AA Note on Universal Bilinear Portfolios
Alex Garivaltis ∗ Northern Illinois UniversityJuly 24, 2019
Abstract
This note provides a neat and enjoyable expansion and application of themagnificent Ordentlich-Cover theory of “universal portfolios.” I generalizeCover’s benchmark of the best constant-rebalanced portfolio (or 1-linear tradingstrategy) in hindsight by considering the best bilinear trading strategy deter-mined in hindsight for the realized sequence of asset prices. A bilinear tradingstrategy is a mini two-period active strategy whose final capital growth factoris linear separately in each period’s gross return vector for the asset market.I apply Cover’s ingenious (1991) performance-weighted averaging technique toconstruct a universal bilinear portfolio that is guaranteed (uniformly for allpossible market behavior) to compound its money at the same asymptotic rateas the best bilinear trading strategy in hindsight. Thus, the universal bilinearportfolio asymptotically dominates the original (1-linear) universal portfolio inthe same technical sense that Cover’s universal portfolios asymptotically domi-nate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact,like so many Russian dolls, one can get carried away and use these ideas to con-struct an endless hierarchy of ever more dominant H -linear universal portfolios. Keywords:
On-Line Portfolio Selection; Universal Portfolios; Robust Pro-cedures; Model Uncertainty; Constant-Rebalanced Portfolios; Asymptotic Cap-ital Growth; Kelly Criterion
JEL Classification Codes:
D81; D83; G11 ˆ W = (cid:90) B ∈B (cid:32) T (cid:89) t =1 x (cid:48) t By t (cid:33) f ( B ) dB (1) ∗ Assistant Professor, Department of Economics, School of Public and Global Affairs, College ofLiberal Arts and Sciences, Northern Illinois University, 514 Zulauf Hall, DeKalb IL 60115. E-mail: [email protected] . Homepage: http://garivaltis.com . ORCID iD: 0000-0003-0944-8517. a r X i v : . [ q -f i n . M F ] J u l e first investigate what a natural goal might befor the growth of wealth for arbitrary marketsequences. For example, a natural goal might beto outperform the best buy-and-hold strategy,thus beating an investor who is given a look at anewspaper n days in the future. We propose amore ambitious goal. —Thomas M. Cover, Universal Portfolios , 1991In 1988, out of the blue, Paul Samuelson wrote aletter to Stanford information theorist ThomasCover. Samuelson had been sent one of Cover’spapers on portfolio theory for review. “If I diduse some of your procedures,” Samuelson wrote,“I would not let that ... bias my portfolio choicetoward choices my alien cousin with log utilitywould make.” He chides Kelly, Latan´e,Markowitz, and “various Ph.D’s who appear withPoisson-distribution probabilities most Junes.”—William Poundstone,
Fortune’s Formula , 2005With four parameters I can fit an elephant, andwith five I can make him wiggle his trunk.—John von Neumann
Note on Universal Bilinear Portfolios
A. Garivaltis
This note contains a nice application and extension of the elegant universal portfoliotheory that was established by Thomas Cover (1991), Cover and Ordentlich (1996),and Ordentlich and Cover (1998).Universal portfolio theory is the on-line analogue of the log-optimal portfolio theory (that is, the theory of asymptotic capital growth), whose brilliant simplicity camedown to us from such illustrative thinkers as John Kelly (1956), Henry Latan´e (1959),Leo Breiman (1961), and card-counter Edward O. Thorp (1969). Under laboratoryconditions where the investor or gambler knows in advance the precise distributionof the profit-and-loss outcomes on which he is betting, the tea leaves say (cf. withMacLean, Thorp, and Ziemba 2011) that log-optimal portfolios (or growth-optimalportfolios ) enjoy tremendous optimality properties, quite apart from the fact thatthey saturate a very specific type of expected utility, as pointed out so many timesby Samuelson (1963, 1969, 1979).Leo Breiman (1961) gave the first substantial results in this direction, namely, thatthe so-called
Kelly gambler will, under general conditions, asymptotically outperformany “essentially different strategy” almost surely by an exponential factor. He alsodemonstrated that, for the sake of goal-based investing, the
Kelly criterion minimizesthe expected waiting time with respect to hitting a distant high-water mark.In a pair of beautiful articles, Robert Bell and Thomas Cover (1980, 1988) estab-lished that, actually, the Kelly rule also possesses very strong short-term competitiveoptimality properties, even for a single period’s fluctuation of a betting or investmentmarket. They considered a static, zero-sum investment φ -game whose payoff kernelis equal to the expected value of an arbitrary increasing function φ ( • ) of the ratioof one trader’s wealth to that of another. Subject to the proviso that, prior to the2 Note on Universal Bilinear Portfolios
A. Garivaltisactual portfolio choice, each contestant is permitted to make a fair randomization ofhis initial dollar (by exchanging it for any random capital whose mean is at most 1),the saddle point of the game amounts to each player using the log-optimal portfolio,together with fair randomizations that depend only on the criterion φ ( • ), and not onany particular characteristic of the underlying investment opportunities.Garivaltis (2018a) showed that the Bell-Cover theorem holds equally well forstochastic differential investment φ -games in continuous time that exhibit state-dependent drift and diffusion; Garivaltis (2019a) generalized this result even further,so as to cover levered investment φ -games over continuous time markets whereby theasset prices follow jump-diffusion processes with compactly-supported jump returns.Some recent work by Curatola (2019) investigates the strategic interaction of two largetraders whose transactions affect not just each other, but also the expected returnsof the entire stock market. For an illuminating discussion of competitive optimalityas it relates to evolutionary contingencies in mathematical biology, consult with Taland Tran (2019).Cover’s universal portfolio theory, which began in earnest with his empirical Bayesstock portfolio (Cover and Gluss 1986), takes its cue from the fact that for stockmarkets with iid returns, the log-optimal portfolio amounts to a certain constant-rebalanced portfolio (CRP); this consists in fixing the correct (growth-optimal) targetpercentages of wealth for each asset, and continuously executing rebalancing tradesso as to counteract allocation drift. However, in the presence of model uncertainty(e.g. for actual stock markets), this particular CRP is completely unknown to thepractitioner.Inspired by the analogies with information theory, Thomas Cover had the brilliantinsight that one should benchmark his on-line investment performance relative tothat of the best constant-rebalanced portfolio determined in hindsight for the actual3 Note on Universal Bilinear Portfolios
