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A reappraisal of fundamental analysis
Doug McLeod
Version: 12th February 2016
[email protected]. For helpful comments and assistance I thank, in chronological order, Garry Emmerson, Robert Kohn (University of NSW), Geoff Kingston (Macquarie University), James Duffy (Oxford University), Kenneth Kasa (Simon Fraser University).
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Abstract
The market for a risky asset is modelled using heterogeneous least squares learning with
multiple sources of data. Because of the linear nature of ordinary least squares regression,
the resulting model is linear and explicit algebraic expressions of its economic properties
can be developed. The price of the security emerges as the product of interlocking
expectations, and the continual revision of investor expectations drives price to the
efficient point. We develop explicit expressions of the profits of different market
participants and do not rely on a noise-trader assumption to support the economics. The
analysis suggests that fundamental analysis is not viable in a one period model. We show
that the price of a security can be regarded not as a single point of information but an
object (as the term is used in computing) with well defined properties and methods, and
operates from a formal standpoint like a set of genes in biology.
Keywords: efficient markets hypothesis, least squares learning, Grossman Stiglitz paradox,
interlocking expectations, objective model, Fisher's fundamental theorem of natural
selection.
JEL classification: G12, G14, G01, D01, D83, D84
1. Introduction: the microeconomic theory of financial markets
This paper reappraises fundamental analysis in the securities market. In this model there
is no fundamental analysis in the sense that black-box (exogenous) methods are used to
derive security value; we dispute that such methods exist either in theory or in practice.
Rather investors are concerned purely with discovering empirical regularities which they
can exploit. We contend that all real-world methods of analysis have such a basis, even
those apparently derived purely from accounting data, because the significance of the
analytical information can only be assessed in the light of experience. For instance,
suppose that the balance sheet of the enterprise underlying a security reveals that the
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enterprise is insolvent. The consequent value of the shares will depend on whether or not
'Chapter 11' style protection is available to the enterprise amongst many other things, and
the investor can only assess the significance of the information by setting it into a broader
context which reflects their experience. Such an assessment can be interpreted as carrying
out regression analysis using similar cases as data points.
It is corollary to this view that every investor in the market for a security uses least squares
learning, and they analyse both price and non-price information. If the reader is prepared
to accept this proposition then the questions which arise naturally are:
can such a market make a price – where do the demand and supply equations cross?
would such a market be efficient? What relationship, if any, does this price have to
true value as defined by the underlying correspondence between data and return?
Let us assume zero net supply and investors can long or short the stock. Positive
profit expectations of investors must be reconciled with a net realization of zero. Can
such a situation be consistent with persistent patterns of behaviour and a viable
market?
If security markets are efficient in that the price of a security reflects its true value (a
proposition which has come under increased scrutiny since the 2007 global financial crisis)
then the economics of security markets are problematic. In physical production, even firms
in perfectly competitive equilibrium earn a 'normal' profit, but in a security market the
output is not goods but information. As an efficient price costlessly communicates
information throughout the market, profits appear hard to come by for fundamental
analysts. These considerations were put into precise form by Grossman and Stiglitz (1980)
and the situation of market impasse is known as the Grossman Stiglitz paradox. Within a
rational expectations model such as this one, the Grossman Stiglitz paradox manifests in
the third question above – how can positive expectations be reconciled with zero net
return? We contend that it has been the failure to provide a structured account of
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fundamental analysis which is responsible for the failure to conceptualize and detect
normal profits in security markets. We make good this deficiency to investigate the
paradox. As investors are using ordinary least squares regression (OLS), the market can be
characterized using linear algebra which allows explicit algebraic expressions to be found
for market characteristics.
A brief summary of previous approaches to the microeconomic theory of the security
market follows. The Grossman and Stiglitz (1980) paper used noise as an artefact to
prevent price from communicating value effectively. Friedman (1953) already considers
noise (random) traders in markets, but argues that because their operations are unprofitable
they will exist only in a transitory and inconsequential way. Black (1986) agrees that noise
traders trade unprofitably, but argues that notwithstanding it is an observable fact that they
do exist and have an important economic role. Their losses subsidize other market
participants and underwrite the viability of fundamental analysis. Some take the argument
even further and claim that noise trading can be profitable. De Long, Shleifer, Summers
and Waldmann (1990) construct a model in which noise traders can make money in the
long term by taking on a large amount of risk. Rational investors do not arbitrage the noise
traders to the true value because it is too risky to do so; they fear the unpredictability of the
noise trader’s behaviour and its effect on price. Interestingly, Chiarella, Gallegati,
Leombruni and Palestrini (2003) get a similar result using a model with fundamental
traders and price watchers. Although the mathematical implementation differs substantially
from the De Long (1990) study, the conclusion that price watchers can make money by
exploiting price dynamics is the same. This is quite different to the Grossman Stiglitz
notion that price watchers make money by tagging along after the fundamental analysts.
The most recent style of paper follows the influential Brock and Hommes (1997) in
emphasizing the role of econometric estimators themselves in generating noise (e.g.
Chiarella and He (2003), Goldbaum (2005), Branch and Evans (2006), Georges (2008)).
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These papers put sophisticated mathematical dynamics at the service of stylized economic
assumptions in order to generate computer simulations which exhibit more complex, and
perhaps more interesting, behaviour than the efficient market hypothesis would suggest.
In all these studies non-price data is treated as a one dimensional variable. The current
paper explores the case where data are multidimensional to discover a range of phenomena
not present in the simple case. The model does not consider price change as a variable and
does not investigate dynamic phenomena driven by price change, but we hope it can
inform the choice of economic assumptions used in this area.
2. Definition of the objective model
2.1. Premises of the objective model
PREMISE 1: BASIC FRAMEWORK. This is a one period model: there is a stock which makes
an uncertain payment of y per unit and retires every period (i.e. similar to a futures or
options contract, but without the margin calls). There is zero net supply and J types of
investor who can offer long or short bids for the stock. The Walrasian auctioneer sets the
market clearing price. Each investor models security return with multivariate OLS
regression on observed returns using an individual data set which may or may not include
price. This assumption is consistent with the Clements and Hendry (2005) finding that
short models (i.e. those which omit some explanatory variables) often outpredict longer
ones, notwithstanding the biased estimation which a short model entails, particularly where
the model specifications differ from the underlying data generation mechanism.
PREMISE 2: INDEPENDENT RE-ESTIMATION. The period between one payment and the next is
referred to as the ‘observation’ period. T observation periods form one ‘estimation’ period
which is the same for every investor. Estimation periods are numbered consecutively into
the past - 0 is the current period, 1 is the previous period, 2 the period before that etc. At
the end of each estimation period, a small proportion w of each type of investor j reestimate
their model using data from that estimation period. The same scheme is found in Georges
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(2008).
