Understanding Cover’s Universal Portfolio Dec 6, 2023

I always wanted to understandt the concept of a universal portfolio from Cover’s. This topic always fascinated me since I first heard about it on my information theory lectures and you know what they say… The best way to learn is to practice. So here’s my attempt …

Folowing the words from the original paper from Cover: We exhibit an algorithm that ourperforms the best stock in the market. For this he exhibit as a goal the wealth Sn that will outperform the best stock in the market and will depend on differents weights alongside different stock options. This will be an online algorithm, this means that the portfolio will be constantly be updated and captures distributions shifts and drifst of the market. Besides that the algorithm is “model agnostic”, in the sense that does not depend on the underlying assumption that you make about the statistics of the log returns. That’s fantastic and seems almost too good to be true.

Here is the link to the original paper: Paper. I’ll not be entering on the math behind the paper but here is my bite-sized summary. Cover simply proved that, with enough time, with a LOT of time, the rebalancing effect across the curve would end up generating an average return across the curve in excess of the return of the better single asset.

This is neither a simple “mean-reversion” nor a “momentum” effect. Aĺl of your portfolios are mean-reverting; but you’re running with their momentum.

The intuition is the arithmetic vs geometric half-variance drag and rebalancing effect. Whatever the two assets’s true return, vol, and correlation, the rebalancing portfolios will be biased to having a better risk-reward than their naked equivalent all across the curve. Cover’ s insight was proving that letting the winning (rebalancing) portfolios run would, in the end, guarantee performance better than the winning asset. In the (VERY) long-run.

If you don’t understand the central anomaly here, ask yourself what is the probability that any market will be X% up before being X% down? If you want to teĺl me it’s a 50:50, then the theoretically optimal bet is to stake a quarter of my wealth it’s down :-) Yes, that’s crazy; but it’s not wrong… market doubles or zeroes every day, buy or sell to hold? The same is true to diminished degrees with “fair bets” in general.

Any fair investment, ie zero long-term expected return, therefore has to have a (SMALL) positive expected return in the short-term.

Cover’ s “Universal Portfolio” is simply an algorithm that exploits this effect it’s “proof” is simply that , with infinite time, the rebalancing gains across the spectrum of weights will end up surpassing the performance differential between the sample assets.

That’s the logic (and proven, assuming traditional assumptions with respect to normality hold true). and a pseudocode that implements the main idea:

# Initialize portfolio weights
initialize_portfolio(weights)  # All weights start with an equal distribution

# Initialize wealth and wealth history
wealth = 1.0
wealth_history = []

# Define the set of available assets
assets = [asset1, asset2, ..., assetN]

# Define the number of time steps
T = number_of_time_steps

# Main loop for T time steps
for t in range(T):
    # Observe returns for all assets at time t
    observe_returns(returns)

    # Calculate the total wealth in the portfolio at time t
    total_wealth = sum(wealth * weight for weight in weights)

    # Calculate the regret for each asset
    regrets = []
    for asset in assets:
        asset_wealth = wealth * weights[assets.index(asset)]
        best_asset_wealth = wealth * max(weights)
        regret = best_asset_wealth - asset_wealth
        regrets.append(regret)

    # Update portfolio weights using exponential weights
    total_regret = sum(regrets)
    for asset in assets:
        weight = weights[assets.index(asset)]
        asset_regret = regrets[assets.index(asset)]
        weight *= (1 + asset_regret) / (1 + total_regret)
        weights[assets.index(asset)] = weight

    # Normalize portfolio weights to sum up to 1
    total_weight = sum(weights)
    weights = [weight / total_weight for weight in weights]

    # Update the portfolio's wealth at time t
    new_wealth = sum(wealth * weight * (1 + asset_return) for weight, asset_return in zip(weights, returns))
    wealth = new_wealth

    # Append the wealth at time t to the wealth history
    wealth_history.append(wealth)

# After T time steps, you have the final portfolio weights in 'weights'

Here is the colab where I tested the ideas: colab