly stock prices, to the release of information, such as news (usually referred toas "announcements") pertaining to earnings and dividends, share repurchase pro-grams, stock splits and dividends, stock and bond sales, stock listings, bond ratingchanges, mergers and acquisitions, and divestitures. Indeed, this area of academic re-search has become a virtual growth industry.4.3.2 looking for PatternsA second method to test for market efficiency is to investigate whether there are anypatterns in security price movements attributable to something other than what onewould expect. Securities can be expected to provide a rate of return over a giventime period in accord with an asset pricing model. Such a model asserts that a se-curitys expected rate of return is equal to a riskfree rate of return plus a "risk pre-mium" whose magnitude is based on the riskiness of the security. Hence, if theriskfree rate and risk premium are both unchanging over time, then securities withhigh average returns in the past can be expected to have high returns in the future.However, the risk premium, for example, may be changing over time, making it dif-ficult to see if there are patterns in security prices. Indeed, any tests that are con-ducted to test for the existence of price patterns are once again joint tests of bothmarket efficiency and the asset pricing model that is used to estimate the risk pre-miums on securities.In recent years, a large number of these patterns have been identified and labeledas "empirical regularities" or "market anomalies." For example, one of the mostprominent market anomalies is theJanuary effect. Security returns appear to be ab-normally high in the month ofJanuary. Other months show no similar type returns.This pattern has a long and consistent track record, and no convincing explanationshave been proposed. (TheJanuary effect and other market anomalies are discussedin Chapter 16.)4.3.3 Examining PerformanceThe third approach is to examine the investment record of professional investors. Aremore of them able to earn abnormally high rates of return than one would expectin a perfectly efficient market? Are more of them able to consistently earn abnormallyhigh returns period after period than one would expect in an efficient market? Again,problems are encountered in conducting such tests because they require the deter-mination of abnormal returns, which, in turn, requires the determination ofjustwhat are "normal returns." Because normal returns can be determined only by as-suming that a particular asset pricing model isapplicable, all such tests are joint testsof market efficiency and an asset pricing model.102 CHAPTER 4 131. . . SUMMARY OF MARKET EFFICIENCY TEST RESULTSMany tests have been conducted over the years examining the degree to which se-curity markets are efficient. Rather than laboriously review the results of those testshere, the most prominent studies are discussed in later chapters, where the specificinvestment activities associated with the studies are considered. However, a few com-ments about the general conclusions drawn from those studies are appropriate here.4.4.1 Weak-Form TestsEarly tests of weak-form market efficiency failed to find any evidence that abnormalprofits could be earned trading on information related to past prices. That is, know-ing how security prices had moved in the past could not be translated into accuratepredictions of future security prices. These tests generally concluded that technicalanalysis, which relies on forecasting security prices on the basis of past prices, was in-effective. More recent studies, however, have indicated that investors may overreactto certain types of information, driving security prices temporarily away from theirinvestment values. As a result, it may be possible to earn abnormal profits buying se-curities that have been "oversold" and selling securities whose prices have been bidup excessively. It should be pointed out, however, that these observations are debat-able and have not been universally accepted.4.4.2 Semistrong-Form TestsThe results of tests of semistrong-form market efficiency have been mixed. Mostevent studies have failed to demonstrate sufficiently large inefficiencies to overcometransaction costs. However, various market "anomalies" have been discovered where-by securities with certain characteristics or during certain time periods appear toproduce abnormally high returns. For example, as mentioned, common stocks inJanuary have been shown to offer returns much higher than those earned in theother 11 months of the year.4.4.3 Strong-Form TestsOne would expect that investors with access to private information would have an ad-vantage over investors who trade only on publicly available information. In general,corporate insiders and stock exchange specialists, who have information not readi-ly available to the investing public, have been shown to be able to earn abnormallyhigh profits. Less clear is the ability of security analysts to produce such profits. Attimes, these analysts have direct access to private information, and in a sense they also"manufacture" their own private information through their research efforts. Somestudies have indicated that certain analysts are able to discern mispriced securities,but whether this ability is due to skill or chance is an open issue.4.4.4 Summary of Efficient MarketsTests of market efficiency demonstrate that U.S. security markets are highly efficient,impounding relevant information about investment values into security prices quick-ly and accurately. Investors cannot easily expect to earn abnormal profits trading onpublicly available information, particularly once the costs of research and transactionsare considered. Nevertheless, numerous pockets of unexplained abnormal returnshave been identified, guaranteeing that the debate over the degree of market effi-ciency will continue.Efficient Markets. Investment Value. and Market Price 103 132. . . SUMMARY1. The forces ofsupply and demand interact to determine a securitys market price.2. An investors demand-to-buy schedule indicates the quantity ofa security that theinvestor wishes to purchase at various prices.3. An investors supply-to-sell schedule indicates the quantity of a security that theinvestor wishes to sell at various prices.4. The demand and supply schedules for individual investors can be aggregated tocreate aggregate demand and supply schedules for a security.5. The intersection of the aggregate demand and supply schedules determines themarket clearing price ofa security.At that price, the quantity traded is maximized.6. The market price of a security can be thought of as representing a consensusopinion about the future prospects for the security.7. An allocationally efficient market is one in which firms with the most promisinginvestment opportunities have access to needed funds.8. Markets must be both internally and externally efficient in order to be alloca-tionally efficient. In an externally efficient market, information is quickly andwidely disseminated, thereby allowing each securitys price to adjust rapidly inan unbiased manner to new information so that it reflects investment value. Inan internally efficient market, brokers and dealers compete fairly so that thecost of transacting is low and the speed of transacting is high.9. A securitys investment value is the present value of the securitys future prospects,as estimated by well-informed and capable analysts, and can be thought of asthe securitys fair or intrinsic value.10. In an efficient market a securitys market price will fully reflect all available in-formation relevant to the securitys value at that time.11. The concept of market efficiency can be expressed in three forms: weak, semi-strong, and strong.12. The three forms of market efficiency make different assumptions about the setof information reflected in security prices.13. Tests of market efficiency are reallyjoint tests about whether markets are efficientand whether security prices are set according to a specific asset pricing model.14. Evidence suggests that U.S. financial markets are highly efficient.QUESTIONS AND PROBLEMS 01. What is the difference between call security markets and continuous securi-ty markets?2. Using demand-to-buy or supply-to-sell schedules, explain and illustrate the effectof the following events on the equilibrium price and quantity traded of FairchildCorporations stock.a. Fairchild officials announce that next years earnings are expected to be sig-nificantly higher than analysts had previously forecast.b. A wealthy shareholder initiates a large secondary offering of Fairchild stock.c. Another company, quite similar to Fairchild in all respects except for beingprivately held, decides to offer its outstanding shares for sale to the public.3. Imp Begley is pondering this statement: "The pattern ofsecurity price behaviormight appear the same whether markets were efficient or security prices bore norelationship whatsoever to investment value." Explain to Imp the meaning ofthe statement.104 . (j--IAPTER 4 133. 