Portfolio Optimization


Internship main project
2021

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Aim

  • Find optimal portfolios over in-sample (IS) time interval
  • Apply weights to out-of-sample (OoS) returns
  • Walk-forward optimization:

  • Weekly rebalancing here
  • Compare with Equally-Weighted (EW) portfolio

Table of contents

Process Data
Data Rebalancing Out-sample returns

Methods
Markowitz Spectral Risk Measures Hierarchical Clustering methods

Additional
Exponential Covariance Shrinkage & Denoising

Guidelines/Previews
Walk-forward optimization Methods parameters Constraints Statistiques Out-Sample Intuitions

Results
Performances Turnovers Box-plots $4$ Strats Study "Boosting"
Ending
Go Further / Ideas Implementation details Bibliography

Data

  • Fund Strategies:
    • Cumulated P&L time series from naïve risk parity allocation
      So strats returns have the same expected volatility
    • Daily from 02/2007 to 06/2021
    • 9 strats aggregated (ex: Bonds Trend, Carry commodities)

  • log-returns are used: $$\tilde{r_{t}} = \log{(1 + r_{t})} \hspace{1cm} \text{with} \hspace{1cm} r_{t} = \frac{PnL_{t} - PnL_{t-1}}{AUM_t}$$
  • $X \in M_{T, N}(\mathbb{R})$ our daily log-return matrix ($T$: number of days, $N$: number of assets)
    • Notes

  • \[\begin{equation} \log{(1 + x)} \approx x \hspace{1cm} \text{for small} \hspace{0.3cm} x \end{equation} \]
  • Fund strategies are not detailed for confidentiality reasons

Rebalancing rules


  • Weekly (can be changed)

  • In-sample windows always end on Friday

  • Gap of two days before rebalancing (practical purpose)

  • Weights are shifted: in order to avoid look-ahead bias

Resample


  • Resample (daily $\rightarrow$ weekly) assets returns every Tuesday

  • Using: \[\begin{equation} \tilde{r}_{week} = \underset{t=Wed}{\overset{Tues}{\sum}}\tilde{r_{t}} = \underset{t=Wed}{\overset{Tues}{\sum}}\log{(1+r_{t})} \end{equation} \]

  • See remarks below
  • In general: \[\begin{aligned} r_{t}^{gen} = \frac{p_t - p_{t-1}}{p_{t-1}} = \frac{p_t}{p_{t-1}} - 1 \\ \text{so} \hspace{1cm} r_{week}^{gen} = \frac{p_{Tues}}{p_{Tues^{-}}} - 1 = \underset{t=Wed}{\overset{Tues}{\prod}}(1 + r_t) - 1 & = e^{\log{\left(\underset{t=Wed}{\overset{Tues}{\prod}}(1 + r_t)\right)}} - 1 \\ & = e^{\tilde{r}_{week}} - 1 \end{aligned} \]
  • But here: \[\begin{equation} r_{week} = \frac{PnL_{Tues} - PnL_{Tues^{-}}}{AUM} = \underset{t=Wed}{\overset{Tues}{\sum}}r_{t} \end{equation} \]

Out-Sample Returns

  • $$R_{OS} = \left[\underset{i=1}{\overset{N}{\sum}}\left(w_{t}^{[i]}r_{t+1}^{[i]}\right)\right]_{t=0,...,T-1}^{T}$$

$W_{opt}$
Date
In-Sample		 Asset$_1$ Asset$_2$
09/02/2007 - 11/01/2008 $w_{0}^{[1]}$ $w_{0}^{[2]}$
16/02/2007 - 18/01/2008 $w_{1}^{[1]}$ $w_{1}^{[2]}$
...
$X_{weekly}$
Date
(Tuesday)		 Asset$_1$ Asset$_2$
15/01/2008 $r_{0}^{[1]}$ $r_{0}^{[2]}$
22/01/2008 $r_{1}^{[1]}$ $r_{1}^{[2]}$
...
  • $$r_{OS}^{22/01/2008} = w_{0}^{[1]} \times r_{1}^{[1]} + w_{0}^{[2]} \times r_{1}^{[2]}$$

    • 15/01/2008: weekly return from Wednesday 09/01/2008 ($\in$ [09/02/2007,11/01/2008]: look-ahead bias)

    22/01/2008: weekly return from Wednesday 16/01/2008 ($\notin$ [09/02/2007,11/01/2008])

    Markowitz

    Max-Sharpe Min-Variance
    $$\underset{w \in \Omega}{max} \left( \frac{\mathbb{E}[R_{P}] - r_f}{\sigma_{P}}\right)$$ $$\underset{w \in \Omega}{min} \left(w\Sigma w^{T}\right)$$

    With:

    • $\sigma_{P} = \sqrt{w\Sigma w^{T}}$ the portfolio volatility and $\Sigma$ the covariance matrix
    • $R_{P} = Xw^{T}$ the portfolio daily log-returns, with $X \in M_{T, N}(\mathbb{R})$
    • $\mathbb{E}[R_{P}]$ the expected return (mean in-sample...)
    • $r_f = 0$ the risk-free rate

    Spectral Risk Measures

    $$M_{\phi}(R_{P}) = - \int_{0}^{1}\phi(p)F_{R_{P}}^{-1}(p)\mathrm{d}p$$
    Min-CVaR $\alpha\%$ "Min-Power" $\alpha \%$
    $$\underset{w \in \Omega}{min} \left( - \frac{1}{\alpha}\int_{0}^{\alpha}F_{R_{P}}^{-1}(p)\mathrm{d}p\right)$$ $$\underset{w \in \Omega}{min} \left( - \frac{1}{\alpha}\int_{0}^{1}(1-p)^{\frac{1}{\alpha}-1}F_{R_{P}}^{-1}(p)\mathrm{d}p\right)$$

    With:

    • $R_{P} = Xw^{T}$ the portfolio daily log-returns, with $X \in M_{T, N}(\mathbb{R})$
    • $F_{R_{P}}^{-1}(p)$ the quantile function, $p \in [0, 1]$
    • $\phi(p)$ the risk-aversion function

    SRM weighting comparison

    Hierarchical Clustering Methods

    • 1 - Create distance matrix from similarity matrix, $\forall (i,j) \in [1,N]^{2}$:

      • Correlation(Pearson) "CVaR" matrix
      $\rho_{i,j} \in [-1,1]$ $c_{i,j}=\frac{CVaR(R_i + R_j)}{CVaR(R_i) + CVaR(R_j)} \in [0,1]$
      $d_{i,j} = \sqrt{\frac{1}{2}(1 - \rho_{i,j})}$ $d_{i,j} = \sqrt{(1 - c_{i,j})}$
    • 2 - Select a linkage method (all but median and weighted here)
    • 3 - Matrix quasi-diagonalisation (see next slide)
    • 4 - Recursion (iterative allocation)
    Methods
    HRP HARP HCP

    Ex: clustering with correlation matrix

    Ex: clustering with "CVaR" matrix

    Recursions

    • Bisection (De Prado's method): split subset into two parts recursively
      • Avoids portfolio concentration
      • But can affect asset to wrong cluster
      • Less dramatic when there are more assets in the portfolio
    • "Topdown": Respect of the dendrogram hierarchy
    • Ex for the first iteration:

    Hierarchical Risk Parity (HRP)
    (De Prado's method) [paper]

    • Pseudo-code:
      • Initialise the weight vector $w = \mathbb{1} = [1,...,1]$
      • For each recursion iteration:
        • For $V_1$, $V_2$ cluster $1$ and cluster $2$ variance (inverse variance portfolio)
        • $\alpha_1 = 1 - \frac{V_1}{V_1 + V_2}$ and $\alpha_2 = 1 - \alpha_1$
        • $w_{cluster_1}: = w_{cluster_1} \times \alpha_1$
        • $w_{cluster_2}: = w_{cluster_2} \times \alpha_2$

    Hierarchical Alpha Risk Parity (HARP)

