A bootstrap-based comparison of portfolio insurance strategies

Abstract

This study presents a systematic comparison of portfolio insurance strategies. We implement a bootstrap-based hypothesis test to assess statistical significance of the differences in a variety of downside-oriented risk and performance measures for pairs of portfolio insurance strategies. Our comparison of different strategies considers the following distinguishing characteristics: static versus dynamic protection; initial wealth versus cumulated wealth protection; model-based versus model-free protection; and strong floor compliance versus probabilistic floor compliance. Our results indicate that the classical portfolio insurance strategies synthetic put and constant proportion portfolio insurance (CPPI) provide superior downside protection compared to a simple stop-loss trading rule and also exhibit a higher risk-adjusted performance in many cases (dependent on the applied performance measure). Analyzing recently developed strategies, neither the TIPP strategy (as an ‘improved’ CPPI strategy) nor the dynamic VaR-strategy provides significant improvements over the more traditional portfolio insurance strategies.

Keywords:

• portfolio insurance strategies
• performance measurement
• bootstrap simulation

JEL Classification Codes:

• G11
• G13

Acknowledgements

We thank two anonymous referees and Chris Adcock (the editor) for valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Brennan and Schwartz (1989), Basak (1995), and Grossman and Zhou (1996) analyze the equilibrium implications of portfolio insurance in an expected utility framework.

2. Annaert, Van Osselaer, and Verstraete (2009) implement block-bootstrap simulations based on the empirical return distribution to incorporate heavy tails and volatility clustering. Bertrand and Prigent (2005) and Zagst and Kraus (2011) also evaluate portfolio insurance performance using stochastic dominance. For example, Zagst and Kraus (2011) derive parameter conditions implying a second- and third-order stochastic dominance of the CPPI strategy against the protective put strategy. However, they assume that the underlying risky asset follows a geometric Brownian motion.

3. Do (2002) compares the CPPI strategy and the synthetic put strategy using Australian market data. While neither strategy can be justified from a loss minimization or a gain participation point of view, he nevertheless reports that the CPPI strategy dominates in terms of floor protection and the costs of insurance.

4. Bookstaber and Langsam (2000) also provide a detailed analysis of the path dependency of portfolio insurance strategies.

5. The skewness of the returns of portfolio insurance strategies must be differentiated conceptually from odd-shaped distributions that can result from fund managers’ “portfolio manipulations”. Goetzmann et al. (2007) describe the strategies managers can use to manipulate Sharpe-like performance measures.

6. Adcock et al. (2014) show that, subject to regularity conditions, all performance measures, which are increasing functions of reward and decreasing functions of risk, are monotonically increasing functions of the Sharpe ratio. By contrast, performance measures which employ upper partial moments or conditional expected excess returns as measures of reward are not robust to differences in distributions. Accordingly, failure of monotonicity is most likely to occur for the Farinelli–Tibiletti ratio and the Rachev ratio.

7. The following description of portfolio insurance strategies is based on Dichtl and Drobetz (2011).

8. For example, with a multiplier of , the risky asset can lose 20% () without violating the floor. When a sudden loss of over 20% occurs, the value of the portfolio falls below the promised minimum value (gambler's ruin). In commercial applications, it is necessary to continuously control the optimal exposure in the risky asset, thus portfolio shifts would have to be executed immediately. In most instances, however, an appropriate trading filter is used.

9. In our implementation of the dynamic VaR-strategy, we use continuously compounded returns. To calculate the continuously compounded portfolio return as a weighted average of stock and cash market positions, we transform the continuously compounded stock and cash market returns into simple returns. The simple portfolio return is transformed back into a continuously compounded return for all following calculations. These elementary transformations are not incorporated in Equation (8) for the purpose of simplification.

10. It is necessary to consider cumulated portfolio performance when implementing the dynamic VaR-strategy. For example, if the target-VaR over a one-year investment horizon is 0% and the portfolio wealth has increased from 100 to 110 after one month, the VaR-budget is −9.09%, thus a loss of up to −9.09% does not violate the original VaR-budget. In contrast, if the portfolio wealth has decreased from 100 to 90, the new VaR-budget becomes +11.11%. The portfolio value must increase by at least 11.11% in order to maintain the original VaR-budget of 0%.

11. Benartzi and Thaler (1995) provide evidence that institutional and private investors tend to use a one-year investment horizon. Therefore, we focus on this time period in our simulation analysis.

12. As a robustness check, we also implement a circular block-bootstrap to avoid an underweighting at the beginning and the end of the original time series. Our results remain qualitatively the same.

13. See Fama and French (2002) and Welch and Goyal (2008) for conceptual problems in estimating the equity risk premium.

14. Reasons for using a simple GARCH(1,1) model instead of more complicated GARCH models are discussed in Jacobsen and Dannenburg (2003).

15. Alternatively, we run the simulations using the original volatility or the Leland (1985) modification in Equation (5a). The simulation results are not sensitive to these changes.

16. In the specific case of a zero target return, the LPM₀ measure indicates the loss probability, and the LPM₁ measure the expected loss.

17. A similar bootstrap-based statistical inference framework is used in Zeisberger, Langer, and Trede (2007).

18. Our bootstrap results are stable with this number of simulation runs.

19. If we implemented a rolling window approach instead of a bootstrap simulation, the resulting return series would suffer from an extremely high degree of serial dependency. In this situation, Efron's (1979) bootstrap methodology (which is based on the i.i.d. assumption) is not adequate. The stationary bootstrap technique proposed by Politis and Romano (1994) – a bootstrap technique which explicitly deals with weakly dependent time series – is also no longer adequate. However, a comparison of our bootstrap simulation results with rolling window simulation results (not shown) confirms that both methodologies provide qualitatively similar results.

20. As emphasized by Eling et al. (2011), the parameters and can be balanced to match the agent's attitude toward the consequences of overperforming or underperforming. The higher and , the higher is the agent's preference for (in the case of expected gains, parameter ) or dislike of (in the case of expected losses, parameter ) extreme events. In particular, if , the investor is risk seeking above the chosen threshold. If , the investor is risk averse below the threshold (see also Zakamouline 2011).

21. This is the simple (not continuously compounded) mean yearly risk-free rate calculated within our simulation period from January 1981 to December 2011. Therefore, this value differs from the (continuously compounded) risk-free rate shown in Table 2, which is calculated over the whole data history ranging from January 1980 to December 2011.

22. We are grateful to an anonymous referee for suggesting this additional analysis.

23. In the exponentially weighted moving average model, we set the decay factor to 0.94 (see J.P. Morgan/Reuters 1996).

24. To enable a comparison between the CPPI and the TIPP strategy, we also set the multiplier for the TIPP strategy to . The deviations in performance measures for both strategies with in Tables 5 and 8 are due to path dependency. Similar deviations across tables are observable for the stop-loss strategy and the dynamic VaR-strategy. However, all deviations are very small, indicating that our simulations exhibit sufficiently accurate convergence properties.

25. This effect is not found for the synthetic put strategy. Perfect volatility forecasts improve the protection quality of the synthetic put strategy (as measured by the LPM₀ and LPM₁ numbers), but returns cannot be enhanced (Tables 3 and 9).

26. In addition to bootstrap simulations, we also run Monte Carlo simulations based on the geometric Brownian motion as a robustness check. With normally distributed stock market returns, the shortfall probabilities (as measured by the LPM₀) are at least as good as or even lower than the expected values determined by the confidence level α. Monte Carlo simulation results confirm the bootstrap results for the comparison of the dynamic VaR-strategy with the synthetic put strategy as well as the CPPI strategy. Most importantly, our systematic comparison indicates that normally distributed returns are an important requirement for the protection quality of the dynamic VaR-strategy.