Maintaining a Normalized Share Process Through Growth Factors

Summary

The document asks how to define portfolio shares whose values sum to one at every time, and how their period-to-period growth factors must relate. It treats each share as a proportion of the total and defines its growth as the ratio of its next-period value to its current value.

The answer gives a necessary and sufficient condition: if the shares sum to one initially, then at each subsequent step the sum of each current share multiplied by its growth factor must equal one. This is a concise accounting identity that constrains the weighted average growth of the shares. The document does not propose a particular share process, discuss how to construct growth factors, or give an example, so applying the condition requires additional modeling choices.

Key ideas

• Shares are required to sum to one at each time step.
• Each share’s growth factor is its next value divided by its current value.
• Starting from shares that sum to one, normalization persists when their current-share-weighted growth factors sum to one at every step.
• The condition describes a constraint and does not prescribe a specific process.

Tags

# How to design a general share process


# How to design a general share process












My question is simple but not easy.

I want to define a share process $s_{i,t}$ that $\sum_i s_{i,t} =1$ at all times $t$.

Moreover, I am interested in the growth of the share processs, $g_{i,t+1}=s_{i,t+1}/s_{i,t}$.

What kind of $g$ or $s$ can satisfy that $\sum_i s_{i,t} =1$?

## Answer by NN2 (score 1)

https://quant.stackexchange.com/a/80964

By definition, given $\sum_{i}s_{i,1} = 1$, the necessary and sufficient condition for $\sum_{i}s_{i,t} = 1$ for all $t\ge 2$ is that $$\sum_{i}g_{i,t+1}\cdot s_{i,t} = 1 \qquad \forall t\ge 2$$

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