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Portfolio Optimizer

Unless specific geographical region requirements apply, the base URL of Portfolio Optimizer is: https://api.portfoliooptimizer.io/.

The current version number of Portfolio Optimizer is v1.

Portfolio Optimizer can be used:

Let be:

The arithmetic return $r_{t+1}$ of the asset over the period from the time $t$ to the time $t+1$, $t=1..T-1$, is defined as $$r_{t+1} = \frac{P_{t+1} - P_{t}}{P_{t}}$$

Let be:

The logarithmic return $r_{t+1}$ of the asset over the period from the time $t$ to the time $t+1$, $t=1..T-1$, is defined as $$r_{t+1} = \log P_{t+1} - \log P_{t}$$

Let be:

The mean return $\overline{r}$ of the asset over the $T$ time periods is defined as the the arithmetic mean of the returns $r_1,...,r_T$, that is $$ \overline{r} = \frac{1}{T} \sum_{t=1}^{T} r_t $$

Let be:

The asset covariance matrix $\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$\Sigma_{i,j} = \sigma_i \sigma_j C_{i,j}, i=1..n, j=1..n$$

Let be:

Alternatively, Let be:

The asset empirical covariance matrix $\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$\Sigma_{i,j} = \frac{1}{T} \sum_{k=1}^T (r_{k,i} - \overline{r_i}) (r_{k,j} - \overline{r_j}), i=1..n, j=1..n$$

Notes:

Let be:

Alternatively, Let be:

The exponentially weighted asset covariance matrix $\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$\Sigma_{i,j} = \frac{1 - \lambda}{1 - \lambda^{T}} \sum_{k=1}^{T} \lambda^{T-k} (r_{k,i} - \overline{r_i}) (r_{k,j} - \overline{r_j}), i=1..n, j=1..n$$

Notes:

Let $\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ be a matrix.

$\Sigma$ is an asset covariance matrix if and only if:

Let be:

The asset correlation matrix $C \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$C_{i,j} = \frac{1}{T} \sum_{k=1}^T \frac{(r_{k,i} - \overline{r_i}) (r_{k,j} - \overline{r_j})}{\sigma_i \sigma_j}, i=1..n, j=1..n$$

Notes:


Alternatively, let be:

The asset correlation matrix $C \in \mathcal{M}(\mathbb{R}^{n \times n})$ extracted from the asset covariance matrix $\Sigma$ is defined by: $$C_{i,j} = \frac{\Sigma_{i,j}}{\sigma_i \sigma_j}, i=1..n, j=1..n$$

Let $C \in \mathcal{M}(\mathbb{R}^{n \times n})$ be a matrix.

$C$ is an asset correlation matrix if and only if:

Let $n$ be the number of assets and $A \in \mathcal{M} \left( \mathbb{R}^{n \times n} \right)$ be an approximate asset correlation matrix (i.e., a matrix with no specific requirements)

Let be:

The nearest correlation matrix $C$ to the matrix $A$ is the solution of the problem: $$ C = \operatorname{argmin} \left\Vert X - A \right\Vert_F \text{ s.t. } X \in S_n \cap \mathcal{E}_n $$

Notes:


Additionally let be:

The nearest correlation matrix $C$ to the matrix $A$, with fixed off-diagonal elements of indexes $\mathcal{N}$, is the solution of the problem: $$ C = \operatorname{argmin} \left\Vert X - A \right\Vert_F \text{ s.t. }X \in S_n \cap \mathcal{E}_n \cap U_n $$

Notes:

Let be:

The arithmetic return of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ \mu {}^t w_p $$


Alternatively, let be:

The arithmetic return of the $p$-th portfolio, $p=1..n_p$, is defined as:$$ \frac{V_{T_p,p} - V_{1,p}}{V_{1,p}}$$

Let be:

The volatility $\sigma$ of the portfolio is defined as: $$ \sigma = \sqrt{ w {}^t \Sigma w} $$


Alternatively, let be:

The volatility $\sigma$ of the portfolio is defined as the standard deviation of the returns $r$: $$ \sigma = \sqrt{\frac{\sum_{t=1}^{T-1} (r_{t} - \overline{r})^2 }{T-1}} $$

, with $\overline{r}$ the average return of the portfolio.

Notes:

Let be:

The Sharpe ratio $SR$ of theportfolio is defined as: $$ SR = \frac{ \mu{}^t w - r_f}{\sqrt{ w {}^t \Sigma w }} $$


Alternatively, let be:

The Sharpe ratio $SR$ of the portfolio is defined as: $$ SR = \frac{\overline{r} - r_f}{\sigma_p} $$

Notes:

Let be:

The diversification ratio $DR$ of the portfolio is defined as: $$ DR = \frac{ \sigma{}^t w}{\sqrt{ w {}^t \Sigma w }} $$


Alternatively, let be:

The diversification ratio $DR$ of the portfolio is defined as: $$ DR = \overline{DR} \; \rho_{P, \overline{P}} $$

, with

Notes:

Let be:

The return contribution of the $i$-th asset to the return of the portfolio, $i=1..n$, is defined as: $$ w_{i} \mu_i $$

Additionally, let be:

The return contribution of the group of assets $\mathcal{N}_k$ to the return of the portfolio, $k=1..n_k$, is defined as: $$ \sum_{j \in \mathcal{N}_k} w_{j} \mu_j $$

Notes:

Let be:

The marginal contribution to risk $MCTR_i(w)$ of the $i$-th asset, $i=1..n$, to the risk of the portfolio $\sigma(w)$ is defined as: $$ MCTR_i(w) = \frac{(\Sigma w)_i}{\sqrt{w {}^t \Sigma w}} $$

The total contribution to risk $TCTR_i(w)$ of the $i$-th asset, $i=1..n$, to the risk of the portfolio $\sigma(w)$ is defined as: $$ TCTR_i(w) = w_{i} MCTR_i(w) $$

Additionally, let be:

The total contribution to risk $TCTR_{\mathcal{N}_k}(w)$ of the $k$-th group of assets $\mathcal{N}_k$, $k=1..n_k$, to the risk of the portfolio $\sigma(w)$ is defined as: $$ TCTR_{\mathcal{N}_k}(w) = \sum_{j \in \mathcal{N}_k} w_{j} \frac{(\Sigma w)_j}{\sqrt{w {}^t \Sigma w}} $$

The marginal contribution to risk $MCTR_{\mathcal{N}_k}(w)$ of the $k$-th group of assets $\mathcal{N}_k$, $k=1..n_k$, to the risk of the portfolio $\sigma(w)$ is defined by the relationship: $$ TCTR_{\mathcal{N}_k}(w) = \left( \sum_{j \in \mathcal{N}_k} w_{j} \right) MCTR_{\mathcal{N}_k}(w) $$

Notes:

Let be:

The continuous mean-variance efficient frontier is the (infinite) set of portfolios whose weights $w \in [0,1]^{n}$ satisfy: $$ w = \operatorname{argmin} \frac{1}{2} w {}^t \Sigma w - \lambda \mu {}^t w \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$ with $\lambda$ a parameter varying in $[0, +\infty[$.

Additionally, let be:

The discretized mean-variance efficient frontier is the (finite) set of $n_p$ portfolios belonging to the continuous mean-variance efficient frontier with equally spaced arithmetic returns.

Notes:

Let be:

The continuous mean-variance minimum variance frontier is the (infinite) set of portfolios whose weights $w \in [0,1]^{n}$ satisfy: $$ w = \operatorname{argmin} \frac{1}{2} w {}^t \Sigma w - \lambda \mu {}^t w \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$ with $\lambda$ a parameter varying in $]-\infty, +\infty[$.

Additionally, let be:

The discretized mean-variance minimum variance frontier is the (finite) set of $n_p$ portfolios belonging to the continuous mean-variance minimum variance frontier with equally spaced arithmetic returns.

