On the diversity of equity markets

Journal of Mathematical Economics 31 Ž1999. 393–417

On the diversity of equity markets Robert Fernholz

)

INTECH, 1 Palmer Square, Suite 303, Princeton, NJ 08540, USA Received 1 May 1997

Abstract This is a study of the diversity of the distribution of capital in an equity market composed of stocks represented by continuous semimartingales. It is shown that for such a market, diversity is not a natural state and some mechanism such as dividend payments is needed to maintain it. Entropy is introduced as a measure of diversity and is used to study the compatibility of market diversity with capital market equilibrium. It is shown that dividend payments must be available to recycle capital from larger companies to smaller companies in order to maintain market diversity under conditions of capital market equilibrium. q 1999 Elsevier Science S.A. All rights reserved. JEL classification: G11; C62 Keywords: Market diversity; Market entropy; Stochastic portfolio theory; Capital market equilibrium

1. Introduction Diversity in the distribution of capital is a prominent characteristic of equity markets; not all the capital is concentrated in a few companies. This is not surprising; Harry Markowitz’s work ŽMarkowitz, 1952. on portfolio theory shows that risk averse investors will hold diversified portfolios and William Sharpe’s work ŽSharpe, 1964. on market equilibrium shows that this will lead to a diverse equity market. Market diversity is a weak stability condition compatible with forms of stationary economic equilibria that have been studied by Grandmont and Hildenbrand Ž1974. and Hellwig Ž1980.. Market diversity allows for shifts of )

Tel.: q1-609-4970442; fax: q1-609-4970441; e-mail: [email protected]

0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 1 8 - 9

394

R. Fernholzr Journal of Mathematical Economics 31 (1999) 393–417

capital from company to company and from economic sector to economic sector, as long as concentration of all the capital in just a few companies is avoided. Diversity appears to be natural in actual equity markets; we shall investigate what is needed to ensure it in hypothetical markets. We shall use a model of stock price processes represented by continuous semimartingales which is fairly standard in continuous-time financial theory Že.g. Karatzas and Shreve, 1991; Duffie, 1992.. We shall consider equity markets composed of such stocks and shall study how market diversity evolves under various hypotheses. We shall show that diversity is not a natural state for such markets and that some mechanism is required to maintain it. We shall introduce entropy as a measurement of market diversity and use it to derive conditions under which market diversity is consistent with capital market equilibrium in the sense of Sharpe Ž1964.. We shall prove that under conditions of capital market equilibrium, capital must be recycled from larger companies to smaller companies, e.g. through dividend payments, in order to maintain market diversity. The theory developed in this paper is applied specifically to equity markets, however it can be modified in a natural manner to apply in other settings such as national wealth distribution or international capital distribution. Throughout this paper we shall use continuous semimartingales to model stock prices and we shall make certain simplifying assumptions. Among these assumptions are: 1. Companies do not merge or break up and the total number of shares of a company remains constant. The list of companies in the market is fixed. 2. Dividends are paid continuously rather than discretely. 3. There are no transaction costs, taxes, or problems with the indivisibility of shares. These assumptions are made for clarity of exposition. The theory presented here can be generalized naturally to include alternative stochastic processes, discrete dividends, changing equity markets and so forth. In particular, jump processes are deliberately excluded because the generality gained would be minimal and none of the results would be significantly affected.

2. Stochastic portfolio theory In this section we develop some of the mathematical machinery we shall need in the later sections. We shall generally follow the definitions and notation used in Karatzas and Shreve Ž1991., some of which we shall review now. Let W s  W Ž t . s Ž W1 Ž t . , . . . , Wn Ž t . . , Ft , t g w 0, ` . 4 be a standard n-dimensional Brownian motion defined on a probability space  V , F , P 4 where  Ft 4 is the augmentation under P of the natural filtration  Ft W s s ŽW Ž s .; 0 F s F t .4 . We say that a process  X Ž t ., Ft , t g w0, `.4 is adapted if

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X Ž t . is Ft-measurable for t g w0, `.. If X and Y are processes defined on  V , F , P 4 , we shall use the notation X s Y if P  X Ž t . s Y Ž t . , t g w 0, ` . 4 s 1. For continuous, square-integrable martingales  M Ž t ., Ft , t g w0, `.4 and  N Ž t ., Ft , t g w0, `.4 , we can define the cross-Õariation process ² M, N :. The crossvariation process is adapted, continuous and of locally bounded variation and the operation ² P , P: is bilinear on the real vector space of continuous, square-integrable martingales. If M s N, we shall use the notation ² M : s ² M, M :; ² M : is called the quadratic Õariation process of M and has continuous, nondecreasing sample paths. The Brownian motion process defined above is a continuous, square-integrable martingale and it is characterized by its cross-variation processes t g w 0, ` . ,

²Wi , Wj :t s d i j t ,

where d i j s 1 if i s j, and 0 otherwise. A continuous semimartingale X s  X Ž t ., Ft , t g w0, `.4 is a measurable, adapted process which has the decomposition, P a.s. t g w 0, ` . ,

X Ž t . s X Ž 0 . q M X Ž t . q VX Ž t . ,

where  M X Ž t ., Ft , t g w0, `.4 is a continuous, square-integrable martingale and  VX Ž t ., Ft , t g w0, `.4 is a continuous, adapted process of locally bounded variation. It can be shown that this decomposition is a.s. unique Žsee Karatzas and Shreve, 1991., so we can define the cross-variation process for continuous semimartingales X and Y by ² X , Y : s ² MX , MY :, where M X and M Y are the martingale parts of X and Y, respectively. Definition 2.1. Let X 0 be a positive number. A stock X s  X Ž t ., Ft , t g w0, `.4 is a process of the form X Ž t . s X 0 exp

ž

t

t

n

H0 g Ž s . d s qH0 Ý j Ž s . dW Ž s . n

n s1

n

/

,

t g w 0, ` . ,

Ž 1.

where g s g Ž t ., Ft , t g w0, `.4 is measurable, adapted, and satisfies H0t < g Ž s .
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so there is no loss of generality in assuming a single share outstanding. The process g is called the growth rate Ž process . of X and, for each n , the process, jn , represents the sensitivity of X to the n th source of uncertainty, Wn . We shall find it convenient to use a logarithmic representation for stocks which follows directly from Definition 2.1 and is more natural when considering long-term behavior Žsee Fernholz and Shay, 1982.. Eq. Ž1. is equivalent to log X Ž t . s log X 0 q

t

t

n

H0 g Ž s . d s qH0 Ý j Ž s . dW Ž s . , n

n

n s1

or, in differential form, n

d log X Ž t . s g Ž t . d t q Ý jn Ž t . dWn Ž t . .

