Life-cycle choices and risk

The optimal savings and investment decisions of households along the life cycle were a central issue in Chapter 5. There, savings decisions were made under various forms of risks.However, we restricted our analysis to three period models owing to the limitations of the numerical all-in-one solution we used. In this chapter we want to take a different approach. Applying the dynamic programming techniques learned so far allows us to separate decision-making at different stages of the life cycle into small sub-problems and therefore increase the number of periods we want to look at enormously. This enables us to take amuchmore detailed look at how life-cycle labour supply, savings, and portfolio choice decisions are made in the presence of earnings, investment, and longevity risk. Unlike in Chapter 9, the models we study here are partial equilibrium models. Hence, all prices as well as government policies are exogenous and do not react to changes in household behaviour. This chapter is split into two parts. The first part focuses on labour supply and savings decisions in the presence of labour-productivity and longevity risk. Insurance markets against these risks are missing, such that households will try to self-insure using the only savings vehicle available, a risk-free asset. This model is a quite standard workhorse model in macroeconomics and a straightforward general equilibrium extension exists, the overlapping generations model, which we study in Chapter 11. In the second part of the chapter, we slightly change our viewpoint and look upon the problem of life-cycle decision-making from a financial economics perspective. We therefore exclude laboursupply decisions, but focus on the optimal portfolio choice of households along the life cycle, when various forms of investment vehicles like bonds, stocks, annuities, and retirement accounts are available. This section is devoted to analysing consumption and savings behaviour when households face uncertainty about future earnings and the length of their life span. We study how households can use precautionary savings in a risk-free asset as a means to selfinsure against the risks they face. While in our baseline model we assume that agents always work full-time, we relax this assumption later on by considering a model with endogenous labour supply as well as a model with a labour-force participation decision of second earners within a family context.

Fortran 90: A simple programming language

Before diving into the art of solving economic problems on a computer, we want to give a short introduction into the syntax and semantics of Fortran 90. As describing all features of the Fortran language would probably fill some hundred pages, we concentrate on the basic features that will be needed to follow the rest of this textbook. Nevertheless, there are various Fortran tutorials on the Internet that can be used as complementary literature. Fortran is pretty old; it is actually considered the first known higher programming language. Going back to a proposal made by John W. Backus, an IBM programmer, in 1953, the term Fortran is derived from The IBM Formula Translation System. Before the release of the first Fortran compiler in April 1957, people used to use assembly languages. The introduction of a higher programming language compiler tremendously reduced the number of code lines needed to write a program. Therefore, the first release of the Fortran programming language grew pretty fast in popularity. From 1957 on, several versions followed the initial Fortran version, namely FORTRAN II and FORTRAN III in 1958, and FORTRAN IV in 1961. In 1966, the American Standards Association (now known as the ANSI) approved a standardized American Standard Fortran. The programming language defined on this standard was called FORTRAN 66. Approving an updated standard in 1977, the ANSI paved the way for a new version of Fortran known as FORTRAN 77. This version became popular in computational economics during the late 80s and early 90s. More than 13 years later, the Fortran 90 standard was released by both the International Organization for Standardization (ISO) and ANSI consecutively. With Fortran 90, the fixed format standard was exchanged by a free format standard and, in addition, many new features like modules, recursive procedures, derived data types, and dynamic memory allocation made the language much more flexible. From Fortran 90 on, there has only been one major revision, in 2003, which introduced object oriented programming features into the Fortran language. However, as object-oriented programming will not be needed and Fortran 90 is by far the more popular language, we will focus on the 1990 version in this book.

Dynamic macro I: Infinite horizon models

In this chapter we apply the principles of dynamic programming to some standard macroeconomic models. For now we stay in the world of infinite horizon models, which are characterized by the fact that they are populated by one or several households with an infinite planning horizon, similar to the previous chapter. There are several justifications for such an assumption. Beneath simplicity, altruism is probably the most famous argument in favour of infinite horizon models. Assume that in a period t there is one generation that dies with certainty after this period.The utility of this generation from its own consumption is u(·). Yet, each generation is altruistic towards its descendants. Consequently, total utility of the generation is Ut = u(·) + βUt+1 where β ≤ 1 can be interpreted as the degree of altruism. All generations together then form a dynasty.

