Defined contribution beta when combined with a defined benefit plan.

JEL G11 * G23

This research is prompted by an analysis of the retirement allocations of civilian employees of the United States Navy. These employees have the option of participating in a Defined Benefit (DB) plan, a Defined Contribution (DC) plan, or both. Since the DB plan is an obligation of the United States government, it is essentially a risk free asset equivalent to United States Treasury bonds.

For any individual, the desired beta of their total retirement portfolio should be the same regardless of the spectrum of plans or assets in plans held:

[beta](D[C.sub.B], DB) = [beta](DC). (1)

That is, the desired beta of any individual's total portfolio is a constant. The beta of the portfolio that the individual holds must be the same whether the person has two assets in their portfolio (both the DB and DC assets), or only one asset (the DC plan asset) in their portfolio. The beta of any portfolio is the weighted average of the asset's beta or:

[beta](D[C.sub.B] DB) = a x [beta](DB) + (1 - a) x [beta](D[C.sub.B]) (2)

where D[C.sub.B] represents the defined contribution assets of a participant in both plans. If we consider the defined benefit plan to be an obligation of the United States government and essentially a risk-free asset composed of United States Treasury securities, then the DB plan will be uncorrelated with the market portfolio:

[beta](DB) = 0 (3)

substituting Eqs. 1 and 3 into Eq. 2 gives: a x (0) + (1 - a) x [beta](D[C.sub.B]) = [beta](DC) (4)

and

[beta](D[C.sub.B])/[beta](DC) = 1/(1 - a). (5)

Since 'a' represents the percent invested in the DB portion of the portfolio of a person in both the DB and DC plans, 'a' must be >0 (or they would only be in the DC plan) and <1 (or they would only be in the DB plan). This result implies that:

1/(1 - a) > 1. (6)

And finally, we have that:

[beta](D[C.sub.B]) > [beta](DC) (7)

Hence, the DC beta of a participant in both plans must be greater than the beta of someone who only participates in the DC plan.

Published online: 28 May 2008

Eugene M. Bland * Robert Trimm

E. M. Bland ([mail])

College of Business, Texas A&M University--Corpus Christi, 6300 Ocean Drive, Corpus Christi, TX 78412, USA

e-mail: eugene.bland@tamucc.edu

R. Trimm

Department of the Navy, Commander Navy Installations Command,

5720 Integrity Drive, Building 457, Millinton, TN 38055-6520, USA