Signal model

Let us denote by \(\mathbf{r}_{t}\) the vector of the returns of \(N\) assets at time \(t\) and suppose they follow an i.i.d. distribution (which is not a totally accurate assumption, but it is widely adopted, nonetheless) with mean \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\).

The portfolio vector \(\mathbf{w}\) denotes the normalized dollar weights allocated to the \(N\) assets, such that \(\mathbf{1}^{T}\mathbf{w}=1\), and the portfolio return is then \(r_{t}^{\sf portf} = \mathbf{w}^{T}\mathbf{r}_{t}\), with expected return \(\mathbf{w}^{T}\boldsymbol{\mu}\) and variance \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\).

Modern Portfolio Theory

In 1952, Markowitz proposed in his seminal paper [1] to find a trade-off between the portfolio expected return and its risk measured by the variance: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}-\lambda\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{w} \ge \mathbf{0}, \quad\mathbf{1}^T\mathbf{w}=1, \end{array}\] where \(\lambda\) is a parameter that controls how risk-averse the investor is.

Markowitz’s portfolio has been heavily critized for over half a century and has never been fully embraced by practitioners, among many reasons because:

• it only considers the risk of the portfolio as a whole and ignores the risk diversification (i.e., concentrates too much risk in few assets, which was observed in the 2008 financial crisis)
• it is highly sensitive to the estimation errors in the parameters (i.e., small estimation errors in the parameters may change the designed portfolio drastically).

Although portfolio management did not change much during the 40 years after the seminal works of Markowitz and Sharpe, the development of risk budgeting techniques marked an important milestone in deepening the relationship between risk and asset management.

From “dollar” to risk diversification

Since the global financial crisis in 2008, risk management has particularly become more important than performance management in portfolio optimization. Indeed, risk parity became a popular financial model after the global financial crisis in 2008 [2], [3].

The alternative risk parity portfolio design has been receiving significant attention from both the theoretical and practical sides [4], [5] because it: (1) diversifies the risk, instead of the capital, among the assets and (2) is less sensitive to parameter estimation errors.

Nowadays, pension funds and institutional investors are using this approach in the development of smart indexing and the redefinition of long-term investment policies.

Risk parity is an approach to portfolio management that focuses on allocation of risk rather than allocation of capital. The risk parity approach asserts that when asset allocations are adjusted to the same risk level, the portfolio can achieve a higher Sharpe ratio and can be more resistant to market downturns.

While the minimum variance portfolio tries to minimize the variance (with the disadvantage that a few assets may be the ones contributing most to the risk), the risk parity portfolio tries to constrain each asset (or asset class, such as bonds, stocks, real estate, etc.) to contribute equally to the portfolio overall volatility.

The term “risk parity” was coined by Edward Qian from PanAgora Asset Management [2] and was then adopted by the asset management industry. Some of its theoretical components were developed in the 1950s and 1960s, but the first risk parity fund, called the “All Weather” fund, was pioneered by Bridgewater Associates LP in 1996. The interest in the risk parity approach has increased since the financial crisis in the late 2000s as the risk parity approach fared better than portfolios designed in traditional fashions.

Some portfolio managers have expressed skepticism with risk parity, while others point to its performance during the financial crisis of 2007-2008 as an indication of its potential success.

The idea of the risk parity portfolio (RPP), aka equal risk portfolio (ERP), is to “equalize” the risk so that the risk contribution of every asset is equal, rather than simply having an equal capital allocation like the equally weighted portfolio (EWP):

Risk parity portfolio

From Euler’s theorem, the volatility of the portfolio \(\sigma\left(\mathbf{w}\right)=\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}\) can be decomposed as \[\sigma\left(\mathbf{w}\right)=\sum_{i=1}^{N}w_i\frac{\partial\sigma}{\partial w_i} = \sum_{i=1}^N\frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{i}}{\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}}.\]

The risk contribution (RC) from the \(i\)th asset to the total risk \(\sigma(\mathbf{w})\) is defined as \[{\sf RC}_i =\frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i}{\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}}\] which satisfies \(\sum_{i=1}^{N}{\sf RC}_i=\sigma\left(\mathbf{w}\right)\).

