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Über dieses Buch

This book presents a consistent and complete framework for studying the risk management of a pension fund. It gives the reader the opportunity to understand, replicate and widen the analysis. To this aim, the book provides all the tools for computing the optimal asset allocation in a dynamic framework where the financial horizon is stochastic (longevity risk) and the investor's wealth is not self-financed. This tutorial enables the reader to replicate all the results presented. The R codes are provided alongside the presentation of the theoretical framework. The book explains and discusses the problem of hedging longevity risk even in an incomplete market, though strong theoretical results about an incomplete framework are still lacking and the problem is still being discussed in most recent literature.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Since the seminal papers of Markowitz (J Financ 7:77–91, 1952) and Merton (Rev Econ Stat 51:247–257, 1969; Merton, J Econ Theory 3:373–413, 1971), the literature about optimal asset allocation and risk management has been developing fast and now takes into account many possible frameworks and applications. Here, we deal with the application of the asset allocation problem to a more recent topic: the pension funds.
Francesco Menoncin

Chapter 2. Decision Theory Under Uncertainty

Abstract
The choice of any agent on the financial market is guided by his/her preferences. Under some technical hypotheses, these preferences can be fully described through a function, the so-called “utility function”. This function is a pure theoretical artifice but allows to use a powerful tool like the calculus for working with preferences and makes computations much easier.
Francesco Menoncin

Chapter 3. Stochastic Processes

Abstract
We have already argued that the analysis of a consistent framework for a pension fund must take into account stochastic variables such as the death time, the force of mortality, the asset prices/returns, the interest rate, the cash flows of contributions and pensions. These stochastic variables follow different paths over time and so they can be modelled through different stochastic processes.
Francesco Menoncin

Chapter 4. The Financial Market

Abstract
The description of a financial market through stochastic processes is fundamental for solving an dynamic optimisation problem. In particular, in this chapter we show how to check whether a market is arbitrage free and complete. We will always assume that the financial market is arbitrage free, since this is an obvious assumption that no model can disregard. Instead, the completeness hypothesis will be technically necessary for solving the optimal portfolio in semi-closed form and, in particular, for using the so-called “martingale method”.
Francesco Menoncin

Chapter 5. The Actuarial Framework

Abstract
In this chapter we show how to use the stochastic tools developed in the previous chapters for measuring the actuarial risk. The main measure that we deal with is the force of mortality. There exist a lot of deterministic models for this particular variable, and after summarising that models, in this chapter we show how to create a stochastic version of them. For this purpose, we use a particular version of the stochastic variables described in the previous chapter. In fact, the force of mortality must diverge over time, since the survival probability must converge towards zero while the agent grows older and older. Nevertheless, we use a mean reverting process since we assume that the force of mortality will stay close to one of its deterministic models. To this purpose, we show a mean reverting process that has a time varying (divergent) equilibrium value and we will show how to calibrate this model to the US actuarial data.
Francesco Menoncin

Chapter 6. Financial-Actuarial Assets

Abstract
In this chapter we show how to price a derivative on human life by using the tools already developed in the previous chapter. In particular, we show three cases: (1) the longevity bond, (2) the out of date Tontine, and (3) the death bond. These particular assets are just examples of a design that could be used to create many other actuarial derivatives whose underlying is the force of mortality. From a theoretical point of view, such assets are very useful for completing the financial market. In fact, it is very difficult to find pure financial assets that have a correlation with the force of mortality which is sufficiently high to make these assets suitable for hedging purposes.
Francesco Menoncin

Chapter 7. Pension Fund Management

Abstract
This chapter is the core of this work. By using all the tools that have been developed in the previous chapters, here we show how to compute the dynamic optimal asset allocation of a pension fund that can invest its wealth in four asset classes: (1) a risk-less asset, (2) a stock index, (3) a rolling zero coupon bond, which is a kind of derivative on the risk-less interest rate, and (4) a rolling longevity bond that can be thought as a derivative on the stochastic force of mortality.
Francesco Menoncin

Chapter 8. A Workable Framework

Abstract
The model we have solved in the previous chapter is very general and is able to accommodate many frameworks with a lot of different dynamics for both the state variables and the asset prices.
Francesco Menoncin

Chapter 9. A Pure Accumulation Fund

Abstract
In this final chapter we compute the optimal portfolio for another institutional investor which deals with neither the mortality nor the longevity risk. Instead, this fund just manages the contributions of a worker and provides him/her with an amount of money at the end of the management period. We assume that the same amount of money is due independently of the agent’s survival or, in other words, the heirs have the right to receive the whole amount due to the original subscriber.
Francesco Menoncin

Backmatter

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