tag:blogger.com,1999:blog-61200007847098699352018-09-17T07:28:48.948+02:00Quant SplintersA blog dedicated to quantitative concepts in investment management - my personal, judgmental, explorative and incomplete work in progress - fragments which are typically hard to remove and sometimes even hurt.Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-6120000784709869935.post-89523022368462093172015-06-18T14:24:00.001+02:002015-06-18T14:25:32.100+02:00We Are Moving!I am publishing on various platforms and have decided to stop adding to this particular blog. If you are interested in my more recent blogs, check on <strong>LinkedIn</strong> by clicking <a href="https://ch.linkedin.com/in/andreassteiner" target="_blank">here</a>. The posts from this blog will most likely get recycled in forthcoming LinkedIn Pulse articles.<br /><br />Thank you in advance,<br />Andreas SteinerAndreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-51715813611129758612014-08-18T22:42:00.003+02:002014-08-18T22:50:32.799+02:00A Practioner's Approach to CVaR Portfolio OptimizationConditional Value-At-Risk (CVaR) is defined as expected loss beyond a certain threshold. On a return distribution, CVaR is the expected return calculated over a certain area of the left tail. It is well known that CVaR has several properties which makes it a much more attractive risk measure than Value-At-Risk (VaR). Especially in the context of portfolio construction, it has been shown that CVaR optimization can be transformed to a linear optimization problem under restrictions. Many algorithms exist to solve such problems, even with thousands of assets.<br />This allows the construction of efficient frontiers in return/CVaR space. This addresses the old concern of practitioners and researchers that volatility might not be a suitable risk measure due to its property that both positive as well as negative surprises increase measured risk. A major advantage of volatility is that portfolio volatility can be calculated from asset exposures and asset risk with a rather simple formula...<br /><br />\$ \sigma_{p}^{2} = w \cdot \Sigma \cdot w' \$<br /><br />With \$ \sigma_{p} \$ as portfolio volatility, \$ w \$ as asset weights and \$ \Sigma \$ as the asset covariance matrix. Asset covariances can be further split into a correlation matrix \$ \Omega \$ and a vector of asset weights \$ \sigma\$.<br />A simple formula for calculating portfolio CVaR from asset tail risk information does not exist. But if we replace the vector of asset volatilities \$ \sigma\$ with a vector of asset CVaR figures and process it with asset correlations \$ \Omega \$, a matrix \$ \Sigma^{*} \$ results which can be interpreted as an asset "coCVaR" matrix. This coCVaR matrix can be used to calculate an approximation for portfolio CVaR which performs surprisingly well in practical applications...<br /><br />\$ CVaR_{p}^{2} \approx w \cdot \Sigma^{*} \cdot w' \$<br /><br />This approximation allows us to calculate return/CVaR efficient frontiers with standard mean variance optimizers or the critical line algorithm with a simple adjustment of our asset covariance matrix.<br />A simple two asset example illustrates the approach: below, we summarize the relevant risk and return characteristics for CGBI WGBI WORLD ALL MATS TR and JPM EMBI+ BRADY BROAD TR, both measured in USD and calculated from monthly data January 1991 to December 2012 (non-annualized)...<br /><div class="separator" style="clear: both; text-align: left;"><a href="http://2.bp.blogspot.com/-gsnYI2FdJnE/U_JYvDttdJI/AAAAAAAAAes/bjoppgq17Ns/s1600/Tab1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-gsnYI2FdJnE/U_JYvDttdJI/AAAAAAAAAes/bjoppgq17Ns/s320/Tab1.png" /></a></div><br /><br />The higher moments of WGBI are close to zero, indicating that it can be approximated with a normal distribution. EMBI, on the other hand, exhibits significant tail risk beyond a normal distribution: negative skewness creates a long tail and positive excess kurtosis a fat tail. Together with the higher volatility, 99% CVaR of EMBI by far exceeds the corresponding WGBI value. <br />The correlation between this WGBI and EMBI data is 12.65%, which indicates that there exist potential diversification benefits between WGBI and EMBI in a portfolio context.<br />Below, we show the composition of the efficient portfolios on the mean/volatility frontier... <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-OGVzMv1c3jk/U_Jhu3OX4kI/AAAAAAAAAe8/UX92cMA1_yI/s1600/Chart2.