Forum for Science, Industry and Business

Sponsored by:     3M 
Search our Site:

 

Are you ready to retire?

22.10.2013
Mathematical models estimate the value of pension plans

There comes a time in each of our lives when we consider starting a pension plan –either on the advice of a friend, a relative, or of our own volition. The plan of choice may depend on various factors, such as the age and salary of the individual, number of years of expected employment, as well as options to retire early or late.

One possible plan is a defined pension plan, where the benefit amount is typically based on the employee’s number of years of service at the time of retirement and the salary and/or average salary over an employment period. For instance, the employee may receive a fraction of the average salary during a certain number of years.

In a paper published last month in the SIAM Journal on Applied Mathematics, authors Carmen Calvo-Garrido, Andrea Pascucci, and Carlos Vázquez present a partial differential equation (PDE) model governing the value of a defined pension plan including the option for early retirement.

“The employer bears the liability of the pension and the value of this liability is understood as the value of the pension plan,” says author Carlos Vázquez. “It is important to develop mathematical models to compute the value of this liability in order to estimate the financial situation of the institution or company that has the obligation with the pension plan member.”

The analysis in the paper uses modeling tools similar to those used in quantitative finance, for instance, for pricing American options.

The model assumes that the wage or salary of an employee at any given time is governed by a stochastic differential equation, which in turn depends on the time of recruitment, current salary of the employee and age of entry. Uncertainty of the salary is assumed to depend only on volatility, which refers to the uncertainty or risk associated with a value or asset. “Models need to reproduce the uncertainties associated with the underlying factors of the plan (salary, interest rate and so on) and should allow one to compute the pension plan price in order to reproduce situations in different scenarios,” author Andrea Pascucci explains.

The authors obtain the value of a defined benefit pension plan including the option for early retirement for the employee, thus computing the value of the pension plan as well as the region of early retirement. “If the pension plan incorporates the option of early retirement by the member, then the additional question arises: when is it optimal to retire? Mathematical modeling tools allow us to pose the problem in terms of partial differential equations,” says Vázquez.

The optimal retirement problem is a “free boundary problem” for the underlying PDE. Most applications of PDEs involve domains with boundaries, and certain boundary conditions need to be satisfied in order to solve the PDEs. Free boundary problems deal with solving PDEs where part of the boundary is unknown in advance, referred to as a free boundary. Thus, in addition to standard boundary conditions, an additional condition must be imposed at the free boundary. The free boundary in this problem is the optimal retirement boundary between the region where it is optimal to retire and the region where it is optimal to continue working.

“The practical solution of the PDE model to obtain pension plan prices from the data requires the use of suitable numerical algorithms to be run on a computer,” says author Carmen Calvo-Garrido. “From the numerical solutions, we can identify at each date, for a given salary and average salary, if it is optimal to retire or not, and also to obtain the value of the pension plan in any case.”

Mathematical analysis provides rigorous justification of the correctness of the model, also proving the expected qualitative properties.

Future directions may involve the application of similar modeling techniques to study the evolution of wages and salaries. “We are working on a more complete model for salaries evolution that includes the possibility of jumps (due to economic crisis, sudden increase or decrease in salaries, etc.),” says Vázquez. “PDE problems including realistic, stochastic interest rate models also present a very challenging topic. The calibration of model parameters is an interesting and difficult problem due to the need of suitable real data.”

Source article:

Mathematical Analysis and Numerical Methods for Pricing Pension Plans Allowing Early Retirement

M. Carmen Calvo-Garrido, Andrea Pascucci, and Carlos Vázquez

SIAM Journal on Applied Mathematics, 73(5), 1747–1767 (Online publish date: September 4, 2013). The source article is available for free access at the link above until February 21, 2014.

About the authors:

M. Carmen Calvo-Garrido is a researcher and Carlos Vázquez is a full professor at the Department of Mathematics in the University of A Coruña in Spain. Andrea Pascucci is a full professor at the Department of Mathematics in the University of Bologna in Italy. This research was partially funded by Spanish Ministery of Science and Innovation and by Galician Regional Government (Xunta de Galicia).

