 # Arbitrage Profit on Bond Futures Contract

Questions 1:

The current annual compounded risk-free rate is 0.30%. Current Data for Futures and Underlying Bond: Futures Contract: Quoted futures price 125.00 Conversion factor 0.90 Time remaining to contract expiration 3 Months Accrued interest over life of futures contract 0 Underlying Bond: Quoted bond price 112.00 Accrued interest since last coupon payment 0.08 Accrued interest at futures contract expiration 0.20

Based on Exhibit 2 and assuming annual compounding, the arbitrage profit on the bond futures contract is closest to: A 0.4158. B 0.5356. C 0.6195. Answer: The no-arbitrage futures price is equal to the following: F0(T) = FV0,T(T)[B0(T + Y) + AI0 – PVCI0,T] F0(T) = (1 + 0.003)0.25(112.00 + 0.08 – 0) F0(T) = (1 + 0.003)0.25(112.08) = 112.1640 The adjusted price of the futures contract is equal to the conversion factor multiplied by the quoted futures price: F0(T) = CF(T)QF0(T) F0(T) = (0.90)(125) = 112.50 Adding the accrued interest of 0.20 in three months (futures contract expiration) to the adjusted price of the futures contract gives a total price of 112.70. This difference means that the futures contract is overpriced by 112.70 – 112.1640 = 0.5360. The available arbitrage profit is the present value of this difference: 0.5360/(1.003)0.25 = 0.5356.

To find the no-arbitrage futures price, why is the accrued interest at futures contract expiration of 0.20 not being subtracted in the formula? Why is it being added to the adjusted price?

Schweser formula for FP = [(Full price - PVC)*(1+rf)T] - Accrued Int at maturity

Can someone please explain the solution?

This is question 1 from Pricing and Valuation of Forward Commitments

Hi all, I have exactly the same question, why is the AI at maturity added, and not left out? As I would think (and as per schweser) the AI at maturity would belong to the seller of the bond, hence this should not be added, perhaps someone can explain this?

The AI at maturity is subtracted from the future price according to the CFAI text too.

it has to be the full price. thats why you added the accrue interest in the beginning, but the AI at the end has to be subtracted

Thank you! This means that the adjusted price of the futures contract should give us a price of 112.30 (not 112.70), correct?

I believe that the quoted price is not included the accrued interest. Thus, you have to add the accrued interest from last coupon, which is 0.08 in this case. Then, you can start applying the formula and subtract the PV of future accrued interest (0.2 in this case)

I think that this question is incorrect in the Curriculum. Could someone more knowledgeable provide some insights? Thanks hans

Bump. Any clarification would be appreciated!

Just wondering if it’s because the future is overvalued while the underlying asset undervalued Hence, the arbitrage transaction would be to “Short future” and “long bond” By short selling the future you’d receive the accrued interest (In this case 0.2)? But this would also mean that the formula given in the curriculum “FP = [(Full price - PVC)*(1+rf)T] - Accrued Int at maturity” indicates the price for a long position? Please correct me for any mistakes I made

Some questions arise for me:

1. Why the 360-day basis, FI Instruments price on 365-basis?

2. Why do they calculate the FP from QFP (given), I would have calculated FP (based on the Bond) and then calculated QFP=FP*1/QF and compared it to the given QFP.

3. Why can I assume that the future value of coupons is zero? Bc there are no AI over the life of the future?

Following the formula I would have gone this way:

FP=[(S0+AIt)*(1+r)T-AIT-FCV]*1/QF=

FP=[(112+0,08)*1,03(90/365)-0,2-0]*1/0,9=125,22

This is a very old post. But I believe it may help candidates preparing for their CFA level 2 this year or the coming years…

The question is asking for a profit not a price. so that’s why we are adding the accrued interest instead of subtracting it. Accrued interest increases our overall profit so to speak but decreases the price as it’s benefit from carrying the underlying.

for an arbitrage opportunity to occur, there must be a price difference between the forward price and the future price of the underlying.
An arbitrager doesn’t use his own money so he borrows the full price of the bond S0 = (112 + .08 AI) at a rate of .30%. he buys the bond for (112 + .08)= 112.08.
which he has to repay after 3 months at T= .25
So the Future value of the bond is 112.08 * (1+.3%)0.25 = 112.164 which is what the arbitrager is going to repay to the lender at time T.

meanwhile, the arbitrager enters a forward contract to sell the bond at T=.25 for F0(T) = 112.5
So, the arbitrager is going to make (112.5 -the forward price- + .2 -we add it back as it has been subtracted when came up with the forward price. it reduces ur price. but increases ur profit.)= 112.7

the arbitrage profit is then the difference between what he made and what he’s going to repay. and come up with the PV of that .

PV (112.7 - 112.164) = .536/(1+.3%)0.25 = .5356

Hope this helps.
Best of Luck.

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