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-OGVzMv1c3jk/U_Jhu3OX4kI/AAAAAAAAAe8/UX92cMA1_yI/s320/Chart2.png" height="241" width="400" /></a></div><br />If we now calculate the mean/CVaR frontier based on the above approximate portfolio CVaR formula, we can compare the true and approximative CVaRs of the frontier portfolios...<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-tFFbmQRdOLE/U_JiSqH2fAI/AAAAAAAAAfE/ZnqfjfacMqM/s1600/Chart1.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-tFFbmQRdOLE/U_JiSqH2fAI/AAAAAAAAAfE/ZnqfjfacMqM/s320/Chart1.png" height="353" width="400" /></a></div><br />As we expect, the approximation works very well. For investment purposes, it is interesting to compare the investment advice implied by volatility risk and tail risk as measured by CVaR. The impact of non-normal tail risk captured by CVaR can be seen by plotting the composition of the efficient portfolios on the mean/CVaR frontier against their volatility...<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-PGdF7gFFK4c/U_JjFJERTDI/AAAAAAAAAfM/6Unm2wMEKA0/s1600/Chart3.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-PGdF7gFFK4c/U_JjFJERTDI/AAAAAAAAAfM/6Unm2wMEKA0/s320/Chart3.png" height="227" width="400" /></a></div><br />Taking into account explicitly the non-normal tail risk characteristics of emerging market debt, investment advice based on CVaR generates lower EM debt exposures despite the rather large diversification potential implied by low correlation values.Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.comtag:blogger.com,1999:blog-6120000784709869935.post-53958300054839197652014-05-27T13:56:00.000+02:002014-05-28T09:32:33.849+02:00Understanding Black/Litterman Posterior ReturnsIn our experience, many people still struggle to understand the Black/Litterman model. The available literature does not really help: it consists of either highly mathematical papers requiring strong priors (pun intended) in Bayesian statistics and matrix algebra or then "intuitive" papers mainly working with examples. We think that the basic idea can be expressed with rather simple algebra. The trick is to assume that the investment universe consists of one asset only.<br /><br />The core of the Black/Litterman model is the equation for posterior returns below...<br /><br />\$ r_{posterior} = [(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P]^{-1}\cdot[(\tau\cdot\Sigma)^{-1}\cdotr_{prior}+P'\cdot\Omega \cdot r_{forecast}] \$<br /><br />...which calculates posterior returns \$ r_{posterior} \$ from numerous input variables and parameters: \$ r_{prior} \$ as prior returns (e.g. implied returns derived from a reverse optimization), \$ \tau \$ as the famous and rather dubious "confidence" parameter, \$ \Sigma \$ as asset covariances, \$ \Omega \$ as estimation risk (expressed as a covariance matrix, P as a view matrix defining the view portfolios, \$ r_{forecast \$ as the forecasted returns of the views.<br /><br />Quite a bit of insight is already gained by reshuffling terms...<br /><br /> \$ r_{posterior} = \frac{(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P}{\tau\cdot\Sigma}\cdot r_{prior}+\frac{P'\cdot\Omega}{(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P}\cdot r_{forecast} \$<br /><br />It becomes immediately clear that posterior returns are a weighted sum of prior and forecast returns. If we now assume that there exists one asset only, the matrix expressions simplify rather dramatically to...<br /><br />\$ r_{posterior} = (1-w) \cdot r_{prior}+ w \cdot r_{forecast} \$<br /><br />\$ w = \frac{\sigma}{(1/\tau) \cdot \sigma_{forecast}+ \sigma} \$<br /><br /><em>Therefore, Black/Litterman posterior returns can be understood as a diversified portfolio consisting of prior and forecast returns, with the relative weight between prior and forecast returns is determined by the percentage of confidence-adjusted forecast risk relative to asset risk.</em> This is the shortest and most precise explanation of Black/Litterman posterior returns possible. It creates the space for the relevant discussion, which should centre around how to specify the inputs (asset risk, asset return priors, asset return forecasts, forecast errors & confidence).Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-9528202819106424582014-04-08T11:29:00.003+02:002014-04-08T11:37:02.368+02:00Diversification is a Second Order EffectDiversification is commonly understood as one of the cornerstones of Modern Portfolio Theory. The most basic MPT models (which are taught in schools and therefore represent the way most people think about MPT), are single-period models in which the inputs (most importantly expected returns, covariance matrix and investor preferences), are assumed to be known in advance and to remain constant over this time period. In this world, diversification is then perceived as the benefit of holding a portfolio of imperfectly correlated assets and is expected to result in improved risk-adjusted returns for the investor.<br /><br />Real-world investing is about running portfolios in a dynamic world characterized by time-varying and asset risk and return characteristics which are subject to significant forecasting uncertainty (as opposed to forecasting risk). The chart below plots the rebased trajectories of the two important asset classes equities and bonds, as measured by two popular market indices (total returns, monthly data 1985-2012, base currency USD)...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-kal2mi6U484/U0PA5mmAA1I/AAAAAAAAAdk/lAeLz8nWg28/s1600/chart1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-kal2mi6U484/U0PA5mmAA1I/AAAAAAAAAdk/lAeLz8nWg28/s1600/chart1.gif" height="240" width="400" /></a></div><span id="goog_704948651"></span><br />In order to visualize the risk dynamics, we calculate annualized rolling 24 month asset volatilities and 24 month rolling asset correlation...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-jEEDaoFy7v4/U0PA8MapugI/AAAAAAAAAeI/G6I5f3o8trk/s1600/chart2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-jEEDaoFy7v4/U0PA8MapugI/AAAAAAAAAeI/G6I5f3o8trk/s1600/chart2.gif" height="240" width="400" /></a></div><br />We see that neither correlations nor volatilities are stable. Plotting correlations against average volatility reveals a relatively week relationship between the two...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-V_iSH-ruL6g/U0PA5uw2_PI/AAAAAAAAAd0/8wltpzjlRtc/s1600/chart3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-V_iSH-ruL6g/U0PA5uw2_PI/AAAAAAAAAd0/8wltpzjlRtc/s1600/chart3.gif" height="240" width="400" /></a></div><br />In order to assess the relative importance of volatility dynamics versus correlation dynamics, we create a 50/50 monthly rebalanced paper portfolio and calculate its 24 month rolling volatility over time and create scatter plots against asset volatilities and asset correlations...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Wv2cLLmncdE/U0PA6X-dJUI/AAAAAAAAAd8/006UcVJAPvg/s1600/chart4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-Wv2cLLmncdE/U0PA6X-dJUI/AAAAAAAAAd8/006UcVJAPvg/s1600/chart4.gif" height="240" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"> </div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-S2A7tp_ud1A/U0PA6tPwm8I/AAAAAAAAAeA/pGPoexw0qII/s1600/chart5.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-S2A7tp_ud1A/U0PA6tPwm8I/AAAAAAAAAeA/pGPoexw0qII/s1600/chart5.gif" height="240" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"> </div>We see that the risk of the 50/50 strategy is driven much more by asset risk dynamics than asset correlations. The difference is so pronounced that we can conclude that time-variations in the diversification potential is a minor second order effect compared to time-variations in asset risk. This stylized fact can be easily reproduced with more sophisticated statistical methods, different asset universes and different historical time periods. This fact has important consequences for real-world investment practice like asset allocation, portfolio construction and risk management.Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-91765557068678328392014-01-24T14:34:00.001+01:002014-06-05T22:10:40.593+02:00Drawdown Risk Budgeting: Contributions to Drawdown-At-Risk and the Drawdown Parity PortfolioSimilar to Value-At-Risk, Drawdown-At-Risk is defined as a point on the drawdown distribution defined by a probability interpreted as a "level of confidence". The well-known risk measure Maximum Drawdown is the 100% Drawdown-At-Risk, i.e. the drawdown which is not exceeded with certainty.<br /><br />The table below shows Drawdown-At-Risk values for the constituents of a specific multi asset class universe (total returns, monthly figures, Jan 2001 to Oct 2011, base currency USD)...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-J57HtWfeIoo/UuJd09r_nyI/AAAAAAAAAbY/feAmvgVyask/s1600/DaR.