PREMISE 3: DATA SET. Data set original origT NX (covering T observation periods for a set
of origN variables) is a compilation which lists every variable used by every investor. *1Ty
is the set of absolute (not percentage) returns per share of the security which is the same for
every investor. Each item of data is available at the start of the observation period and the
corresponding return is measured at the end of the observation period. We suppose that the
independent variables are fixed, i.e. originalX is the same in every estimation period. Now
data set originalX has a normalized orthogonal basis X containing N vectors, N T , which
is also the same in each period:
*N NX X = I (1)
PREMISE 4: INVESTOR STRATEGIES. The information which an investor uses is referred to as
their data strategy. Each investor type j implicitly employs jK strategy vectors 1jk N a
to get jK linearly independent variables 1jk T X assembled in one matrix j T jKX .
jk jkX X a (2)
and j jX X a where 1 2j j j jKN jK a a a a contains all jka (3)
RESULT 1: COMPLETENESS. All investor strategy matrices taken together span N space.
PROOF: 1 2original
JX X a a a by (3), where 1 2 Ja a a a assembles all ja (4)
so 1 2original
Jrank rank N a a a X by Premise 3 # (5)
PREMISE 5: PARTICULAR PROPERTIES OF PRICE. At least one investor includes the price of
the security 1T p as a variable in their data set jX .
PREMISE 6: GENERATION OF RETURNS. It is assumed that the return per period per unit (one
share, not one dollar’s worth) of the security is generated according to:
y Xμ + u where E( ) 0u , 2*E T T uu I (6)
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This relation is an empirical regularity for the particular data set X and does not purport to
be causative; if investors had a different data set available to them then a different relation
would apply. Nonetheless we suppose that the return vector μ has the character of a
parameter in that it is stable over time.
PREMISE 7: DEMAND PROPORTIONAL TO EXPECTED RETURN. The standard risk-theoretic
development given at Grossman 1976 pp 574-576 results in linear demand functions:
ˆt t tj j jBq r (7)
1tj T q : shares demanded in period 0 by investor type j who estimated in period t
ˆ 1tj T r : net return per share predicted in 0 by type j who estimated in t
scalar 0tjB : amount of stock per unit of net return demanded in 0 by j estimating in t
1
t t tj j j
j t
b B B
: 'weight of money' at time 0 of investor type j who estimated in t (8)
1
tj j
t
b b
: total weight of money of investor type j : constant in each period (9)
We see 0jb and 1jj
b (10)
2.2. Preliminary results
NOTATION: Regression coefficients, period of estimation is denoted by superscript:
ˆ 1tj jK β : data regression coefficient estimated in period t by type j, if j uses data
ˆ scalartj : price regression coefficient estimated in period t by type j, if j uses price
Net coefficients summed over j, period of estimation is denoted by superscript:
ˆN 1t tj j j
j
b β a β : data coefficient estimated in period t summed over all types j (11)
ˆscalart tj j
j
b : price coefficient estimated in period t summed over all types j (12)
Net coefficients summed over j and t, current period subscripted:
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01
ˆ1 t tj j j
j t
N b
β a β : data coefficient as at 0 for all investors & estimation times (13)
01
ˆscalar t tj j
j t
b
: price coefficient as at 0 for all investors & estimation times (14)
RESULT 2: WEIGHT OF MONEY FORMULA. Here an additional subscript is employed for tjb :
1
0 1tt
j jb b w w (15)
where 0tjb , 1
tjb : weight of money in periods 0, 1 of type j who estimated in period t
PROOF: Trivial, available on request.
RESULT 3: COEFFICIENT UPDATE RULE.
10 11w w β β β (16)
10 11w w (17)
PROOF: Trivial, available on request.
2.3. Demand and price equations
RESULT 4: DEMAND EQUATION. Weighted predicted return is zero.
1
ˆ 0t tj j
j t
b
r (18)
PROOF: 11 1 1
ˆ ˆt t t t tT j j j j j
j t j t j t
B b
0 q r r by Premise 1, Premise 5, (7), (8) # (19)
RESULT 5: PRICE THEOREM. Price can be expressed as an exact linear function π of X :
0 0 0p X π (20)
where 00
0
=
β
π is referred to as the price equation. (21)
PROOF: 0 0 0 0
ˆ ˆˆ
ˆ ˆ
t tj jt
j j jt tj j
β βr X p X a p by (3) (22)
0 0 1ˆ ˆt t t
j j j j Tj t
b X a β p 0 substituting in (18) using (22) (23)
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00 0 0
0
ˆ
ˆ
t tj j j
j t
t tj j
j t
b
b
a ββ
p X X by (13), (14) # (24)
RESULT 6: HETEROGENEITY THEOREM. Investors must be heterogeneous.
PROOF OUTLINE. An investor using price must omit some non-price variable, since price is
collinear by (20). That variable must be used by someone or it would not be in the dataset. #
3. The basic objective model
3.1. New coefficients
RESULT 7: ESTIMATION. The estimates created in estimation period t are given by:
1
ˆ = tj jt jt jt t t
β a a a μ - π + e (25)
where 1t t tN e X u , E =e 0 , 2E N N ee I also denoted N NΩ (26)
ˆ
ˆ 1ˆ
tjt aug
j tj
jK
ββ : augmented regression coefficient estimated by j in t (27)
augjt N jKa : augmented strategy matrix for type j and can consist of: (28)
jt j t a a π price and data used by j, jt ja a just data, jt ta π just price (29)
augjK denotes 1jK if investor type j uses the price variable, jK otherwise.
PROOF: Net absolute realized return in estimation period t is given by:
= t t t t t t t r y p X μ + u X π applying (6), (20) (30)
so 1ˆ
ˆ
tj
jt t jt t jt t ttj
β= X p X p X p r standard OLS formula (31)
1
j t t t j t j t t t t t t
= a π X X a π a π X X μ - X π + u by (3), (30) (32)
1
= jt jt jt t t
a a a μ - π + e by (1), (28), (26) (33)
Observe t t te X u , so 2
*t t t t t t N NE E e e X u u X I # (34)
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Error variance matrix Ω is given here as 2 I but it can be defined more generally as a
diagonal matrix for the results to follow. We see the interesting and important result that
regression of return against data is equivalent to regression of the underlying parameters
μ - π against strategy matrix jta . In a sense the investor’s problem does not involve data
at all but is a test of how well the investor’s data coefficients capture the return coefficients
μ - π . We have abstracted the problem from ‘data space’ T to ‘coefficient space’ N .