    • Pseudo-code:
      • Initialise the weight vector $w = \mathbb{1} = [1,...,1]$
      • For each recursion iteration:
        • For $V_1$, $V_2$ cluster $1$ and cluster $2$ variance (inverse variance portfolio)
        • For $\bar{R_1}$, $\bar{R_2}$ cluster $1$ and cluster $2$ score return (mean of assets score)
        • $\alpha_1 = 1 - \frac{V_1}{V_1 + V_2}$ and $\alpha_2 = 1 - \alpha_1$
        • $\beta_1 = \frac{\bar{R_1}}{\bar{R_1} + \bar{R_2}}$ and $\beta_2 = 1 - \beta_1$
        • $w_{cluster_1}: = w_{cluster_1} \times $($\lambda$$\alpha_1$$ + (1 - \lambda)\beta_1$) $\hspace{0.4cm}$ $\lambda$: risk-aversion parameter ($=0.5$)
        • $w_{cluster_2}: = w_{cluster_2} \times $($\lambda$$\alpha_2$$ +(1 - \lambda)\beta_2$)

    Inverse-Variance Portfolio

    $$\underset{w \in \Omega}{min} \left(w\Sigma w^{T} \right) \hspace{0.5cm} u.c \hspace{0.5cm} w^{T}\mathbb{1} = 1$$
    Optimal (Min-Variance) If $\Sigma$ is diagonal
    $w_{MV} = \frac{\Sigma^{-1}\mathbb{1}}{\mathbb{1}^{T}\Sigma\mathbb{1}}$ $w_{MV} = w_{IV} = \frac{diag\left[\Sigma\right]^{-1}}{tr\left[ diag\left[\Sigma\right]^{-1}\right]}$ and $w_{i} = \frac{\Sigma^{-1}_{i,i}}{\underset{j=1}{\overset{N}{\sum}}\Sigma^{-1}_{j,j}}$
    • When $N=2$ assets: $w_1 = \frac{\frac{1}{\Sigma_{1,1}}}{\frac{1}{\Sigma_{1,1}} + \frac{1}{\Sigma_{2,2}}} = 1 - \frac{\Sigma_{1,1}}{\Sigma_{1,1} + \Sigma_{2,2}}$
    • HRP logic is back
    • Note: in HRP the cluster variance is $V = \tilde{w}\tilde{\Sigma}\tilde{w}^{T}$ where $\tilde{\Sigma}$ is the
      subset (cluster) covariance and $\tilde{w} = \frac{diag\left[\tilde{\Sigma}\right]^{-1}}{tr\left[ diag\left[\tilde{\Sigma}\right]^{-1}\right]}$

    Returns (asset mean in-sample) to score

    • Logistic rescale preferred to a MinMaxScaler: $\hspace{0.5cm} l_{rank} = \frac{1}{1 + e^{-(a\times rank + b)}}$
    • Penalizes the lower ranked and boosts the higher ranked

    Hierarchical Clustering Portfolio (HCP)
    [paper]

    • "Hierarchical Equally-Weighted portfolio", straightforward
    • Favours non-correlated assets
    • Only topdown recursion

    Exponential Covariance [link]

    • More interest in recent observations
    • Standard ($\forall (i,j) \in [1,N]^{2}$): $$Cov(R^{[i]}, R^{[j]}) = \frac{1}{T-1}\underset{t=1}{\overset{T}{\sum}}\left( r^{[i]}_{t} - \bar{R}^{[i]} \right)\left( r^{[j]}_{t} - \bar{R}^{[j]} \right)$$
    • Exponential: $$Cov_{exp}(R^{[i]}, R^{[j]}) = \frac{\alpha}{T-1}\underset{t=1}{\overset{T}{\sum}}(1 - \alpha)^{t-1}\left( r^{[i]}_{t} - \bar{R}^{[i]} \right)\left( r^{[j]}_{t} - \bar{R}^{[j]} \right)$$
    • $(r_1^{[i]}, r_1^{[j]})$ the more recent observations

