SØK460 Finance Theory, Diderik Lund, 7 February 2001
Derivation of zero-beta CAPM: Efficient portfolios
• Assumptions as CAPM, except Rf does not exist.
• Argument which leads to Capital Market Line is invalid.
• (No straight line through Rf , tilted up as far as possible.)
• Everyone chooses on upper half of hyperbola.
• But different preferences imply different choices along curve.
• Thus two differences from CAPM:
— Not same portfolio of risky assets for everyone.
— Not same MRS between σ and µ for everyone.
• Market portfolio is convex combination of agents’ portfolios.
• (Convex combination is linear combination with weights ∈ [0, 1].)
• Corollary 5: That convex combination also on upper half.
• Market portfolio is efficient.
• Importance of this: One efficient portfolio is observable.
• (Under the assumptions of the model.)
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Derivation of zero-beta CAPM: CAPM equation
• Derivation of CAPM equation follows previous argument.
• (See lecture of 30 January.)
• Argument works with any efficient portfolio, not only M.
• Combine one efficient pf., e, and one other asset (or pf.), j.
• Weight a in j, 1 − a in e. Let a vary.
• Combinations form “little” hyperbola in (σ, µ) plane.
• Little hyperbola goes through (σe, µe) and (σj , µj ).
• At (σe , µe ), little and big hyperbola have same slope.
• At (σe , µe ), little hyperbola has the slope
¯
µj − µe
dµ ¯¯¯
=
.
¯
dσ ¯a=0 (σje − σe2)/σe
• Corollary 3A: Slope of big hyperbola at (σe , µe ) is
µe − µz(e)
,
σe
where z(e) means frontier portfolio uncorrelated with e.
• Equality of these:
µe − µz(e)
µj − µe
σje
=
=
µ
+(µ
)
.
⇐⇒
µ
−µ
j
e
z(e)
z(e)
(σje − σe2)/σe
σe
σe2
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Zero-beta CAPM: Conclusion, application?
• Rewritten in more explicit notation:
E(R̃j ) = E(R̃z(e)) +
cov(R̃j , R̃e )
[E(R̃e ) − E(R̃z(e))].
var(R̃e)
• Specifically: Replace e with M, market pf.
• This gives us “zero-beta CAPM,” in Roll’s notation
rj = rz + βj (rm − rz ).
• Asset z has βz = 0, thus z is short for zero-beta.
• Only difference from CAPM equation: rz replaces rF .
• If rF exists, it is uncorrelated with everything.
• Thus the standard CAPM is special case of zero-beta CAPM.
• How to apply the zero-beta CAPM? More complicated.
• E(R̃z(M) ) is not immediately observable.
• May be estimated from time-series data if stable over time.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Estimation, the usefulness of zero covariance
In Copeland and Weston type notation:
E(R̃j ) = E(R̃z(e)) + βj [E(R̃e ) − E(R̃z(e))],
or, with E(ε̃) = 0,
R̃j = R̃z(e)(1 − βj ) + βj R̃e + ε̃.
• For estimation: Natural to think of this as linear regression.
• Is it possible to estimate βj by OLS on the equation?
• In general, OLS with two non-constant regressors gives something else.
• Gujarati, eq. 7.4.8: The estimated coefficient for Re is
cov(R̃j , R̃e) var(R̃z(e) ) − cov(R̃j , R̃z(e)) cov(R̃z(e), R̃e )
,
βˆj =
2
var(R̃z(e)) var(R̃e) − [cov(R̃z(e), R̃e)]
or rather, the sample moments corresponding to this.
• Now we use the fact that cov(R̃z(e), R̃e) = 0, and observe that
(asymptotically, as the sample moments approach the theoretical moments) we get
cov(R̃j , R̃e) var(R̃z(e)) cov(R̃j , R̃e )
=
βˆj =
var(R̃z(e)) var(R̃e )
var(R̃e)
• Conclusion: This is an estimate of the model’s βj .
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Roll’s main text: Testing the CAPM?
• Test must be based on time series data.
• (Only way to observe E(R̃j ), variances, covariances)
• (Relies on assumption of stability over time.)
• Data define sample means, sample variances, s. covariances.
• These sample moments define an ex post portfolio frontier.
• Results about ex ante frontier also hold for ex post frontier.
• If m is ex post efficient, then
rj = rz + βj (rm − rz )
holds for ex post data. But if not, equation will not hold.
• Trying to test whether equation holds in data?
• Amounts to testing whether m is ex post efficient.
• If not: Many researchers would reject model.
• But the valid conclusion is only: Bad data for m.
• m portfolio we observed, was not ex post efficient.
• If enough data: Ex post distribution → ex ante distn.
• Traditional rejection results from m portfolio ex ante inefficient.
• Or traditional rejection may result from too few data.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
What can be tested?
• Question whether CAPM can be tested at all.
• Roll mentions four hypotheses:
(H1) Investors consider as optimal those portfolios which are
mean-variance efficient.
(H2) The market portfolio is ex ante efficient.
Also: If rF exists:
(H3) Investors can borrow and lend at a risk free rate, rF .