3.2. How estimation impacts the price
RESULT 8: PRICE CHANGE LEMMA
1
1 1 11
1 1
1w
w
dπ β π (35)
PROOF: 1 0 0 0N 0 β π restating (21) price equation (36)
1 1*1 0 1 0 11N w w 0 β π β π by(16), (17) (37)
We differentiate (37) twice with respect to w and evaluate derivatives at 0w .
Diff (37) wrt w: 1 1 1*1 1 1 11N w w
dw
dπ0 β π β π (38)
Diff (38) wrt w: 2
1 1 1*1 1 1 12
1N w wdw dw dw dw
dπ dπ d π dπ
0 (39)
(38) at 0w : 1 1
1
1dw
dπ β π
noting at 0w , 1 1 *1N β π 0 and 0 1π π (40)
(39) at 0w : 2 1
12
1
2dw dw
d π dπ (41)
Now 2
22
0 0
1
2w w
w wdw dw
dπ d π
dπ two terms of Taylor expansion (42)
1
21
1
w wdw
dπ by (41) and apply (40) for result # (43)
RESULT 9: PRICE CHANGE PRINCIPLE
10
1
11 1 1
1 1
1w
w
dπ = H μ - π + e (44)
where 1
1 1 1 1 1j j j j jj
N N b
H a a a a is referred to as the 'estimation matrix'. (45)
PROOF: 1 1 1 11 1
ˆ ˆj j j jj
b β π a β π by (11) (46)
11ˆ
j j jj
b a β by (28), (27) (47)
1
1 1 1 1 1 1j j j j jj
b
a a a a μ π + e by (25). Apply (45), (35) for result. # (48)
Note what this result does not say – the price change is given by the change in regression
coefficients - but rather, the price change is determined by new estimates alone, so price π
is pushed towards the mean μ .
11
1
w
dπ β β intuitive but wrong! (49)
1
1
w
dπ β correct! Applying (47) to (35) (50)
3.3. Properties of the estimation matrix
It is useful to introduce a normalized version jtα of the augmented strategy matrix jta .
RESULT 10: NORMALIZED STRATEGY MATRIX. Estimation matrix 1H can be expressed as:
1 1 1j j jj
b H α α where 1aug
j N jKα contains orthogonalized unit vectors (51)
PROOF: 1
1 1 1 1j j j j a a a
a a a a Λ λ Λ projection matrix, orthogonal eigenvectors aΛ (52)
1*1 2 1 1
2
jKaug jKaug
ΛI 0= Λ Λ = Λ Λ
0 0 Λ placing unity eigenvalues first (53)
Take 1 1jα = Λ . We see 1 1 1 1 *j j jKaug jKaug α α = Λ Λ = I as required. # (54)
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RESULT 11: NORMALIZED ESTIMATION MATRIX. Estimation matrix 1H is positive definite
and 1 1 1 1H Λ λ Λ where eigenvectors in 1 are orthogonal, eigenvalues 1 0 . (55)
PROOF: 1H is real symmetric so 1 1 1 1H Λ λ Λ where 1λ is real, 1Λ is real orthogonal. (56)
By (5), jka spans the space N , so for all 1Nx , there exists jka such that 0.jk x a (57)
Without loss of generality choose jka as one of the eigenvectors in 1 in (53) and so 1jα .
Then 1 0 0 0 0
0
jk
j j j j j j j jkj
b b b
a x
x H x = x α α x x α α x x a # (58)
RESULT 12. The eigenvalues 1 of the estimation matrix 1H are such that
10 1 (59)
PROOF: 21 1 v v v H H v where v is eigenvector of 1H for eigenvalue . (60)
ˆ ˆi j i ji j
b b v v i.e. 1 1 1 ˆj j j j jj j
b b H v α α v v , ˆ jv is projection j (61)
i ji j
b b v v ˆ ˆ ˆ ˆgiven i j i j v v v v v v (62)
v v noting 1i ji j
b b . Divide both sides by v v . # (63)
3.4. The Efficient Market Theorem
Estimation matrix 1H is a function of augmented strategy matrix 1ja which is a function
of price 1π . Since price varies, it is useful to develop an approximation for 1 1H μ - π .
RESULT 13: FIRST ORDER APPROXIMATION. A first order approximation for 1 1H μ - π is
1 1 1H μ - π H μ - π where H denotes tH evaluated at tπ = μ , i.e. π μ
H . (64)
PROOF: 1 1 1j jk jkj k
b H μ π α α μ π by (51) where columnjk j kα α (65)
12
1 1
1 1jk jk
j jk jk jk jkj k
b
α αdH μ π
α μ π α μ π α αdπ π π
(66)
1 11 1 1 1
π μπ μ
dH μ πH μ π H μ π π μ
dπ first order Taylor (67)
1 1 1N j N N N N jk jkj k
b
π μ
0 0 0 α α π μ H π μ by (66) # (68)
RESULT 14: CONVERGENCE MATRIX. Matrix w
H I H can be expressed as:
w
H I λ where λ , are the eigenmatrices of H (69)
and 0E denotes the mean value of the price coefficient (14). (70)
PROOF: Expand (69) #. The eigenvalue matrix of convergence matrix H is denoted λ .
PREMISE 8: STABILITY CONDITION. The reader may feel that market stability is a property
which should be implicit within the model, but this presupposes that real world markets are
stable and this is not always the case. We specify the following stability condition:
max
2
w (71)
RESULT 15. Given stability condition (71), eigenvalues of matrix H satisfy:
1 1 (72)
Proof is straightforward. We now demonstrate that the model is efficient and stable.
RESULT 16: EFFICIENT MARKET THEOREM. If stability condition (71) is satisfied then:
lim nn
E
π μ (73)
PROOF: 0 1 1 1 11
w
μ - π μ - π + H μ - π + e by (44), discarding 2nd order factor (74)
0 1 1
wE E E
μ - π μ - π H μ - π as 1 1 1
1
w w
H μ π H μ π by (64) (75)
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1
wE
= I H μ - π factorizing (76)
1E= ΛλΛ μ - π substituting ΛλΛ for w
I H by (69) (77)
22E= Λλ Λ μ - π etc. as Λ is orthogonal. Apply (72) for result # (78)
FIG. 1. Convergence of price to return in the objective model. Investor j regresses the net
return μ - π against their regressors jα . This corresponds to the projection of the net
return vector onto the regressor vector. The investor buys according to their estimated
return, which moves price along the line of the regressor – this is shown by the unbolded
arrows. Total impact on price by all investors is the vector sum shown by the bold arrow.
