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

Let be:

The average arithmetic return $\overline{r}$ of the asset over the $T$ time periods is defined as the the arithmetic average of the arithmetic 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: $$\Sigma_{i,j} = \frac{1}{T} \sum_{k=1}^T (r_{k,i} - \overline{r_i}) (r_{k,j} - \overline{r_j}), i=1..n, j=1..n$$

Alternatively, 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:

The exponentially weighted asset covariance matrix $\Sigma \in \mathcal{M}(\mathbb{R}^{n \times n})$ is defined by: $$\Sigma_{i,j} = \frac{1 - \lambda}{1 - \lambda^{T}} \sum_{k=0}^{T-1} \lambda^{k} (r_{T-k,i} - \overline{r_i}) (r_{T-k,j} - \overline{r_j}), i=1..n, j=1..n$$

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: $$C_{i,j} = \frac{1}{T} \sum_{k=1}^T \frac{(r_{k,i} - \overline{r_i}) (r_{k,j} - \overline{r_j})}{\sigma_i \sigma_j}, i=1..n, j=1..n$$

Alternatively, let be:

The asset correlation matrix $C \in \mathcal{M}(\mathbb{R}^{n \times n})$ 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: $$ C = \operatorname{argmin} \left\Vert X - A \right\Vert_F \text{ s.t. } X \in S_n^\delta \cap \mathcal{E}_n $$

Notes:

Let be:

Additionally, let be:

Then, for $p=1..n_p$ and $t=1..T$ : $$ V_{t, p} = V_{1, p} \sum_{i=1}^{n} w_{p,i} \frac{P_{t,i}}{P_{1,i}}$$

Notes:

Let be:

Additionally, let be:

Then, for $p=1..n_p$ and $t=2..T$ : $$ V_{t, p} = V_{t-1, p} \sum_{i=1}^{n} w_{p,i} \frac{P_{t,i}}{P_{t-1,i}}$$

Notes:

Let be:

Additionally, let be:

Then, for $p=1..n_p$ and $t=2..T $: $$ V_{t, p} = V_{t-1, p} \sum_{i=1}^{n} w_{t-1,p,i} \frac{P_{t,i}}{P_{t-1,i}} $$

with $w_{t, p} \in [0,1]^{n}$ the vector of the $p$-th portfolio weights at the time $t$, $p=1..n_p$, $t=1..T$, generated at random and satisfying: $$ \begin{cases} 0 \leqslant w_{t,p,i} \leqslant 1, i = 1..n \newline \sum_{i=1}^{n} w_{t,p,i} = 1 \end{cases} $$

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 $\sigma_p$ of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ \sigma_p = \sqrt{ w_p {}^t \Sigma w_p} $$

Alternatively, let be:

The volatility $\sigma_p$ of the $p$-th portfolio, $p=1..n_p$, is defined as the standard deviation of the arithmetic returns $r_p$: $$ \sigma_p = \sqrt{\frac{\sum_{t=1}^{T_p-1} (r_{p,t} - \overline{r_p})^2 }{T_p-1}} $$

With $\overline{r_p}$ the average arithmetic return of the $p$-th portfolio.

Notes:

Let be:

The Sharpe ratio $SR_p$ of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ SR_p = \frac{ \mu{}^t w_p - r_f}{\sqrt{ w_p {}^t \Sigma w_p }} $$

Alternatively, let be:

The Sharpe ratio $SR_p$ of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ SR_p = \frac{\overline{r_p} - r_f}{\sigma_p} $$

Let be:

The Diversification ratio of the $p$-th portfolio $p=1..n_p$, is defined as: $$ \frac{ \sigma{}^t w_p}{\sqrt{ w_p {}^t \Sigma w_p }} $$

Let be:

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

Additionally, let be:

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

Notes:

Let be:

The risk contribution of the $i$-th asset to the risk of the $p$-th portfolio, $i=1..n$ and $p=1..n_p$, is defined as: $$ w_{p,i} \frac{(\Sigma w_p)_i}{\sqrt{w_p {}^t \Sigma w_p}} $$

Additionally, let be:

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

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 $p$ vectors of portfolio weights $w_p \in [0,1]^{n}$, $p=1..n_p$, are generated at random and satisfy: $$ \begin{cases} l \leqslant w_p \leqslant u \newline w_{min} \leqslant \sum_{i=1}^{n} w_{p,i} \leqslant w_{max} \end{cases} $$

Let be:

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

Notes:

Let be:

The 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:

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

Notes:

Let be:

The 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 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 tracking error of the $p$-th portfolio, $p=1..n_p$, is defined as the volatility of the difference of the returns of the portfolio and of the returns of the benchmark over the $T$ time periods: $$ \sqrt{\frac{\sum_{t=1}^{T} (r_{b,t} - r_{p,t})^2 }{T}} $$

Notes:

Let be:

The returns $R_{res, i} \in \mathcal{R}^{T}$ 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 $\beta_p \in \mathcal{R}^{m}$ of the $p$-th portfolio to the $m$ factors, $p=1..n_p$, are defined as the unique solution of minimum euclidean norm of the linear least squares problem: $$ \operatorname{argmin_{(\alpha_p \in \mathcal{R}, \beta_p \in \mathcal{R}^{m})}} \lVert r_p {}^t - \alpha_p - \beta_p {}^t X \rVert_2^2 $$

Notes:

Let be:

The Jensen's alpha $\alpha_p \in \mathbb{R}$ of the $p$-th portfolio, $p=1..n_p$, is defined as the intercept of the regression equation in the Capital Asset Pricing Model: $$ r_{p,t} - r_{f,t} = \alpha_p + \beta_p (r_{b,t} - r_{f,t}) + \epsilon_t $$, with $t=1..T$

Notes:

Let be:

The beta $\beta_p \in \mathbb{R}$ of the $p$-th portfolio, $p=1..n_p$, is defined as the slope of the regression equation in the Capital Asset Pricing Model: $$ r_{p,t} - r_{f,t} = \alpha_p + \beta_p (r_{b,t} - r_{f,t}) + \epsilon_t $$, with $t=1..T$

Notes:

Let be:

The (linear) shrinkage of the matrix $C$ toward the matrix $C_T$ is done through the computation of the convex linear combination: $$ C_S = (1-\lambda) C_T + \lambda T $$

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:

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 arithmetic return $\overline{r_p}$ of the $p$-th portfolio, $p=1..n_p$, is defined as the arithmetic average of the arithmetic returns $r_p$: $$ \overline{r_p} = \frac{\sum_{t=1}^{T_p-1} r_{p,t}}{T_p-1} $$

Let be:

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

Let be:

The Ulcer Performance Index $UPI_p$ of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ UPI_p = \frac {\overline{r_p} - r_f}{UI_p} $$

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 conditional value at risk with confidence level $\alpha$, $CVaR_{\alpha}$, of the $p$-th portfolio, $p=1..n_p$, is the average of the distribution of the $p$-th portfolio losses $l_p$ over all the losses larger than the value at risk with confidence level $\alpha$.

Notes:

Let be:

The value at risk with confidence level $\alpha$, $VaR_{\alpha}$, of the $p$-th portfolio, $p=1..n_p$, is the $\alpha$-quantile of the distribution of the $p$-th portfolio losses $l_p$.

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:

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 $d_T$ of the assets over the period $T$ is defined as:

$$ d_T = \frac{1}{n} (r_T - \mu_{T'}) {}^t \Sigma{_{T'}}^{-1} (r_T - \mu_{T'}) $$

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 subsets to generate $n_B$, with $1 \le n_B$
  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 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 subsets to generate $n_B$, with $1 \le n_B$
  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 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 subsets to generate $n_B$, with $1 \le n_B$
  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 weights $w^* \in [0,1]^{n}$ of a 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 random subset of assets, with $2 \le n_S \le n$
  2. Determine the number of random subsets of assets $n_B$ to generate, with $1 \le n_B$
  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 a 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 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 keeping the correlations between all the other assets fixed and while 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 cosine distance $\mathcal{d}_{corr}$ between the matrices $C$ and $C_R$ is defined as: $$ \mathcal{d}_{corr}\left( C, C_R \right) = 1 - \frac{\mathrm{tr}(CC_R)}{\left\Vert C \right\Vert_F \left\Vert C_R \right\Vert_F } $$

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 the benchmark and can be found 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: