St. Petersburg Spring School in Risk Management, Insurance, and Finance 2015

The Fourth School in Risk Management, Insurance, and Finance took place on March 30 — April 1, 2015. Contrary to previous Schools this one had three days of lectures centred on one common topic: Options and Guarantees in Life Insurance and Pension Products. Ermanno Pitacco and Anna Rita Bacinello (both from the University of Trieste, Italy) spoke about «Building up the Post-Retirement Income», «Risk Management in a Longevity Risk Scenario», and «Valuation of Life Insurance Contracts in a Contingent-Claims Framework». The School benefited from a generous support from JTI.


The Programme

Many modern insurance and pension products are designed as packages, whose items may be either included or not in the product actually purchased by the client. For example: the endowment insurance which can include various rider benefits and options, the Universal Life insurance, the Variable Annuities, the presence of possible LTC benefits in pension products. Appropriate modelling tools are then needed for valuating these products, and appropriate actions are required for managing the insurance portfolios.
The School first aims at presenting the main features of various insurance and pension products (Insurance Day). Special attention is then placed on the assessment and the management of risks inherent in a life annuity portfolio or pension plan (Risk Management Day). Finally, valuation of life insurance contracts in a contingent-claims framework is discussed, with a special focus on variable annuity products (Finance day).



March 30th, 2015: Insurance day (6 hours)


Building up the Post-Retirement Income


Professor Ermanno Pitacco, University of Trieste, Italy


Summary of the Course

The benefits provided by many modern life insurance and pension products imply a wide range of “guarantees” and hence risks borne by the insurance company (or the pension fund). Guarantees and inherent risks clearly emerge in recent scenarios, in particular because of volatility in the financial markets and trends in mortality / longevity. Appropriate modeling tools are then needed for pricing and reserving. Hence, a progressive shift from expected present values, and their prominent role in life insurance (and pension) calculations, to more modern and complex approaches, in particular the Enterprise Risk Management (ERM) based approach, is currently updating the actuarial toolkit.

However the implementation of complex mathematical methods often constitutes, on the one hand, an obstacle on the way towards sound pricing and reserving principles. On the other hand, facing the risks by charging very high premiums trivially reduces the insurer’s market share. Then, alternative solutions can be suggested by an appropriate product design which aims at sharing risks between the insurer and the policyholders. Interesting examples are provided by the design of life annuities as regards the longevity risk, and by profit participation mechanisms as regards the financial market risks.

In this course, we first focus on various arrangements which aim at building the post-retirement income, and involve either the accumulation phase, or the payout phase, or both. Various products are available on financial and insurance markets, each product with a specific guarantee structure (Conventional Life Annuities either immediate or deferred, Variable annuities, withdrawal plans, etc.). We then shift to a range of specific annuity products, stressing the relevant features: Advanced Life Delayed Annuity (ALDA), Ruin Contingent Life Annuity (RCLA), Variable Annuities (VA). Finally, we focus on some arrangements for the payout phase: the life annuity with a guarantee period, the value-protected life annuity (that is, providing “capital protection”), progressive annuitization schemes, life annuities combined with Long Term Care (LTC) benefits, longevity-linked life annuities. We conclude with a short introduction to the special-rate annuities (e.g. the impaired-life annuities).


1.1    Introduction
1.2    Packaging guarantees and options in insurance products
1.3    Product design: sharing the market risk in participating endowment insurance
1.4    Standard life annuities
1.5    Propensity to annuitize
1.6    Life annuity and death benefits
1.7    Last survivor annuities
1.8    Annuitization strategies
1.9    Guarantees in life annuities: basic structures
1.10    Non-standard life annuities (ALDA, RCLA, long-term care benefits, variable annuities, longevity-linked life annuities)
1.11    Underwritten (special-rate) life annuities


March 31st, 2015: Risk Management day (6 hours)


Risk Management in a Longevity Risk Scenario


Professor Ermanno Pitacco, University of Trieste, Italy


Summary of the Course

The calculation procedures proposed by the traditional actuarial mathematics rely, in a modern perspective, on deterministic models, because only expected values are usually addressed. The need for a sound assessment of the insurer's risk profile (as, in particular, emerges from new solvency standards) conversely suggests a comprehensive approach to the formal representation of the life insurance and annuity business, more general than that provided by traditional actuarial mathematics.

