Extending Relative Signal Allocation to Multiple Countries
Summary
The document considers how to convert independent country signals into equity portfolio weights while preserving a neutral allocation and bounding each country's deviation. Its proposed construction treats the signal vector as a point in a multidimensional space and identifies the nearest point on the line where all country signals are equal. This line represents balanced signals, so common movement across all signals does not by itself shift relative weights.
The vector from that balanced line to the observed signal point becomes a displacement direction for adjusting the portfolio. In two dimensions, this is consistent with using the difference between country signals; in higher dimensions, it generalizes that idea through projection. The answer does not specify the final mapping from displacement to weights. A scaling or other rule is still needed to enforce the sum-to-one constraint and individual allocation bounds, so this is a framework for designing an allocation method rather than a fully specified solution.
Key ideas
- Equal country signals lie on a neutral axis and imply no relative allocation shift.
- Project the signal vector onto the subspace perpendicular to the equal-signal line to obtain relative displacement.
- In two countries, the displacement direction corresponds to comparing the signals.
- A separate mapping and scaling rule must convert the displacement into weights that satisfy portfolio constraints.
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Full text
# Country allocation -optimization 3 countries
# Country allocation -optimization 3 countries
I have the following problem: an equity portfolio allocated to 3 countries. Each country has an independent indicator (signal) which takes values from -4 to 4. The allocation for each country at neutral is 1/3 (33.33%) and based on the indicator can oscillate -20%/+20%.
the weights are: w1 +w2 +w3 =100% constraint w1,w2,w3 =33.33% (at neutral) w1,w2,w3 e [13.33%;53,33%] constraint the indicators : i1,2,3 e [-4;4]
the allocation can be linear, proportional. How can this be solved for 3 countries? because for 1 is easy: Like -4 represents -20% in a country. 0 represents neutral allocation , keeping 33.33% in that specific country. For 2 I made a compound indicator like the difference between countries indicators. But for 3+ countries I cannot find the solution.
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/41527
Here's a thought.
In 2-dim your score $(x, y) \in [-4,4]^2$ is best characterised as the minimal distance from the line $y=x$, along which your portfolio is balanced. I.e. wherever $y=x$ either at $(0,0), (-1,-1) (4,4)$ the weight is 50-50, since there is no minimal displacement vector. As a further example $(0,2)$ has minimal distance from the point $(1,1)$, whose displacement vector is $(-1,1)$. Note this is consistent with a difference mentality: $(0,2)$ having the same displacement as say $(1,3)$ or $(-2,0)$. Then you define some topology that maps a displacment vector to a new portfolio position.
In n-dim your score $\mathbf{x}=(x_1, ..., x_n) \in [-4,4]^n$ is reclassed as the minimum distance from the line $\mathbf{r} = t \mathbf{1} $ (i.e $x=y=z$ in 3-dim). And the vector defining the position from that central axis to your specific point score defines the displacement of the portfolio from neutrality. Note that in this case the minimum distance vector from the line to the point score is $\mathbf{x} - (\mathbf{x}\cdot \frac{\mathbf{1}}{||\mathbf{1}||})\frac{ \mathbf{1}}{||\mathbf{1}||}$. (https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line)
Yes, OK, you still need to define the topology, but this a mathematical construct from which to develop consistency in multiple dimensions. Presumably some scaling to satisfy your constraints.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.