Value of a Continuously Rebalanced Equal-Weight Stock Portfolio

Summary

The document formulates the value of a portfolio that keeps equal weights in a set of stocks through continuous trading. For two stocks, it represents portfolio value as a function of their share prices and proposes partial-derivative conditions: each price sensitivity equals the portfolio’s fractional holding of that stock. It generalizes the conditions to an arbitrary number of stocks.

Under the stated frictionless assumptions, these conditions imply that portfolio value is proportional to the geometric mean of the stock prices, with the constant set by the portfolio’s initial scale. This gives a mathematical characterization of the continuously rebalanced portfolio. The setup assumes trading in arbitrarily small amounts with no transaction costs and does not account for market frictions, discrete rebalancing, dividends, or changes to the stock universe. The document asks how to define the function and provides no empirical comparison or practical implementation.

Key ideas

• Equal weighting requires each stock’s value sensitivity to match its fractional portfolio holding.
• The proposed partial-derivative conditions extend from two assets to any number of stocks.
• Under the stated conditions, portfolio value is proportional to the geometric mean of the stock prices.
• The model assumes continuous trading, fractional shares, and zero transaction costs.
• The formulation omits practical frictions and does not test realized portfolio performance.

Tags

# Value of continuously rebalanced stock portfolio


# Value of continuously rebalanced stock portfolio












I'm thinking about what a theoretical continuously re-balanced stock portfolio could look like, in which the portfolio is uniformly distributed over a selection of stocks at all times.

For example, if my portfolio consists only of 2 stocks the value of the portfolio would be described by some function $f(x,y)$ of the values of the share-price of the stocks X and Y, and at all times my portfolio shall be 50% x-stock and 50% y-stock, under the assumption that shares of X and Y can be continuously bought and sold with zero additional cost (transaction fees, etc.) in infinitely small amounts while share prices chance, in order to keep the portfolio balanced.

Here, $x(t)$ and $y(t)$ shall be functions of time, representing the current share price of the stocks. I arrive at the following differential equations: $\frac{\partial f}{\partial x}(x,y)=\frac{1}{2}\frac{f(x,y)}{x}$, respectively $\frac{\partial f}{\partial y}(x,y)=\frac{1}{2}\frac{f(x,y)}{y}$, because those are the amounts of (fractional) shares of the stock in the portfolio at any time.

In the general case of n stocks that would then be $\frac{\partial f}{\partial x_k}(x_1,...,x_n)=\frac{1}{n}\frac{f(x_1,...,x_n)}{x_k}$ for all k.

How could f be defined?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.