Universal Portfolios with Side Information Brian Chen 6.975 - PDF document

Universal Portfolios with Side Information Brian Chen 6.975 October 16, 2002

General Investment Problem General Problem: Allocate wealth among m assets (stocks). Variations: • Performance measure (score): return vs. risk • Stock returns model: probabilistic vs. de- terministic • Rules, restrictions: – short sales – one-period vs. n -period – constant rebalanced vs. sequential – side information 2

Cover and Ordentlich Version Ref: T.M. Cover and E. Ordentlich, “Universal Portfolios with Side Information,” IEEE Trans. on Info. Thy. , Mar. 1996. • Performance measure: wealth growth rate (return), no notion of risk • Stock returns model: deterministic (more precisely, consider all possible stock returns sequences) • Rules, restrictions: – no short sales allowed – n -period – constant rebalanced with perfect hind- sight and sequential without hindsight – side information 3

Notation • x i = [ x i 1 · · · x im ] T ∈ ℜ m + is vector of price relatives (ratio between ending and starting price) for period i . x n ∆ = ( x 1 , · · · , x n ). • y i ∈ { 1 , ..., k } is a side information state for period i . • b i = [ b 1 · · · b m ] T ∈ B , where   m  b ∈ ℜ m :   � B = b j = 1 , b j ≥ 0  , j =1 is portfolio vector indicating fraction of wealth invested in each stock during period i . 4

Notation (cont.) • S i is wealth at end of period i . S 0 = 1. n � b T � b T � S n = S n − 1 = n x n i x i i =1 • W n is average “exponential” growth rate (log-return) over n periods: n 1 S i = 1 � W n = log n log S n n S i − 1 i =1 = exp( nW n ) S n 5

Side Information • Can model information such as interest rates, etc. • Could even identify best stock on trading day i (maximally informative). However, no mechanism for learning this relationship sequentially. 6

Constant Rebalanced (CRB) Portfolios • Same portfolio each period, i.e. , ∀ i b i = b (constant) b i ( y ) = b ( y ) , ∀ y ∈ { 1 , . . . , k } (state-constant) • Note: different from buy-and-hold. During i -th period, b j → x ij b j / ( b T x i ) . • Motivations – Achieves optimal growth rate for iid price relatives. – Sequential compound Bayes decision prob- lem of Robbins, Hannan, et. al. (Goal is to exhibit sequential player strategy which approximates performance of best constant strategy.) 7

Universal Portfolios (wrt. State-CRB Portfolios) Def.: Sequential portfolio with same asymp- totic exponential growth rate of wealth for ev- ery x n and y n as the best state-constant rebal- anced portfolio chosen in hindsight. Formally, • Sequential portfolio is defined by n func- tions b i = b i ( x i − 1 , y i ) , i = 1 , . . . , n with ending wealth n S n ( x n | y n ) = b T x i − 1 , y i � � � x i . i i =1 • Best State-CRB for given x n and y n is b ∗ ( · ) = arg max b ( · ) ∈B k S n ( b ( · ) , x n | y n ) with ending wealth S ∗ n ( x n | y n ) = max b ( ·∈B k S n ( b ( · ) , x n | y n ) . 8

Universal Portfolios (cont.) � x n − 1 , y n � • Sequential portfolio b 1 () , . . . , b n is universal if n log S ∗ 1 n x n ,y n ( W ∗ n →∞ sup lim = n →∞ sup lim n − W n ) S n x n ,y n = 0 ( ≤ 0 ???) Hmmm... • Surprising that universal portfolios exist? (What can one learn from the past when sequence is arbitrary?) • Universal portfolio itself is not a state-CRB portfolio. Possible for sequential to outper- form state-CRB? Consider universal over some other class. 9

Main Result Universal portfolio wrt. state-CRB portfolios exists with growth rate ˆ W n such that W n ) ≤ d 2 n log( n + 1) + k x n ,y n ( W ∗ n − ˆ sup n log 2 , where d = k ( m − 1). • d is the number of degrees of freedom of state-CRB algorithm. • “Cost” of universality per degree of free- 1 dom is essentially 2 n log n , similar to that arising in data compression. 10

Word of Caution Suppose best state-CRB in hindsight yields mil- lions of dollars after n periods. Then, universal portfolio allows one to make ˆ S n 1 = ( n + 1) d/ 2 2 k − → 0 , as n → ∞ , S ∗ n within a polynomial factor of millions of dol- lars. (Retire for a polynomial fraction of each workday?) Note: Conventional IT wisdom suggests need ˆ S n W n → W ∗ ˆ large n where n , but actually n is S ∗ maximum for n = 1. 11

