I am just confused about the conversion factor when calculating arbitrage profit. Why divide by the conversion factor for the bond value we estimate (Adjusted full price x 1/.90) (assuming this is to get the cheapest to deliver version of that bond) and then essentially multiply the conversion factor back to that bond (no net effect on estimated bond) when finding the arbitrage profit AND multiplying the already-quoted bond by the conversion factor (e.g. [125 - 124.4] x 90)? In other words, why does the market-quoted bond ($125) end up getting multiplied by .9 while the estimated (112.16 - .20) is essentially multiplied by 1/.90 and then multiplied by .90 when taking the arbitrage profit?

The conversion factor tells you what price you will actuall receive for the bond.

Take the futures price x CF, adjusted for AI, to get price received.

With the approach you have used you have calculaated the Arb profit (125 - 124.4) based on futures pricing but you are not getting that amount. What you get will be based on actual pricing of the bond so need to multiple by CF. You scalled bond price to match futures, calculated and profit so you need to scale back to what you actually receive.

Easier approach.

- What is the fair FULL PRICE of the actual bond at expiry of futures (ignore CF)

FP = Full price x (1 +r)^t - FVC = 112.164 - What will I receive for this bond if I shorted futures

= (125 x 0.9) + 0.2 = 112.70 - Difference = 112.70 - 112.1639 = 0.5360
- Discount back to now 0.5360/1.003^0.25 = 0.5356

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