Successive iterations move price along the path of the bold arrows to converge with return.
Perhaps contrary to intuition, market efficiency does not require strategy switching, i.e.
changes in the weight of money jb towards the most relevant or profitable data.
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4. The second order objective model
4.1. Stationary distribution for price
The most basic requirement of a market model is that it is capable of making a price. Prima
facie it appears that regression of net return y p Xμ Xπ against price Xπ will yield a
price coefficient close to zero, but a zero or positive price coefficient will not work in price
equation (21). We find the condition under which the price coefficient is negative. We derive
the return which is expected by different classes of investor and show that the results are
consistent with demand equation (18). Finally we validate the results by simulation.
price at time of estimation
new price π0
Heavy arrows show the summation of the two price impact vectors, which becomes the price move. Price converges to return.
arrows show the price impact of each investor, given by dw*bj*jjπ1) and proportional to the projection vectors
Coefficient 1
Coefficient 2
Coefficient space (RN space ) statistical variation
e1 left out of this diagram
gross return regressors
of investor 2
regressors of investor1
projection vector for investor 1, ie projection of the net return vector π1
onto the regressors of investor 1, given by:π1)
projection vector for investor 2
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The process defined by (44) can be viewed as a Markov chain in discrete time and
continuous state space, which is the coefficient space N . The efficient market theorem
(73) demonstrates that the process converges to a part of the state space, the part around
mean μ , and regions sufficiently far from the mean can be neglected. It possible for price to
move from one point in coefficient space to any other point with a suitably chosen error term,
so the process is strongly irreducible and aperiodic, and converges to a stationary
distribution.
PREMISE 9: PRICE DISTRIBUTION. Price has a stationary distribution. Present values are now
calculated conditional on that distribution rather than on the initial value of price as in the
previous section.
PREMISE 10: SPECIALIZATION. Assume that all traders can be divided into data analysts
and price analysts with none looking at both. The case with combined price-and-data
analysts is essentially the same but more complicated notationally so is omitted at this
stage. The proportion of investors using price is denoted Pb .
For clarity and without loss of generality, variables are stated in terms of coordinate axes
specified by eigenvector matrix for estimation matrix
π μ
H H . So , H λ H λ . (79)
4.2. Four simplifications for analytical tractability
SIMPLIFICATION 1: ALIGNED EIGENVECTORS. Standard matrix H can be approximated by a
similar matrix *H which has mean μ as an eigenvector.
We describe a scheme by which a suitable *H can be generated. Specify some eigenvectors
* which include μ . Apply H to each of these eigenvectors v to get estimate ˆ v Hv , and
define 2R as the proportion of eigenvector v which is explained by v - found by projecting
estimate v onto v to get ˆv (this same definition applies to 2R in the OLS case).
RESULT 17: 2R GENERATED BY ESTIMATION MATRIX. The 2R value for the estimate of vector
v obtained from matrix H is a weighted average of the eigenvalues of H . Each eigenvalue
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is weighted by the coordinate of vector v in that direction. Sum of weights is unity.
PROOF: 2 2ˆ
ˆ ˆ ˆN
i ii
R v v vv v v v Hv v v λv
v v noting v =1 (80)
Form the replacement standard matrix *H using the 2R values:
21
* * 2 *2
2N
R
R
R
H , (81)
the rationale being that this matrix *H will give the same 2R for each of the eigenvectors v
as the original matrix H. Given that eigenvalues of original matrix H lie between 0 and 1,
matrix *H will give approximately the same results as matrix H.
RESULT 18: The sum of the eigenvalues of matrices H, *H is the same.
PROOF: * 2 * * * *tr tr trN N N
i i i H i H H Hi i i
R v λ v λ Λ λ Λ λ by (80) # (82)
SIMPLIFICATION 2: In the limit as price variance goes to zero ( 2 0 ), price approaches
return and orthogonal variations in price become inconsequential. Create an approximation
for the price regression coefficient by substituting return μ for price π , subtracting the
mean of the new estimator and adding back the correct mean.
RESULT 19: PRICE REGRESSION COEFFICIENT APPROXIMATION.
1 ˆˆ tj t t μ μ μ π - π e (83)
PROOF: 1ˆ tj t t μ μ μ μ - π e mean of this estimator + correct mean (84)
1 1 ˆt t μ μ μ μ - π e μ μ μ μ - π (85)
1 ˆt t μ μ μ π - π e . # (86)
SIMPLIFICATION 3: CONSTANT ESTIMATION MATRICES. The estimation matrices H and H are
assumed to be constant when used to model the dynamic process (but not when used as part
of the price change principle (44) to find the expected price at (123) below).
17
Then it is easy to show using the price change principle (44) and the lag operator L that:
1
0 0 0L L
μ π e I H I e (87)
The problem is that the expectation of the RHS, a series of error terms, is zero whereas the
LHS has positive expectation. This is a consequence of assuming H and constant when
in fact these vary with price π . We form an approximate version of the price change
principle suitable for dynamic contexts by using expected price π rather than return μ .
SIMPLIFICATION 4: 0 1 1
w
dπ H π π e (88)
4.3. Notation
The component of a vector collinear to the mean μ , or variance calculated for such a vector,
is denoted , and components orthogonal to the mean are denoted . Collinear
components are placed first. Decompose standard matrix H :
1j X
j j j j j Pj
b b
-1H a a a a μ μ μ μ dividing into data and price parts (89)
X P H H where XH is data analyst's part of H , PH is price watcher's part (90)
1
2 20
0
X P X P
X X
XN XN
b
Λ Λ Λ Λ Λ Λ (91)
where 1X X , P Pb , and total collinear eigenvalue T is given by T P X . (92)
4.4. Six preliminary results
RESULT 20: NET PRICE COEFFICIENT RESULT.
ˆP where ˆ ˆ t
jE (93)
PROOF: 1
ˆˆ ˆt t tj j P j P
j t
E b b E
by (70), (14), (92) # (94)
The following result derives price variance from error variance in straightforward manner.
18
RESULT 21: PRICE VARIANCE. Variance of price N NΣ is given by :
12
Σ I λ 1 + λ where *N N E Σ π π π π (95)
2 Σ HΣ H this intermediate result is used at (118) (96)
where or w
H λ I H H defines a complementary matrix to H (97)
PROOF: 0 1 1
w w
π - π I + H π - π + He restating (74) (98)
0 1 1 π - π Ι λ π π λe restating with eigenvalues (99)
2 22 N N Σ Ι λ Σ Ι λ 0 λ (99) by itself, take expectations (100)
so 0, ij i ij j i j taking element ij, i j , recalling 1 (101)
0 1 i j ij ij given 0 , 1i j by (72) (102)
so 2 2 Σ I Ι λ Ι λ λ (100), diagonal matrices ,Σ λ commutative (103)
2 Σ I λ λ (97). Rearranging with (79) gives results. # (104)
RESULT 22: RETURN VARIANCE.
