Reverse Optimization: finding the returns that satisfy specific weights given one known return

Question

Here is the premise: I have a three asset portfolio, I know the assets covariance, the client's risk aversion and the expected return of one of the assets. I also have a desired set of weights.

So, 1) how do I find the expected returns for the two assets I don't know that gives me my desired set of weights and 2) if I reverse the function can I get the desired weights back i.e. test the results?

Here's my Python code:

import numpy as np

# Define the function
def get_weight(expected_return, asset_covariance, risk_aversion):
   ER = expected_return
   S = asset_covariance
   L = risk_aversion
   return np.linalg.inv(L * S) @ ER

   # Given data
   risk_aversion = 1.0 # example value for L
   cov_matrix = np.array([[0.01, 0.002, 0.001], [0.002, 0.03, 0.004], [0.001, 0.004, 0.02]]) 
   # example covariance matrix

   # Desired weights
   desired_weights = np.array([0, 0.55, 0.45])

   # Solve for expected returns
   expected_returns = risk_aversion * cov_matrix @ desired_weights
   expected_returns[0] = 0.009 # given return 

   # Calculate weights using the original function
   calculated_weights = get_weight(expected_returns, cov_matrix, risk_aversion)

   # Output the results
   expected_returns, calculated_weights

Here's the output I get. You clearly see it's not circular. I expect the weights array to be 0.00,0.55,0.45. I can understand some rounding but the result is quite a bit off.

Where am I going wrong? Please reference Python in your comments if you can.

Answer 1

OP takes Black-Litterman reverse optimization expected returns, overrides one of the values with a different expected return, and finds that the resulting portfolio didn't match the original one. This is not surprising.

The answer referenced by @KaiSqDst is nearly sufficient to give an analytic solution (this is very close to a repeat of that question).

That answer (at least when I am viewing it) has the formula

$$ \gamma\Sigma(w^*-w_0)=\left(\mathbf{I}-\frac{1}{a}\mathbf{1}\mathbf{1^T}\Sigma^{-1}\right)\mu $$

and correctly notes that the problem is underdetermined and you cannot invert the right hand side matrix. However, it is possible, especially for a simple problem like this but perhaps not in the general case, to take a subset of the matrix and solve that. For your purpose, this would be to exclude the first asset that is assumed to have an expected return of 0.009. This would be like solving for the excess returns of the other assets. From the excess returns, you would add back 0.009 to each to get actual returns. The result is a 0.009 for asset 1, 0.09275 for asset 2, and 0.05725 for asset 3.

When double-checking the results, you need to make sure to assume the weights sum to 1. It isn't "circular" with unconstrained mean-variance.

Answer 2

Referencing the original answer to the general problem and the definitions wherein, we remember the optimal investment decision which we can rewrite as:

$$ \begin{align} w^*&=w_0+\frac{b}{\gamma}\left(w_M-w_0\right)\\ \Rightarrow \gamma \Sigma \left(w^*-w_0\right)&=\mu-\frac{b}{a}\mathbf{1} \end{align} $$ We do know all elements of the vector on the lhs and we know one element of $\mu$, say $\mu_1$. Hence, we can trivially solve:

$$ \begin{align} \left(\gamma \Sigma \left(w^*-w_0\right)\right)_{(1)}&=\mu_1-\frac{b}{a}\\ \Rightarrow b&=a\mu_1 - a\left(\gamma \Sigma \left(w^*-w_0\right)\right)_{(1)} \end{align} $$

Given knowledge of $b$, we obtain:

$$ \mu=\gamma\Sigma\left(w^*-w_0\right)+\frac{b}{a}\mathbf{1} $$