A. Garivaltis(realized) sequence of asset prices. The hindsight-optimized wealth can be interpretedas a financial derivative that is susceptible of exact pricing and replication in the(complete) continuous time market of Black and Scholes (1973). On that score,Ordentlich and Cover (1998) priced the rebalancing option at time-0 for unleveredhindsight optimization over a single risk asset; their work sat unfinished for twentyyears, until it was completed by Garivaltis (2019b), who demonstrated how to priceand replicate Cover’s (levered) rebalancing option at any time t , for any number ofcorrelated stocks in geometric Brownian motion.In discrete time, the empirical Bayes stock portfolio (Cover and Gluss 1986), theDirichlet-weighted universal portfolio (Cover and Ordentlich 1996), and the minimaxuniversal portfolio (Ordentlich and Cover 1998) are all notable in that they guaran-tee to achieve a high percentage of the final wealth of the best constant-rebalancedportfolio in hindsight, uniformly for all possible sequences of asset prices. On accountof the fact that this percentage (or competitive ratio ) converges to zero at a slow(polynomial) rate, the excess compound (logarithmic) growth rate of the best CRPin hindsight (over and above that of the on-line portfolio) converges uniformly to zero.Thus, universal portfolios succeed in matching the performance of the best CRP inhindsight “to first order in the exponent.”The original universal portfolios (inspired as they were by iid stock markets) sufferfrom the defect that they fail to recognize and exploit even very simple types of serialdependence in the individual sequence of asset returns. For example, consider a two-asset market whereby asset 2 is cash (that pays no interest), and asset 1 is a “hotstock” whose price alternately doubles in odd periods and gets cut in half in evenperiods. Naturally, one should hope that his portfolio selection algorithm is capableof detecting such a trivial pattern, thereby learning to (asymptotically) double itscapital every two periods. But the original universal portfolios, when applied to4 Note on Universal Bilinear Portfolios
A. Garivaltisthis particular sequence of asset prices, merely learn to use the constant-rebalancedportfolio that puts 50% of its wealth into the stock and holds the rest in cash at thestart of each investment period; this generates asymptotic capital growth at a rate oflog(9 /
8) = 11 .
8% every two periods, compounded continuously — a far cry from thelog 2 = 69 .
3% that accrues to perfect trading.One way out of this conundrum is the use the universal portfolio with side infor-mation (Cover and Ordentlich 1996) along with a “signal” that indicates, say, whetheror not the current period is odd. The obvious objection here is that the efficacy of thisparticular signal (as opposed to any other piece of side information) will only ever be-come apparent in hindsight. Accordingly, this paper tackles the problem differently:we consider an expanded parametric family of mini 2-period active trading strategiescalled bilinear portfolios , which explicitly generalize the constant-rebalanced portfo-lios (here called ). Accordingly, we apply the Ordentlich-Cover tech-niques to design a universal bilinear portfolio that compounds its money at the sameasymptotic rate as the best bilinear trading strategy in hindsight (thereby learningto trade perfectly in the motivating example). Thus, the universal bilinear portfoliowill be shown to asymptotically dominate the universal 1-linear portfolio in the sametechnical sense (cf. with Cover and Thomas 2006) that the universal 1-linear portfo-lio asymptotically dominates all constant-rebalanced portfolios and all buy-and-holdstrategies. Once this is done, it will become readily apparent just how one can goabout constructing an endless hierarchy of ever more dominant universal H -linearportfolios , for all possible mini-horizons H ∈ { , , , ... } .5 Note on Universal Bilinear Portfolios
A. Garivaltis
We start by defining the concept of a bilinear trading strategy (or bilinear portfolio ),which is a simple 2-period active strategy that generalizes the notion of a constant-rebalanced portfolio (CRP). To this end, we assume that there are m assets called i, j ∈ { , ..., m } ; we let x i ≥ of a $1 investment in asset i in period 1, and similarly we let y j ≥ j in period 2.We let x := ( x , ..., x m ) (cid:48) ∈ R m + − { } denote the gross return vector in period 1, andin the same vein, y := ( y , ..., y m ) (cid:48) ∈ R m + − { } is the gross return vector in period 2. Definition 1. A bilinear trading strategy is a square matrix B := [ b ij ] m × m ofnon-negative weights that sum to one. After two investment periods, the bilineartrading strategy B multiplies the initial dollar by a factor of Two-Period Capital Growth Factor := x (cid:48) By = m (cid:88) i =1 m (cid:88) j =1 b ij x i y j . (2) The set of all bilinear trading strategies is denoted B := { B ∈ Mat m,m ( R ) : B ≥ and (cid:48) B = 1 } , (3) where := (1 , ..., (cid:48) is an m × vector of ones. Proposition 1.
The bilinear final wealth x (cid:48) By is uniquely replicated by the follow-ing 2-period active trading strategy: in period 1, we use the initial portfolio p :=( p , ..., p m ) (cid:48) = B , where p i = m (cid:80) j =1 b ij is the initial fraction of wealth that will be e.g. if x i := 1 .
05 then asset i appreciated 5% in period 1; if x i := 0 .
98, then asset i lost 2% ofits value in period 1, etc. Bilinearity (cf. with Serge Lang 1987) refers to the fact that the capital growth factor x (cid:48) By islinear separately in each of the vectors x and y . When viewed jointly as a function of ( x, y ), thebilinear form x (cid:48) By is a homogeneous quadratic polynomial in the 2 m variables x , ..., x m , y , ..., y m . Note on Universal Bilinear Portfolios
A. Garivaltis invested in asset i ; in period 2, we must use the portfolio q ( x ) := ( q ( x ) , ..., q m ( x )) (cid:48) = B (cid:48) xp (cid:48) x = B (cid:48) xx (cid:48) B , (4) e.g. q j ( x ) = m (cid:80) i =1 b ij x im (cid:80) i =1 m (cid:80) k =1 b ik x i . (5) Proof.
We start with the functional equation( p (cid:48) x ) · ( q ( x ) (cid:48) y ) = x (cid:48) By, (6)e.g. the two-period growth factor is equal to the product of the individual growthfactors that were achieved in periods 1 and 2. To start, we substitute y := =(1 , ..., (cid:48) and x := e i = (0 , ..., , i , , ..., (cid:48) , which is the i th unit basis vector for R m .There lies p i = e (cid:48) i B = m (cid:80) j =1 b ij , as promised. Next, in the identity q ( x ) (cid:48) y = x (cid:48) Byp (cid:48) x , (7)we put y := e j . This leaves us with q j ( x ) = m (cid:80) i =1 b ij x im (cid:80) i =1 m (cid:80) j =1 b ik x i , (8)which is the desired result. In order to be logically complete, we must substitute ourexpressions for p and q ( x ) into equation (6) so as to verify that they turn it into an7 Note on Universal Bilinear Portfolios
A. Garivaltisidentity. Here you go: ( B ) (cid:48) x · (cid:18) B (cid:48) xx (cid:48) B (cid:19) (cid:48) y = x (cid:48) By. (9) (cid:4)
Example 1.
Every constant-rebalanced portfolio (cf. with Thomas Cover 1991) c :=( c , ..., c m ) (cid:48) amounts to a bilinear trading strategy that is represented by the outerproduct B := cc (cid:48) , e.g. b ij := c i c j for all i, j ∈ { , ..., m } . Here, the constant-rebalancedportfolio c resolves to maintain the constant fraction c i of wealth in each asset i at alltimes , where c i ≥ and m (cid:80) i =1 c i = 1 . Example 2.
More generally, consider the trading strategy that always uses the port-folio c := ( c , ..., c m ) (cid:48) ∈ ∆ m in period 1 and then always uses the portfolio d :=( d , ..., d m ) (cid:48) ∈ ∆ m in period 2 (regardless of the observed value of x ), where ∆ m := (cid:26) c ∈ R m + : m (cid:80) i =1 c i = 1 (cid:27) denotes the unit portfolio simplex in R m + . This scheme is a bi-linear trading strategy that corresponds to the outer product B := cd (cid:48) , e.g. b ij := c i d j for all i, j ∈ { , ..., m } . Example 3.
Every buy-and-hold strategy (that buys some initial portfolio c := ( c , ..., c m ) (cid:48) and holds it for two periods, without rebalancing) amounts to a bilinear trading strat-egy that is represented by the diagonal matrix B := diag( c , ..., c m ) . Inspired by Ordentlich and Cover (1998) and Cover and Thomas (2006), we notethat the concept of a bilinear trading strategy admits the following simple and lucidinterpretation. Let an extremal strategy be defined by the simple trading scheme:in period 1, we put 100% of wealth into asset i , and then in period 2, we take allthe proceeds and roll them over into asset j . Hence, there are m different extremal On account of allocation drift, e.g. the fact that some constituent assets will outperform theportfolio each period (and some assets will underperform), a CRP must generally trade each periodso as to restore the target allocation c := ( c , ..., c m ) (cid:48) . Literally, an extreme point of B . Note on Universal Bilinear Portfolios
A. Garivaltisstrategies ( i, j ) ∈ { , ..., m } × { , ..., m } ; since the ( i, j ) th extremal strategy yields acapital growth factor of x i y j , it therefore amounts to to the bilinear trading strategy B := e i e (cid:48) j , which is an extreme point of B . The general bilinear portfolio B := [ b ij ] m × m is uniquely representable as a convex combination B = m (cid:88) i =1 m (cid:88) j =1 b ij e i e (cid:48) j (10)of extremal strategies; this means that the practitioner of B has elected to invest thefraction b ij of his initial dollar into each extremal strategy ( i, j ). Thus, after the elapseof two periods, the investor’s total wealth will be equal to m (cid:80) i =1 m (cid:80) j =1 b ij x i y j = x (cid:48) By.
We now consider the on-line learning of the asymptotically dominant (or growth-optimal) bilinear portfolio. To this end, we assume that there are T basic investmentperiods t ∈ { , ..., T } , each of which is divided into a “first half” (during which thegross return vector is x t ) and a “second half” (during which the gross return vector is y t .) We let x t := ( x , ..., x t ) ∈ (cid:0) R m + − { } (cid:1) t denote the history of returns in the firsthalves of periods 1 , ..., t , and, likewise, we let y t := ( y , ..., y t ) ∈ (cid:0) R m + − { } (cid:1) t denotethe return history for the latter halves of periods 1 , ..., t . Thus, we have the transitionlaws x t +1 := ( x t , x t +1 ) and y t +1 := ( y t , y t +1 ), where x and y denote empty histories.We let W B ( x t , y t ) := t (cid:89) s =1 x (cid:48) s By s (11)9 Note on Universal Bilinear Portfolios
A. Garivaltisdenote the final wealth function of the bilinear trading strategy B against thereturn history ( x t , y t ); similarly, we write W B ( x t , y t − ) := (cid:32) t − (cid:89) s =1 x (cid:48) s By s (cid:33) × ( x (cid:48) t B ) = W B (cid:0) x t , ( y t − , ) (cid:1) (12)if period t has only been half-completed. We will consider sequential investmentstrategies ˆ B ( • , • ) that, at the start of each period t , select some bilinear portfolioˆ B ( x t − , y t − ) := (cid:104) ˆ b ij ( x t − , y t − ) (cid:105) m × m that is conditioned on the observed return his-tory ( x t − , y t − ); this bilinear portfolio will be used for the entire duration of period t . The capital growth factor achieved by an investment scheme ˆ B ( • , • ) against thehistory ( x t , y t ) is equal to ˆ W ( x t , y t ) := t (cid:89) s =1 x (cid:48) s ˆ B ( x s − , y s − ) y s , (13)and, if period t is only half-finished, we writeˆ W ( x t , y t − ) := (cid:34) t − (cid:89) s =1 x (cid:48) s ˆ B ( x s − , y s − ) y s (cid:35) × (cid:104) x (cid:48) t ˆ B ( x s − , y s − ) (cid:105) = ˆ W (cid:0) x t , ( y t − , ) (cid:1) . (14)Within a given period t , the on-line behavior of ˆ B ( • , • ) amounts to the portfoliovectors ˆ p ( x t − , y t − ) := ˆ B ( x t − , y t − ) andˆ q ( x t , y t − ) := ˆ B ( x t − , y t − ) (cid:48) x t x (cid:48) t ˆ B ( x t − , y t − ) . (15)In order to have a practical benchmark for the on-line performance of ˆ B ( • , • )after the elapse of t complete investment periods, we will consider the best bilinear The initial monetary deposit into B is equal to the empty product W B ( x , y ) := $1. Note on Universal Bilinear Portfolios
A. Garivaltis trading strategy in hindsight for the individual sequence ( x t , y t ): B ∗ ( x t , y t ) := arg max B ∈B W B ( x t , y t ) (16)and B ∗ ( x t , y t − ) := arg max B ∈B W B ( x t , y t − ) = B ∗ (cid:0) x t , ( y t − , ) (cid:1) . (17)The final wealth that accrues to B ∗ ( x t , y t ) is a path-dependent financial derivative,with payoff D ( x t , y t ) := max B ∈B W B ( x t , y t ) = W B ∗ ( x t ,y t ) ( x t , y t ) (18)and D ( x t , y t − ) := max B ∈B W B ( x t , y t − ) = D (cid:0) x t , ( y t − , ) (cid:1) . (19) Proposition 2.
