Convexity adjustment, part 2

Key words: Convexity Bias, convexity adjustment, FRA, EuroDollars, Interest Rates
PG

As it was written in the previous article “Futures and forward convexity adjustment”, there is a systematic advantage to being short EuroDollar futures relative to FRAs.
This advantage is characterized as a convexity bias and appropriate methods exist to adjust Eurodollar futures prices to eliminate the difference between the futures and forward LIBOR rates.

The article explained that marking-to-market is one of the sources of convexity bias:

– When interest rates rise, the borrower receives a payment that can be invested at the higher interest rate.
– When interest rates fall, the borrower makes a payment that can be funded at the lower interest rate.

However, the settlement structure of the Eurodollar contract is another reason for convexity bias.
The following examples illustrate the main difference between the settlement structures of FRA and Eurodollar futures.

1. In Novemeber, we buy FRA @1.45% 3-month LIBOR rate in order to guarantee a 1.45% borrowing rate for a $500M loan for 3 months from March to June.

FRA payoff

The FRAs payoff is well describe in the article “FRA” and following the same logic we have:

– If interest rate rise up to 1.6%, we make $738 188 to be settled by the seller in March.
– If interest rate is 1.2%, we suffer a loss of -$1 235 177 to be settled by us to the seller in March.

2. Let’s consider again the example in which we wish to guarantee a 1.45% borrowing rate for a $500M loan for 3 months.

Taking into account that if we bought the FRA in the previous example in order to hedge our interest rate exposure against interest rate rise, we need to short 500 Eurodollar futures contracts (the loan is $500M and thus 500 contracts have notional of $500M) in order to make the same hedge.

Suppose, in Novemeber, the March Eurodollar implied 3-month LIBOR is 1.45% and thus the futures price is 94.2, i.e., (100 − 94.2) / (4 * 100) = 1.45% over 3 months.

According to the article “EuroDollars”, the payoff at expiration is:
(94.2 − (100 − LIBOR at expiration)) * 100 * $25.

– If the 3-month LIBOR rate in March is 1.2%, the Eurodollar futures price will be 95.2 (100 – 1.2% * 4)
The payment is: ((94.2 − 95.2) * 100 * $25) * 500= −$1.25M

– If the 3-month LIBOR rate in March is 1.6%, the Eurodollar futures price will be 93.6 (100 – 1.6% * 4)
The payment is: ((94.2 − 93.6) * 100 * $25) * 500= $0.75M

This is like the payment on the FRA paid in June (in-arrears) except that the futures contract settles in March due to its specification, i.e., the FRAs result in June = ED futures result in March as it follows from the table below.

The results are the same but the months are different. If we compare the results in March, we see the FRA result differs from the result of Eurodollar futures. Clearly, if we want to fix the same interest rate of @1.45%, we must adjust the number of Eurodollar futures accordingly.

Put it differently, the rate implied by the Eurodollar contract cannot equal the prevailing FRA rate for the same loan.

FRA vs Eurodollar futures

In order to have the same @1.45% interest rate (both for FRA and Eurodollar futures), we adjust the number of Eurodollar futures by shorting fewer than 500 contracts. Using the implied 3-month Eurodollar rate of 1.45% as the discount factor, we receive:
500 / (1+ 0.0145) = 492.85 contracts to short.

Now, we recalculate the results in March, taking into account 492.85 contracts instead of 500.

– If the 3-month LIBOR rate in March is 1.6%, the Eurodollar futures price will be 93.6 (100 – 1.6% * 4). The payment in March is:
((94.2 − 93.6) * 100 * $25) * 492.85= $739 275.
This sum is equal $739 275 * 1.016 = $751 103 to be paid in June

– If the 3-month LIBOR rate in March is 1.2%, the Eurodollar futures price will be 95.2 (100 – 1.2% * 4). The payment in March is:
((94.2 − 95.2) * 100 * $25) * 492.85= −$1 232 125.
This sum is equal −$1 232 125 * 1.012 = −$1 246 910 to be paid in June

FRA and Eurodollar futures

Now, if we compare the results for each month, we see the difference exists but it is relatively small. One more conclution we can make is that the settlement structure of the Eurodollar contract works systematically in favor of the borrower. Put differently, we can verify that it systematically works against a lender.

As it follows from the table above, Eurdollar futures has an advantage over FRAs, which is called a convexity bias:

In June, if quarterly LIBOR is 1.6%, we pay $8M (500M * 1.6%) in borrowing cost and:
Eurodollar contract earns $751 103 on the , a net borrowing expense is $8M – $751 103= $7 248 897

A net borrowing with the FRA contract will result in $8M – $750 000 = $7 250 000

Therefore, totally, we make a profit, relative to an FRA, of $7 250 000 − $7 248 897 = $1 103

In June, if quarterly LIBOR is 1.2%, we pay $6M (500M * 1.2%) in borrowing cost and:

Eurodollar contract loses -$1 246 910, a net borrowing expense is $6M + $1 246 910= $7 246 910

A net borrowing with the FRA contract will result in $6M + $1 250 000 = $7 250 000

Therefore, totally, we make a profit, relative to an FRA, of $7 250 000 − $7 246 910 = $3 090

The examples above show that Eurodollar futures contracts have an advantage over FRA and thus Eurodollar futures prices should be lower than their so-called fair values. Put differently, the 3-month interest rates implied by Eurodollar futures prices should be higher than the 3-month forward rates to which they are tied.

The difference between the FRA rate and the Eurodollar rate, known as convexity bias, is adjusted by the formula discussed in the article “Futures and forward convexity adjustment”.


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