The natural mathematics arising in information theory andinvestment

Thomas Cover

Stanford University

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Felicity of mathematics

We wish to maximize the growth rate of wealth.

There is a satisfactory theory. The strategy achieving this goal is controversial.(Probably because the strategy involves maximizing the expected logarithm.)

Why is π fundamental? π = C/D,∑

n1

n2 = π2

6, φ(x) = 1√

2πe−

x2

2 .

Recall from physics the statement that the laws of physics have a strangely felicitousrelation with mathematics. We shall try to establish the reasonableness of the theory ofgrowth optimality by presenting the richness of the mathematics that describes it andby giving a number of problems having growth optimality as the answer.

A theory is natural if it fits and has few “moving parts”. Ideally, it should “predict” otherproperties.

The new or unpublished statements will be identified.

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Outline

Setup

Mean variance theory

Growth optimal portfolios for stochastic markets

Properties:

Stability of optimal portfolioExpected Ratio OptimalityCompetitive optimalitySn/S∗

n Martingale

S∗n

.= enW∗

(AEP)

Growth optimal portolios for arbitrary markets

Universal portfolios

Sn/S∗n ≥ 1

2√

n+1for all xn

Amplification

Relationship of growth optimality to information theory

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Portfolio Selection

Stock X:X = (X1, X2, . . . , Xm) ∼ F (x)

X ≥ 0

Xi = price-relative of stock i

Portfolio b:b = (b1, b2, . . . , bm), bi ≥ 0,

∑bi = 1

proportion invested

Wealth Relative S: Factor by which wealth increases

S =m∑

i=1

biXi = btX

Find the “largest” S.

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Mean-Variance Theory.Markowitz, Tobin, Sharpe, . . .

Choose b so that (Var S, ES) is undominated. S = btX.

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Conflict of mean-variance theory and growth rate.

Portfolio selection:

Maximize growth rate of wealth.

Sn(X1, X2, . . . , Xn)·= 2nW

Efficient portfolio is not necessarily growth optimal (E.Thorp)

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Consider the stock market process Xi:

Xi ∈ Rm,

Portfolios bi(·):m∑

j=1

bij(xi−1) = 1

for each time i = 1, 2, ... and for every past xi−1 = (x1, x2, ...,xi−1).

Note: bij < 0 corresponds to shorting stock j on day i. Shorting cash is calledbuying on margin.

Goal: Given a stochastic process Xi with known distribution, find portfoliosequence bi(·) that “maximizes”

Sn =n∏

i=1

bti(X

i−1)Xi

.

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1. Asymptotic Growth Rate of Wealth

X1,X2, . . . i.i.d. ∼ F (x)

Wealth at time n:

Sn =n∏

i=1

btXi

= 2(n 1n

∑log b

tXi)

= 2n(E log btX+o(1)), a.e.

Definition: Growth rate

W (b, F ) =

∫log btx dF (x)

W ∗ = maxb

W (b, F )

Sn.= 2nW∗

.

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Example

Cash vs. Hot Stock

X =

(1, 2), prob 12

(1, 1

2

), prob 1

2

b = (b1, b2)

E log S =1

2log(b1 + 2b2) +

1

2log(b1 +

1

2b2)

b∗ = (1

2,1

2)

W ∗ =1

2log

9

8

S∗n

.=

(9

8

)n/2.= (1.06)n

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Live off fluctuations

n

s

Cash

Hot stock

S∗n

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Calculation of optimal portfolio

X ∼ F (x)

Log Optimal Portfolio b∗:maxb

E log btX = W ∗

Log Optimal Wealth:S∗ = b∗tX

∂

∂biE lnbtX = E

Xi

btX

Kuhn-Tucker conditions:

b∗ : E Xi

b∗tX= 1, b∗i > 0≤ 1, b∗i = 0

Consequence: ES/S∗ ≤ 1, for all S.

Theorem E ln SS∗ ≤ 0,∀S ⇔ E S

S∗ ≤ 1, ∀S

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Properties of growth rate W (b, F )

.Theorem W (b, F ) is concave in b and linear in F .

Let bF maximize W (b, F ) over all portfolios b :∑m

i=1 bi = 1.W ∗(F ) = W (bF , F )

W (b, F )

b

0 1

Theorem W ∗(F ) is convex in F .

Question: Let W (b) =∫

lnbtx dF (x). Is W (b) a transform?

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2. Stability of b∗: Expected proportion remains constant

b∗ is a stable point

Let b = (b1, b2, ..., bm) denote the proportion of wealth in each stock.

The proportions held in each stock at the end of the trading day are

b = (b1X1

btX,b2X2

btX, ...,

bmXm

btX)

Then b is log optimal if and only if

b = Eb

i.e. bi = E biXi

btX, i = 1, 2, ...,m, i.e. the expected proportions remain unchanged.

This is the counterpart to Kelly gambling.

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Generalization to arbitrary stochastic processes Xn

Xn: arbitrary stochastic process:

Wealth from bi(·) : Sn =n∏

i=1

btiXi, bi = bi(X

i−1)

Let S∗n =

n∏

i=1

b∗ti Xi, b∗

i = b∗i (Xi−1)

where b∗i is conditionally log optimal . Thus

b∗i (Xi−1) : max

b

ElnbtXi|Xi−1

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Optimality for arbitrary stochastic processes Xn

Theorem For any market process Xi,

ESn+1/S∗n+1|Xn ≤ Sn/S∗

n.

Sn/S∗n is a nonnegative super martingale with respect to Xn

Sn/S∗n −→ Y, a.e.

EY ≤ 1.

Corollary:Prsup

n

Sn

S∗n

≥ t ≤ 1/t,

by Kolmogorov’s inequality. So Sn cannot ever exceed S∗n by factor t with probability

greater than 1/t. Same as fair gambling.

Theorem If Xi is ergodic, then 1n

log S∗n −→ W , a.e.

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3. Value of Side Information

Theorem: Believe that X ∼ g, when in fact X ∼ f . Loss in growth rate:

∆(f‖g) = Ef logbt

fX

btgX

≤ D(f ||g) =

∫f log

f

g.

Mutual information: I(X; Y ) =∑

p(x, y) logp(x, y)

p(x)p(y)

Value of side information:

W (X) = maxb

E lnbtX, W (X|Y) = maxb(·)

E lnbt(Y)X

W (X) → W (X|Y )

b∗ b∗(y)

∆(X; Y ) = Increase in growth rate for market X.

Theorem: (A.Barron ,T.C.)∆(X;Y ) ≤ I(X;Y ).

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4. Black-Scholes option pricing

Cash: 1

Stock: Xi =

1 + u, w.p. p

1 − d, w.p. q

Option: Pay c dollars today for option to buy at time n the stock at price K.

c →

(Xn − K), Xn ≥ K

0, Xn < K

Black, Scholes idea:Replicate option by buying and selling Xi, at times i = 1, 2, ..., n.Example: Option expiration date n = 1. Strike price K. Initial wealth = c.

c1 + c2X = (X − K)+. c = c1 + c2.

If it takes c dollars to replicate option, then c is a correct price for the option.

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Black-Scholes option pricing

Growth optimal approach:(

1, X,(X − K)+

c

)

Best portfolio without option:

maxb1+b2=1E ln (b1 + b2X)

Growth optimal wealth:X∗ = b∗1 + b∗2X

Add option:

maxb

E ln

((1 − b)X∗ + b

(X − K)+

c

)

d

dbE ln

((1 − b)X∗ +

b(X − k)+

c

)∣∣∣∣b=0

= E

(X−K)+

c− X∗

X∗ ≥ 0,

or E(X − K)+

X∗ ≥ c.

Critical price:

c∗ = E(X − K)+

X∗ .

But this is the same critical option price c∗ as the Black Scholes theory.Note: c∗ does not depend on probabilities, only on u and d.

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5. Asymptotic Equipartition Principle

AEPX1, X2, ..., Xn i.i.d. ∼ p(x),

1

nlog

1

p(X1, X2, ...,Xn)→ H.

AEP for marketsWealth:

Sn =n∏

i=1

btXi.

1

nlog Sn → W.

Proof:1

nlog Sn =

1

nlog

n∏

i=1

btXi =1

n

n∑

i=1

log btXi → W.

p(X1, X2, ...,Xn).= 2−nH

Sn(X1, X2, ...,Xn).= 2nW

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Asymptotic Equipartition Principle: Horse race

b = (b1, b2, ..., bm),

X = (0, 0, ...,0, m︸︷︷︸, 0, ...,0), with probability pi,

b∗ = (p1, p2, ..., pm) Kelly gambling

Proof:

W = E log S

=m∑

i=1

pi log bim

= log m +∑

i

pi logbi

pi+

∑

i

pi log pi

≤ log m − H(p1, ..., pm),

with equality if and only if bi = pi, for i = 1, 2, ...,m.

Conservation law

W + H = log m

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Comparisons

Information Theory Investment

Entropy Rate Doubling RateH = −

∑pi log pi W ∗ = maxb E log btX

AEPp(X1, X2, ...,Xn)

.= 2−nH S∗(X1, X2, ...,Xn)

.= 2nW∗

Universal Data Compression Universal Portfolio Selectionl∗∗(X1, X2, ...,Xn)

.= nH S∗∗(X1, X2, ...,Xn)

.= 2nW∗

W ∗ + H ≤ log m

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6. Competitive optimality

X ∼ F (x). Consider the two-person zero sum game:

Player 1: Portfolio b1. Wealth S1 = W1bt1X.

Player 2: portfolio b1. Wealth S2 = W2bt2X.

Fair randomization: EW1 = EW2 = 1, Wi ≥ 0.

Payoff: PrS1 ≥ S2V = max

b1,W1

minb2,W2

PrS1 ≥ S2

Theorem (R.Bell, T.C.) The value V of the game is 1/2. Optimal strategy for player1 is b1 = b∗, where b∗ is the log optimal portfolio. W1 ∼ unif[0, 2].

Comment: b∗ is both long run and short run optimal.

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7. Universal portfolio selection

Market sequencex1,x2, . . . , xn

Sn(b) =n∏

i=1

btxi

S∗n = max

b

Sn(b) =n∏

i=1

b∗tXi.

Investor:bi(x1,x2, . . . ,xi−1)

Sn =n∏

i=1

btixi

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Minimax regret universal portfolio

Minimax regret for horizon n is defined as

R∗n = min

b(·)maxxn,b

∏ni=1 btxi∏n

i=1 bi(xi−1)xi

= minb

maxxn

S∗n

Sn

Theorem: (Erik Ordentlich, T.C.)

R∗n =

1

Vn,

where Vn =∑ ( n

n1,...,nm

)2−nH(

n1n

,..., nmn

)

Note: For m = 2 stocks,

Vn =∑n

k=0

(nk

)2−nH( k

n) ∼

√2

πn

Vn ≤ 2√n+1

Corollary: For m = 2 stocks, there exists bi(xi−1) such that

Sn ≥ 2S∗n√

n + 1, for every sequence x1, . . . , xn.

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Achieving R∗n: Universal Portfolio for horizon n

Portfolio bi(Xi−1) :Invest

b(jn) =1

Vn

(n1(jn)

n

)n1(jn) (n2(jn)

n

)n2(jn)

· · ·(

nm(jn)

n

)nm(jn)

in “plunging” strategy jn and let it ride, where jn ∈ 1, 2, ...,mn.

Example For horizon n = 2. For m = 2.

X1 = (X11, X12)

b1 = ( 12, 12)

b2(X1) = (45

X11+ 15

X12

X11+X12,

15

X11+ 45

X12

X11+X12)

b(11) = 4/10

b(12) = 1/10

b(21) = 1/10

b(22) = 4/10

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8. Accelerated Performance

Stock x ∈ Rm+ , requires b ∈ Rm

+ , so that btx ≥ 0.

Let X(α) = x ∈ Rm: xi ≥ α,

m∑

i=1

xi = 1

B(α) = b ∈ Rm :m∑

i=1

bi = 1, btx ≥ 0, ∀x ∈ X(α)

B(α) is polar cone to X (α): B(α) = X⊥(α).

B(α) allows short selling and buying on margin.

Thus x ∈ X (α), b ∈ B(α) yields S = btx ≥ 0.Let Ω = Rm

+ , X (α) = AΩ, B(α) = A−1Ω.

A =

(α 1 − α

1 − α α

)A

−1 =1

2α − 1

(α −(1 − α)

−(1 − α) α

)

b ∈ Ω,X ∈ Ω. b = A−1b ∈ B(α), X = AX ∈ X (α).

btX = bt(A−1

)tAX = btX

α 1 − α

X (α)

B(α)

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Accelerated Performance

Theorem (Acceleration (Erik Ordentlich, T.C., to appear))

m = 2 stocks. The short selling investor can come within factor Vn(α) of the bestlong-only investor given hindsight:

maxbi(·)∈B(α)

minx∈Xn(α),

b∈B(0)

∏ni=1 bt

ixi∏ni=1 btxi

= Vn(α),

where [x] = x rounded off to interval [α, α].

Vn(α) =n∑

k=0

(n

k

) [k

n

]k [n − k

n

]n−k

Note: Vn(α) ր. Vn(0) ∼√

2π

1√n

. Vn( 12) = 1.

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Accelerated Performance

0 50 100 150 200 250 300900

1000

1100

1200

1300

1400

1500

160028−Sep−07 till 14−Oct−08

Time

S&

P50

0

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Accelerated Performance

−6 −4 −2 0 2 4 60

0.5

1

1.5

2

2.5

b

Sn

Sn*

Sn**

9/28/07 – 10/14/08, n = 263.S∗

n: Wealth of best long-only constant rebalanced portfolio in hindsight.S∗∗

n : Wealth of best short selling and margin constant rebalanced portfolio in hindsight.

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Accelerated Performance

−6 −4 −2 0 2 4 60

0.5

1

1.5

2

2.5

b

Sn

α=0.45 Sn^= 1.0475

Sn*

Sn**

Sn^

9/28/07 – 10/14/08, n = 263.S∗

n: Wealth of best long-only constant rebalanced portfolio in hindsight.S∗∗

n : Wealth of best short selling and margin constant rebalanced portfolio in hindsight.Sn: Wealth of universal portfolio.

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Comparisons with Information Theory

General Market Horse Race Market

X ∼ F (x) X = mei, pi

b∗ : Eb∗i Xi

b∗tX= b∗i bi = pi Kelly gambling

W ∗ = Eb∗tX W ∗ = log m − H(p), H =entropy

Wrong distribution G(x):

∆(F ||G) =∫ b

tF x

btG

xdF (x) ∆ =

∑pi ln pi

gi= D(p||g), relative entropy

Side information (X, Y) ∼ f(x, y):

∆ =∫

lnb

tf(x|y)x

btf(x)

xf(x,y)dxdy ∆ =

∑p(x, y) ln

p(x,y)p(x)p(y)

= I(X; Y ), mutual information

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Comparisons

General Market Horse Race Market

Asymptotic growth rateXi stationary:

W ∗ = maxb ElnbtX0|X−1−∞ W ∗ = log m − H(X0|X−1

−∞)= log m − H(X ), H(X ) = entropy rate

AEP for ergodic processes:

1n

log S∗n → W ∗, a.e. − 1

nlog p(Xn) → H(X ), a.e.

S∗n

·= 2nW∗

p(Xn)·= 2−nH

Page 36 of 40

Comparisons

Universal portfolio (individual sequences):

General Market Horse Race Market

x1,x2, ...,xn ∈ Rm+ x1, x2, ...,xn ∈ e1, ..., em

Sn(b, xn) =∏n

i=1 btxi Sn(b, xn) =∏m

i=1 bni(x

n)i

Sn(bn,xn) =∏n

i=1 bt(xi−1)xi Sn(bn, xn) = b(xn)

Vn Vn

Same cost of universality for both.

Vn = minb(·)

maxb,xn

Sn(bn,xn)

Sn(b, xn)

=∑ ( n

n1, ..., nm

)2−nH(

n1n

,...,nmn

)

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Concluding remarks

Growth optimal portfolios have many properties:

Long run optimality

Martingale property

Competitive optimality

Asymptotic equipartition property

Universal achievability

Black-Scholes

Amplification

Relationship with information theory

Page 38 of 40

References

Algoet Barron Bell BorodinCover Erkip Gluss GyorfiHakansson Iyengar Jamshidian LugosiMathis Merton Ordentlich PlatenSamuelson Shannon Thorp VajdaWarmuth Ziemba Markowitz SharpeDuffie

Page 39 of 40

References

R. Bell and T. Cover, “Game-Theoretic Optimal Portfolios,” Management Science,34(6):724-733, June 1988.

T. Cover, “Universal Portfolios,” Mathematical Finance, 1(1):1-29, January 1991.

T. Cover and E. Ordentlich, “Universal Portfolios with Side Information,”IEEETransactions on Information Theory, 42(2):348-363, March 1996.

E. Ordentlich and T. Cover, “The Cost of Achieving the Best Portfolio in Hindsight,”Mathematics of Operations Research, 23(4):960-982, November 1998.

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