The first overlay is the one every allocator uses. A portfolio of value V_P with beta β_P can be moved to a target beta with
N = (β_target − β_P) × V_P / (F × m)
where F is the futures price and m the multiplier or lot size. A negative N is a short. Setting β_target = 0 gives the full hedge; setting it above β_P gives a long futures position that levers market exposure without buying a share.
Run it in each market so the shape of the residual is visible in both. A $48,000,000 US portfolio at beta 1.15, moved to 0.60 with the E-mini at 5,866.00, needs (0.60 − 1.15) × 48,000,000 / 293,300 = −90.010228 contracts, so 90 short. The rounding leaves an effective beta of 0.600062, an error of 0.000062, which on this book is genuinely nothing. An ₹8,40,00,000 Indian portfolio at beta 1.18, moved to 0.50 with the Nifty contract at ₹18,64,875, needs −30.629399 contracts on the 75-unit lot, so 31 short, and the rounding overshoots to an effective beta of 0.491772 while leaving ₹6,91,125 of notional over-hedged. Same formula, and the granularity differs by 3.63 times: one E-mini is 0.6110% of the US portfolio while one Nifty lot is 2.2201% of the Indian one, so the worst-case beta error from rounding is 0.0031 in the first case and 0.0111 in the second.
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How do you equitise cash with futures?
Equitising cash is the same formula with β_P = 0. A fund holding $12,000,000 of cash that it wants exposed to the index buys 12,000,000 / (F × m) contracts and keeps the cash earning the money-market rate. The combination replicates an index position, and the reason it works is the cost-of-carry relation: the futures price already contains the financing cost and the forgone dividend, so a long future plus cash earning the financing rate is, by construction, a long index position.
Except that it is not, exactly, and the gap has three named parts. Worked Example 4 computes a tracking difference of −$7,739.33 on a $12m balance, or −6.45 basis points, and splits it three ways. Minus $4,817.50 because the contract was bought 2.35 points above its fair value. Plus $860.88 from rounding the position up to a whole 41 contracts. And minus $3,782.71 because the overlay was sized at inception and never rebalanced, so as the index rose the futures position carried less exposure than the cash it was standing in for. That third term has a closed form, Cash × (r − q)T × (1 − S_T / F_fair), and it reproduces the numerical residual to the cent. Naming it is the difference between an overlay you can audit and an overlay you can only hope about.
What residuals does an overlay leave?
Those three residuals generalise to every other overlay in Exchange-Traded Derivatives in Practice. Rounding, because contracts are integers. Basis, because a futures price is not the index and converges only at expiry. Drift, because the hedge ratio was computed at a market level that has since moved. A professional states all three before trading and re-states them at each roll.
An overlay whose residual has no name has not been measured.