Miller-Orr on the Stochastic Residual

The model sets an upper limit $H$, a return point $\mathit{RP}$, and a lower limit $L$ such that when the operating-cash balance hits $H$, the excess sweeps into the higher-yielding sleeve, and when it hits $L$, the sleeve is liquidated back to $\mathit{RP}$:

with three inputs that must be measured correctly:

Variance of residual cash flow, ${\sigma }^2$.

Compute over a year of data using only the unpredictable transactions. Strip out recurring bills, scheduled tax payments, salary deposits, and any other line item you knew in advance. The remainder — the squared deviation of those residual transactions from their daily average — is the variance the model needs. Using a year captures seasonality (a quiet quarter offsets a noisy one); using less can mislead.

Flat transaction cost, $C$.

The fee to move cash in or out of the investment sleeve. Transferring between a brokerage cash account and a Treasury money market fund (MMF) via ACH is typically $0; a structured wire might run $15–$25; a CD broken early includes the early-withdrawal penalty as the cost. Do not count foregone interest here — that is captured separately by the daily rate $r$ in the denominator. Counting both double-charges the opportunity cost and inflates the spread.

Daily after-tax interest differential, $r$.

After-tax yield on the investment sleeve, converted from annual to daily. For a Treasury-only MMF at 4.3% nominal at the California stack from section “Consider Taxation”, the after-tax rate is roughly 2.55% annual, or $r\approx 0.0255∕365\approx 0.0000699$ per day.

$L$ is the operating-cash floor: typically four to six weeks of deterministic recurring spend, sized to the survival-floor test.

Worked example, residual-only. A household with deterministic recurring and known-lump cash flow fully calendared into separate sinking funds. The residual: roughly 30 unpredictable transactions a year averaging $1,500 each, irregular dates. Strip the recurring activity from the account history and the residual variance computed over a year is roughly ${\sigma }^2\approx 80,000$ — two orders of magnitude smaller than the $2.96M figure the all-transactions calculation would produce. With $C=\$25$ for the occasional wire and $r=0.0000699$:

For $L=\$10,000$ (one month of deterministic spend): $\mathit{RP}\approx \$12,780$, $H\approx \$18,340$. Above $18,340 in operating cash, sweep $5,560 into the Treasury MMF; below $10,000, redeem $2,780 of MMF or T-bill to return to the return point.

The $C\to 0$ degenerate case. For most readers moving cash between accounts at the same broker, $C$ is effectively zero and instantaneous. The spread collapses to zero, $H=L=\mathit{RP}$, and the model degenerates to a single instruction: pin the operating-cash balance at $L$ and sweep everything above it continuously. This is exactly what modern brokerage cash-sweep features do without asking for an opinion. The Miller-Orr math becomes a sanity check on whether the sweep thresholds your broker chose for you are in the right neighborhood, not a number you compute every month.

Where the model adds real value is in identifying the cases where $C$ is non-trivial (broken CDs, structured wires, capital calls into illiquid funds with notice periods) and the residual variance is also non-trivial (founder-style irregular distributions, partnership K-1 distributions, performance fees). Those readers should compute their own ${\sigma }^2$ from a year of residual cash flow and use the formula above rather than the generic “three months of expenses in checking” heuristic.