About SIAM
The Society for Industrial and Applied Mathematics (SIAM), headquartered in Philadelphia, Pennsylvania, is an international society of over 14,000 individual members, including applied and computational mathematicians and computer scientists, as well as other scientists and engineers. Members from 85 countries are researchers, educators, students, and practitioners in industry, government, laboratories, and academia. The Society, which also includes nearly 500 academic and corporate institutional members, serves and advances the disciplines of applied mathematics and computational science by publishing a variety of books and prestigious peer-reviewed research journals, by conducting conferences, and by hosting activity groups in various areas of mathematics. SIAM provides many opportunities for students including regional sections and student chapters. Further information is available at www.siam.org.

Karthika Muthukumaraswamy | EurekAlert!
Further information:
http://www.siam.org

More articles from Social Sciences:

nachricht Fixating on faces
26.01.2017 | California Institute of Technology

nachricht Internet use in class tied to lower test scores
16.12.2016 | Michigan State University

All articles from Social Sciences >>>

The most recent press releases about innovation >>>

Die letzten 5 Focus-News des innovations-reports im Überblick:

Im Focus: Breakthrough with a chain of gold atoms

In the field of nanoscience, an international team of physicists with participants from Konstanz has achieved a breakthrough in understanding heat transport

In the field of nanoscience, an international team of physicists with participants from Konstanz has achieved a breakthrough in understanding heat transport

Im Focus: DNA repair: a new letter in the cell alphabet

Results reveal how discoveries may be hidden in scientific “blind spots”

Cells need to repair damaged DNA in our genes to prevent the development of cancer and other diseases. Our cells therefore activate and send “repair-proteins”...

Im Focus: Dresdner scientists print tomorrow’s world

The Fraunhofer IWS Dresden and Technische Universität Dresden inaugurated their jointly operated Center for Additive Manufacturing Dresden (AMCD) with a festive ceremony on February 7, 2017. Scientists from various disciplines perform research on materials, additive manufacturing processes and innovative technologies, which build up components in a layer by layer process. This technology opens up new horizons for component design and combinations of functions. For example during fabrication, electrical conductors and sensors are already able to be additively manufactured into components. They provide information about stress conditions of a product during operation.

The 3D-printing technology, or additive manufacturing as it is often called, has long made the step out of scientific research laboratories into industrial...

Im Focus: Mimicking nature's cellular architectures via 3-D printing

Research offers new level of control over the structure of 3-D printed materials

Nature does amazing things with limited design materials. Grass, for example, can support its own weight, resist strong wind loads, and recover after being...

Im Focus: Three Magnetic States for Each Hole

Nanometer-scale magnetic perforated grids could create new possibilities for computing. Together with international colleagues, scientists from the Helmholtz Zentrum Dresden-Rossendorf (HZDR) have shown how a cobalt grid can be reliably programmed at room temperature. In addition they discovered that for every hole ("antidot") three magnetic states can be configured. The results have been published in the journal "Scientific Reports".

Physicist Dr. Rantej Bali from the HZDR, together with scientists from Singapore and Australia, designed a special grid structure in a thin layer of cobalt in...

All Focus news of the innovation-report >>>

Anzeige

Anzeige

Event News

Booth and panel discussion – The Lindau Nobel Laureate Meetings at the AAAS 2017 Annual Meeting

13.02.2017 | Event News

Complex Loading versus Hidden Reserves

10.02.2017 | Event News

International Conference on Crystal Growth in Freiburg

09.02.2017 | Event News

 
Latest News

Switched-on DNA

20.02.2017 | Materials Sciences

Second cause of hidden hearing loss identified

20.02.2017 | Health and Medicine

Prospect for more effective treatment of nerve pain

20.02.2017 | Health and Medicine

VideoLinks
B2B-VideoLinks
More VideoLinks >>>