The relative risk contribution (RRC) is a normalized version: \[{\sf RRC}_i = \frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i}{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}\] so that \(\sum_{i=1}^{N}{\sf RRC}_i=1\).

The risk parity portfolio (RPP) attemps to “equalize” the risk contributions: \[{\sf RC}_i = \frac{1}{N}\sigma(\mathbf{w})\quad\text{or}\quad{\sf RRC}_i = \frac{1}{N}.\]

More generally, the risk budgeting portfolio (RBP) attemps to allocate the risk according to the risk profile determined by the weights \(\mathbf{b}\) (with \(\mathbf{1}^T\mathbf{b}=1\) and \(\mathbf{b}\ge \mathbf{0}\)): \[{\sf RC}_i = b_i \sigma(\mathbf{w})\quad\text{or}\quad{\sf RRC}_i = b_i.\]

In practice, one can express the condition \({\sf RC}_i = \frac{1}{N}\sigma(\mathbf{w})\) in different equivalent ways such as \[w_i(\Sigma \mathbf{w})_{i} = w_j(\Sigma \mathbf{w})_{j}, \quad\forall i, j.\] The budget condition \({\sf RC}_i = b_i \sigma(\mathbf{w})\) can also be expressed as \[w_i (\Sigma \mathbf{w})_i = b_i \mathbf{w}^{T}\Sigma\mathbf{w}, \quad\forall i.\]

Naive diagonal formulation

Assuming that the assets are uncorrelated, i.e., that \(\boldsymbol{\Sigma}\) is diagonal, and simply using the volatilities \(\boldsymbol{\sigma} = \sqrt{{\sf diag(\boldsymbol{\Sigma})}}\), one obtains \[\mathbf{w} = \frac{\boldsymbol{\sigma}^{-1}}{\mathbf{1}^T\boldsymbol{\sigma}^{-1}}\] or, more generally, \[\mathbf{w} = \frac{\sqrt{\mathbf{b}}\odot\boldsymbol{\sigma}^{-1}}{\mathbf{1}^T\left(\sqrt{\mathbf{b}}\odot\boldsymbol{\sigma}^{-1}\right)}.\] However, for non-diagonal \(\boldsymbol{\Sigma}\) or with other additional constraints or objective function terms, a closed-form solution does not exist and some optimization procedures have to be constructed. The previous diagonal solution can always be used and is called naive risk budgeting portfolio.

Vanilla convex formulation

Suppose we only have the constraints \(\mathbf{1}^T\mathbf{w}=1\) and \(\mathbf{w} \ge \mathbf{0}\). Then, after the change of variable \(\mathbf{x}=\mathbf{w}/\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}\), the equations \(w_i (\Sigma \mathbf{w})_i = b_i \mathbf{w}^{T}\Sigma\mathbf{w}\) become \(x_i\left(\boldsymbol{\Sigma}\mathbf{x}\right)_i = b_i\) or, more compactly in vector form, as \[\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}\] with \(\mathbf{x} \ge \mathbf{0}\) and we can always recover the portfolio by normalizing: \(\mathbf{w} = \mathbf{x}/(\mathbf{1}^T\mathbf{x})\).

At this point, one could use a nonlinear multivariate root finder for \(\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}\). For example, in R we can use the package rootSolve.

With the goal of designing risk budget portfolios, Spinu proposed in [6] to solve the following convex optimization problem: \[\underset{\mathbf{x}\ge\mathbf{0}}{\textsf{minimize}} \quad \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \sum_{i=1}^{N}b_i\log(x_i),\] where the portfolio can be recovered as \(\mathbf{w} = \mathbf{x}/(\mathbf{1}^T\mathbf{x})\).