The final wealth function W B ( x T , y T ) is a multilinear form in thevectors x , y , x , y , ..., x T , y T , e.g. it is linear separately in each vector x t and alsoin each vector y t , for ≤ t ≤ T. Consequently, the hindsight-optimized final wealth D ( x T , y T ) is convex and positively homogeneous separately in each x t and also in each y t .Proof. The multi-linearity of W B ( • , • ) follows easily from the definition, e.g. W B ( x T , y T ) = (cid:18) t − (cid:81) s =1 x (cid:48) s By s (cid:19) · ( x (cid:48) t By t ) · (cid:18) T (cid:81) s = t +1 x (cid:48) s By s (cid:19) is clearly additive and homogeneous in x t andalso in y t . If we write D ( x t ) and view D ( • , • ) as a function of x t alone, then the con-vexity and homogeneity with respect to x t (or with respect to y t ) follow from the factthat the mapping x t (cid:55)→ D ( x t ) is a pointwise maximum of a family of linear functions,namely, ( W B ( x t )) B ∈B . (cid:4) For obvious reasons, the hindsight-optimized payoff D ( • , • ) is not achievable byany causal (or non-anticipating) investment strategy ˆ B ( • , • ); however, it is possible11 Note on Universal Bilinear Portfolios
A. Garivaltisto achieve any average ˆ W ( x t , y t ) := (cid:90) B ∈B W B ( x t , y t ) f ( B ) dB, (20)where f ( • ) is a continuous density function over B . That is, inspired by ThomasCover (1991) and Cover and Ordentlich (1996), we make the following definition. Definition 2.
The universal bilinear portfolio (that corresponds to the priordensity f ( • ) ) is a performance-weighted average of all bilinear-trading strategies: ˆ B ( x t , y t ) := (cid:82) B ∈B B · W B ( x t , y t ) f ( B ) dB (cid:82) B ∈B W B ( x t , y t ) f ( B ) dB = E f [ B · W B ( x t , y t )] E f [ W B ( x t , y t )] . (21)So-defined, the matrix ˆ B ( • , • ) is indeed a valid bilinear portfolio, on account of thefact that ˆ B ( x t , y t ) ≥ (cid:48) ˆ B ( x t , y t ) = 1. The initial bilinear portfolio ˆ B ( x , y )is equal to the center of mass (cid:82) B ∈B Bf ( B ) dB = E f [ B ] that is induced by the priordensity f ( • ). Proposition 3.
After T complete investment periods, the universal wealth ˆ W ( x T , y T ) is equal to the average value ˆ W ( x T , y T ) = (cid:90) B ∈B W B ( x T , y T ) f ( B ) dB = E f (cid:2) W B ( x T , y T ) (cid:3) . (22) By the way, if a discrete-time payoff D ( x t , y t ) = ˆ W ( x t , y t ) can be exactly replicated (or hedged)by some causal (non-anticipating) trading strategy, then that strategy is necessarily be unique. Wehave encountered this phenomenon already vis-´a-vis the bilinear payoff x (cid:48) By . Note on Universal Bilinear Portfolios
A. Garivaltis
Proof.
The gross return of the universal bilinear portfolio in period t is given by x (cid:48) t ˆ B ( x t − , y t − ) y t = (cid:82) B ∈B ( x (cid:48) t By t ) · W B ( x t − , y t − ) f ( B ) dB (cid:82) B ∈B W B ( x t − , y t − ) f ( B ) dB = (cid:82) B ∈B W B ( x t , y t ) f ( B ) dB (cid:82) B ∈B W B ( x t − , y t − ) f ( B ) dB . (23)Taking the (telescopic) product of both sides of equation (23) for t := 1 , ..., T , andbearing in mind that W B ( x , y ) = 1 = (cid:82) B ∈B f ( B ) dB , we arrive at the desired result:ˆ W ( x T , y T ) = (cid:82) B ∈B W B ( x T , y T ) f ( B ) dB . (cid:4) Following Cover (1991) and Cover and Thomas (2006), the intuition behind theuniversal bilinear portfolio is just this: we distribute the initial dollar (according to f ( • )) among all the bilinear trading strategies B ∈ B , whereby the bilinear port-folios in the neighborhood of a given B receive f ( B ) dB dollars to manage (fromnow until kingdom come). After the elapse of t complete investment periods, thebilinear strategies in this locale have grown their bankroll to W B ( x t , y t ) f ( B ) dB ; theinvestor’s aggregate wealth is thereby equal to (cid:82) B ∈B W B ( x t , y t ) f ( B ) dB. With this intu-ition in hand, the formula for ˆ B ( x t , y t ) can be written down immediately, on accountof the fact that the locale of a given B is responsible for managing the fraction φ ( B ) dB := W B ( x t , y t ) f ( B ) dB (cid:30) (cid:82) B ∈B W B ( x t , y t ) f ( B ) dB of the aggregate wealth.Hence, the overall bilinear portfolio is just the convex combination ˆ B = (cid:82) B ∈B B · φ ( B ) dB . Over long periods of time, the bilinear trading strategies in the neighbor-hood of B ∗ ( x t , y t ) will come to control an ever-greater share of the aggregate wealth,on account of their superior exponential growth rate, namely (1 /t ) log D ( x t , y t ). Thus,the aggregate bankroll will (asymptotically) compound itself at this same rate; that13 Note on Universal Bilinear Portfolios
A. Garivaltisis, we have the relationlim t →∞ (Excess Growth Rate of the Best Bilinear Portfolio in Hindsight)= lim t →∞ (cid:34) log D ( x t , y t ) t − log ˆ W ( x t , y t ) t (cid:35) = 0 , (24)regardless of the individual return sequence ω := ( x t , y t ) ∞ t =1 . The remainder of thepaper is concerned with fleshing out the necessary details. On that score, we makethe definition: Definition 3.