Notes:

Let be:

Additionally, let be:

The discretized resampling-based mean-variance efficient frontier is computed through the following procedure:

  1. Determine the number of efficient frontiers to simulate $n_B$, with $n_B \ge 1$
  2. For $b = 1..n_B$ do:
    1. Generate a vector of asset arithmetic returns $\mu' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\mu_l)_1, (\mu_u)_1\right] \times ... \times \left[(\mu_l)_n, (\mu_u)_n\right]$
    2. Generate a vector of asset standard deviations $\sigma' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\sigma_l)_1, (\sigma_u)_1\right] \times ... \times \left[(\sigma_l)_n, (\sigma_u)_n\right]$
    3. Generate an asset correlation matrix $C' \in \mathcal{M}(\mathbb{R}^{n \times n})$ asymptotically uniformly at random over the space of positive definite correlation matrices whose distance in terms of max norm to $C$ is at most equal to $\delta$
    4. Generate an asset covariance matrix $\Sigma'\in \mathcal{M}(\mathbb{R}^{n \times n})$ with $\Sigma' = Diag(\sigma') C' Diag(\sigma')$
    5. Compute the mean-variance efficient frontier associated to the asset arithmetic returns $\mu'$ and covariance matrix $\Sigma'$, taking into account the applicable constraints
    6. Discretize the computed efficient frontier into a (finite) set of portfolios:
      • If the discretization is based on the portfolio returns (return rank-association), the computed efficient frontier is discretized into $n_p$ portfolios with equally spaced arithmetic returns
      • If the discretization is based on the risk tolerance parameter ($\lambda$-association) (default), the computed efficient frontier is discretized into an automatically determined number of portfolios
  3. Aggregate the portfolios belonging to each of the $n_B$ efficient frontiers into $n_p$ aggregated portfolios:
    • If the discretization is based on the portfolio returns (return rank-association), the weights of the $r$-th aggregated portfolio are obtained by averaging the weights of each of the $r$-th return ranked portfolio over the $n_B$ efficient frontiers, $r = 1..n_p$
    • If the discretization is based on the risk tolerance parameter ($\lambda$-association) (default), the weights of the $r$-th aggregated portfolio are obtained by averaging the weights of each mean-variance efficient portfolio belonging to each of the $n_B$ efficient frontiers with a target return automatically determined so as to result in $n_p$ aggregated portfolios with equally spaced arithmetic returns

Notes:

Let be:

A random portfolio is a portfolio whose asset weights $w \in [0,1]^{n}$ are generated at random and satisfy: $$ \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_{i} \leqslant w_{max} \newline n_{min} \le \lvert \{ i=1..n, w_i \ne 0 \} \rvert \le n_{max} \end{cases} $$

Notes:

Let be:

The evolution of the value of a random portfolio over time is defined as the portfolio values $V_{t} \in \mathbb{R}^{+,*}$, $t=1..T$, simulated from:

Notes:

Let be:

The evolution of the value of the portfolio over time is defined as the portfolio values $V_{t} \in \mathbb{R}^{+,*}$, $t=1..T$, simulated from:

Notes:

Let be:

The index-tracking portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w = \operatorname{argmin} \frac{1}{T} \lVert X w - r_b \rVert_2^2 \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Notes:

Let be:

The sparse index-tracking portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w \approx \operatorname{argmin} \frac{1}{T} \lVert X w - r_b \rVert_2^2 \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \newline \lvert \{ i=1..n, w_i \ne 0 \} \rvert \le n_{max} \end{cases} $$

Notes:

Let be:

The equal-weighted portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w_i = \frac{1}{n}, i=1..n$$

Let be:

The inverse variance-weighted portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w_i = \frac{1/\sigma_i^2}{\sum_{j=1}^{n} 1/\sigma_j^2}, i=1..n$$

Let be:

The inverse volatility-weighted portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w_i = \frac{1/\sigma_i}{\sum_{j=1}^{n} 1/\sigma_j}, i=1..n$$

Notes:

Let be:

The market capitalization-weighted portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w_i = \frac{mktcap_i}{\sum_{j=1}^{n} mktcap_j}, i=1..n$$

Let be:

The equal Sharpe ratio contributions portfolio weights $w \in [0,1]^{n}$ satisfy:

Notes:

Let be:

The maximum return portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmax} w {}^t \mu \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Notes:

Let be:

The minimum variance portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} w {}^t \Sigma w \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Notes:

Let be:

The risk budgeting portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} \sqrt{ w {}^t \Sigma w} - \frac{\lambda}{n} \sum_{i=1}^{n} b_i \ln(w_i) \newline \textrm{s.t. }l \leqslant w \leqslant u $$ with $\lambda \in \mathbb{R}^{+,*}$ a parameter to be determined such that $\sum_{i=1}^{n} w_i = 1$.

Notes:

Let be:

The equal risk contributions portfolio is a specific risk budgeting portfolio, where the asset risk budgets are all identical and equal to $\frac{1}{n}$, hence its name.

Notes:

Let be:

The clustering-based equal risk contributions portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. For $i = 1..k$ do
    1. Compute the weights $w_{i}^* \in [0,1]^{n}$ of the equal risk contributions portfolio within each group of assets $\mathcal{N}_{i}$, without taking into account asset weight constraints
  2. Compute the weights $w_c^* \in [0,1]^{k}$ of the equal risk contributions portfolio across all groups of assets $\mathcal{N}_1,...,\mathcal{N}_{k}$ - each considered as a synthetic asset whose weights in terms of the original assets are $w_{i}^*$, $i=1..k$ -, taking into account asset weight constraints
  3. Compute the weights $w^*$ as the matrix-vector product $$ w^* = \begin{pmatrix} w_1^* \\ w_2^* \\ ... \\ w_k^* \end{pmatrix}^t w_c^* $$

Notes:

Let be:

The maximum decorrelation portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmax} 1 - w {}^t C w \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Let be:

The maximum Sharpe ratio portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmax} \frac{w {}^t \mu - r_f }{\sqrt{ w {}^t \Sigma w}} \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Notes:

Let be:

The minimum correlation portfolio is a portfolio where the assets are weighted proportionally to their average correlation with each other.

In more details:

Let be:

The most diversified portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmax} \frac{w {}^t \sigma }{\sqrt{ w {}^t \Sigma w}} \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

Let be:

A mean-variance efficient portfolio is a portfolio whose weights $w^* \in [0,1]^{n}$ satisfy: $$ \exists \lambda \in [0, +\infty[, w^* = \operatorname{argmin} \frac{1}{2} w {}^t \Sigma w - \lambda \mu {}^t w \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

The parameter $\lambda$ is usually called the risk tolerance parameter, and is uniquely determined by an additional constraint on the portfolio:

Notes:

Let be:

The investable portfolio weights $w \in [0,1]^{n}$ closest to the desired portfolio weights $w_t$ satisfy: $$ w = \operatorname{argmin} \frac{1}{2} \lVert w - w_t \rVert_2^2 \newline \textrm{s.t. } \begin{cases} \sum_{i=1}^{n} w_i \leq 1 \newline w_i = \frac{k_i nl_i P_i}{TV}, k_i \in \mathbb{N}, i=1,...,n \newline k_i \neq 0 \implies k_i nl_i \geq ml_i, i=1,...,n \newline k_i \neq 0 \implies k_i nl_i P_i \geq mv_i, i=1,...,n \end{cases} $$

Unfortunately, the above optimization problem is computationally intractable due to the integer constraints, so that it is only possible to compute an approximate solution.

Notes:

Let be:

The post-processed portfolio weights $w^{**}$ with rounding increment $inc$ satisfy: $$ w^{**} = \operatorname{argmin} \lVert w - w^* \rVert_{\infty} \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \newline w_i = k_i inc, k_i \in \mathbb{N}, i=1..n \end{cases} $$

Notes:

Let be:

The empirical tracking error $ETE$ of the portfolio is defined as the mean of the squared differences between the returns of the portfolio and the returns of the benchmark over the $T$ time periods: $$ ETE = \frac{1}{T} \sum_{t=1}^{T} \left( r_{b,t} - r_{t} \right)^2 $$


Alternatively, let be:

The empirical tracking error $ETE$ of the portfolio of weights $w$ is defined as the mean of the squared differences between the returns of the portfolio and the returns of the benchmark over the $T$ time periods: $$ ETE = \frac{1}{T} \lVert X w - r_b \rVert_2^2 $$

Notes:

Let be:

The tracking error variance $TEV$ of the portfolio is defined as the variance of the differences between the returns of the portfolio and the returns of the benchmark over the $T$ time periods: $$ TEV = \frac{1}{T} \sum_{t=1}^{T} \left( \left( r_{b,t} - r_{t} \right) - \left( \mu_{r_b} - \mu_{r} \right) \right)^2 $$, with $ \mu_{r_b} = \frac{1}{T} \sum_{t=1}^T r_{b,t} $ and $ \mu_{r} = \frac{1}{T} \sum_{t=1}^T r_{t} $.

