How does the conversion factor determine the cheapest-to-deliver bond in Treasury futures, and when does it break down?

The conversion factor (CF) system in Treasury futures is designed to make different deliverable bonds roughly equivalent, but it systematically favors certain bonds depending on the yield environment. Understanding this bias is critical for anyone hedging or trading Treasury futures.\n\nConversion Factor Mechanics:\n\nThe CF for a deliverable bond is calculated as the bond's clean price per dollar of par if it were priced to yield exactly 6% (the historical notional coupon). This means:\n\n- A bond with a coupon above 6% has CF > 1\n- A bond with a coupon below 6% has CF < 1\n- A bond with exactly 6% coupon at a round maturity has CF = 1\n\nDelivery Invoice:\n\nInvoice = Futures Settlement Price x CF + Accrued Interest\n\nThe short position delivers the bond and receives this invoice. The CTD is the bond that maximizes:\n\nDelivery Profit = Invoice Price - Market Price of Bond\n\nOr equivalently, the bond with the lowest basis net of carry (BNOC).\n\nWhy the CF System Fails to Equalize:\n\nThe CF assumes a flat 6% yield curve. When actual yields differ from 6%, the CF over- or under-compensates:\n\n| Yield Environment | CTD Tends To Be | Reason |\n|---|---|---|\n| Yields > 6% | Longest duration, lowest coupon | CF undercompensates duration; high-duration bonds are cheaper |\n| Yields < 6% | Shortest duration, highest coupon | CF overcompensates duration; short-duration bonds are cheaper |\n| Yields = 6% | All roughly equivalent | CF works as designed |\n\nWorked Example:\nCrestwood Fixed Income is short 200 ten-year Treasury futures contracts. Three bonds are deliverable:\n\n| Bond | Coupon | Maturity | CF | Market Price |\n|---|---|---|---|---|\n| Bond A | 3.875% | 8.5 years | 0.8412 | 92.15 |\n| Bond B | 4.625% | 9.2 years | 0.8987 | 97.80 |\n| Bond C | 5.250% | 10.0 years | 0.9534 | 103.45 |\n\nFutures settlement: 109.50\n\nInvoice prices:\n- Bond A: 109.50 x 0.8412 = 92.11 vs. market 92.15 --> basis = 0.04\n- Bond B: 109.50 x 0.8987 = 98.41 vs. market 97.80 --> basis = -0.61\n- Bond C: 109.50 x 0.9534 = 104.40 vs. market 103.45 --> basis = -0.95\n\nBond C has the most negative basis (highest delivery profit), making it the CTD.\n\nCTD Switching:\nAs yields fall below 6%, Bond A (lowest duration) becomes CTD. This switch creates convexity risk for futures hedgers -- the futures contract's effective duration changes abruptly, requiring position rebalancing.\n\nPractice CTD identification in our FRM question bank.