Ž 2.

ns1

It follows that log X Ž t . is a continuous semimartingale with bounded variation part t

H0 g Ž s . d s and martingale part t

n

H0 Ý j Ž s . dW Ž s . . n

n

n s1

By Ito’s ˆ Lemma Žsee Itoˆ Ž1951. or Karatzas and Shreve Ž1991.., d X Ž t . s X Ž t . d log X Ž t . q 12 X Ž t . d²log X :t , where n

d²log X :t s Ý jn2 Ž t . d t , ns1

so X is also a continuous semimartingale. If we define the rate of return Ž process .

a Ž t. sg Ž t. q

1

n

Ý jn2 Ž t . ,

Ž 3.

2 ns1

then we have the classical model for a stock price process with n

d X Ž t. sa Ž t. X Ž t. dtqX Ž t.

Ý jn Ž t . dWn Ž t .

Ž 4.

ns1

Žsee Merton Ž1990. or Karatzas and Shreve Ž1991... Suppose that we have a family of stocks X i , i s 1, . . . , n X i Ž t . s X 0i exp

ž

t

t

n

H0 g Ž s . d s qH0 Ý j i

n s1

in

/

Ž s . dWn Ž s . , t g w 0, ` . .

Ž 5.

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Consider the matrix valued process j defined by j Ž t . s Ž j i n Ž t ..1 F i, n F n and define the coÕariance process s where s Ž t . s j Ž t . j T Ž t .. The cross-variation processes for log X i and log X j are related to s by ²log X i , log X j :t s

t

H0 s

ij

Ž s . d s, t g w 0, ` . .

Ž 6.

Definition 2.2. A market is a family M s  X i , . . . , X n 4 of stocks, each defined as in Eq. Ž5., for which there is a number ´ ) 0 such that xs Ž t . xT G ´ 5 x 5 2 ,

x g R n , t g w 0, ` . , a.s.

Ž 7.

Since the processes j i n are assumed to be bounded in w0, `., the same will hold for si j . Hence there exists a positive number M such that xs Ž t . xT F M 5 x 5 2 ,

x g R n , t g w 0, ` . , a.s.

Ž 8.

Definition 2.3. Let M be a market of n stocks. A portfolio in M is a measurable, adapted process p s p Ž t . s Žp 1Ž t ., . . . , pnŽ t .., Ft , t g w0, `.4 such that p Ž t . is bounded on w0, `. = V and t g w 0, ` . , a.s.

p 1 Ž t . q PPP qpn Ž t . s 1,

The processes p i represent the respective proportions, or weights, of each stock in the portfolio. A negative value for p i Ž t . indicates a short sale. Suppose Zp Ž t . represents the value of an investment in p at time t. Then the amount invested in the ith stock X i will be

p i Ž t . Zp Ž t . , so if the price of X i changes by d X i Ž t ., the induced change in the portfolio value will be

p i Ž t . Zp Ž t .

d Xi Ž t . Xi Ž t .

.

Hence the total change in the portfolio value at time t will be n

d Zp Ž t . s Ý p i Ž t . Zp Ž t . is1

d Xi Ž t . Xi Ž t .

,

or, equivalently, d Zp Ž t . Zp Ž t .

n

s Ý pi Ž t. is1

d Xi Ž t . Xi Ž t .

.

Ž 9.

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We wish to study the nature of solutions to Eq. Ž9.. For background regarding solutions to stochastic differential equations, see Karatzas and Shreve Ž1991... Proposition 2.1. Let p be a portfolio and let n

gp Ž t . s Ý p i Ž t . g i Ž t . q gp) Ž t . ,

Ž 10 .

is1

where

gp) Ž t . s

1 2

n

ž

n

Ý p i Ž t . si i Ž t . y Ý is1

/

p i Ž t . p j Ž t . si j Ž t . .

i , js1

Ž 11 .

Then, for any positiÕe initial Õalue Z0 p , the process Zp defined by Zp Ž t . s Zp0 exp

t

žH

gp Ž s . d s q

0

t

n

H0 Ý p Ž s . j i

in

/

Ž s . dWn Ž s . , t g w 0, ` . ,

i , n s1

Ž 12 . is a strong solution of Eq. Ž9.. Proof. The process Zp defined by Eq. Ž12. is adapted, Zp Ž t . ) 0 for all t g w0, `., a.s., and Zp Ž0. s Zp0 . We must check if Zp Ž t . satisfies Eq. Ž9.. It follows immediately from Eq. Ž12. that n

d log Zp Ž t . s gp Ž t . d t q

p i Ž t . j i n Ž t . dWn Ž t . .

Ý i , n s1

The properties of gp , g i , p i and j i n imply that log Zp is a continuous semimartingale. By Ito’s ˆ Lemma, d Zp Ž t . s Zp Ž t . d log Zp Ž t . q 12 Zp Ž t . d²log Zp :t , so d Zp Ž t . Zp Ž t .

n

s gp Ž t . d t q 12 d²log Zp :t q

Ý

p i Ž t . j i n Ž t . dWn Ž t . .

i , n s1

Now, n

d²log Zp :t s

Ý

n

p i Ž t . p j Ž t . d²log X i , log X j :t s

i , js1

Ý

p i Ž t . p j Ž t . si j Ž t . d t ,

i , js1

by Eq. Ž6.. Since by definition, n

gp Ž t . s Ý p i Ž t . g i Ž t . q is1

1 2

n

ž

n

Ý p i Ž t . si i Ž t . y Ý is1

i , js1

/

p i Ž t . p j Ž t . si j Ž t . ,

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therefore, d Zp Ž t . Zp Ž t .

n

s Ý p i Ž t . gi Ž t . d t q is1

1

n

Ý p i Ž t . si i Ž t . 2 is1

n

q

Ý

p i Ž t . j i n Ž t . dWn Ž t .

i , n s1 n

s Ý p i Ž t . ai Ž t . d t is1 n

q

p i Ž t . j i n Ž t . dWn Ž t . ,

Ý

Ž 13 .

i , n s1

where a i Ž t . is defined as in Eq. Ž3.. Eq. Ž13. is equivalent to Eq. Ž9. with each X i Ž t . expressed in the form Eq. Ž4.. I Corollary 2.1. Let p be a portfolio and Zp be its Õalue process. Then n

d log Zp Ž t . s Ý p i Ž t . d log X i Ž t . q gp) Ž t . d t.

Ž 14 .

is1

Proof. The proof follows immediately from Proposition 2.1 and Eq. Ž2..

I

The process gp in Eq. Ž10. is called the portfolio growth rate Ž process . of the portfolio p and gp) in Eq. Ž11. is called the excess growth rate Ž process .. If we let spp be the process defined by

spp Ž t . s p Ž t . s Ž t . p T Ž t . ,

Ž 15 .

then ²log Zp :t s

t

H0 s

pp

Ž s . d s.

Eq. Ž11. is equivalent to

gp) Ž t . s

1 2

n

žÝ

is1

/

p i Ž t . si i Ž t . y sp p Ž t . .

Ž 16 .

For each i, si i Ž t . represents the rate of variation of log X i Ž t . and sp p Ž t . represents the rate of variation log Zp Ž t ., so gp) Ž t . is half the difference between the weighted average rate of variation of the stock prices and the rate of variation of the portfolio value. We shall show that gp) Ž t . is positive for portfolios which hold more than a single stock and have no short sales, so, heuristically, gp) Ž t . can be regarded as a measure of the efficacy of portfolio diversification in reducing the volatility of Zp compared to that of its component stocks. This suggests that

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gp) Ž t . might be used as an indicator of market diversity and we shall investigate this possibility in Section 3. The portfolio growth rate gp Ž t . characterizes the long-term behavior of Zp Ž t .. Proposition 2.2. Let p be a portfolio with growth rate gp . Then

½

P lim ty1 log Ž Zp Ž t . rZp0 . y

ž

t™`

t

H0 g

p

/ 5

Ž s . d s s 0 s 1.

Ž 17 .

Proof. By Proposition 2.1 log Ž Zp Ž t . rZp0 . s

t

t

H0 g

p Ž s . d s qH

n

Ý

0 i , n s1

p i Ž s . j i n Ž s . dWn Ž s . .

Ž 18 .

Let ÕŽ t. s

t

H0 s

pp

Ž s. d s

where spp is defined in Eq. Ž15.. The components of p Ž t . sum to 1 and p is bounded on w0, `. = V , so ny1 F 5p Ž t . 5 2 F N,

t g w 0, ` . ,

for some number N. It follows from Eqs. Ž7. and Ž8. that there exist numbers ´ , M ) 0 such that

´ ny1 F spp Ž t . F MN,

t g w 0, ` . , a.s.

Therefore, Õ Ž t . is continuous and strictly increasing with

Ž ´rn . t F Õ Ž t . F NMt, t g w 0, ` . , a.s., so Õ has a continuous inverse function Õy1 defined on w0, `. such that 0 F Õy1 Ž t . F Ž nr´ . t ,

t g w 0, ` . , a.s.

Let B be the process defined by BŽ t. s

H0Õ

y1 Ž

t.

n

Ý

p i Ž s . j i n Ž s . dWn Ž s . ,

t g w 0, ` . .

Ž 19 .

i , n s1

Then B is a continuous, square-integrable martingale with ² B :t s t ,

t g w 0, ` . ,

and it follows that B is a Brownian motion process with initial value B Ž0. s 0 Žsee Karatzas and Shreve, 1991.. From Eqs. Ž18. and Ž19. we have B Ž Õ Ž t . . s log Ž Zp Ž t . rZp0 . y

t

H0 g

p

Ž s . d s.

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401

The strong law of large numbers for Brownian motion Žsee Karatzas and Shreve, 1991. states that P lim ty1 B Ž t . s 0 s 1,

½

5

t™`

so, since Õ Ž t . ™ ` as t ™ ` almost surely,

½

P lim t™`

1 ÕŽ t.

ž

log Ž Zp Ž t . rZp0 . y

t

H0 g

/ 5

p

Ž s . d s s 0 s 1,

p

Ž s . d s s 0 s 1.

Since 0 - Õ Ž t . - MNt,

½

P lim t™`

1 MNt

ž

log Ž Zp Ž t . rZp0 . y

t

H0 g

/ 5

Ž 20 .

and the proposition follows. I Corollary 2.2. Let X be a stock with growth rate g . Then

½

ž

P lim ty1 log Ž X Ž t . rX 0 . y t™`

t

H0 g Ž s . d s

/ 5

s 0 s 1.

Ž 21 .

Proof. Apply Proposition 2.2 to a portfolio with initial value X 0 in which the weight corresponding to X is 1 and all the other weights are 0. I For any stock X i and portfolio p we can consider the quotient process X irZp defined by log Ž X i Ž t . rZp Ž t . . s log X i Ž t . y log Zp Ž t . .

Ž 22 .

This process is a continuous semimartingale with ²log Ž X irZp . , log Ž X jrZp . :t s ²log X i , log X j :t y ²log X i , log Zp :t y ²log X j , log Zp :t q ²log Zp :t .

Ž 23 .

If we define the process sip by n

s ip Ž t . s Ý p j Ž t . s i j Ž t . , js1

for i s 1, . . . , n, then ²log X i , log Zp :t s

t

H0 s

ip

Ž s . d s.

Define the relatiÕe coÕariance Ž process . t p to be the matrix valued process

t p Ž t . s Ž t ipj Ž t . . 1Fi , jFn ,

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402

where

t ipj Ž t . s si j Ž t . y sip Ž t . y sjp Ž t . q sp p Ž t . ,

Ž 24 .

for i, j s 1, . . . , n. Then for all i and j, ²log Ž X irZp . , log Ž X jrZp . :t s

t p

H0 t

ij

Ž s . d s.

Ž 25 .

In the case that i s j, we know that ²logŽ X irZp .:t is non-decreasing, so

t ipi Ž t . G 0, t g w 0, ` . , a.s. Let h s h Ž t . s Žh1Ž t ., . . . , hnŽ t .., Ft , t g w0, `.4 be a portfolio. Then the relatiÕe Õariance Ž process . of h and p is defined by T

p thh Ž t . s Ž h Ž t . y p Ž t . . s Ž t . Ž h Ž t . y p Ž t . . s h Ž t . t p Ž t . hT Ž t . .

Ž 26 . From Eq. Ž26. and the condition in Eq. Ž7., it follows that

h Ž t . t p Ž t . hT Ž t . s 0 if and only if h Ž t . s p Ž t ., so rank t p Ž t . s n y 1, a.s., with the null space of t p Ž t . spanned by p Ž t .. The following lemmas relate the value of the excess growth rate of a portfolio to the values of its weights, and will be useful later. They also show that the excess growth rate of a portfolio with no short sales is non-negative, and is positive if it has positive weights in more than one stock. Lemma 2.1. Let p and h be portfolios. Then 1 n gh) Ž t . s Ý h Ž t . t ipi Ž t . y thhp Ž t . . 2 is1 i

ž

/

Proof. The proof is a direct calculation using Eqs. Ž24. and Ž26.. Lemma 2.2. Let p be a portfolio. Then 1 n gp) Ž t . s Ý p i Ž t . t ipi Ž t . . 2 is1

Ž 27 . I

Ž 28 .

Proof. Let h in Lemma 2.1 be the same portfolio as p . Since p Ž t .t p Ž t .p T Ž t . s 0, Eq. Ž27. reduces to Eq. Ž28.. I Lemma 2.3. Let p be a portfolio such that for i s 1, . . . , n, p i Ž t . G 0, for all t g w0, `.. Then

gp) Ž t . G 0,

t g w 0, ` . , a.s.

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Proof. Since t ipi Ž t . G 0, t g w0, `., a.s., for all i, all of the terms on the right hand side of Eq. Ž28. will be non-negative, a.s. I Lemma 2.4. Let p be a portfolio such that for i s 1, . . . , n, 0 F p i Ž t . - 1, for all t g w0, `.. Then there exists a number ´ ) 0 such that for i s 1, . . . , n,

gp) Ž t . G ´ Ž p i Ž t . y p i2 Ž t . . ,

t g w 0, ` . , a.s.

Ž 29 .

Proof. Choose an integer k,1 F k F n. Since p k Ž t . - 1, we can define

hi Ž t . s

½

p i Ž t . r Ž1 y p k Ž t . . 0

if i / k , if i s k ,

Ž 30 .

for t g w0, `., i s 1, . . . , n. Then Žh1Ž t ., . . . , hnŽ t .. defines a portfolio h with non-negative weights. Now n

n

2gp) Ž t . s Ý p i Ž t . si i Ž t . y is1

Ý

p i Ž t . p j Ž t . si j Ž t .

i , js1 n

s p k Ž t . s k k Ž t . q Ž 1 y p k Ž t . . Ý hi Ž t . si i Ž t . y p k2 Ž t . s k k Ž t . is1 n

y 2p k Ž t . Ž 1 y p k Ž t . . Ý hi Ž t . si k Ž t . is1

y Ž1 y p k Ž t . .

n

2

Ý hi Ž t . hj Ž t . si j Ž t . i , js1

s Ž p k Ž t . y p k2 Ž t . . Ž s k k Ž t . y 2 s kh Ž t . q shh Ž t . . n

q Ž1 y p k Ž t . .

ž

Ý hi Ž t . si i Ž t . y shh Ž t . is1

/

G Ž p k Ž t . y p k2 Ž t . . Ž s k k Ž t . y 2 s kh Ž t . q shh Ž t . . ,

Ž 31 .

since 1 y p k Ž t . ) 0 and n

Ý hi Ž t . si i Ž t . y shh Ž t . s 2gh) Ž t . G 0, is1

by Lemma 2.3. Let ´ X ) 0 be chosen such that xs Ž t . xT G ´X 5 x 5 2 ,

x g R n , t g w 0, ` . , a.s.

as in Definition 2.2. If we let x s Ž h1 Ž t . , . . . , y1, . . . , hn Ž t . . ,

Ž 32 .

404

R. Fernholzr Journal of Mathematical Economics 31 (1999) 393–417

with the y1 in the k th position, then

s k k Ž t . y 2 s kh Ž t . q shh Ž t . s x s Ž t . x T G ´ X 5 x 5 2 ,

t g w 0, ` . , a.s.

Since 5 x 5 2 G 1, it follows that 2gp) Ž t . G ´ X Ž p k Ž t . y p k2 Ž t . . ,

t g w 0, ` . , a.s.

Since k, 1 F k F n was arbitrary, Eq. Ž29. follows with ´ s ´ Xr2.

I

Lemma 2.5. Let p be a portfolio such that for i s 1, . . . , n, 0 F p i Ž t . - 1, for all t g w0, `.. Then there exists a number ´ ) 0 such that for i s 1, . . . , n,

p i Ž t . F 1 y ´gp) Ž t . ,

t g w 0, ` . , a.s.

Ž 33 .

Proof. Let M ) 0 be chosen such that xs Ž t . xT F M 5 x 5 2 ,

x g R n , t g w 0, ` . , a.s.,

Ž 34 .

as in Eq. Ž8.. It follows that for i s 1, . . . , n, si i Ž t . F M, t g w0, `., a.s. Choose k, 1 F k F n and let h be defined as in Eq. Ž30.. Then n

2gh) Ž t . s Ý hi Ž t . si i Ž t . y shh Ž t . is1 n

F Ý hi Ž t . si i Ž t . F M ,

t g w 0, ` . , a.s.

Ž 35 .

is1

Let x be defined as in Eq. Ž32.. Then 5 x 5 2 F 2, so for k s 1, . . . , n,

s k k Ž t . y 2 s kh Ž t . q shh Ž t . s x s Ž t . x T F 2 M ,

t g w 0, ` . , a.s.

Ž 36 .

From Eq. Ž31., we have for k s 1, . . . , n, 2gp) Ž t . s Ž p k Ž t . y p k2 Ž t . . Ž s k k Ž t . y 2 s kh Ž t . q shh Ž t . . q 2 Ž 1 y p k t . . gh) Ž t . F Ž1 y p k Ž t . . Ž 2 M q M .

t g w 0, ` . , a.s.,

by Eqs. Ž35. and Ž36., and Eq. Ž33. follows with ´ s 2r3M.

I

3. Market diversity Heuristically speaking, a market is ‘diverse’ if the capital is spread among a reasonably large number of stocks. In this section we shall give a formal definition

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of market diversity and then characterize market diversity in terms of the excess growth rate. We shall then use the excess growth rate to study the diversity of a market in which all the stocks are assumed to have the same growth rate. It would naively seem that in such a market, diversity would naturally be maintained, but we shall see that this is not so. We shall introduce dividends into our model and investigate how dividends have the potential to maintain market diversity. We shall assume from now on that the market is M s  X i , . . . , X n4 , with n stocks. Definition 3.1. The portfolio

p s  p Ž t . s Ž p 1 Ž t . , . . . , pn Ž t . . ,

Ft , t g w 0, ` . 4 ,

where

pi Ž t. s

Xi Ž t . X 1 Ž t . q PPP qX n Ž t .