Alternatively, let be:

The tracking error variance $TEV$ of the portfolio of weights $w$ is defined as the variance of the differences between the returns of the portfolio and the returns of the benchmark over the $T$ time periods: $$ TEV = Var \left( \sum_{i=1}^n w_i X_{i,1} - r_{b,1}, ..., \sum_{i=1}^n w_i X_{i,T} - r_{b,T} \right) $$

Let be:

The returns $R_{res, i} \in \mathbb{R}^{T}$ of the residualized factor $i \in {1..m}$ are defined as: $$ R_{res, i} {}^t = X_i - \alpha - \beta {}^t X_{-i} $$ where:

Let be:

The exposures of the portfolio to the $m$ factors $F_1,...,F_m$ are defined as the coefficients $\beta_1, ..., \beta_m \in \mathbb{R}$ in the linear regression equation: $$ r_p = \alpha + \sum_{i=1}^m \beta_i r_{F_i} + \epsilon $$, where $\epsilon \in \mathbb{R}^{T}$ is an error term.

Notes:

Let be:

The Jensen's alpha $\alpha \in \mathbb{R}$ of the portfolio is defined as the intercept of the regression equation in the Capital Asset Pricing Model: $$ r_p - r_f = \alpha + \beta \left( r_b - r_f \right) + \epsilon $$, where $\epsilon \in \mathbb{R}^{T}$ is an error term.

Notes:

Let be:

The beta $\beta \in \mathbb{R}$ of the portfolio is defined as the slope of the regression equation in the Capital Asset Pricing Model: $$ r_p - r_f = \alpha + \beta \left( r_b - r_f \right) + \epsilon $$, where $\epsilon \in \mathbb{R}^{T}$ is an error term.

Notes:

Let be:

The correlation matrix $C_S$ defined by $$ C_S = (1-\lambda) C + \lambda C_T $$ corresponds to the (linear) shrinkage of the matrix $C$ toward the matrix $C_T$ with parameter $\lambda$.

Notes:

Let be:

A (linearly) shrunk empirical asset covariance matrix is the asset covariance matrix $\Sigma_S$ defined by $$ \Sigma_S = (1-\lambda) \Sigma + \lambda \Sigma_T $$.

Notes:

Let be:

A clustering-based (linearly) shrunk empirical asset covariance matrix is the asset covariance matrix $\Sigma_S$ defined by $$ \Sigma_S = (1-\lambda) \Sigma + \lambda \Sigma_T $$.

Notes:

Let be:

The hierarchical risk parity portfolio is a portfolio blending graph theory and machine learning techniques where similar assets are first grouped together thanks to a hierarchical clustering algorithm and asset weights are then computed through a recursive top-down bisection of the resulting hierarchical tree.

Notes:

Let be:

The hierarchical clustering-based risk parity portfolio is a portfolio building on the hierarchical risk parity portfolio, where similar assets are first grouped together thanks to an early stopped hierarchical clustering algorithm and asset weights are then computed through a recursive top-down division into two parts of the resulting hierarchical tree.

Notes:

Let be:

A random correlation matrix is a matrix $C \in \mathcal{M}(\mathbb{R}^{n \times n})$ generated uniformly at random over the space of positive definite correlation matrices, which is defined as $$ \mathcal{E}_n = \{ C \in \mathcal{M}(\mathbb{R}^{n \times n}) : C {}^t = C, C_{i,i} = 1, i=1..n, x {}^t C x > 0, \forall x \in\mathbb{R}^n \} $$

Notes:

Let be:

A randomly perturbed version of the asset correlation matrix $C$ is an asset correlation matrix $C^{'}$ whose coefficients are "close" to those of $C$, the meaning of "close" being described in the reference.


Alternatively, let be:

A randomly perturbed version of the asset correlation matrix $C$ with maximum noise level $\epsilon_{max}$ is an asset correlation matrix $C^{'}$ whose coefficients satisfy $$ \left| C_{i,j} - C^{'}_{i,j} \right| \leq \epsilon_{max}, i=1..n, j=1..n $$


Alternatively, let be:

A randomly perturbed version of the asset correlation matrix $C$ with (exact) noise level $\epsilon$ is an asset correlation matrix $C^{'}$ whose coefficients satisfy $$ \left| C_{i,j} - C^{'}_{i,j} \right| \leq \epsilon, i=1..n, j=1..n $$, with equality for at least one pair of coefficients.

Notes:

Let be:

The effective rank of the matrix $\Sigma$ is defined as $$ \textrm{erank}(\Sigma) = e^{- \sum_{i=1}^{n} \rho_i \ln(\rho_i)} $$

Notes:

Let be:

The effective rank of the matrix $C$ is defined as $$ \textrm{erank}(C) = e^{- \sum_{i=1}^{n} \rho_i \ln(\rho_i)} $$

Notes:

Let be:

The average return $\overline{r}$ of the portfolio is defined as the arithmetic average of the returns $r$: $$ \overline{r} = \frac{1}{T} \sum_{t=1}^{T} r_{t} $$

Let be:

The Ulcer Index $UI$ of the portfolio is defined as: $$ UI = \sqrt{\frac{\sum_{t=1}^{T} \left(100 * \left(\frac{V_{t}}{\max_{t'=1..t} V_{t'}} - 1\right)\right)^2 }{T}} $$

Let be:

The Ulcer Performance Index $UPI$ of the portfolio is defined as: $$ UPI = \frac {\overline{r} - r_f}{UI} $$

Notes:

Let be:

The maximum Ulcer Performance Index portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmax} \frac{w {}^t \overline{r} - r_f }{UI(w)} \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

With $UI(w)$ the Ulcer Index of the portfolio with returns $\langle w , y_t \rangle$, $t=1..T$, that is:

$$\text{UI}(w) =\sqrt{\frac{1}{T}\sum_{k=1}^{T} \left ( \max_{j = 1..k} \left ( \langle w , y_j \rangle \right ) - \langle w , y_k \rangle \right ) ^2}$$

Notes:

Let be:

The minimum Ulcer Index portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} UI(w) \newline \textrm{s.t. } \begin{cases} l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

With $UI(w)$ the Ulcer Index of the portfolio with returns $\langle w , y_t \rangle$, $t=1..T$, that is:

$$\text{UI}(w) =\sqrt{\frac{1}{T}\sum_{k=1}^{T} \left ( \max_{j = 1..k} \left ( \langle w , y_j \rangle \right ) - \langle w , y_k \rangle \right ) ^2}$$

Let be:

The empirical conditional value at risk at a confidence level $\alpha$% of the portfolio, $CVaR_{\alpha}$, is the opposite of the average of the distribution of the portfolio returns $r$ over the worst $(1 - \alpha)$% portfolio returns.

When the distribution of the portfolio returns is continuous $$ CVaR_{\alpha} = - \textrm{E} [ r | r \leq - VaR_{\alpha} ] $$ , where $VaR_{\alpha}$ is the the empirical value at risk of the portfolio.

