Fulvio Ortu

Pricing equity-linked life insurance with endogenous minimum guarantees

Abstract This paper analyses the problem of pricing insurance contracts in which the benefits are linked to the realization of a portfolio of equities and a minimum amount guaranteed is provided. Building on the models of Brennan and Schwartz (1976, 1979) and Delbaen (1990) for endowment policies, we extend them to the case in which the minimum guarantees are endogenous , i.e. they are functions of the premium(s) paid. In this framework we give sufficient conditions for both the single and the periodic premium to be well defined and present some significant examples of endogenous minimum guarantees satisfying these conditions. We also consider the problem of pricing insurance contracts different from the endowment one, and conclude with some numerical results.

Single and Periodic Premiums for Guaranteed Equity-Linked Life Insurance under Interest-Rate Risk: The “Lognormal + Vasicek” Case

Interest-rate risk, while significantly affecting the pricing of almost all life-insurance products, has been up to now disregarded in the analysis of equity-linked policies for which a minimum-amount-guaranteed provision operates. The purpose of the present paper is to build on the work of Brennan and Schwartz(1976,1979a,b) and Delbaen(1990) to show how uncertainty in interest rates influences both single and periodic premiums for equity-linked life insurance. To this end, we consider a model in which the unit price of the fund to which benefits are referred follows a lognormal process, while the spot rate of interest is described as in Vasicek(1977), and we employ the martingale approach to contingent-claims pricing introduced by Harrison and Kreps(1979) to obtain pricing formulae for guaranteed equity-linked policies that account for interest-rate risk. The paper includes a detailed comparative static analysis of our extended formulae, as well as some numerical examples.

Fixed income linked life insurance policies with minimum guarantees: Pricing models and numerical results

Abstract We propose a pricing model for life insurance policies in which the benefits are linked to the performance of a portfolio of interest rate sensitive assets (reference fund), and a minimum guarantee provision is present. The model is cast in the celebrated term structure framework developed by Cox, Ingersoll and Ross (1985). As for the behaviour of the investment component, we analyse two polar cases. In the first one the payments due on the reference fund when the contract is still “alive” are not reinvested, while in the second case we propose a reinvestment policy. We show how to obtain a closed form solution for the single premium in the no-reinvestment case, and how to implement a simulation approach to calculate numerically the single premium in the reinvestment case. We illustrate our analysis with numerical results that help in understanding the comparative static properties of the models proposed.

A spectral estimation of tempered stable stochastic volatility models and option pricing

A characteristic function-based method is proposed to estimate the time-changed Levy models, which take into account both stochastic volatility and infinite-activity jumps. The method facilitates computation and overcomes problems related to the discretization error and to the non-tractable probability density. Estimation results and option pricing performance indicate that the infinite-activity model performs better than the finite-activity one. By introducing a jump component in the volatility process, a double-jump model is also investigated.

Numeraires, Equivalent Martingale Measures and Completeness in Finite Dimensional Securities Markets

Abstract A numeraire is a portfolio that, when securities, prices and dividends are expressed in its units, admits an equivalent martingale measure transforming any gain process into a martingale. We show that the set of equivalent martingale measures of a numeraire is one-to-one with a subset of Arrow-Debreu state prices, which becomes the whole set if and only if the numeraire is self-financing. Hence our result extends those (e.g. Harrison and Kreps ( Journal of Economic Theory , 1979, 20, 381–408) Dothan ( Prices in Financial Markets , Oxford Univ. Press, New York, 1990)) stated for specific self-financing numeraires. We also identify markets admitting self-financing numeraires, and characterize completeness, in terms of equivalent martingale measures, without requiring that specific securities be traded.

Implications of Return Predictability across Horizons for Asset Pricing Models

Two broad classes of consumption dynamics - long-run risks and rare disasters - have proven successful in explaining the equity premium puzzle when used in conjunction with recursive preference. We show that bounds a-la Gallant, Hansen and Tauchen (1990) that restrict the volatility of the Stochastic Discount Factor by conditioning on a set of return predictors constitute a useful tool to discriminate between these alternative dynamics. In particular we document that models that rely on rare disasters meet comfortably the bounds independently of the forecasting horizon and the asset classes used to construct the bounds. However, the specific nature of disasters is a relevant characteristic at the 1-year horizon: disasters that unfold over multiple years are more successful in meeting the predictors-based bounds than one-period disasters. Instead, over a longer, 5-year horizon, the sole presence of disasters - even if one-period and permanent - is sufficient for the model to satisfy the bounds. Finally, the bounds point to multiple volatility components in consumption as a promising dimension for long-run risks models.

A Persistence-Based Wold-Type Decomposition for Stationary Time Series

If the aggregate response of the economy to an exogenous shock is a superposition of effects which develop over different time scales, then the statistical estimation of low frequency components is difficult. In fact highly persistent shocks have generally low instantaneous volatility and are hidden by those shocks with high instantaneous volatility and fast decay. We refer to this situation as heterogeneity of persistence levels phenomenon. This paper introduces a new spectral approach which is applicable to the analysis of time series in the presence of persistence heterogeneity. A new linear decomposition of a time series is introduced which generalizes the Wold decomposition for stationary time series and the Beveridge-Nelson permanent transitory decomposition for non stationary integrated ones. In order to prove the relevance of this new methodology for financial valuation, we apply it to clarify some open issues which arise in the empirical analysis of gdp and inflation forecasting. JEL Classification Codes: E32, E43, E44, G12.The Classical Wold Decomposition Theorem allows to split a weakly stationary time series x into a non-deterministic component, driven by uncorrelated innovations, and a deterministic term. This decomposition is a special case of the Abstract Wold Theorem, which deals with isometric operators defined on Hilbert spaces. As the lag operator is isometric on the Hilbert space H_t(x) spanned by the sequence {x_{t-k}_k}, the Classical Wold Decomposition for time series obtains. Moreover, the \emph{scaling operator} is isometric on the Hilbert space H_t(e), spanned by the classical Wold innovations of x, and it provides an Extended Wold Decomposition. Thus, the process x may be seen as a sum, across scales, of uncorrelated components that explain different layers of persistence, from temporary fluctuations to low-frequency shocks. Multiscale impulse response functions are, then, defined. Conversely, the sum of suitable uncorrelated components delivers a weakly stationary process. This decomposition fruitfully applies to ARMA and fractional ARIMA processes.

Envelope theorems in Banach lattices

We derive envelope theorems for optimization problems in which the value function takes values in a general Banach lattice, and not necessarily in the real line. We impose no restriction whatsoever on the choice set. Our result extend therefore the ones of Milgrom and Segal (2002). We apply our results to discuss the existence of a well-defined notion of marginal utility of wealth in optimal consumption-portfolio problems in which the utility from consumption is additive but possibly state-dependent and, most importantly, the information structure is not required to be Markovian. In this general setting, the value function is itself a random variable and, if integrable, takes values in a Banach lattice so that our general results can be applied.

Effective Vs. Efficient Securities in Arbitrage-Free Markets with Bid-Ask Spreads: A Linear Programming Characterization

We consider a securities market with bid-ask spreads at any period, including liquidation. Although the minimum-cost super-replication problem is non-linear, we introduce an auxiliary problem that allows us to characterize no-arbitrage via linear programming techniques. Since no-arbitrage per se does not bound the bid-ask spread of a newly traded security, we introduce the notion of effective new security. We show that effectiveness restricts the no-arbitrage bid and ask prices of a new security to the interval defined by the minimum-cost problem. We discuss in details the cases in which the boundaries of this interval can be reached without violating no-arbitrage. We also compare effectiveness to efficiency as discussed in Jouini and Kallal (2001). We show that effectiveness is not sufficient for efficency, but is equivalent to the weaker notion of zero inefficiency cost.

Generic Existence and Robust Nonexistence of Numeraires in Finite Dimensional Securities Markets

A numeraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numeraires in a finite dimensional setting. We then characterize the arbitrage-free prices and dividends for which the absence of numeraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numeraires, or their nonexistence under no arbitrage.