,

Ž 37 .

for i s 1, . . . , n, is called the market portfolio Ž process .. It is clear that the p i defined by Eq. Ž37. satisfy the requirements of Definition 2.3. If we let Z Ž t . s X 1 Ž t . q PPP qX n Ž t . ,

Ž 38 .

it is easily verified that ZŽ t . satisfies Eq. Ž9. with proportions p i Ž t . given by Ž37.. Hence, the value of the market portfolio represents the combined capitalization of all the stocks in the market. From this point on, we shall let p exclusively represent the market portfolio and ZŽ t . in Eq. Ž38. represent its value. We shall use the notation g ) to represent gp) , the excess growth rate process for the market portfolio p and we shall use t i j to represent the ijth component of the relative covariance process t p . Definition 3.2. The market M is diÕerse if there exists a number d ) 0 such that the market portfolio weights satisfy

pi Ž t. F1yd ,

t g w 0, ` . , a.s.,

for i s 1, . . . , n. By this definition, a market is diverse if no single stock accounts for almost the entire market capitalization. This is a fairly weak requirement and it is clear that ‘real’ equity markets are diverse according to this definition. The results of Section 2 allow us to characterize market diversity in terms of the market excess growth rate process g ) Ž t .. Proposition 3.1. The market M is diÕerse if and only if there is a d ) 0 such that

g ) Ž t. Gd ,

t g w 0, ` . , a.s.

Ž 39 .

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Proof. Suppose g ) Ž t . G d X ) 0, t g w0, `., a.s. Choose ´ ) 0 as in Lemma 2.5, then for i s 1, . . . , n,

p i Ž t . F 1 y ´d X ,

t g w 0, ` . , a.s.,

so Definition 3.2 is satisfied with d s ´d X and hence M is diverse. Now suppose M is diverse, so there is a d X ) 0 such that

pi Ž t. F1ydX ,

t g w 0, ` . , a.s.,

for i s 1, . . . , n. Let ´ ) 0 be chosen as in Lemma 2.4. At least one p i Ž t . must be greater than or equal to ny1 , so by Lemma 2.4,

g ) Ž t.

G ´ ny1 Ž 1 y p i Ž t . . , y1 X

G´ n d ,

t g w 0, ` . ,

a.s.,

t g w 0, ` . ,

a.s.,

y1 X

and Eq. Ž39. follows with d s ´ n d .

I

Suppose that all the stocks have the same growth rate. In this case Eq. Ž10. reduces to

gp Ž t . s g Ž t . q g ) Ž t . . Corollary 2.2 states that the long-term behavior of a stock is characterized by its growth rate and since ZŽ t . is the simple sum of X i Ž t ., all of which have the same growth rate g Ž t ., it is apparent that ZŽ t . must in some sense share this common growth rate. It follows that over the long term the contribution of g ) Ž t . must be minimal. Proposition 3.2. Suppose that all the stocks in the market M haÕe the same growth rate process. Then

½

P lim ty1 t™`

t

H0 g

)

5

Ž s . d s s 0 s 1.

Ž 40 .

Proof. Let g be the common growth rate process. By Corollary 2.2,

½

P lim ty1 log Ž X i Ž t . rX 0i . y t™`

ž

t

H0 g Ž s . d s

/ 5

s 0 s 1,

for i s 1, . . . , n. Since lim t ™` ty1 log X 0i s 0 for all i, this is equivalent to

½

ž

P lim ty1 log X i Ž t . y t™`

t

H0 g Ž s . d s

/ 5

s 0 s 1,

for i s 1, . . . , n. Hence,

½

ž

P lim ty1 log min X i Ž t . y t™`

ž

1FiFn

/

t

H0 g Ž s . d s

/ 5

s 0 s 1,

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and the same holds for max 1 F i F n X i Ž t .. Now, n min X i Ž t . F X 1 Ž t . q PPP qX n Ž t . F n max X i Ž t . , 1FiFn

1FiFn

so

ž

/

log n q log min X i Ž t . F log Z Ž t . F log n q log max X i Ž t . . 1FiFn

ž

1FiFn

/

Since lim t ™` ty1 log n s 0, it follows that

½

ž

P lim ty1 log Z Ž t . y t™`

t

H0 g Ž s . d s

/ 5

s 0 s 1.

Ž 41 .

By Proposition 2.2

½

ž

P lim ty1 log Ž Z Ž t . rZ Ž 0 . . y t™`

t

H0 g

p

/ 5

Ž s . d s s 0 s 1.

Since gp Ž t . s g Ž t . q g ) Ž t . and lim t ™` ty1 log ZŽ0. s 0, we have

½

ž

P lim ty1 log Z Ž t . y t™`

t

t

H0 g Ž s . d s yH0 g

Eqs. Ž41. and Ž42. imply Eq. Ž40..

)

/ 5

Ž s . d s s 0 s 1.

Ž 42 .

I

Proposition 3.2 shows that over the long term, the average excess growth rate of this market is vanishingly small. But this has implications regarding the diversity of the market. Corollary 3.1. If all the stocks in the market M haÕe the same growth rate process, then M is not diÕerse. Proof. If M is diverse, Proposition 3.1 implies that there exists a d ) 0 such that g ) Ž t . G d for all t g w0, `.. In this case ty1

t

H0 g

)

Ž s. d s Gd ,

for all t g w0, `.. But this contradicts Proposition 3.2.

I

Corollary 3.2. If all the stocks in the market M haÕe constant growth rate processes, then M is not diÕerse. Proof. Corollary 2.2 implies that all stocks except those that share the highest growth rate process will represent a negligible part of the market value in the long term. But then the Žsub.market composed of the stocks that share the highest growth rate process satisfies the hypotheses of the previous corollary and hence will not be diverse. I

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Corollaries 3.1 and 3.2 show that a common growth rate among the stocks in a market is not sufficient to maintain market diversity. However, we should be able to maintain diversity if we allow companies to redistribute capital in some way, e.g. in the form of dividends. A diÕidend rate Ž process . is a process d s  d Ž t ., Ft , t g w0, `.4 which is measurable, adapted and satisfies H0t < d Ž s .
r Ž t. sg Ž t. qd Ž t. ,

Ž 43 .

where g is the growth rate process of X. Sometimes we shall assume that d Ž t . G 0, but this need not hold in general. If d Ž t . s 0 for all t g w0, `. then r s g . We shall henceforth assume that all stocks have dividend rates, even if in some cases the dividend rate is identically zero. We define the total return process Xˆ for a stock X by Xˆ Ž t . s X Ž t . exp

t

žH

/

d Ž s. d s .

