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

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Cover and Ordentlich Version Ref: T.M. Cover and E. Ordentlich, “Universal Portfolios with Side Information,” IEEE Trans.

• n 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

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Notation

• xi = [xi1 · · · xim]T ∈ ℜm

+ is vector of price

relatives (ratio between ending and starting price) for period i. xn ∆ = (x1, · · · , xn).

• yi ∈ {1, ..., k} is a side information state for

period i.

• bi = [b1 · · · bm]T ∈ B, where

B =

  b ∈ ℜm :

m

• j=1

bj = 1, bj ≥ 0

   ,

is portfolio vector indicating fraction of wealth invested in each stock during period i.

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Notation (cont.)

• Si is wealth at end of period i. S0 = 1.

Sn = Sn−1

• bT

nxn

• =

n

• i=1

bT

i xi

• Wn is average “exponential” growth rate

(log-return) over n periods: Wn = 1 n

n

• i=1

log Si Si−1 = 1 n log Sn Sn = exp(nWn)

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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.

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Constant Rebalanced (CRB) Portfolios

• Same portfolio each period, i.e., ∀i

bi = b

(constant)

bi(y) = b(y), ∀y ∈ {1, . . . , k}

(state-constant)

• Note: different from buy-and-hold. During

i-th period, bj → xijbj/(bTxi).

• 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.)

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Universal Portfolios (wrt. State-CRB Portfolios) Def.: Sequential portfolio with same asymp- totic exponential growth rate of wealth for ev- ery xn and yn as the best state-constant rebal- anced portfolio chosen in hindsight. Formally,

• Sequential portfolio is defined by n func-

tions

bi = bi(xi−1, yi),

i = 1, . . . , n with ending wealth Sn (xn|yn) =

n

• i=1

bT

i

• xi−1, yi

xi.

• Best State-CRB for given xn and yn is

b∗(·) = arg max

b(·)∈Bk Sn (b(·), xn|yn)

with ending wealth S∗

n (xn|yn) = max

b(·∈Bk Sn (b(·), xn|yn) .

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Universal Portfolios (cont.)

• Sequential portfolio b1(), . . . , bn
• xn−1, yn

is universal if lim

n→∞ sup

xn,yn

1 n log S∗

n

Sn = lim

n→∞ sup

xn,yn(W ∗

n − Wn)

= (≤ 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.

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Main Result Universal portfolio wrt. state-CRB portfolios exists with growth rate ˆ Wn such that sup

xn,yn(W ∗

n − ˆ

Wn) ≤ d 2n log(n + 1) + k 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-

dom is essentially

1 2n log n, similar to that

arising in data compression.

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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 Sn S∗

n

= 1 (n + 1)d/2 2k − → 0, as n → ∞, within a polynomial factor of millions of dol-

• lars. (Retire for a polynomial fraction of each

workday?) Note: Conventional IT wisdom suggests need large n where ˆ Wn → W ∗

n, but actually ˆ Sn S∗

n is

maximum for n = 1.

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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 rth expert for period i and let S(r)

n

denote wealth factor achieved. Universal portfolio: allocate fraction 1/m′ of wealth to each expert’s portfolio. Then, ˆ Sn = (1/m′)

m′

• r=1

S(r)

n

≥ (1/m′) max

r

S(r)

n

= (1/m′) S∗

n

Thus, ˆ Wn ≥ W ∗

n − 1

n log m′.

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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”. ˆ

bi =

m′

r=1 S(r) i−1b(r) i

m′

r=1 S(r) 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
• ught to perform asymptotically as well (in

terms of growth rate) as best member of class.

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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.

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Universal Portfolio Construction (cont.) Resulting “µ-weighted” portfolio is ˆ

bi(y) =

• B bSi−1(b|y) dµ(b)
• B Si−1(b|y) dµ(b)

for y = 1, . . . , k and i = 1, . . . , n, where

• B dµ(b) = 1,

and Si(b|y) =

• j≤i:yj=y

bTxj

is wealth obtained by CRB portfolio b along subsequence {j ≤ i : yj = y}.

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Choices for µ Wealth generated by µ-weighted portfolio is ˆ Sn =

n

• i=1

ˆ

bT

i (yi) xi

=

k

• y=1
• B Sn(b|y) dµ(b).
• Uniform distribution results in

ˆ Sn(xn|yn) ≥ S∗

n

(n + 1)d.

• Dirichlet(1/2, . . . , 1/2) distribution corresponds

to dµ(b) = Γ(m/2) [Γ(1/2)]m

m

• j=1

b−1/2

j

db and results in ˆ Sn(xn|yn) ≥ S∗

n

(n + 1)d/2 2k. Also, allows for recursive, exact computa- tion of ˆ Sn and ˆ

bn.

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Connection with Universal Data Compression Can write S∗

n(xn)

ˆ Sn(xn) ≤ max

jn∈{1,...,m}n n

• i=1

b∗

ji

• B

n

• i=1

bji dµ(b)

• View b = (b1, . . . , bm) as probabilities on

stock indices j ∈ {1, . . . , m}.

• Numerator is probability of jn under iid dis-

tribution b∗, and denominator is probability

• f jn under mixture of all iid distributions.
• Logs of probabilities are codeword lengths

(to within 1 bit) assigned to jn by codes for respective distributions.

• log S∗

n/Sn gives worst case (over jn) redun-

dancy of mixture code over b∗ code.

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Universal Data Compression (cont.)

• Earlier expressions for ˆ

Wn, ˆ Sn 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 wealth =

i returni

− → (eW)n file size =

i bitsi

− → Wn Thus, log of wealth ← → total file size

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Key Points

• Weighted sum (of nonnegative elements)

grows like largest element.

• Add to problem-solving “toolkit”: consider

algorithm obtained by averaging over class

• f 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.

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