Indeed, Spinu realized in [6] that precisely the risk budgeting equation \(\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}\) corresponds to the gradient of the convex function \(f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}^T\log(\mathbf{x})\) set to zero: \[\nabla f(\mathbf{x}) = \boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}/\mathbf{x} = \mathbf{0}.\]

Thus, a convenient way to solve the problem is by solving the following convex optimization problem: \[\underset{\mathbf{x}\ge\mathbf{0}}{\textsf{minimize}} \quad \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}^T\log(\mathbf{x})\] which has optimality condition \(\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}\).

Such solution can be computed using a general-purpose convex optimization package, but faster algorithms such as the Newton method and the cyclical coordinate descent method, employed in [6] and [7], are implemented in this package. (Yet another convex formulation was proposed in [8].)

General nonconvex formulation

The previous methods are based on a convex reformulation of the problem so they are guaranteed to converge to the optimal risk budgeting solution. However, they can only be employed for the simplest risk budgeting formulation with a simplex constraint set (i.e., \(\mathbf{1}^T\mathbf{w}=1\) and \(\mathbf{w} \ge \mathbf{0}\)). They cannot be used if

• we have other constraints like allowing shortselling or box constraints: \(l_i \le w_i \le u_i\)
• on top of the risk budgeting constraints \(w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i = b_i \;\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\) we have other objectives like maximizing the expected return \(\mathbf{w}^T\boldsymbol{\mu}\) or minimizing the overall variance \(\mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}\).

For those more general cases, we need more sophisticated formulations, which unfortunately are not convex. The idea is to try to achieve equal risk contributions \({\sf RC}_i = \frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i}{\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}}\) by penalizing the differences between the terms \(w_{i}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{i}\).

There are many reformulations possible. For illustrative purposes, one such formulation is \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \sum_{i,j=1}^{N}\left(w_{i}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{i}-w_{j}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{j}\right)^{2} \color{blue}{- \;F(\mathbf{w})}\\ \textsf{subject to} & \mathbf{w} \ge \mathbf{0}, \quad\mathbf{1}^T\mathbf{w}=1, \quad\color{blue}{\mathbf{w}\in\cal{W}} \end{array}\] where \(F(\mathbf{w})\) denotes some additional objective function and \(\cal{W}\) denotes an arbitrary convex set of constraints. More expressions for the risk concentration terms are listed in Appendix I.

The way to solve this general problem is derived in [9], [10] and is based on a powerful optimization framework named successive convex approximation (SCA) [11]. See Appendix II for a general idea of the method.

RPP with additional expected return term

Let us explore, for instance, the effect of including the expected return as an additional objective in the optimization problem. The problem can be formulated as \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & R(\mathbf{w}) - \lambda_{\mu} \mathbf{w}^{T}\boldsymbol{\mu}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1, \mathbf{w} \ge \mathbf{0}, \end{array}\] where \(R(\mathbf{w}) = \sum_{i=1}^{N}\left(w_{i}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i - b_i\mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}\right)^{2}\) is the risk concentration function or risk parity function, \(\mathbf{w}^{T}\boldsymbol{\mu}\) is the expected return, and \(\lambda_{\mu}\) is a trade-off parameter.

library(ggplot2)
library(riskParityPortfolio)

N <- 10
V <- matrix(rnorm(N^2), nrow = N)
Sigma <- cov(V)
mu <- runif(N)

lmd_sweep <- 10^seq(-5, 2, .25)
mean_return <- c()
risk_concentration <- c()
for (lmd_mu in lmd_sweep) {
  rpp <- riskParityPortfolio(Sigma, mu = mu, lmd_mu = lmd_mu,
                             formulation = "rc-over-sd vs b-times-sd")
  mean_return <- c(mean_return, rpp$mean_return)
  risk_concentration <- c(risk_concentration, rpp$risk_concentration)
}

ggplot(data.frame(risk_concentration, mean_return),
       aes(x = risk_concentration, y = mean_return)) +
  geom_line() + geom_point() +
  labs(title = "Expected Return vs Risk Concentration",
       x = "Risk Concentration", y = "Expected Return")

RPP with additional variance term

Similarly, the riskParityPortfolio package allows us to include the variance \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\) as an objective term: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & R(\mathbf{w}) + \lambda_{\sf var} \mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1, \mathbf{w} \ge \mathbf{0}, \end{array}\] where \(\lambda_{\sf var}\) is a trade-off parameter.

