Corporations tender for their own shares for a variety of reasons. Some stock tenders are made for strategic purposes—to prevent a take-over, to raise the market price of the stock, or simply because the stock represents ‘a good investment.’ For discussion of tendering in these situations, see the articles of Ellis [2] and Guthart [1]. In addition, there may be tactical reasons for a stock tender; one such reason is to reduce bookkeeping and shareholder servicing costs. In this instance, the argument runs roughly as follows: “The annual cost of servicing a holding is independent of the number of shares; consequently, the cost per share of servicing small holdings is relatively great. Let us reduce these high per-share costs by buying up small holdings.” Typical procedure is to then mail out an offer to buy holdings of less than a certain size directly, thus permitting the shareholder to dispose of his holding without paying the usual brokerage and odd-lot fees. Frequently no premium is offered except for the avoidance of brokerage fees. If one were to consider the premium offered as a controllable variable, it would be surprising to discover that its optimal value were exactly zero. One also recognizes that the maximum shareholding tendered for may be another decision variable available for optimization. See the appendix for data on tenders of this sort made in recent years. The variety of policies seems to indicate an almost complete absence of systematic application of the ideas presented here.