An Illuminating Example (Finite Class) Consider universal portfolio over m ′ experts, each with their own portfolio strategy. Let b ( r ) i and represent portfolio of r th expert for period i and let S ( r ) denote wealth factor achieved. n allocate fraction 1 /m ′ of Universal portfolio: wealth to each expert’s portfolio. Then, m ′ S ( r ) (1 /m ′ ) ˆ � S n = n r =1 S ( r ) (1 /m ′ ) max ≥ n r (1 /m ′ ) S ∗ = n Thus, n − 1 W n ≥ W ∗ n log m ′ . ˆ 12

Finite Class Example (cont.) Notes: • Key insight: average of exponentials grows as fast as largest exponential • Universal portfolio: put fraction of wealth in each portfolio of class and “ride along with the winner”. � m ′ r =1 S ( r ) i − 1 b ( r ) i ˆ b i = . r =1 S ( r ) � m ′ i − 1 • Caution, again: a constant factor of 1 /m ′ in wealth may make a big difference at re- tirement! • Suggests an index for a class of portfolios ought to perform asymptotically as well (in terms of growth rate) as best member of class. 13

Universal Portfolio Construction (State-CRB Case) Extend finite class case to state-CRB case, i.e. , construct average of portfolios in class. • Sum over m ′ experts becomes integral over the set of CRB portfolios B . • Weight CRB portfolio b by dµ ( b ). • Handle side information y by splitting stock sequence into k subsequences, one for each possible value of y . 14

Universal Portfolio Construction (cont.) Resulting “ µ -weighted” portfolio is � B b S i − 1 ( b | y ) dµ ( b ) ˆ b i ( y ) = � B S i − 1 ( b | y ) dµ ( b ) for y = 1 , . . . , k and i = 1 , . . . , n , where � B dµ ( b ) = 1 , and b T x j � S i ( b | y ) = j ≤ i : y j = y is wealth obtained by CRB portfolio b along subsequence { j ≤ i : y j = y } . 15

Choices for µ Wealth generated by µ -weighted portfolio is n b T ˆ ˆ � S n = i ( y i ) x i i =1 k � � = B S n ( b | y ) dµ ( b ) . y =1 • Uniform distribution results in S ∗ n S n ( x n | y n ) ≥ ˆ ( n + 1) d . • Dirichlet(1 / 2 , . . . , 1 / 2) distribution corresponds to m Γ( m/ 2) b − 1 / 2 � dµ ( b ) = d b j [Γ(1 / 2)] m j =1 and results in S ∗ n S n ( x n | y n ) ≥ ˆ ( n + 1) d/ 2 2 k . Also, allows for recursive, exact computa- S n and ˆ tion of ˆ b n . 16

Connection with Universal Data Compression Can write n b ∗ � j i S ∗ n ( x n ) i =1 max S n ( x n ) ≤ n ˆ j n ∈{ 1 ,...,m } n � � b j i dµ ( b ) B i =1 • View b = ( b 1 , . . . , b m ) as probabilities on stock indices j ∈ { 1 , . . . , m } . • Numerator is probability of j n under iid dis- tribution b ∗ , and denominator is probability of j n under mixture of all iid distributions. • Logs of probabilities are codeword lengths (to within 1 bit) assigned to j n by codes for respective distributions. • log S ∗ n /S n gives worst case (over j n ) redun- dancy of mixture code over b ∗ code. 17

Universal Data Compression (cont.) • Earlier expressions for ˆ W n , ˆ S n bound redun- dancy and are independent of b ∗ . • Thus, codes corresponding to uniform and Dirichlet(1 / 2 , . . . , 1 / 2) distributions for µ are pointwise universal for iid sources. Summarizing portfolio weights ← → source probabilities CRB ← → iid wealth ratios probability ratios ← → growth rates compression rates ← → Key difference → ( e W ) n wealth = � i return i − file size = � i bits i − → Wn Thus, log of wealth ← → total file size 18

Key Points • Weighted sum (of nonnegative elements) grows like largest element. • Add to problem-solving “toolkit”: consider algorithm obtained by averaging over class of algorithms. • Universal portfolios are only average, but average is almost as good as the best (for the right definition of “good”). • Reality check: polynomial (and even con- stant) factors sometimes matter. 19