Fund Replication

Replication of the DAX with global Smart Beta / Factor ETFs

Most investment products are disadvantageous (too expensive or capital inefficient) repackagings of elementary portfolio components, such as global equity & fixed income exposure. To check this, I have compiled this Jupyter Notebook, which works as follows:

Optimizer

First, I defined a function weight_optimization that determines a parameter vector $\beta $, in this case, the weights of the portfolio components that are to be used for replication. The following parameters can be specified:

Loss Function

Here are four common loss functions available:

• Mean Squared Error (mse):

$$ \min_{\boldsymbol{\beta}} \frac{1}{n} \sum_{i=1}^{n} (y_i - X_i \boldsymbol{\beta})^2 $$

• Mean Absolute Error (mae):

$$ \min_{\boldsymbol{\beta}} \frac{1}{n} \sum_{i=1}^{n} |y_i - X_i \boldsymbol{\beta}| $$

• Tracking Error (te):

$$ \min_{\boldsymbol{\beta}} \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (y_i - X_i \boldsymbol{\beta})^2} $$

• Quantile Loss (q):

$$ \min_{\boldsymbol{\beta}} \frac{1}{n} \sum_{i=1}^{n} \rho_{\tau}(y_i - X_i \boldsymbol{\beta}) $$

Weight Constraints

Here, the following self-explanatory optimization constraints can be specified:

• Long-Only Constraint (if long_only_constraint is True):

  $$\beta_j \geq 0 \quad \forall j \in{1,2, \ldots, m}$$

• Leverage Constraint (if leverage_constraint is True):

  $$0 \leq \sum_{j=1}^m \beta_j \leq 1$$

Number of Assets Contraint

It's possible to limit the number of considered assets by the paramater max_assets. This means that a large number of potential assets can be provided, and the function will use the subset of the desired cardinality that produces the best result. However, since the function accomplishes this through iterative combinatorial means, the runtime may potentially be slightly extended.

Optimization Method

The optimization is performed using the Sequential Least Squares Programming (SLSQP) method or any other multivariate method from the scipy library specified by the optimizer parameter. However, not all methods reliably find the global optimum. I recommend sticking with SLSQP.

Application

Here, the respective tickers assets and target of the securities of interest, as they are listed on Yahoo Finance, can be specified, and the above-defined function can be applied. The results are Portfolio Weights $\beta $, the Coefficient of Determination of the linear fitting, the Correlation of the replicating portfolio to the fund, and the Tracking Error, which represents the sample standard deviation of the differences in returns.

If the fund cannot be well replicated, the notebook includes both a bootstrap test of the fund's Sharpe ratio relative to the Sharpe ratio of the replicating portfolio and a 1-factor regression to determine the alpha. This can be used to examine how statistically significant the differences are.

The bootstrap test produces the following plot and a p-value for the right-side alternative hypothesis $H_1 : SR_{Fund} > SR_{Portfolio} $:

Null Distribution of Replication Portfolio Sharpe Ratios for the DAX

Benchmark Identification

This notebook can also be used to find an appropriate benchmark, for example, by passing an extensive list of potential alternatives to the weight_optimization function and setting the max_assets parameter to 1.