CPPI Strategies in Discrete Time

Abstract

In general, the purpose of portfolio insurance strategies is to limit the downside risk of risky portfolios. The constant proportion portfolio insurance (CPPI) is a prominent example of a portfolio insurance strategy. Based on a dynamic trading rule, the CPPI provides payoffs greater than some minimum wealth level at some specified time horizon. The great advantage of the CPPI is its particularly simple trading rule, which basically only requires the knowledge of the current portfolio value and thus makes the CPPI applicable to any kind of risky portfolio. Under the assumption of a complete financial market where trading takes place in continuous time, it is well known that the payoffs provided by the CPPI are greater than a pre-specified minimum wealth level with certainty. In this thesis we are concerned with various sources of market incompleteness. One source of market incompleteness are trading restrictions. Restricting the possibility of making changes to the portfolio to a fixed set of trading dates allows for payoffs below the minimum wealth level. The associated risk is called gap risk. The assumption of a fixed set of trading dates is well suited for the derivation of various risk-measures related to gap risk. Analyzing the gap risk is important with respect to the effectiveness of the CPPI if trading in continuous time is not possible. One natural reason for the assumption of trading restrictions are transaction costs. However, in the presence of transaction costs the frequency of monitoring the portfolio is generally larger than the willingness to rebalance the portfolio. With respect to transaction costs it is reasonable only to rebalance the portfolio upon relevant changes in the portfolio value or the underlying assets. This rationale leads to the notion of triggered trading dates. It turns out that triggered trading dates are also better suited with respect to analyzing modifications of the CPPI. The basic CPPI exhibits at least three structural problems. First, it requires the assumption of unlimited borrowing which can be explicitly modelled with the introduction of a borrowing constraint. Second, in the case of a good performance of the portfolio, it is well possible that the minimum wealth level becomes insignificant in comparison to the portfolio value. This can be modelled by increasing the minimum wealth level upon good performances of the portfolio. Third, the exposure to the underlying risky assets can become arbitrarily small such that portfolio may basically only consist of riskless assets. Explicitly defining a minimum on the exposure to the risky assets provides another modification. All modifications can be analyzed in a setup with triggered trading dates. While the use of triggered trading dates allows for the modelling of transaction costs also for the modifications of the CPPI, choosing small triggers allows for approximations of the continuous-time case for which analytic expressions for the modifications are not known in the literature so far either.