The competitive ratio R ( x T , y T ) measures the percentage of hindsight-optimized bilinear wealth that was actually achieved by the universal bilinear portfolio,e.g. R ( x T , y T ) := ˆ W ( x T , y T ) D ( x T , y T ) = Average Value of W B ( x T , y T ) Maximum Value of W B ( x T , y T ) . (25) Lemma 1.
The competitive ratio R ( • , • ) is always ≤ ; it is homogeneous of degree0 and quasi-concave separately in each vector x t and also in each vector y t .Proof. The fact that R ( x T , y T ) ≤ W B ( x T , y T )) B ∈B cannot exceedtheir maximum. The homogeneity of degree 0 follows from the fact that W B ( • , • )and D ( • , • ) are both linearly homogeneous (of degree 1) in each vector x t or y t . Themulti-quasi-concavity obtains from the fact that, when viewed as a function R ( x t ) of x t alone (or of y t alone), we are dealing with the ratio of a positive linear function(namely, ˆ W ( x t )) to a positive convex function (viz., D ( x t )). That is, if we consider Not just almost everywhere; but everywhere , for all possible ω ∈ (cid:16)(cid:0) R m + − { } (cid:1) (cid:17) N . Note on Universal Bilinear Portfolios
A. Garivaltisthe upper contour sets U α := (cid:8) x t ∈ R m + : R ( x t ) ≥ α (cid:9) = (cid:110) x t ∈ R m + : ˆ W ( x t ) − αD ( x t ) ≥ (cid:111) , (26)then we see that U α is a convex set for all α ∈ R . For, if α ≤
0, then U α = R m + ,which is convex; if α ≥
0, then U α is convex because it is an upper contour set of theconcave function x t (cid:55)→ ˆ W ( x t ) − αD ( x t ). (cid:4) On account of the (multi-) homogeneity of degree 0, the competitive ratio onlycares about the directions of the vectors x t or y t — their lengths do not affect therelative performance of the universal bilinear portfolio. Thus, we are free to scaleeach x t (resp. y t ) by a factor of λ := 1 / || x t || (resp. 1 / || y t || ), so that the coordinatesof x t (resp. y t ) sum to one, e.g. we may assume that each x t or y t belongs to the unitsimplex ∆ m . Hence, we have the relation R ( x T , y T ) := inf ( x T ,y T ) ∈ ( R m + −{ } ) T R ( x T , y T ) = min ( x T ,y T ) ∈ ∆ Tm R ( x T , y T ) , (27)e.g. the worst-case relative performance R ( x T , y T ) obtains over the product of sim-plices ∆ Tm . Even better, since R ( • , • ) is multi-quasi-concave, its minimum value mustin fact be realized at some extreme point ( x T , y T ) ∈ { e , ..., e m } T , e.g. a return his-tory whereby all x t , y t are unit basis vectors. This happens on account of the factthat when R ( • , • ) is viewed as a function solely of x t ∈ ∆ m (or solely of y t ∈ ∆ m ),we have R ( x t ) = R ( x t e + · · · + x tm e m ) ≥ min { R ( e ) , ..., R ( e m ) } = R ( e i ∗ ) , (28) Come what may — for all possible market behavior ( x T , y T ). Note on Universal Bilinear Portfolios
A. Garivaltisso that the competitive ratio can always be reduced by replacing any x t or y t by anappropriate unit basis vector e i ∗ .In what follows, we will consider sequences of unit basis vectors x T := ( e i , ..., e i T )and y T := ( e j , ..., e j T ), where i T := ( i , ..., i T ) ∈ { , ..., m } T and j T := ( j , ..., j T ) ∈{ , ..., m } T . For the sake of simplicity, we will abuse notation by writing the (self-evident) expressions R ( i T , j T ), ˆ W ( i T , j T ), and D ( i T , j T ). Sequences of unit basisvectors will hereby be referred to as extremal sequences , or Kelly horse racesequences , on account of the fact that they correspond to betting markets (say, horseraces or prediction markets) whereby only one of the m assets has a positive grossreturn. For a given Kelly sequence ( i T , j T ), we will require the counts, or relativefrequencies n ij ( i T , j T ) := (number of times ( i t , j t ) = ( i, j )) = (cid:88) { t :( i t ,j t )=( i,j ) } , (29)so that n ij ≥ m (cid:80) i =1 m (cid:80) j =1 n ij = T . Lemma 2.
For any Kelly sequence ( i T , j T ) , the final wealth of the best bilinear tradingstrategy in hindsight is equal to D (cid:16) [ n ij ] mi,j =1 (cid:17) = (cid:81) ( i,j ): n ij > ( n ij /T ) n ij ; the universalwealth ˆ W ( i T , j T ) admits the minorant ˆ W ( i T , j T ) ≥ f ( T + m − m (cid:89) i =1 m (cid:89) j =1 n ij ! , (30) where f := min B ∈B f ( B ) is the minimum weight assigned to any bilinear portfolio by theprior density f ( • ) .Proof. Against the Kelly sequence ( i T , j T ), the final wealth of the bilinear trading16 Note on Universal Bilinear Portfolios
A. Garivaltisstrategy B is given by W B ( i T , j T ) = T (cid:89) t =1 b i t j t = (cid:89) ( i,j ): n ij > b n ij ( i T ,j T ) ij . (31)Maximization of this quantity with respect to B amounts to a standard Cobb-Douglasoptimization problem over the unit simplex in R m + . Lagrange’s multipliers yield thesolution b ∗ ij = n ij /T , so that D ( i T , j T ) = (cid:81) ( i,j ): n ij > ( n ij /T ) n ij .The stated minorant for ˆ W ( i T , j T ) will be gotten by direct integration of W B ( i T , j T )over the set of bilinear trading strategies. To this end, we will identify B with thesolid region (cid:110) ( b , ..., b m , b , ..., b m , ..., b m , ..., b m,m − ) ∈ R m − : b + · · · + b m,m − ≤ (cid:111) , (32)where b mm = 1 − b − · · · − b m,m − is not a free variable. Thus, we must evaluate the( m − (cid:90) b =0 1 − b (cid:90) b =0 ··· − b − ··· − b m,m − (cid:90) b m,m − =0 (cid:89) ( i,j ) (cid:54) =( m,m ) b n ij ij − (cid:88) ( i,j ) (cid:54) =( m,m ) b ij n mm f ( B ) db m,m − ··· db . (33)17 Note on Universal Bilinear Portfolios
A. GarivaltisUsing the fact that f ( B ) ≥ f , and recalling the general identity (cid:90) z =0 1 − z (cid:90) z =0 · · · − z −···− z k − (cid:90) z k − =0 z α z α · · · z α k − k − (1 − z − z − · · · − z k − ) α k dz k − · · · dz dz = Γ( α + 1)Γ( α + 1) · · · Γ( α k + 1)Γ( α + α + · · · + α k + k ) , (34)where Γ( • ) is the gamma function, we put k := m and obtainˆ W ( i T , j T ) ≥ f · m (cid:81) i =1 m (cid:81) j =1 Γ( n ij + 1)Γ (cid:32) m + m (cid:80) i =1 m (cid:80) j =1 n ij (cid:33) = f · m (cid:81) i =1 m (cid:81) j =1 n ij !( T + m − , (35)as promised. (cid:4) Corollary 1.