4. We all know that investors have widely diverse opinions about the future courseof the economy and earnings forecasts for various industries and companies.How then is it possible for all these investors to arrive at an equilibrium price forany particular security?5. Distinguish between the three forms of market efficiency.6. Does the fact that a market exhibits weak-form efficiency necessarily imply thatit is also strong-form efficient? How about the converse statement? Explain.7. Consider the following types of information. If this information is immediatelyand fully reflected in security prices, what form of market efficiency is implied?a. A companys recent quarterly earnings announcementb. Historical bond yieldsc. Deliberations of a companys board of directors concerning a possible merg-er with another companyd. Limit orders in a specialists booke. A brokerage firms published research report on a particular companyf. Movements in the Dow Jones Industrial Average as plotted in The WallStreetJournal8. Would you expect that fundamental security analysis makes security marketsmore efficient? Why?9. Would you expect that NYSEspecialists should be able to earn an abnormal prof-it in a semistrong efficient market? Why?10. Is it true that in a perfectly efficient market no investor would consistently be ableto earn a profit?11. Although security markets may not be perfectly efficient, what is the rationalefor expecting them to be highly efficient?12. When a corporation announces its earnings for a period, the volume of trans-actions in its stock may increase, but frequently that increase is not associated withsignificant moves in the price of its stock. How can this situation be explained?13. What are the implications of the three forms of market efficiency for technicaland fundamental analysis (discussed in Chapter I)?14. In 1986 and 1987 several high-profile insider trading scandals were exposed.a. Is successful insider trading consistent with the three forms of market effi-ciency? Explain.b. Play the role of devils advocate and present a case outlining the benefits tofinancial markets of insider trading.15. The text states that tests for market efficiency involving an asset pricing modelare really joint tests. What is meant by that statement?16. In perfectly efficient markets with transaction costs, why should analysts be ableto find mispriced securities?17. When investment managers present their historical performance records to po-tential and existing customers, market regulators require the managers to qual-ify those records with the comment that "past performance is no guarantee offuture performance." In a perfectly efficient market, why would such an admo-nition be particularly appropriate?CFA EXAM QUESTIONS18. Discuss the role of a portfolio manager in a perfectly efficient market.19. Fairfax asks for information concerning the benefits of active portfolio man-agement. She is particularly interested in the question of whether active man-agers can be expected to consistently exploit inefficiencies in the capital marketsto produce above-average returns without assuming higher risk.The semistrong form of the efficient market hypothesis (EMH) asserts thatall publicly available information is rapidly and correctly reflected in securitiesEfficient Markets, Investment Value, and Market Price lOS 134. E N D N O T E S K E Y TERMSprices. This assertion implies that investors cannot expect to derive above-aver-age profits from purchases made after the information has become public be-cause security prices already reflect the informations full effects.a. (i) Identify and explain two examples of empirical evidence that tend tosupport the EMH implication stated above.(ii) Identify and explain two examples of empirical evidence that tend to re-fute the EMH implication stated above.b. Discuss two reasons why an investor might choose an active manager even ifthe markets were, in fact, semistrong-form efficient.I A burgeoning field of research in finance is known as market microstructure, which involvesthe study of internal market efficiency.2 Actually, not all investors need to meet these three conditions in order for security prices toequal their investment values. Instead, what is needed is for marginal investors to meet theseconditions, since their trades would correct the mispricing that would occur in their absence.~ Eugene F. Fama, "Efficient Capital Markets: A Review ofTheory and Empirical Work,"Jour-nal oj Finance25, no. 5 (May 1970): 383-417.4 Some people assert that daily stock prices follow a random walk, meaning that stock pricechanges (say, from one day to the next) are independently and identically distributed. Thatis, the price change from day t to day t + 1 is not influenced by the price change from day/ - I to day t, and the size of the price change from one day to the next can be viewed asbeing determined by the spin ofa roulette wheel (with the same roulette wheel being usedevery day) . Statistically, in a random walk p, =P-I + e, where e, is a random error termwhose expected outcome is zero but whose actual outcome is like the spin of a roulettewheel. A more reasonable characterization is to describe daily stock prices as following a ran-dom walk with drift, where p, = P-I + d + e,where d is a small positive number, since stockprices have shown a tendency over time to go up. However, since it can be reasonably arguedthat the random error term e,is not independent and identically distributed over time (thatis, evidence suggests that the same roulette wheel is not used every period) , the most accu-ate characterization of stock prices is to say they follow a submnrtingale process. This simplymeans that a stocks expected price in period t + I, given the information that is availableat time t, is greater than the stocks price at time t.:;Sanford]. Grossman andJoseph E. Stiglitz, "On the Impossibility oflnformationally EfficientMarkets," AmericanEconomicReoieto 70, no . 3 Uune 1980).REFERENCESdemand-to-buy schedulesupply-to-sell scheduleallocationally efficient marketexternally efficient marketinternally efficient marketinvestment valueefficient marketweak-form efficientsemistrong-form efficientstrong-form efficientrandom walkrandom walk with drift1. Discussion and examination of the demand curves for stocks are contained in :Andrei Shleifer, "Do Demand Curves Slope Down?" Journal oj Finance 41, no. 3 Uuly1986): 579-590.106 CHAPTER 4 135. Lawrence Harris and Eitan Curel, "Price and Volume Effects Associated with Changes inthe S&P 500: New Evidence for the Existence of Price Pressures," journal of Finance 41,no.4 (September 1986): 815-829.Stephen W. Pruitt and K. C.John Wei, "Institutional Ownership and Changes in the S&P500 ," journal ofFinance 44, no . 