Notes:

Let be:

The empirical value at risk at a confidence level $\alpha$% of the portfolio, $VaR_{\alpha}$, is the $\alpha$%-quantile of the distribution of the portfolio losses $-r$, or equivalently the opposite of the $(1 - \alpha)$%-quantile of the distribution of the portfolio returns $r$, that is $$ VaR_{\alpha} = - \inf_{x} \left\{ x \in \mathbb{R}, P(r \leq x)\geq 1-\alpha \right\} $$

Notes:

Let be:

The diversified maximum return portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} \sum_{i=1}^{n} w_i^2 \newline \textrm{s.t. } \begin{cases} \sqrt{ w {}^t \Sigma w} \leqslant \sigma^* (1 + \delta_{\sigma}) \newline \mu^* (1 - \delta_{\mu}) \leqslant w {}^t \mu \newline l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

where:

Let be:

The diversified maximum Sharpe ratio portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} \sum_{i=1}^{n} w_i^2 \newline \textrm{s.t. } \begin{cases} \sqrt{ w {}^t \Sigma w} \leqslant \sigma^* (1 + \delta_{\sigma}) \newline \mu^* (1 - \delta_{\mu}) \leqslant w {}^t \mu \newline l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

where:

Notes:

Let be:

The diversified minimum variance portfolio weights $w^* \in [0,1]^{n}$ satisfy: $$ w^* = \operatorname{argmin} \sum_{i=1}^{n} w_i^2 \newline \textrm{s.t. } \begin{cases} \sqrt{ w {}^t \Sigma w} \leqslant \sigma^* (1 + \delta_{\sigma}) \newline \mu^* (1 - \delta_{\mu}) \leqslant w {}^t \mu \newline l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

where:

Let be:

The weights $w^* \in [0,1]^{n}$ of a diversified mean-variance efficient portfolio satisfy:

$$ w^* = \operatorname{argmin} \sum_{i=1}^{n} w_i^2 \newline \textrm{s.t. } \begin{cases} \sqrt{ w {}^t \Sigma w} \leqslant \sigma^* (1 + \delta_{\sigma}) \newline \mu^* (1 - \delta_{\mu}) \leqslant w {}^t \mu \newline l \leqslant w \leqslant u \newline Gw \leqslant u_g \newline w_{min} \leqslant \sum_{i=1}^{n} w_i \leqslant w_{max} \end{cases} $$

where:

Let be:

The turbulence index $d$ of the assets over the period $T$, using the period $T'$ as reference, is defined as:

$$ d = \frac{1}{n} (r - \mu) {}^t \Sigma^{-1} (r - \mu) $$

Notes:

Let be:

The absorption ratio $AR$ of the assets is defined as:

$$ AR = \frac{\sum_{i=1}^N \sigma_{E_i}^2}{\sum_{i=1}^n \sigma_{E_i}^2} $$

Notes:

Let be:

The subset resampling-based maximum return portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of assets $n_S$ to include in each subset, with $2 \le n_S \le n$
  2. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  3. For $b = 1..n_B$ do
    1. Generate uniformly at random without replacement a subset of $n_S$ assets from the original set of $n$ assets
    2. Compute the weights $w_b^*$ of the maximum return portfolio associated to the generated subset of $n_S$ assets, taking into account the applicable constraints
  4. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    5. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The resampling-based maximum return portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  2. For $b = 1..n_B$ do
    1. Generate a vector of asset arithmetic returns $\mu' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\mu_l)_1, (\mu_u)_1\right] \times ... \times \left[(\mu_l)_n, (\mu_u)_n\right]$
    2. Compute the weights $w_b^*$ of the maximum return portfolio associated to the asset arithmetic returns $\mu'$, taking into account the applicable constraints
  3. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    4. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The resampling-based maximum Sharpe ratio portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  2. For $b = 1..n_B$ do
    1. Generate a vector of asset arithmetic returns $\mu' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\mu_l)_1, (\mu_u)_1\right] \times ... \times \left[(\mu_l)_n, (\mu_u)_n\right]$
    2. Generate a vector of asset standard deviations $\sigma' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\sigma_l)_1, (\sigma_u)_1\right] \times ... \times \left[(\sigma_l)_n, (\sigma_u)_n\right]$
    3. Generate an asset correlation matrix $C' \in \mathcal{M}(\mathbb{R}^{n \times n})$ asymptotically uniformly at random over the space of positive definite correlation matrices whose distance in terms of max norm to $C$ is at most equal to $\delta$
    4. Generate an asset covariance matrix $\Sigma'\in \mathcal{M}(\mathbb{R}^{n \times n})$ with $\Sigma' = Diag(\sigma') C' Diag(\sigma')$
    5. Compute the weights $w_b^*$ of the maximum Sharpe ratio portfolio associated to the asset arithmetic returns $\mu'$ and covariance matrix $\Sigma'$, taking into account the applicable constraints
  3. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    4. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The subset resampling-based maximum Sharpe ratio portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of assets $n_S$ to include in each subset, with $2 \le n_S \le n$
  2. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  3. For $b = 1..n_B$ do
    1. Generate uniformly at random without replacement a subset of $n_S$ assets from the original set of $n$ assets
    2. Compute the weights $w_b^*$ of the maximum Sharpe ratio portfolio associated to the generated subset of $n_S$ assets, taking into account the applicable constraints
  4. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    5. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The resampling-based minimum variance portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  2. For $b = 1..n_B$ do
    1. Generate a vector of asset standard deviations $\sigma' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\sigma_l)_1, (\sigma_u)_1\right] \times ... \times \left[(\sigma_l)_n, (\sigma_u)_n\right]$
    2. Generate an asset correlation matrix $C' \in \mathcal{M}(\mathbb{R}^{n \times n})$ asymptotically uniformly at random over the space of positive definite correlation matrices whose distance in terms of max norm to $C$ is at most equal to $\delta$
    3. Generate an asset covariance matrix $\Sigma'\in \mathcal{M}(\mathbb{R}^{n \times n})$ with $\Sigma' = Diag(\sigma') C' Diag(\sigma')$
    4. Compute the weights $w_b^*$ of the minimum variance portfolio associated to the asset covariance matrix $\Sigma'$, taking into account the applicable constraints
  3. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    4. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The subset resampling-based minimum variance portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of assets $n_S$ to include in each subset, with $2 \le n_S \le n$
  2. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  3. For $b = 1..n_B$ do
    1. Generate uniformly at random without replacement a subset of $n_S$ assets from the original set of $n$ assets
    2. Compute the weights $w_b^*$ of the minimum variance portfolio associated to the generated subset of $n_S$ assets, taking into account the applicable constraints
  4. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    5. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The resampling-based mean-variance efficient portfolio weights $w^* \in [0,1]^{n}$ are computed through the following procedure:

  1. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  2. For $b = 1..n_B$ do
    1. Generate a vector of asset arithmetic returns $\mu' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\mu_l)_1, (\mu_u)_1\right] \times ... \times \left[(\mu_l)_n, (\mu_u)_n\right]$
    2. Generate a vector of asset standard deviations $\sigma' \in \mathbb{R}^{n}$ uniformly at random over the $n$-dimensional box $\left[(\sigma_l)_1, (\sigma_u)_1\right] \times ... \times \left[(\sigma_l)_n, (\sigma_u)_n\right]$
    3. Generate an asset correlation matrix $C' \in \mathcal{M}(\mathbb{R}^{n \times n})$ asymptotically uniformly at random over the space of positive definite correlation matrices whose distance in terms of max norm to $C$ is at most equal to $\delta$
    4. Generate an asset covariance matrix $\Sigma'\in \mathcal{M}(\mathbb{R}^{n \times n})$ with $\Sigma' = Diag(\sigma') C' Diag(\sigma')$
    5. Compute the weights $w_b^*$ of the mean-variance efficient portfolio associated to the asset arithmetic returns $\mu'$ and covariance matrix $\Sigma'$, taking into account the applicable constraints
  3. Aggregate the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    4. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The weights $w^* \in [0,1]^{n}$ of the subset resampling-based mean-variance efficient portfolio are computed through the following procedure:

  1. Determine the number of assets $n_S$ to include in each subset, with $2 \le n_S \le n$
  2. Determine the number of portfolios to simulate $n_B$, with $n_B \ge 1$
  3. For $b = 1..n_B$ do
    1. Generate uniformly at random without replacement a subset of $n_S$ assets from the original set of $n$ assets
    2. Compute the weights $w_b^*$ of the mean-variance efficient portfolio associated to the generated subset of $n_S$ assets, taking into account the applicable constraints
  4. Combine the $n_B$ portfolio weights $w_1^*,..,w_{n_B}^*$ by averaging them through the formula:
    5. $$ w^* = \frac{1}{n_B} \sum_{b=1}^{n_B} w_b^* $$

Notes:

Let be:

The lower bounds $L \in \mathcal{M}(\mathbb{R}^{n \times n})$ and the upper bounds $U \in \mathcal{M}(\mathbb{R}^{n \times n})$ of the asset correlation matrix $C$ associated to the selected group of assets $\mathcal{B}$ are correlation matrices containing respectively the lowest and the highest possible values among which the correlations of the selected group of assets can linearly vary together while both keeping the correlations between all the other assets fixed and ensuring that the resulting correlation matrix is a valid correlation matrix.