0

Ž 44 .

If d Ž t . ' 0, then Xˆ s X. It follows from Eq. Ž44. that XˆŽ0. s X Ž0. and that d log Xˆ Ž t . s d log X Ž t . q d Ž t . d t. Let M be a market of stocks X 1 , . . . , X n with respective dividend rates d 1 , . . . , dn and total growth rates r 1 , . . . , rn . Suppose h is a portfolio defined in M . Then we define the portfolio diÕidend rate Ž process . dn by n

dh Ž t . s Ý hi Ž t . d i Ž t . ,

t g w 0, ` . ,

is1

and the portfolio total growth rate Ž process . by

rh Ž t . s gh Ž t . q dh Ž t . ,

t g w 0, ` . .

The total return Ž process . Zˆh for h is defined by Zˆh Ž t . s Zh Ž t . exp

t

žH

/

dh Ž s . d s ,

0

so, as for individual stocks above, d log Zˆh Ž t . s d log Zh Ž t . q dh Ž t . d t. The process Zˆh represents the value of a portfolio with the same proportions h1Ž t ., . . . , hnŽ t . as h , but in which all dividends are reinvested proportionally across the entire portfolio according to the weight of each stock. Hence the reinvestment of the dividends modifies the value of Zˆh while preserving the weights of the portfolio h.

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Let us consider now a market in which there are non-negative dividend rates and all the stocks have the same total growth rate. This models a situation in which all the stocks have the same potential for capital growth, but some of the companies elect to distribute part of their capital in the form of dividends rather than reinvesting in themselves. Proposition 3.3. Suppose that all the stocks in the market M haÕe non-negatiÕe diÕidend rates and the same total growth rate process. Then

½

P lim inf ty1 t™`

t

H0 Ž d

p

5

Ž s . y g ) Ž s . . d s G 0 s 1.

Ž 45 .

Proof. Let r be the common total growth rate process. Define the process W by W Ž t . s Xˆ1 Ž t . q PPP qXˆn Ž t . ,

t g w 0, ` . .

W represents the value of a portfolio with W Ž0. s ZŽ0. in which the dividends of each stock are reinvested in the same stock. Since all the Xˆi have the same total growth rate r , the same steps as in the proof of Proposition 2.2 will establish that

½

ž

P lim ty1 log Ž W Ž t . rZ Ž 0 . . y t™`

t

H0 r Ž s . d s

/ 5

s 0 s 1.

Since for all i, X i Ž t . F Xˆi Ž t ., it follows that ZŽ t . F W Ž t . , and hence,

½

P lim inf ty1 t™`

t

žH

/ 5

r Ž s . d s y log Ž Z Ž t . rZ Ž 0 . . G 0 s 1.

0

Ž 46 .

By Proposition 2.2,

½

ž

P lim ty1 log Ž Z Ž t . rZ Ž 0 . . y t™`

t

H0 g

p

/ 5

Ž s . d s s 0 s 1.

Ž 47 .

Eqs. Ž46. and Ž47. imply that

½

P lim inf ty1 t™`

t

H0 Ž r Ž s . y g

Now,

rp Ž t . s r Ž t . q g ) Ž t . and also

rp Ž t . s gp Ž t . q dp Ž t . ,

p

5

Ž s . . d s G 0 s 1.

Ž 48 .

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410

so

r Ž t . y gp Ž t . s dp Ž t . y g ) Ž t . . Therefore, Eq. Ž48. is equivalent to

½

P lim inf ty1 t™`

t

H0 Ž d

p

5

Ž s . y g ) Ž s . . d s G 0 s 1,

and the proposition is proved. I Proposition 3.3 shows that in a market in which the stocks have the same total growth rate, the average dividend rate must at least equal the average excess growth rate of the market over the long term. Since the excess growth rate is related to market diversity by Proposition 3.1, at least some dividends must be paid for a market of this type to remain diverse. Corollary 3.3. Suppose that all the stocks in the market M haÕe non-negatiÕe diÕidend rates and the same total growth rate process. If M is diÕerse, then there exists a d ) 0 such that

½

P lim inf ty1 t™`

t

H0 d

p

5

Ž s . d s G d s 1.

Ž 49 .

Proof. If M is diverse, then by Proposition 3.1 there is a d ) 0 such that g ) Ž t . G d for all t g w0, `., a.s. By Proposition 3.3 0 Flim inf ty1 t™`

Flim inf ty1 t™`

slim inf ty1 t™`

t

H0 Ž d

p

t

H0 Ž d

p

t

H0 d

p

Ž s . y g ) Ž s . . d s, a.s.

Ž s . y d . d s, a.s.

Ž s . d s y d , a.s.,

and the corollary follows. I Although it is not explicit in Corollary 3.3, presumably the dividend rates of the larger stocks must be greater than those of the smaller stocks in order to maintain diversity. We shall study this in more detail in the following sections, where we shall relax the requirement that all the stocks have the same growth rate or total growth rate.

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4. Market entropy In this section we introduce entropy as a measure of market diversity. Entropy is a measure of the uniformity of a distribution; it was first used in thermodynamics and statistical mechanics and more recently has been used as a measure of randomness and information in probability theory and information theory Žsee Shannon, 1948.. Entropy has appeared in the financial and economic literature, but apparently not in the context that we shall study it here Že.g. Cowell, 1980; Majumder and Chakravarty, 1990; Schweizer, 1995; Karatzas and Kou, 1996.. Definition 4.1. Let p be the market portfolio. Then the market entropy (process) S is defined by n

S Ž t . s y Ý p i Ž t . log p i Ž t . . is1

It follows from this definition that S is a continuous semimartingale and that 0 - SŽ t . F log n, for all t g w0, `., a.s. Definition 3.2 provided a criterion by which to determine whether or not a market is diverse; entropy provides a measure of the degree of the diversity in the market. Let us see how these two concepts are related. Proposition 4.1. The market M is diÕerse if and only if there is a number ´ ) 0 such that SŽ t . G ´ ,

t g w 0, ` . , a.s.

Ž 50 .

