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 !
