The Mayers-Rice conjecture

Journal of Fmanctal Economics

8 (1980) 87400

0 North-Holland

Pubhshmg

Company

THE MAYER!+RICE CONJECTURE A Counterexample Robert

E VERRECCHIA*

Untverslty of Chtcago, Chrcago, IL 60637, USA

Received September

1979, final version received February 1980

Mayers and Rice coqe-cture that an Investor with better mformatlon wdl on average plot above the security market hne as drawn by unmformed Investors Thus paper demonstrates that thus coqecture IS false m general, by constructmg a counterexample However, the Mayer+Rice coqecture IS really part of a much broader hypothesis concernmg whether Increases m expected returns correspond to mcreases m expected utlhtles It IS shown that this latter hypothesis IS true when the Investor has an exponential or logarlthmlc utlhty function

1. Introduction In response to Roll’s (1978) crltlclsm of the capital asset prlcmg model, which called m question (among other things) the assumption that superior performance implies and IS implied by plotting above the security market line, Mayers and Rice (1979) made the following observation They defined an mdlvldual as superior whenever he had better mformatlon than other mdlvlduals, and conjectured that If investor Z’s [the superior mdlvldual] probability beliefs are correct, he ~111 on average plot above the security market hne as drawn by the uninformed investors An equivalent way of stating this would be that investor Z expects to plot above the security market line drawn by unmformed investors Symbohcally, the statement of the theorem IS

where E’(R,) IS the informed Investor’s uncondltlonal expected return from his portfolio, R, ISthe risk free rate, E"(RM)ISthe uninformed’s *The author gratefully acknowledges the assistance of Edward M Rice, who orlgmally brought this problem to the author’s attentton, and whose assistance m the preparation of this manuscript was Invaluable Any remammg errors, however, are solely the responslblhty of the author

88

RE

Verrecchq

The Mayers-Race

conjecture

A counterexample

expectation of the return on the market portfoho, cov (R,, R,)/var (RJ as assessed by the uninformed

and &’ 1s the

The two proofs of this theorem offered by Mayers and Rice rely on circumstances that are suficlently hmltmg as to call mto question whether the theorem 1s generally true The proofs require either that the informed mdlvldual learns nothing about the market return, or that the receipt of superior mformatlon and correspondmg portfolio choice leave marginal expected utility unchanged These circumstances, especially the second, must be regarded as extremely unlikely for many (potentially) Informed mdlvlduals Thus, we refer to the theorem as the Mayers-Rice conlecture The Mayers-Rice conjecture IS couched m terms of the capital asset pricing model, which suggests that it only has slgmlicance wlthm that paradigm This IS not the case, however The Mayers-Rice conjecture actually motivates a much broader hypothesis which, to avoid confusion, we refer to as the expected return conjecture If an informed investor receives superior mformatlon, and d receipt of the mformatlon has no effect on prices, then It IS well known that the mformatlon will increase his expected utlhty (or, m any event, not decrease It) vu-$-vls what it would be m the absence of that mformatlon An hypothesis which we call the expected return conjecture, and which 1s suggested by the Mayers and Rice dlscusslon, 1s that increases m expected utility (due to the receipt of superior mformatlon) ~111on average also show up as increases m expected returns That IS, the superior investor ~111 on average achieve a greater return than the uninformed investors expect In effect, the fundamental proposltlon conveyed by the expected return conjecture 1s that mcreases m expected utility from superior mformatlon manifest themselves m mcreases m expected returns above the market ‘equlhbrmm’ rates (I e, the expected rates of return on the basis of some equlhbrmm model of the market) The subtlety which makes this proposition less than self-evident IS that risk-averse investors do not necessarily choose portfolios which exhibit the greatest expected return, since such portfolios may entail considerable risk The importance of this fundamental proposltlon should not be underemphasized Much of portfolio analysis, even when not using the security market lme of the capital asset prlcmg model, assesses superior performance on the basis of excess rates of return That is, the analysis uses a pricing model to assess an equlhbrmm expected rate of return on a portfolio, and deviations from this rate of return are used as measures of performance 1 If the expected return conjecture 1s false, this entlre method of analysis 1s called mto question It suggests that more than the mean of the returns ‘See Cornell (1979) for an example of such an alternatlve portfoho analysw techmque

