# Optimal weights in portfolio after rebalancing

I have a quite simple question but while looking for answers in research papers I couldn't find anything. The question can be summarized as : if you expect a shock on an asset, why don't you rebalance your portfolio homogeneously ? On what basis do you buy some assets and don't buy others ? Does it reflect the "substitutability" of assets in portfolios ?

Imagine you have a portfolio composed of 4 risky asset $$x_1, x_2, x_3$$ and $$x_4$$ with respective weights $$w_i$$.

The portfolio returns is thus given by the combination of each assets weights and their returns : $$r_p = w_1 r_1 + w_2 r_2 + w_3 r_3 + w_4 r_3$$

Lets give some random weights so that the sum of all weights are 1:

$$r_p = 0.5 r_1 + 0.3 r_2 + 0.1 r_3 + 0.1 r_3$$.

If an investor expect a negative shock on $$r_1$$, he/she will reduce its exposure to $$x_1$$, say that $$w_1$$ will decrease of 50%. But what will be the impact on $$w_2$$, $$w_3$$ and $$w_4$$ ? In other words, I'm looking for determinants of $$\frac{\Delta w_2}{\Delta w_1}$$, the elasticity of $$x_2$$ weights following a shock on $$x_1$$ weights.

My intuition (that I explain below) is that investors attribute weights based on relative performance (risk-return) of each assets. Two assets with similar weights are more similar (in terms of risks and returns) than two assets with different weights. In the example, $$x_3$$ and $$x_4$$ and more similar because their weights are similar (so they are more substitutable). Following a shock on $$x_1$$, the most similar asset is $$x_2$$, thus my intuitions leads to the fact that $$\frac{\Delta w_2}{\Delta w_1} > \frac{\Delta w_3}{\Delta w_1} = \frac{\Delta w_4}{\Delta w_5}$$ under some assumptions (the shock is purely idiosyncratic, correlations are 0, budget allocation is the same, no short-selling, constant risk-aversion...)

Does anyone can tell me if this makes sense ? Is there a paper that explicitly speaks about the elasticities of weights ?

Here, I will explain more formally my "intuition", with three assets for simplicity.

$$R_p = r_A w_A+r_Bw_B +r_Cw_C$$ or, in variation : $$\Delta R_p = \Delta r_A \Delta w_A+\Delta r_B\Delta w_B +\Delta r_C\Delta w_C$$

Let a negative shock hit asset $$A$$ returns. The investor wants to keep the same portfolio returns, thus $$\Delta R_p =0$$, we get :

$$-\Delta r_A \Delta w_A = \Delta r_B \Delta w_B +\Delta r_C\Delta w_C$$

$$-\Delta r_A = \Delta r_B \frac{\Delta w_B}{\Delta w_A} +\Delta r_C\frac{\Delta w_C}{\Delta w_A}$$

isolating $$\frac{\Delta w_B}{\Delta w_A}$$, we get

$$\frac{\Delta w_B}{\Delta w_A} = -\frac{\Delta r_A}{\Delta r_B} - \frac{\Delta r_C}{\Delta r_B}.\frac{\Delta w_C}{\Delta w_A}$$

Here, the elasticity of change in weights of asset $$B$$ following a shock on asset $$A$$ decreases with the "distance" between $$A$$ and $$B$$ characteristics (in terms of returns) (or their relative returns), and with the "distance" between $$C$$ and $$B$$ returns, times the elasticity of change in weights of asset $$C$$ following a shock in $$A$$.

For $$n$$ assets, we have :

$$\frac{w_B}{w_A} = \frac{-\Delta r_A}{\Delta r_B} - \frac{1}{\Delta r_B \Delta w_A} \sum_{i=3}^{n}\Delta r_i \Delta w_i$$

However, returns are not the only characteristics of assets, and I'd like to find something similar with the portfolio's variance given by :

$$Var(R_p) = \sigma^2_A w_A^2+ \sigma^2_B w_B^2 + \sigma^2 w_C^2 + \sigma_A\sigma_B w_A w_B \rho_{AB} + \sigma_A\sigma_C w_A w_C \rho_{AC} + \sigma_C\sigma_B w_C w_B \rho_{CB}$$

Where $$\sigma_i\sigma_j w_i w_j \rho_{ij}$$ is the covariance between $$i$$ and $$j$$, $$\rho_{ij}$$ is the correlation between $$i$$ and $$j$$. However, I find something that is much less elegant (and much less interpretable) :

$$\frac{w_B}{w_A}= \frac{\Delta w^2_A\Delta \sigma^2_A- \Delta \sigma_A \Delta \sigma_B \Delta w_B \Delta w_A \Delta \rho_{AB} - \Delta \sigma_A \Delta \sigma_C \Delta w_A \Delta w_C \Delta \rho_{AC} - \Delta \sigma^2_B w_B^2 - \sigma_C^2 w_C^2 }{\Delta \sigma_B \Delta \sigma_C \Delta w_C \Delta \rho_{BC}}$$

And I'm stuck here !

• Upvoting this post because I think it is an interesting question and I would like to know the answer too. Commented Mar 25 at 13:16
• Hi, interesting question. These 2 papers here come to my mind: jstor.org/stable/2962107 and jstor.org/stable/2632559. Is this what you are looking for? Or at least the right direction?
– T123
Commented Mar 25 at 15:39
• Thanks @T123 ! Those papers are interesting, and I think that they go in the right direction. However, I think that they study more the inherent flaws of the Markowitz model (weights are "too" sensitive to input changes) rather than the more general elasticity of substitution between assets ! Commented Mar 25 at 17:07
• If change is driven by a common factor, you substitute high beta stocks by low beta stocks. If it is idiosyncratic, you substitute that asset for something with better outlook but smaller vols. If you want the same risk and return, there will be many, many candidates satisfying this condition if choice of assets is high enough. Commented Mar 25 at 19:10

$$\max_{w \geq 0} w^\top x \text{ subject to } \mathbf{1}^\top w$$
This is a standard linear program, and the effect of changing any component of $$x$$ is well-studied (and unsurprising). In short, you reallocate into the next best asset.