Info, Bubbles, and the Efficient Markets Hypothesis (Econ 308, Tesfatsion) (2024)

Information, Bubbles, and the Efficient Markets Hypothesis

Last Updated: 4 March 2024

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Econ 308 Web Site:

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The notes, below, provide basic background discussion on information,bubbles, and the efficient markets hypothesis. For a more extensive set ofnotes relating to these and other financial economics topics prepared for anundergraduate course, visit the home page for Econ 353 (Money, Banking, and Financial Institutions)

1. Overview
2. Analytical Representation for the Efficient Markets Hypothesis
3. Bubbles and Fundamental Values in Asset Pricing: A Simple Illustration
4. Empirical Testing of the Efficient Markets Hypothesis
5. References
6. Appendix: Technical Remarks

A. Overview:

The concept of "efficiency" has a variety of related but distinctmeanings in economics. An economy can be efficient in the sense ofproductive efficiency, meaning non-wastage of physical resources. Aneconomy can also be efficient in the sense of Pareto-efficiency,meaning non-wastage of utility. Finally, an economy can be efficient in thesense of informational efficiency, meaning non-wastage of information.

In finance, the term "efficiency" has come to be used specifically inthe last sense. Roughly speaking, a market for financial assets is said toexhibit (informational) efficiency if all available information isoptimally used in the determination of asset prices at each point in time.This has been formalized as the so-called "efficient markets hypothesis."

The Efficient Markets Hypothesis (EMH): The contention that, at eachtime t, the current prices of financial assets reflect all availableinformation relevant for judging the future returns of those assets.

The intuitive idea is that rational individual traders process theinformation that is available to them and take optimal positions in assets onthe basis of this information. The market price for an asset then aggregatesthis diverse trader information, and in this sense "reflects" the availableinformation.

What is the relationship among these concepts of productive, informational,and Pareto efficiency? Roughly speaking, productive and informationalefficiency can be thought of as requirements for an economy to be Paretoefficient. If either physical resources or information are being "wasted,"then (essentially by definition of "wasted") there must exist a way toimprove the use of resources and/or information to make at least one personbetter off without hurting anyone else.

B. Analytical Representation for the Efficient Markets Hypothesis

An asset is said to be risky relative to a particular time periodif its return rate over this time period is not completely known or assured;otherwise it is said to be risk free. Intuitively, if a market forrisky financial assets over period [t,t+1) is to be "efficient," then it mustbe the case that traders in this market exploit all available profitopportunities, leaving no room for further profits to be made in net terms.

Assuming traders are risk neutral (i.e., they only care aboutexpected return rates, not the volatility of return rates), and that at leastone risk-free asset is held, this implies the following arbitragecondition for each risky asset A in this market: At the margin, thereturn rate that a trader expects to obtain over the period [t,t+1) byinvesting one last dollar in asset A at time t should just equal the returnrate that the trader could obtain by instead investing this dollar in a best(most profitable) available risk-free asset (e.g., U.S. Treasury bills).

REMARK: As noted above, this use of the risk-free return rateas the "opportunity cost" used by traders to judge the desirability ofinvesting in risky financial assets assumes that traders are risk neutral, sothat only the expected return rate is of importance to them in comparingthe purchase of alternative financial assets. Moreover, which particularasset is to be designated as "the" risk-free asset determining theopportunity cost of traders has also been controversial. Following Fama(1970), the empirical finance literature has come to recognize that EMH testsactually represent joint tests of the EMH, the degree of risk aversionexhibited by traders, and the precise specification of the opportunity cost.

Let the return rate on the best available risk-free asset over period[t,t+1) be denoted by r_F(t,t+1), and let r_A(t,t+1) denote the return rateon some risky asset A over period [t,t+1). Also, let I(t) denote theinformation available to investors at time t, where I(t) includes the valueof r_F(t,t+1). Then the EMH relative to the sequence of information sets{I(t)} states that the expected value of r_A(t,t+1), conditional on I(t),must equal the risk-free return rate r_F(t,t+1). Formally:

(B.1) E[r_A(t,t+1)| I(t) ] = r_F(t,t+1) , t = 0, 1, ... .

Let P_A(t) and P_A(t+1) denote the market prices of asset A at time t andt+1, respectively. Also, let D(t,t+1) denote the amount of paymentsgenerated by asset A over the period [t,t+1), assumed to be paid out at timet+1. Then the return rate r_A(t,t+1) for asset A over [t,t+1) is given by

(B.2) r_A(t,t+1) = [P_A(t+1) - P_A(t) + D(t,t+1)]/P_A(t) .

If the value of P_A(t) is included in the information set I(t), conditions(B.1) and (B.2) together imply that

(B.3) E[P_A(t+1)+D(t,t+1)|I(t)] = [1+r_F(t,t+1)]P_A(t) ,

or

(B.4) E[P_A(t+1)+D(t,t+1)|I(t)] --------------------------- = P_A(t) . 1 + r_F(t,t+1)

Roughly speaking, the EMH implies that the market for information aboutasset returns is competitive in the following sense. Investors cannot expectto earn persistent monopolistic returns -- i.e., returns in excess of thosenecessary to induce the investor to hold an asset -- by collectinginformation about past asset prices and trading volumes and basing theircurrent trades on this information. The reason for this is that the very actof trading in the asset tends to reveal the investor's information. Thus,a dramatic implication of the EMH is that it denies thepossibility of successful technical trading schemes. Given the current priceof an asset, no clever use of information concerning the past priceand trade volume history of an asset can improve one's chances of predictingits future prices.

The fundamental value of an asset is defined to be the present valueof its (possibly infinite) payment stream, discounted by the risk-free rater_F. If the price of an asset deviates from its fundamental value, the assetis said to exhibit a (price) bubble.

Some economists (e.g., Arthur et al. (1997)) analytically represent theEMH in stronger terms than (B.4). They equate the EMH with the absence ofbubbles in the pricing of financial assets. That is, they argue that, in afinancial market consisting of rational fully-informed traders, the price ofeach financial asset will equal the asset's fundamental value. In the nextsection it is shown that the arbitrage condition (B.4) is not sufficient, perse, to rule out the existence of price bubbles.

C. Bubbles and Fundamental Values in Asset Pricing: A Simple Illustration

Consider an economy for which government issued money ("dollars") andgovernment issued debt instruments are the only available financial assets.Suppose the latter take two forms: (1) Risk-free Treasury bills paying aconstant risk-free return rate r_F in each period t = [t,t+1); and (2) riskyconsols, where each consol promises the holder a coupon payment of onedollar at the end of each period in perpetuity (i.e., with no maturity date).

The first question to be addressed concerns the relation of the price ofa consol to its return rate. Let P_C(t) denote the nominal price of aconsol at the beginning of any period t, i.e., the price measured in dollars.The nominal return from purchasing a consol at time t at price P_C(t),holding the consol for one period, and selling the consol at time t+1 at aprice P_C(t+1), is then given by

(C.1) P_C(t+1) - P_C(t) + $1 per consol Return from Price Appreciation Coupon Payment (Measured in dollars per consol)

Consequently, dropping the "per consol" designations for ease of notation,the nominal return rate attained by holding a consol from time t totime t+1 is given by

 [P_C(t+1) - P_C(t) + $1](C.2) r_C(t,t+1) = -------------------------- P_C(t)

Now suppose that the behavior of investors in each period t can becaptured by the behavior of a single risk-neutral infinitely-lived"representative" investor whose portfolio contains both Treasury billsearning the constant risk-free return rate r_F and and consols earning thereturn rate (C.2). For each time t, letting I(t) denote the time-tinformation set of this investor, suppose the EMH (B.4) holds for consols,implying that

(C.3) E[r_C(t,t+1)|I(t)] = r_F .

Finally, suppose for each time t that I(t) includes the value of P_C(t), thevalue of r_F, and the value of the $1 payments promised by each consol, andthat successive information sets I(t) and I(t+1) are "nested" in the sensethat I(t) is included in I(t+1).

Using (C.2) to substitute out for r_C(t,t+1) in (C.3), and multiplyingthrough the resulting expression by P_C(t) and solving for P_C(t), one obtainsthe following reformulation of condition (C.3):

 $1 E[P_C(t+1)|I(t)](C.4) P_C(t) = ------- + ---------------- (1+r_F) (1+r_F)

By assumption, relation (C.4) holds for every time t. Consequently, updating(C.4) by one time period, one gets an expression for P_C(t+1) that can beused to substitute out for the far-right appearance of P_C(t+1) in theoriginal expression (C.4). Using the assumption that I(t) includes thevalues of the $1 coupon payments and the risk-free return rate r_F, and thatsuccessive information sets are nested, this substitution can be shown togive

 $1 $1 E[P_C(t+2)|I(t)](C.5) P_C(t) = ------- + --------- + ---------------- (1+r_F) (1+r_F)^2 (1+r_F)^2

Continuing to expand the right side of (C.5) by successive substitution outfor the price term, one obtains a representation for P_C(t) in terms of adiscounted sum of $1 coupon payments plus a remainder term. For example,given any period L greater than t, recursing forward the far-right price termin (C.4) for L periods into the future yields the following representationfor the nominal price of a consol at the beginning of period t:

 L-t $1 E[P_C(L)|I(t)](C.6) P_C(t) = SUM --------- + -------------- . j=1 (1+r_F)^j (1+r_F)^(L-t)

Suppose the following transversality condition holds: thefar-right price term in (C.6) approaches 0 as L approaches infinity. A basicmathematical lemma states that, for any constant b with absolute value lessthan 1, the infinite sum of b raised to successively higher powers (startingwith b^0 = 1) is 1/[1-b]. Applying this lemma to the sum in (C.6) as Lapproaches infinity, with b equal to 1/(1+r_F), and imposing thetransversality condition, it follows that

 +oo $1 $1(C.7) P_C(t) = SUM --------- = ----- . j=1 (1+r_F)^j r_F

It is important to keep in mind that (C.7) was derived under two specialassumptions: (i) EMH: For each t, the expected value of the period-treturn rate (C.2) for the consol, conditional on the information set I(t), isequal to the (constant) risk-free return rate r_F; and (ii) TransversalityCondition: For each t, the far-right price term in (C.6) approaches zeroas L approaches infinity.

By definition, the yield to maturity for a financial asset in someperiod t is the particular fixed one-period interest rate R which, if used todiscount the financial asset's future payment stream to the holder, wouldyield a present value for this payment stream that is just equal to thefinancial asset's period-t market value. For a consol in period t, thepayment stream ($1,$1,...) simply consists of periodic $1 payments inperpetuity. Thus, the yield to maturity R for a consol bond in period t isthe particular fixed one-period interest rate R that satisfies

 +oo $1 $1(C.8) P_C(t) = SUM ------- = --- . j=1 (1+R)^j R

As previously noted, the return rate r_C(t,t+1) defined for the consol by(C.2) -- which only accounts for the $1 payment and the capital gain or lossover the holding period [t,t+1) -- can in general differ from the yield tomaturity R defined by relation (C.8), which takes into account the entire(infinite) payment stream for the consol starting from time t.

The fundamental value F(t) of the consol in period t is definedto be the present value of the payment stream ($1,$1,...), where thediscounting of the payment stream to present value is carried out using therisk-free return rate r_F. Thus,

 +oo $1 $1 .(C.9) F(t) = SUM --------- = ---- j=1 (1+r_F)^j r_F

Recall from Section B that the consol is said to exhibit a pricebubble at time t if its period-t price P_C(t) deviates from its period-tfundamental value F(t). Comparing (C.6) through (C.9), it is seen that theabsence of price bubbles is guaranteed for consols if two conditions hold:(i) the EMH in form (C.3); and (ii) the transversality condition (convergenceto zero of the far-right price term in (C.6)).

Intuitively, it might seem that price bubbles must be absent on consolsin order for the market for consols to be in equilibrium in each period t.If P_C(t) is greater than F(t) (hence R is less than r_F), why wouldn'tsavers switch towards the risk-free asset and away from consols in order tocapture the higher yield r_F, a move that would tend to lower P_C(t) in thedirection of F(t). Conversely, if P_C(t) is less than F(T) (hence R isgreater than r_F), why wouldn't savers switch away from the risk-free assetand towards consols to capture the higher yield R, a move that would tend toraise P_C(t) in the direction of F(t)? Thus, F(t) appears to provide aperfectly rational way to determine the equilibrium market price of a consolin period t.

However, this intuitive reasoning implicitly assumes the existence of aninfinitely-lived risk neutral "representative investor." Equilibriumrequires the absence of perceived exploitable profit opportunities. Thus,given such a risk-neutral investor, equilibrium requires that the EMH in form(C.3) must hold. Moreover, the transversality condition must also hold,since otherwise the investor could take advantage of the difference betweenP_C(t) and F(t) by implementing a "buy and hold forever" strategy in thedirection of the asset with the greater yield (either R on consols or r_F onTreasury bills).

Absent the existence of a single risk-neutral infinitely-lived investor,however, the existence of a positive price bubble on the consol is notnecessarily irrational. Suppose, for example, that the market containssuccessive finite-lived traders with successive finite holding periods forbonds. These traders could be expecting the price P_C of the consol toaccelerate over time at a rate at least equal to (1+r_F), implying that thefar-right price term in (C.6) fails to converge to zero in the forwardrecursion. Note that these expectations of an appreciating price for consolswould tend to be self-fulfilling, at least for a while, since the resultingincreased demand for the consols would tend to raise the price of the consolsin actuality. No single trader could take advantage of the resulting pricebubble through a "buy and hold forever" strategy. Thus, the tradingactivities of such traders would not necessarily drive the consol priceP_C(t) to its fundamental value F(t). In short, with successive finite-livedtraders, the transversality condition could fail to hold even though the EMHin form (C.3) is satisfied in every period.

Recall the "dot.com bubble" in the U.S. during the late 1990s, in whichmany stocks for web-based companies paying zero dividends (and havingnegative net earnings) were nevertheless able to command a substantiallypositive market price on the anticipation of future price increases. For awhile, until the bubble burst, lots of investors made lots of money. Werethey irrational? The difficulty comes in trying to form a "rational"prediction regarding when (and if) a price bubble will burst. The issue ofbubbles is currently a hot topic in economics. See, for example, thesymposium on bubbles in the Journal of Economic Perspectives (Volume4, Spring 1990).

D. Empirical Testing of the Efficient Markets Hypothesis

The EMH is given empirical content by specifying the information setsI(t) used by traders in a financial market M at each time t to determine theperiod-t prices of risky financial assets in accordance with relation (B.4).Fama (1970) describes a hierarchy of nested categories of information sets.As one moves down the hierarchy from the largest to the smallest set,efficiency is required with respect to ever decreasing amounts ofinformation, and hence in an ever weaker sense. Since the categories ofinformation sets are nested, rejection of any one type implies the rejectionof all stronger forms.

• Strong-Form Efficiency for market M asserts that the traders' informationsets I(t) contain all available information which could possibly be relevantfor the pricing of assets in market M. Not only is all publicly availableinformation embodied in these prices, but all privately held information aswell.
• Semistrong-Form Efficiency for market M asserts that the traders'information sets I(t) contains all publicly available information. Incontrast to strong-form efficiency, the information sets I(t) are not assumedto contain privately held information (i.e., information that has not beenmade public). [NOTE: It is not always easy to draw the line betweenprivately and publicly held information.]
• Weak-Form Efficiency for market M asserts that the traders'information sets I(t) at any current time t only contain the current and pastprice histories of the assets in market M, as well as the current risk-freereturn rate r_F(t,t+1).

