Thank you in advance,

Andreas Steiner

A 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.

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 **LinkedIn** by clicking here. The posts from this blog will most likely get recycled in forthcoming LinkedIn Pulse articles.

Thank you in advance,

Andreas Steiner

Thank you in advance,

Andreas Steiner

Conditional 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.

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...

\$ \sigma_{p}^{2} = w \cdot \Sigma \cdot w' \$

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\$.

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...

\$ CVaR_{p}^{2} \approx w \cdot \Sigma^{*} \cdot w' \$

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.

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)...

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.

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.

Below, we show the composition of the efficient portfolios on the mean/volatility frontier...

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...

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...

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.

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...

\$ \sigma_{p}^{2} = w \cdot \Sigma \cdot w' \$

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\$.

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...

\$ CVaR_{p}^{2} \approx w \cdot \Sigma^{*} \cdot w' \$

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.

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)...

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.

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.

Below, we show the composition of the efficient portfolios on the mean/volatility frontier...

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...

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...

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.

In 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.

The core of the Black/Litterman model is the equation for posterior returns below...

\$ 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}] \$

...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.

Quite a bit of insight is already gained by reshuffling terms...

\$ 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} \$

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...

\$ r_{posterior} = (1-w) \cdot r_{prior}+ w \cdot r_{forecast} \$

\$ w = \frac{\sigma}{(1/\tau) \cdot \sigma_{forecast}+ \sigma} \$

*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.* 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).

The core of the Black/Litterman model is the equation for posterior returns below...

\$ 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}] \$

...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.

Quite a bit of insight is already gained by reshuffling terms...

\$ 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} \$

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...

\$ r_{posterior} = (1-w) \cdot r_{prior}+ w \cdot r_{forecast} \$

\$ w = \frac{\sigma}{(1/\tau) \cdot \sigma_{forecast}+ \sigma} \$

Diversification 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.

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)...

In order to visualize the risk dynamics, we calculate annualized rolling 24 month asset volatilities and 24 month rolling asset correlation...

We see that neither correlations nor volatilities are stable. Plotting correlations against average volatility reveals a relatively week relationship between the two...

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...

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.

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)...

In order to visualize the risk dynamics, we calculate annualized rolling 24 month asset volatilities and 24 month rolling asset correlation...

We see that neither correlations nor volatilities are stable. Plotting correlations against average volatility reveals a relatively week relationship between the two...

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...

Similar 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.

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)...

In portfolio analytics, fully additive contributions to risk can be derived from Euler's homogeneous function theorem for*linear* homogeneous risk measures. Portfolio volatility and tracking error are examples of risk measure which are linear homogeneous in constituent weights.

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)...

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...

The full details of the risk decomposition for the equal-weighted portfolio...

...in comparsion with the minimum 95% DaR portfolio...

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)...

In portfolio analytics, fully additive contributions to risk can be derived from Euler's homogeneous function theorem for

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)...

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...

The full details of the risk decomposition for the equal-weighted portfolio...

...in comparsion with the minimum 95% DaR portfolio...

Additive contributions to portfolio Drawdown-At-Risk open up the door for **drawdown risk budgeting**. For example, the **Drawdown Parity Portfolio** can be calculated as the portfolio with equal constituent contributions to portfolio drawdown risk...

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.

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.

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...

\$ MDD_{e} = (0.63519 + 0.5 \cdot ln(T) + ln(\frac{mu}{sigma})) \cdot \frac{sigma^2}{mu} \$

Expected maximum drawdown is function in investment horizon (+), volatility (+) and expected return (-).

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.

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.

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).

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.

The chart below plots the resampled portfolios of 16 portfolios on a particular mean/variance efficient frontier...

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).

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.

The chart below plots the resampled portfolios of 16 portfolios on a particular mean/variance efficient frontier...

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.

The chart below plots the same frontier with the same number of simulations, but a much larger number of generated observations...

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.
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?

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?

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