9.10 Cash Optimization (Baumol) Model: The Mathematics
While the firm wishes to minimize both its transaction and opportunity costs, it cannot do both simultaneously. In order to minimize transaction costs, it would have to infrequently order large quantities of cash; in so doing, it would increase its opportunity costs and vice versa. The relationship between transaction and opportunity costs is inverse. In the end, this shall be a mathematical problem involving the optimization of two inversely related functions: Transaction Costs + Holding (Opportunity) Costs. We shall formalize the foregoing notions mathematically:
Total Cost of Average Cash Balances = Transaction Costs + Holding (Opportunity) Costs
Next, we shall break down the formula into its components and related objectives:
- Minimize Transaction Costs of converting short-term assets (i.e., money market instruments) to cash
Transaction Cost = (T/C) × F
T = total amount of cash needed yearly, for all corporate business transactions
C = amount of cash provided (“ordered”) by (either) selling ST securities (or borrowing); also, total outlay required per sub-annual period; the “order quantity”
T/C = number of transactions over the course of the year
F = Fixed Transaction costs – for either selling a ST security or getting a loan; this excludes variable component having to do with size
- Minimize Opportunity Cost of interest income forgone from not having invested in ST securities
C/2 = average cash balance held during period (alternatively, beg. and end. cash balance ÷ 2)
(C/2) × i = opportunity cost of holding this balance, where i is the interest rate foregone
- Total Cost = Transaction + Opportunity Costs = (T/C × F) + (C/2 × i)
Given the inverse relationship of the components, we shall try to minimize total costs, i.e., to optimize cash.