A New Twist To Dollar Offset

By Louis Schleifer
Senior Product Manager, SunGard Treasury Systems

The dollar-offset method of assessing FAS 133 effectiveness is inarguably the simplest approach available. By all accounts, it is also the one most commonly offered by system vendors and, as anecdotal evidence suggests, most commonly used by corporate hedgers. However, the basic dollar offset method has one serious drawback, namely its sensitivity to small price changes. Lou Schleifer, of SunGard Treasury Systems, argues that this flaw should not necessarily force companies to turn to alternative, statistical methods. Rather, he has developed an algorithm that modifies the dollar-offset method so as to filter-out the noise associated with small price changes.

Introduction

The dollar offset is inarguably the most straightforward way to approach the assessment of retrospective and prospective effectiveness under FAS 133. But this simplicity does not come “free of charge.”

In particular, the original dollar offset approach reacts aggressively to small changes in prices, creating the potential for unwarranted noise and potential ineffectiveness. Apparently some companies and vendors are using this as a reason to abandon dollar offset entirely, in favor of statistical measures. But, it may be worthwhile to modify the dollar offset instead, and thereby retain some of its advantages (e.g., simplicity and its reliance on existing pricing data), while at the same time eliminating the possibility that immaterial price moves will trigger an ineffective result.

This sort of approach, when discussed and determined with senior management and the company’s auditors, could be used to illustrate the hedging company’s risk management approach, and could be codified into its assessment of effectiveness, in a manner consistent with FAS 133.

Simple, but…

The simplicity of the dollar offset method is immediately evident in its definition (Equation 1):

Dollar Offset = [Change in Fair Value of Hedge] / [Change in Fair Value of Hedged Item]

In addition to its simplicity, the dollar-offset method has some other key advantages, including the following:

It relies upon calculations—namely change in fair value calculations—that are already a required part of FAS133 Accounting.
It relies upon data—namely Fair Values of the Hedge and of the Hedged Item—that must already be captured for FAS 133 accounting; it does not rely upon externally supplied, historical data series, as many statistical methods do.
It is similar to a hedge ratio calculation., but one based on observed-market values instead of projected-future-market values (for retrospective effectiveness, that is),
It is sensitive to mismatches in size between the hedge and the hedged item, unlike most statistical methods.
It can be easily duplicated.

Unfortunately, though, there is one serious flaw with this algorithm: It exhibits unwarranted behavior when market rates stay relatively static, and therefore prices change very little over the period in question.

In this situation, intuition tells us that the price changes observed are financially immaterial—because of their relatively small size—and represent nothing more than statistical noise from an otherwise-perfect hedging relationship. As a result, one would certainly expect to have the effectiveness algorithm—whatever form it might take—return a result very close to 100 percent, as this represents a perfect value of retrospective/prospective effectiveness.

But this is not the case for dollar offset. As both the numerator and denominator in equation 1 (see above) approach zero, it is not too hard to see that their ratio can vary widely unless they coincidentally happen to approach zero in lockstep (a very unlikely occurrence indeed).

Rebuilding dollar offset from the ground up

It seems that the standard reaction to this state of affairs has been to abandon the dollar-offset approach altogether, and find a different calculation (e.g., regression-analysis, or another statistical method). But there is an alternative, with some mathematical care (and flair), it is possible to fix the dollar offset algorithm to ensure that it behaves properly all of the time—even when the observed price changes are miniscule.

To this end, consider the concept of allowing the hedger to mathematically quantify his/her definition of financial immateriality (or “noise”) via a noise-threshold parameter. In other words, hedgers could determine, mathematically, the level at which price changes constitute a material change in fair value. Once this is done, the new-and-improved dollar-offset algorithm (see below) can compare the actual changes in fair value to the hedger’s noise-threshold in order to see how relevant these price changes really are in computing FAS 133 retrospective/prospective effectiveness. Depending upon the result of this comparison, the new algorithm’s behavior can be split into one of three possible regimes:

· Regime #1—Small Changes: The observed price changes are small compared to the user-defined noise-threshold, so the new algorithm should return a result very close to 100 percent (i.e., perfect effectiveness).

· Regime #2—Transition Period: The observed price changes are on the order of the user-defined noise-threshold, so the new algorithm should give a result somewhere between what it would give for Regimes #1 and #3.

· Regime #3—Large Changes: The observed price changes are large compared to the user-defined noise-threshold, and so the new algorithm should return a result very close to the dollar-offset algorithm (see equation 1, above).

To better quantify the concepts just introduced, the following definitions are made:

DFV {Financial Instrument} = Change in Fair Value of the specified Financial Instrument over the accounting period in question.
NT_N = user-defined value of Noise Threshold (Normalized value), quoted in basis points. This can be any positive, integral value. This variable lets the user quantitatively define his level of financial materiality.
M_P = Magnitude of the Prices of the financial instruments being considered. We will define this in greater detail below.
NT_A = Noise Threshold (Absolute value), computed as follows (Equation 2):

NT_A = M_P * (NT_N / 10,000)

Defining M_P

As one can see from equation 2, above, M_P measures the size of the financial instruments included in a hedging relationship. As a first approximation, one would probably think to define this size as the notional amount of the hedges (or the hedged item). But this is too primitive since it misses the impact that instrument tenor/characteristics and market rates have on defining the fair value. So, as a better approximation to measuring size, one might rather think to take the fair value of the hedges.

Unfortunately, this alternative is even worse. To see why, consider that each leg of an at-market $10 million notional Interest-Rate Swap will probably have a fair value somewhere in the ballpark range of $1 million to $5 million, but these values perfectly offset each other to result in a net fair value of zero. This exact—or almost-exact—offsetting of two large fair values is typical of many derivatives, including IR Swaps, CCIR Swaps, and FX Forwards.

The ideal approximation to measuring “size” could well be the present-value of one leg of the derivative. However, consider the complications that this definition would entail should a hedging relationship chance to have many derivatives moving into and out of it over time: This measure of size would have to be computed on one leg of each one of these derivatives, and then be prorated for the time that the respective derivative actually resided in the hedging relationship.

The best compromise is to look to the hedged item. More specifically, consider using the present-value of only that portion of the hedged item’s cash flows that are being hedged with the derivative. This should serve as a good first-order approximation to the present-value of one leg of the hedging instrument, without too much computational aggravation. Here is how to do this:

For an IR Swap: M_P = Present-Value of the coupons only on the hedged item
For CCIR Swap: M_P = Present-Value of all cash flows on the hedged item
For FX Forward or FX Option: M_P = Present-Value of the foreign-currency leg only of the hedged item

Looking Good

With these definitions in place, the stage is now set for the “first take” on a new dollar offset algorithm. For purposes of comparison, the reader should recall the definition of the standard dollar offset algorithm, reiterated here with the new nomenclature introduced above (Equation 3):

Dollar Offset = DFV {Hedge} / DFV {Hedged-Item}

Contrast the above with the new, single-variable, dollar-offset algorithm, presented below (and developed by William Lipp [w.b.lipp@ieee.org], an independent consultant hired by SunGard):

Lipp Modulated Dollar Offset (Equation 4)=

[DFV{Hedge} + NT_A] / [DFV{Hedged-Item} + NT_A]

Technical Note: An implicit assumption inherent in equation (4) is that all three variables involved are nonnegative. We can assure this by taking the absolute value of each variable prior to invoking the equation. Of course, even before taking this step we must check to ensure that DFV{Hedge} and DFV{Hedged-Item} have opposite signs. If they don’t, then there’s no point to even calculating effectiveness, as we have added to our risk, not hedged it. In this case, we would simply return with an error condition (e.g., Effectiveness = 0).

Note that in the “Small Changes” regime, the ratio in equation approaches

NT_A/ NT_A = 1 = 100%

Moreover, in the “Large Changes” regime, the ratio approaches

DFV {Hedge} / DFV {Hedged-Item}

…which is nothing more than the standard dollar offset, as given in equation 3. [Equation 4 is currently supported in STS’s GTM DAS module (“GTM”=“Global Treasury Management”, and “DAS”=“Derivative Accounting System”).]