10 0 0 0
2E
w
π π e π π e λ Σ (105)
PROOF: 0 0 0 0E π π e π π e Σ Ω 0π and 0e independent (106)
1 Σ I λ I λ Σ by (95) (107)
1
12 2
w
I λ Σ λ Σ by (69) (108)
To understand market characteristics we need to relate the current price to the prices which
preceded it. We apply the price change principle recursively to get the next three results.
RESULT 23: ERROR LEMMA. Price deviation for the current period can be expressed as:
19
10 1 2...
t tt t
π π He H He H He H π π (109)
PROOF: 0 1 0 π π π π dπ definition 0dπ (110)
1 1 1
w
π π H π π e by (88) (111)
1 1 H π π He on rearranging (112)
2 2 1 H H π π He He iterating # (113)
RESULT 24: TEMPORAL CROSS CORRELATION.
10 0
tt tE
π π e π π e H Σ where 0t (114)
PROOF: 11 2...
t tt t t tLHS E
He HHe H He H π π π π e by (109) (115)
1t tt t t tE
H He H π π π π e discarding uncorrelated terms (116)
1t tt t t tE
H He e H π π π π expanding and discarding (117)
1 1 2 1t t t t H HΩ H Σ H HΣ H H Σ using (96) # (118)
RESULT 25: TOTAL TEMPORAL CROSS CORRELATION.
1 10 0
1
1t
t tt
E w w
π π + e π π e H Σ (119)
PROOF: 1 1
1
1t t
t
LHS w w
H Σ by (114) (120)
11w w
I I H Σ sum geometric progression (121)
1
1 ww w
H Σ H Σ taking limit as 0dw # (122)
4.5. Expected price and expected price coefficient
We can now use the results derived above together with the price change principle (44) to
20
develop an accurate expression for expected price. Second order considerations result in a
slight but important downward bias in the expected value of price.
RESULT 26: EXPECTED PRICE THEOREM
2PC
T
π μ μ (123)
where 2 var / π μ μ : variance of collinear price relative to magnitude of μ (124)
2 var / π μ μ : variance of orthogonal price relative to magnitude of μ (125)
2 2 2var / π μ μ : variance of price relative to magnitude of μ (126)
2 2 2C : a modified variance referred to as 'conjugate' variance (127)
PROOF: 2
11 1 1 1 1 1 12
1 1
w w
dπ H μ - π + e H μ - π + e (44) restated (128)
1 1 1 1 1 1 12
2 21
1 1 1 1 1 1 12 21 1
w w
w w
H μ - π + e H μ - π + e
H μ - π + e H μ - π + e
approx for 1 (129)
Consider first term of (129). As error 2 0 , projection of mean μ onto price π yields:
ˆ 1
π πμ μ
μ μ by simple geometric argument (130)
ˆ μ π likewise (131)
Define 1
t t t t t
M π π π π projection matrix for price. 1t (132)
1
M μ μ μ μ projection matrix for return (133)
so ˆ ˆt t t t t M μ π μ μ π noting ˆt t M μ μ , t M π π (134)
1 t tt t t
π πμ π π π
μ μ by (130),(131), t t t π π π (135)
21
t tt
π πM μ μ M π
μ μ noting μ M μ , t tπ M π (136)
and 1 1 1 1X PE E H μ - π H M μ π noting 1 1X P H H M (137)
2X P H μ π M μ π μ taking expected value of (136) (138)
2P H μ π μ noting X P H H M by (89) (139)
Consider the second term of (129): 1 is formed from estimates up to period 2
2
1 1 122
ˆ2 1t t
jj t
wnd b w w
H μ - π + e (14),(15),(64) (140)
2
2 1
1 121
t
P t tt
ww w
H μ - π + e μ μ μ π π e (92),(83),(93) (141)
2
2 1
1 121
t
P t tt
ww w
H μ π + e π π e μ μ μ (142)
112
2 P
wE nd
H H Σ μ μ μ (119), μ π component goes out (143)
2
122
2
0
0P
N
w
Σ
Σ μ μ
Σ
0
0
μ standard coordinates (144)
2P
w
μ recall 2
22
Σ
μ
by (124) (145)
Consider the third term of (129):
2
11 12
ˆ3 j jj
wE rd E b
H μ π + e by (12), (64) (146)
2
1
1 1 1 12 P
wE
H μ π + e π π e μ μ μ by (92), (83), (93) (147)
2
112
2P
w
dw
H λ Σ μ μ μ by (105) (148)
22
22 P
w
μ as (145), using (79) (149)
Finally the fourth term of (129) is a higher order term in w and can be disregarded:
1 1 124
wE th E w
H μ π e 1st factor = 2nd term (129) (150)
2P
ww
μ by (145), this is a term in 2w (151)
Now 1NE dπ 0 , i.e. expected price change is zero at system equilibrium. (152)
Apply expectation to (129), use (152), (139), (145), (149) and (151) to get:
2 2 21 12N P P P N
w w w
0 H μ π μ μ μ 0 (153)
2 2P P
T T
π μ μ μ noting 1
T H μ μ by (91) # (154)
RESULT 27: EXPECTED PRICE COEFFICIENT.
2ˆ XC
T
(155)
2X PC
T
(156)
PROOF: Recall
μ0
. Set *
μ0
and 1
2
1*
y
π
π x (157)
1
1
2
1 1 11 1
1 2
y yy y
y y
π π π μ x xx 0 x x
(158)
22 2(1 ) 1 2 2y y y y y x x x x approximating denominator (159)
21 y y x x expanding and discarding all 3rd order and higher terms (160)
so 1 21 1
ˆ 1 1E E y y π π π μ π e x x (161)
23
2
11 2 22 2
E E
π π noting 1y
(162)
2 2 2 2 2P PC C C
T T
(123),(124),(125). Use (93) for (156) # (163)
It appears that the natural tendency of an unbiased price vector to generate a downward
biased price coefficient (because the price vector is not perfectly correlated with return, i.e.
does not point the same way) is offset to some degree by the downward bias in expected
price which causes a corresponding upward adjustment in the estimated price coefficient.
However the value of the price coefficient cannot be determined by (156) alone because
the conjugate variance 2C itself depends on the price coefficient via (95). We need to solve
these equations simultaneously to confirm that the price coefficient can be negative. A
negative price coefficient confirms that the 'conjugate variance' 2C is indeed positive.