R E Verrecchta, The Mayers-Race coyecture

A counterexample

89

dlstrlbutlon must be investigated It may even be that superior performance 1s not detectable without specific knowledge of mvestor utility functions In this paper we demonstrate that the expected return conjecture 1s valid for two popular types of utlhty functions, the (negative) exponential and the logarlthmlc These utility functions imply, respectively, constant risk aversion, and constant relative risk aversion However, we also demonstrate that the conjecture 1s not generally true by provldmg a counterexample m the case of the quadratic utility function This, m turn, implies that the Mayers-Rice conjecture 1s not generally true smce it 1s simply a particular form of the expected return conjecture related specifically to the quadratic utlhty function and the capital asset pricing model In brief, anomalies exist to the general suggestion that expected returns above the market equlhbrlum rates can serve as reliable surrogates for increases in expected utlhty due to the receipt of superior mformatlon A dlscusslon of the seriousness of these anomalies 1s reserved for the concludmg section of this paper The sole purpose of this analysis 1s to offer proofs and a counterexample Therefore, the reader 1s advlsed to reference Mayers and Rice for a detailed dlscusslon of salient points to which we only briefly allude 2. Informed expectations

versus consensus beliefs

To begin, we assume there exists an informed receives one of L possible messages each period , q,J IS a set of messages 9 The vector q = (ql, It is assumed that the probability that one of the

investor m the market who with probability q, from a vector of possible messages messages occurs 1s one, 1 e ,

and each message has some posltlve probability associated occurrence, i e , for all J E 9, q, > 0 Upon receipt of message informed mvestor’s beliefs are represented as a vector 11”=(ny, that for all s E ./lr, X: > 0, and

with J E 3,

,n;),

Its the such

n” represents the probablhtles he asslgned to N possible states of the world, and JV represents the set of all possible states To avoid a case which would be otherwise unmterestmg, we assume that there exists at least some pair of messages k, I E A?, which imply dlstmct probablhty assignments Mayers and Rice assume that every other investor m the market IS

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R E Verrecchra, The MayersRwe

conlecture

A counterexample

unmformed with homogeneous beliefs represented by Jo”=(ny, . , 7~:) We prefer to characterize n” as the ‘consensus behef implied by the marketclearing prices; that is, n” IS the probablhty belief which, if held by all investors, would result m the same prices as those currently exlstmg.’ A key assumption m the Mayers-Rice conjecture 1s that the uncondltlonal probablhty dlstrlbutlon of the Informed investor IS identical to the consensus behef, i.e., n”=q

IF,

where 7~‘~IS an L x N matrix whose Jth row is the probablhty assignment 7~” This Implies that for each SEJCT, L

a,U=1 q'%'J j=l

We refer to this as the ‘Jomt rationality’ assumption Mayers and Rice Justify this assumption by arguing that m its absence, either the expectations of the Informed investor, or the expectations of the market as implied by the consensus belief, are not rational To digress briefly, it 1s important to emphasize that m the absence of this assumption, the Mayers-Rice coqecture IS false Although we do not provide any here, counterexamples to the conjecture can be constructed If we assume that the expectations of an rnformed investor are ipso facto ratlonal, this allows us to suggest a better focal point for Roll’s crltlclsm of the capital asset pricing model Even if we assume the Mayers-Rice conjecture 1s vahd, If the expectations of the market are not rational (as we have narrowly defined the term), an Informed investor may not on average plot above the security market lme 3. A characterization of the conjecture We assume that the informed investor’s wealth IS unaffected by which message he receives In the market, the informed investor exchanges his wealth for units of consumption contingent on the occurrence of the possible states of the world 3 Assummg a complete market,4 any portfoho P can be represented as a *SeeVerrecchla

(1979) 30ne moddicatlon we make to the Mayers and Rice dIscussIon 1s to ehmmate the posslbdlty of current consumption This amehorates the notatlonal burden wrthout restrlctmg the g&erabty of the results m any way 41t has been pomted out by the referee that the existence of complete markets IS also a crItIca assumption m the analysis Spec&cally, counterexamples to the coqecture can be constructed m the absence of complete markets

RE Verrecchra, The Mayers-Race conjecture A counterexample

91

vector YP=(Y?,

9m

where Y,’ represents the umts of consumption received by holding portfolio P if state s occurs When the informed Investor receives message 1~9, his expected utility for holding portfolio P 1s represented by

where U IS the informed Investor’s utility for units of consumption It 1s assumed that U IS a contmuous, real-valued utility function with contmuous first and second derivatives U’(Y) and v”(Y), respectively, furthermore, V’(Y) 1s a positive function and V’(Y) 1s a negative function Utility functions of this form exhibit stnct global risk aversion Let o, be the price of a contingent claim on a unit of consumption m state s, where it IS assumed that o, >O for all SE N, it 1s also assumed that w, 1s unaffected by which message the informed investor receives. Then, m the market, the informed investor assembles a portfolio P, say, which maxlmlzes his expected utility SUbJeCtto the constraint that E,“=, o,Yf not exceed hts endowed wealth For example, suppose that the informed investor receives message J, and, in consequence, assembles portfolio YJ= (Y/,, , YiN) to maxlmlze his expected utility sub@ to the constraint imposed by his wealth Then, the informed investor’s expectation of his portfoho return 1s

Using the capital asset pricing model, the expectation on the basis of the consensus belief 1s

of the same portfolio

Thus, the Mayers-Rice conJecture IS that

or

(1)

R E Verrecchaa, The Mayer*Race

92

conlecture

A counterexample

Although Mayers and Rice discuss then conjecture wlthm the context of the capital asset pricing model, we interpret eq (1) more broadly as a mathematical, or logical, representation of the expected return conjecture That is, eq. (1) asserts that, on the basis of an equlhbrmm model of the market, the rnformed investor will on average achieve a greater rate of return than that anticipated by the consensus belief We use eq (1) to examine the validity of the expected return conjecture without restricting ourselves to assumptions inherent m the capital asset pricing model, such as the one requiring use of the quadratic utility function

4. Some preliminary results The analysis will demonstrate that the expected return conJecture 1s valid for the exponential and logarlthmlc utility functions, but not generally true for the quadratic Prehmmary to this, we mtroduce a lemma This lemma 1s very mechanical and simply reduces (1) to a more convenient form Lemma

1

A suflcient

condltlon that (1) hold IS that for all messages

and for some messages k, 1E 9 the inequality

J, 1~9,

rn (2) 1s strict ’

Proof Because the informed mvestor’s utility function exhibits nonsatlatlon (since U is strictly monotonically increasing), the denominator &o,Yi, will be equal to the informed investor’s endowed wealth for all messages JEY (In effect, he will allocate all of his endowed wealth to future consumption ) Consequently, the denominator 1s invariant across messages Thus, we focus exclusively on the sign of the numerator m (1) Recall that by the assumption of Joint rationality, for any (dummy) index 1 and for every s E Jy,

Henceforth, we drop the ‘I’ superscript and subscript from the expressions $ 5We note that Lemma 1 IS also a necessary condltlon m the sense that If there exists some message pair J, reY such that the expresslon xb, (nj-z:)( Y: - Y;) IS negative, a counterexample to the Mayers-Rice conjecture IS easdy constructed as follows Select a q such that qk =e/(L-2) for k # J, I, and qt= (l-&)/2 for k=~. I Then q,qr approaches zero as E becomes Thus, from the relatIonshIp small for all k, [E P, except q,q,, which approaches one-fourth expressed m (3), the Mayers-Rice conjecture IS false

R E Verrecchta, The Mayers-Rice

conlecture

A counterexample

93

and Y;,, respectively, as it IS clear that both relate to the informed investor. Thus,

s=l

J=ll=l

= i J=ln=~+l

f

qJq,

2 (n;-n:)(Y:-Y:)

s=1

(3) Recall that by definition q,>O for all J E 49, thus, a sufficient condltlon that the expression on the right-hand side of (3) be positive IS that for all J, ZEN,

and for some k, IE 14 the mequahty 1s strict This completes the proof Before proceeding, we require some addltlonal notation Let RN, designate the N-fold Cartesian product of the posltlve orthant, and let Int (F) designate the interior of the N- 1 dimensional simplex F= {n= (?rl, ,%)I+,=1 and 1~~20, s=l, 9N) The portfolio an informed investor chooses on the basis of some message can be interpreted as a function whose domam 1s Int (F), and whose range 1s Rr (We assume that each message implies a portfolio m which the informed investor contracts for some consumption m each state In effect, this limits the discussion to equlhbrla m the mterlor of Rr A necessary, but not suffklent, condltlon that equlhbrla be m the interior of Ry IS that the informed investor associates some posltlve probability to the occurrence of every state, regardless of the message he receives This 1s the reason why we require that each message imply a probablhty assignment which IS m the interior of F ) That is, the vector Y= {Y,, , Y,} represents a vector of functions x, each of which maps elements of 1r~1nt (F) mto the posltlve-half real lme through the requirement that there exist J > 0 such that for all s E N,

v’(r,(n))n,=Aw,, D

(4)

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R.E Verrecchta, The Mayers-Rice

conlecture

A counterexample

and (5) where W designates the (endowed) wealth of the Informed investor Now, we define a function f which maps elements of Int (F) x Int (F) mto the real lme by the expression

f(n,fo= f (%-mr,(~)-s=l

W’))

To show that the relatlonshlp m eq (1) holds, we must demonstrate that for any fixed, arbitrary z’~Int (F), f 1s non-negative for all z lint (F), thus, f IS non-negative for any conceivable message pair J, I E 9 Furthermore, we must show that f 1s zero only when all messages imply an identical probablhty assignment, however, because of the Joint rationality assumption dlscussed previously, when all messages imply an ldentlcal probablhty assignment the informed investor always has a belief ldentlcal to the consensus behef Thus, m effect, he 1s not informed Therefore, if an informed investor is truly Informed, there must exist some message pair k, IE 2’ which imply dlstmct probability assignments, this, m turn, implies that f 1s posltlve Thus, the Mayers-Rice conjecture 1s vahdated through Lemma 1, which establishes Its validity whenever f 1snon-negative for all message paw, and positive for at least one pan 5. A proof for the exponential utility function To prove the validity of the expected return conjecture for the exponential utility function, we consider eqs (4) and (5) The exponential utility function has U(Y)= -ceVY/‘, and U’(Y)=e-“‘, where c 1s some posltlve parameter (see table 1) Thus, (4) becomes e-

ydC7r, = Aq,

or x = c ln(n,/lw,),

(6)

where In denotes the natural logarithm For our specific case, (6) rmphes (7)

R E Verrecchaa, The Mayers-Rice

conJecture

A counterexample

95

and K(i) = c ln(n:/l’o,),

(8)

where 1 and A’ are the (possibly) distinct expected margmal utlhtles lmphed by the (possibly) distinct probablhty assignments 1c and a’ Note, however, that whichever probability assignment the informed Investor makes, prices remain the same This is implied by the Mayers-Rice assumption that the informed investor has ‘zero weight’ m the market Substltutmg for (7) and (8) m the defimtlon forf, we have f(7c, i) = t (R, - 7ci){cln(n$Ao,) - c ln(n~A’w,)} s=l N =c

,sl

(ns -

n:)(ln(7r,) - In(A)- ln(o,) - ln(7c:)+ ln(nl) + ln(o,)}

= c F (R, - 7t:)ln(7@:) + c ln(nl/A) 5 (IL,- n:) = c F (xs - 7rl)ln(a&), S=l

(9)

since Z, II’E Int(F) lmphes that Z,“=I (II, - $) =0 Furthermore, for each s E .,V, if 7~~2~1, then this implies ln(n&) 20, and, slmdarly, if n, < ~1, then ln(n&) < 0 Therefore, for each s E JV, we have

consequently, the function f as expressed m (9) 1s non-negative Finally, we observe that f 1s zero if and only if 7c,= n: for all s E JV, 1 e , the probability assignments are identical From our previous dlscusslon, this establishes through Lemma 1 the validity of the expected return conjecture for the exponential utility function 6. A proof for the logarithmic utility function The proof for the logarlthmlc utility function 1s similar to the previous one The logarlthmlc utlhty function has U( Y)=ln( Y - b), and U’(Y)= (Y-b)-‘, where b 1s some posltlve parameter considered to be a subsistence level, and such that Y > b for all Y (see table 1) Eq (4) implies

or y, = b + (n,/ko,)

(10)

R E Verrecchza, The Mayers-Rice

96

conJecture

A counterexample

Table 1 Types of utdlty functions under conslderatlon

of

and their charactenstlcs

Type utd1ty function

Domam restrlctlon

Functional representation

FunctIonal representation of margmal utdlty

FunctIonal representation of lmphed demand m state s at eqmhbrmm

Exponential

None

-ce-‘1;

e-Y/C

c In (7rJk.0~)

L.ogarlthmlc

Y>b

In(Y-b)

(Y-b)-’

b + (d=w

)

QuadratIc

Y
bY-)Y2

b-Y

b - W/z,

)

For convemence, we assume that prices are properly normalized such that E,“=1o, = 1 Then, multlplymg both sides of (10) by o,, and summing over s, we have

2 o,y,= 5 o,b+

s=l

a=1

since C,“=1 or,= 1 whenever implies from (11) that

;

(a,/rZ)=b+(l/A),

(11)

S=l

x E Int(F)

Furthermore,

X,“=1 o,Y, = W which

2=(W-by’

(12)

Thus, for our specific case, (10) and (12) imply

K(~)=~+w~,)(~--b),

(13)

r,(lcz)=b+(n~cos)(W-b)

(14)

Note that (12) implies that the logarlthmlc utility function yields the special case of constant expected marginal utlhty This 1s because the informed investor’s endowed wealth, W IS mvarlant with respect to which message he receives, since receipt of the various messages has no effect on prices, once again, the latter 1s implied by the Mayers-Rice ‘zero weight’ assumption Substltutmg for (13) and (14) m the defimtlon forf, we have f(a,ff’)=

5 (IL,--~){(b+(IL,/O,)CW--])-((b+(n~~~)[W--])} s=l = $ (RS-W:){(~~--71:)C(W--b/0,1}

s=l

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Verrecchra, The Mayers-Rwe

conjecture

A counterexample

97

Observe that o,>O and x-b >O by assumption, the latter implies W-b > 0 Thus, f, as determined m (15), 1s non-negative, and zero d and only d 71, =‘lt: for all SEX, 1e , the probability assignments are identical As m the case of the exponential, this establishes the validity of the expected return conjecture for the logarlthmlc utility function The fact that the expected return conjecture IS true for two remarkably tractable utility functions should not be interpreted as a suggestion that the conjecture generally holds For example, m the next section we demonstrate with a carefully chosen counterexample that the conjecture 1s false for a utility function regarded to be as tractable as the exponential or logarlthmlc the quadratic 6 7. A counterexample to the conjecture For the quadratic utility function, U(Y)= U’(Y)=b-

bY -3 Yz and

Y,

where b IS some arbitrarily large positive parameter considered to be a goal level, and such that b 1s always greater than Y (see table 1) This implies, from eq (4), that for all s E N, (b - Y,)a, = IIq,

or (16)

r,=b-@q/n,)

However, since ICf=1 co, Y, = W from (5), (16) yields

w=

5 o,y,= s=l

f s=1

o,b-A

5

(o,z/nn,)

(17)

s=l

We choose (0, , s = 1, , IV} such that prices are properly normahzed, that IS, o, > 0 for all SE N, and IS:=1 co,= 1 Then, C:=, o,b = b and (17) implies

or, substltutmg for 2 m (16),

“On the basis of a more elaborate analysis of this problem, the detads of which are not Included m this manuscript, we posit that the results for the exponential and logarlthmlc utdlty functions are exceptlonal That IS, m most cucumstances, and for most utd:ty functions, counterexamples can be constructed However, thts should bc welghed agamst the fact that counterexamples must be very carefully conitructed, which suggests that they may represent a pathology1 See our remarks m the concludmg sectlon of thts manuscript

