# A Note on Universal Bilinear Portfolios

## Abstract

**:**

We first investigate what a natural goal might be for the growth of wealth for arbitrary market sequences. For example, a natural goal might be to outperform the best buy-and-hold strategy, thus beating an investor who is given a look at a newspaper n days in the future. We propose a more ambitious goal.—Thomas M. Cover, Universal Portfolios, 1991

In 1988, out of the blue, Paul Samuelson wrote a letter to Stanford information theorist Thomas Cover. Samuelson had been sent one of Cover’s papers on portfolio theory for review. “If I did use some of your procedures,” Samuelson wrote, “I would not let that … bias my portfolio choice toward choices my alien cousin with log utility would make”. He chides Kelly, Latané, Markowitz, and “various Ph.D’s who appear with Poisson-distribution probabilities most Junes”.—William Poundstone, Fortune’s Formula, 2005

With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.—John von Neumann

## 1. Introduction; Literature Review

## 2. Bilinear Trading Strategies

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**extremal strategy**4 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 all the proceeds and roll them over into asset j. Hence, there are ${m}^{2}$ different extremal strategies $(i,j)\in \{1,...,m\}\times \{1,...,m\}$; since the ${(i,j)}^{th}$ extremal strategy yields a capital growth factor of ${x}_{i}{y}_{j}$, it therefore amounts to the bilinear trading strategy $B:={e}_{i}{e}_{j}^{\prime}$, which is an extreme point of $\mathcal{B}$. The general bilinear portfolio $B:={\left[{b}_{ij}\right]}_{m\times m}$ is uniquely representable as a convex combination

## 3. Universal Bilinear Portfolios

**final wealth function**5 of the bilinear trading strategy B against the return history $({x}^{t},{y}^{t})$; similarly, we write

**best bilinear trading strategy in hindsight**for the individual sequence $({x}^{t},{y}^{t})$:

**Proposition**

**2.**

**Proof.**

**Definition**

**2.**

**universal bilinear portfolio**(that corresponds to the prior density $f(\u2022)$) is a performance-weighted average of all bilinear-trading strategies:

**Proposition**

**3.**

**Proof.**

**Definition**

**3.**

**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.,

**Lemma**

**1.**

**Proof.**

**extremal sequences**, or

**Kelly horse race sequences**, on account of the fact that they correspond to betting markets (say, horse races or prediction markets) whereby only one of the m assets has a positive gross return. For a given Kelly sequence $({i}^{T},{j}^{T})$, we will require the counts, or relative frequencies

**Lemma**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### Resolution of the Motivating Example

## 4. Summary and Conclusions

#### Disclosures

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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1. | e.g., if ${x}_{i}:=1.05$ then asset i appreciated $5\%$ in period 1; if ${x}_{i}:=0.98$, then asset i lost $2\%$ of its value in period 1, etc. |

2. | Bilinearity (cf. with Serge Lang (1987)) refers to the fact that the capital growth factor ${x}^{\prime}By$ is linear separately in each of the vectors x and y. When viewed jointly as a function of $(x,y)$, the bilinear form ${x}^{\prime}By$ is a homogeneous quadratic polynomial in the $2m$ variables ${x}_{1},...,{x}_{m},{y}_{1},...,{y}_{m}$. |

3. | On account of allocation drift, e.g., the fact that some constituent assets will outperform the portfolio each period (and some assets will underperform), a CRP must generally trade each period so as to restore the target allocation $c:={({c}_{1},...,{c}_{m})}^{\prime}$. |

4. | Literally, an extreme point of $\mathcal{B}$. |

5. | The initial monetary deposit into B is equal to the empty product ${W}_{B}({x}^{0},{y}^{0}):=\$1$. |

6. | By the way, if a discrete-time payoff $D({x}^{t},{y}^{t})=\widehat{W}({x}^{t},{y}^{t})$ can be exactly replicated (or hedged) by some causal (non-anticipating) trading strategy, then that strategy is necessarily be unique. We have encountered this phenomenon already vis-á-vis the bilinear payoff ${x}^{\prime}By$. |

7. | Not just almost everywhere; but everywhere, for all possible $\omega \in {\left({\left({\mathbb{R}}_{+}^{m}-\left\{0\right\}\right)}^{2}\right)}^{\mathbb{N}}$. |

8. | Come what may—for all possible market behavior $({x}^{T},{y}^{T})$. |

9. | This identity follows by direct evaluation of the iterated integral (33). In order to accomplish this, one must repeatedly invoke the special case $k:=2$, e.g., $\underset{z=0}{\overset{1}{\int}}}{z}^{\alpha}{(1-z)}^{\beta}dz=\mathsf{\Gamma}(\alpha +1)\mathsf{\Gamma}(\beta +1)/\mathsf{\Gamma}(\alpha +\beta +2),$ which is the beta function, or Euler integral of the first kind (cf. with David Widder (1989)). |

10. | That is, per complete investment period (both halves). |

11. | The practitioner of the universal bilinear portfolio must hope against hope that the individual return sequence $\omega :={({x}_{t},{y}_{t})}_{t=1}^{\infty}$ has this pleasant feature. |

12. | Here, we have used the uniform prior density $g\left(c\right)\equiv 1$ over the unit interval $[0,1]$. |

13. | One of which can be cash, or a risk-free bond. |

**Figure 1.**Geometric depiction of the set $\mathcal{B}$ of all possible bilinear trading strategies $B:={\left[{b}_{ij}\right]}_{2\times 2}$ over two assets. The defining relations are $B\ge 0;\phantom{\rule{0.166667em}{0ex}}{b}_{11}+{b}_{12}+{b}_{21}\le 1;\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{b}_{22}:=1-{b}_{11}-{b}_{12}-{b}_{21}.$ The volume of this tetrahedron is $1/6$.

**Figure 2.**Superior performance of the universal bilinear portfolio against the individual return sequence ${x}_{t}:\equiv {(2,1)}^{\prime}$ and ${y}_{t}:\equiv {(0.5,1)}^{\prime}$. Asset 2 is cash (that pays no interest); asset 1 is a “hot stock” that doubles in the first half of each investment period and loses $50\%$ of its value in the latter half of each investment period. Note that in the bottom plot, we have ${\mathrm{lim}}_{t\to \infty}{\widehat{b}}_{12}({x}^{t},{y}^{t})=1$ and ${\widehat{b}}_{11}({x}^{t},{y}^{t})\equiv {\widehat{b}}_{22}({x}^{t},{y}^{t})\sim 1/t\to 0$.

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Garivaltis, A.
A Note on Universal Bilinear Portfolios. *Int. J. Financial Stud.* **2021**, *9*, 11.
https://doi.org/10.3390/ijfs9010011

**AMA Style**

Garivaltis A.
A Note on Universal Bilinear Portfolios. *International Journal of Financial Studies*. 2021; 9(1):11.
https://doi.org/10.3390/ijfs9010011

**Chicago/Turabian Style**

Garivaltis, Alex.
2021. "A Note on Universal Bilinear Portfolios" *International Journal of Financial Studies* 9, no. 1: 11.
https://doi.org/10.3390/ijfs9010011