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Using Black‐Scholes to determine an optimal funding term

Roger Gay (Department of Accounting and Finance, Faculty of Business and Economics, Monash University, Clayton, Australia)

Managerial Finance

ISSN: 0307-4358

Article publication date: 27 September 2011



The purpose of this paper is to examine use of the Black‐Scholes (BS) risky asset model to determine choice of optimal investment term in a reinvestment chain model.


An extension of Tobin's separation theorem is used to establish a mean‐variance efficient strategy for lump sum conversion to an income stream over any fixed term; two criteria involving the BS model are then applied to determine optimal investment term in a perpetual chain of reinvestment. The first criterion selects the term to maximize the value of a call option on excess of a market portfolio accumulation over the indexed value of the original lump sum. The second criterion selects term to maximize the expected present value of this excess without the no‐arbitrage assumption.


It is found that both criteria lead to useful but different income stream funding strategies. Annual returns data for the All Ordinaries Accumulation Index for years 1900‐2009 are used for an empirical assessment of the relative usefulness of the two criteria. Empirical evidence favours use of the criterion without the no‐arbitrage assumption.


Mean‐variance efficiency of the lump sum conversion strategy has been described elsewhere, but it has not previously been recognized as an extension of the Tobin theorem. Determination of optimal reinvestment term in this context is new and crucial to practical application of the model. One application of universal significance is for retirees emerging from defined contribution pension schemes with lump sums to provide for retirement in the face of longevity risk.



Gay, R. (2011), "Using Black‐Scholes to determine an optimal funding term", Managerial Finance, Vol. 37 No. 11, pp. 985-994.



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