Notes:

Let be:

The effective number of bets of the $p$-th portfolio, $p=1..n_p$, is defined as $$ \mathcal{N}_{Ent,p} = e^{- \sum_{i=1}^{n} {d_p}_i \ln({d_p}_i)} $$, where $d_p \in [0,1]^{n}$ is the diversification distribution of the $p$-th portfolio defined as $$d_p = \frac{ \left( \left( \mathring{l} {}^t \right)^{-1} w_p \right) \circ \left( \mathring{l} \Sigma w_p \right) }{w_p {}^t \Sigma w_p}$$

Notes:

Let be:

The theory-implied correlation matrix associated with a theoretical hierarchical classification of a universe of assets - like the MSCI Global Industry Classification Standard for stocks - and an empirical asset correlation matrix is computed thanks to a machine learning technique. Similar assets are first grouped together thanks to a hierarchical clustering algorithm constrained to match the hierarchical classification of the assets and theory-implied asset correlations are then derived from the resulting hierarchical tree.

Notes:

Let be:

The Euclidean distance $\mathcal{d}_F$ between the matrices $C$ and $C_R$ is defined as: $$ \mathcal{d}_F \left( C, C_R \right) = \left\Vert C - C_R \right\Vert_F $$

The correlation matrix distance $\mathcal{d}_{corr}$ between the matrices $C$ and $C_R$ is defined as: $$ \mathcal{d}_{corr}\left( C, C_R \right) = 1 - \frac{< C, C_R>}{\left\Vert C \right\Vert_F \left\Vert C_R \right\Vert_F } $$

The Bures distance $\mathcal{d}_{Bures}$ between the matrices $C$ and $C_R$ is defined as: $$ \mathcal{d}_{Bures}^2\left( C, C_R \right) = \mathrm{tr} \left( C \right) + \mathrm{tr} \left( C_R \right) -2 \mathrm{tr} \left( C^{\frac{1}{2}} C_R C^{\frac{1}{2}} \right)^{\frac{1}{2}} $$

Notes:

Let be:

A bootstrap simulation of the original $n$ asset returns over $T'$ time periods is defined as the sampling with replacement of $T'$ cross-sectional returns from the returns $r_{t,i}, i=1..n, t=1..T$ using one of the bootstrap methods described in the references.

Notes:

Let be:

Then, if $SR^* \in \mathbb{R}$ is a benchmark Sharpe ratio, the minimum track record length $MinTRL(SR^*)$ of the portfolio is defined as the (floating point) number of arithmetic returns $T^*$ that are required to ensure that the probabilistic Sharpe ratio of the portfolio $PSR(SR^*)$ is greater than or equal to $(1 - \alpha)$%, that is $$ MinTRL(SR^*) = T^* \textrm{ such that } PSR(SR^*) \geq 1 - \alpha $$


Alternatively, if $B_t \in \mathbb{R}^{+,*}$ is the value of a benchmark at time $t$, $t=1..T+1$, the minimum track record length $MinTRL(B)$ of the portfolio is defined as the (floating point) number of arithmetic returns $T^*$ that are required to ensure that $PSR(B)$ is greater than or equal to $(1 - \alpha)$%, that is $$ MinTRL(B) = T^* \textrm{ such that } PSR(B) \geq 1 - \alpha $$

Notes:

Let be:

Then, if $SR^* \in \mathbb{R}$ is a benchmark Sharpe ratio, the probabilistic Sharpe ratio $PSR(SR^*)$ of the portfolio is defined as the probability that $SR$, considered as a statistical estimator subject to estimation error, is greater than or equal to $SR^*$, with formula $$ PSR(SR^*) = \Phi\left( \frac{SR - SR^*}{ \sqrt{\frac{1 - \kappa SR + (\gamma - 1) \frac{SR^2}{4}}{T}} } \right ) $$


Alternatively, if $B_t \in \mathbb{R}^{+,*}$ is the value of a benchmark at time $t$, $t=1..T+1$, the probabilistic Sharpe ratio $PSR(B)$ of the portfolio is defined as the probability that $SR$, considered as a statistical estimator subject to estimation error, is greater than or equal to the Sharpe ratio of the benchmark $SR_B \in \mathbb{R}$, also considered as a statistical estimator subject to estimation error, with formula $$ PSR(B) = \Phi\left( \frac{SR - SR_B}{ \sqrt{\frac{1 - \kappa SR + (\gamma - 1) \frac{SR^2}{4} + 1 - \kappa_B SR_B + (\gamma_B - 1) \frac{SR_B^2}{4} + ...}{T} } } \right ) $$, where $\kappa_B \in \mathbb{R}$ is the skewness of the arithmetic returns of the benchmark, $\gamma_B \in \mathbb{R}$ is the kurtosis of the arithmetic returns of the benchmark and $...$ depends on the multivariate central moments of the arithmetic returns of the portfolio and of the benchmark as described in the first reference.

In both cases, $\Phi$ is the cumulative distribution function of the standard normal distribution.

Let be:

The Sharpe ratio adjusted for small sample bias of the portfolio is defined as $ \frac{SR}{\left( 1 + \frac{1}{4} \frac{\gamma - 1}{T} \right)} $.

Let be:

A confidence interval at a confidence level $(1 - \alpha)$% for the Sharpe ratio of a portfolio, considered as a statistical estimator subject to estimation error, is a real interval whose values are not statistically significantly different from $SR$ at a confidence level $(1 - \alpha)$%.

Notes:

Let be:

The denoised asset correlation matrix is computed by altering the empirical asset correlation matrix $C$ using one of the methods described in the references.

Notes:

Let be:

The informativeness of $C$ is defined as its normalized distance - belonging to the interval $[0,1]$ - to the set of equicorrelation matrices $\mathcal{N}$, c.f. the first reference.

Notes:

Let be:

The equal volatility-weighted portfolio weights $w \in [0,1]^{n}$ satisfy: $$ w_i = \frac{\sigma_i}{\sum_{j=1}^{n} \sigma_j}, i=1..n$$

Let be:

The correlation spectrum of the portfolio is defined as the vector $\rho(w) \in [-1,1]^{n}$ with components $$ \rho(w)_i = \frac{ \left( \Sigma{} w \right)_i }{\sigma_p \sigma_i} $$


Alternatively, let be:

The correlation spectrum of the portfolio is defined as the vector $\rho(w) \in [-1,1]^{n}$ with components $$ \rho(w)_i = \rho_{p,i} $$

, with $\rho_{p,i} \in [-1,1]$ the correlation of the arithmetic returns of the portfolio with the arithmetic returns of the asset $i, i=1..n$.

