A Stochastic Linear Programming Model for Asset Liability Management: The Case of an Indian Insurance Company

Asset - Liability management is one of the most critical tasks for any financial institution for determining its cushion against the risk and the net returns. The problem of asset liability management for an insurance company requires matching the cash inflows from premium collections and investment income with the cash outflows due to casualty and maturity claims. Thus, what is required is a prudent investment strategy such that the returns earned on the assets match the liability claims at all points of time in future. Conventionally, the asset allocation has been done using the Mean Variance approach due to Markowitz (1952, 1959). While such a strategy ensures that the asset value always match or are greater than the liability for the next year, it does not maximise the net worth of the firm nor does it take care of all the cash inflows and outflows over a long term period. A stochastic linear programming model (on the lines of Pirbhai, 2004) maximises the net worth of the firm and also takes care of the uncertainties. While there are instances of stochastic linear programming being applied for ALM in financial institutions in developed markets, no such practical application has been reported in this area in Indian context as yet. In this paper, we describe the development of a multi stage stochastic linear programming model for insurance companies. The multi-stage stochastic linear programming model was developed on the modelling language AMPL (Fourer, 2002).

Implementation of new regulatory rules in a multistage ALM model for Dutch pension funds

This paper discusses the implementation of new regulatory rules in a multistage recourse ALM model for Dutch pension funds. The new regulatory rules, which are called the ?Financieel Toetsingskader?, are effective as of January 2007 and have deep impact on the issues of valuation of liabilities, solvency, contribution rate, and indexation. Multistage recourse models have proved to be valuable for pension fund ALM. The ability to include the new regulatory rules would increase the practical value of these models.

Joined-Up Pensions Policy in the UK: An Asset-Libility Model for Simultaneously Determining the Asset Allocation and Contribution Rate

The trustees of funded defined benefit pension schemes must make two vital and inter-related decisions - setting the asset allocation and the contribution rate. While these decisions are usually taken separately, it is argued that they are intimately related and should be taken jointly. The objective of funded pension schemes is taken to be the minimization of both the mean and the variance of the contribution rate, where the asset allocation decision is designed to achieve this objective. This is done by splitting the problem into two main steps. First, the Markowitz mean-variance model is generalised to include three types of pension scheme liabilities (actives, deferreds and pensioners), and this model is used to generate the efficient set of asset allocations. Second, for each point on the risk-return efficient set of the asset-liability portfolio model, the mathematical model of Haberman (1992) is used to compute the corresponding mean and variance of the contribution rate and funding ratio. Since the Haberman model assumes that the discount rate for computing the present value of liabilities equals the investment return, it is generalised to avoid this restriction. This generalisation removes the trade-off between contribution rate risk and funding ratio risk for a fixed spread period. Pension schemes need to choose a spread period, and it is shown how this can be set to minimise the variance of the contribution rate. Finally, using the result that the funding ratio follows an inverted gamma distribution, shortfall risk and expected tail loss are computed for funding below the minimum funding requirement, and funding above the taxation limit. This model is then applied to one of the largest UK pension schemes - the Universities Superannuation Scheme

Employees Provident Fund (EPF) Malaysia: Generic models for asset and liability management under uncertainty

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We describe Employees Provident Funds (EPF) Malaysia. We explain about Defined Contribution and Defined Benefit Pension Funds and examine their similarities and differences. We also briefly discuss and compare EPF schemes in four Commonwealth countries. A family of Stochastic Programming Models is developed for the Employees Provident Fund Malaysia. This is a family of ex-ante decision models whose main aim is to manage, that is, balance assets and liabilities. The decision models comprise Expected Value Linear Programming, Two Stage Stochastic Programming with recourse, Chance Constrained Programming and Integrated Chance Constraints Programming. For the last three decision models we use scenario generators which capture the uncertainties of asset returns, salary contributions and lump sum liabilities payments. These scenario generation models for Assets and liabilities were developed and calibrated using historical data. The resulting decisions are evaluated with in-sample analysis using typical risk adjusted performance measures. Out- of- sample testing is also carried out with a larger set of generated scenarios. The benefits of two stage stochastic programming over deterministic approaches on asset allocation as well as the amount of borrowing needed for each pre-specified growth dividend are demonstrated. The contributions of this thesis are i) an insightful overview of EPF ii) construction of scenarios for assets returns and liabilities with different values of growth dividend, that combine the Markov population model with the salary growth model and retirement payments iii) construction and analysis of generic ex-ante decision models taking into consideration uncertain asset returns and uncertain liabilities iv) testing and performance evaluation of these decisions in an ex-post setting.This stuyd is funded by the Universiti Teknologi MARA Malaysia

Population ageing and fiscal sustainability in Finland: a stochastic analysis

This study analyses the fiscal sustainability of the Finnish public sector using stochastic projections to describe uncertain future demographic trends and asset yields. While current tax rates are unlikely to yield sufficient tax revenue to finance public expenditure with an ageing population, if developments are as expected, the problem will not be very large. However, there is a small, but not negligible, probability that taxes will need to be raised dramatically, perhaps by over 5 percentage points. Such outcomes, if realised, could destabilise the entire welfare state. The study also analyses three policy options aimed at improving sustainability. Longevity adjustment of pension benefits and introduction of an NDC pension system would reduce the expected problem and narrow the sustainability gap distribution. Under the third option, pension funds would invest more in equities and expect to get higher returns. This policy also limits the sustainability problem, but only under precondition that policymakers in the future can live with substantially larger variation in the value of the funds without adjusting tax rules or benefits.public finance; fiscal sustainability; uncertainty; stochastic simulations

Optimoidut strategiat varojen dynaamiseen kohdentamiseen

Modern portfolio theory is a widely used framework in the financial industry. It has a solid theoretical background, and has been successfully employed by the practitioners for decades. Traditional models based on Harry Markowitz's portfolio theory, and its further improved versions, have one significant shortcoming: they are single-period models by definition, and are not able to accommodate multi-period considerations. In this thesis, instead of modern portfolio theory and mean-variance optimisation, we use stochastic programming. To employ stochastic programming as a technique to find the optimal allocations, we need to develop scenarios, or scenario trees that describe the stochastic variables and their distributions. To generate the scenarios, we employ a methodology called moment matching, where the relevant properties of stochastic variables in our generated scenarios are fitted to counterparts estimated by means of time series analysis and econometric modelling. These stochastic factors are also called market invariants in this context. Market invariants are then translated into asset returns, which make it possible to find optimal asset allocations in each stage of the scenario tree. An illustrative asset allocation example is presented in this thesis to demonstrate how the dynamic allocation strategy performs compared to a fixed allocation decision. The results are rather intuitive, and as expected, the dynamic allocation strategy outperforms the fixed strategy in the scenarios generated. A comparison to traditional mean-variance framework is conducted, and it is seen that the resulting allocations for both dynamic and fixed strategy are close to being mean-variance efficient. Further research topics include changing the scenario generation methodology, and more sophisticated modelling of interest bearing instruments. An interesting direction for further development would be constructing the entire term structure of a yield curve, which would allow flexible valuation of assets and liabilities based on their present values.Moderni portfolioteoria on rahoitusalalla yleisesti käytetty. Sillä on vahva teoreettinen pohja, ja sen sovelluksia on käytetty onnistuneesti vuosikymmenien ajan. Harry Markowitzin kehittämän portfolioteorian, ja siitä kehitettyjen parannettujen versioiden yksi ilmeinen heikkous on kuitenkin se, että ne ovat rakenteeltaan yksiperiodisia malleja. Ne eivät näin ollen sovellu moniperiodiseen tarkasteluun. Tässä diplomityössä portfolioteorian perinteisten mallien sijaan sovelletaan stokastista ohjelmointia optimaalisten omaisuuslajiallokaatioiden löytämiseen. Jotta stokastista ohjelmointia voisi hyödyntää, on ensin kehitettävä skenaariot, jotka kuvaavat satunnaismuuttujat ja niiden jakaumat, joiden perusteella optimointi voidaan tehdä. Skenaarioiden luomiseksi käytämme momenttien sovittamiseksi kutsuttua menetelmää, jossa ongelman kannalta relevantit satunnaismuuttujien ominaisuudet sovitetaan skenaarioissa aikasarja-analyysin ja muiden ekonometristen menetelmien avulla estimoituihin vastineisiin. Stokastisia muuttujia kutsutaan tässä yhteydessä markkinainvarianteiksi, ja ne voidaan muuntaa omaisuuslajien tuotoiksi, joiden perusteella voidaan laskea optimaalinen omaisuuslajiallokaatio skenaariopuun jokaisessa haarassa. Työssä esitellään havainnollistava esimerkki dynaamisen allokaatiostrategian ja kiinteän allokaatiostrategian vertailua varten. Tulokset ovat intuitiivisia ja kuten odotettua, dynaaminen strategia pärjää kiinteää paremmin. Tehty vertailu perinteiseen Markowitzin mallin mukaiseen optimointiin osoitti, että sekä dynaaminen että kiinteä stokastisen optimoinnin strategia ovat lähellä Markowitzin mallin mukaista tehokasta rintamaa. Jatkotutkimuskohteita ovat skenaarioiden generointiin käytetyt menetelmät ja korkoperustaisten sijoituslajien tarkempi mallintaminen. Kiinnostava tutkimussuunta olisi koko korkokäyrän mallintaminen, joka mahdollistaisi mielivaltaisten tase-erien markkina-arvostamisen nykyarvoonsa

Essays on Multistage Stochastic Programming applied to Asset Liability Management

Uncertainty is a key element of reality. Thus, it becomes natural that the search for methods allows us to represent the unknown in mathematical terms. These problems originate a large class of probabilistic programs recognized as stochastic programming models. They are more realistic than deterministic ones, and their aim is to incorporate uncertainty into their definitions. This dissertation approaches the probabilistic problem class of multistage stochastic problems with chance constraints and joint-chance constraints. Initially, we propose a multistage stochastic asset liability management (ALM) model for a Brazilian pension fund industry. Our model is formalized in compliance with the Brazilian laws and policies. Next, given the relevance of the input parameters for these optimization models, we turn our attention to different sampling models, which compose the discretization process of these stochastic models. We check how these different sampling methodologies impact on the final solution and the portfolio allocation, outlining good options for ALM models. Finally, we propose a framework for the scenario-tree generation and optimization of multistage stochastic programming problems. Relying on the Knuth transform, we generate the scenario trees, taking advantage of the left-child, right-sibling representation, which makes the simulation more efficient in terms of time and the number of scenarios. We also formalize an ALM model reformulation based on implicit extensive form for the optimization model. This technique is designed by the definition of a filtration process with bundles, and coded with the support of an algebraic modeling language. The efficiency of this methodology is tested in a multistage stochastic ALM model with joint-chance constraints. Our framework makes it possible to reach the optimal solution for trees with a reasonable number of scenarios.A incerteza é um elemento fundamental da realidade. Então, torna-se natural a busca por métodos que nos permitam representar o desconhecido em termos matemáticos. Esses problemas originam uma grande classe de programas probabilísticos reconhecidos como modelos de programação estocástica. Eles são mais realísticos que os modelos determinísticos, e tem por objetivo incorporar a incerteza em suas definições. Essa tese aborda os problemas probabilísticos da classe de problemas de multi-estágio com incerteza e com restrições probabilísticas e com restrições probabilísticas conjuntas. Inicialmente, nós propomos um modelo de administração de ativos e passivos multi-estágio estocástico para a indústria de fundos de pensão brasileira. Nosso modelo é formalizado em conformidade com a leis e políticas brasileiras. A seguir, dada a relevância dos dados de entrada para esses modelos de otimização, tornamos nossa atenção às diferentes técnicas de amostragem. Elas compõem o processo de discretização desses modelos estocásticos Nós verificamos como as diferentes metodologias de amostragem impactam a solução final e a alocação do portfólio, destacando boas opções para modelos de administração de ativos e passivos. Finalmente, nós propomos um “framework” para a geração de árvores de cenário e otimização de modelos com incerteza multi-estágio. Baseados na tranformação de Knuth, nós geramos a árvore de cenários considerando a representação filho-esqueda, irmão-direita o que torna a simulação mais eficiente em termos de tempo e de número de cenários. Nós também formalizamos uma reformulação do modelo de administração de ativos e passivos baseada na abordagem extensiva implícita para o modelo de otimização. Essa técnica é projetada pela definição de um processo de filtragem com “bundles”; e codifciada com o auxílio de uma linguagem de modelagem algébrica. A eficiência dessa metodologia é testada em um modelo de administração de ativos e passivos com incerteza com restrições probabilísticas conjuntas. Nosso framework torna possível encontrar a solução ótima para árvores com um número razoável de cenários

Numerical study of discretizations of multistage stochastic programs

summary:This paper presents a numerical study of a deterministic discretization procedure for multistage stochastic programs where the underlying stochastic process has a continuous probability distribution. The discretization procedure is based on quasi-Monte Carlo techniques originally developed for numerical multivariate integration. The solutions of the discretized problems are evaluated by statistical bounds obtained from random sample average approximations and out-of-sample simulations. In the numerical tests, the optimal values of the discretizations as well as their first-stage solutions approach those of the original infinite-dimensional problem as the discretizations are made finer

Applications of biased randomised algorithms and simheuristics to asset and liability management