RESULT 28: PRICE COEFFICIENT EXISTENCE CONDITION. The mean price coefficient is
2 2
2X P e T e
T
w
which exists if 2
2
e
e T
(164)
where denotes an average orthogonal eigenvalue, defined as the solution to:
1 12 2
2
N
e i i eii
I I I I
(165)
and 2e ,
2e , 2
ei , 2eC derive from error matrix Ω (as distinct from price matrix Σ ).
PROOF:
2 2
2 2T T
e eX P Tw w
(95),(69),(156),(165) (166)
Simplify. Variables , w are both small so high order terms can be discarded to leave:
3 2 24 2 0T e T e X P w rearrange for result # (167)
4.6. Satisfaction of the demand equation
We tie the analysis together by confirming that the results satisfy the expectation of demand
24
equation (18); a vector diagram is handy in demonstrating the weightings of the price and
data components of expected return.
RESULT 29: PRICE RETURN LEMMA. The return on price, 0ˆ t π , can be approximated by:
1
0 0ˆˆ t
t t π π π - π e μ μ μ μ (168)
PROOF: 0
0
ˆ ˆ
ˆˆ ˆt
t d
π μ π note: ˆd π is not a time period increment (169)
0ˆ ˆˆ td μ μ dπ calculus product theorem (170)
0ˆ ˆ td π μ noting 0 0
ˆ ˆ ˆ μ dπ π (171)
Now 1ˆˆ ˆt tt td π - π e μ μ μ by (83); sub in (171) for result # (172)
RESULT 30: MODEL CHECK. The approximate results satisfy demand equation (18).
PROOF: Expected data return is given by:
1ˆ
X Xt tj j j j t t j j j j
j t j t
E b b E
β a μ π e a a a a using (25) (173)
X μ π H simplifying and taking expectation (174)
2PC X
T
μ using (123), note μ π is μ -collinear (175)
Expected price return is given by:
1
0 0ˆˆ
P Pt t tj j t t
j t j t
E b b E π π π π e μ μ μ μ by (168) (176)
2 2X PP C C
T T
μ μ (155), (123), 1
0t tE π - π e μ μ μ μ (177)
2X PC
T
μ discarding higher order terms. (175), (177) are opposite # (178)
4.7. Simulation testing
25
Because the results derived here rely on various approximations, simulation testing was
carried out to validate them. The simulation is set in two dimensional parameter space (not
data space). The market is defined according to orthogonal error 20, errorN e ,
collinear error 20, errorN e , coefficient estimation formula (25), coefficient update
formulae (16), (17), and price equation (21). Theoretical price coefficient is found
iteratively using (95), (156). Values of the parameters are: return 1 0μ , data
eigenvalues 1 2 0.5X X , price eigenvalue 0.2P , orthogonal standard deviation
0.5error . Collinear standard deviation takes five different values (0%, 25%, 50%, 75%
and 100% of orthogonal standard deviation). The price change principle, which is a
derivation, is not used. The market is considered stable if price coefficient 0t over
32,000 iterations.
The explicit formula (164) for the price coefficient was verified by calculation, as was the
attached existence condition (164). Here the existence condition takes specific form
2 2 0.5 0.7e e so we do not expect the price coefficient to exist where standard
deviation ratio 1e e .
TABLE 1: RESULTS OF SIMULATION. Theoretical values are constant, the empirical price
coefficients are the averaged results of ten trials. The actual price coefficient shows good
agreement with the theoretical value right across the stable range and converges at the
limit. There are cases where stability condition (71) was satisfied but the simulation was
unstable. Serial correlation is present in the price coefficient and local means for can
be somewhat less than the mean value . In such situations the stability condition is
breached locally. Spreadsheet is available online.
26
Estimation rate Ratio of collinear error standard deviation to orthogonal error standard deviation
dw 0 0.25 0.50 0.75 1.00
Theoretical price coefficient. Empirically stable region shaded dark grey.
0.000001 - 0.000095 - 0.000090 - 0.000076 - 0.000043 undefined
0.0001 -0.000957 -0.000915 -0.000772 -0.000427 undefined
0.01 - 0.010781 - 0.010299 - 0.008559 undefined undefined
0.02 - 0.016095 - 0.015363 - 0.012474 undefined undefined
0.03 - 0.020540 - 0.019585 - 0.014891 undefined undefined
0.04 - 0.024548 - 0.023373 undefined undefined undefined
Theoretical stability coefficient2
Tw
from (71).
0.000001 0% 0% 0% 1%
0.0001 4% 4% 5% 8%
0.01 32% 34% 41%
0.02 43% 46% 56%
0.03 51% 54% 71%
0.04 57% 60%
Empirical price coefficient, as a ratio of the theoretical price coefficient.
0.000001 100.0% 100.0% 100.0% 100.0%
0.0001 99.2% 98.9% 98.9% 94.4%
0.01 92.7% 92.3%
0.02 90.6% 90.9%
0.03 90.0%
0.04
5. The economics of the objective model
5.1. Profit calculus
Profit is the product of quantity and net return. We sum the profit of every observation
period in an estimation period by taking the inner product of quantity and realized return to
arrive at a representation of profit within coefficient space.
RESULT 31: PROFIT EXPRESSIONS. Profit is given by:
27
1
0 0t X tj j t t j j j jb
μ π e a a a a μ π e data users (179)
1
0 0 0 0 0 0ˆt P t t tj j t t t t t jb b
μ π e π π π π μ π e π μ π e price users (180)
where scalartj : profit in period 0 of investor type j using estimates from period t
PROOF 0 0 0 0 0 0 0
ˆ ˆˆˆ
ˆ ˆ
t tj jt t
j j j j jt tj j
β βr X a p X a X π X a β (22),(28),(27) (181)
so 0 0ˆt t t tj j j jb q r r r by (7),(8) (182)
1
0 0 0 0 0 0tj j jt jt jt t tb
X a a a a μ π e X μ X π u (181),(25),(30) (183)
1
0 0 0 0 0 0 0 0tj t t jt jt jt jb
μ π e a a a a X X μ X X π X u (184)
1
0 0 0tj t t jt jt jt jb
μ π e a a a a μ π e (1),(26). Apply price, data # (185)
RESULT 32: ZERO PROFIT. The aggregate profit across investors is zero.
PROOF: 0 1* 0 0t tj j T
j t j t
q r 0 r (182), Premise 1, 0r same for all j # (186)
Here realized return 0r can refer to total return 0 0 μ π e , a long run component generated
by the expected value of realized return μ π , or a fluctuating short term component
generated by the disturbance component 0 0 π π e . In each case profits sum to zero.
5.2. Profit analysis for each category
We now use the technical results derived in section 4.4 to derive profit in each category.
RESULT 33: EXPECTED SHORT TERM DATA PROFIT.
22 2X SR X
T
E
μ where X SR denotes short run data profit (187)
PROOF: 1
1
0 01
1X
tX SRj t t j j j j
j t
b dw dw
μ π e a a a a π π e (179) (188)
28
1
0 01
1t
t t Xt
dw dw
μ π e H π π e by (90) (189)
1
0 01
tr 1t
X t tt
dw dw
H π π e μ π e trace commutativity (190)
1trX SRXE H H Σ by (119). μ π component goes out (191)
1 2
222 2 2
2
trX T
X X
XN XN N
μ
(192)
22 2X
T
μ noting 2 2 2 22 3 ... N # (193)
We give an intuitive interpretation of this negative short term profit for data watchers.
Analyze the SR profit equation for an investor who reestimated in period 1:
11 1 0 0
X SRj X
μ π e H π π e (194)
1 1 1 0 1 1X
dw
μ π e H π - π + e + H π - π + e using (88) (195)
1 1 1 0 1 1 1 1 1 1X X X μ - π + e H π - π + e μ - π + e H λ π - π μ - π + e H λ e (196)
The first term of (196) is what the investor would receive if the price remained at 1π and it
is positive in line with the investor's expectations. The second term is the movement of
price towards the mean in response to the investor's demand following estimation. It is
negative but probably not so large as to eliminate the expected profit. The third term
isolates the effect of the error term 1e , which causes the investors' estimates to be
inaccurate and a price move in the direction of the inaccuracy. Because the error term is
relatively large, this term is strongly negative for the investor.
RESULT 34: EXPECTED SHORT TERM PRICE PROFIT.
22 2P SR X
T
E
μ where P SR is short run price profit (197)
29
PROOF: 0 0 0ˆt P SR t tj jb π π π e short run component (180) (198)
1
0 0 0ˆt
j t tb
π π - π e μ μ μ μ π π e substituting (168) (199)
1
0 0 0 0 0ˆt t
j j t tb b π π π e π - π e μ μ μ μ π π e (200)
2 22 2ˆP SR P
PT
E
μ μ 2nd term as previous proof (201)
2 22 2 2P
T
μ μ by (156),(126). Rearrange # (202)
RESULT 35: EXPECTED LONG RUN DATA PROFIT.
22X LR P
CT
E
μ where X LR is long run data profit (203)
PROOF: 1
t X LR tj j t t j j j jb
μ π e a a a a μ π long run component (179) (204)
X LRXE μ π H μ π noting 0t tE π π e (205)
2 220 0 0
0
X
P PC X C
T TXN
using (123) (206)
22 2 2X P P P
C C CT T T
μ μ μ using (156) (207)
RESULT 36: EXPECTED LONG RUN PRICE PROFIT.
22P LR P
CT
E
μ where P LR is long run price profit (208)
PROOF: 0ˆt P LR t tj jb π μ π by (180) (209)
1
0ˆt
j t tb
π π - π e μ μ μ μ μ π using (168) (210)
0ˆP LR
P t t PE E b π μ π π - π e H μ π sum over ,j t (211)
30
2 2 20P P PC C C
T T T
π μ μ μ μ by (123) # (212)
Expected return Data analysts Price watchers Total
Long run 2 2P P
T T
2 2P P
T T
0
Short run 2 2X
T
2 2X X
T
0
Total 2 2XX
T
2 2X
T
0
TABLE 2: Expected return per estimation period, classified by type of trader and
component of return - long term (average) or short term (fluctuating). Entries must be
multiplied by the positive scaling factor 2 μ . These results show that price analysis
makes money, data analysis loses money. Results have been verified by simulation.
6. Learning as a genetic algorithm
6.1. A cognate biological model
The point of this paper is that the price of a security (and indeed any good) is a vector
rather than a single point of information. We argue that this is an abiding principle by
constructing a genetic algorithm and showing that it operates in the same way on genotype
as learning acts on price. The similarity of the processes does not depend on the
profitability and survival of the economic agents, which is the traditional avenue for
constructing an analogy of the ‘economic Darwinism’ type. Nor does it rely on the
selection of particular strategies, which is the version of this concept found in rational
expectations literature (e.g. Marimon and McGrattan 1995). Rather it is the information
processing itself – least squares learning - which is analogous to natural selection.
Let creature type j be represented by an N*1 vector jg in ‘gene space’. Each gene
represents a separate dimension of variation, variations (technically speaking the 'alleles' of
31
the gene) are ranked in some physical order, and there are N separate genes so gene space
is N . We suppose that reproductive capability is a smooth concave function of genes g,
with maximum at genotype μ + e . The stochastic component e is set by prevailing
environment conditions, the situation with predator/prey species etc. For the species this
presents as a hill climbing problem.
jb proportion of creatures of type j. 1jj
b (213)
1 j jj
bπ = g is the mean genotype in period 1 (214)
1j j α g π is creature type j's genetic variation. j jj
b α = 0 (215)
1 1 1 r μ + e π population fitness vector, fitness measured by its length (216)
1 j μ e g fitness vector of creature type j (217)
PREMISE 1. Relative fitness is measured by the difference in length of the fitness vectors.
1 1 1 1j j jrelative fitness μ e g μ e π α r where jα not too large (218)
PREMISE 2. The number of offspring of a creature is equal to one for a replacement
creature plus a loading which is proportional at rate 0w to its relative fitness 1jα r .
11j descendents j j j jb b no of offspring b w α r (219)
Observe 1 11 1j descendents j j j j jj j j j
b b w b w b
α r α r by (215) (220)
PREMISE 3. The average genotype g of the descendents equals the genotype of the parent:
j descendents jg g (221)
RESULT 37: COGNATE PROCESSES THEOREM. Premises 1-3 imply:
1 1w dπ = H μ π + e equivalent to (44) price change principle (222)
32
where j j jj
b H = α α equivalent to (51) estimation matrix H (223)
PROOF: 0 j descendents j descendentsj
bπ = g (224)
1 11j j jj
b w = π +α α r by (221),(214),(219) (225)
1 1 1 1j j j j j j j jj j j j
b b w b b w = π + π α r + α + α α r (226)
1 1j j j jj j
b w b = π + 0 + 0 + α α r by (215) (227)
1 1 1w = π H μ π e using (216), (223) # (228)
This result can be developed as per the efficient market theorem (73) to show convergence
to optimal genotype μ . The three premises can also be regarded as applying to investor
strategy selection, so strategy selection can be represented by an estimation matrix H.
