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Managing and monitoring tail risk is not just about insuring against extreme losses. Boryana Racheva-Iotova describes the potential for expected tail loss measures to feed into tactical portfolio optimisation where variance is traditionally deployed

The crisis of 2008 drew attention to tail-risk management and subsequent crises - Greece, Japan and the Middle East to name a few - have solidified a foundation for implementing tail risk-hedging strategies to mitigate extreme loss. These defensive tactics are always associated with cost and savings, rather than profit and reward.

Tail-risk hedging, what-ifs and stress-testing are indeed invaluable tools in managing market extremes and no doubt help you mitigate disastrous scenarios. But the presence of fat-tails, tail dependence, volatility clustering and the so-called ‘correlations go to one' effect in financial returns should not be discounted during normal and not-so-extreme market conditions. Indeed, these phenomena offer tantalising opportunities for significant performance gains when optimising tail risk and return, regardless of the market regime.

Traditional portfolio optimisation is performed by minimising the variance for a desired rate of return - the so-called Markowitz mean-variance portfolio. But this Nobel Prize-winning concept is only optimal in cases of idealised normal markets. In reality, variance and thus standard deviation are insufficient to describe the risk of an asset, as they cannot differentiate between upside gains and downside losses. As well, they cannot adequately describe the risk of large low-probability events.

The expected tail loss (ETL) measure, also known as conditional VaR or expected shortfall, aims to rectify this. ETL measures the average possible loss when the loss is higher than a threshold that can be breached at a selected probability. Now that we are focusing solely on downside loss, separating it from upside return, it seems then quite natural to construct portfolios by minimising ETL - rather than variance.

However, it is not as simple as enhancing traditional optimisation and asset-allocation engines with proper downside risk measures. It is also about accurately modelling for fat-tails, skewness, volatility clustering and dependency asymmetry when measuring the risk. Moreover, as demonstrated in a previous IPE article, ‘The emergency room' (April 2010), these phenomena are dynamic. They change over time and across assets. Proper estimation of these parameters can be predictive of changing market environments and serve as early warning indicators. These are the model components that bring steady and consistent performance gains.

We construct an example portfolio of 10 Vanguard indices - seven equities, two fixed income and one real estate. The equities represent a cross section of US domestic markets, along with one international fund. Both a long-term bond fund and inflation-protected securities fund make up the fixed income component. Real assets are represented by the Vanguard REIT fund.

Accounting for fat-tails and skewness, the Stable Paretian model is employed. Tail dependence is captured by applying a subordinated model that assigns non-zero tail dependence between two indices based on the probability of observing extreme events occurring simultaneously in a given period. GARCH explicitly models the volatility clustering.

We optimise the portfolio in out-of-sample backtests with monthly rebalancing between the period 1 July 2002 and 31 January 2011, fitting the models using a rolling 500-day time window.

Rather than just comparing the Markowitz mean-variance model against a single mean-ETL model, five models are run, each with an added component of complexity, with the aim of demonstrating how each component drives improved performance:

• Model 1 - Markowitz mean-variance (normal distribution)
• Model 2 - Min-ETL + Normal distribution
• Model 3 - Min-ETL + Fat-tailed (Stable) distribution without tail dependency
• Model 4 - Min-ETL + Fat-tailed (Stable) distribution + Dependent Tails (Subordinated Model)
• Model 5 - Min-ETL + Fat-tailed (Stable) distribution + Dependent Tails + GARCH.


The results over 8.5 years in figure 1 clearly show the added value of each component with one exception - using ETL under a normal distribution provides zero added performance. The Stable min-ETL with dependent tail and GARCH outperforms the Markowitz model by 70% over the full backtest period. But the ‘white swans' are found in the details. Examining the allocations of the models in different periods before and around the 2008 crisis demonstrates the tail-diversification premium that can be extracted in seemingly calm times, as well as during periods of strong correlation and volatility.

Assessing the tail-fatness parameters of each of the funds over time  provides insight into the performance drivers. The tail-fatness parameter is two minus the tail index (alpha) of the stable distribution. A tail-fatness of zero indicates a normal distribution; the higher it is, the fatter the distribution tail.

First noticing the period from July 2004 through Jan 2007 , most of the equity markets are normal and the inflation protected fund is significantly more fat-tailed - even more so than the long-term bond fund. Taking advantage of this diversification opportunity, the fat-tailed model significantly underweights the inflation-protected fund compared with the Markowitz model while adding to the long-term bond and the equity indices. Added performance over Markowitz is consistently realised, even in this relatively calm period.

Another interesting period is the lead up to and during the 2008 crisis. The tail-fatness parameter of equities begins to soar 20 months before the 2008 crisis, warning of increased risk of extreme events, while the long-term bond and inflation-protected funds are stable for some time. Just prior to September 2008, during this time of seemingly extreme correlation, the tail-fatness parameter of the long-term bond and inflation-protected funds actually drops, and significantly so. They become less fat-tailed, offering a tail diversification opportunity. This leads to significant allocation differences for the 2007-08 period between the partial Stables models with and without tail dependence: the dependent tail model underweights domestic equities and allocates more into the long-term bond and inflation-protected funds.

Clearly tail risk management can serve mightier purposes beyond risk mitigation and hedging. Dramatic performance gains can be realized with optimized portfolios based on minimising ETL in both calm and turbulent markets. But this can only be accomplished with the presence of predictive fat-tailed models that capture dynamic correlations and volatility clustering.

Boryana Racheva-Iotova is president at FinAnalytica

 

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