Tensions in Practice: The average path is not the average ending¶
Two random proportional changes to a toy quantity
Start with 1 unit. At each of two independent steps, the exposed part is multiplied by 1.8 or 0.4 with equal probability. Compare exposing the whole current quantity with exposing half and retaining half unchanged, rebalancing before each step. The latter makes the total factor 1.4 or 0.7. Each of the four ordered paths has probability one quarter. Their endings can be averaged across imagined runs, but a single run follows one product.
Retain the higher mean ending
Keep full exposure to the stipulated favorable average multiplier.
Moderate the lower endings
Reduce each step’s proportional loss by retaining an unexposed part.
Why these aims pull against each other
Full exposure has the higher ensemble mean, yet three of the four two-step endings are below the starting 1. Partial exposure reduces those losses while also reducing the upper ending and the mean.
Choose an arrangement to see what changes and what remains difficult.
Up and down name the two equiprobable shocks. × means multiply the current quantity. Each row has probability 1/4; the initial quantity is 1, so the last cell is the product of its two factors.
What this choice protects
What it costs
When it fits
Compare the arrangements
Expose all
Multiply the whole current quantity at both steps. The four endings average to 1.21; the two central endings are both 0.72.
| Step 1 | Step 2 | Ending | |
|---|---|---|---|
| Up, up | ×1.8 | ×1.8 | 3.24 |
| Up, down | ×1.8 | ×0.4 | 0.72 |
| Down, up | ×0.4 | ×1.8 | 0.72 |
| Down, down | ×0.4 | ×0.4 | 0.16 |
- What it protects
- Under this exact random law, the expected ending is 1.21 and the upper path reaches 3.24.
- What it costs
- Three paths end below 1, and the lower path leaves only 0.16. Expected level does not describe the ending on each run.
- When it fits
- Plausible when the stated expected-level objective is appropriate and these lower outcomes are tolerable.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
Expose half each step
Before each step, expose half the current quantity and retain the other half unchanged. The combined step factors are 1.4 and 0.7; the four endings average to 1.1025.
| Step 1 | Step 2 | Ending | |
|---|---|---|---|
| Up, up | ×1.4 | ×1.4 | 1.96 |
| Up, down | ×1.4 | ×0.7 | 0.98 |
| Down, up | ×0.7 | ×1.4 | 0.98 |
| Down, down | ×0.7 | ×0.7 | 0.49 |
- What it protects
- The lower ending rises to 0.49 and the central endings to 0.98 under the same up/down paths.
- What it costs
- The upper ending falls to 1.96 and the expected ending falls to 1.1025. Rebalancing and isolating half must be feasible.
- When it fits
- Plausible when moderating the lower endings matters more than maximizing the mean under this stipulated model.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
What this illustration does—and does not—establish
The source establishes the structural tension; the concrete alternatives and their conditional costs are editorial synthesis. No arrangement is a universal recommendation.
- This is an invented positive-state process, not an investment recommendation or estimate of real growth. Isolation and costless rebalancing are explicit simplifying assumptions.
- Two steps establish only these four paths. No long-run distribution, extinction probability or empirical typical growth rate is inferred.
- Independence and equal step probabilities are essential to the stated equal path weights; different probabilities or correlated steps change the means.
Source entries
Multiplicative Random Growth
Multiplicative random growth Arithmetic mean versus typical path supplies the local tension. The setting, alternative arrangements, and stipulated consequences are editorial applications.
Arithmetic mean versus typical path
Rare large realizations can lift the level mean while most paths grow more slowly or decline. Diagnostic: Is the claim about expected level, median level, or expected log growth?