Index-tracking portfolio selection with background risk

We develop an index tracking portfolio model while incorporating “background risks” that households cannot insure against or avoid. Examples of such risks are variations in proprietary income, investments in real estate, and unexpected health care expenses. We characterize pertinent properties of the optimal portfolio of the proposed mean-enhanced index tracking model under background risk in comparison to the optimal index tracking portfolio without background risk, and study factors that contribute to their differences. We show that the presence of a background risk, and its capital allocation, can significantly impact the composition of the optimal portfolio. It develops that ignoring background risk can result in significant mis-assessment of tracking error and variance risk in optimal tracking portfolio. We provide numerical illustrations, as well as a case study using the U.S. markets, to highlight the importance of our findings and sensitivity on the wealth allocated to the background asset.