Multivariate Density Modeling for Retirement Finance

CHRISTOPHER J. ROOK

ABSTRACT

Prior to the financial crisis mortgage securitization models increased in sophistication as did products built to insure against losses. Layers of complexity formed upon a foundation that could not support it and as the foundation crumbled the housing market followed. That foundation was the Gaussian copula which failed to correctly model failure-time correlations of derivative securities in duress. In retirement, surveys suggest the greatest fear is running out of money and as retirement decumulation models become increasingly sophisticated, large financial firms and robo-advisors may guarantee their success. Similar to an investment bank failure the event of retirement ruin is driven by outliers and correlations in times of stress. It would be desirable to have a foundation able to support the increased complexity before it forms however the industry currently relies upon similar Gaussian (or lognormal) dependence structures. We propose a multivariate density model having fixed marginals that is tractable and fits data which are skewed, heavy-tailed, multimodal, i.e., of arbitrary complexity allowing for a rich correlation structure. It is also ideal for stress-testing a retirement plan by fitting historical data seeded with black swan events. A preliminary section reviews all concepts before they are used and fully documented C/C++ source code is attached making the research self-contained.
Lastly, we take the opportunity to challenge existing retirement finance dogma and also review some recent criticisms of retirement ruin probabilities and their suggested replacement metrics.

TABLE OF CONTENTS Introduction ………………………………………………………………………………………………………………………….. 1  I. Literature Review …………………………………………………………………………………………………………….. 2  II. Preliminaries …………………………………………………………………………………………………………………… 3  III. Univariate Density Modeling …………………………………………………………………………………………. 29  IV. Multivariate Density Modeling w/out Covariances …………………………………………………………… 37  V. Multivariate Density Modeling w/Covariances …………………………………………………………………. 40  VI. Expense-Adjusted Real Compounding Return on a Diversified Portfolio ……………………………. 47  VII. Retirement Portfolio Optimization ………………………………………………………………………………… 49  VIII. Conclusion ……………………………………………………………………………………………………………….. 51  References ………………………………………………………………………………………………………………………….. 52  Data Sources/Retirement Surveys ………………………………………………………………………………………….. 55  IX. Appendix with Source Code ………………………………………………………………………………………….. 56 

Keywords: variance components, EM algorithm, ECME algorithm, maximum likelihood, PDF, CDF, information criteria, finite mixture model, constrained optimization, retirement decumulation, probability of ruin, static/dynamic glidepaths, financial crisis Contact: cjr5@njit.edu

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A financial security that is purchased for $pt-1 at time t-1 with all distributions reinvested yields a value at time t called the adjusted price, say $Pt, for t = 1,2,…,T. The total return at time t is Rt = (Pt - pt-1)/pt-1 and the total compounding return is 1+Rt = Pt/pt-1 so that $pt-1(1+Rt) = $Pt. If the inflation rate between times t-1 and t is It then 1+Rt = (1+It)(1+rt), where rt = (1+Rt)/(1+It) – 1 is the real return at time t. The real price at time t is the value $pt such that (pt-pt-1)/pt-1 = rt, which upon solving yields $pt = $Pt/(1+It). In an efficient market real prices are governed by a geometric random walk (GRW), that is, ln($pt) = ln($pt-1) + St, where St ~ N(μ,σ2). A value of μ > 0 represents a drift and is the expected log-scale price increase sufficient to compensate the investor for risk (σ2) between times t-1 and t. In a random walk, the next value is the current value plus a random normal step, St, and the best predictor of it is the current value + μ. Exponentiating both sides of the GRW model yields the alternative form $pt = ($pt-1)eୗ౪, where eୗ౪= (1+rt) ~ lognormal(μ,σ2). Under strict conditions, the normally distributed step, St, can be justified. Decompose the

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