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Portfolio Optimizer

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The current version number of Portfolio Optimizer is v1.

Portfolio Optimizer can be used:

Let be:

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

Let be:

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

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

$\Sigma$ is a covariance matrix of $n$ assets if and only if:

Let be:

The assets 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 a correlation matrix of $n$ assets 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 assets 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 of the $p$-th portfolio, $p=1..n_p$, is defined as: $$ \sqrt{ w_p {}^t \Sigma w_p} $$

Alternatively, let be:

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

Let be:

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

Alternatively, let be:

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

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: $$ w_i = \frac{1/\sigma_i}{\sum_{j=1}^{n} 1/\sigma_j}, i=1..n$$

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 $T$ is done through the computation of the convex linear combination: $$ S = (1-\lambda) C + \lambda T $$

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 assets weights are then computed through a recursive top-down bisection 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: