# Clearing, the Evolution of a Price, and the Impact of Discounting

When presenting the history behind the choice of OIS versus LIBOR discounting, we have shown that, because of the small spread between the two rates, the impact was very small. We shall now discuss whether, in general, discounting has a large impact on a swap price.

The life of a swap can be very long. This statement of the obvious is to stress that we need to view the evolution of the price of a swap and basically observe two situations, when the swap price is close to par and when it is not. (Borrowing the terminology of options, we can call a par swap an at-the-money [ATM] swap, and otherwise, when it is not close to par, an out-of-the-money [OTM] swap.)

In Table 2.1 we see the cash flows of two swaps, an at the money and an out of the money. The two swaps are identical in characteristics in that they both pay a fix rate annually and a floating rate semiannually. We assume that we have built our curve using a certain amount of market information from which we have obtained the discount factors given in the first column and the forward rates which are given in the, identical, third and fifth columns. We can see that the difference between the two swaps lies in the fixed rate: the ATM swap pays 3.2668% and the OTM swap pays 4.5%.

In finance as in science it is very important, when observing an effect, to be able to say whether it is large or small. An effective way of doing this is to make it dimensionless, to scale it by an accepted quantity. In finance we do so by comparing any impact to the bid-offer spread of the relevant market data. To test the impact of a shift in discount factor we are going to take four steps. First, we are going to shift the discount factor by a certain amount; for simplicity this is going to be an artificial shift but it could realistically

TABLE 2.1 Cash flows in a par swap (ATM) and in an out-of-the-money swap (OTM).

 DT Zero rate equiv. (%) ATM swap: fix CF (%) ATM swap: floating CF (%) OTM swap: fix CF (%) OTM swap: floating CF (%) 0.9852 2.9780 3.0000 3.0000 0.9706 3.0249 3.2268 3.0500 4.5000 3.0500 0.9558 3.0734 3.1000 3.1000 0.9412 3.1221 3.2268 3.1500 4.5000 3.1500 0.9264 3.1727 3.2000 3.2000 0.9118 3.2253 3.2268 3.2500 4.5000 3.2500 0.8970 3.2762 3.3000 3.3000 0.8826 3.3251 3.2268 3.3200 4.5000 3.3200 0.8681 3.3718 3.3300 3.3300 0.8540 3.4170 3.2268 3.3500 4.5000 3.3500 MTM (in %) 0.0 -5.80

correspond to, for example, a choice of instrument used to build the discount curve. Second, we are going to translate the discount factor into a zero rate. A zero rate has the same definition of cash deposit with the difference that it is not a traded instrument. We are doing this purely for illustration purposes. Almost everyone has more familiarity with the concept of rate expressed in percentages rather than as discount factor. Third, we are going to recalculate the swap price with the new discount factors and show the impact in the MTM. Finally, in order to gauge whether the impact is a large one, we are going to calculate the equivalent impact in the swap rate, that is, that shift in the swap rate that would have given the same impact in MTM. (By a very rough rule of thumb, this is given by the MTM impact divided by the length in years of the swap.)

The first and second steps are shown in Table 2.2. For simplicity the shifts (–0.02 and –0.03) are constant throughout the discount factor's curve, meaning that they will have a greater impact on a short dated discount factor than a longer dated one. This can be seen by the fact that the difference between the zero rates shown in the second and fourth columns of Table 2.2 and the second column of Table 2.1 is very large for the first cash flow and it tapers out.

In Table 2.3 we see that the impact of the first shift on the MTM is 0.20 bps for the ATM swap and 12.93 bps for the OTM swap. The ratio of 60 times greater is reflected in the equivalent swap rate impact, which is 0.05 bps for the ATM swap and 2.90 bps for the OTM one. Not surprisingly, for the second shift the impact is even greater with an MTM impact of 0.30 bps for the ATM swap and a 19.40 impact for the OTM swap. The impact on the swap rate is 0.30 bps for the ATM swap and 4.40 bps

TABLE 2.2 Shifts in discount factors and zero rate equivalent.

 DT 1st shift Zero rate equiv. (%) DT 2nd shift Zero rate equiv. (%) 0.9652 7.1501 0.9552 9.3017 0.9506 5.1896 0.9406 6.3064 0.9358 4.5587 0.9258 5.3255 0.9212 4.2747 0.9112 4.8700 0.9064 4.1234 0.8964 4.6147 0.8918 4.0430 0.8818 4.4667 0.8770 4.0015 0.8670 4.3766 0.8626 3.9816 0.8526 4.3215 0.8481 3.9747 0.8381 4.2869 0.8340 3.9783 0.8240 4.2692

TABLE 2.3 Impact of shifts in discount factors on MTM of ATM and OTM swaps.

 Impact on MTM of ATM swap (bps) Impact on swap rate of ATM swap (bps) Impact on MTM of OTM swap (bps) Impact on swap rate of OTM swap (bps) 1st shift 0.20 0.05 12.93 2.90 2nd shift 0.30 0.07 19.40 4.40

for the OTM swap. To put the impact into context, the bid-offer spread on USD interest rate swaps is 3 bps. This shows how, at inception when a swap is at par,[1] the impact of discounting matters very little. It is when a swap is already in the middle of its life (exemplified here by a fixed rate very different from the par swap rate) that two discount factors have very different effects on the swap price. This is not difficult to understand: the forward rates shown in the fourth column of Table 2.1 are fairly close to each other (we call this a rather flat swap curve), the par swap rate paid as fix rate in the third column of the same table is also a number roughly of the same magnitude. This means that on both sides the numbers we are going to multiply the discount factors by are very close, reducing the impact of the discount factors themselves. The same cannot be said of an OTM swap.[2]

Let us now remind ourselves why two parties need to agree on the MTM of a swap. This is not for actual settlement reasons: payments are linked to the rate itself, which sets officially and about which there is no disagreement. It is for collateral purposes: according to the value of the MTM, the party that sees it as negative needs to post collateral to the other party. Different discounting choices lead to collateral disputes when two parties disagree on the MTM of the swaps. Collateral disputes are frequent and tiresome processes rendered even less clear by the fact that, because of the netting of positions we mentioned in Section 2.4.1, a party will argue with different degrees of forcefulness according to whether certain transactions will result in a net lower or higher MTM (the ideal situation being one of lower MTM, hence, less collateral to be posted).

After the financial crisis there is a general wish to move simple swaps (the one we have seen would fall in that category) to exchanges (for an interesting description of the time line of the issue and how exchanges try to deal with it, see Whittall [84]). Exchange-traded derivatives, such as interest rate futures, are not only fully standardized contracts, they are also contracts whose price is published by an exchange, that is, there is an official price. Crucially this price is not only official on the day of entering the contract (this would not differ very much from what takes place already with a broker publishing such a price), but also throughout the life of the trade. This process is known as clearing where an exchange (acting as a clearinghouse) becomes a third party and arbiter to a transaction between two market participants. Essentially the two market participants would not be facing each other but instead would each have a position with the clearinghouse. The exchange- decides on an indisputable MTM and as a consequence it decides on the collateral that needs to be posted. This collateral would be in the form of margin calls, similar to other exchange-traded contracts such as futures. We immediately see that in this situation there is no more argument as to what is the correct choice of instruments to build a discount curve on, the hopefully intelligent choice made by the exchange becomes the correct choice. It is up to each of the two financial institutions facing the exchange in this swap triangle to value it in a manner similar to the exchange, in which case it would avoid seeing internal surprises, or value it differently and accept the fact that the value obtained internally could be dramatically different from the official one.

The move, challenging as it may be for all the parties involved (clearinghouses included), is very important. Since we have seen how curve construction is a challenging numerical process that may elude some of the less sophisticated firms, there was always the risk that some institutions could have built large positions marked incorrectly, which, at the moment of reckoning, could have led to potential catastrophes.

• [1] Of course one can trade swaps that start out of the money, but it is a more unusual situation, and here, would fall in the same category as an already existing swap.
• [2] There is also the less common situation of a swap built on a set of forward rates increasing not as smoothly as in Table 2.1. That would be the case of a steepening swap curve and the impact falls in between the two cases presented here.