# How to apply risk-parity portfolio construction to a dollar-neutral portfolio?

Long-only risk-parity portfolios have proliferated in recent years. An optimized long-only risk-parity portfolio requires that the asset weight * marginal contribution to risk of the asset is identical for all securities.

One way to implement this idea is to find the solution to a dual-problem. For a long-only risk-parity solution one can find the weights that minimize the variance of each assets weights ($w$) * marginal contribution to risk ($\text{MCTR}=\frac{\partial \sigma(w)}{\partial w_i}=\partial_i \sigma(w)$).

In this manner, formally the problem is to choose weights (constrained to sum to one) using your favorite optimizer:

$\underset{w}{\arg \min} \quad \text{Risk} = \text{Var}( w_1 * \partial_1 \sigma(w) , w_2 * \partial_2 \sigma(w), ... , w_n * \partial_n \sigma(w))$

My question -- are there any research articles or insights for constructing risk-parity portfolios assuming dollar-neutral (instead of long-only) weight constraints?

Implementing risk-parity in a dollar-neutral portfolio is not as trivial as applying the same objective function as above and simply changing the weight constraints. For example, because variance is symmetrical two solutions would be produced: optimal weights and -1*optimal weights. (Of course, a more complex objective function that included a maximize alpha objective would not result in symmetrical solutions.)

Also, convergence in the long-only case is fairly rapid whereas in the dollar-neutral case the objective function conflicts with the constraint that i) cash weight + long weight + short weight = 1, and ii) long weight = -short weight. The combination of i) and ii) implies cash weight = 1 in dollar-neutral.

To flesh out the second point, intuitively, the optimizer objective function is minimized when $w_n * \partial_n \sigma(w)$ is identical for all securities (i.e. the variance is zero). However, this is impossible when some weights must be positive and other weights must be negative to satisfy constraint (ii), and where nearly all securities have a positive MCTR.

Perhaps there is a more suitable choice of objective function to minimize in the dollar-neutral case, or another way to construct a risk-parity portfolio in a dollar-neutral context?

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how do you define exactly dollar-neutral? $\sum w_i = 0$? – SRKX Dec 28 '11 at 9:29
1) Sum of weights always = 1 (in particular the weight in cash = 1, and all long positions are financed by shorts. 2) Sum of long weights = sum of short weights. – Ram Ahluwalia Dec 28 '11 at 14:05
@SRKX - thanks for the edit! – Ram Ahluwalia Dec 28 '11 at 14:23
There are various conventions for weights in long-short. My favorite is the sum of absolute values equals 1. Using this convention, dollar neutral is the same as the sum of weights equals zero. – Patrick Burns Dec 28 '11 at 15:44
@QuantGuy : I would have done the following: $\underset{w}{\arg \min} \sum_{i=1}^N [\frac{\sqrt{w^T \Sigma w}}{n} - w_i \partial_i \sigma (w) ]^2$ – SRKX Dec 28 '11 at 17:59

The goal function I suggest for this optimization is the following.

$$\underset{w}{\arg \min} \sum_{i=1}^N [\frac{\sqrt{w^T \Sigma w}}{N} - w_i\partial_i\sigma(w)]^2$$

I added the square root compared to the comment as you are actually using the euler decomposition on $\sigma$ (not on $\sigma^2$) as follows:

$$\sigma(w)=\sqrt{w^T \Sigma w} = \sum_{i=1}^N w_i \partial_i \sigma(w)$$

All the properties of this setup are discussed in details in this paper, but mainly with the assumption of $w_i \geq 0 \quad \forall i$.

For the long-only case, it works, I'm pretty sure. I think we might have to add an absolute value for the general case though....

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+1. I'll try this formulation of the objective and let you know how it goes. I think this should work... Thanks for the link to the paper – Ram Ahluwalia Dec 29 '11 at 17:40
Where is your formula in the paper? Formula 6 and 7 in the paper (see page 7 of the pdf) do not match the goal function described above. – Ram Ahluwalia Dec 29 '11 at 19:51
@QuantGuy: Actually I did not get the goal from the paper, I "did it myself" when I tried out ERC. I implemented it in Matlab and the problem is indeed found using fmincon (I checked the risk contributions and the constraints) – SRKX Dec 29 '11 at 22:43
Just a heads up that there are multiple optimal solutions when weights are allowed to go negative. In particular, the -1*weights is also a valid solution with equal utility. I have added a minimize transaction cost objective into my utility function to minimize the churn that might result. – Ram Ahluwalia Jan 2 '12 at 15:46

I played around with this a little using Portfolio Probe. The way to get risk parity portfolios (in the sense you are using) with that is to constrain the fractions of variance for each asset to be slightly more than one over the number of assets. Slightly more because trading is done in integer amounts.

I took 20 assets and tried forming dollar neutral portfolios. Constraining risk fractions to less than .051 was on the edge of feasible (for long-only .0501 is okay). The dollar neutral constraint seems to be in conflict with the risk parity constraint (net was constrained to be -100 to 100 dollars with a gross of 1e6).

Obviously switching longs for shorts gets you another solution -- perhaps a bunch of other solutions. But generating random portfolios with the looser constraint of .055 shows much more diversity than that in the size of positions.

I'm not convinced this is of practical interest, but it is intellectually interesting.

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Not sure how you can constrain the "fraction of variance to be slightly more than 1/N". The fraction of variance is weight * MCTR. If weights are negative (as some must be in dollar-neutral), then this constraint is impossible to satisfy. This approach would work only in the long-only case. +1 for the exploration though, I appreciate it – Ram Ahluwalia Dec 28 '11 at 16:28
Further explanation is at portfolioprobe.com/features/constraints/… In long-short risk fractions are typically positive or only slightly negative. The target that SRKX suggested is aiming at what I'm doing, and it would get reasonably small. Having all the risk fractions positive is certainly feasible. One of my attempts has a range of 0.04460133 to 0.05099992 with 20 stocks. – Patrick Burns Dec 28 '11 at 19:18