Let be:

Now, let be:

The asset returns $r_t, t=1..T$ are partitioned into $m+1$ partitions using the following thresholding procedure:

  1. Convert the turbulence thresholds $tt_i$ into turbulence scores $ts_i$, $i=1..m$
  2. For each asset return vector $r_t$, $t=1..T$
    1. Compute the unnormalized turbulence index value $ d(r_t) = n d_t$
    2. For each turbulence score $ts_i$, $i=1..m$
      • If $d(r_t) \leq ts_i$, $r_t$ is classified as belonging to the $i$-th partition of asset returns
    3. If $d(r_t) > ts_i, \forall i=1..m$, $r_t$ is classified as belonging to the default $m+1$-th partition of asset returns

Alternatively, let be:

The asset returns $r_t, t=1..T$ are partitioned into $m$ partitions using the following clustering procedure:

  1. Compute the unnormalized turbulence index values $ d(r_t) = n d_t$, $t=1..T$, for all asset return vectors $r_t$
  2. Perform an exact 1d k-means clustering algorithm on the turbulence index values $ d(r_t) $, $t=1..T$, with $k = m$
  3. For each asset return vector $r_t$, $t=1..T$
    1. $r_t$ is classified as belonging to the partition of asset returns corresponding to the k-means partition of its turbulence index value $ d(r_t) $

Notes:

Let be:

A Monte Carlo simulation of a Gaussian distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed random variables $X_i, i=1..T$ each following a Gaussian distribution with a mean equal to $\mu$ and a variance equal to $\sigma^2$, that is $$ X_i \sim \mathcal{N} \left( \mu, \sigma^2 \right), i = 1..T $$

Additionally, if an exact sample mean and (biased) sample variance constraint is enforced, the i.i.d. random variables $X_i = 1..T$ are generated so that:

Notes:

Let be:

A Gaussian distribution fitting the asset returns $r_t$, $t=1..T$ is a Gaussian distribution $ \mathcal{N} \left( \mu, \sigma^2 \right) $ with a mean equal to $\mu$ and a variance equal to $\sigma^2$.

Let be:

A Monte Carlo simulation of a Gaussian mixture distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed random variables $X_i, i=1..T$ each following a Gaussian mixture distribution made of $m$ Gaussian components, each with weight $p_j$ and with Gaussian distribution $\mathcal{N} \left( \mu_j, \sigma_j^2 \right)$, $j=1..m$.

Notes:

Let be:

A Gaussian mixture distribution fitting the asset returns $r_t$, $t=1..T$ is a Gaussian mixture distribution made of $m$ Gaussian components, each with weight $p_j$ and with Gaussian distribution $ \mathcal{N} \left( \mu_j, \sigma_j^2 \right) $ with a mean equal to $\mu_j$ and a variance equal to $\sigma_j^2$, $j=1..m$, where:

Notes:

Let be:

A Monte Carlo simulation of a multivariate Gaussian distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed $n$-dimensional random variables $X_i, i=1..T$ each following a multivariate Gaussian distribution with a mean vector equal to $\mu$ and a covariance matrix equal to $\Sigma$, that is $$ X_i \sim \mathcal{N} \left( \mu, \Sigma \right), i = 1..T $$

Additionally, if an exact sample mean vector and (biased) sample covariance variance matrix constraint is enforced, the i.i.d. random variables $X_i = 1..T$ are generated so that:

Notes:

Let be:

A multivariate Gaussian distribution fitting the vectors of asset returns $r_i$, $i=1..n$ is a multivariate Gaussian distribution $ \mathcal{N} \left( \mu, \Sigma \right) $ with a mean vector equal to $\mu$ and a covariance matrix equal to $\Sigma$.

Let be:

A Monte Carlo simulation of a multivariate Gaussian mixture distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed random variables $X_i, i=1..T$ each following a multivariate Gaussian mixture distribution made of $m$ multivariate Gaussian components whose multivariate Gaussian distribution is $\mathcal{N} \left( \mu_j, \Sigma_j \right)$, $j=1..m$.

Notes:

Let be:

A Monte Carlo simulation of a Cornish-Fisher distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed random variables $X_i, i=1..T$ each following a Cornish-Fisher distribution with a mean parameter equal to $\mu$, a variance parameter equal to $\sigma^2$, a skewness parameter equal to $\kappa$ and an excess kurtosis parameter equal to $\gamma$, that is $$ X_i \sim \mu + \sigma \left( Z + (Z^2 - 1) \frac{\kappa}{6} + (Z^3-3Z) \frac{\gamma}{24} -(2Z^3-5Z)\frac{\kappa^2}{36} \right), i = 1..T $$, where $ Z $ follows a Gaussian distribution with zero mean and unit variance, that is, $ Z \sim \mathcal{N} \left(0, 1\right) $, c.f. the references.

Notes:

Let be:

A Cornish-Fisher distribution fitting the asset returns $r_t$, $t=1..T$ is a Cornish-Fisher distribution with a mean parameter equal to $\mu$, a variance parameter equal to $\sigma^2$, a skewness parameter equal to $\kappa$ and an excess kurtosis parameter equal to $\gamma$.

Let be:

A Monte Carlo simulation of a corrected Cornish-Fisher distribution over $T$ time periods is defined as the generation of $T$ independent and identically distributed random variables $X_i, i=1..T$ each following a Cornish-Fisher distribution with a mean parameter equal to $\mu_*$, a variance parameter equal to ${\sigma_*}^2$, a skewness parameter equal to $\kappa_*$ and an excess kurtosis parameter equal to $\gamma_*$, that is $$ X_i \sim \mu_* + \sigma_* \left( Z + (Z^2 - 1) \frac{\kappa_*}{6} + (Z^3-3Z) \frac{\gamma_*}{24} -(2Z^3-5Z)\frac{{\kappa_*}^2}{36} \right), i = 1..T $$, where:

, c.f. the references.

Notes:

Let be:

A corrected Cornish-Fisher distribution fitting the asset returns $r_t$, $t=1..T$ is a Cornish-Fisher distribution with a mean parameter equal to $\mu_*$, a variance parameter equal to ${\sigma_*}^2$, a skewness parameter equal to $\kappa_*$ and an excess kurtosis parameter equal to $\gamma_*$, where $\mu_*, {\sigma_*}^2, \kappa_*, \gamma_*$ are defined such that the mean, the variance, the skewness and the excess kurtosis of the Cornish-Fisher distribution are exactly equal to $\mu, {\sigma}^2, \kappa, \gamma$, c.f. the references.

Notes:

Let be:

The Gaussian conditional value at risk at a confidence level $\alpha$% of the portfolio, $GCVaR_{\alpha}$, is defined as $$ GCVaR_{\alpha} = - \mu + \sigma \frac{1}{1-\alpha} \Phi \left( z_{1-\alpha} \right) $$ , where:

Notes:

Let be:

The Gaussian mixture conditional value at risk at a confidence level $\alpha$% of the portfolio, $GmCVaR_{\alpha}$, is defined as $$ GmCVaR_{\alpha} = - \frac{1}{1-\alpha} \sum_{i=1}^m p_i \left( \mu_i \Phi \left( - h_i \right) - \sigma_i \phi \left( - h_i \right) \right) $$ , where:

Notes:

Let be:

The Cornish-Fisher conditional value at risk at a confidence level $\alpha$% of the portfolio, $CFCVaR_{\alpha}$, is defined as $$ CFCVaR_{\alpha} = -\mu + \sigma y_{\alpha} \left[ 1 - \nu_{\alpha} \frac{\kappa}{6} + (1 - 2 \nu_{\alpha}^2) \frac{\kappa^2}{36} + (-1 + \nu_{\alpha}^2) \frac{\gamma}{24} \right] $$ , where:

Notes:

Let be:

The corrected Cornish-Fisher conditional value at risk at a confidence level $\alpha$% of the portfolio, $cCFCVaR_{\alpha}$, is defined as $$ cCFCVaR_{\alpha} = -\mu_* + \sigma_* y_{\alpha} \left[ 1 - \nu_{\alpha} \frac{\kappa_*}{6} + (1 - 2 \nu_{\alpha}^2) \frac{\kappa_*^2}{36} + (-1 + \nu_{\alpha}^2) \frac{\gamma_*}{24} \right] $$ , where:

Notes:

Let be:

The Gaussian value at risk at a confidence level $\alpha$% of the portfolio, $GVaR_{\alpha}$, is defined as $$ GVaR_{\alpha} = - \mu - \sigma z_{1-\alpha} $$ , where $z_{1-\alpha}$ is the $1 - \alpha$%-quantile of the standard Gaussian distribution.

Notes:

Let be:

The Gaussian mixture value at risk at a confidence level $\alpha$% of the portfolio, $GmVaR_{\alpha}$, is defined as the solution of the equation $$ \sum_{i=1}^m p_i \Phi \left( - \frac{ GmVaR_{\alpha} + \mu_i }{\sigma_i} \right) = 1 - \alpha $$ , where $\Phi$ is the cumulative distribution function of the standard normal distribution.