An implication of weak-form efficiency (hence of any of the stronger forms ofefficiency), is that asset prices from periods prior to period t will not beof any help in predicting asset prices for periods t+1 and beyond, since thispast price information is already fully reflected in period-t asset prices.Literally, condition (B.4) states that the only part of the information setI(t) that is relevant for forming return rate predictions for asset A inperiod t is the current period-t price of asset A together with the risk-freereturn rate.

Given certain additional assumptions, another testable implication ofthe weak-form EMH (and hence all stronger forms of the EMH) is that thereturn rates of an asset A are not serially correlated over time. Roughly,this means that the successive return rates for asset A in successive timeperiods do not exhibit any systematic relationship, either positive (atendency to move together) or negative (a tendency to move in oppositedirections). For example, if the return rate for asset A happens to behigher than its average level in the current period, one cannot infer that itwill be higher or lower than this average level in the following period. (Seethe Appendix to these notes for a more detailed discussion of thisimplication of weak-form EMH.)

The Empirical Bottom Line [See Sheffrin (1991, 133-137), Fama (1991)]

Throughout much of the 1970's, many economists seemed to accept thepresumption that financial markets are efficient, i.e., that the EMH wassatisfied, at least in its weaker forms. Nevertheless, careful reviews ofthe efficient market literature have always pointed out that tests of the EMHare #joint# tests of a particular asset pricing model together with theassumption that market participants make efficient use of information indetermining their expectations for asset prices. Such was the faith in theEMH through the 1970's, however, that test failures generally led researchersto modify their asset pricing models rather than reject the EMH per se.

At least some researchers (e.g., Ross 1989) have concluded from theempirical literature that markets appear to be consistent with efficiency, atleast in the weak-form sense. However, a growing number of researchers nowtake a more cautious stance. Fama (1991), for example, concludes that --with regard to firm-specific events -- the adjustment of stock prices to newinformation appears to be efficient. As for other tests of efficiency, however,he reports mixed results and does not draw a conclusion. What happened?

The growing pessimism about the EMH seems to have begun in the early1980's with the work of Shiller (1981,1989) on the "excess volatility" offinancial markets. Shiller formulated statistical tests of the EMH based onthe volatility of stock market prices relative to dividend volatility whichhe claimed were more powerful than traditional regression-based tests of theEMH. He found that stock prices diverged systematically from fundamentalvalues, and he interpreted this finding as a violation of the EMH. This wasthe start of the "bubble literature" -- see the symposium on bubbles in the1990 Journal of Economic Perspectives.

In a well-known study, Fama and French (1988) concluded that stockreturns exhibit high negative correlation, implying there is a slowmean-reverting movement in stock returns. As just discussed, this violatesthe weak-form EMH implication that asset return rates should be seriallyuncorrelated.

One interpretation of mean-reversion offered by Fama and French is thatexpectations are still rational, but that stock returns are subject to randomshocks. These shocks do not affect future dividends, hence stock returnseventually revert to their "normal" long-run value. Another interpretation,however, is that investors become inexplicably bullish or bearish. If thesefads persist for a while before eventually dying out, then stock marketprices will exhibit "bubbles" (divergence from fundamental value) in theshort run and mean reversion in the long run; see Summers (1986).

The notion that asset prices can diverge from their fundamental values isalso an implication of the "noise trader" theory developed by Black (1986).A noise trader is a market participant with incorrect information whoimplements trades on the basis of this information under the false beliefthat the information is correct. Black argued that the presence of noisetraders was necessary to explain the large volume of trading activity thatoccurs in financial markets. Without noise traders, there would be virtuallyno trade in individual shares. Rational investors trading with each otherwould realize that any other trader willing to pay a higher price for theasset must have superior information about the asset's return. But without alarge volume of trade in individual shares, how does the market determine theappropriate market prices for assets?

The presence of noise traders provides rational traders with an incentiveto gather information. Sophisticated traders can bring correct information toa market and exploit the profit opportunities created by the presence of noisetraders. Sophisticated traders tend to move asset prices toward fundamentalvalues. However, the continual presence of noise traders (particularly, thecontinual entrance of new noise traders) makes it difficult for other tradersto discern who is a noise trader and who is acting on correct information. Thiscan give a sophisticated trader a chance to make profits from his superiorinformation for an extended period of time.

Do sophisticated traders drive noise traders from a market? Notnecessarily. For example, Shleifer and Summers (1990) argue that -- if noisetraders (unwittingly) take larger risky positions on average, because oferroneous beliefs, some noise traders can still profit in the market despitetheir erroneous beliefs. The essential idea is that some noise traders holdinglarge risky positions with high expected returns are lucky and manage to earna high enough return rate on their wealth to enable them to remain in themarket. In addition, the presence of noise traders actually benefitssophisticated traders who take direct advantage of them in trades, as well asother agents (brokers, dealers, etc.) who feed off noise traders indirectlyby providing financial services to them.

Along similar lines, Fama (1991, p. 1607) reports that some researchershave concluded that insiders can and do profit from trading on the basis oftheir inside knowledge, but outsiders cannot profit from public informationabout this insider trading. Thus, the market may be characterized by "noisyrational expectations" (Grossman and Stiglitz, 1980) in the sense that thereis private information which is not fully reflected in asset prices, but allinvestors nevertheless exhibit rational behavior.

All of this work on noise traders requires asymmetricinformation, i.e., different market participants engaging in trades witheach other on the basis of different information sets.

E. References

Journal of Economic Perspectives (Spring 1990), Symposium on Bubbles.

George A. Akerlof, "The Market for Lemons," Quarterly Journal ofEconomics 84 (August 1970), 488-500.

W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and PaulTaylor, "Asset Pricing Under Endogenous Expectations in an Artificial StockMarket," pp. 15-44 in W. B. Arthur, S. Durlauf, and D. Lane (eds.), TheEconomy as an Evolving Complex System, II, Volume XXVII, Addison-Wesley,1997.

Fischer Black, "Noise," Journal of Finance 41 (1986), 529-543.

Eugene Fama, "Efficient Capital Markets: A Review of Theory and EmpiricalWork," Journal of Finance 25 (1970), 383-423.

Eugene Fama, "Efficient Capital Markets: II,"Journal of Finance 46(December 1991), 1575-1617.

Eugene Fama and K. French, "Permanent and Temporary Components of StockPrices," Journal of Political Economy 96 (1988), 246-273.

Mark Gertler, "Financial Structure and Aggregate Economic Activity: AnOverview," Journal of Money, Credit, and Banking 20 (1988), 559-588.

Sanford Grossman and Joseph Stiglitz, "On the Impossibility of InformationallyEfficient Markets," American Economic Review 70 (1980), 393-408.

Stephen LeRoy, "Efficient Capital Markets and Martingales," Journal ofEconomic Literature 27 (December 1989), 1583-1621.

Steven Ross, "Finance," pp. 326-329 in the New Palgrave Dictionary ofEconomics, J. Eatwell, M. Milgate, and P. Newman, eds., W. W. Norton andCo., New York, 1989, in four volumes.

Steven Sheffrin, The Making of Economic Policy, Blackwell, 1991.

Robert Shiller, "Do Stock Prices Move Too Much to be Justified by SubsequentChanges in Dividends?" American Economic Review 71 (June 1981),421-436.

Robert Shiller, Market Volatility, MIT Press, Cambridge, 1989.