2 (june 1989) : 509-513.2. For articles presenting arguments that securities are "overpriced" owing to short sale re-strictions, see:Edward M. Miller, "Risk, Uncertainty, and Divergence of Opinion," journal ofFinance 32,no.4 (September 1977): 1151-1168.Douglas W. Diamond and Robert E. Verrecchia, "Constraints on Short-Selling and AssetPrice Adjustment to Private Information," journal of Financial Economics 18, no. 2 (june1987) : 277-311.3. Allocational, external, and internal market efficiency are discussed in:Richard R. West, "Two Kinds of Market Efficiency," Financial Analysts journal 31, no. 6(November/December 1975): 30-34.4. Many people believe that the following articles are the seminal pieces on efficient markets:Paul Samuelson, "Proof That Properly Anticipated Prices Fluctuate Randomly," Industri-al Managemnu Reuieut6 (1965): 41-49.Han) V. Roberts, "Stock Market Patterns and Financial Analysis: Methodological Sug-gestions,"./ournal of Finance 14, no. 1 (March 1959): 1-10.Eugene F. Fama, "Efficient Capital Markets: A Reviewof Theory and Empirical Work,"jour-nal ofFinance 25, no . 5 (May 1970) : 383-417.Eugene F. Fama, "Efficient Capital Markets: II," journal of Finance 46, no . 5 (December1991): 1575-1617.5. For an extensive discussion of efficient markets and related evidence, see:Ceorge Foster, Financial Statement Analysis (Englewood Cliffs, NJ: Prentice Hall, 1986),Chapters 9 and 11.Stephen F. LeRoy, "Capital Market Efficiency: An Update," Federal Reserve Bank of SanFranciscoEconomic Review no. 2 (Spring 1990): 29-40. A more detailed version of this papercan be found in: Stephen F. LeRoy, "Efficient Capital Markets and Martingales," journalof Economic Literature 27, no. 4 (December 1989): 1583-1621.Peter Fortune, "Stock Market Efficiency: An Autopsy?" New England Economic Review(March/April 1991) : 17-40.Richard A. Brealey and Stewart C. Myers, Principles of Corporate Finance (New York: Mc-Craw-Hill, 1996), Chapter 13.Stephen A. Ross, Randolph W. Westerfield, and jeffrey F.Jaffe, CorporateFinance (Boston:Invin /McCraw Hill, 1996) , Chapter 13.6. Interesting overviews of efficient market concepts are presented in:Robert Ferguson, "An Efficient Stock Market? Ridiculous!" journal ofPortfolio Management9, no . 4 (Summer 1983) : 31-38.Bob L. Boldt and Harold L. Arbit, "Efficient Markets and the Professional Investor," Fi-nancial Analystsjournal 40, no. 4 (july/August 1984): 22-34.Fischer Black, "Noise," journal ofFinance41, no . 3 Guly 1986): 529- 543.Keith C. Brown, W. V. Harlow, and Seha M. Tinic, "How Rational Investors Deal with Un-certainty (Or, Reports of the Death of Efficient Markets Theory Are Greatly Exaggerat-ed)," journal ofApplied CorporateFinance 2, no. 3 (Fall 1989) : 45-58.Ray Ball, "The Theory of Stock Market Efficiency: Accomplishments and Limitations,"journal ofApplied CorporateFinance 8, no. I (Spring 1995) : 4-17.Efficient Markets. Investment Value. and Market Price 107 136. CHAPTER 5THE VALUATIONOF RISKLESS SECURITIESA useful first step in understanding security valuation is to consider riskless secu-rities, which are those fixed-income securities that are certain of making theirpromised payments in full and on time. Subsequent chapters will be devoted to valu-ing risky securities, for which the size and timing of payments to investors are un-certain. Whereas common stocks and corporate bonds are examples of riskysecurities, the obvious candidates for consideration as riskless securities are the se-curities that represent the debt of the federal government. Because the governmentcan print money whenever it chooses, the promised payments on such securities arevirtually certain to be made on schedule. However, there is a degree of uncertaintyas to the purchasing power of the promised payments. Although government bondsmay be riskless in terms of their nominal payments, they may be quite risky in termsof their real (or inflation-adjusted) payments. This chapter begins with a discussionof the relationship between nominal and real interest rates.Despite the concern with inflation risk, it will be assumed that there are fixed-income securities whose nominal and real payments are certain. Specifically, it willbe assumed that the magnitude of inflation can be accurately predicted. Such an as-sumption makes it possible to focus on the impact of timeon bond valuation. The in-fluences of other attributes on bond valuation can be considered afterward.II1II NOMINAL VERSUS REAL INTEREST RATESModern economies gain much of their efficiency through the use of money-a gen-erally agreed-upon medium of exchange. Instead of trading present corn for a futureToyota, as in a barter economy, the citizen of a modern economy can trade his or hercorn for money (that is, "sell it"), trade the money for future money (that is, "investit"), and finally trade the future money for a Toyota (that is, "buy it") . The rate atwhich he or she can trade present money for future money is the nominal (or "mon-etary") interest rate-usually simply called the interest rate.In periods of changing prices, the nominal interest rate may prove a poor guideto the real return obtained by the investor. Although there is no completely satis-factory way to summarize the many price changes that take place in such periods, mostgovernments attempt to do so by measuring the cost of a specified bundle of major108 137. items. The "overall" price level computed for this representative combination ofitems is usually called a cost-of-living index or consumer price index.Whether the index is relevant for a given individual depends to a major extenton the similarity of his or her purchases to the bundle of goods and services used toconstruct the index. Moreover, such indices tend to overstate increases in the cost ofliving for people who do purchase the chosen bundle of items for two reasons. First,improvements in quality are seldom taken adequately into account. Perhaps more im-portant, little or no adjustment is made in the composition of items in the bundleas relative prices change. The rational consumer can reduce the cost of attaining agiven standard ofliving as prices change by substituting relatively less expensive itemsfor those that have become relatively more expensive.Despite these drawbacks, cost-of-living indices provide at least rough estimates ofchanges in prices. And such indices can be used to determine an overall real rate ofinterest. For example, assume that during a year in which the nominal rate of inter-est is 7%, the cost-of-living index increases from 121 to 124. This means that the bun-dle of goods and services that cost $100 in some base year and $121 at the beginningof the year now cost $124 at year-end. The owner of such a bundle could have soldit for $121 at the start of the year, invested the proceeds at 7% to obtain $129.47(= $121 X 1.07) at the end of the year, and then immediately purchased 1.0441(= $129.47/$124) bundles. The real rate of interest was, thus, 4.41% (= 1.0441 - 1).Effectively then, the real interest rate represents the percentage increase in an in-vestors consumption level from one period to another.These calculations can be summarized in the following formula:Go(l + NIR) = 1 + RIRGwhere:Co =:level df the cost~f-livingindex at the beginning of the yearC ~ level of the cOllt~f..Jivingindex at the end of the yearNIR = the nomlnal interest rateRlR = the real-interest rateAlternatively, Equation (5.1) can be written as:1 + NIR = 1 + RIR1 + GGL(5.1 )(5.2)where GCL equals the rate ofchange in the cost of living index, or (G - Go)/ Co. Inthis case, CCL = .02479 = (124 - 121)/121, so prices increased by about 2.5%.For quick calculation, the real rate of interest can be estimated by simply sub-tracting the rate of change in the cost of living index from the nominal interestrate:RIR == NIR - GGL (5.3)where == means "is approximately equal to." In this case the quick calculation re-sults in an estimate of 4.5% (= 7% - 2.5%), which is reasonably close to the truevalue of4.41 %.Equation (5.2) can be applied on either a historical or a forward-looking basis.The economist Irving Fisher argued in 1930 that the