Beta-Neutral Portfolio Adjustment
You have a long basket $L$ with weight vector $\mathbf{w}_L$ and a short basket $S$ with weight vector $\mathbf{w}_S$, each normalized so that the gross exposure of each side is
You want to adjust the portfolio to be exactly beta-neutral while preserving dollar neutrality and keeping gross exposure at
$\mathbf{w}(\lambda) = \lambda \, \mathbf{w}_L - (1 - \lambda) \, \mathbf{w}_S$
where $\lambda \in (0, 1)$ controls how much capital is allocated to each side.
- Write the portfolio beta as a function of $\lambda$, $\beta_L = \mathbf{w}_L^\top \boldsymbol{\beta}$, and $\beta_S = \mathbf{w}_S^\top \boldsymbol{\beta}$.
- Solve for the $\lambda^{*}$ that makes the portfolio beta exactly zero.
- What is the resulting gross exposure, and under what condition does $\lambda^{*} \in (0, 1)$?
Hints
- Portfolio beta is linear in the weights, and the weights are linear in $\lambda$ -- so portfolio beta is a simple linear function of $\lambda$.
- Write the portfolio beta as $\beta_P(\lambda) = \lambda \beta_L - (1 - \lambda) \beta_S$ and set it to zero to solve for $\lambda^{*}$.
- Check that the solution $\lambda^{*} = \frac{\beta_S}{\beta_L + \beta_S}$ lies in $(0,1)$ by considering the signs of $\beta_L$ and $\beta_S$.
Worked Solution
How to Think About It: You have a long/short equity portfolio and you want to hedge out market risk. The naive approach -- equal dollar on each side -- only works if both baskets have the same beta. In practice the long side might be higher-beta growth stocks and the short side might be lower-beta value names. You need to tilt the capital allocation to compensate. This is a one-dimensional root-finding problem: find the $\lambda$ that zeros out the portfolio beta.
Key Insight: Since portfolio beta is linear in the weights, and the weights are linear in $\lambda$, the portfolio beta is a linear function of $\lambda$. Setting it to zero gives a unique closed-form solution.
The Method:
Define the weighted betas of each side: - Long-side beta: $\beta_L = \sum_i (w_L)_i \beta_i = \mathbf{w}_L^\top \boldsymbol{\beta}$ - Short-side beta: $\beta_S = \sum_i (w_S)_i \beta_i = \mathbf{w}_S^\top \boldsymbol{\beta}$
Step 1: Portfolio beta as a function of $\lambda$.
The portfolio weight vector is $\mathbf{w}(\lambda) = \lambda \, \mathbf{w}_L - (1-\lambda) \, \mathbf{w}_S$. The portfolio beta is:
$\beta_P(\lambda) = \mathbf{w}(\lambda)^\top \boldsymbol{\beta} = \lambda \, \beta_L - (1 - \lambda) \, \beta_S$
Expanding:
$\beta_P(\lambda) = \lambda (\beta_L + \beta_S) - \beta_S$
**Step 2: Solve $\beta_P(\lambda^{*}) = 0$.**
Setting the expression to zero:
$\lambda^{*} (\beta_L + \beta_S) = \beta_S$
$\lambda^{*} = \frac{\beta_S}{\beta_L + \beta_S}$
This is the fraction of gross exposure allocated to the long side. The short side gets
Step 3: Gross exposure and feasibility.
The gross exposure is $\lambda^{*} \sum_i |(w_L)_i| + (1-\lambda^{*}) \sum_i |(w_S)_i|$. Since $\mathbf{w}_L$ is normalized so $\|\mathbf{w}_L\|_1 = 1/2$, the long gross is $\lambda^{*} \cdot (1/2) \cdot 2 = \lambda^{*}$ and similarly the short gross is
For $\lambda^{*} \in (0, 1)$ we need $\beta_L$ and $\beta_S$ to have the same sign (both positive), which is the normal case for equity portfolios. Specifically, $\beta_L > 0$ and $\beta_S > 0$ ensures a feasible solution.
Practical Considerations: - If $\beta_L = \beta_S$, then $\lambda^{*} = 1/2$ -- equal capital on each side, the naive case. - If $\beta_L > \beta_S$ (long side is higher beta), then $\lambda^{*} < 1/2$: you shrink the long side and expand the short side to compensate. - If $\beta_S > \beta_L$ (short side is higher beta), then $\lambda^{*} > 1/2$: you expand the long side. - This adjustment changes your dollar exposure to each name, which affects alpha capture. A large beta mismatch forces large tilts that may dilute the signal.
Answer: The beta-neutral allocation is $\lambda^{*} = \frac{\beta_S}{\beta_L + \beta_S}$, which zeros out portfolio beta while maintaining gross exposure of
Intuition
Beta-neutrality is one of the first risk constraints a systematic equity fund implements. The idea is simple: you want your P&L to come from stock selection (alpha), not from whether the market goes up or down (beta). The one-parameter rescaling is the simplest way to achieve this -- you are just rebalancing how much capital sits on each side of the book. The formula $\lambda^{*} = \beta_S / (\beta_L + \beta_S)$ has a clean interpretation: if your short basket has higher beta, you need more capital on the long side to match it. In practice, portfolio managers often start with a dollar-neutral, beta-neutral book and then layer on additional constraints (sector neutrality, factor neutrality). The tension is always between tighter risk controls and preserving the alpha signal -- every constraint you add potentially dilutes the positions your model likes best.