MARGINAL CONTRIBUTION TO TRACKING ERROR

Since tracking error arises from various bets (some intentional and some unintentional) placed by the manager through overweights and underweights relative to the benchmark index, it would be useful to understand how sensitive the tracking error is to small changes in each of these bets.

Suppose, for example, a portfolio initially has an overweight of 3% in the semiconductor industry relative to its benchmark index, and that the tracking error is 6%. Suppose that the tracking error subsequently increases to 6.1% due to the semiconductor industry weight in the portfolio increasing by 1% (and hence the overweight goes to 4%). Then, it can be said that this industry adds 0.1% to tracking error for every 1% increase in its weight. That is, its marginal contribution to tracking error is 0.1%. This would hold only at the margin, that is, for a small change, and not for large changes.

Marginal contributions can be also calculated for individual stocks. If the risk analysis employs a multifactor risk model, then similar marginal contribution estimates can be obtained for the risk factors also.

Generally, marginal contributions would be positive for overweighted industries (or stocks) and negative for underweighted ones. The reason is as follows. If a portfolio already holds an excess weight in an industry, then increasing this weight would cause the portfolio to diverge further from the benchmark index. This increased divergence adds to tracking error, leading to a positive marginal contribution for this industry. Suppose, however, the portfolio has an underweight in an industry. Then, increasing the portfolio weight in this industry would make the portfolio converge towards the benchmark, thus reducing tracking error. This leads to a negative marginal contribution for this industry.

An analysis of the marginal contributions can be useful for a manager who seeks to alter the portfolio tracking error. Suppose a manager wishes to reduce the tracking error, then she should reduce portfolio overweights in industries (or stocks) with the highest positive marginal contributions. Alternatively, she can reduce the underweights (that is, increase the overall weights) in industries (or stocks) with the most negative marginal contributions. Such changes would be most effective in reducing the tracking error while minimizing the necessary turnover and the associated expenses.