Notes:

Let be:

The Cornish-Fisher value at risk at a confidence level $\alpha$% of the portfolio, $CFVaR_{\alpha}$, is defined as $$ CFVaR_{\alpha} = - \left( \mu +\sigma \left[ z_{1-\alpha} + (z_{1-\alpha}^2 - 1) \frac{\kappa}{6} + (z_{1-\alpha}^3-3z_{1-\alpha}) \frac{\gamma}{24} -(2z_{1-\alpha}^3-5z_{1-\alpha})\frac{{\kappa}^2}{36} \right] \right)$$ , where $z_{1-\alpha}$ is the $1 - \alpha$%-quantile of the standard Gaussian distribution.

.

Notes:

Let be:

The corrected Cornish-Fisher value at risk at a confidence level $\alpha$% of the portfolio, $cCFVaR_{\alpha}$, is defined as $$ cCFVaR_{\alpha} = - \left( \mu_* +\sigma_* \left[ z_{1-\alpha} + (z_{1-\alpha}^2 - 1) \frac{\kappa_*}{6} + (z_{1-\alpha}^3-3z_{1-\alpha}) \frac{\gamma_*}{24} -(2z_{1-\alpha}^3-5z_{1-\alpha})\frac{{\kappa_*}^2}{36} \right] \right)$$ , where

Notes:

Let be:

The asset Gerber correlation matrix $G \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$G_{i,j} = \frac{n_{i,j}^{UU} + n_{i,j}^{DD} - n_{i,j}^{UD} -n_{i,j}^{DU}}{T - n_{i,j}^{NN}}, i=1..n, j=1..n $$, where:

Notes:

Let be:

The asset Gerber covariance matrix $\Sigma_G \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$\left( \Sigma_{G} \right)_{i,j} = g_{i,j} \, \sigma_i \, \sigma_j, i=1..n, j=1..n $$, where $g_{i,j}$ is the Gerber correlation between assets $i$ and $j$.

Notes:

Let be:

Under this arrangement of the assets, missing asset returns are backfilled (i.e., simulated) as follows:

  1. The assets are grouped into $1 \leq J \leq n$ groups whose length of returns history differ, as described in the second reference, with asset 1 belonging to the 1st group and asset $n$ belonging to the $J$-th group
  2. The procedure described in the first reference is first applied to the group 1 and 2:
    1. The average return of each asset belonging to the 2nd group is estimated by maximum-likelihood over the $T$ time periods $1..T$
    2. The covariance matrix between assets belonging to the 2nd group and assets belonging to the 1st group is estimated by maximum-likelihood over the $T$ time periods $1..T$
    3. Based on these maximum-likelihood estimates, missing returns for the assets belonging to the 2nd group are inferred through a backfilling procedure
  3. The procedure described in the first reference is applied next to the union of the groups 1 and 2 (which includes backfilled asset returns) and to the group 3
  4. ...
  5. The procedure described in the first reference is applied last to the union of the groups $1,2,...,J-1$ (which includes backfilled asset returns) and to the group $J$

Notes:

Let be:

The asset close-to-close volatility $\sigma_{cc} \left( T \right)$ over the time period is defined as the (unbiased) sample standard deviation of the asset close-to-close logarithmic returns, that is $$ \sigma_{cc} \left( T \right) = \sqrt{\frac{1}{T-2} \sum_{i=2}^T \left( \ln \frac{C_i}{C_{i-1}} - \mu_{cc} \right)^2} $$, where $\mu_{cc} = \frac{1}{T-1} \sum_{i=2}^T \ln \frac{C_i}{C_{i-1}} $ is the arithmetic average of these returns.

Additionally, the asset close-to-close volatility assuming a zero average (log) return $\sigma_{cc,0} \left( T \right)$ over the time period is defined as the standard deviation of the asset close-to-close logarithmic returns with a mean assumed to be equal to zero, that is $$ \sigma_{cc,0} \left( T \right) = \sqrt{\frac{1}{T-1} \sum_{i=2}^T \left( \ln \frac{C_i}{C_{i-1}} \right)^2} $$

Notes:

Let be:

The asset Parkinson volatility $\sigma_{P} \left( T \right)$ over the time period is defined as $$ \sigma_{P} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{\frac{1}{4 \ln 2} \sum_{i=1}^T \left( \ln \frac{H_i}{L_i} \right) ^2} $$

Notes:

Let be:

The asset jump-adjusted Parkinson volatility $\sigma_{P,a} \left( T \right)$ over the time period is defined as $$ \sigma_{P,a} \left( T \right) = \sqrt{ \sigma_{co}^2 + \sigma_{P}^2 } $$, where

Notes:

Let be:

The asset Garman-Klass volatility $\sigma_{GK} \left( T \right)$ over the time period is defined as $$ \sigma_{GK} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{ \sum_{i=1}^T \frac{1}{2} \left( \ln\frac{H_i}{L_i} \right) ^2 - \left( 2 \ln2 - 1 \right) \left( \ln\frac{C_i}{O_i} \right )^2 } $$

Notes:

Let be:

The asset jump-adjusted Garman-Klass volatility $\sigma_{GK,a} \left( T \right)$ over the time period is defined as $$ \sigma_{GK,a} \left( T \right) = \sqrt{ \sigma_{co}^2 + \sigma_{GK}^2 } $$, where

Notes:

Let be:

The asset original Garman-Klass volatility $\sigma_{GKo} \left( T \right)$ over the time period is defined as $$ \sigma_{GKo} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{ \sum_{i=1}^T 0.511 \left( \ln\frac{H_i}{L_i} \right) ^2 - 0.019 \left[ \ln\frac{C_i}{O_i} \left( \ln\frac{H_i}{O_i} + \ln\frac{L_i}{O_i} \right) - 2 \ln\frac{H_i}{O_i} \ln\frac{L_i}{O_i} \right] - 0.383 \left( \ln\frac{C_i}{O_i} \right )^2 } $$

Notes:

Let be:

The asset jump-adjusted original Garman-Klass volatility $\sigma_{GKo,a} \left( T \right)$ over the time period is defined as $$ \sigma_{GKo,a} \left( T \right) = \sqrt{ \sigma_{co}^2 + \sigma_{GKo}^2 } $$, where

Notes:

Let be:

The asset Rogers-Satchell volatility $\sigma_{RS} \left( T \right)$ over the time period is defined as $$ \sigma_{RS} \left( T \right) = \sqrt{\frac{1}{T}} \sqrt{ \sum_{i=1}^T \ln\frac{H_i}{C_i} \ln\frac{H_i}{O_i} - \ln\frac{L_i}{C_i} \ln\frac{L_i}{O_i} } $$

Notes:

Let be:

The asset jump-adjusted Rogers-Satchell volatility $\sigma_{RS,a} \left( T \right)$ over the time period is defined as $$ \sigma_{RS,a} \left( T \right) = \sqrt{ \sigma_{co}^2 + \sigma_{RS}^2 } $$, where

Notes:

Let be:

The asset Yang-Zhang volatility $\sigma_{YZ} \left( T \right)$ over the time period is defined as $$ \sigma_{YZ} \left( T \right) = \sqrt{ \sigma_{ov}^2 + k \sigma_{oc}^2 + (1-k) \sigma_{RS}^2 } $$, where:

Notes:

Let be:

The asset one-step-ahead simple moving average volatility forecast $\hat{\sigma}_{T+1}$ is defined as $$ \hat{\sigma}_{T+1} = \sqrt{ \frac{1}{w} \sum_{i=0}^{w-1} \tilde{\sigma}^2_{T-i} } $$

The asset $h$-step-ahead simple moving average volatility forecast $\hat{\sigma}_{T+h}, h \geq 2$ is defined by recursive substitution as $$ \hat{\sigma}_{T+h} = \sqrt{ \frac{ \sum_{i=1}^{k-h+1} \tilde{\sigma}^2_{T+1-i} + \sum_{i=1}^{h-1} \hat{\sigma}^2_{T+h-i}}{k} } $$

The asset aggregated volatility forecast over the forecast horizon $h$, $\hat{\sigma}_{T+1:T+h}, h \geq 1$, is defined as $$ \hat{\sigma}_{T+1:T+h} = \sqrt{ \sum_{i=1}^h \hat{\sigma}_{T+i}^2 } $$

Notes:

Let be:

The asset one-step-ahead exponentially weighted moving average volatility forecast $\hat{\sigma}_{T+1}$ is defined as $$ \hat{\sigma}_{T+1} = \sqrt{ \frac{1 - \lambda}{1 - \lambda^{T}} \sum_{k=1}^{T} \lambda^{T-k} \tilde{\sigma}^2_{k} } $$

The asset $h$-step-ahead exponentially weighted moving average volatility forecast $\hat{\sigma}_{T+h}, h \geq 2$ is defined as $$ \hat{\sigma}_{T+h} = \hat{\sigma}_{T+1} $$

The asset aggregated volatility forecast over the forecast horizon $h$, $\hat{\sigma}_{T+1:T+h}, h \geq 1$, is defined as $$ \hat{\sigma}_{T+1:T+h} = \sqrt{ \sum_{i=1}^h \hat{\sigma}_{T+i}^2 } $$

Notes:

Let be:

The asset one-step-ahead GARCH(1,1) volatility forecast $\hat{\sigma}_{T+1}$ is defined recursively as $$ \hat{\sigma}_{T+1} = \sqrt{ \gamma \tilde{\sigma}^2 + \alpha \tilde{\sigma}^2_{T} + \beta \hat{\sigma}_{T}^2 } $$, with:

The asset $h$-step-ahead GARCH(1,1) volatility forecast $\hat{\sigma}_{T+h}, h \geq 2$ is defined as $$ \hat{\sigma}_{T+h} = \sqrt{ \tilde{\sigma}^2 + \left( \alpha + \beta \right)^{h-1} \left( \hat{\sigma}_{T+1} - \tilde{\sigma}^2 \right) } $$

The asset aggregated volatility forecast over the forecast horizon $h$, $\hat{\sigma}_{T+1:T+h}, h \geq 1$, is defined as $$ \hat{\sigma}_{T+1:T+h} = \sqrt{ \sum_{i=1}^h \hat{\sigma}_{T+i}^2 } $$

Notes:

Let be:

The asset one-step-ahead HAR volatility forecast $\hat{\sigma}_{T+1}$ is defined as $$ \hat{\sigma}_{T+1} = \sqrt{ \beta + \beta_d \tilde{\sigma}^2_{T} + \frac{\beta_w}{5} \sum_{i=1}^5 \tilde{\sigma}^2_{T+1-i} + \frac{\beta_m}{22} \sum_{i=1}^{22} \tilde{\sigma}^2_{T+1-i} } $$, with:

The asset $h$-step-ahead HAR volatility forecast $\hat{\sigma}_{T+h}, h \geq 2$ is defined by recursive substitution as $$ \hat{\sigma}_{T+h} = \sqrt{ \beta + \beta_d \hat{\sigma}_{T+h-1}^2 + \frac{\beta_w}{5} \left( \sum_{i=1}^{5-h+1} \tilde{\sigma}^2_{T+1-i} + \sum_{i=1}^{h-1} \hat{\sigma}^2_{T+h-i} \right) + \frac{\beta_m}{22} \left( \sum_{i=1}^{22-h+1} \tilde{\sigma}^2_{T+1-i} + \sum_{i=1}^{h-1} \hat{\sigma}^2_{T+h-i} \right) } $$

The asset aggregated volatility forecast over the forecast horizon $h$, $\hat{\sigma}_{T+1:T+h}, h \geq 1$, is defined as $$ \hat{\sigma}_{T+1:T+h} = \sqrt{ \sum_{i=1}^{h} \hat{\sigma}^2_{T+i} } $$

Notes:

Let be:

The asset one-step-ahead HExp volatility forecast $\hat{\sigma}_{T+1}$ is defined as $$ \hat{\sigma}_{T+1} = \sqrt{ \beta + \beta_d ExpVP_T^{\lambda(1)} + \beta_w ExpVP_T^{\lambda(5)} + \beta_mExpVP_T^{\lambda(25)} } $$, with:

The asset $h$-step-ahead HExp volatility forecast $\hat{\sigma}_{T+h}, h \geq 2$ is defined by recursive substitution as $$ \hat{\sigma}_{T+h} = \sqrt{ \beta + \beta_d ExpVP_{T+h-1}^{\lambda(1)} + \beta_w ExpVP_{T+h-1}^{\lambda(5)} + \beta_m ExpVP_{T+h-1}^{\lambda(25)} } $$, with:

The asset aggregated volatility forecast over the forecast horizon $h$, $\hat{\sigma}_{T+1:T+h}, h \geq 1$, is defined as $$ \hat{\sigma}_{T+1:T+h} = \sqrt{ \sum_{i=1}^{h} \hat{\sigma}^2_{T+i} } $$

Notes:

Let be:

Alternatively, Let be:

The forecasted averaged asset covariance matrix over the horizon $h$ $\hat{\Sigma}_{T+1:T+h}$ is defined as $ \hat{\Sigma}_{T+1:T+h} = \frac{1}{h} \sum_{i=1}^{h} \hat{\Sigma}_{T+i} $, with:

Notes:

Let be:

Alternatively, Let be:

The forecasted averaged asset correlation matrix over the horizon $h$ $\hat{C}_{T+1:T+h}$ is defined as $\hat{C}_{T+1:T+h} = \frac{1}{h} \sum_{i=1}^h \hat{C}_i$, with $C_{T+1}$,$C_{T+2}$,...,$C_{T+h}$ the correlation matrices associated to the covariance matrices forecasted using the logic described in the SMA covariance matrix forecast endpoint.

Notes:

Let be:

Alternatively, Let be:

The forecasted averaged asset covariance matrix over the horizon $h$ $\hat{\Sigma}_{T+1:T+h}$ is defined as $ \hat{\Sigma}_{T+1:T+h} = \frac{1}{h} \sum_{i=1}^{h} \hat{\Sigma}_{T+i} $, with:

Notes:

Let be:

Alternatively, Let be:

The forecasted averaged asset correlation matrix over the horizon $h$ $\hat{C}_{T+1:T+h}$ is defined as $ \hat{C}_{T+1:T+h} = \frac{1}{h} \sum_{i=1}^h \hat{C}_i $, with $C_{T+1}$,$C_{T+2}$,...,$C_{T+h}$ the correlation matrices associated to the covariance matrices forecasted using the logic described in the EWMA covariance matrix forecast endpoint.

Notes:

Let be:

Fast threshold clustering group similar assets together using the average correlation of each asset to all other assets.

Notes:

Let be:

Hierarchical clustering group similar assets together thanks to a hierarchical tree, cut at a certain height, computed from the asset correlation matrix $C$.

Notes:

Let be:

Asset returns estimates $\mu_1,...,\mu_n$ consistent with ordering information about these assets returns - as provided in the matrix $D$ - corresponds to the geometric centroid $\mu \in \mathbb{R}^n$ of the consistent cone $\{ r \in \mathbb{R}^n, Dr \geq 0 \}$ under any probability distribution radially symmetric about the origin, c.f. the reference.

Notes:

Let be:

The moment-matched asset returns $\tilde{R} \in \mathbb{R}^{T \times n}$ are defined as:

$$ \tilde{R} = \begin{pmatrix} 1 \\ 1 \\ ... \\ 1 \end{pmatrix} \bar{\mu}^t + \left( R - \begin{pmatrix} 1 \\ 1 \\ ... \\ 1 \end{pmatrix} \mu^t \right) A_* $$, with $A_* = { \bar{\Sigma}^{\frac{1}{2}} }^t \left( { \bar{\Sigma}^{\frac{1}{2}} } \Sigma { \bar{\Sigma}^{\frac{1}{2}} }^t \right)^{-\frac{1}{2}} { \bar{\Sigma}^{\frac{1}{2}} }$

Notes: