# Determining the Optimum Cash Balance
A firm should hold an optimum cash balance — enough for day-to-day operations plus a safety buffer, but neither too much (opportunity cost) nor too little (risk of being unable to pay). The right level depends on the risk–return trade-off.
Mathematical models help find this optimum so that cash neither lies idle nor runs short. They fall into two categories:
1. Inventory-type models — used when cash flows are predictable / certain → e.g. Baumol's EOQ model.
2. Stochastic models — used when demand for cash is random / not known in advance → e.g. Miller-Orr model.
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## A. William J. Baumol's EOQ Model (1952)
Applies Wilson's economic-order-quantity logic to cash, under conditions of certainty / predictable cash flows.
Idea: Optimum cash is the level where total cost is minimum, i.e. where carrying cost = transaction cost.
- Carrying cost = cost of holding cash = opportunity cost / interest foregone on marketable securities.
- Transaction cost = cost of converting marketable securities into cash.
Formula:
$$C = \sqrt{\dfrac{2U \times P}{S}}$$
Where:
- C = Optimum cash balance (amount converted from securities to cash each time)
- U = Annual (or monthly) cash disbursement
- P = Fixed cost per transaction
- S = Opportunity cost of one rupee p.a. (or p.m.)
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## B. Miller-Orr Cash Management Model (1966)
Used when net cash flow is completely stochastic (random). It is a control-limit model that decides the timing and size of transfers between the cash account and an investment (marketable securities) account.
Three control limits:
- h = Upper limit
- z = Return point
- 0 = Lower limit
How it works:
- When cash hits the upper limit (h) → invest cash equal to (h − z) into marketable securities, bringing balance back down to z.
- When cash hits the lower limit (0) → sell securities to bring balance back up to z.
- While cash stays between the limits (anywhere in (0, h)) → do nothing; no transfers are made.
Setting the limits depends on:
- The fixed (transaction) cost of securities transactions,
- The opportunity cost of holding cash, and
- The degree of fluctuation (variance) in cash balances.
Why it's more realistic than Baumol: it allows the cash balance to fluctuate freely within a lower and upper band, rather than assuming a smooth, predictable drawdown. The finance manager sets the band according to the firm's liquidity needs (minimum and maximum cash).