I am stuck at this question… for so long… desperately need help!
Quoted futures price = 125
Conversion factor = 0.90
Time remaining to contract expiration = 3 months
AI over the life of futures contract = 0
Quoted bond price = 112
AI since last coupon payment = 0.08
AI at futures contract expiration = 0.2
Current annual risk-free rate = 0.30%
What is arbitrage profit on the bond futures contract?
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In the solution, they recalculate the adjusted price of the futures (125 x 0.9) and then add 0.20. Shouldn’t the 125 already include AI at time T? Anyway the final answer is 0.5356.
I was attempting to convert the quoted bond price to quoted futures price and compared with the given quoted futures price of 125 to find the arbitrage value.
My calculation:
Formula used, F0(T) = [(Price of Bond + Accrued Interest at time 0) – PVCI] – (Accrued Interest at time T) = 111.88
Therefore, quoted F0(T) = 111.88/0.9 = 124.31
Value of the arbitrage = (125 – 124.31)/ [1 + (0.0003) (3/12)] = 0.68948
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I would have thought whether you compare between futures price or quoted futures price, the arbitrage value should be the same. but clearly i am wrong! Masters, teach me!
If the price is clean, they likely will just say its the quoted price. If they give you full price, they will say its the dirty price or full price. Sometimes they will say the price includes the AI.
Why do you compare futures price calculated at t=0 and futures price calculated at t instead of quoted futures price at t=0 and quoted future price at t?
Why is it in the calculation of the futures price based on bond price does not include AI at time T but in the quoted futures price of 125, it was multiply with 0.9 (which I understand why) and then plus (0.2 (which I understood as because the initial quoted price was a clean price)?
1.) The conversion factor makes it more complicated to compare. I think you can, but the algebra is a bit more messy. Its easier to remove the conversion factor. Either way, you want to compare apples to apples. A
2.) I’m not sure my reasoning is right but I added .2 to equalize the equation. FP+AI_T = (full price)(1+Rf)T
Either you add .2 to te left side, or you subtract .2 from the right side. The outcome is the same: FP = (full price)(1+Rf)T - AI_T
Again, this is comparing apples to apples. Without adding +.2 to the left side, you would be comparing the FP without the AI_T while comparing the right side with AI_T.
i. Quoted futures price = clean price. Given the question gives the following: “Accrued interest at futures contract expiration = 0.2”, the given AI is added to the quoted futures price to make it “dirty” and is only applicable to the quoted futures priice.
ii. The question does not provide AI at time T. Therefore, the full formula of counting the futures price based on the quoted bond price is F(T) = ( Bond price - AI at time 0 - PVCI) (1 + r)(3/12). This is the dirty price based on the given bond price with PVCI = 0 and AI at time T for this bond = 0.
The AI at time T is given as .2. That is true for the your statement i and ii.
Your statement ii equation is incorrect. It’s missing -AI_T. Full equation is FP = FV(bond price -AI_0 -pvc) - AI_T
You can compare dirty to dirty or clean to clean. If you go with dirty (which is how the problem appears to solve), you would add .2 to FP and compare it with FV(bond price -AI_0 -pvc) - AI_T + AI_T.
112.7 -112.1639 =0.5361
You can compare clean to clean by just taking FP and compare to FV(bond price -AI_0 -pvc) - AI_T.