nominal interest rate ought tobe related to the expected real interest rate and the expected inflation rate through anequation that became known as the Fisher equation. That is:The Valuation of Riskless Securities 109 138. ~ INSTITUTIONAL ISSUES. .Almost Riskfree SecuritiesR iskfree securities playa central role in modern fi-nancial theory, providing the baseline against whichto evaluate risky investment altern atives. It is perhapssurprising therefore that no risklree financial assethas historically been available to U.S. investors. Re-cent developments in the U.S. Treasury bond market,however, have made important strides toward bring-ing that deficiency to an end.A riskless security provides an investor with aguarante(~d (certain) return over the investors timehorizon. A~ the investor is ultimately interested in thepurchasing power of his or her investments, the risk-less securitys return should be certain, not just on anominal but on a real (or inflation-adjusted) basis.Although U.S. Treasury securities have zero riskof default, even they have not traditionally providedriskless real returns. Their principal and interest pay-ments are not adjusted for infl ation over the securi-ties lives. As a result, unexpected inflation mayproduce real returns quite different from those ex-pected at the time the securities were purchased.However, assume for the moment that U.S. in-flation remains low and fairly predictable so that wecan effectively ignore inflation risk. Will Treasury se-curities provide investors with riskless returns? Theanswer is generally no.An investors time horizon usually will not coin-cide with the life of a particular Treasury security. Ifthe investors time horizon is longer than the securi-tys life, then the investor must purchase another Trea-sury security when the first security matures. If interestrates have changed in the interim, the investor willearn a different return than he or she originally an-ticipated. (This risk is known as reinvestment-rate risk;see Chapter 8.)If the Treasury securitys life exceeds the in-vestors time horizon, then the investor will have tosell the security before it matures. If interest rateschange before the sale, the price of the security willchange, thereby causing the investor to earn a dif-ferent return from the one originally anticipated.(This risk is known as interest-rate or price risk; seeChapter 8.)Even if the Treasury securitys life matches theinvestors time horizon, it generally will not provide ariskless return. With the exception of Treasury billsand STRIPS, all Treasury securities make periodic in-terest payments (see Chapter 13), which the investormust reinvest. If interest rates change over the secu-ritys life, then the investors reinvestment rate willchange, causing the investors return to differ fromthat originally anticipated at the time the security waspurchased.Clearly, what investors need are Treasury securi-ties that make only one payment (which includes prin-cipal and all interest) when those securities mature.Investors could then select a security whose lifematched their investment time horizons. These se-curities would be truly riskless, at least on a nominalbasis.A fixed-income security that makes only one pay-ment at maturity is called a zero-coupon (or pure-dis-count) bond. Until the 1980s, however, zero-couponTreasury bonds did not exist, except for Treasury bills(which have a maximum maturity of one year). How-ever, a coupon-bearing Treasury security can beviewed as a portfolio of zero-coupon bonds, with eachinterest payment, as well as the principal, considereda separate bond. In 1982, several brokerage firmscame to a novel realization: A Treasury securitys pay-ments could be segregated and sold piecemealthrough a process known as coupon stripping.For example, broke rage firm XYZ pu rchases anewly issued 20 -year Treasury bond and depositsthe bond with a custodian bank. Assuming semi-annual interest payments, XYZ creates 41 separatezero-coupon bonds (40 interest payments plus oneprincipal repayment) . Naturally, XYZ can createlarger zero-coupon bonds by buying and deposit-ing more se curities of the same Treasury issue.NIR = [1 + E(RlR)] X [1 + E(CCL)] - 1where E( ) indicates the expectation of the variable.If the real interest rate is fairly stable over time and relatively small in size, thenthe nominal interest rate should increase approximately one-for-one with changes inthe expected inflation rate. Unfortunately, the precise magnitude of inflation is hardto forecast much in advance. Some people believe that with the U.S. Treasury De-partments recent introduction of Treasury Inflation Indexed Securities it will be110 CHAPTER 5 139. These zero-coupon bonds, in turn, are sold to in-vestors (for a fee, of course) . As the Treasury makesits required payments on the bond, the custodianbank remits the payments to the zero-eoupon bond-holders of the appropriate maturity and effective-ly retires that particular bond. The processcontinues until all interest and principal paymentshave been made and all of the zero-coupon bondsassociated with this Treasury bond have beenextinguished.Brokerage firms have issued zero-eoupon bondsbased on Treasury securities under a number ofexoticnames, such as LIONs (Lehman Investment Oppor-tunity Notes), TIGRs (Merrill Lynchs Treasury In-vestment Growth Receipts), and CATs (SalomonBrothers Certificates of Accrual on Treasury Securi-ties) . Not surprisingly, these securities have becomeknown in the trade as "animals."Brokerage firms and investors benefit fromcoupon stripping. The brokerage firms found thatthe sum of the parts was worth more than the whole,as the zero-coupon bonds could be sold to investorsat a higher combined price than could the sourceTreasury security. Investors benefited from the cre-ation of a liquid market in riskless securities.The U.S. Treasury only belatedly recognized thepopularity of stripped Treasury securities. In 1985,the Treasury introduced a program called STRIPS(Separate Trading of Registered Interest and Princi-pal Securities). This program allows purchasers ofcer-tain interest-bearing Treasury securities to keepwhatever cash payments they wan t and to sell the rest.The stripped bonds are "held" in the Federal ReserveSystems computer (called the book entry system) , andpayments on the bonds are made electronically. Anyfinancial institution that is registered on the FederalReserve Systems computer may participate in theSTRIPS program. Brokerage firms soon found that itwas much less expensive to create zero-eoupon bondsthrough STRIPS than to use bank custody accounts.They subsequently switched virtually all of their ac-tivity to that program.The in troduction of stripped Treasuries still didnot solve the inflation risk problem faced by investorsseeking a truly riskfree security. Finally, in 1997 theU.S. Treasury offered the first U.S. securities whoseprincipal and interest payments were indexed to in-flation , officially named Treasury Inflation IndexedSecurities. With the introduction of these securities,the United States joined Australia, Canada, Israel,New Zealand, Sweden, and the United Kingdom inissuing debt that is indexed to the rate of inflation.To date, the Treasury has issued three maturities(5, 10, and 30 years). The principal amounts of these"indexed" bonds are adjusted semiannually on thebasis of the cumulative change in the ConsumerPrice Index (CPI) since issuance of the bonds. Afixed rate of interest, determined at the time of thebonds issuance, is paid on the adjusted principal,thereby producing inflation-adjusted interest pay-ments as well. Coupon stripping is allowed so thatinvestors can purchase zero-coupon inflation-pro-tected U.S. bonds-apparently the real McCoy ofriskfree securities.Only a nitpicker would note several flaws in thisarrangement. The bonds issuer (the U.S. govern-ment) controls the definition and calculation of theCPI. In light of concerns over whether the CPI accu-rately measures inflation, there is