Cointegration is a dependence concept which is based on the observation that a linear combination of non-stationary assets that move together can be stationary. Cointegration has been used in the analysis of financial time series as well as in portfolio construction. The cointegration portfolio has been defined as the linear combination of assets which bests tracks a certain benchmark in the sense that its active returns are the most stationary.

Diversification can be understood as being invested in assets that move in opposite directions, as much as possible. Is the most diversified portfolio therefore the least stationary portfolio?

## Tuesday, 31 December 2013

## Saturday, 21 December 2013

### Topography of UPM/LPM Space

Cumova/Nawrocki's 2013 "Portfolio Optimization in an Upside Potential and Downside Risk Framework" is an interesting summary of the upside and downside (pun intended) of portfolio construction based on partial moments.

The issues are non-trivial. For example, given the following multi-asset class universe and specifying the upper and lower partial moments with thresholds of both 9% and degrees of both 0.5 (implying an S-shape utility function in the spirit of Kahneman/Tversky)...

...the resulting investment opportunity sets in return/volatility, return/LPM and UPM/LPM space look like this...

The opportunity spaces were generated with 20'000 long-only no-leverage random portfolios. Portfolio constituent weights are not equally distributed, but biased towards the vertexes in order to have enough points on the "efficient frontier", defined as convex hulls of the opportunity sets.

Note how the efficient frontiers in Return/LPM and UPM/LPM space are not very smooth.

If we increase both thresholds to 2%, the topography changes dramatically...

It would be interesting to use the above visuals in a Rorschach Test. Anyway...

As we are calculating the partial moments based on historical data with a limited number of 120 observations, one explanation for the results is estimation risk. But looking at the data above and below the thresholds, this this only part of the story...

The issues are non-trivial. For example, given the following multi-asset class universe and specifying the upper and lower partial moments with thresholds of both 9% and degrees of both 0.5 (implying an S-shape utility function in the spirit of Kahneman/Tversky)...

...the resulting investment opportunity sets in return/volatility, return/LPM and UPM/LPM space look like this...

Note how the efficient frontiers in Return/LPM and UPM/LPM space are not very smooth.

If we increase both thresholds to 2%, the topography changes dramatically...

It would be interesting to use the above visuals in a Rorschach Test. Anyway...

As we are calculating the partial moments based on historical data with a limited number of 120 observations, one explanation for the results is estimation risk. But looking at the data above and below the thresholds, this this only part of the story...

We calculated the "exact" endogenous portfolio lower and upper partial moments, by the way.

## Tuesday, 17 December 2013

### How to Quanitify Mean Reversion and Momentum?

An unsorted risk of mean reversion / momentum indicators...

- Persistence of Regimes as Measured by Regime Transition Probabilities - mean reversion: prob(current state prevails) < prob(different state in next period), momentum: prob(current state prevails) > prob(different state in next period). See Samuleson(1991).
- Hurst Coefficient - <more details will follow at a later point>
- ...

## Monday, 16 December 2013

### Tackling the Indeterminacy of Asset Allocation in Factor Investing

Factor investing is about managing exposures relative to economic value and risk drivers.

Factor models are typically linear multiple regressions defining risk (to keep it simple, take volatility) and return of factor portfolios...

\$ r_{a} = B \cdot r_{f} \$

\$ r_{p} = w_{a} \cdot r'_{a} = w_{a} \cdot B \cdot r'_{f} = w_{f} \cdot r'_{f} \$

With $ r_{a} $ as a vector of asset returns, $ B $ as a matrix of factor exposures of all assets, $ r_{f} $ as a vector of factor returns, $ r_{p} $ as portfolio return and $ w_{a} $ as a vector of asset weights in the portfolio.

We assume that the factor model is "complete" in the sense that it fully explains the variability in the investment universe without residuals (i.e. idiosyncratic risk). An example of a complete model is a statistical factor model derived from a Principal Component Analysis (PCA) on the correlation matrix of asset returns.

The factor portfolio selection problem is an entirely unrestricted mean-variance problem in factor space...

\$ \max_{w_{f}} w_{f} \cdot r'_{f} - \frac{1}{2 \cdot \lambda} \cdot w_{f} \cdot \Omega_{f} \cdot w'_{f} \$

As factor exposures are not directly investable, the big question is how to convert optimal factor allocations into investable asset allocations. Formally, this is a linear problem...

\$ w_{a} = B^{-1} \cdot w_{f} \$

Unfortunately, it is a linear problem with multiple solutions in most cases...

A solution can be found with a pseudo-inverse matrix operation, e.g. Moore-Penrose matrix inversion. With \$ B^+ \$ being the pseudo-inverse of B...

\$ w_{a} = B^+ \cdot w_{f} \$

Additional solutions can be found with...

\$ w_{a} = B^+ \cdot w_{f} + ( I - B^+ \cdot B ) \cdot x \$

x is a vector with arbitrary real numbers.

Portfolio leverage is a linear equality constraint. Example: fully invested portfolio.

\$ w_{a} = B^+ \cdot w_{f} \\ \mathbf{1} \cdot w_{f} = 1 \$

Equality constraints can be handled by including "dummy variables" in \$ w_{a} \$ and $ B^+ \$, no problem.

Restrictions on asset weights are inequality constraints (example: long-only portfolios). Inequalities can be handled with so-called slack variables. An alternative might be using a constrained generalized inverse, e.g. the Bott-Duffin inverse.

~~An illustrative Excel spreadsheet with sample calculations is available ~~~~upon request~~~~.~~ This topic has potential. More about it at a later point in time.

For real-world investment purposes, the big issue is the stability of the factor betas over time, market regimes and asset universes.

Meucci(2007): "Risk contributions from generic user-defined factors" proposes a solution if the factor model is not complete: the asset weights are determined such that the residual risk is minimized. In this approach, the asset weights are determined as regression coefficients with a simple formula. Unfortunately, the approach is not able to handle restrictions.

Roncalli/Weisang(2012): "Risk Parity Portfolios with Risk Factors" discuss the issue in the context of risk budgeting with risk factor exposures. They interpret $ ( I - B^+ \cdot B ) \cdot x $ as an idiosyncratic risk component due to the fact that the pseudo-inverse is itself a least squares solution to a system of linear equations.

Factor models are typically linear multiple regressions defining risk (to keep it simple, take volatility) and return of factor portfolios...

\$ r_{a} = B \cdot r_{f} \$

\$ r_{p} = w_{a} \cdot r'_{a} = w_{a} \cdot B \cdot r'_{f} = w_{f} \cdot r'_{f} \$

With $ r_{a} $ as a vector of asset returns, $ B $ as a matrix of factor exposures of all assets, $ r_{f} $ as a vector of factor returns, $ r_{p} $ as portfolio return and $ w_{a} $ as a vector of asset weights in the portfolio.

We assume that the factor model is "complete" in the sense that it fully explains the variability in the investment universe without residuals (i.e. idiosyncratic risk). An example of a complete model is a statistical factor model derived from a Principal Component Analysis (PCA) on the correlation matrix of asset returns.

The factor portfolio selection problem is an entirely unrestricted mean-variance problem in factor space...

\$ \max_{w_{f}} w_{f} \cdot r'_{f} - \frac{1}{2 \cdot \lambda} \cdot w_{f} \cdot \Omega_{f} \cdot w'_{f} \$

As factor exposures are not directly investable, the big question is how to convert optimal factor allocations into investable asset allocations. Formally, this is a linear problem...

\$ w_{a} = B^{-1} \cdot w_{f} \$

Unfortunately, it is a linear problem with multiple solutions in most cases...

*1. Case "No restrictions on portfolio leverage, no restrictions on asset weights"*A solution can be found with a pseudo-inverse matrix operation, e.g. Moore-Penrose matrix inversion. With \$ B^+ \$ being the pseudo-inverse of B...