98

RB

Verrecchaa, The Mayers-Rice

coyecture

A counterexample

We show there exists a circumstance m which

or, equivalently, that

Suppose that the informed investor has access to two messages, message one and message two, and the receipt of each 1s an equally likely event, (1e , q, = q2 =3). We assume there are three possible states of the world’ (1e , N = 3) When the Informed investor receives message one, he assigns the followmg probablhtles to each of the three states n1 = {l/4,5/8,1/8} When the informed investor receives message two, he asslgns the following probablhtles to each of the three states: 1~’= {l/8,3/8,1/2} This lmphes that under the assumption of Joint ratlonahty, n” = (3/16,1/2,5/16} We select for prices Wt = 0 4995,

01 =o 4995,

w,=OOOl

Let us further suppose that our particular informed mdlvldual has W=3

and

b=4

m his utlhty function Then, by (18), given message one, he will select (07)2+(049$5)”

r; =4-(4-3)

I (000;)~}-‘~~;~5}

{ =2 570 Similarly, Y:=3428

and

Y:=3994

‘An mterestmg pomt IS the fact that the expected return conjecture concave utlhty funcbon whenever there are only two states of the world prove this, but It Seems to have no slgndicance

IS always vahd for any It IS an easy exercise to

R.E Verrecchzo, The MayersRace

Given

conlecture

A counterexample

99

message two, from (18) again, r: = 2.4985,

Now, the informed

Y; = 3.4995,

mdlvldual

E’(R,)=f(3[*(2

expects his return

570)+3(3

+3[$(2

r;=3999

428)+$(3

4985)+&3

4995)+*(3

to be, on average,

994)] 999)]} - 1

=o 1514 But, an unmformed

mdlvldual

would expect the return

3 428;34995

2570;24985

5 +iz (

3944+3999 2

to be

-1 >}

=0 1520 Thus, E'(R,)-E"(R,)
8. Conclusion The expected return conjecture IS a subtle proposltlon which should not be dlsmlssed as self-evident It 1s valid for two popular, and remarkably wellbehaved, types of utlhty functions, the exponential and logarlthmlc But the conjecture does not hold m general, as demonstrated m the case of the quadratic utility function On the other hand, we should not overstate our results Even for the quadratic utility function, the Mayers-Rice conJecture 1s correct much of the time That is, counterexamples exist, but they must be very carefully constructed, like the one above, and it IS unclear that they correspond to any mformatlon structure actually exlstmg m the market On the other hand, since we do not have empirical evidence on mformatlon structures, these anomalies may be very important To insure against errors m assessing these informed investors as inferior, it seems necessary to look at parameters of the return dlstrlbutlon other than the mean Unfortunately, comparing returns dlstrlbutlons then may require

100

R E Verrecchua, The Mayers-Race conlecture

A counterexample

knowledge of utlhty functions Such knowledge 1s nearly lmposslble to obtam Thus, measuring excess (over unmformed) average rates of return may be the best we can practically do Mayers and Rice show this will be valid for investors with quadratic utlhty under certam circumstances We have shown that it will always be valid for Investors with logarlthmlc or exponential utility The care necessary to find a counterexample to the conjecture furthkr reinforces its reasonableness References Cornell, Bradford, 1979, AsymmetrIc mformatlon and portfoho performance measurement, Journal of Fmanclal EconomIca, forthcoming Flemmg, Wendell H , 1965, Functions of several variables (Addison-Wesley, Readmg, MA) Mayers, David and Edward Rice, 1979, Portfoho performance, residual analysis and capital asset prlcmg model tests, Journal of Fmanclal Economics 7, 3-28 Roll, Richard, 1977, A crltlque of the asset prlcmg theory tests, Part I On past and potential testabdlty of the theory, Journal of Fmanclal Economics 4, 129-176 Roll, Richard, 1978, Amblgmty when performance IS measured by the securmes market Ime, Journal of Fmance 33, 1051-1067 Verrecchla, Robert E, 1979, A proof of the existence of ‘consensus behefs’, Journal of Fmance 34,957-963