some risk that thegovernment might change the index in ways detri-mental to inflation-indexed securityholders (althoughthe Treasury could adjust the bond terms to com-pensate for such an adverse change). Further, for tax-able investors, the adjustments to the bonds principalvalues immediately become taxable income, reduc-ing (perhaps significantly, depending on the investorstax rate) the inflation protection provided by thebonds. Despite these complications, it appears thatthe Treasury has finally introduced a security that cantruly claim the title of (almost) riskfree.much easier to arrive at a consensus forecast of inflation in the United States by sim-ply looking at their market prices. (These securities are discussed further in "Insti-tutional Issues: Almost Riskfree Securities" and in Chapter 12.) Further considerationof real versus nominal interest rates and forecasting the rate of inflation is deferreduntil Chapter 12. Suffice it to say that it may be best to view the expected real inter-est rate as being determined by the interaction of numerous underlying economicforces, with the nominal interest rate approximately equal to the expected real in-terest rate plus the expected rate of change in prices.The Valuation of Riskless Securities 111 140. _ _ YIELD-TO-MATURITYThere are many interest rates, not just one. Furthermore, there are many ways thatinterest rates can be calculated. One such method results in an interest rate that isknown as the yield-to-maturity. Another results in an interest rate known as the spotrate, which will be discussed in the next section.5.2.1 Calculating Yield-to-MaturityIn describing yields-to-maturity and spot rates, three hypothetical Treasury securi-ties that are available to the public for investment will be considered. Treasury se-curities are widely believed to be free from default risk, meaning that investors haveno doubts about being paid fully and on time. Thus the impact of differing degreesof default risk on yields-to-rnaturity and spot rates is removed.The three Treasury securities to be considered will be referred to as bonds A, B,and C. Bonds A and B are called pure-discount bonds because they make no inter-im interest (or "coupon") payments before maturity. Any investor who purchasesthis kind of bond pays a market-determined price and in return receives the princi-pal (or "face") value of the bond at maturity. In this case, bond A matures in a year,at which time the investor will receive $1,000. Similarly, bond B matures in two years,at which time the investor will receive $1,000. Finally, bond C is not a pure-discountbond. Rather, it is a coupon bond that pays the investor $50 one year from now andmatures two years from now, paying the investor $1,050 at that time. The prices atwhich these bonds are currently being sold in the market are:Bond A (the one-year pure-discount bond): $934.58Bond B (the two-year pure-discount bond): $857.34Bond C (the two-year coupon bond): $946.93The yield-to-maturity on any fixed-income security is the single interest rate (withinterest compounded at some specified interval) that, ifpaid by a bank on the amountinvested, would enable the investor to obtain all the payments promised by the se-curity in question. It is simple to determine the yield-to-maturity on a one-year pure-discount security such as bond A. Because an investment of $934.58 will pay $1,000one year later, the yield-to-maturity on this bond is the interest rate Ttl that a bankwould have to pay on a deposit of $934.58 in order for the account to have a bal-ance of$I,OOO after one year. Thus the yield-to-maturity on bond A is the rate Ttl thatsolves the following equation:(1 + Ttl) X $934.58 = $1,000 (5.4)which is 7%.In the case of bond B, assuming annual compounding at a rate Tn, an accountwith $857.34 invested initially (the cost of B) would grow to (1 + Tn) X $857.34in one year. If this total is left intact, the account will grow to (1 + Tn) X[( 1 + Tn) X $857.34] by the end of the second year. The yield-to-maturity is the rateTn that makes this amount equal to $1,000. In other words, the yield-to-maturity onbond B is the rate Tn that solves the following equation:which is 8%.112(1 + Tn) X[(I + Tn) X$857.34] = $1,000 (5.5)CHAPTER 5 141. For bond C, consider investing $946.93 in a bank account. At the end of oneyear, the account would grow in value to (1 + Te) X $946.93. Then the investorwould remove $50, leaving a balance of [( 1 + Te) X $946.93] - $50. At the endof the second year this balance would have grown to an amount equal to(1 + Tc) X{[(I + Tr.) X$946.93] - $50} . The yield-to-maturity on bond Cis therate rc that makes this amount equal to $1,050:(1 + Tc) X{[(I + Te) X$946.93] - $50} = $1,050 (5.6)which is 7.975%.Equivalently, yield-to-maturity is the discount rate that makes the present valueof the promised future cash flows equal in sum to the current market price of thebond.I When viewed in this manner, yield-to-maturity is analogous to internal rate ofreturn, a concept used for making capital budgeting decisions and often describedin introductory finance textbooks. This equivalence can be seen for bond A by di-viding both sides of Equation (5.4) by (1 + Ttl), resulting in:$934.58 = $1,000(1 + Ttl)(5.7)Similarly, for bond B both sides of Equation (5.5) can be divided by (1 + Tfj)2, re-sulting in:$1,000$857.34 = ( )21 + Tnwhereas for bond C both sides of Equation (5.6) can be divided by (1 + Ter:(5.8)$50 $1,050$946.93 - ( )(1 + Ter1 + Teor:$50 + $1,050(5.9)$946 .93 = (1 + TC) (1 + Tc)2Because Equations (5.7), (5.8), and (5.9) are equivalent to Equations (5.4), (5.5),and (5.6), respectively, the solutions must be the same as before, with Til = 7%,TB = 8%, and Te = 7.975%.For coupon-bearing bonds, the procedure for determining yield-to-maturity in-volves trial and error. In the case of bond C,a discount rate of 10% could be tried ini-tially,resulting in a value for the right-hand side of Equation (5.9) of$913.22, a valuethat is too low.This result indicates that the number in the denominator is too high,so a lower discount rate is used next-say 6%. In this case, the value on the right-handside is $981.67, which is too high and indicates that 6% is too low. The solution issomewhere between 6% and 10%, and the search could continue until the answer.7.975%, is found .The Valuation of Riskless Securities 113 142. (5.10)(5.11)Fortunately, computers arc good at trial-and-error calculations. One can entera very complex series of cash flows into a properly programmed computer and in-stantaneously find the yield-to-maturity. In fact, many hand-held calculators andspreadsheets come with built-ill programs to find yield-to-maturity; you simply enterthe number of days to maturity, the annual coupon payments, and the current mar-ket price, then press the key that indicates yield-to-maturity.11I:II SPOTRATESA spot rate is measured at a given point in time as the yield-to-maturity on a pure-discount security and can be thought of as the interest rate associated with a spotcontract. Such a contract, when signed, involves the immediate loaning of moneyfrom one party to another. The loan, along with interest, is to be repaid in its en-tirety at a specific time in the future. The interest rate that is specified in the con-tract is the spot rate.Bonds A and B in the previous example were pure-discount securities, meaningthat an investor who purchased either one would expect to receive only one cashpayment from the issuer. Accordingly, in this example the one-year spot rate is 7%and the two-year spot rate is 8%. In general, the t-year spot rate s, is the solution tothe following equation:MP = I, (1 + s,)where P, is the current market price ofa pure-discount bond that matures in tyearsand has a maturity value of MI For example, the values of P, and M, for bond B wouldbe $857.34 and $1,000, respectively, with t = 2.Spot rates can also be determined in another manner if only coupon-bearingTreasury bonds are available for longer maturities