Retirement Income Recipes in R

Frontmatter

Chapter 1. Setting Expectations and Deviations

Abstract

This brief chapter provides an overview and outline of the book, with a particular emphasis on the specific topics covered and the recipes described within each individual chapter.

Moshe Arye Milevsky

Chapter 2. Loading and Getting to Know R

Abstract

The objective of this chapter is to carefully explain how to download and install R together with R-studio, which is the front-end I’ll be using. This chapter will teach you how to run and compile basic calculations, create some basic functions, and plot some basic figures. If you are already familiar with and/or have worked with R-studio, then you can skip ahead to the next chapter.

Moshe Arye Milevsky

Chapter 3. Coding the (Simple) Financial Life-Cycle Model

Abstract

This chapter continues the introductory process of learning to work and compute within R-studio. The pedagogical objective is to build a set of (simple) R-functions that implement various aspects of the financial life-cycle model, which (in the opinion of this author) is the core foundation of retirement income planning. In particular, recipes are provided for the optimal consumption (or savings) rates, the optimal multiple of salary that should be accumulated at any age, as well as the optimal amount of financial capital (a.k.a. nest egg) required at retirement. Emphasis here is on the word optimal. In other words, I solve for them and they aren’t imposed exogenously (i.e. from outside). Being that we are just starting the journey, this chapter assumes no income taxes or pre-existing pensions, and absolutely no randomness in (1) investment returns, (2) salary and wages, or (3) mortality and longevity.

Moshe Arye Milevsky

Chapter 4. Data in R: The Family Balance Sheet

Abstract

The technical objective of this chapter is to learn how to import and analyze (large) sets of data within R-studio. This really is the primary purpose and core strength of R, statistical data manipulation. The chapter begins by explaining how to import a simulated dataset of numbers representing a hypothetical family balance sheet (FBS). The underlying variables are consistent with the financial life-cycle model presented in the prior chapter. Then, using some (simple) statistical tools in R, the chapter investigates and discusses the following questions. Does the typical family (individual) in this dataset have a healthy FBS? What fraction of the population as represented by the dataset could be considered to be financially secure? What fraction of the population might be at risk of not maintaining their standard of living during retirement?

Moshe Arye Milevsky

Chapter 5. Portfolio Longevity: Deterministic and Stochastic

Abstract

This chapter formally introduces the notion of portfolio longevity (PL) within the context of retirement income planning. It explains how to define, measure, and simulate PL in a simple deterministic as well as stochastic (random) environment. This chapter also explains how to generate (simple) Monte Carlo simulations of forward-looking portfolio returns and its connection to asset allocation. Finally, the PL metric is used to analyze the (infamous) 4% rule of retirement income planning, and concludes by discussing some of its shortfalls.

Moshe Arye Milevsky

Chapter 6. Modeling the Risk of Sequence-of-Returns

Abstract

This chapter is focused on a phenomenon known by professionals in the retirement income business, as the sequence-of-returns effect. Broadly speaking—and using terminology introduced in the previous chapter—this relates to the disproportionate sensitivity of portfolio longevity to realized investment returns in the early stages of retirement withdrawals. More specifically this chapter proposes some formal metrics that measure the extent and magnitude of the risk using statistical correlation and regression methodologies. The chapter concludes by analyzing some derivative-based strategies, using put and call options, that can be used to mitigate the risk of sequence-of-returns.

Moshe Arye Milevsky

Chapter 7. Modeling Human Longevity and Life Tables

Abstract

Up to this point in the book, I have assumed a great fiction, namely that human longevity is known and finite. The success or failure of a retirement income plan was monitored and measured until a finite, e.g. 30 year, horizon. This chapter is the first to focus on the uncertainty or randomness in human longevity versus portfolio longevity. It begins with a detailed description and analysis of (historical) cohort life tables from the Human Mortality Database. The cohort life tables are used to extract population survival and death rates, which are then used to reconstruct life tables. The chapter concludes with a high-level discussion of mortality projections and improvements for future birth cohorts. The main emphasis is on gaining familiarity with the basic atomic structure (q _x) of actuarial life-science.

Moshe Arye Milevsky

Chapter 8. Life and Death in Continuous Time: Gompertz 101

Abstract

Continuing from where the prior Chap. 7 left off, this chapter explains how to construct and work with (a.k.a. cook) the remaining lifetime random variable: T _x. The approach to lifetime randomness is based on the underlying mortality hazard rate λ _x, which is the continuous-time (and probabilistic) analog of the 1-year death rate q _x. This chapter models and constructs T _x variables for a variety of given mortality hazard rates λ _x. This then sets the stage for the main intellectual objective, which is to introduce (and justify) the Benjamin Gompertz law of mortality. That important and famous law is experienced via a number of simulation exercises and experiments in R. The Gompertz model is the computational backbone for many of the subsequent computations (and recipes) in the book.

Moshe Arye Milevsky

Chapter 9. The Lifetime Ruin Probability (LRP)

Abstract

This chapter returns to the realm of portfolio longevity and focuses on computational algorithms for success and failure rates associated with various retirement income strategies, but accounting for longevity risk. The chapter begins by defining the so-called lifetime ruin probability (LRP), which is the simplest retirement risk metric, widely used by practitioners. After reviewing the underlying probability concepts, the chapter provides a number of analytic expressions and simulation-based recipes for computing, interpreting, and understanding the limitations of the LRP.

Moshe Arye Milevsky

Chapter 10. Life Annuities: From Immediate to Deferred

Abstract

This chapter develops a methodology for valuing simple cash-flow streams that last a lifetime, which are part of most Defined Benefit (DB) pensions. The focus is on the longevity-contingent building blocks of: (1) immediate, (2) temporary, and (3) deferred income annuities. The chapter begins with a discussion of the value of a longevity-contingent claim and how it differs from the market price versus the manufacturing cost of the product. The algorithms and user-defined R functions are mostly based on the Gompertz law of mortality, although a number of alternative continuous and discrete mortality models are discussed as well. The chapter concludes with a mathematical derivation and implementation of a closed-form expression for the Gompertz Annuity Valuation Model.

Moshe Arye Milevsky

Chapter 11. Intelligent Drawdown Rates

Abstract

This chapter, which could have been called rational decumulation, introduces dynamic risk-adjusted approaches to spending during retirement. The intelligent drawdown philosophy is contrasted with static approaches, such as the 4% rule and its variants, the focus of prior chapters. The material begins with a light-hearted game that develops an intuition for how longevity uncertainty should affect retirement spending as well as a discussion of the benefits from risk pooling. Moving on to the technical content, after a brief crash course on utility theory, the chapter develops algorithms for (1) computing optimal withdrawal rates based on risk aversion preferences and (2) adjusting ongoing spending based on realized financial and longevity variables. One of the surprising aspects of an intelligent approach to retirement spending is that planning to deplete one’s liquid wealth at some advanced age is “rationale” if you have ample pension annuity income.

Moshe Arye Milevsky

Chapter 12. Pensionization: From Benefits to Utility

Abstract

This chapter discusses the economic rationale for defined benefit (DB) pension plans, such as government social security programs and corporate retirement plans, which are schemes that implicitly provide longevity insurance by pooling participants. The chapter begins by contrasting such collective plans with do-it-yourself programs and then goes on to discuss the underlying concept of pensionization in greater detail, including the implicit wealth depletion time (WDT). The optimal amount of pension annuity income is linked to the WDT and illustrated with a detailed case study. The chapter concludes by presenting a simple metric for measuring the utility-based benefits of annuitization.

Moshe Arye Milevsky

Chapter 13. Biological (and Other) Ages

Abstract

This chapter examines the implications of mortality heterogeneity, that is the dispersion of longevity prospects within the population. It begins by discussing the extended Gompertz–Makeham model, as well as the compensation law of mortality, linking moments of the remaining lifetime random variable. It then introduces non-chronological measures of age, such as biological age and (especially) longevity risk-adjusted age to illustrate its dispersion. This chapter illustrates how true age can differ around the world and even within countries, based on wealth and income. The main computational implication of this chapter is that the human longevity random variable T _x, depends on (much) more than just chronological age x. Such heterogeneity must be accounted for in any intelligent drawdown methodology or pensionization scheme.

Moshe Arye Milevsky

Chapter 14. Exotic Annuities for Longevity Risk

Abstract

This chapter motivates—and offers recipes for valuing—a unique type of life annuity that is contingent on the performance of a stock market index or portfolio. It is christened a ruin-contingent life annuity (RCLA), a contingent claim that is also the foundation of all variable annuities (VAs) with guaranteed withdrawal benefits. Like an advanced life delayed annuity (ALDA), the RCLA begins making payments at a later date, but it is only triggered when an underlying reference portfolio hits zero, a.k.a. ruin. This chapter explains why an RCLA might be a desirable financial instrument for retirees and compares its (cheaper) theoretical value to (more expensive) life annuities.

Moshe Arye Milevsky

Chapter 15. Very Last Thoughts

Abstract

This chapter very briefly discusses some of the topics that were not covered in this book, but that nevertheless are important for retirement income planning. References to relevant research work are noted where appropriate and interested readers are directed to those sources for further information.

Moshe Arye Milevsky

Backmatter