(H4) rz(m) = rF .
• If everyone has same beliefs (about probability distributions),
then (H2) follows from (H1). Identical beliefs also necessary for
(H4).
• (H3) and (H4) are only relevant if rF exists.
• Roll: Difficult to test (H1) and (H3) directly.
• Thus we are left with two testable hypotheses: (H2) and (H4).
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Roll’s critical comments
• Roll considers some tests from early 70’s.
• In particular Fama and MacBeth (1973).
• They presented three separate hypotheses:
(C1) E(R̃j ) depends linearly on βj .
(C2) Other risk measures than βj do not influence E(R̃j ).
(C3) Risk aversion implies E(R̃m) > E(R̃z(m)).
• Roll points out: If m efficient, then (C1) and (C2) will hold.
• Also: (C3) will hold independently of risk aversion.
• Thus left with one hypothesis: Market portfolio is efficient.
• Points out misleading statement in Fama and MacBeth.
• “To test (C1)—(C3), need identify efficient m portfolio.”
• But if identify efficient m pf., no need to test (C1)—(C3).
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Problems with “approximately correct” tests
• What if data for approximate m portfolio available?
• Assume that pf. has high correlation with actual m.
• Roll: But may then reject even if model is true.
• Example: May reject (H4) even when true.
• Happens if m is on strongly concave part of hyperbola.
• Consider small deviation in approximate m.
• Still high correlation with actual m.
• But tangent’s intersection with vertical axis strongly affected.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Market efficiency
• Normally “efficiency” refers to allocational efficiency.
• (Cannot produce more of one good without reducing other.)
• (Nor improve someone’s utility without reducing someone else’s.)
• But here, another interpretation.
• How do markets reflect new information?
• In particular, financial markets (bonds, shares, options, . . . ).
• What kinds of information are reflected in prices?
• (Perhaps also:) How quickly is information reflected?
Standard definition has three forms of efficiency:
1. Weak form: No investor can earn excess returns by developing
trading rules based on historical prices (or rates of return) on
the same asset.
2. Semi-strong form: No investor can earn excess returns from
trading rules based on any publicly available information.
3. Strong form: No investor can earn excess returns using any
information, publicly available or not.
Consequence of market efficiency: Changes in prices reflect new
information, i.e., the arrival of new information to the market, and
nothing else.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Market efficiency: Information sets
• Definition based on three information sets:
1. Asset’s own history of prices (and returns).
2. All publicly available information.
3. All existing information.
• The first is subset of the second, which is subset of third.
Example:
• Firm makes new invention, cheaper production technique.
• First some engineers get the news. Info restricted to these.
• They realize that invention increases earnings potential.
• May be easy to estimate value increase if cost saving risk free.
• If engineers do not trade in shares, share price unaffected.
• Info exists, but has no effect. Not strong efficiency.
• If engineers start buying shares: Share price increases.
• Strong efficiency conflicts with laws against insider trading?
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Semi-strong efficiency
• If not: Some trading rule will give “excess returns.”
• Using publicly available info only.
• Perhaps info on many shares, macroeconomic variables, . . .
• (No conflict with laws against insider trading.)
• “Excess returns” requires comparison with something.
• To disprove semi-strong eff.: Come up with trading rule.
• But if you know such a rule: Would perhaps prefer to use it?
• May exist lots of evidence which is never published.
Weak efficiency
• To disprove: Trading rule based on one time series.
• Known as technical analysis. See newspapers.
• No known evidence that this works.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Testing for “excess returns”
• Empirical tests for market efficiency: Excess returns?
• Market efficiency rules out systematic excess returns.
• “Excess” means “above normal,” “extraordinary.”
• How to test for this empirically?
• Could look for obviously very large returns.
— E.g., rates of return of 50 per cent per year.
— Look if these appear systematically in some situations.
— If so, doubt market efficiency.
• More commonly, compare to model predicted rates of return.
— Typically CAPM or some extension of it.
— Then: Joint hypothesis: Model and market efficiency.
— If rejected: Either model or market efficiency or both.
— Not desirable. Would prefer not to rely on specific model.
• Third alternative: Look for other phenomena.
— Phenomena which occur in price time series, but which
cannot be explained (rationally) by information arriving in
market.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Information paradox
If markets are efficient, there is no reason for anyone to use
resources to gather information about firms’ earnings potentials
(or “fundamental values”), since these will be reflected in market prices already. But then, no one will gather information,
and it is hard to believe that markets are efficient.
• Non-efficiency also paradox: Then it pays to gather info.
• If it pays, many will do it.
• But then one moves towards efficiency.
How to describe an equilibrium situation?
• Assume: Costly to gather information.
• Cost equal to expected gain from gathering it.
• Model by Cornell and Roll (1981), see C & W, sect. 10.E.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Model of costly information gathering (“analysis”)
Model assumptions:
• Each trade is between two distinct people.
• Buyer may or may not have analyzed information.
• Seller may or may not have analyzed information.
• For simplicity: Either “analyze” or not.
• (Impossible to “analyze a little.”)
• Gain from having more info than opposite party.
• Express gains and costs in terms of rates of return.
• Cost of trade is c1 if no analysis is done.
• Cost of trade is c2 if analysis is done.
• Gross rate of return r if opposite party equally informed.
• Gross rate of return dr for a better informed party.
• Gross rate of return r/d for less informed party.
Table entries show payoffs for player A for each of the four possible
pairs of combinations of strategies for players A and B:
B Analyze Not analyze
A
Analyze
Not analyze
r − c2
r/d − c1
dr − c2
r − c1
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Solution to Cornell and Roll model
• At the outset, everyone is equal in model.
• Thus: Likely that everyone chooses same strategy.
• Would bring us back to the paradox.
• Alternative solution concept: Mixed strategy equilibrium.
• In equilibrium each trader indifferent between strategies.
• Let p be probability of meeting an analyst.
• Let 1 − p be probability of meeting a non-analyst.
• Indifference in terms of expected returns if
E(payoff to analysis) = E(payoff to non-analysis)
r
⇐⇒ p(r − c2) + (1 − p)(dr − c2) = p( − c1) + (1 − p)(r − c1)
d
r(1 − d) + c2 − c1
.
⇐⇒ p =
2r − rd − r/d
• Can show that when d > 1, r > 0 then p ∈ (0, 1) requires
r(d − 1 > c2 − c1) and r(1 − 1/d) < c2 − c1.
• Equilibrium if
— People meet at random.
— A fraction p chooses analysis, 1 − p not.
• One interpretation: Opportunities to collect info occur randomly, at different times to different people.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Anomalies
• Anomalies: Irrational patterns in (share) prices.
• Cannot be explained under rationality and market efficiency.
• Example: Prices fall around weekends.
• Actually observed in some markets in some periods.
• When recognized by traders: Possible to earn excess returns.
• Trading rule: Buy Friday, sell Tuesday.
• Obstacle to this trading rule: Transaction costs.
• Typically brokers demand fees both for buying and selling.
• If transaction costs large, they can explain many anomalies.
• Or at least, anomalies then compatible with efficiency.
• Transaction costs must be small if anomalies used as proof.
• “Excess returns” in definition should exceed transaction costs.
Large sale of shares
• If someone sells many shares, often interpreted as bad news.
• Implies permanent downward shift in share price.
• But in some markets in some periods: Temporary dip as well.
• Buying during temporary dip should give excess returns.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Scholes’s study of large sales of shares
• Used data for 345 large sales in the U.S. in 1960’s.
• Excluded emissions, i.e., sales by corporation of own shares.
• Emissions may signal new project, high returns.
• Compared returns with those predicted by simple CAPM version:
Rjt = at + bj Rmt + εjt.
• Estimated on time series data. Abnormal return is
ejt = Rjt − (âj + b̂j Rmt).
• Conclusion: Yes, abnormally low return around large sale.
• But the drop in share price is permanent.
• Does not contradict market efficiency.
• Then Scholes divided observations according to sellers.
• Five different groups of sellers: Investment companies/mutual
funds, corporation’s own employees, banks/insurance companies, individuals, estates and trusts.
• Of these, the first two are supposed to have more information.
• Large sales by these two groups had stronger signal effects.
• Effect not immediate, since seller not known immediately.
• Conclude: Support market efficiency.
• Later studies: Temporary effect, disappears in five minutes.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
(Ir)rational interpretation of accounting info
• If corporation reduces future earnings, share price should fall.
• But some corporations seem interested in “appearance.”
• “Appearance” is, e.g., accounting, book values.
• Accounting information sometimes have no economic content.
• Example: Choice of inventory accounting method. Inflation.
• Inventory with many identical items bought at different times.
• When taking one item out, what was it bought for?
• Either FIFO (first in, first out). Bought at low price.
• Or LIFO (last in, first out). Bought at higher price.
• LIFO gives lower book profits, lower taxes.
• Real economic effect of LIFO: Higher after-tax profits.
• When accounts made public, choice FIFO/LIFO is revealed.
• Change from LIFO to FIFO should imply lower firm value.
• Could be explained by importance of appearance.
• Studies show that FIFO chosen when profits very low.
• Important not to show negative book profits.
• Also: Negative profits imply no tax that year.
• If no tax, nothing to lose from increasing book profits.
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SØK460 Finance Theory, Diderik Lund, 7 February 2001
Stock splits
• Another event used to test market efficiency.
• Stock splits: Corporations’ decisions to split own shares.
• Simply give out, e.g., 2 or 3 new shares for one old.
• Convenient: Minimum number of shares to trade is common.
• Apart from improved divisibility, no real effect.
• Psychological: Perhaps signal of maintaining high value?
• Fama et al.: Stock splits preceded by high returns.
• Nothing abnormal after the (announcement of the) splits.
• But indeed higher dividend/price ratios after splits.
• 71 per cent got higher D/P , 29 per cent got lower D/P .
• Splitting the sample according to higher or lower D/P .
• Those with higher D/P continued high returns after split.
• Those with lower D/P got low returns after split.
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