Benchmarking Through Ex-Ante Tracking Error

  print  

Market risk, the impact of systematic market volatility, has dramatically increased over the past few years and has forced market participants to rethink risk management. The complexity of asset classes and the exposures to derivatives and currencies are forcing market participants to look for non-traditional methodologies for risk assessment. Value-at-Risk (VaR) has shown itself to be an effective tool for managers to quantify market risk on a given portfolio or security. When contextualized through a benchmark framework, VaR is even more tangible and meaningful. Just as investment performance is naturally contrasted to a benchmark, so should the associated risk of a portfolio. Relative VaR refers to the VaR of a portfolio compared to the VaR of the benchmark.¹ The appeal of Relative VaR is its simplicity in expressing a portfolio's risk of potentially incurring greater losses than those of its benchmark. Typically, the risk is stated in dollar terms. However, it can also be stated as a percentage of total assets, in which case it is referred to as ex-ante tracking error.

Ex-Post Tracking Error

In the past, management employed separate risk models for the analysis of dissimilar asset classes. Fixed income securities and sectors have been closely tied to duration measures, while equity sectors have been viewed in beta and alpha terms. The Greeks - deltas and gammas - have been extensively used on derivative instruments.

In assessing the risk of the entire portfolio, one typically relied on the mean-variance approach. Standard deviation, tracking error, information ratio, and Sharpe ratio are categorized as ex-post, or after the fact, risk measures. These traditional methods look back in the past through the statistical variance of an asset's return relative to its mean. Active and passive strategies have made use of ex-post tracking error, the standard deviation of excess returns, to determine the portfolio risk relative to the benchmark. The risk of a portfolio underperforming its point of reference, the benchmark, increases as the tracking error rises. The mean-variance approach is a good measure of the historical performance of a manager. However, it does not necessarily indicate future risk, nor does it accommodate the asymmetric return distribution of derivative instruments.

Ex-Ante Tracking Error

Value-at-Risk has become the standard for risk measurement and analysis. VaR statistically measures the probability of potential losses of the current portfolio given a certain probability, or confidence level, over a certain time frame, or analysis horizon. Since VaR is based on the probability distribution of asset returns, and all sources of risk are factored into probability distribution, then theoretically VaR incorporates all market risk.

Relative VaR measures risk by calculating the potential loss of the deviations of the portfolio in respect to a benchmark through historical returns for the assets (equity, foreign exchange rates, and interest rates). VaR statistics, when expressed as a percentage of total assets comparative to a benchmark, yields the portfolio's ex-ante, or forward looking, tracking error. Instead of computing the volatility of the portfolio's absolute returns (which is appropriate for active mandates), Relative VaR computes the volatility of the portfolio's return relative to those of the benchmark. This relative expression of risk can be used for both active as well as passive/indexing mandates. The methodology directly compares the risk of the managed portfolio with that of the short portfolio. Relative VaR is calculated by creating a portfolio consisting of two sub-portfolios: one being the actual managed portfolio (long portfolio) and the second being the various corresponding benchmarks (short positions). Relative VaR is the VaR of the deviations of the portfolio with respect to the benchmark.

A Practical Example of Relative VaR

Exhibit 1 demonstrates ex-ante tracking error in action for a hypothetical portfolio with an enhanced index mandate. Suppose the monthly VaR of a portfolio indicates that it may lose more then $14,500 (145 basis points) in 1 out of 20 months, and that the largest individual contribution to the potential loss would come from global equities.

A risk manager would also assess the Relative VaR, which indicates that the portfolio may under-perform the benchmark by $1,700 (17 basis points) in 1 out of 20 months. Relative VaR also shows that global equities would contribute only 4 basis points to the under-performance versus the benchmark.

The distinction revealed in global equities is essential - a stand-alone position in global equities may have a high VaR, but the asset class within the portfolio compared to its benchmark has a relatively low probability of under-performance.

Exhibit 1 (Click to enlarge)

Benefits of Relative VaR

Relative VaR addresses the two key deficiencies of ex-post tracking error: first, that mean-variance analysis takes into consideration both under- and over-performance, and second, that it looks just at the past and not toward the future.

The mean-variance framework examines both tails (over-performance and under-performance) equally with normality of return distribution. Consequently, as empirical evidence suggests, asset returns tend to have higher than normal kurtosis (Mandelbrot, 1963; Pagan, 1996). The implication is that the asset returns exhibit non-normal distribution. This is especially problematic since derivative instruments, which are becoming more and more prevalent in today's diversified portfolios, usually exhibit asymmetrical non-linear returns. Value-at-Risk, on the other hand, enables managers to analyze asset classes with non-symmetrical returns.² VaR and Relative VaR are solely focused on under-performance, or the downside risk. Managers who are appropriately concerned with maximum losses in comparison to a benchmark may find relative VaR invaluable.

Unlike ex-post tracking error, ex-ante tracking error forecasts the risk of underperforming a benchmark. The latter approach utilizes present holdings to estimate future return streams and their projected variability based upon statistical assumptions. In other words, ex-post tracking error assumes consistency in management style (no style drift), while ex-ante tracking error estimates possible future under-performance based on current holdings. Relative VaR assesses the exposures to assets, currencies, derivatives, interest rates, and the potential impact of market movements on the current portfolio. The statistical calculations incorporate the deviations of the portfolio with respect to the benchmark by using the historical returns or simulated returns, all the while maintaining the current composition of the actual long portfolio and the conversing short portfolio.

Stress Testing

Value-at-Risk is inherently limited by the assumptions that comprise it - for example, that future risk can be projected from historical distribution of returns and that normal market conditions will hold. Certainly, history has demonstrated that historical assumptions will not always apply and that low probability events will occur. To bridge this gap, the implementation of supplementary stress tests and scenario analysis is vital to assess risk in abnormal market environments and answer the question, "How much could my portfolio lose if the stress scenario occurred?" With Relative VaR and the test scenario, a risk manager would examine the potential under-performance of the portfolio to the benchmark if market conditions were to drastically change. This combined approach of Relative VaR and stress testing provides an all-encompassing picture of risk in any conjured market environment.

Exhibit 2 demonstrates various stress tests for a hypothetical portfolio with an enhanced index mandate. Under normal market conditions, the portfolio may lose more than $14,500 (145 basis points) and under-perform the benchmark by $1,700 (17 basis points) in 1 out of 20 months. In extreme market conditions, however, the portfolio may lose much more. In this example, the portfolio may experience the steepest losses under the conditions that prevailed on September 11, 2001; however, relative to its benchmark, the under-performance will be less than under the other scenarios tested.

Exhibit 2 (Click to enlarge)

Conclusion

The latest debt crisis has brought risk management into the forefront of consideration. Countless debates and discussions are revolving around risk assessment and control. To accurately assess exposure and mitigate downside risk, ex-ante statistical models accompanied by low probability market event stress testing is essential. Relative VaR expresses the associated risk of a portfolio sustaining losses greater than its predefined benchmark. Meaningful and regular stress testing incorporating current holdings and utilizing a variety of scenarios exposes excessive losses in abnormal market environments.

The beauty of Relative VaR lies in its simplicity of comprehension and interpretation - it makes dissemination of risk information to senior management and inexperienced parties straightforward: "Given a 95% confidence level, our portfolio has a 5% chance of losses exceeding those of the benchmark over the next year." Fundamental discussions and preemptive investment adjustments should then follow. The complementary approach of Relative VaR and scenario testing offers a robust depiction of risk far broader than any mean-variance analysis could provide. Fittingly, benchmarking through ex-ante tracking error offers risk managers a concise statistical model to compare their portfolio to the associated benchmark.