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