In the code, that can be done by passing a positive value to the parameter lmd_var. Let’s check the following illustrative example that depicts the trade-off between volatility and risk parity:

library(ggplot2)
library(riskParityPortfolio)

load("Sigma_mu.RData")

lmd_sweep <- 10^seq(-5, 5, .25)
variance <- c()
risk_concentration <- c()
for (lmd_var in lmd_sweep) {
  rpp <- riskParityPortfolio(Sigma, lmd_var = lmd_var,
                             formulation = "rc-over-sd vs b-times-sd")
  variance <- c(variance, rpp$variance)
  risk_concentration <- c(risk_concentration, rpp$risk_concentration)
}
volatility <- sqrt(variance)

ggplot(data.frame(risk_concentration, volatility),
       aes(x = risk_concentration, y = volatility)) +
  geom_line() + geom_point() +
  labs(title = "Volatility vs Risk Concentration",
       x = "Risk Concentration", y = "Volatility")

RPP with general linear constraints

In version 2.0, we added support for general linear constraints, i.e., riskParityPortfolio is now able to solve the following problem: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & R(\mathbf{w}) + \lambda F(\mathbf{w})\\ \textsf{subject to} & \mathbf{C}\mathbf{w} = \mathbf{c},~~\mathbf{D}\mathbf{w} \leq \mathbf{d}. \end{array}\]

Users interested in the details of the algorithm used to solve this problems are refered to Section V (Advanced Solving Approaches) of [9]. In summary, the algorithm fits well within the SCA framework, while preserving speed and scalability.

It was recently mentioned by [12] that the problem of designing risk parity portfolios with general constraints is harder than it seems. Indeed, [12] shows that, after imposing general linear constraints, the property of equal risk contributions (ERC) is unlikely to be preserved among the assets affected by the constraints.

Let’s check out a numerical example from [12]. Consider we have a universe of eight assets and we would like to design a risk parity portfolio \(\mathbf{w}\) satisfying the following constraints: \[ w_5 + w_6 + w_7 + w_8 \geq 30\%, \] \[ w_2 + w_6 \geq w_1 + w_5 + 5\%, \] and \[ \sum_i w_i = 100\% . \]

# compute the covariance matrix
vol <- c(0.05, 0.05, 0.07, 0.1, 0.15, 0.15, 0.15, 0.18)
Corr <- rbind(c(100,  80,  60, -20, -10, -20, -20, -20),
              c( 80, 100,  40, -20, -20, -10, -20, -20),
              c( 60,  40, 100,  50,  30,  20,  20,  30),
              c(-20, -20,  50, 100,  60,  60,  50,  60),
              c(-10, -20,  30,  60, 100,  90,  70,  70),
              c(-20, -10,  20,  60,  90, 100,  60,  70),
              c(-20, -20,  20,  50,  70,  60, 100,  70),
              c(-20, -20,  30,  60,  70,  70,  70, 100)) / 100
Sigma <- Corr * (vol %o% vol)

# define linear constraints
Dmat <- matrix(0, 2, 8)
Dmat[1, ] <- c(rep(0, 4), rep(-1, 4))
Dmat[2, ] <- c(1, -1, 0, 0, 1, -1, 0, 0)
dvec <- c(-0.30, -0.05)

# design portfolio
rpp <- riskParityPortfolio(Sigma, Dmat = Dmat, dvec = dvec)

# plot portfolio weights
barplotPortfolioRisk(rpp$w, Sigma)

As we can observe, the risk contributions are somewhat clustered according to the relationship among assets defined by the linear constraints, as mentioned by [12].

The linear constraints are obviously satisfied:

# equality constraints
print(sum(rpp$w))
#> [1] 1

# inequality constraints
print(Dmat %*% rpp$w)
#>       [,1]
#> [1,] -0.30
#> [2,] -0.05

The results obtained by our implementation agree with those reported by [12].