Proof. The proof is elementary and depends on the continuity of the function n

S Ž x . s y Ý x i log x i is1

when extended to the simplex

D s  x : x 1 q PPP qx n s 1; 0 F x i F 1, i s 1, . . . , n4 ; R n . S is non-negative on the compact set D and SŽ x . s 0 only on the vertices. I We can define a portfolio associated with the market entropy process S. Definition 4.2. Let p be the market portfolio. The portfolio process

h s  h Ž t . s Ž h1 Ž t . , . . . , hn Ž t . . , Ft , t g w 0, ` . 4 , where

hi Ž t . s

yp i Ž t . log p i Ž t . SŽ t .

,

for i s 1, . . . , n, is called the entropy weighted portfolio (process).

Ž 51 .

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412

It can be verified that h satisfies the conditions of Definition 2.3. Note that the ratio of hi Ž t . to p i Ž t . decreases with increasing p i Ž t .. Hence h will be less concentrated then p in those stocks with the greatest market weights. Theorem 4.1. Let p be the market portfolio and h be the entropy weighted portfolio and let Z and Zh be their portfolio Õalue processes, respectiÕely. Then g ) Ž t. d log S Ž t . s d log Ž Zh Ž t . rZ Ž t . . y d t. Ž 52 . SŽ t . Proof. The weight process p i is a quotient process satisfying log p i Ž t . s log Ž X i Ž t . rZ Ž t . . , as in Eq. Ž22.. From Eq. Ž25. it follows that d²log p i , log p j :t s t i j Ž t . d t ,

Ž 53 .

so by Ito’s ˆ Lemma, dp i Ž t . s p i Ž t . d log p i Ž t . q 12 p i Ž t . t i i Ž t . d t.

Ž 54 .

Another application of Ito’s ˆ Lemma gives us d Ž p i Ž t . log p i Ž t . . s p i Ž t . d log p i Ž t . q log p i Ž t . dp i Ž t . q d²p i , log p i :t s dp i Ž t . q log p i Ž t . dp i Ž t . q 12 p i Ž t . t i i Ž t . d t , by Eq. Ž54. and the fact that d²p i , log p i :t s p i Ž t .t i i Ž t .d t. Therefore n

dS Ž t . s y Ý log p i Ž t . dp i Ž t . y g ) Ž t . d t , is1

by Lemma 2.2, since Ý nis1 dp i Ž t . s 0. By Eq. Ž54., we have n

dS Ž t . s y Ý p i Ž t . log p i Ž t . d log p i Ž t . is1

1 y

n

Ý p i Ž t . log p i Ž t . t i i Ž t . d t y g ) Ž t . d t , 2

Ž 55 .

is1

so, dS Ž t . SŽ t .

n

s Ý hi Ž t . d log p i Ž t . q is1

1

n

Ý hi Ž t . t i i Ž t . d t y

2 is1

g ) Ž t. SŽ t .

Now, from Eq. Ž55. it follows that n

d²S:t s

Ý

p i Ž t . p j Ž t . log p i Ž t . log p j Ž t . d²log p i , log p j :t

i , js1 n

s

Ý i , js1

p i Ž t . p j Ž t . log p i Ž t . log p j Ž t . t i j Ž t . d t ,

d t.

Ž 56 .

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413

as in Eq. Ž53.. By Ito’s ˆ Lemma, d log S Ž t . s s

dS Ž t .

1 y

SŽ t . dS Ž t . SŽ t .

2

2S Ž t . 1 y

d²S:t

n

Ý hi Ž t . hj Ž t . t i j Ž t . d t

2 i , js1

n

s Ý hi Ž t . d log p i Ž t . q gh) Ž t . d t y is1

g ) Ž t. SŽ t .

dt,

by Eq. Ž56. and Lemma 2.1. Now n

n

Ý hi Ž t . d log p i Ž t . s Ý hi Ž t . d log X i Ž t . y d log Z Ž t . , is1

is1

and n

d log Zh Ž t . s Ý hi Ž t . d log X i Ž t . q gh) Ž t . d t , is1

by Corollary 2.1, so d log S Ž t . s d log Zh Ž t . y d log Z Ž t . y

g )Ž t. SŽ t .

d t.

I Since market entropy is related to market diversity by Proposition 4.1, Theorem 4.1 now relates the behavior of a portfolio, the entropy weighted portfolio, to market diversity. Corollary 4.1. Let p be the market portfolio and h be the entropy weighted portfolio, and suppose that the market M is diÕerse. Then

½

5

P lim Z Ž t . rZh Ž t . s 0 s 1. t™`

Ž 57 .

Proof. Theorem 4.1 implies that log Ž Zh Ž t . rZ Ž t . . s log Ž Zh Ž 0 . rZ Ž 0 . . q log Ž S Ž t . rS Ž 0 . . q

t

H0

g ) Ž s. SŽ s .

d s.

Now, SŽ t . F log n and by Propositions 3.1 and 4.1 there are d 1 , d 2 ) 0 such that g ) Ž t . G d 1 and SŽ t . G d 2 . Hence, d1 t log Ž Zh Ž t . rZ Ž t . . G log Ž Zh Ž 0 . rZ Ž 0 . . q log d 2 y log n q , log n and this diverges to ` as t ™ `.

I

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414

Corollary 4.2. Let p be the market portfolio and h be the entropy weighted portfolio and let Zˆ and Zˆh be their total return processes, respectiÕely. Then d log Zˆh Ž t . rZˆŽ t . s d log S Ž t . q dh Ž t . y dp Ž t . q

ž

ž

/

g ) Ž t. SŽ t .

/

d t.

Ž 58 .

Proof. The proof follows immediately from Theorem 4.1 and the definition of the total return processes Zˆ and Zˆh . I Corollary 4.2 relates dividend rates and entropy change to the total return processes Zˆ and Zˆh . Capital market equilibrium has certain implications regarding the relative behavior of Zˆ and Zˆh and we shall use this to draw conclusions about dp and dh . 5. Market diversity and CAPM Capital market equilibrium in the form of the capital asset pricing model ŽCAPM. of Sharpe Ž1964. implies that a rational, risk-averse investor will prefer a combination of a riskless asset and the market portfolio to any other combination of riskless asset and stocks. More recent versions of CAPM have similar conclusions Žsee Merton, 1990.. We can use this characterization of market equilibrium to impose conditions on the behavior of the entropy weighted portfolio relative to the market portfolio and then apply the results of the previous section to derive conclusions regarding market diversity. Definition 5.1. Let h and j be portfolios with total return processes Zˆh and Zˆj , respectively. Then h dominates j if there is a t ) 0 such that P Zˆh Ž t . rZˆh Ž 0 . ) Zˆj Ž t . rZˆj Ž 0 . s 1.