by using a procedure known asbootstrapping. The one-year spot rate (SI) generally is known, as there typically is aone-year pure-discount Treasury security available for making this calculation. How-ever, it may be that no two-year pure-discount Treasury security exists. Instead, onlya two-year coupon-bearing bond may be available for investment, having a currentmarket price of P2 , a maturity value of M2 , and a coupon payment one year fromnow equal to C1 In this situation, the two-year spot rate (S2) is the solution to the fol-lowing equation:Po CI + M22=(I+sd (l+s2)2Once the two-year spot rate has been calculated, then it and the one-year spot ratecan be used in a similar manner to determine the three-year spot rate by examininga three-year coupon-bearing bond. By applying this procedure iteratively in a simi-lar fashion over and over again, one can calculate an entire set of spot rates from aset of coupon-bearing bonds.For example, assume that only bonds A and Cexist. In this situation it is knownthat the one-year spot rate, SI is 7%. Now Equation (5.I I) can be used to determinethe two-year spot rate, S2 where P2 = $946.93, CI = $50, and M2 = $1,050:$946.93 = $50 + $1,050(I + .07) (1 + S2?The solution to this equation is S2 = .08 = 8%. Thus, the two-year spot rate is de-termined to be the same in this example, regardless of whether it is calculated di-rectly by analyzing pure-discount bond B or indirectly by analyzing coupon-bearing114 CHAPTER 5 143. (5.12)bond G in conjunction with bond A. Although the rate will not always be the samein both calculations when actual bond prices are examined, typically the differencesare insignificant._ _ DISCOUNT FACTORSOnce a set of spot rates has been determined, it is a straightforward matter to de-termine the corresponding set of discount factors. A discount factor d, is equivalentto the present value of $1 to be received t years in the future from a Treasury secu-rity and is equal to:1d=---, (1 + s,)The set of these factors is sometimes referred to as the market discountfunction, and it changes from day to day as spot rates change. In the example,d, = 1/(1 + .07)1 = .9346; and d2 = 1/(1 + .08)2 = .8573.Once the market discount function has been determined, it is fairly straightfor-ward to find the present value of any Treasury security (or, for that matter, any de-fault-free security). Let G/ denote the cash payment to be made to the investor atyear t by the security being evaluated. The multiplication of G/ by d, is termed dis-counting: converting the given future value into an equivalent present value. Thelatter is equivalent in the sense that P present dollars can be converted into G, dol-lars in year tvia available investment instruments, given the currently prevailing spotrates. An investment paying G, dollars in year twith certainty should sell for P = d.C,dollars today. If it sells for more, it is overpriced; if it sells for less. it is underpriced.These statements rest solely on comparisons with equivalent opportunities in themarketplace. Valuation of default-free investments thus requires no assessment ofindividual preferences; instead, it requires only careful analysis of the available op-portunities in the marketplace.The simplest and, in a sense, most fundamental characterization of the marketstructure for default-free bonds is given by the current set of discount factors, re-ferred to earlier as the market discount function. With this set of factors, it is a sim-ple matter to evaluate a default-free bond that provides more than one payment, forit is, in effect, a package of bonds, each of which provides a single payment. Eachamount is simply multiplied by the appropriate discount factor, and the resultantpresent values are summed.For example, assume that the Treasury is preparing to offer for sale a two-yearcoupon-bearing security that will pay $70 in one year and $1,070 in two years. Whatis a fair price for such a security? It is the present value of$70 and $1,070. How canthis value be determined? By multiplying the $70 and $1,070 by the one-year andtwo-year discount factors, respectively. Doing so results in ($70 X .9346) +($1,070 X .8573), which equals $982.73.No matter how complex the pattern of payments, this procedure can be used todetermine the value of any default-free bond of this type. The general formula for abonds present value (PV) is:(5.13)where the bond has promised cash payments G, for each year from year 1 throughyear n.At this point, it has been shown how spot rates and, in turn. discount factors canbe calculated. However, no link between different spot rates (or different discountThe Valuation of Riskless Securities 115 144. (5.14)factors) has been established. For example, it has yet to be shown how the one-yearspot rate of 7% is related to the two-year spot rate of8%. The concept of forward ratesmakes that link.11III FORWARD RATESIn the example, the one-year spot rate was determined to be 7%. This means that themarket has determined that the present value of$1 to be paid by the United StatesTreasury in one year is $1/1.07, 01 $.9346. That is, the relevant discount rate for con-verting a cash flow one year from now to its present value is 7%. Because, as was alsonoted, the two-year spot rate was 8%, the present value of $1 to he paid by the Trea-sury in two years is $1/1.082, or $.8573.An alternative view of $1 to be paid in two years is that it can be discounted intwo steps. The first step determines its equivalent one-year value. That is, $1 to be re-ceived in two years is equivalent to $1/( 1 + ii. 2) to be received in one year. The sec-ond step determines the present value of this equivalent one-year amount bydiscounting it at the one-year spot rate of 7%. Thus its current value is:$1/( 1 + ii.2)(1 +.07)However, this value must be equal to $.8573, as it was mentioned earlier that, ac-cording to the two-year spot rate, $.8573 is the present value of $1 to be paid in twoyears. That is,$1/( 1 + ii.2) =$.8573(1 +.07)which has a solution for ii.2 of 9.0 1%.Th e discount rate ii.2 is known as the forward rate from year one to year two.That is, it is the discount rate for determining the equivalent value one year from nowof a dollar that is to be received two years from now. In the example, $1 to be re-ceived two years from now is equivalent in value to $1/1.0901 = $.9174 to be re-ceived one year from now (in turn, note that the present value of $.9174 is$.9174/1.07 = $.8573).Symbolically, the link between the one-year spot rate, two-year spot rate, andone-year forward rate is:(5.15)which can be rewritten as:(5.16)or:(5.17)Figure 5.1 illustrates this relation by referring to the example and then generaliz-ing from it.More generally, for year t - 1 and year t spot rates, the link to the forward ratebetween years t - 1 and tis:116(5.18)CHAPTER 5 145. Now 1 Year 2 Years1----1----1An Example:.