Asset and Liability Management (ALM) has captured the attention of academics and financial researchers over the last few decades. On the one hand, we need to try to maximise our wealth by taking advantage of the financial market and, on the other hand, we need to cover our payments (liabilities) over time. The purpose of ALM is to give investors a series of resources or techniques to select the appropriate assets on the financial market that respond to the aforementioned two key factors: cover our liabilities and maximise our wealth. This thesis presents a set of techniques that are capable of tackling realistic financial problems without the usual requirement of considerable computational resources. These techniques are based on heuristics and simulation. Specifically, a biased randomised metaheuristic model is developed that has a direct application in the way insurance companies usually operate. The algorithm makes it possible to efficiently select the smallest number of assets, mainly fixed income, on the balance sheet while guaranteeing the company's obligations. This development allows for the incorporating of the credit quality of the issuer of the assets used. Likewise, a portfolio optimisation model with liabilities is developed and solved with a genetic algorithm. The portfolio optimisation problem differs from the usual one in that it is multi-period, and incorporates liabilities over time. Additionally, the possibility of external financing is included when the entity does not have sufficient cash. These conditions give rise to a complex problem that is efficiently solved by an evolutionary algorithm. In both cases, the algorithms are improved with the incorporation of Monte Carlo simulation. This allows the solutions to be robust when considering realistic market situations. The results are very promising. This research shows that simheuristics is an ideal method for this type of problem.La gestión de activos y pasivos (asset and liability management, ALM) ha acaparado la atención de académicos e investigadores financieros en las últimas décadas. Por un lado, debemos tratar de maximizar nuestra riqueza aprovechando el mercado financiero, y por otro, debemos cubrir nuestros pagos (pasivos) a lo largo del tiempo. El objetivo del ALM es dotar al inversor de una serie de recursos o técnicas para seleccionar los activos del mercado financiero adecuados para obedecer a los dos factores clave mencionados: cumplir con nuestros pasivos y maximizar nuestra riqueza. Esta tesis presenta un conjunto de técnicas que son capaces de abordar problemas financieros realistas sin la necesidad habitual de considerables recursos computacionales. Estas técnicas se basan en la heurística y la simulación. En concreto, se desarrolla un modelo metaheurístico sesgado que tiene una aplicación directa en la operación habitual de inmunización de las compañías de seguros. El algoritmo permite seleccionar eficientemente el menor número de activos, principalmente de renta fija, en el balance y garantizar las obligaciones de la compañía. Este desarrollo permite incorporar la calidad crediticia del emisor de los activos utilizados. Asimismo, se desarrolla un modelo de optimización de la cartera con el pasivo y se resuelve con un algoritmo genético. El problema de optimización de la cartera difiere del habitual en que es multiperiodo e incorpora los pasivos a lo largo del tiempo. Además, se incluye la posibilidad de financiación externa cuando la entidad no tiene suficiente efectivo. Estas condiciones dan lugar a un problema complejo que se resuelve eficientemente mediante un algoritmo evolutivo. En ambos casos, los algoritmos se mejoran con la incorporación de la simulación de Montecarlo. Esto permite que las soluciones sean robustas cuando consideramos situaciones de mercado realistas. Los resultados son muy prometedores. Esta investigación demuestra que la simheurística es un método ideal para este tipo de problemas.La gestió d'actius i passius (asset and liability management, ALM) ha acaparat l'atenció d'acadèmics i investigadors financers les darreres dècades. D'una banda, hem de mirar de maximitzar la nostra riquesa aprofitant el mercat financer, i de l'altra, hem de cobrir els nostres pagaments (passius) al llarg del temps. L'objectiu de l'ALM és dotar l'inversor d'una sèrie de recursos o tècniques per seleccionar els actius del mercat financer adequats per obeir als dos factors clau esmentats: complir els passius i maximitzar la nostra riquesa. Aquesta tesi presenta un conjunt de tècniques que són capaces d'abordar problemes financers realistes sense la necessitat habitual de recursos computacionals considerables. Aquestes tècniques es basen en l'heurística i la simulació. En concret, es desenvolupa un model metaheurístic esbiaixat que té una aplicació directa a l'operació habitual d'immunització de les companyies d'assegurances. L'algorisme permet seleccionar eficientment el menor nombre d'actius, principalment de renda fixa, al balanç i garantir les obligacions de la companyia. Aquest desenvolupament permet incorporar la qualitat creditícia de l'emissor dels actius utilitzats. Així mateix, es desenvolupa un model d'optimització de la cartera amb el passiu i es resol amb un algorisme genètic. El problema d'optimització de la cartera difereix de l'habitual en el fet que és multiperíode i incorpora els passius al llarg del temps. A més, s'inclou la possibilitat de finançament extern quan l'entitat no té prou efectiu. Aquestes condicions donen lloc a un problema complex que es resol eficientment mitjançant un algorisme evolutiu. En tots dos casos, els algorismes es milloren amb la incorporació de la simulació de Montecarlo. Això permet que les solucions siguin robustes quan considerem situacions de mercat realistes. Els resultats són molt prometedors. Aquesta recerca demostra que la simheurística és un mètode ideal per a aquesta mena de problemes.Tecnologías de la información y de rede