The natural mathematics arising in information theory and investment Thomas Cover Stanford University Page 1 of 40

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.) � 2 π e − x 2 n 2 = π 2 1 1 2 . Why is π fundamental? π = C/D , 6 , φ ( x ) = √ n Recall from physics the statement that the laws of physics have a strangely felicitous relation with mathematics . We shall try to establish the reasonableness of the theory of growth optimality by presenting the richness of the mathematics that describes it and by 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” other properties. The new or unpublished statements will be identified. Page 2 of 40

Outline Setup Mean variance theory Growth optimal portfolios for stochastic markets Properties: Stability of optimal portfolio Expected Ratio Optimality Competitive optimality S n /S ∗ n Martingale = e nW ∗ (AEP) . S ∗ n Growth optimal portolios for arbitrary markets Universal portfolios S n /S ∗ ˆ 2 √ n +1 for all x n 1 n ≥ Amplification Relationship of growth optimality to information theory Page 3 of 40

Portfolio Selection Stock X : X = ( X 1 , X 2 , . . . , X m ) ∼ F ( x ) X ≥ 0 X i = price-relative of stock i Portfolio b : � b = ( b 1 , b 2 , . . . , b m ) , b i ≥ 0 , b i = 1 proportion invested Wealth Relative S : Factor by which wealth increases m � b i X i = b t X S = i =1 Find the “largest” S . Page 4 of 40

Mean-Variance Theory. Markowitz, Tobin, Sharpe, . . . S = b t X . Choose b so that (Var S, ES ) is undominated. Page 5 of 40

Conflict of mean-variance theory and growth rate. Portfolio selection: Maximize growth rate of wealth. S n ( X 1 , X 2 , . . . , X n ) · = 2 nW Efficient portfolio is not necessarily growth optimal (E.Thorp) Page 6 of 40

Consider the stock market process { X i } : X i ∈ R m , Portfolios b i ( · ) : m � b ij ( x i − 1 ) = 1 j =1 for each time i = 1 , 2 , ... and for every past x i − 1 = ( x 1 , x 2 , ..., x i − 1 ) . Note: b ij < 0 corresponds to shorting stock j on day i . Shorting cash is called buying on margin. Goal: Given a stochastic process { X i } with known distribution, find portfolio sequence b i ( · ) that “maximizes” n � b t i ( X i − 1 ) X i S n = i =1 . Page 7 of 40

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1. Asymptotic Growth Rate of Wealth X 1 , X 2 , . . . i.i.d. ∼ F ( x ) Wealth at time n : n � b t X i S n = i =1 � log b t X i ) 2 ( n 1 = n 2 n ( E log b t X + o (1)) , = a.e. Definition: Growth rate � log b t x dF ( x ) W ( b , F ) = W ∗ = max W ( b , F ) b 2 nW ∗ . . S n = Page 10 of 40

Example Cash vs. Hot Stock prob 1 (1 , 2) , 2 X = b = ( b 1 , b 2 ) � � 1 , 1 prob 1 , 2 2 E log S = 1 2 log( b 1 + 2 b 2 ) + 1 2 log( b 1 + 1 2 b 2 ) b ∗ = (1 2 , 1 2) W ∗ = 1 2 log 9 8 � 9 � n/ 2 . . S ∗ = (1 . 06) n = n 8 Page 11 of 40

Live off fluctuations s S ∗ n Hot stock Cash n Page 12 of 40

Calculation of optimal portfolio X ∼ F ( x ) Log Optimal Portfolio b ∗ : E log b t X = W ∗ max b Log Optimal Wealth: S ∗ = b ∗ t X ∂ E ln b t X = E X i b t X ∂b i Kuhn-Tucker conditions: b ∗ : E X i b ∗ = 1 , i > 0 b ∗ t X b ∗ ≤ 1 , i = 0 Consequence: ES/S ∗ ≤ 1 , for all S. Theorem E ln S E S S ∗ ≤ 0 , ∀ S ⇔ S ∗ ≤ 1 , ∀ S Page 13 of 40

Properties of growth rate W ( b , F ) . Theorem W ( b , F ) is concave in b and linear in F . Let b F maximize W ( b , F ) over all portfolios b : � m i =1 b i = 1 . W ∗ ( F ) = W ( b F , F ) W ( b , F ) b 0 1 Theorem W ∗ ( F ) is convex in F . � Question: Let W ( b ) = ln b t x dF ( x ) . Is W ( b ) a transform? Page 14 of 40

2. Stability of b ∗ : Expected proportion remains constant b ∗ is a stable point Let b = ( b 1 , b 2 , ..., b m ) denote the proportion of wealth in each stock. The proportions held in each stock at the end of the trading day are b = ( b 1 X 1 b t X , b 2 X 2 b t X , ..., b m X m � b t X ) Then b is log optimal if and only if b = E � b i.e. b i = E b i X i b t X , i = 1 , 2 , ..., m, i.e. the expected proportions remain unchanged. This is the counterpart to Kelly gambling. Page 15 of 40

Generalization to arbitrary stochastic processes { X n } X n : arbitrary stochastic process: n � b t b i = b i ( X i − 1 ) Wealth from b i ( · ) : S n = i X i , i =1 n � b ∗ t i ( X i − 1 ) Let S ∗ b ∗ i = b ∗ n = i X i , i =1 where b ∗ i is conditionally log optimal . Thus i ( X i − 1 ) : max E { ln b t X i | X i − 1 } b ∗ b Page 16 of 40

Optimality for arbitrary stochastic processes { X n } Theorem For any market process { X i } , E { S n +1 /S ∗ n +1 | X n } ≤ S n /S ∗ n . S n /S ∗ n is a nonnegative super martingale with respect to { X n } S n /S ∗ n − → Y, a.e. EY ≤ 1 . Corollary: S n Pr { sup ≥ t } ≤ 1 /t, S ∗ n n by Kolmogorov’s inequality. So S n cannot ever exceed S ∗ n by factor t with probability greater than 1 /t . Same as fair gambling. Theorem If { X i } is ergodic, then 1 n log S ∗ n − → W , a.e. Page 17 of 40

3. Value of Side Information Theorem: Believe that X ∼ g , when in fact X ∼ f . Loss in growth rate: � b t f X f log f ∆( f � g ) = E f log ≤ D ( f || g ) = g . b t g X � p ( x, y ) log p ( x, y ) Mutual information: I ( X ; Y ) = p ( x ) p ( y ) Value of side information: E ln b t X , b ( · ) E ln b t ( Y ) X W ( X ) = max W ( X | Y ) = max b 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 ) . Page 18 of 40

4. Black-Scholes option pricing Cash: 1 � 1 + u, w.p. p Stock: X i = 1 − d, w.p. q Option: Pay c dollars today for option to buy at time n the stock at price K . � ( X n − K ) , X n ≥ K c → 0 , X n < K Black, Scholes idea: Replicate option by buying and selling X i , at times i = 1 , 2 , ..., n . Example: Option expiration date n = 1 . Strike price K . Initial wealth = c . c 1 + c 2 X = ( X − K ) + . c = c 1 + c 2 . If it takes c dollars to replicate option, then c is a correct price for the option. Page 19 of 40

Black-Scholes option pricing Growth optimal approach: � � 1 , X, ( X − K ) + c Best portfolio without option: max b 1 + b 2 =1 E ln ( b 1 + b 2 X ) Growth optimal wealth: X ∗ = b ∗ 1 + b ∗ 2 X Add option: � � (1 − b ) X ∗ + b ( X − K ) + max E ln c b �� � ( X − K ) + − X ∗ (1 − b ) X ∗ + b ( X − k ) + � d � c db E ln = E ≥ 0 , � X ∗ c b =0 E ( X − K ) + or ≥ c. X ∗ 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 . Page 20 of 40

5. Asymptotic Equipartition Principle AEP X 1 , X 2 , ..., X n i.i.d. ∼ p ( x ) , 1 1 n log p ( X 1 , X 2 , ..., X n ) → H. AEP for markets Wealth: n � b t X i . S n = i =1 1 n log S n → W. Proof: n n � � n log S n = 1 1 b t X i = 1 log b t X i → W. n log n i =1 i =1 . 2 − nH p ( X 1 , X 2 , ..., X n ) = . 2 nW S n ( X 1 , X 2 , ..., X n ) = Page 21 of 40

Asymptotic Equipartition Principle: Horse race b = ( b 1 , b 2 , ..., b m ) , = (0 , 0 , ..., 0 , m ���� , 0 , ..., 0) , with probability p i , X b ∗ = ( p 1 , p 2 , ..., p m ) Kelly gambling Proof: W = E log S m � = p i log b i m i =1 � � p i log b i = log m + + p i log p i p i i i ≤ log m − H ( p 1 , ..., p m ) , with equality if and only if b i = p i , for i = 1 , 2 , ..., m . Conservation law W + H = log m Page 22 of 40

Comparisons Information Theory Investment Entropy Rate Doubling Rate H = − � p i log p i W ∗ = max b E log b t X AEP p ( X 1 , X 2 , ..., X n ) . S ∗ ( X 1 , X 2 , ..., X n ) . = 2 nW ∗ = 2 − nH Universal Data Compression Universal Portfolio Selection l ∗∗ ( X 1 , X 2 , ..., X n ) . S ∗∗ ( X 1 , X 2 , ..., X n ) . = 2 nW ∗ = nH W ∗ + H ≤ log m Page 23 of 40

6. Competitive optimality X ∼ F ( x ) . Consider the two-person zero sum game: Player 1: Portfolio b 1 . Wealth S 1 = W 1 b t 1 X . Player 2: portfolio b 1 . Wealth S 2 = W 2 b t 2 X . Fair randomization: EW 1 = EW 2 = 1 , W i ≥ 0 . Payoff: Pr { S 1 ≥ S 2 } V = max min Pr { S 1 ≥ S 2 } b 1 ,W 1 b 2 ,W 2 Theorem (R.Bell, T.C.) The value V of the game is 1 / 2 . Optimal strategy for player 1 is b 1 = b ∗ , where b ∗ is the log optimal portfolio. W 1 ∼ unif[0 , 2] . Comment: b ∗ is both long run and short run optimal. Page 24 of 40

7. Universal portfolio selection Market sequence x 1 , x 2 , . . . , x n n � b t x i S n ( b ) = i =1 n � S ∗ b ∗ t X i . n = max S n ( b ) = b i =1 Investor: ˆ b i ( x 1 , x 2 , . . . , x i − 1 ) n � ˆ ˆ b t S n = i x i i =1 Page 25 of 40

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