The competitive ratio has the following (uniform) bounds, for all x T , y T : ≥ R ( x T , y T ) ≥ f ( T + 1)( T + 2) · · · ( T + m − ∼ fT m − . (36) Hence, the excess continuously-compounded per-period growth rate of the best bilin-ear portfolio in hindsight (namely, − (1 /T ) log R ( x T , y T ) ) is sandwiched by ≤ Excess Growth Rate ≤ log (cid:0) /f (cid:1) T + 1 T m − (cid:88) j =1 log( T + j ) . (37) This identity follows by direct evaluation of the iterated integral (34). In order to accomplishthis, one must repeatedly invoke the special case k := 2, e.g. (cid:82) z =0 z α (1 − z ) β dz = Γ( α + 1)Γ( β +1) / Γ( α + β + 2) , which is the beta function, or Euler integral of the first kind (cf. with David Widder1989). The relation ∼ signifies that the two sequences are asymptotically equivalent , e.g. a n ∼ b n meansthat lim n →∞ a n /b n = 1. That is, per complete investment period (both halves). Note on Universal Bilinear Portfolios
A. Garivaltis
That is, at worst, the excess growth rate is asymptotically equivalent to the quantity ( m −
1) log( T ) /T .Proof. For any Kelly sequence ( i T , j T ), Lemma 1 implies that R ( i T , j T ) = ˆ W ( i T , j T ) (cid:81) ( i,j ): n ij > ( n ij /T ) n ij ≥ f · T T ( T + m − m (cid:89) i =1 m (cid:89) j =1 n ij ! n n ij ij , (38)where the right-hand side makes use of the convention that 0 := 1. Now, note thatthe integer program min (cid:40) [ n ij ] ≥ m (cid:80) i =1 m (cid:80) j =1 n ij = T (cid:41) m (cid:89) i =1 m (cid:89) j =1 n ij ! n n ij ij (39)is solved by setting any entry of the matrix [ n ij ] m × m to T and setting all the otherentries to zero, e.g. we have the well-known inequality (cf. with Cover and Ordentlich1996) m (cid:89) i =1 m (cid:89) j =1 n ij ! n n ij ij ≥ T ! T T . (40)Hence, there lies R ( x T , y T ) ≥ min ( i T ,j T ) ∈{ ,...,m } T R ( i T , j T ) ≥ f ( T + 1)( T + 2) · · · ( T + m − . (41) (cid:4) Theorem 1.
The universal bilinear portfolio asymptotically dominates the original(1-linear) universal portfolio in precisely the same technical sense that the universal1-linear portfolio asymptotically dominates all constant-rebalanced portfolios and allbuy-and-hold strategies.If it turns out that the best bilinear trading strategy in hindsight sustains a higherasymptotic capital growth rate than the best constant-rebalanced portfolio in hindsight, Note on Universal Bilinear Portfolios
A. Garivaltis then the universal bilinear portfolio will asymptotically outperform the universal 1-linear portfolio by an exponential factor.Proof.
We letˆ S ( x t , y t ) := (cid:90) c ∈ ∆ m (cid:34) t (cid:89) s =1 ( c (cid:48) x s ) ( c (cid:48) y s ) (cid:35) g ( c ) dc = E g (cid:34) t (cid:89) s =1 ( c (cid:48) x s ) ( c (cid:48) y s ) (cid:35) (42)denote the wealth of the universal 1-linear portfolio (cf. with Thomas Cover 1991 andCover and Ordentlich 1996) after the elapse of t complete investment periods, where∆ m is the unit portfolio simplex in R m + and g ( • ) is a prior density over ∆ m . The finalwealth of the best constant-rebalanced portfolio in hindsight will be denoted S ∗ ( x t , y t ) := max c ∈ ∆ m t (cid:89) s =1 ( c (cid:48) x s ) ( c (cid:48) y s ) . (43)On account of the lower boundˆ W ( x t , y t )ˆ S ( x t , y t ) = ˆ W ( x t , y t ) D ( x t , y t ) · D ( x t , y t ) S ∗ ( x t , y t ) · S ∗ ( x t , y t )ˆ S ( x t , y t ) ≥ f m − (cid:81) j =1 ( t + j ) · D ( x t , y t ) S ∗ ( x t , y t ) · , (44)we can minorize the asymptotic excess growth rate (of the universal bilinear portfoliorelative to the universal 1-linear portfolio) as follows:lim inf t →∞ (cid:34) log ˆ W ( x t , y t ) t − log ˆ S ( x t , y t ) t (cid:35) ≥ lim inf t →∞ (1 /t ) log (cid:32) f (cid:30) m − (cid:89) j =1 ( t + j ) (cid:33) + lim inf t →∞ (cid:20) log D ( x t , y t ) t − log S ∗ ( x t , y t ) t (cid:21) = lim inf t →∞ (cid:20) log D ( x t , y t ) t − log S ∗ ( x t , y t ) t (cid:21) ≥ , (45)20 Note on Universal Bilinear Portfolios
A. Garivaltiswhere we have made use of the fact that the relations S ∗ ( x t , y t ) ≥ ˆ S ( x t , y t ) and D ( x t , y t ) ≥ S ∗ ( x t , y t ) hold for all x t and all y t .Thus, we have shown that even the smallest subsequential limit of the excessgrowth rate (1 /t ) log (cid:16) ˆ W / ˆ S (cid:17) is non-negative; if the best bilinear trading strategyin hindsight happens to achieve a higher asymptotic growth rate than the bestconstant-rebalanced portfolio in hindsight (in the sense that the smallest subsequen-tial limit of (1 /t ) log ( D/S ∗ ) is strictly positive), then the universal bilinear portfoliowill asymptotically outperform the universal 1-linear portfolio by an exponential fac-tor. (cid:4) To close out the paper, this subsection provides exact formulas for the behavior ofthe universal bilinear portfolio in the context of our original motivating example (asdiscussed in the introduction) for the case of m := 2 assets. Accordingly, we