    • Same for in-sample $\mathbb{E}[R_P]$ (instead of standard mean)
    • $\alpha \in [0, 1]$
      • higher $\alpha \rightarrow$ more interest in recent obs
      • lower $\alpha \rightarrow$ tend to arithmetic mean
    • Consistent coeffs distribution: $\alpha = \frac{5}{T}$ ($T$: nb days in-sample)
    • Ex:

    Covariance Shrinkage/Denoising



    • Ledoit-Wolf (LW) [link] (source code modified for exp. cov)

    • Oracle Approximation Shrinkage (OAS) [link] (source code modified for exp. cov)

    • Denoising (Random Matrix Theory, see below)

    • Denoising Target Shrinkage (see below)

    Random Matrix Theory

    • Let $X \in M_{T, N}(\mathbb{R})$ be the observation matrix of a process with zero mean and variance $\sigma^{2}$
    • $C = \frac{1}{T}X^{T}X$ eigenvalues converge asymptotically ($T \rightarrow +\infty$ and $N \rightarrow +\infty$) to the Marcenko-Pastur distribution \[ f(\lambda) = \begin{cases} \frac{T}{N}\frac{\sqrt{(\lambda_{+} - \lambda)(\lambda - \lambda_{-})}}{2\pi\lambda\sigma^{2}} & \text{if $\lambda \in [\lambda_{-}, \lambda_{+}]$} \\ 0 & \text{if $\lambda \notin [\lambda_{-}, \lambda_{+}]$} \end{cases} \]
    • With $\lambda_{-} = \sigma^{2}\left(1 - \sqrt{\frac{N}{T}}\right)^{2}$ and $\lambda_{+} = \sigma^{2}\left(1 + \sqrt{\frac{N}{T}}\right)^{2}$

    Random Matrix Theory

    Eigenvalues Associated with
    $\lambda \in [0, \lambda_{+}]$ noise
    $\lambda \notin [0, \lambda_{+}]$ signal

    Process

    • Denoising: Let $i$ be the position of the eigenvalue such that $i > \lambda_{+}$ and $i+1 \leq \lambda_{+}$. Replace: $$\lambda_{j} = \frac{1}{N-i}\underset{k=i+1}{\overset{N}{\sum}}\lambda_{k} \hspace{0.5cm} \forall j \in [i+1,N]$$
    • Denoising Target Shrinkage: Eigenvectors associated with noise are known, apply shrinkage only to these eigenvectors
    • Covariance matrix is retrieved by PCA
    • More: [code snippets/blog] (exactly the same snippets as in this book)

    Criticism

    • These methods are normally applied to a (very) large number of assets...
      (Here we have 9 strategies)
    • Intuition: Because correlations are more likely to be noisy if the ratio of $T$ to $N$ is small (i.e if the finite time series length is small compared to the number of assets)
    • But let's see the results in our case

    Walk Forward Optimisation Guidelines


    • For all methods and declinaisons (see next for methods summary)

    • For rolling in-sample window sizes $l \in [1M, 36M]$

    • rolling out-sample window size always $1W$

    • Same out-sample time series for all methods for fair comparisons:

      • Starting $\hspace{0.2cm} \tilde{t}_0 = t_0 + 36M + 1W$, we have: $\hspace{0.2cm} T_{OS} = T - \tilde{t}_0$

    Methods Summary: parameters variants
    (X: involved)


    Method Cov:
    Classic + Exp.    		 Shrinkages:
    w/o + Others    		 Clusters:
    Correl + CVaR    		 Linkage:
    ward + single + ...    		 Recursion:
    Bisection + Topdown    		 $\alpha$:
    $5\%$ + $10\%$
    Max-Sharpe X X
    Min-Variance X X
    Min-CVaR X
    Min-Power X
    HRP X X X X X
    HARP X X X X X
    HCP X X X X

    Constraints $\Omega$

    • All constraints are the same for each method
    • Full investment: \[\begin{equation} \underset{i=1}{\overset{N}{\sum}}w_i = 1 \end{equation} \]
    • Long-only and min/max allocation (pratical purpose): \[\begin{equation} w_i \in [5\%, 35\%] \hspace{1cm} \forall i \in [1, N] \end{equation} \]

