How Portfolio Optimization Determines Reallocation After an Asset Shock

# Optimal weights in portfolio after rebalancing


# 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 !

## Answer by msantama (score 2)

https://quant.stackexchange.com/a/79193

It depends what your optimization problem is. The simplest would be return maximization:

$$\max_{w \geq 0} w^\top x \text{ subject to } \mathbf{1}^\top w$$

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.

As the papers linked in the comments mention, things get more interesting when your optimization problem becomes more sophisticated.