willassume that asset 2 is cash (which pays no interest) and that asset 1 is a “hot stock”that always doubles in the first half of each investment period and then loses 50% ofits value in the latter half of each investment period. Thus, we have the individualreturn sequence defined by x t : ≡ (2 , (cid:48) and y t : ≡ (1 / , (cid:48) . As depicted by Figure 1,the set of all bilinear trading strategies now consists in the tetrahedron B := (cid:8) ( b , b , b ) ∈ R : b + b + b ≤ (cid:9) , (46)where the variable b is bound by the relation b := 1 − b − b − b .Analogous to Thomas Cover (1991), we will use the uniform prior density f ( b , b , b ) ≡ The practitioner of the universal bilinear portfolio must hope against hope that the individualreturn sequence ω := ( x t , y t ) ∞ t =1 has this pleasant feature. Note on Universal Bilinear Portfolios
A. Garivaltis b b The Set of Bilinear Trading Strategies Over Two Assets b b + b + b b ij b := 1 ! b ! b ! b : Figure 1:
Geometric depiction of the set B of all possible bilineartrading strategies B := [ b ij ] × over two assets. The defining relationsare B ≥ b + b + b ≤ and b := 1 − b − b − b . The volume of thistetrahedron is / .
6, e.g. the volume of the tetrahedron B is given byVolume( B ) = (cid:90) b =0 1 − b (cid:90) b =0 1 − b − b (cid:90) b =0 db db db = 16 . (47)During each (complete) investment period, the (intra-period) capital growth factorachieved by the bilinear trading strategy B amounts to x (cid:48) t By t = (cid:20) (cid:21) b b b − b − b − b / = 1 + b − b , (48)so that W B ( x t , y t ) = (1 + b + b / t . Thus, the universal wealth ˆ W ( x t , y t ) thatobtains after the elapse of t complete investment periods is found by evaluating the22 Note on Universal Bilinear Portfolios
A. Garivaltistriple integral6 (cid:90) b =0 1 − b (cid:90) b =0 1 − b − b (cid:90) b =0 (cid:18) b − b (cid:19) t db db db = 2 t +5 − t + 2) − − t ( t + 1)( t + 2)( t + 3) ∼ t · t . (49)The best bilinear trading strategy in hindsight is obviously B ∗ ( x t , y t ) ≡ , (50)e.g. the extremal strategy that bets the ranch on the stock in the first half ofeach investment period, and then cashes out completely in the latter half of eachinvestment period. This (perfect trading) yields the hindsight-optimized wealth D ( x t , y t ) = D ( x t , y t − ) = 2 t , which corresponds to the asymptotic growth ratelim t →∞ (1 /t ) log D ( x t , y t ) = log 2 = 69 .
3% per complete investment period, compoundedcontinuously. The competitive ratio after t full periods is equal to R ( x t , y t ) = 32 − t + 2)2 − t − − t ( t + 1)( t + 2)( t + 3) ∼ t . (51)Note well that Corollary 1 promised us the minorant R ( x t , y t ) ≥ t + 1)( t + 2)( t + 3) , (52)which is indeed correct; we of course have lim t →∞ (1 /t ) log R ( x t , y t ) = 0, so that theuniversal bilinear portfolio compounds its money at the same asymptotic rate as thebest bilinear trading strategy in hindsight.23 Note on Universal Bilinear Portfolios
A. GarivaltisAgainst this individual return sequence, the universal bilinear portfolio finds itsexpression in the triple integral6ˆ W ( x t , y t ) (cid:90) b =0 1 − b (cid:90) b =0 1 − b − b (cid:90) b =0 (cid:18) b − b (cid:19) t b b b − b − b − b db db db . (53)With some effort, one can explicitly evaluate the on-line bilinear weights, as follows:ˆ b ( x t , y t ) = 2 · t +2 − · t ( t + 5 t + 10)( t + 4)[4 t +2 − · t +1 ( t + 2) + 1] ∼ t → , (54)ˆ b ( x t , y t ) = 2 t +4 (3 t −
4) + 18( t + 4) + 2 − t t + 4)[2 t +4 − t + 2) − − t ] → , (55)ˆ b ( x t , y t ) = 2 t +6 − t + 1) − − t (3 t + 19)3( t + 4)[2 t +4 − t + 2) − − t ] ∼ t → , (56)ˆ b ( x t , y t ) = ˆ b ( x t , y t ) ∼ t → . (57)Notice that the (1 ,
1) and (2 ,
2) extremal strategies (which both amount to buy-and-hold strategies) are assigned equal weights by the universal bilinear portfolio (in thesense that ˆ b = ˆ b ); this happens on account of the fact that both assets produceidentical results for a buy-and-hold investor over any complete investment period.Thus, the universal bilinear portfolio learns to trade perfectly in as much aslim t →∞ ˆ B ( x t , y t ) = . (58)The same cannot be said for the universal 1-linear portfolio, which achieves the capital24 Note on Universal Bilinear Portfolios
A. Garivaltisgrowth factor ˆ S ( x t , y t ) = (cid:90) c =0 (1 + c ) t (1 − c/ t dc = t (cid:88) k =0 t − k (cid:88) k =0 (cid:18) tk , k , t − k − k (cid:19) ( − k k + k ( k + 2 k + 1) . (59)After t complete investment periods, the best constant-rebalanced portfolio in hind-sight is equal to (1 / , / (cid:48) , which corresponds to the (sub-optimal) bilinear tradingstrategy B = / / / / . The final wealth of the best constant-rebalanced portfo-lio in hindsight is thereby S ∗ ( x t , y t ) = (9 / t . Thus, the excess asymptotic growthrate of the universal bilinear portfolio (over and above that of the universal 1-linearportfolio) is log 2 − log(9 /
8) = 57 .