Enterprise Risk Management (ERM) suggests a comprehensive approach and provides a sound framework for dealing with life insurance and annuity business, and in particular with the related technical issues. In this framework, risk identification, risk assessment and impact assessment constitute basic steps towards the choice of appropriate tools for managing the risk themselves. Besides the traditional tools given by reinsurance and capital allocation, a risk management perspective suggests as effective tools, for example, a careful product design, an appropriate hedging involving opposite exposures to mortality/longevity risks, etc.

In this course we mainly focus on the application of ERM principles to life annuity and pension business. Starting from biometric issues (the longevity dynamics and the related uncertainty), we then describe a set of risk management actions for a life annuity portfolio or a pension fund. Several numerical examples illustrate the impact of longevity dynamics and the effect of capital allocation. We conclude stressing the need for monitoring mortality experience, and presenting a model for Bayesian inference on future mortality trends.


2.1    Introduction
2.2    Biometric assumptions for life annuities: projected life tables
2.2    The longevity dynamics: individual vs aggregate longevity risk
2.3    Modelling issues: from expected present values to ERM
2.4    Risk management actions
2.5    Risk assessment and impact assessment for pensions and life annuities
2.6    Monitoring: inference from longevity experience


April 1st, 2015: Finance day (6 hours)


Valuation of Life Insurance Contracts in a Contingent-Claims Framework


Professor Anna Rita Bacinello, University of Trieste, Italy


Summary of the Course

The object of life and pension insurance is to provide an adequate level of financial protection against uncertain events affecting the life of the insured. In particular, this protection can be addressed to the family of the insured and cover the case of her (early) death, or it can be addressed to the insured herself in order to get an adequate income after retirement. Of course both these objectives can also be pursued together. Especially in the second case, the protection takes the form of a long-term contract, where the financial component is predominant. In order to face the more and more aggressive competition with financial markets, the insurance industry offers specific and very flexible products such as, e.g., variable annuities, that allow the policyholders to benefit from high yields in financial markets without taking (all) the risk connected with them. Hence the problem of adequately assessing the value of the guarantees, that usually take the form of embedded derivatives, is very crucial, along with the necessity of singling out suitable hedging strategies for them.

In this course we start by recalling some generalities on life insurance contracts. Then we pass to the analysis of a unit-linked policy with minimum guarantees that was first studied by Boyle, Brennan and Schwartz in the second half of the seventies of the twentieth century. In particular, these authors were the first who recognised that a guaranteed benefit can be decomposed in terms of European options, once the mortality risk is diversified away, and merged together the traditional actuarial techniques with the then recent results on option pricing theory initiated by Black, Merton and Scholes in the first half of the same decade. After that we pass to the analysis of some extensions of the previous model that allow for different contracts, or benefit structures, or types of guarantees, as well as other embedded options such as, e.g., the surrender option and the guaranteed annuity option.

We focus, in particular, on the surrender option, a typical American-style contingent-claim that entitles the policyholder to early terminate a life insurance contract receiving a cash amount, called surrender value. As for any kind of American derivative, this option does not admit a closed-form valuation formula, so that a numerical approach is called for. We provide a very general valuation framework for a life insurance contract embedding such an option, based on the Least Squares Monte Carlo technique.

Then we pass to the analysis of variable annuities, that package different types of options and guarantees. Basically, these products are unit-linked investment policies providing a post-retirement income. The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type `x'), include minimum benefits both in case of death and survival. After providing a detailed description of the most common guarantees offered by variable annuities, we introduce the alternative assumptions underlying the policyholder behavior (static, mixed and dynamic approach) and propose a unifying framework for treatment and valuation of variable annuities under quite general model assumptions. Then we concentrate on the valuation of Guaranteed Minimum Withdrawal Benefits with the dynamic approach, and formulate the valuation model as a discrete stochastic control problem which is solved with a dynamic programming algorithm.