-Generalization:Figure 5.1Spot and Forward Ratesor:$2(1.07) (1 +11,2) = (1.08)211,2= [(1.08)2/0.07)] -1 =9.01%0+ sJ) (1 +11,2) = 0 + S2)2Iis = [(1 + S2)2/0 + sl)] - 1(5.19)There is another interpretation that can be given to forward rates. Consider acontract made now wherein money will be loaned a year from now and paid back twoyears from now. Such a contract is known as aforward contract. The interest rate spec-ified on the one-year loan (note that the interest will be paid when the loan maturesin two years) is known as the forward rate.It is important to distinguish this rate from the rate for one-year loans that wiIIprevail for deals made a year from now (the spot rate at that time) . A forward rateapplies to contracts made now but relating to a period "forward" in time. By the na-ture of the contract the terms are certain now, even though the actual transaction willoccur later. If instead one were to wait until next year and sign a contract to borrowmoney in the spot market at that time, the terms might turn out to be better or worsethan todays forward rate, because the future spot rate is not perfectly predictable.In the example, the marketplace has priced Treasury securities such that a rep-resentative investor making a two-year loan to the government would demand an in-terest rate equal to the two-year spot rate, 8%. Equivalently, the investor would bewilling to simultaneously (1) make a one-year loan to the government at an interestrate equal to the one-year spot rate, 7%, and (2) sign a forward contract with thegovernment to loan the government money one year from now, being repaid twoyears from now with the interest rate to be paid being the forward rate, 9.01 %.When viewed in this manner, forward contracts are implicit. However, forwardcontracts are sometimes made explicitly. For example, a contractor might obtain acommitment from a bank for a one-year construction loan a year hence at a fixed rateof interest. Financial futures markets (discussed in Chapter 20) provide standard-ized forward contracts of this type. For example, in September an investor couldcontract to pay approximately $970 in December to purchase a 90-day Treasury billthat would pay $1,000 in the following March. .The Valuation of Riskless Securities , 17 146. 11IIII FORWARD RATES AND DISCOUNT FACTORSIn Equation (5.12) it was shown that a discount factor for t years could be calculat-ed by adding one to the t-year spot rate, raising this sum to the power t, and then tak-ing the reciprocal of the result. For example, it was shown that the twa-year discountfactor associated with the two-year spot rate of8% was equal to 1/ (I + .08)2 = .8573.Equation (5.17) suggesL~ an equivalent method for calculating discount factors.In the case of the two-year factor, this method involves multiplying one plus the one-year spot rate by one plus the forward rate and taking the reciprocal of the result:(5.20)which in the example is:d2= (1 + .07) X (1 + .0901)= .8573.More generally, the discount factor for year t that is shown in Equation (5.12) canbe restated as follows:dl = ( )1-1 ( )1 + 5/-1 X 1 +it-I,t(5.21 )Thus, given a set of spot rates, one can determine the market discount functionin either of two ways, both ofwhich will provide the same figures. First, the spot ratescan be used in Equation (5.12) to arrive at a set of discount factors. Alternatively,the spot rates can be used to determine a set of forward rates, and then the spot ratesand forward rates can be used in Equation (5.21) to arrive at a set of discount factors._ COMPOUNDINGThus far, the discussion has concentrated on annual interest rates by assuming thatcash flows are compounded (or discounted) annually. This assumption is often ap-propriate, but for more precise calculations a shorter period may be more desirable.Moreover, some lenders explicitly compound funds more often than once each year.Compounding is the payment of "interest on interest." At the end of each com-pounding interval, interest is computed and added to principal. This sum becomesthe principal on which interest is computed at the end of the next interval. Theprocess continues until the end of the final compounding interval is reached.No problem is involved in adapting the previously stated formulas to com-pounding intervals other than a year. The simplest procedure is to count in units ofthe chosen interval. For example, yield-to-maturity can be calculated using any cho-sen compounding interval. If payment of P dollars now will result in the receipt ofF dollars 10 years from now, the yield-to-maturity can be calculated using annualcompounding by finding a value Tn that satisfies the equation:CHAPTER 5 147. P(1 + T,yn = F (5.22)since Fwill be received ten annual periods from now. The result, Ta , will be expressedas an annual rate with annual compounding.Alternatively, yield-to-maturity can be calculated using semiannual compound-ing by finding a value T, that satisfies the equation:p(1 + T,rn= F (5.23)since F will be received 20 semiannual periods from now. The result, Tn will be ex-pressed as a semiannual rate with semiannual compounding. It can be doubled togive an annual rate with semiannual compounding. Alternatively, the annual ratewith annual compounding can be computed for a given value of TJ by using the fol-lowing equation:(21 + Ttl = 1 + T,) (5.24)For example, consider an investment costing $2,315.97 that will pay $5,000 tenyears later. Applying Equations (5.22) and (5.23) to this security results in, respectively:( )10$2,315.97 1 + T" = $5,000and:( )20$2,315.97 1 + T, = $5,000where the solutions are T" = 8% and T, = 3.923%. Thus this security can be describedas having an annual rate with annual compounding of 8%, a semiannual rate withsemiannual compounding of 3.923%, and an annual rate with semiannual com-pounding of7.846% (= 2 X 3.923%).2Semiannual compounding is commonly used to determine the yield-to-maturi-ty for bonds, because coupon payments are usually made twice each year. Most pre-programmed calculators and spreadsheets use this approach.P_ THE BANK DISCOUNT METHODOne time-honored method to summarize interest rates is a procedure called thebank discount method. If someone "borrows" $100 from a bank, to be repaid a yearlater, the bank may subtract the interest payment of, for instance, $8, and give the bor-rower $92. According to the bank discount method, this is an interest rate of 8%. How-ever, this method understates the effective annual interest rate that the borrowerpays. That is, the borrower receives only $92, for which he or she must pay $8 in in-terest after one year. The true interest rate must be based on the money the bor-rower actually gets to use, which in this case is $92, making the true interest rate8.70% (= $8/$92).It is simple to convert an interest rate quoted on the bank discount method toa true interest rate. (In this situation, the true interest rate is often called the bondequivalent yield.) If the bank discount rate is denoted BDR, the true rate is simplyBDR/(l - BDR). Because BDR > 0, the bank discount rate understates the truecost of borrowing [that is, BDR < BDR/(l - BDR)] . The previous example pro-vides an illustration: 8.70 % = .08/0 - .08). Note how the bank discount rate of8% understates the true cost of borrowing by .70% in this example._ YIELD CURVESAt any point in time, Treasury securities will be priced approximately in accord withthe existing set ofspot rates and the associated discount factors. Although there haveThe Valuation of Riskless Securities 119 148. been times when all the spot rates are roughly equal in size, generally they have dif-ferent values. Often the one-year spot rate is less than the twa-year spot rate, whichin turn is less than the three-year spot rate, and so on (that is, s, increases as t in-creases). At other times, the one-year spot rate is greater than the twa-year spot rate,which in turn is greater than the three-year spot rate, and so on (that is, s/ decreas-es as t increases). It is wise for the security analyst to know which case currently pre-vails, as this is a useful starting point in valuing fixed-income securities.Unfortunately, specifying the current situation is easier said than done. Only thebonds of the U.S. government are clearly free from default risk. However, governmentbonds differ in tax treatment, as well as in callability, liquidity, and other features. De-spite these problems, a summary of the approximate relationshi p between yields-to-maturity on Treasury securities ofvarious terms-to-maturity is presented in each issueof the Treasury Bulletin. This summary is given in the form of a graph illustrating thecurrent yield curve. Figure 5.2 provides an example.A yield curve is a graph that shows the yields-to-maturity (on the vertical axis) forTreasury securities ofvarious maturities (on the horizontal axis) as of a particular date.This graph provides an estimate of the current term structure of interest rates andwill change daily as yields-to-maturity change. Figure 5.3 illustrates some of the com-monly observed shapes that the yield curve has taken in the past."Although not