5% per (complete) investment period, compoundedcontinuously.For the sake of visualization, Figure 2 plots the bankroll of the universal bilinearportfolio in comparison to that of the universal 1-linear portfolio and the wealthachieved by a perfect trader. The lower panel illustrates the parameter learning thatobtains from the performance-weighted average of all bilinear trading strategies.
In this note, we constructed a neat application and extension of the brilliantly lucidOrdentlich-Cover theory of “universal portfolios.” The original (1-linear) universalportfolios guarantee to achieve a high percentage of the final wealth that would haveaccrued to the best constant-rebalanced portfolio in hindsight for the actual (realized) Here, we have used the uniform prior density g ( c ) ≡ , Note on Universal Bilinear Portfolios
A. Garivaltissequence of asset prices.The constant-rebalanced portfolios constitute a very simple parametric family ofactive trading strategies, where the “activity” amounts to continuously executingrebalancing trades so as to restore the portfolio to a given target allocation. Inspiredby the fact that a constant-rebalanced portfolio is a (horizon-1) trading strategy whosecapital growth factor in any given period is a linear function of the market’s grossreturn vector, we decided to consider the wider class of bilinear trading strategies (or bilinear portfolios ), which are mini 2-period active strategies whose capital growthfactors are linear separately in the two gross return vectors.Accordingly, we hit upon the more powerful benchmark of the best bilinear trad-ing strategy in hindsight for the actual sequence of asset prices. This led us to applyCover’s ingenious (1991) performance-weighted averaging technique to this new sit-uation, e.g. the universal bilinear portfolio is a performance-weighted average of allpossible bilinear trading strategies.Applying Cover and Ordentlich’s elegant (1996) methodology, we showed that forany financial market with m assets , at worst, the percentage of hindsight-optimizedwealth achieved by the universal bilinear portfolio will tend to zero like the quantity T − ( m − as T → ∞ , where T denotes the number of complete (bipartite) invest-ment periods. Consequently, the universal bilinear portfolio succeeds in matchingthe performance of the best bilinear trading strategy in hindsight to “first order inthe exponent,” e.g. the excess continuously-compounded per-period capital growthrate of the best bilinear trading strategy in hindsight converges (uniformly) to zero,regardless of the individual sequence of asset prices.Thus, we showed that the universal bilinear portfolio asymptotically dominatesthe universal 1-linear portfolio in the same technical sense that the universal 1-linear One of which can be cash, or a risk-free bond. Note on Universal Bilinear Portfolios
A. Garivaltisportfolio asymptotically dominates all constant-rebalanced portfolios and all buy-and-hold strategies. The universal bilinear portfolio will beat the universal 1-linearportfolio by an exponential factor, provided that the individual sequence of assetprices enjoys the property that the best bilinear trading strategy in hindsight achievesan asymptotic growth rate that is strictly greater than that of the best constant-rebalanced portfolio in hindsight.Analogously, we can get carried away and define the concept of a trilinear tradingstrategy B := ( b ijk ) mi,j,k =1 , whose (horizon-3) capital growth factor in any (tripartite)period t is equal to the trilinear form (cid:104) x t , y t , z t (cid:105) B := m (cid:88) i =1 m (cid:88) j =1 m (cid:88) k =1 b ijk x ti y tj z tk , (60)where b ijk ≥ m (cid:80) i =1 m (cid:80) j =1 m (cid:80) k =1 b ijk = 1. This leads to a universal trilinear portfolio whose worst-case competitive ratio behaves like T − ( m − as T → ∞ . In general, an H -linear trading strategy (cf. with Garivaltis 2018b) divides each period t into H sub-periods, wherein the gross return vectors are denoted ( x t , x t , ..., x ht , ..., x Ht ) = ( x ht ) Hh =1 .Intra-period capital growth is now generated by the H -linear form (cf. with SergeLang 1987) (cid:104) x t , ..., x Ht (cid:105) B := (cid:88) ( i ,...,i H ) ∈{ ,...,m } H (cid:40) B ( i , ..., i H ) H (cid:89) h =1 x hti h (cid:41) , (61)where B ( i , ..., i H ) ≥ (cid:80) ( i ,...,i H ) ∈{ ,...,m } H B ( i , ..., i H ) = 1; the attendant universal H -linear portfolio asymptotically achieves, at worst, the fraction T − ( m H − of the finalwealth of the best H -linear trading strategy in hindsight.Hence, one can use this method to construct an endless hierarchy of ever more27 Note on Universal Bilinear Portfolios
A. Garivaltisdominant universal portfolios. If the horizon H is an integer multiple of the horizon H , say H := q · H , then the act of repeating a given H -linear portfolio B for q times in succession constitutes a special type of H -linear portfolio; the universal H -linear portfolio thereby asymptotically outperforms the universal H -linear portfolio“to first order in the exponent,” ´a la Cover. Northern Illinois University
Disclosures.
This paper is solely the work of the author, who declares that he has no conflicts ofinterest; the work was funded entirely through his regular academic appointment atNorthern Illinois University.
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Note on Universal Bilinear Portfolios
A. Garivaltis
Number of Full Investment Periods ( t ) N u m b e r o f C a p i t a l D o ub li n g s Performance Against x t := (2 ; and y t := (0 : ; for all t log (Stock Price)log (Wealth of 50-50 CRP)log (Wealth of Perfect Trader)log (Universal Bilinear Port.)log (Cover's (1991) U.P.) Number of Full Investment Periods ( t ) W e i g h t o n E x t r e m a l S t r a t e g y Evolution of the Universal Bilinear Portfolio^ b ; ^ b ^ b ^ b Figure 2: Superior performance of the universal bilinear portfolio against the individ-ual return sequence x t : ≡ (2 , (cid:48) and y t : ≡ (0 . , (cid:48) . Asset 2 is cash (that pays no in-terest); asset 1 is a “hot stock” that doubles in the first half of each investment periodand loses 50% of its value in the latter half of each investment period. Note that in thebottom plot, we have lim t →∞ ˆ b ( x t , y t ) = 1 and ˆ b ( x t , y t ) ≡ ˆ b ( x t , y t ) ∼ /t →→