The overlapping generations model

In discussing the life-cycle model, we focused on the individual-choice problem without taking into account the interaction between households, the production sector of the economy, and the government. In this chapter we take a broader perspective and embed the life-cycle model into a general equilibrium framework. In this framework, prices adjust in order to balance supply and demand in goods and factor markets and the government has to operate under some balanced-budget rules.As in the previous chapter, individuals save in order to smooth consumption over the life cycle. However now, individual savings behaviour endogenously determines the capital stock. This is the central difference from the static general equilibrium model discussed in Chapter 3. Since in our equilibrium framework we have to distinguish households within a given period according to their age or birth year, the models we study are called overlapping generations (OLG) models. In this chapter we introduce the most basic version of the OLG model and discuss the computation of a transition path and the intergenerational welfare effects of policy reforms. In Chapter 7 we extend this baseline model version in various directions. This subsection sketches the economic environment used in this chapter and Chapter 7. We describe the lifetime of people who inhabit the economy as well as their consumption decisions. Then we move on to the production side and the government structure. Finally, the equilibrium conditions for goods and factor markets which close the model are derived. Demographics As in Chapter 5 we assume that households in the model live for three periods. For simplicity we do not account for income and lifespan uncertainty. However, now in each successive period t a new cohort is born, where the number of households Nt in this cohort grows at a rate np,t, i.e. Nt = (1 + np,t)Nt−1. From Figure 6.1 one can understand why this demographic structure is called ‘overlapping generations’. In each period t a cohort Nt is born, but this ‘new’ cohort overlaps with the two cohorts Nt−1 and Nt−2 born in the previous two periods.

Topics in finance and risk management

This chapter introduces basic concepts of modern finance theory and demonstrates how to apply them in complex real-world problems. Financial deals and investment decisions are typically determined under uncertainty.Therefore, although this chapter is self-contained, we have to expect some theoretical background in individual decisionmaking and optimal investment under uncertainty. We organize our discussion into four central sections. The starting point is a portfolio choice problem, where an investor has to choose between different assets with specific risk and return characteristics. We then move on to some option pricing applications. We first derive analytical formulas and then evaluate numerical procedures for pricing European and American options as well as more exotic option products. The third section elaborates on credit risk measurement and management using a corporate bond portfolio as example. In the last section we discuss mortality risk and the optimal portfolio structure of a life insurance company. This section provides different numerical approaches to find an optimal portfolio structure with many risky assets. It begins with simple measures of risk and return of a single asset and then develops decision rules to choose optimal portfolios that maximize expected utility of wealth in worlds without and with riskless borrowing and lending opportunities. The purpose of this section is to optimize a portfolio of equity shares and a risk-free investment opportunity. The investor faces the most basic two-period investment choice problem: He buys assets in the first period and these assets pay off in the next period. The problem of the investor is to choose from i = 1, . . . ,N risky assets which may be shares, bonds, real estate, etc. The gross return of each asset i is denoted by rit = qit/qit−1 − 1, where qit−1 is the first-period market price and qit − qit−1 the second-period payoff.

Extending the OLG model

In Chapter 6 we used our basic OLG model to discuss the welfare and efficiency effects of various policy reforms. Of course, we have to be cautious in drawing robust conclusions fromsuch a policy analysis. In the basic model households only decide on their intertemporal consumption allocation. Hence, public policy solely distorts the savings decision and, consequently, most of the policy reforms hardly impact on economic efficiency but only redistribute across cohorts. Our analysis could be much more instructive when decisions of economic agents are multidimensional, so that various distortions induced by public policy interact. In this chapter we therefore introduce an extended individual decision process. Households not only decide on their savings, but also on their time use. Given a specific time endowment (say a day or a year), agents can either work in the market (and earn income), go to school (and acquire human capital for future income generation), or consume leisure. Public policy may distort all of these decisions. A good policy thus has to create a balance between intertemporal and intratemporal distortions. Finally, we study the implications of lifespan uncertainty and missing annuity markets, asking how public policy can improve the allocation of resources by providing insurance against longevity risk. In this section we allow households to decide how many hours to work in each period. The remaining time is used for leisure consumption which now features in household utility. Leisure demand in each period of the life cycle strongly depends on the respective value of human capital hj, which measures the value of the time endowment in terms of labour market productivity. Hence agents may work the same number of hours, but they may be differently productive, so that they earn a different wage per time unit. Whenever the wage a household earns in the labour market is very small, the household might want to consume more leisure than the actual time endowment. In order to guarantee that the time endowment is met, we calculate a so-called shadow wage μj,s. The shadow wage is added to the regular wage of the household and calculated such that the household’s optimal decision consists in consuming the household’s total endowment of time as leisure.

Introduction to dynamic programming

Dynamic optimization is widely used in many fields of economics, finance, and business management. Typically one searches for the optimal time path of one or several variables that maximizes the value of a specific objective function given certain constraints. While there exist some analytical solutions to deterministic dynamic optimization problems, things become much more complicated as soon as the environment in which we are searching for optimal decisions becomes uncertain. In such cases researchers typically rely on the technique of dynamic programming. This chapter introduces the principles of dynamic programming and provides a couple of solution algorithms that differ in accuracy, speed, and applicability. Chapters 8 to 11 show how to apply these dynamic programming techniques to various problems in macroeconomics and finance. To get things started we want to lay out the basic idea of dynamic programming and introduce the language that is typically used to describe it. The easiest way to do this is with a very simple example that we can solve both ‘by hand’ and with the dynamic programming technique. Let’s assume an agent owns a certain resource (say a cake or a mine) which has the size a0. In every period t = 0, 1, 2, . . . ,∞ the agent can decide how much to extract from this resource and consume, i.e. how much of the cake to eat or how many resources to extract from the mine.We denote his consumption in period t as ct. At each point in time the agent derives some utility from consumption which we express by the so-called instantaneous utility function u(ct). We furthermore assume that the agent’s utility is additively separable over time and that the agent is impatient, meaning that he derives more utility from consuming in period t than in any later period.We describe the extent of his impatience with the time discount factor 0 < β < 1.

Numerical solution methods

In this chapter we develop simple methods for solving numerical problems. We start with linear equation systems, continue with nonlinear equations and finally talk about optimization, interpolation, and integration methods. Each section starts with a motivating example from economics before we discuss some of the theory and intuition behind the numerical solution method. Finally, we present some Fortran code that applies the solution technique to the economic problem. This section mainly addresses the issue of solving linear equation systems. As a linear equation system is usually defined by a matrix equation, we first have to talk about how to work with matrices and vectors in Fortran. After that, we will present some linear equation system solving techniques.

Dynamic macro II: The stochastic OLG model

In Chapters 6 and 7 we discussed how to compute overlapping generations models and how to use them for policy analysis. The models developed there are completely deterministic in that they exclude both income and investment risk. While this ensured analytical tractability of the household problem and greatly facilitated computation, it certainly limit the scope of policy analysis. Consequently these chapters centred around clarifying the impact of public policy on the labour-supply and savings decisions of households and around evaluating its consequences for intergenerational redistribution. In practice, however, households face all kinds of risks that cannot be insured perfectly by the market. This opens up an additional channel through which the government could increase households’ welfare, namely by providing public insurance. In Chapter 10, we studied individual behaviour in an uncertain world, where individuals face idiosyncratic labour income and mortality risk as well as aggregate capital-market risk. The models therein are partial equilibrium models, meaning that prices are fixed and there is no need for the government to operate a balanced budget. In this chapter, we embed a household’s decision model with idiosyncratic labour-productivity risk and endogenous labour-supply decisions into a general equilibrium framework, which leads us to the stochastic OLG model. In this setup, factor prices respond to changes in individual behaviour and the government will be an explicit entity that collects revenue from taxes to finance its expenditure. Such a setup allows us to analyse both the distortionary and the risk-sharing effects of public policies. This chapter is organized in three main sections. The first two closely follow the setup of Chapter 6. We first explain the general structure of the stochastic OLG model with all its actors and conduct some steady-state policy analysis. We then discuss how to extend the model to include a transition path between steady states and to compute aggregate efficiency effects. In the last section, Section 11.3, we provide some policy applications where we analyse optimal tax schedules and the optimal size of the social-security system in more detail. In the following, we extend the life-cycle model with variable labour supply from Section 10.1.2 to a full general equilibrium setup with overlapping generations.

The life-cycle model and intertemporal choice

The discussion in the Chapters 3 and 4 centred around static optimization problems.The static general equilibrium model of Chapter 3 features an exogenous capital stock and Chapter 4 discusses investment decisions with risky assets, but in a static context. In this chapter we take a first step towards the analysis of dynamic problems. We introduce the life-cycle model and analyse the intertemporal choice of consumption and individual savings. We start with discussing the most basic version of this model and then introduce labour-income uncertainty to explain different motives for saving. In later sections, we extended the model by considering alternative savings vehicles and explain portfolio choice and annuity demand. Throughout this chapter we follow a partial equilibrium approach, so that factor prices for capital and labour are specified exogenously and not determined endogenously as in Chapter 3. This section assumes that households can only save in one asset. Since we abstract from bequest motives in this chapter, households do save because they need resources to consume in old age or because they want to provide a buffer stock in case of uncertain future outcomes.The first motive is the so-called old-age savings motive while the second is the precautionary savings motive. In order to derive savings decisions it is assumed in the following that a household lives for three periods. In the first two periods the agent works and receives labour income w while in the last period the agent lives from his accumulated previous savings. In order to derive the optimal asset structure a2 and a3 (i.e. the optimal savings), the agent maximizes the utility function . . . U(c1, c2, c3) = u(c1) + βu(c2) + β2u(c3) . . . where β denotes a time discount factor and u(c) = c1−1/γ /1−1/γ describes the preference function with γ ≥ 0 measuring the intertemporal elasticity of substitution.