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I am very happy with the following equivalent formulation for the risk budgeting problem (as presented in Bruder, Roncalli, 2012, Managing Risk Exposures using the Risk Budgeting Apporach): Let $b_i$, $\Sigma_{i=1}^n b_i =1$ be the risk budgets, $y_i$ the unscaled portfolio weights and $S$ the variance covariance matrix and $c$ arbitrary. $$y^* = \text{... 6 The Risk Parity portfolio will be equal weighted if the assets have uniform correlation and equal variance. This would be the case for the shrunk covariance matrix if the shrinkage coefficient used equals unity. In sklearn, you can check the shrinkage coefficient for the Ledoit-Wolf shrinkage after fitting it from the instance's .shrinkage_ attribute. If the ... 5 This is a very good question. It can be argued that risk parity is one example of a smart beta strategy. Yet it is important to understand that both are coming from two different directions: risk parity is basically a form of risk management (in the sense of risk-adjustment) because its basic approach lies in diversification - like the alternative methods ... 4 Another approach to construct a risk parity portfolio would be to use the formulation proposed by Spinu [1]:$$\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}\Sigma\mathbf{w} - \sum_{i=1}^{N}b_i\log(w_i)\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1. \end{array}$$where \mathbf{w} is the vector of portfolio ... 4 For question 1), lets add the topic of positive homogeneity to the discussion: Whenever a risk measure is positively homogeneous, we can calculate risk contributions. A risk measure is positively homogeneous of degree \lambda, if$$R(cx)= c^{\lambda} R(x),\quad \text{with}\ x \in \mathbb{R}^n$$If then, \lambda>0, this is equivalent to the Euler ... 3 Apologies in advance for being hyper-critical. I have somewhat strong feelings about this =P The purpose of risk parity is to improve portfolio efficiency via achieving better diversification. (We won't delve into philosophical debates about whether or not this is true here...) Mechanically, by leveraging up low risk assets (e.g., US Treasuries) and ... 3 Your question seems very simple. The \rho_{ij} are the correlations between asset i and asset j, in other words these are the elements of the correlation matrix. This notation is very standard in portfolio optimization problems. The number of securities n, the n-by-n correlation matrix R and the n vector of \sigma_j's are the main inputs of a risk parity ... 2 I had tried something similar to this in the past. It's much easier when there's an analytical formula for CVaR than when using simulations because it's much easier to calculate the derivatives you need to calculate the marginal contribution. However, if you're doing it this way, then you're probably assuming a multivariate normal distribution. Calculating ... 2 Here is my take on trying to answer this question. My backtest goes as far as December 1979 (just before the great bond bull run). I used daily total return index for Wilshire 5000 and 10-year constant maturity treasury rates. Remember you can proxy total bond return for Yield - Duration*\Delta Rate. Nevertheless, I agree with Helin that this is a less-... 2 Let us intuitively understand the risk parity algorithm. In this algorithm, the important point to consider is it allocates more capital for the assets which has lower risk and less capital to the assets which has higher risks. For example, consider two assets where the risk of asset1 is 9% and the risk of asset2 is 5%. Then, the amount of capital allocated ... 2 As mentioned in the short comment above, this particular allocation comes from an interview documented in the book Money: Master the Game. Dalio explained that "in his All Weather strategy, they use very sophisticated investment instruments, and they also use leverage to maximize returns." The interviewer then followed up by requesting the weights that an "... 1 Let \mathbb{1} denote a vector of ones. With the definition of risk parity in the question, we have$$ Sw=c\mathbb{1} $$with c some constant, thus$$ w=cS^{-1}\mathbb{1} $$As \mathbb{1}^Tw=1, we have$$ c\mathbb{1}^TS^{-1}c\mathbb{1}=1 \Rightarrow c=\frac{1}{\mathbb{1}^TS^{-1}\mathbb{1}} $$and hence$$ w=\frac{S^{-1}\mathbb{1}}{\mathbb{1}^TS^{-1}\...

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What horizon are you using to calculate your "equal risk contribution" weights? I am guessing it's 1 year, maybe 5 year... versus Dalio looking over decades. Plus, where are TIPS? When the Taper Tantrum hit in 2013, Bridgewater was on the record being 180% long of these! Gold obviously shares characteristics/correlation with TIPS; but BW holds truck loads of ...

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your goal is to find the weight vector, $w$, which minimized your "utility" function $\sum_{i}^{N} [\frac{\sqrt{w^{T}\Sigma w}}{N} - w_i\cdot c(w_i))] ^{2}$. A general approach is to use gradient descent algo to find the optimal vector. What gradient descent does is it assumed finding best possible solution on each direction (in your case in each $w_i$) is ...

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I figured out the issue, in the risk_budget_objective_error(weights,*args) function, I used pre-defined variables to figure calculate portfolio_stdev and assets_risk_contribution, which is the optimizer kept spitting back the initial weights after only one iteration. Instead, I should have used portfolio_stdev = calculate_portfolio_stdev(weights,...

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Further to my comment is it because your functions are returning matrix's which are deprecated? Why not re-write your functions using ndarrays; import numpy as np ca_cov = np.array([[ 5.28024463e-06, 3.29734889e-07, -7.04781216e-08], [ 3.29734889e-07, 1.32373854e-05, 3.71807979e-08], [-7.04781216e-08, 3.71807979e-08, 3.50845569e-05]]) ca_ini_weights = np....

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I think this paper gives a really good overview about risk parity link. As it points out, risk parity is a alternative to traditional mean variance portfolio construction.

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Smart Beta refers a trend in making well known quantitative strategies more accessible to investors. Simple examples for equities include Value, Momentum, Quality, and Low Volatility. Fixed income might include Carry and Credit. Risk parity is a strategy that incorporates several sources of return that may include some of the smart beta strategies mentioned ...

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They are not the same as in they are equal, but risk parity can be considered a smart beta strategy. Smart beta is this opaque term that covers anything that can be put into a factor, regressed against returns and adjusted for, but also a host of other non-factor strategies that aim to create a mechanical, non-stock index weighting scheme that is ...

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