3.1    Introduction
3.2    Unit-linked products with minimum guarantees
3.3    The Boyle-Brennan-Schwartz model and related extensions
3.4    The surrender option
3.5    The structure of variable annuities: guarantees and options
3.6    Valuation of variable annuities under alternative policyholders’ behaviors

References for the Course

The three courses are mainly based on the following material:

A. R. Bacinello, F. Ortu (1993a): «Pricing equity-linked life insurance with endogenous minimum guarantees», Insurance: Mathematics and Economics, vol. 12, n. 3, p. 245–258

A. R. Bacinello (2005): «Endogenous model of surrender conditions in equity-linked life insurance», Insurance: Mathematics and Economics, vol. 37, n.2, p. 270–296

A. R. Bacinello (2008): «A full Monte Carlo approach to the valuation of the surrender option embedded in life insurance contracts», Mathematical and Statistical Methods for Insurance and Finance, C. Perna and M. Sibillo (eds.), Springer, p. 19–26

A. R. Bacinello, E. Biffis, P. Millossovich (2009): «Pricing life insurance contracts with early exercise features», Journal of Computational and Applied Mathematics, vol. 233, n.1, p. 27–35

A. R. Bacinello, E. Biffis, P. Millossovich (2010): «Regression-based algorithms for life insurance contracts with surrender guarantees», Quantitative Finance, vol. 10, n. 9, p. 1077–1090

A. R. Bacinello, P. Millossovich, A. Olivieri, E. Pitacco (2011): «Variable annuities: a unifying valuation approach», Insurance: Mathematics & Economics, vol. 49, p. 285–297

A. R. Bacinello, P. Millossovich, A. Olivieri, E. Pitacco (2012): «Variable Annuities as Life Insurance Packages: A Unifying Approach to the Valuation of Guarantees», Actuarial and Financial Mathematics Conference. Interplay between Finance and Insurance, M. Vanmaele et al (eds.), Koninklijke Vlaamse Academie van Belgie voor Wetenschappen en Kunsten, Brussel, p. 3–15

A. R. Bacinello, P. Millossovich, A. Montealegre (2014): «The Valuation of GMWB Variable Annuities under Alternative Fund Distributions and Policyholder Behaviours», Scandinavian Actuarial Journal, Available at:

A. Olivieri, E. Pitacco (2008): «Assessing the cost of capital for longevity risk», Insurance: Mathematics & Economics, vol. 42, n. 3, p. 1013–1021

A. Olivieri, E. Pitacco (2009): «Solvency requirements for life annuities allowing for mortality risks: internal models versus standard formulas», in: M. Cruz (Ed.), The Solvency II Handbook. Developing ERM frameworks in insurance and reinsurance companies, Risk Books: p. 371–397

A. Olivieri, E. Pitacco (2009): «Stochastic mortality: the impact on target capital», ASTIN Bulletin, vol. 39, n. 2, p. 541–563. doi: 10.2143 /AST.39.2.2044647

A. Olivieri, E. Pitacco (2011): «Life tables in actuarial models: from the deterministic setting to a Bayesian approach», AStA Advances in Statistical Analysis, vol. 96, p. 127–153

E. Pitacco (2007): «Mortality and longevity: a risk management perspective», Invited lecture at the 1st IAA Life Section Colloquium, Stockholm (available at:

E. Pitacco, M. Denuit, S. Haberman, A. Olivieri (2009): Modelling longevity dynamics for pensions and annuity business, Oxford University Press

E. Pitacco (2012): «From benefits to guarantees: looking at life insurance products in a new framework», CEPAR Working Paper 2012 / 26, University of New South Wales, Sydney. Available at:

E. Pitacco (2013): «Biometric risk transfers in life annuities and pension products: A survey», CEPAR Working Paper 2013 / 25, University of New South Wales, Sydney, Available at:


The Organising Committee


Partners and Supporters

The School greatly benefited from generous donations from

JTI logoInformation about the School was circulated via



Photo Report of the School 2015