Andrei Shleifer and Lawrence Summers, "The Noise Trader Approach toFinance," pp. 19-33, in Symposium on Bubbles, Journal ofEconomic Perspectives 4 (Spring 1990).

Lawrence Summers, "Does the stock Market Rationally Reflect FundamentalValues?" Journal of Finance 41 (1986), 591-601.

Hal Varian, Microeconomic Analysis, W. W. Norton and Co., ThirdEdition, 1992.

F. Appendix: Technical Remarks

As noted in Section D, above, given certain additional assumptions,another testable implication of the weak-form EMH (and hence all strongerforms of the EMH) is that the return rates of an asset A should not exhibitserial correlation.

This claim will now be shown for an asset A that distributes a dividendpayment D(t,t+1) at the end of each period t = [t,t+1). The demonstrationwill use a famous "iterated expectation lemma" in the following special form.Let X and Y be two real-valued random variables X and Y with joint probabilitydensity function p(x,y) and with marginal probability density functions p(x)and p(y). Letting INT denote integration and INT INT denote doubleintegration, suppose H(x,y) is any function such that the expectations

 EH(X,Y) = INT INT H(x,y)p(x,y)dxdy ; E[H(X,Y)|Y=y] = INT H(x,y)p(x|y)dx ,

exist and are finite for each y. Then, using p(x,y) = p(x|y)p(y) for each xand y,

(1) EH(X,Y) = INT INT H(x,y)p(x,y)dxdy = INT [ INT H(x,y)p(x|y)dx ]p(y)dy = INT E[H(X,Y)|Y=y]p(y)dy = E[ E(H(X,Y)|Y] ] .

Now consider the covariance between the rates of return on asset A over any two consecutive time periods [t,t+1) and [t+1,t+2). Using straightforward calculations, as well as the iterated expectations lemma 1)with X = r_A(t+1,t+2) and Y = r_A(t,t+1), one obtains

(2) Cov( r_A(t+1,t+2), r_A(t,t+1) ) = E[(r_A(t+1,t+2)-E[r_A(t+1,t+2)])(r_A(t,t+1)-E[r_A(t,t+1)])] = E[(r_A(t+1,t+2))(r_A(t,t+1)-E[r_A(t,t+1)])] = E[E[(r_A(t+1,t+2))(r_A(t,t+1)-E[r_A(t,t+1)])|r_A(t,t+1)]] = E[E[(r_A(t+1,t+2))|r_A(t,t+1)](r_A(t,t+1)-E[r_A(t,t+1)])].

Suppose for each t that the risk-free interest rate r_F(t,t+1) is independentof the rate of return r_A(k,k+1) on asset A for each k less than t. Underweak-form efficiency, the time t+1 information set I(t+1) contains allcurrent and past prices of asset A. Suppose I(t+1) also contains thedividend payment D(t,t+1). Then I(t+1) contains the return rate for asset Aduring period t: namely,

 r_A(t,t+1) = [p_A(t+1) + D(t,t+1) - p_A(t)]/p_A(t) .

Making use of the iterated expectation lemma (1), as well as the EMHimplication (B.1), it follows that

(3) E[r_A(t+1,t+2)|r_A(t,t+1)] = E[E[r_A(t+1,t+2)|I(t+1)]|r_A(t,t+1)] = E[r_F(t+1,t+2)|r_A(t,t+1)] = E[r_F(t+1,t+2)].

Consequently, combining (2) and (3),

(4) Cov( r_A(t+1,t+2), r_A(t,t+1) ) = E[E[(r_A(t+1,t+2))|r_A(t,t+1)](r_A(t,t+1)-E[r_A(t,t+1)])] = E[E[r_F(t+1,t+2)](r_A(t,t+1)-E[r_A(t,t+1)])] = E[r_F(t+1,t+2)]E[r_A(t,t+1)-E[r_A(t,t+1)] = 0 .

In other words, the return rates for asset A are serially uncorrelated.

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