6.2. Fisher's Fundamental Theorem of Natural Selection
It is natural to ask whether the preceding result is known to biologists; the biologist and
statistician Ronald Fisher (1930) deemed this principle so important that he awarded it the
grand title of the Fundamental Theorem of Natural Selection (FTNS). To quote Grafen
(2003): "It may be an understatement to say Fisher believed the fundamental theorem to be
his main contribution to evolutionary biology. I suspect he believed it to be the main
contribution that anyone would ever make to evolutionary biology." Price's reformulation
(1972) is regarded as clearer, we give Andersen's (2004) version of the reformulation.
cov ,j j j jr A r A E r A (229)
where j : the population of one particular gene is divided into j types (alleles)
jA : quantitative characteristic of interest, different for each type j
1 0,j jn n : population of type j in periods 1, 0. Recall period 1 precedes period 0 here.
0 1j j jr n n : reproduction coefficient (230)
33
A : mean across j of characteristic A
0 1A A A :change in the mean between periods. Similarly 0 1j j jA A A (231)
Additionally, j js n n : share of the population of type j. (232)
RESULT 38: COGNATE RESULTS. The FTNS as restated by Price (229) yields (222) above.
PROOF: Price1j j jA g π α superscript 'Price' denotes Price's variables (233)
Price Price Price1j j j j
j j
A s A b g π by (214) (234)
so Price0 1A π π dπ (235)
Here PricejA 0 by (221) (236)
and Price Price0 0Price 1
1Price Price0 1 1
1j jj j
j j
n bnr g g g w
n n b α r by (219) where
Price0Price1
ng
n (237)
and Price1Price Price
1Price1
1jj j j
nr r b g w g
n
α r by (237),(220) (238)
and Price Price Price Price Price Price Pricecov ,j j j j jj
r A s A A r r
1 1 11j j jj
b g w g π α π α r (233),(234),(237),(238) (239)
1 1 1j j jj
gw b gw α α r H μ π e by (216),(223) (240)
so substituting for each expression in (229) yields:
1 1 11 jg gw E g w dπ H μ π e α r 0 (238),(235),(240),(237),(236)# (241)
The matrix j j jb H = α α is the genetic variance so we see that result (222) states that
change is proportional to genetic variance and this was Fisher's preferred interpretation.
There are three differences between the biological model presented in Section 6.1 and the
FTNS: this model makes no provision for changes PricejA in the average genotype of type j;
34
this model is a multivariate model whereas the FTNS considers one gene in isolation; and
this model employs the concept of a fitness landscape, introduced by Fisher's rival Sewall
Wright and explicitly rejected by Fisher.
7. Conclusion
7.1. Economics of the efficient market
We have modelled fundamental analysis using heterogeneous least squares learning and
developed a mathematic approach referred to as the objective model. The model, while not
exact, has been demonstrated to work to within 10% accuracy across the range of a typical
parameter set. We have seen that the market converges to true value even when information
is dispersed, but that the efficient market is a journey and not a destination. Price
approaches true value but does not coincide with it. We have uncovered a general principle
of organization referred to here as the price change principle, and identified that principle
with R. A. Fisher's fundamental theorem of natural selection.
It appears from this model that the analysis of data is not viable and that users should
gravitate to price analysis. The Grossman Stiglitz paradox is based on the costs of
processing data; the analysis of market processes presented here suggests that there are
deeper reasons why fundamental analysis will be unprofitable. This is a contingent result
and does not consider some of the factors which may boost the return to data analysts - in
particular we have not considered a price change variable, and we have assumed that the
data set is fixed in each period. We conjecture that in a multi-period model, agents who use
a price change variable would cannibalize price-watchers' profit in the same way as price
watchers take data analysts' profit. A variable data set in each period would also inhibit
the ability of price watchers to freeride.
7.2. The fragile nature of financial market processes
The process of forming a price in financial markets is inherently fragile because it relies on
the generation of a negative price coefficient to act as denominator, and it turns out that the
35
price coefficient is close to zero. Within the objective model it is only a fortuitous interplay
of parameters which allows the market to operate at all, and it is not surprising that markets
are susceptible to occasional malfunction for a variety of reasons. Markets will be unstable
if either the stability condition (71) or the price coefficient existence condition (164) are
violated. This can happen either because the rate w with which investors update their
estimates is too great, or because investors move away from data analysis into price
watching which decreases the eigenvalue ratio T . A bubble environment is precisely
where one might expect both these things, so there are positive feedback mechanisms for
instability. We note Brock and Hommes (1997) also identify 'intensity of choice',
equivalent to dw here, as the key parameter determining market stability. While the model
presented here is not fully developed, we suggest that its explicit treatment of the price
coefficient may prove useful in understanding real world instability.
7.3. The coefficient substrate and the reinterpretation of price as an object
We have constructed a 'coefficient space' which abstracts the problem from the original
'data space'. Viewed from within coefficient space, price is not a single value which is
immediately dependent on demand and supply conditions, but a vector with a persistent
value. Indeed within the objective model price can be regarded as an object in the
computing sense, viz. it is an independent entity with its own properties and methods. Its
property is a collection π of information. Its methods are those of a computer memory: it
stores information through the price equation (21), makes it available to investors via
estimation (25) and updates according to the price change principle (44). Heterogeneous
least squares learning operating on the price vector is formally identical to natural selection
operating on the genotype within a biological system. The correspondence may be seen as
an analogy or isomorphism, but the stronger interpretation is that the same underlying
process, a blind multipronged hillclimbing algorithm, is expressed in two different self-
organizing systems. The reification of price puts analytical clothing on the bones of
36
Dawkins’ (1976) meme concept – purportedly an equivalent of the gene in social systems.
7.4. The investor’s life in the bush of ghosts
Within the objective model, investor strategies support each other in a self-sustaining
equilibrium. Each investor relies on the pattern of investment of the other investors for the
accuracy of their own estimations, and no part can be removed without disrupting the other
parts. Notwithstanding the system is 'reducibly complex' – it can bootstrap itself up over
time from a low information starting point to a state of specialization and dependency.
A fundamental theme of economics going back to Adam Smith’s invisible hand is that the
whole is greater than the sum of the parts. The objective model identifies the parts which
have previously been obscured. Behind the outward appearance of a market – the trades
and current market price - is a hidden substrate, in which price is not a single piece of
information but an object with its own independent existence as an economic entity. The
object stores information and makes it available to traders in a same way as a set of genes
in biology. The price change principle is a genetic operator which moves the price object
to the point of maximum explanatory power.
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