Treasury & Corporate Finance
Baumol-Tobin Optimal Cash Replenishment Size and Opportunity Cost Audit
Balance fixed securities-liquidation fees against foregone short-term yield using the Baumol-Tobin square-root cash model C* = SQRT(2 * b * T / i) and verify cost symmetry.
Predict the exact cash transfer lot and cost equality by hand
Suppose an operating account pays T_ann = 1,440,000 in steady annual disbursements (B2), each sweep from short-term T-bills costs a fixed wire/brokerage fee b_fee = 50 (B3), and T-bills earn annual yield i_rate = 0.04 (B4). Holding huge cash balances avoids transfer fees but sacrifices 4% interest on average cash C/2; sweeping tiny amounts daily minimizes idle cash but multiplies fixed fees (T_ann / C) * b_fee.
Test a naive once-a-month sweep of 120,000 first: 12 transfers * 50 = 600 in fees, plus (120,000 / 2) * 0.04 = 2,400 in foregone interest, totaling 3,000. Now compute Baumol-Tobin C* = SQRT(2 * 50 * 1,440,000 / 0.04) = SQRT(3,600,000,000) = 60,000 (24 sweeps/year): transfer fees = 24 * 50 = 1,200 and foregone interest = 30,000 * 0.04 = 1,200, cutting total cost to 2,400.
| Policy | Transfer Lot C | Annual Fees + Foregone Yield = Total Cost |
|---|---|---|
| Naive Monthly Sweep | 120,000 (12x/yr) | 600 + 2,400 = 3,000 (600 excess cost) |
| Baumol-Tobin Optimum C* | 60,000 (24x/yr) | 1,200 + 1,200 = 2,400 (Exact minimum) |
| Higher Yield i = 9% | 40,000 (36x/yr) | 1,800 + 1,800 = 3,600 (Square-root scaling) |
Optimal Cash Replenishment Lot C* Formula
Guard non-positive opportunity rates or disbursement inputs before taking the square root.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Cash Input", ROUND(SQRT(2 * B3 * B2 / B4), 2))Total Annual Cash Management Cost & Hint-First Check
Compute total annual cost (T_ann / C) * b_fee + (C / 2) * i_rate for any candidate lot C so you can compare C* directly against ad-hoc treasury schedules. If you ask the in-app Pro assistant for learning help, try: “Give me a hint on how to verify that my transfer size is optimal by comparing the two cost components, without writing the formula for me.” When annual yield rises 2.25x from 0.04 to 0.09, C* shrinks by 1 / 1.5 to 40,000.
=LET(t_ann, B2, b_fee, B3, i_rate, B4, c_opt, SQRT(2 * b_fee * t_ann / i_rate), ROUND((t_ann / c_opt) * b_fee + (c_opt / 2) * i_rate, 2))Try the example
Use the original values or change them to test your understanding.