Numeric example for synthetic FRA

Hi, I am reading the FRA & synthetic FRA in the curriculum but the book explains in graph and text only. Could anyone provide me a numeric example so I could understand more about FRA & synthetic FRA? I did a search on our forum but found nothing similar yet, maybe this issue is quite easy for others (but not for me :sweat_smile:). Thank you!

FRAs

A long position in the FRA means to lock in a forward borrowing rate.

Assuming entering into $100, 9x12 FRA (begins in 9 months, ends in 12 months, and covers a 3 month period) at 5.4054% but an actual LIBOR in 3-months (the settlement date) is 5.6%, therefore, the right to borrow was bought at 5.4054% but the actual is 5.6%. Thus, this long position should receive (by using actual/360 day-count convention):

\frac{(0.056-0.054054)\times{90\over360}\times{100}}{(1+{0.056\times{90\over360}})} = {${0.048}}

Synthetic FRAs : to create the same effect of the FRA, you can use zero coupon bonds here.

Letโ€™s assume there are two zero coupon bonds maturing at T = 360 and T= 270:

  • Zero coupon bond maturing in 270 days at the rate of return = 3.6%
  • Zero coupon bond maturing in 360 days at the rate of return = 5%

At T=0:

  • you will buy $96.5251 zero coupons maturing at t = 270. PV = {100\over{(1+0.036)}}
  • you will sell $96.5251 zero coupons maturing at t = 360. PV = {101.3514\over{(1+0.05)}}

Therefore, the total cost is zero by both buying and selling two bonds at T=0.

At T=270

  • you receive the face value of zero coupons maturing at t = 270 at 100.
  • you pay or close the position of zero coupons maturing at t = 360 at -99.9521. {{PV}_{t_{270}}} = {-101.3514\over{(1+0.056\times{90\over360})}} . This means that you close your position at T=270 before its maturity (T=360). Therefore, you have to discount back to T=270 by using the same rate as FRA example above which is 5.6%

Therefore, the total payoff is 100 - 99.9521 = $0.048.

This shows that on the synthetic positions on zero coupon bonds, you can replicate the same payoff as the FRA.

Also, you can opt to create the synthetic position by using Eurodollar Future contracts:

  • Long 360-day Eurodollar contract
  • Short 270-day Eurodollar contract

At T=270, you would close the long position of 360-day Eurodollar in order to synthetically create the exposure to a 90-day FRA. A Eurodollar future is a futures contract, it is a cash settlement meaning the settlement of the futures will be settled at the beginning of the loan (upfront) and it is not discounted.

However, for level 1 just donโ€™t worry about FRAs too much though, just try to understand the concept. Then for level 2 the devil will reveal itself. :japanese_ogre:

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Thank you so much Pyng!!!

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