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-J57HtWfeIoo/UuJd09r_nyI/AAAAAAAAAbY/feAmvgVyask/s1600/DaR.JPG" height="212" width="640" /></a></div><br />In portfolio analytics, fully additive contributions to risk can be derived from <a href="http://en.wikipedia.org/wiki/Euler%27s_homogeneous_function_theorem#Positive_homogeneity" target="_blank">Euler's homogeneous function theorem</a> for <em>linear</em> homogeneous risk measures. Portfolio volatility and tracking error are examples of risk measure which are linear homogeneous in constituent weights.<br /><br />Non-linear homogeneous risk measures can be approximated (e.g. Taylor series expansions, using the total differential as a linear approximation and so on). In the chart below, we show how the 95% DaR of an equal-weighted portfolio varies with variations in individual constituent weights (we make the assumptions that exposures are booked against a riskfree cash account with zero return)...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-R5qYhjmfJGo/UuJigLt3TMI/AAAAAAAAAbo/5jPM5-sBq2w/s1600/DaRProfile2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-R5qYhjmfJGo/UuJigLt3TMI/AAAAAAAAAbo/5jPM5-sBq2w/s1600/DaRProfile2.JPG" height="358" width="400" /></a></div><br /><br />This chart is called "the Spaghetti chart" by certain people. In the case of the minimum 95% DaR portfolio, i.e. the fully invested long-only portfolio with minimum 95% Drawdown-At-Risk, all spaghettis must point downwards...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-bvuh-l9I7Ko/UuJjhg3IlQI/AAAAAAAAAbw/WC3myCHtujs/s1600/MinDaRSpag.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-bvuh-l9I7Ko/UuJjhg3IlQI/AAAAAAAAAbw/WC3myCHtujs/s1600/MinDaRSpag.JPG" height="361" width="400" /></a></div><br />The full details of the risk decomposition for the equal-weighted portfolio...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-R_kjMHdzaWI/UuJnDLJjv-I/AAAAAAAAAcA/qY9jodQu09I/s1600/DetailsEqualWeights.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-R_kjMHdzaWI/UuJnDLJjv-I/AAAAAAAAAcA/qY9jodQu09I/s1600/DetailsEqualWeights.JPG" height="116" width="640" /></a></div><br />...in comparsion with the minimum 95% DaR portfolio...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-qw-MWbazRgI/UuJnTky7TuI/AAAAAAAAAcI/Vzo0xa-c0Ss/s1600/DetailsMinDaR.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-qw-MWbazRgI/UuJnTky7TuI/AAAAAAAAAcI/Vzo0xa-c0Ss/s1600/DetailsMinDaR.JPG" height="118" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"> </div><div class="separator" style="clear: both; text-align: left;">Additive contributions to portfolio Drawdown-At-Risk open up the door for <strong>drawdown risk budgeting</strong>. For example, the <strong>Drawdown Parity Portfolio</strong> can be calculated as the portfolio with equal constituent contributions to portfolio drawdown risk...</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-wiDAYb4bOIE/UuJrr65EgqI/AAAAAAAAAcU/V01TxDzs0-8/s1600/DetailsEqualDaRContrPortfolio.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-wiDAYb4bOIE/UuJrr65EgqI/AAAAAAAAAcU/V01TxDzs0-8/s1600/DetailsEqualDaRContrPortfolio.JPG" height="102" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;"></div><div class="separator" style="clear: both; text-align: left;">Due to the residual, the DaR contributions are not perfectly equalized. Taking into account estimation risk and other implementation issues, this is acceptable for practical purposes.</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">Being able to calculate additive contributions to drawdown-at-risk is useful for descriptive ex post or ex ante risk budgeting purposes. The trade risk charts are useful indicators providing information on a) the risk drivers in the portfolio and b) the directions to trade.</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">Budgeting drawdown risk is really budgeting for future drawdowns ("ex ante drawdown"). This involves estimating future drawdowns. Whether future drawdowns can be estimated with the required precision is an empirical question. In order to assess what this task might involve, it is interesting reviewing certain findings in the theoretical literature related to the expected maximum drawdown for geometric Brownian motions (see for example "An Analysis of the Expected Maximum Drawdown Risk Measure" by Magdon-Ismail/Atyia. More recently, analytical results have been derived for return generating processes with time-varying volatility). In the long-run, the expected maximum drawdown for a geometric Brownian motion is...</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">\$ MDD_{e} = (0.63519 + 0.5 \cdot ln(T) + ln(\frac{mu}{sigma})) \cdot \frac{sigma^2}{mu} \$</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">Expected maximum drawdown is function in investment horizon (+), volatility (+) and expected return (-).