# A variational inequality approach to financial valuation of retirement benefits based on salary

## Author

Listed:
• Avner Friedman

() (University of Minnesota, Department of Mathematics, Minneapolis, MN 55455, USA)

• Weixi Shen

() (Department of Mathematics, Fudan University, Shanghai 200433, China Manuscript)

## Abstract

We consider a pension plan with the option of early retirement, and paid benefits $\Psi (S,t)$ based on salary S at the time of retirement, but with guaranteed minimum; $S=S(t)$ is a Markov process. Denote by V(S,t) the financial value of the retirement benefits; its formal definition is given in (1.16). Then $\Psi (S,t) = V(S,t)$ at the end period T, while $\Psi (S,t)\leq V(S,t)$ if early retirement is exercised. We prove that V is the unique solution of a variational inequality, and that the set $\{\Psi = V\}$, which corresponds to the optimal time to retire, consists of either one or two continuous curves $S = S_i(t)$, depending on the parameters of the model.

## Suggested Citation

• Avner Friedman & Weixi Shen, 2002. "A variational inequality approach to financial valuation of retirement benefits based on salary," Finance and Stochastics, Springer, vol. 6(3), pages 273-302.
• Handle: RePEc:spr:finsto:v:6:y:2002:i:3:p:273-302
as

As the access to this document is restricted, you may want to search for a different version of it.

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Moshe A. Milevsky & Kristen S. Moore & Virginia R. Young, 2006. "Asset Allocation And Annuity-Purchase Strategies To Minimize The Probability Of Financial Ruin," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 647-671.
2. Moore, Kristen S., 2009. "Optimal surrender strategies for equity-indexed annuity investors," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 1-18, February.

### Keywords

Retirement benefits; variational inequality; free boundary; stochastic differential equations; optimal time;

### JEL classification:

• G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:finsto:v:6:y:2002:i:3:p:273-302. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: http://www.springer.com .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

We have no references for this item. You can help adding them by using this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.