½

5

Since there are no trading costs, a rational investor will replace his portfolio with another portfolio which dominates it if he can. This allows us to define a weak form of market equilibrium. Definition 5.2. The market M is weakly in equilibrium if there is no portfolio in M which dominates the market portfolio. Corollary 4.2 relates the value of the entropy weighted portfolio to the value of the market portfolio. If the market is diverse, market entropy is bounded, so over the long term the dominant term on the right hand side of Eq. Ž58. will be

ž

dh Ž t . y dp Ž t . q

g ) Ž t. SŽ t .

/

d t.

Ž 59 .

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415

The total contribution of Eq. Ž59. must be bounded from above, otherwise the entropy weighted portfolio will dominate the market portfolio. This imposes certain conditions on the dividend rates dp and dh . Theorem 5.1. Let p be the market portfolio and h be the entropy weighted portfolio, and suppose that the market M is diÕerse and weakly in equilibrium. Then for any ´ ) 0, there is a T ) 0 such that for t ) T,

½

P ty1

t

t

y1 H p Ž s . y dh Ž s . . d s q ´ ) t

H0 Ž d

g ) Ž s.

0

SŽ s .

5

d s ) 0.

Ž 60 .

Proof. Choose ´ ) 0. Corollary 4.2 implies that log Ž ZˆŽ t . rZˆŽ 0 . . s log Zˆh Ž t . rZˆh Ž 0 . y log Ž S Ž t . rS Ž 0 . .

ž

/

t

H0

q

ž

dp Ž s . y dh Ž s . y

g ) Ž s. SŽ s .

/

d s.

Since M is weakly in equilibrium, for all t ) 0, P log Ž ZˆŽ t . rZˆŽ 0 . . G log Zˆh Ž t . rZˆh Ž 0 .

½

ž

/ 5 ) 0,

so P

t

½ž H0

dp Ž s . y dh Ž s . y

g ) Ž s. SŽ s .

/

5

d s G log Ž S Ž t . rS Ž 0 . . ) 0.

Now, since M is diverse, Proposition 4.1 implies that there is a d ) 0 such that SŽ t . G d , for all t g w0, `., a.s. Therefore, for all t ) 0, log Ž S Ž t . rS Ž 0 . . G log d y log log n,

a.s.

It follows that for all t ) 0, t

½ Hž

P ty1

dp Ž s . y dh Ž s . y

0

g ) Ž s. SŽ s .

/

5

d s G ty1 Ž log d y log log n . ) 0.

If we choose T G ´y1 Žlog log n y log d ., then for t ) T, t

½ Hž y1

P t

0

dp Ž s . y dh Ž s . y

g ) Ž s. SŽ s .

/

5

d s ) y´ ) 0.

I

Since in a diverse market g ) Ž t .rSŽ t . is bounded away from zero, Theorem 5.1 implies that for the market to be weakly in equilibrium, it must be possible for the market portfolio to have a higher average dividend rate then the entropy weighted portfolio. Since the market portfolio is more concentrated in the larger stocks than the entropy weighted portfolio is, this means that under certain circumstances the dividend rates of the larger stocks must exceed those of the smaller stocks. In particular, not all stocks can share the same dividend rate.

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Corollary 5.1. Suppose that the market M is diÕerse and weakly in equilibrium. Then not all the stocks in M haÕe the same diÕidend rate. Proof. If all the stocks in M have the same dividend rate, then dh s dp . Since M is diverse, Proposition 3.1 implies that there exists a d ) 0 such that g ) Ž s. d G , s g w 0, ` . , a.s. SŽ s . log n If dh s dp , then for large enough t, Eq. Ž60. cannot be satisfied and we have a contradiction. I The weak equilibrium condition of Definition 5.2 is sufficient to imply that dividends must exist as a mechanism for maintaining market diversity. However, most versions of CAPM impose stronger equilibrium conditions than Definition 5.2 and with these stronger conditions it is possible to derive quantitative results regarding dividend rates. Example 5.1 (Intertemporal CAPM). See Merton Ž1990., ch. 15. Suppose we have a riskless asset the value of which follows the process X Ž t . s exp

t

žH Ž . /

r s ds ,

t g w 0, ` . ,

0

where r s  r Ž t ., Ft , t g w0, `.4 is called the interest rate Ž process . and is measurable, adapted, and bounded. We can define the rate of return (process) for a portfolio j by sj 2 Ž t . aj Ž t . s rj Ž t . q , Ž 61 . 2 where sj Ž t . s Ž sjj Ž t ..1r2 . This is similar to the rate of return process for a stock defined in Eq. Ž3.. If p is the market portfolio and j is any other portfolio, CAPM implies that in equilibrium, aj Ž t . y r Ž t . ap Ž t . y r Ž t . F , Ž 62 . sj Ž t . sp Ž t . for t g w0, `. Žsee Merton, 1972.. We say that the market is in entropic equilibrium if the process log SŽ t . is a martingale. Under entropic equilibrium, Corollary 4.2 implies that g ) Ž t. rh Ž t . y rp Ž t . s dh Ž t . y dp Ž t . q , Ž 63 . SŽ t . for t g w0, `.. If we combine Eqs. Ž61. – Ž63., we have g ) Ž t. dp Ž t . y dh Ž t . G q 12 Ž sh2 Ž t . y sp2 Ž t . . SŽ t . sh Ž t . y 1y Ž 64 . Ž ap Ž t . y r Ž t . . , sp Ž t .

ž

/

R. Fernholzr Journal of Mathematical Economics 31 (1999) 393–417

417

for t g w0, `.. The terms in Eq. Ž64. are either observable or, with the possible exception of ap Ž t ., can probably be estimated statistically Žsee Malkiel and Xu, 1995.. For theoretical reasons, ap Ž t . y r Ž t . ) 0 Žsee Merton, 1990..

6. Conclusions The results of this paper show that the payment of dividends is an important factor in maintaining the diversity of a market composed of stocks modeled by continuous semimartingales. Even if all the stocks have the same growth rate, some mechanism is needed to maintain market diversity. Dividend payments are needed to recycle capital from larger companies to smaller companies for market diversity to be maintained under conditions of capital market equilibrium.

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