</div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">While we have time series models with proven high predictive power to estimate volatility risk (e.g. GARCH), the estimation of maximum drawdown is a much more challenging task because it involves estimating expected returns, which is known to be subject to much higher estimation risk.</div><script type="text/javascript">MathJax.Hub.Queue(["Typeset",MathJax.Hub]);</script>Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-43093919942742160252014-01-06T13:55:00.002+01:002014-01-06T15:09:46.233+01:00Resampling the Efficient Frontier - With How Many Observations?Since optimizer inputs are stochastic variables, it follows that any efficient frontier must be a stochastic object. The efficient frontier we usually plot in mean/variance space is the expected efficient frontier. The realized efficient frontier will almost always deviate from the expected frontier and will lie within certain confidence bands.<br />Several attempts have been made to illustrate the stochastic nature of the efficient frontier, the most famous one probably being the so-called "Resampled Efficient Frontier" (tm) by Michaud/Michaud(1998).<br />Resampling involves setting the number of simulations as well as setting the number of observations to generate in each simulation. The importance of the latter decision is typically underestimated.<br />The chart below plots the resampled portfolios of 16 portfolios on a particular mean/variance efficient frontier...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ERm2F768IpA/UsqmhoQ5uqI/AAAAAAAAAbE/jp9jDk9j7VQ/s1600/JustAboutRight.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-ERm2F768IpA/UsqmhoQ5uqI/AAAAAAAAAbE/jp9jDk9j7VQ/s1600/JustAboutRight.jpg" height="256" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: left;">The larger density of points at the bottom left end of the frontier is a result from the fact that there exist two very similar corner portfolios in this area of the curve.</div><div class="separator" style="clear: both; text-align: left;">The chart below plots the same frontier with the same number of simulations, but a much larger number of generated observations...</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-dYG5JkXEZbg/Usqm1pCt6CI/AAAAAAAAAbI/2TqQSmoEGTA/s1600/TooMuch.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-dYG5JkXEZbg/Usqm1pCt6CI/AAAAAAAAAbI/2TqQSmoEGTA/s1600/TooMuch.jpg" height="255" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;"></div>As the confidence bands, average weights or any risk and return characteristics are largely determined by the choice of number of simulations and number of observations in each simulation, it is worth keeping an eye on these modelling decisions when using relying on a resampling approach for investment purposes.Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-4608988415208944052013-12-31T15:24:00.002+01:002013-12-31T15:24:34.660+01:00How Diversified is the Minimum Cointegration Portolio?Cointegration is a dependence concept which is based on the observation that a linear combination of <a href="http://en.wikipedia.org/wiki/Covariance-stationary" target="_blank">non-stationary</a> assets that move together can be <a href="http://en.wikipedia.org/wiki/Covariance-stationary" target="_blank">stationary</a>. 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.<br />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?Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-57577813534473544582013-12-21T07:00:00.000+01:002013-12-21T08:45:12.833+01:00Topography of UPM/LPM SpaceCumova/Nawrocki's 2013 "<a href="http://dx.doi.org/10.1016/j.jeconbus.2013.08.001" target="_blank">Portfolio Optimization in an Upside Potential and Downside Risk Framework</a>" is an interesting summary of the upside and downside (pun intended) of portfolio construction based on partial moments.<br /><br />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)...<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-k2n9JfH96qI/UrUoQCexuHI/AAAAAAAAAZA/hCYWHb7GYwc/s1600/table1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="292" src="http://3.bp.blogspot.com/-k2n9JfH96qI/UrUoQCexuHI/AAAAAAAAAZA/hCYWHb7GYwc/s640/table1.jpg" width="640" /></a></div><br />...the resulting investment opportunity sets in return/volatility, return/LPM and UPM/LPM space look like this...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-p8VgzwIOf1I/UrUpykOH3CI/AAAAAAAAAZQ/cl8MyZVVciY/s1600/mv.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="http://4.bp.blogspot.com/-p8VgzwIOf1I/UrUpykOH3CI/AAAAAAAAAZQ/cl8MyZVVciY/s400/mv.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-T9xUAJkT2DA/UrUpyUj2XyI/AAAAAAAAAZM/oILDkQOM7r4/s1600/mlpm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="http://2.bp.blogspot.com/-T9xUAJkT2DA/UrUpyUj2XyI/AAAAAAAAAZM/oILDkQOM7r4/s400/mlpm1.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-gs43pkxD3TY/UrUpyZBe0kI/AAAAAAAAAZU/CF4gK7xUmos/s1600/mupm1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="http://1.bp.blogspot.com/-gs43pkxD3TY/UrUpyZBe0kI/AAAAAAAAAZU/CF4gK7xUmos/s400/mupm1.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"> </div>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.<br /><br />Note how the efficient frontiers in Return/LPM and UPM/LPM space are not very smooth.<br /><br />If we increase both thresholds to 2%, the topography changes dramatically...<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-HB_hJUAcJhU/UrUq4wTwUeI/AAAAAAAAAZ4/eYvwcZjbwno/s1600/mlpm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="http://1.bp.blogspot.com/-HB_hJUAcJhU/UrUq4wTwUeI/AAAAAAAAAZ4/eYvwcZjbwno/s400/mlpm2.jpg" width="400" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-VpNTyvn79OI/UrUq4wPuSqI/AAAAAAAAAZ8/SRYzzOVRjhQ/s1600/mupm2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="http://2.bp.blogspot.com/-VpNTyvn79OI/UrUq4wPuSqI/AAAAAAAAAZ8/SRYzzOVRjhQ/s400/mupm2.jpg" width="400" /></a></div><br />It would be interesting to use the above visuals in a <a href="http://en.wikipedia.org/wiki/Rorschach_test" target="_blank">Rorschach Test</a>. Anyway...<br /><br />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...<br /><br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-DOtDZAOr4n8/UrVGvEXBGnI/AAAAAAAAAao/XqWiaAgk7uk/s1600/table2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="99" src="http://2.bp.blogspot.com/-DOtDZAOr4n8/UrVGvEXBGnI/AAAAAAAAAao/XqWiaAgk7uk/s640/table2.jpg" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"> </div><div class="separator" style="clear: both; text-align: left;"> </div><div class="separator" style="clear: both; text-align: left;">We calculated the "exact" endogenous portfolio lower and upper partial moments, by the way.</div>Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-30397012002824328092013-12-17T09:45:00.001+01:002013-12-17T09:46:35.375+01:00How to Quanitify Mean Reversion and Momentum?An unsorted risk of mean reversion / momentum indicators...<br /><ul><li>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).</li><li>Hurst Coefficient - <more details will follow at a later point></li><li>...</li></ul>Please <a href="mailto:email@andreassteiner.net" target="_blank">contact us</a> if you know additional measures which are not on this list.Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-54349427721297606472013-12-16T10:46:00.000+01:002014-06-06T09:08:39.595+02:00Tackling the Indeterminacy of Asset Allocation in Factor InvestingFactor investing is about managing exposures relative to economic value and risk drivers.<br /><br />Factor models are typically linear multiple regressions defining risk (to keep it simple, take volatility) and return of factor portfolios...<br />\$ r_{a} = B \cdot r_{f} \$<br />\$ r_{p} = w_{a} \cdot r'_{a} = w_{a} \cdot B \cdot r'_{f} = w_{f} \cdot r'_{f} \$<br />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.<br /><br />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.<br /><br />The factor portfolio selection problem is an entirely unrestricted mean-variance problem in factor space...<br />\$ \max_{w_{f}} w_{f} \cdot r'_{f} - \frac{1}{2 \cdot \lambda} \cdot w_{f} \cdot \Omega_{f} \cdot w'_{f} \$<br />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...<br />\$ w_{a} = B^{-1} \cdot w_{f} \$<br />Unfortunately, it is a linear problem with multiple solutions in most cases...<br /><strong><em></em></strong><br /><strong><em>1. Case "No restrictions on portfolio leverage, no restrictions on asset weights"</em></strong><br />A solution can be found with a <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse" target="_blank">pseudo-inverse</a> matrix operation, e.g. Moore-Penrose matrix inversion. With \$ B^+ \$ being the pseudo-inverse of B...<br />\$ w_{a} = B^+ \cdot w_{f} \$<br />Additional solutions can be found with...<br />\$ w_{a} = B^+ \cdot w_{f} + ( I - B^+ \cdot B ) \cdot x \$<br />x is a vector with arbitrary real numbers.<br /><strong><em></em></strong><br /><strong><em>2. Case "Restrictions on portfolio leverage, no restrictions on asset weights</em></strong>"<br />Portfolio leverage is a linear equality constraint. Example: fully invested portfolio.<br />\$ w_{a} = B^+ \cdot w_{f} \\ \mathbf{1} \cdot w_{f} = 1 \$<br />Equality constraints can be handled by including "dummy variables" in \$ w_{a} \$ and $ B^+ \$, no problem.<br /><div style="text-align: left;"><em><strong></strong></em><br /><em><strong>3. Case "Restrictions on portfolio leverage, restrictions on asset weights"</strong></em><br />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 <a href="http://en.wikipedia.org/wiki/Bott%E2%80%93Duffin_inverse" target="_blank">constrained generalized inverse</a>, e.g. the Bott-Duffin inverse.<br /><br /><br /><div style="text-align: center;"><...></div><br /><strike>An illustrative Excel spreadsheet with sample calculations is available </strike><a href="mailto:email@andreassteiner.net" target="_blank"><strike>upon request</strike></a><strike>.</strike> This topic has potential. More about it at a later point in time.<br /><br />For real-world investment purposes, the big issue is the stability of the factor betas over time, market regimes and asset universes.<br /><br />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.<br /><br />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. <!--[endif]--><script type="text/javascript">MathJax.Hub.Queue(["Typeset",MathJax.Hub]);</script></div>Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-45863302927892719152013-12-14T14:30:00.000+01:002013-12-16T15:04:05.303+01:00Why Modelling Power-Law Tails?It is rather well known among investment quants that the <a href="http://en.wikipedia.org/wiki/Central_limit_theorem" target="_blank">Central Limit Theorem</a> 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 <a href="http://en.wikipedia.org/wiki/Stable_distribution" target="_blank">stable distribution</a> $ f(x;\alpha,0,c,0)$.<br /><br />Most return data generated by financial markets is clearly not normally distributed. But then, <em><strong>I have never come across an asset with infinite variance</strong></em>. Models are abstract realities, they are always wrong in the sense that they do not exactly reproduce observable reality. The abstraction is the result of the assumptions made. Assumptions are chosen to yield a model which is relevant for a particular purpose or clarify a particular point. Assuming normally distributed (continuous or discrete) asset returns is wrong, as wrong as assuming power-law tails. But I do not see any relevant purpose or unique reason to work with power-law tails. If you do, I would be interested in learning more. <script type="text/javascript">MathJax.Hub.Queue(["Typeset",MathJax.Hub]);</script><br /> Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0tag:blogger.com,1999:blog-6120000784709869935.post-33385349066570748172013-12-13T13:30:00.000+01:002013-12-16T15:51:21.698+01:00Diversification MeasuresBelow a (growing) list of diversification measures...<br /><ul><li>N - number of positions in a portfolio</li><li>$ 1 / \sum_{} w^2_i $ - number of "effective positions"</li><li>Herfindahl Index = $ \sum_{} w^2_i $ - concentration in exposures</li><li>Gini Index - another way to express concentration in exposures</li><li>Standard deviation of portfolio constituent weights - dispersion of exposures</li><li>Shannon entropy of portfolio constituent weights - since weights have the properties of a probability, Shannon entropy can be used as a dispersion measure</li><li>Shannon entropy of Diversification index - see A. Meucci</li><li>Diversification Ratio - portfolio volatility calculated with a correlation matrix of ones divided by actual portfolio volatility</li><li>Diversification Index - inverse of the diversification ratio, Tasche(2008).</li><li>Degree of Diversification - portfolio volatility of the global minimum variance portfolio divided by actual portfolio volatility</li><li>% of idiosyncratic risk - residual risk measured in the context of a single- or multiple-factor model</li><li>% of total variability explained by first principal component - dependence on a single-most important factor</li></ul>Please <a href="mailto:email@andreassteiner.net" target="_blank">contact us</a> if you know diversification measures which are not on the above list.<script type="text/javascript">MathJax.Hub.Queue(["Typeset",MathJax.Hub]);</script>Andreas Steinerhttp://www.blogger.com/profile/10016331791465556615noreply@blogger.com0