shown in Figures 5.2 and 5.3, this relationship between yields andmaturities is less than perfect. That is, not all Treasury securities lie exactly on the yieldcurve. Part of this variation is because of the previously mentioned differences intax treatment, callability, liquidity, and the like. Part is because of the fact that theyield-to-maturity on a coupon-bearing security is not clearly linked to the set ofspotrates currently in existence. Because the set of spot rates is a fundamental determi-nant of the price of any Treasury security, there is no reason to expect yields to lieexactly on the curve. Indeed, a more meaningful graph would be one on which spotrates instead of yields-to-maturity are measured on the vertical axis. With this caveatin mind, ponder these two interesting questions: Why are the spot rates of differentmagnitudes? And why do the differences in these rates change over time, in thatlong-term spot rates usually are greater than short-term spot rates but sometimes theopposite occurs? Attempts to answer such questions can be found in various termstructure theories.IIIIZ!I TERM STRUCTURE THEORIESFour primary theories are used to explain the term structure of interest rates. In dis-cussing them, the focus will be on the term structure of spot rates, because theserates (not yields-to-maturity) are critically important in determining the price of anyTreasury security.5.10.1 The Unbiased Expectations TheoryThe unbiased expectations theory (or pure expectations theory, as it is sometimescalled) holds that the forward rate represents the average opinion of what the ex-pected future spot rate for the period in question will be. Thus, a set ofspot rates thatis rising can be explained by arguing that the marketplace (that is, the general opin-ion of investors) believes that spot rates will be rising in the future. Conversely, a setof decreasing spot rates is explained by arguing that the marketplace expects spotrates to fall in the future."120 CHAPTER 5 149. 7-.----------------------------------,5--1-,---,----,----r-----r--,---,---..,---r-----r---.-----,----,--...---,---I12/31 /97 99 01 03 05 07 09 II 13 15 17 19 21 23 25 12/31 /27YearFigure 5.2Yield Curve of Treasury Securities. September 30, 1997 (based on closing bid quotations inpercentages)Source: Tre"""ry Bulletin, December 1997, p. 51Note: The curve is based only on the most actively traded issues. Market yields on coupon issues due in less than 3months are excluded.(a) Upward SlopingMaturityFigure 5.3Typical Yield CurveShapesUpward-Sloping Yield Curves(b) Flat"0li:>:1----------Maturity(c) Downward SlopingMaturityIn order to understand this theory more fully, consider the earlier example in whichthe one-year spot rate was 7% and the two-year spot rate was 8%. The basic questionis this: Why are these two spot rates different? Equivalently, why is the yield curveupward-sloping?The Valuation of Riskless Securities 121 150. Consider an investor with $1 to invest for two years (for ease of exposition, it willbe assumed that any amount of money can be invested at the prevailing spot rates).This investor could follow a "maturity strategy," investing the money now for the fulltwo years at the two-year spot rate of 8%. With this strategy, at the end of two yearsthe dollar will have grown in value to $1.1664 (= $1 X 1.08 X 1.08) . Alternatively,the investor could invest the dollar now for one year at the one-year spot rate of 7%,so that the investor knows that one year from now he or she will have $1.07(= $1 X 1.07) to reinvest for one more year. Although the investor does not knowwhat the one-year spot rate will be one year from now, the investor has an expectationabout what it will be (this expected future spot rate will hereafter be denoted eSt. 2)If the investor thinks that it willbe 10%, then his or her $1 investment has an expectedvalue two years from now of$1.177 (= $1 X 1.07 X 1.10). In this case , the investorwould choose a "rollover strategy," meaning that the investor would choose to investin a one-year security at 7% rather than in the two-year security, because he or shewould expect to have more money at the end of two years by doing so($1.177 > $1.1664).However, an expected future spot rate of 10% cannot represent the general viewin the marketplace. If it did, people would not be willing to invest money at the two-year spot rate, as a higher return would be expected from investing money at theone-year rate and using the rollover strategy. Thus the two-year spot rate would quick-ly rise as the supply of funds for two-year loans at 8% would be less than the demand.Conversely, the supply of funds for one year at 7% would be more than the demand,causing the one-year rate to fall quickly. Thus, a one-year spot rate of7%, a two-yearspot rate of8%, and an expected future spot rate of 10% cannot represent an equi-librium situation.What if the expected future spot rate is 6% instead of 10%? In this case, accord-ing to the rollover strategy the investor would expect $1 to be worth $1.1342(= $1 X 1.07 X 1.06) at the end of two years. This is less than the value the dollarwill have if the maturity strategy is followed ($1.1342 < $1.664), so the investorwould choose the maturity strategy. Again, however, an expected future spot rate of6% cannot represent the general view in the marketplace because if it did, peoplewould not be willing to invest money at the one-year spot rate.Earlier, it was shown that the forward rate in this example was 9.01%. What if theexpected future spot rate was of this magnitude? At the end of two years the value of$1 with the rollover strategy would be $1.1664 (= $1 X 1.07 X 1.0901), the same asthe value of$l with the maturity strategy. In this case, equilibrium would exist in themarketplace because the general view would be that the two strategies have the sameexpected return. Accordingly, investors with a two-year holding period would nothave an incentive to choose one strategy over the other.Note that an investor with a one-year holding period could follow a maturitystrategy by investing $1 in the one-year security and receiving $1.07 after one year.Alternatively, a "naive strategy" could be followed, where a two-year security could bepurchased now and sold after one year. If that were done, the expected selling pricewould be $1.07 (= $1.1664/1.0901), for a return of7% (the security would have a ma-turity value of $1.1664 = $1 X 1.08 X 1.08, but since the spot rate is expected tobe 9.01% in a year. its expected selling price isjust the discounted value of its matu-rity value). Because the maturity and naive strategies have the same expected return,investors with a one-year holding period would not have an incentive to choose onestrategy over the other.Thus the unbiased expectations theory asserts that the expected future spot rateis equal in magnitude to the forward rate. In the example, the current one-year spotrate is 7%, and, according to this theory, the general opinion is that it will rise to arate of 9.01 % in one year. This expected rise in the one-year spot rate is the reason122 CHAPTER 5 151. behind the upward-sloping term structure where the two-year spot rate (8%) is greaterthan the one-year spot rate (7%).EquilibriumIn equation form, the unbiased expectations theory states that in equilibrium theexpected future spot rate is equal to the forward rate:(5.25)Thus Equation (5. I7) can be restated with esJ, 2 substituted for [i, 2 as follows:(5.26)which can be conveniently interpreted to mean that the expected return from a ma-turity strategy must equal the expected return on a rollover strategy, "The previous example dealt with an upward-sloping term structure; the longerthe term, the higher the spot rate. It is a straightforward matter to deal with a down-ward-sloping term structure, where the longer the term, the lower the spot rate.Whereas the explanation for an upward-sloping term structure was that investors ex-pect spot rates to rise in the future, the reason for t