\$ w_{a} = B^+ \cdot w_{f} \$

Additional solutions can be found with...

\$ w_{a} = B^+ \cdot w_{f} + ( I - B^+ \cdot B ) \cdot x \$

x is a vector with arbitrary real numbers.

**"***2. Case "Restrictions on portfolio leverage, no restrictions on asset weights*Portfolio leverage is a linear equality constraint. Example: fully invested portfolio.

\$ w_{a} = B^+ \cdot w_{f} \\ \mathbf{1} \cdot w_{f} = 1 \$

Equality constraints can be handled by including "dummy variables" in \$ w_{a} \$ and $ B^+ \$, no problem.

**3. Case "Restrictions on portfolio leverage, restrictions on asset weights"**Restrictions on asset weights are inequality constraints (example: long-only portfolios). Inequalities can be handled with so-called slack variables. An alternative might be using a constrained generalized inverse, e.g. the Bott-Duffin inverse.

<...>

For real-world investment purposes, the big issue is the stability of the factor betas over time, market regimes and asset universes.

Meucci(2007): "Risk contributions from generic user-defined factors" proposes a solution if the factor model is not complete: the asset weights are determined such that the residual risk is minimized. In this approach, the asset weights are determined as regression coefficients with a simple formula. Unfortunately, the approach is not able to handle restrictions.

Roncalli/Weisang(2012): "Risk Parity Portfolios with Risk Factors" discuss the issue in the context of risk budgeting with risk factor exposures. They interpret $ ( I - B^+ \cdot B ) \cdot x $ as an idiosyncratic risk component due to the fact that the pseudo-inverse is itself a least squares solution to a system of linear equations.

## Saturday, 14 December 2013

### Why Modelling Power-Law Tails?

It is rather well known among investment quants that the Central Limit Theorem is based on random variables with finite second moments, such as the normal distribution. This assumption is violated when random variables exhibit power-law tail distributions as $ |x|^{−α−1} $ where $ 0 < \alpha < 2 $ and therefore having infinite variance. The sum or arithmetic mean of such random variables will converge to a stable distribution $ f(x;\alpha,0,c,0)$.

Most return data generated by financial markets is clearly not normally distributed. But then,

Most return data generated by financial markets is clearly not normally distributed. But then,

*. Models are abstract realities, they are always wrong in the sense that they do not exactly reproduce observable reality. The abstraction is the result of the assumptions made. Assumptions are chosen to yield a model which is relevant for a particular purpose or clarify a particular point. Assuming normally distributed (continuous or discrete) asset returns is wrong, as wrong as assuming power-law tails. But I do not see any relevant purpose or unique reason to work with power-law tails. If you do, I would be interested in learning more.***I have never come across an asset with infinite variance**## Friday, 13 December 2013

### Diversification Measures

Below a (growing) list of diversification measures...

- N - number of positions in a portfolio
- $ 1 / \sum_{} w^2_i $ - number of "effective positions"
- Herfindahl Index = $ \sum_{} w^2_i $ - concentration in exposures
- Gini Index - another way to express concentration in exposures
- Standard deviation of portfolio constituent weights - dispersion of exposures
- Shannon entropy of portfolio constituent weights - since weights have the properties of a probability, Shannon entropy can be used as a dispersion measure
- Shannon entropy of Diversification index - see A. Meucci
- Diversification Ratio - portfolio volatility calculated with a correlation matrix of ones divided by actual portfolio volatility
- Diversification Index - inverse of the diversification ratio, Tasche(2008).
- Degree of Diversification - portfolio volatility of the global minimum variance portfolio divided by actual portfolio volatility
- % of idiosyncratic risk - residual risk measured in the context of a single- or multiple-factor model
- % of total variability explained by first principal component - dependence on a single-most important factor

Subscribe to:
Posts (Atom)