A Linear Regression Approach to Modeling Mortality Rates of Different Forms

Abstract

In this article, we propose a linear regression approach to modeling mortality rates of different forms. First, we repeat to fit a mortality sequence for each of K years (called the fitting years) with another mortality sequence of equal length for some year (called the base year) differing by t_j years (j = 1, …, K) using a simple linear regression. Then we fit the sequences of the estimated slope and intercept parameters of length K, respectively, with the sequence of {t_j} by each of the simple linear regression and random walk with drift models. The sequences of the fitted slope and intercept parameters can be used for forecasting deterministic and stochastic mortality rates. Forecasting performances are compared among these two approaches and the Lee-Carter model. The CBD model is also included for comparisons for an elderly age group. Moreover, we give a central-death-rate–linked security to hedge mortality/longevity risks. Optimal units, purchased from the special purpose vehicle, which maximize the hedge effectiveness for life insurers and annuity providers, respectively, are derived and can be expressed in terms of the cumulative distribution function of the standard normal random variable. A measure with hedge cost involved, called hedge effectiveness rate, for comparing risk reduction amount per dollar spent among mortality models is proposed. Finally, numerical examples are presented for illustrations.

Discussions on this article can be submitted until October 1, 2015. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

APPENDIX. DERIVATIONS OF THE EXPRESSIONS FOR CALCULATING AND

Recall that if W ∼ N(μ, σ²) for some μ and σ², then Y = e^W ∼ LN(μ, σ²) with and where Y∧y₀ = min (Y, y₀), and Φ and F_Y are the cumulative distribution functions of the standard normal random variable and Y, respectively.

To calculate , firstly with Y = [(Y − y₀)₊] + [Y∧y₀] we have Next, and lead to and Therefore, Finally, Y [(Y − y₀)₊] = [(Y − y₀)₊]² + y₀ [(Y − y₀)₊] yields (A.1) To calculate , by [(y₀ − Y)₊] − [(Y − y₀)₊] = y₀ − Y and [(y₀ − Y)₊]² + [(Y − y₀)₊]² = (Y − y₀)² we have and Also, applying Y [(y₀ − Y)₊] = y₀ [(y₀ − Y)₊] − [(y₀ − Y)₊]² gives (A.2) Note that since Y ∼ LN(μ, σ²). Therefore, in (4.2(4.2) ) and in (4.4(4.4) ) can be expressed in terms of the cumulative distribution function of the standard normal random variable.

To prove using Y [(Y − y₀)₊] = [(Y − y₀)₊]² + y₀ [(Y − y₀)₊], Y [(y₀ − Y)₊] = y₀ [(y₀ − Y)₊] − [(y₀ − Y)₊]², [(Y − y₀)₊]² + [(y₀ − Y)₊]² = (Y − y₀)² and [(y₀ − Y)₊] − [(Y − y₀)₊] = y₀ − Y produces which is Var(Y − y₀) = σ²[Y].