Portfolio Optimization¶

Key Words¶

Variables¶

Variance of Portfolio¶

If the portfolio has one unit of each security whose prices are tracked in the Covariance matrix, the portfolio variance is the sum of the items in the covariance matrix.

If set of positions \(X=\{ x_1, x_2, \dots \}\), then the variance of the portfolio is given by \(\hat \sigma_p^2 = X' \text{Cov}_{XX} X\)

Markowitz Mean-Variance Analysis¶

Limitations¶

• Variance is not ideal risk measurement since it penalizes both unwanted high losses and desired low losses
• Solution: VaR, CVaR, etc
• Sensitive to estimated parameters \(\mu\) and \(\sum\)
• Solution: Robust optimization

Von Neumann-Morgenstern Utility Theory¶

• Rational portfolio choice must apply preferences based on expected utility

• Optimal pro folio solves the expected utility max problem

• Wealth after one period: \(W = W_0 (1+R_p)\)

• Expected utility: \(E[u(W)] = E[ \ u \Big( W_0(1+R_p) \Big) \ ]\)

Index Tracking/Benchmark Replication¶

Portfolio compression strategy aimed at mimicking the risk/return profile of a financial instrument, by focusing on a reduced basket of representative assets

Intuitively similar to L1 regularization $$ \begin{aligned} \text{Tracking error TE}(w) &= {\vert\vert r_b - Xw \vert\vert}_2 \ \implies \min \text{TE}(w) & + \lambda {\vert\vert w \vert\vert}_0 \end{aligned} $$ where

• \(r_b \in R^T\) : returns of benchmark instrument in the past T days
• \(X = [r_1, \dots r_T]^T \in R^{T \times N}\) : returns of \(N\) stocks in the past T days

Pairs Trading Portfolio¶

Spread \(z_t = y_{1t} - \gamma y_{2t}\) with weights \(w = \begin{bmatrix} 1 \\ -\gamma \end{bmatrix}\)

Use VECM modelling of the universe of stocks

From the parameter \(\beta\) contained in the low-ranked matrix \(\Pi = \alpha \beta^T\), one can simply use any/all column(s) of \(\beta\)

\(\beta\) defines a co-integration subspace and we can then optimize the portfolio within that con integration subspace

Conversion from Yield to Price¶

Fixed-income securities (such as bonds) trade as yield (ROI) $$ \text{Price} = \text{PV01} \cdot \text{Close} \cdot 100 $$ “PV01” of a portfolio of assets is the sensitivity of the total scheme assets to a one basis point (or 0.01 per cent) change in interest rates

Duration vs DV01¶

Either measure is fine, but be mindful of units

Spread PV01¶

For credit-risky securities, we should distinguish b/w interest rate risk & credit risk

Credit spread takes default (and recovery) into consideration

If recovery = 0, PV01 = CSPV01

Different sources of spread

• Calculated
• CDS
• Asset Swap Spreads

Larger the credit spread, higher the probability of credit defaults

Game Theory¶

When designing your portfolio, you need to incorporate external factors and others’ ideas as well (kinda like Game Theory)

Kelly Criterion¶

Simulation for Optimization¶

• Simulate the validation prices series

• Even a simple AR(1) is fine

• Naive Benchmark

• Buy if expected log return > \(k \sigma_0\)

• Sell if expected log return < \(-k \sigma_0\)

• Flatten, otherwise

• Find trading parameters that

• maximizes the average Sharpe Ratio over all simulated price series

  • \(\implies\) Solving HJB Equation

• or

  maximizes the average Sharpe Ratio over all simulated series

  • \(\implies\) Solving MLE

Sharpe Ratio¶

Limitations¶

Selection bias of strategies results in false-positives regarding the success of a strategy

Deflated Sharpe Ratio¶

Probability that SR is statistically-significant, after controlling for inflationary effect of

• No of independent trials with the strategy \(k\)
• List all the returns of all strategies

• Find the independent series

• Data Dredging \(V \left[ \widehat{\text{SR}}_k \right]\)

• Non-normality of returns: \(\hat y_3, \hat y_4\)
• Length of time series \(T\)

Can help identify if the

Calmar Ratio¶

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