A client of NEXT Financial Group (a top brokerage firm) seeks to buy 250,000 shares(x=250,000) in smaller parcels to minimize his total execution costs B. These execution costs B include both order processing costs (i.e., F per transaction) and market impact (slippage) costs that are increasing in the size c of each of the n equal-size transactions.

NEXT Financial Group uses a market impact model that estimates the slippage cost S of each of the n equal-sized executions c=X/n with function S=vce", where the coefficient v and exponent m are related to the liquidity of the security to be purchased or sold. In this case, the per-transaction cost B/n is estimated to have a fixed component F(order processing costs) and an increasing variable cost component vcm = v(x/n)m (this is the market impact component of the execution costs): /n B/=F+v%)

Suppose that broker analysts have estimated F to be $0.02 per order, based on knowledge of order processing costs, and v and m to be 0.00005 and 1.2, respectively

What is the optimal number of transactions?