    • Optimization: constraints as an argument of the optimiser


    • Otherwise, redistribution as ($N$ number of assets):
      • $$\text{if} \hspace{0.2cm} \exists i \hspace{0.2cm} w_i > w_{max}, \forall j \ne i, \hspace{0.2cm} w_j: = w_j + \frac{w_i - w_{max}}{N-1} \hspace{0.2cm} \text{and} \hspace{0.2cm} w_i = w_{max}$$
      • $$\text{if} \hspace{0.2cm} \exists i \hspace{0.2cm} w_i < w_{min}, \forall j \ne i, \hspace{0.2cm} w_j:=w_j - \frac{w_{min} - w_i}{N-1} \hspace{0.2cm} \text{and} \hspace{0.2cm} w_i=w_{min}$$

    Statistics Out-Sample

    Annualized Sharpe Concentration (Herfindahl index)
    $$S_{ann} = \frac{\mathbb{E}[R_{OS}]}{\sigma_{R_{OS}}}\sqrt{52}$$ $$N(w) = \frac{1}{\underset{i=1}{\overset{N}{\sum}}w_{i}^{2}}$$
    Turnover(Weekly) Maximum-Drawdown
    $$TR = \frac{1}{T_{OS}-1}\underset{t=\tilde{t}_0}{\overset{T-1}{\sum}}\underset{i=1}{\overset{N}{\sum}}\left|w_{i,t+1} - w_{i,t}\right|$$ (Computed on the refactorized performance (see after))
  • Deflated Sharpe Ratio [code snippets] to add

    • Intuitions


    • Bisection:

      • Can affects assets to wrong clusters so doesn't make sense

      • (The more assets in the portfolio, the less dramatic it is)

    • Markowitz:

      • Known to be unstable out-sample (because of forecast returns deviations)
      • Covariance matrix easier to estimate, so risk-based like Min-Variance is generally preferred for stability

    • Exp. Covariance:

      • More reactive ? If yes with more turnovers...

    Performances

    • Graphics perfs: Returns out-sample refactorized with vol objective of $12\%$ $$Perf_{OS} = \underset{\tilde{t}_0}{\overset{T}{\prod}}\left( 1 + r_{t}\times\frac{12\%}{\sigma_{R_{OS}}} \right)$$

    • $1\,000\,000$ out-sample perfs with random weights to bound Sharpe ratio:
      • Plot $1$ (leverage) Plot $2$ Plot $3$ (our constraints)
      $w \sim \mathbb{U}[0,1]$ $w \sim \mathbb{U}[0,1]$ $w \sim \mathbb{U}[5\%,35\%]$ $\in M_{T_{OS}, N}(\mathbb{R})$
      $\underset{i=1}{\overset{N}{\sum}}w_i \ne 1$ $\underset{i=1}{\overset{N}{\sum}}w_i = 1$ $\underset{i=1}{\overset{N}{\sum}}w_i = 1$

    • Then plots of all performances strategies (without shrinkage)

    Random Weights: Plot $1$ (leverage)

    $w \in \mathcal{M}^{T_{OS}\times N} \sim \mathbb{U}[0,1]$, $\underset{i=1}{\overset{N}{\sum}}w_i \ne 1$, $1\,000\,000$ strats

    Notes

    Random Weights: Plot $2$

    $w \in \mathcal{M}^{T_{OS}\times N} \sim \mathbb{U}[0,1]$, $\underset{i=1}{\overset{N}{\sum}}w_i = 1$, $1\,000\,000$ strats

    Notes

    Random Weights: Plot $3$ (our constrains)

    $w \in \mathcal{M}^{T_{OS}\times N} \sim \mathbb{U}[5\%,35\%]$, $\underset{i=1}{\overset{N}{\sum}}w_i = 1$, $1\,000\,000$ strats

