Antje Brigitte Mahayni

HOW GOOD ARE PORTFOLIO INSURANCE STRATEGIES

Portfolio insurance strategies are designed to achieve a minimum level of wealth while at the same time participating in upward moving markets. The most prominent examples of dynamic versions are option based strategies with synthetic put and constant proportion portfolio insurance strategies. It is well known that, in a Black/Scholes type model setup, these strategies can be achieved as optimal solution by forcing an exogenously given guarantee into the expected utility maximization problem of an investor with CRRA utility function. The CPPI approach is attained by the introduction of a subsistence level, the OBPI approach stems from an additional constraint on the terminal portfolio value. We bring these results together in order to explain when and why OBPI strategies are better than CPPI strategies and vice versa. We determine the utility losses which are caused by introducing a terminal guarantee into the unconstrained maximization approach. In addition, we focus on utility losses which are due to market frictions such as discrete–time trading, transaction costs and borrowing constraints.

Performance evaluation of optimized portfolio insurance strategies

We use S&P 500 index return data for the time period 1985–2013 to evaluate the performance of portfolio insurance strategies. We shed light on the question if the performance of a constant proportion portfolio insurance (CPPI) strategy can be improved by means of a time-varying multiplier which depends on the estimated future volatility. Neglecting any inter-temporal hedging demand, the theoretical foundation of the strategies is given by maximizing the expected utility of a HARA investor in a diffusion model setup. If the risk premium is assumed to be proportional to the variance, the optimal strategy is a CPPI strategy. Otherwise, the multiple is time-varying (PPI). It turns out that even time-varying multiple strategies based on a rolling window of historical volatility estimates give a significant improvement of CPPI strategies. The out-performance is robust w.r.t. alternative performance measures and is also true for proportional transaction costs and adequate trigger trading.

Return Guarantees with Delayed Payment

Abstract A unit-linked insurance contract can be formulated in terms of a guaranteed amount together with a fraction of a positive excess return of a benchmark portfolio. Normally, the excess return is determined annually and accumulated until the maturity of the contract. The accumulation factor that is granted with respect to the delayed payments can either be deterministic or equal to the (stochastic) bank account. It turns out that the common choice of a deterministic accumulation factor gives rise to problems concerning the pricing and the risk management of the insurance contract.

Primal--dual linear Monte Carlo algorithm for multiple stopping—an application to flexible caps

Abstract In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primal–dual linear Monte Carlo algorithm that allows for efficient simulation of the lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of the primal–dual Monte Carlo algorithm for standard Bermudan options proposed by Schoenmakers et al. [SIAM J. Finance Math., 2013, 4, 86–116] to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers. At each level, the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. The algorithm also provides approximate continuation functions that may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull and White model setup. The simple model choice allows for comparison of the computed price bounds with the exact price obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multi-dimensional problems and is tractable to implement.

Minimum Return Guarantees, Investment Caps, and Investment Flexibility

We study the merits of capped retirement products with guarantee for investors who have the flexibility to dynamically adjust their investment strategy. All contracts under consideration are fairly priced such that the net profit of the provider is zero. Without the rider, an expected utility maximizing CRRA investor does not want an investment cap. Here, she commits herself to a strategy a priori. With the flexibility rider, the optimization problem changes and the optimal strategy is a response to an exogenously set price. A fair pricing then anticipates the optimal response of the investor. We show that the maximum expected utility of the investor can, for anticipated fairly priced products, be obtained for a finite cap. Thus, a capped product design can give a Pareto improvement to the otherwise uncapped contract version.

Wertschöpfung durch Versicherungen: Die Bedeutung von Versicherungen für eine effiziente Risikoallokation

Der Beitrag von Versicherungen zur wirtschaftlichen Effizienz umfasst die Verbesserung der Risikoallokation, den Schutz des bestehenden Vermogens, die Kapitalakkumulation, die Mobilisierung von finanziellen Ressourcen, die Kontrolle des Unternehmensverhaltens und die Entlastung des Staates. Ein besonderer Fokus dieses Beitrags liegt auf dem zuerst genannten Punkt, d.h. der Verbesserung der Risikoallokation. Hierzu geben wir eine einfache Darstellung des Gesetzes der grosen Zahlen und dessen Bedeutung fur den Ausgleich im Kollektiv bzw. das Pooling von Risiken. Wahrend unter „idealen Bedingungen“ eine entsprechend des erwarteten Schadens kalkulierte (Risiko-)Pramie zum Erwerb des Versicherungsschutzes ausreichen wurde, beinhaltet die Risikopramie in Realitat Aufschlage, welche sich (zumindest teilweise) durch die so genannten versicherungstechnischen Risiken rechtfertigen lassen. Zusatzlich zu der Risikopramie muss der Versicherungsnehmer Verwaltungskosten etc. entrichten. Im Rahmen des Bernoulli-Prinzips erlautern wir, warum Versicherungsnehmer bereit sind, eine uber den erwarteten Schaden hinausgehende Pramie zu zahlen und trotzdem vom Versicherungsschutz profitieren. Die Zusammenhange werden anhand einer Schaden- und einer Rentenversicherung illustriert.

Time-Consistency of Optimal Investment under Smooth Ambiguity

We study portfolio choice in a Black-Scholes world under drift uncertainty. Preferences towards risk and ambiguity are modeled using the smooth ambiguity approach under a double power utility assumption and a normal distribution assumption on the unknown drift. Optimal investment in this setting is time-inconsistent: While utility is maximized by a precommitment strategy resembling the classical Merton solution, the investor’s future selves prefer to constantly increase the riskiness of the strategy. In contrast, the optimal dynamically consistent investment strategy accounts for variations in the perceived severity of drift uncertainty, thus increasing the riskiness of the strategy gradually over time. We provide a detailed comparative analysis of the mechanics and interplay of ambiguity, myopia and optimal decisions in this setting. We show that an investor who pre-commits will regret that decision from some time point onwards, wishing that she had followed the dynamically consistent strategy. This “point of regret” always lies near the middle of the investment horizon.

Natural Hedging with Fix and Floating Strike Guarantees

The paper analyzes minimum return rate guarantees (MRRGs) including fixed guarantee rates prevailing for the whole contract horizon as well as floating guarantee rates which are linked to the interest rate evolution. In a complete arbitrage free market where the asset and bond price dynamics are given by Gaussian processes, we obtain closed form pricing solutions for both guarantee schemes. Differences in the guarantee costs are then explained by the difference of the arbitrage free values of the fix and floating rate guarantees and the difference between cumulated volatilities resulting from forward and simple volatilities. We then consider the perspective of the asset liability management, i.e. we analyze the sensitivities of the asset and liability side against changes in the interest rate. We show that a combination of fix price and floating strike guarantees enables a natural hedge against changes in the interest rate.

Precautionary Saving and Insurance Under Generalized Mean-Variance Preferences

We analyze the optimal insurance demand in a dynamic setup with two periods. In addition to the possibility to insure, the investor is allowed to transfer wealth between the two periods, i.e. she can save. While it is difficult to interpret the optimal saving and insurance decisions without disentangling time and risk preferences, we do so in a generalized mean variance setup. In this dynamic setup we state a natural way to separate between time and risk preferences by means of a variance decomposition. We show that we are in- deed able to disentangle the preferences. While the variance within the period where a loss can occur determines the optimal insurance level, the aversion against the variance between the expected wealth at different times gives the optimal savings decision. The results are tractable and easy to interpret.

Minimum Return Rate Guarantees Under Default Risk - Optimal Design of Quantile Guarantees

The paper analyzes the design of participating life insurance contracts with minimum return rate guarantees. Without default risk, the insured receives the maximum of a guaranteed rate and a participation in the investment returns. With default risk, the payoff is modified by a default put implying a compound option. We represent the yearly returns of the liabilities by a portfolio of plain vanilla options. In a BS model, the optimal payoff constrained by a maximal shortfall probability can be stated in closed form. Due to the completeness of the market, it can be implemented for any equity to debt ratio.