Financial Stability and Regulatory Compliance
Robust Hedging in Incomplete Markets*
Sally Shen, Research Associate, Global Risk Institute
Antoon Pelsser, Professor, Department of Finance, Maastricht University
Peter Schotman, Professor, Department of Finance, Maastricht University
After the 2008 global financial crisis, the performance of U.S. pension funds has remained depressed. The poor solvency situation has been driven by a declining discount rate and also a fall in equity prices. Since 2012 funding ratios (asset values divided by projected benefit obligations) of the top 100 largest U.S. corporate defined-benefit pension plans have not rebounded. More importantly, projected future funding ratios show a wide range of uncertainty for the next two years. This raises the question of how to price and hedge downside risks when confronted with fragile beliefs about the likelihood of different funding ratio scenarios.
The pricing and hedging of pension or insurance liabilities faces two obstacles. First, the market is incomplete. Liability risks are typically not – or not actively - traded in financial markets. The second problem concerns model parameter uncertainty in hedging liability risks. On the liability side of the balance sheet, longevity has seen unprecedented and largely unpredictable improvements over the past few decades. On the other side of the balance sheet, the expected asset return is notoriously difficult to estimate from historical data.
The paper aims to develop a hedging strategy for an agent (a pension fund) who faces uncertainty about the expected return on the assets as well as uncertainty about the expected growth in liabilities. We employ robust control theory to deal with the concerns around of model uncertainty. The agent who worries about model misspecification looks for a prudent policy that is resilient to fragile beliefs about the likelihood of the state variables. The robust optimal hedging strategy that we propose takes both downside risks as well as market incompleteness into account for an agent who fears parameter uncertainty. The robust agent is assumed to minimize the shortfall between assets and liabilities under a statistically plausible worst-case scenario by means of solving a min-max robust optimization problem.
When the funding ratio is low, robustness reduces the demand for risky assets. However, in attempts to cover the liabilities, a substantial risk exposure is still optimal. A longer investment horizon or a higher funding ratio weakens the investor’s fear of model misspecification. If the expected equity return is overestimated, the initial capital requirement for hedging can be decreased by following the robust strategy.
We consider a pension fund that needs to hedge uncertain long-term liabilities. We model the pension fund as a robust investor facing an incomplete market and fearing model uncertainty for the evolution of its liabilities. The robust agent is assumed to minimize the shortfall between the assets and liabilities under an endogenous worst case scenario by means of solving a min-max robust optimization problem. When the funding ratio is low, robustness reduces the demand for risky assets. However, cherishing the hope of covering the liabilities, a substantial risk exposure is still optimal. A longer investment horizon or a higher funding ratio weakens the investor's fear of model misspecication. If the expected equity return is overestimated, the initial capital requirement for hedging can be decreased by following the robust strategy.
[*] This version: January 18, 2018.
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