    Perfs Out-sample

    No Shrinkage $l \in [1M, 36M]$ $2988$ strats

  • We can see graphically the added value compared to the perfs with random weights

    • Perfs Out-sample

    No Shrinkage No Max-Sharpe No Bisection No Exp. Cov $l \in [12M, 36M]$ $875$ strats

  • We can discard the strategies (w/o shrinkage here) that are worst than EW by following our intuitions
    (Overall on Max-Sharpe and Bisection, not Exp. covariance)

    • $TR = f(l)$

  • See Intuitions

    • $TR = f(l)$
    Clustering

    Box-plots analysis


    • The goal is to detect patterns while keeping the most informations in the plots
    • Each box-plot represents the different windows lengths $l$
    • For each method involved, without shrinkage (for the moment), comparison of:
      • Covariance matrix: Classic vs. Exponential
      • Clustering: Correlation matrix vs. "CVaR" matrix
      • Recursion (hierarchical): Bisection vs. Topdown

    • Then checkpoint and conclution
    • Then see if the shrinkage/denoising adds value in our case ? (we have just $9$ assets in portfolio)

    Box-plots analysis
    Covariance: Classic vs. Exponential

    Classic Covariance looks better ? (Press down)

    Box-plots analysis
    Covariance: Classic vs. Exponential

    Exp. Cov better for $l \in [1M, 11M]$ (Press down)

    Box-plots analysis
    Covariance: Classic vs. Exponential

    Classic Cov better for $l \in [12M, 36M]$

    Box-plots analysis
    Clustering: Correlation vs. "CVaR" similarity matrix

    Correlation matrix / "CVaR" matrix for clusters: hard to differenciate

    Box-plots analysis
    Recursion:Topdown vs. Bisection

    Topdown better, see Intuitions

    Checkpoint: first conclusions

    • Exponential covariance only for $l \in [1M, 11M]$ (but bigger turnovers)
    • Classic covariance only for $l \in [12M, 36M]$
    • We discard the bisection recursion, otherwise it will reduce the boxes of clustering methods
    • See next slide all strategies boxs-plots, with shrinkage/denoising and SRMs methods

    Box-plots analysis
    All strategies with Exp. Cov, $l \in [1M, 11M]$, Correlation similarity

    Denoising decreases Sharpe (maybe not adapted with Exp. Cov)

    Box-plots analysis
    All strategies with Classic Cov, $l \in [12M, 36M]$, Correlation similarity

    Shrinkage doesn't add value

    Cases Study


    Spider Charts

    • Statistics for the $4$ strategies aggreg by mean for:
      • $l \in [1M, 11M]$ if Exp. Cov as parameter
      • $l \in [12M, 36M]$ otherwise

    • Opposite of TR and Vol before scaling (the less the better)

    HCP

    Statistics

    HARP

    Statistics

    Min-Variance

    Statistics

    Min-CVaR $5\%$

    Statistics

    "Boosting"



    • Why not considering the methods as "weak learners" and combine the weights ?
      [wisdom of the crowd]


    • More: Why not adding the weights into another model ? [Stacking]

    "Boosting"

    "Boosting"

    More consistent

    Go Further / Ideas

    • Imperative to carry out this study on several investment universes before generalising
    • Boostrapping:
      • For each method and each in-sample window, repeat the optimization over different parts of the window and average the resulting weights
    • Walk-forward expanding optimisation with market states ? [paper]
    • Causality instead of correlation as similarity for clusters ? [paper]
      See also [causaLens]
    • Deep Learning framework (my crush): Sharpe ratio, or mixed metrics as cost function isn't so cool ? [paper]
      See also [deepdow]
    • Reinforcement Learning ! (Inevitable nowadays)
      • Can be mixed with Deep Learning for the agent's action
      • Market states directly incorporated as the environment ?

    Implementation details

    • All in python
    • Packages used:
      • Mandatory: numpy, pandas
      • ML/Optim: scikit-learn, scipy
      • Plots: matplotlib, seaborn, plotly
      • Others: itertools, tqdm, pickle, warnings

    Bibliography