During the Transition Period, meanwhile, equation 4 exhibits a smooth transition between these two regimes. To get a better feel for the behavior of Lipp’s method, consider Figure 1: The graph therein illustrates the case of a hypothetical hedging relationship that satisfies the following criteria across a large range of possible changes in fair value (from miniscule to gargantuan):

DFV{Hedge}= 2 *DFV {Hedged-Item},

From the above, it is easy to see that:

Dollar Offset = 2 = 200%

In other words, the hypothetical hedger, in this case inexplicably used exactly twice as much hedging vehicle as was required to properly hedge the underlying risk. Although this would be an egregious mistake were it actually to occur, nevertheless, it represents an excellent test for Lipp’s method, since it poses this question: At what range of price changes should the algorithm first begin to notice this over-hedging?

Figure 1 plots three examples of equation 4, each with a distinct value of NT_N. Viewing these graphs from left-to-right, the corresponding values of NT_N are 10, 500, and 15,000.

The first important observation is that each of these curves makes a very smooth transition between the Small Changes regime (where effectiveness is close to 100 percent) and the Large Changes regime (where effectiveness is close to 200 percent).

But the second observation is that the user now has a “degree-of-freedom” to play with when defining this curve: The parameter NT_N allows the hedger to define at what level of price changes he wants the transition to take place. [We note that the X-Axis in Figure 1 gives a new measure of the size of the observed price changes; this measure is defined below in equation 5.]

And Looking Even Better

If one had to find fault with the Lipp method (equation 4), it might be the following: Although it does give the user control over defining the onset of the transition between the two regimes, it doesn’t give him any control over how fast the transition occurs. The solution is to introduce a second user-controlled variable into equation 4, as well as an additional variable that is required by the calculation:

· S_T: The speed of the transition, as defined by the user. Must be a decimal value that is strictly greater than -1.

· M_DP: The Magnitude of the observed Price Changes. This is defined as follows (Equation 5):

M_DP = [DFV {Hedge} ^ 2 + DFV {Hedged-Item} ^ 2 ] ^ (1/2)

Make sure not to confuse M_P with M_DP: The former is a measure of the size of the instruments contained in our hedging relationship, whereas the latter is a measure of the size of the price changes of these same instruments over a period of time. With these new variables in place, it is now posble to transform Lipp’s algorithm into a full-fledged, two-variable modification to the standard dollar offset algorithm. The result (a modification to Lipp’s algorithm discovered by the author) is:

Schleifer-Lipp Modulated Dollar Offset (Equation 6)=

[DFV{Hedge} * (M_DP/ NT_A) ^ S_T + NT_A ] /
[DFV{Hedged-Item} * (M_DP/ NT_A) ^ S_T + NT_A ]

The only qualification to be made on this result
is the following (Equation 7):

S_T > -1

Note: Equation 6 is currently supported in STS’s GTM DAS module.

Technical Note: The same proviso that was made on equation (4) is equally applicable here: namely, we must take the absolute value of the same three variables that appeared in equation (4), prior to invoking equation (6). But, once again, we wouldn’t even bother to use equation (6) if we first saw that DFV{Hedge} and DFV{Hedged-Item} had the same sign. In this case, we would simply return with an error condition (e.g., Effectiveness = 0).

If one sets S_T = 0 in equation 6, the algorithm obviously degenerates to equation (4), the first approach at modifying dollar offset. Given this relationship between the equations, it should come as no surprise that equation 6 also allows the user to “place” the transition via the parameter NT_N--just as was possible in equation 4—for any permissible value of S_T.

However, should the user choose to start trying non-zero values of S_T in equation 6, he will quickly see that doing so gives him a new, surprising measure of control over the shape of the transition period.

As a first observation, if one starts with a value of S_T = 0 and begins increasing it, the transition period will start to shrink—i.e., it will be compressed into continually smaller intervals along the X-axis. This is the same as saying that the Small Changes regime and the Large Changes regime begin to converge on each other. In fact, as S_T continues to increase without bound, the Transition period starts to disappear completely, as equation (6) mathematically approaches the discontinuous curve that satisfies these constraints:

If M_DP < NT_A, Then Effectiveness = 100%

If M_DP > NT_A, Then Effectiveness = Dollar Offset, as computed via equation 1

For the second observation, note that if one starts with a value of S_T = 0 and begins to decrease it towards the value of –1, the transition period will start to lengthen. This is equivalent to saying that the Small Changes and Large Changes regimes start receding from each other. But there is no value to actually setting S_T = -1, as doing so in equation 6 results in an expression that is independent of NT_A. This means the graph would be independent of the size of the price changes observed, and would therefore be perfectly flat!

Figure 2 aptly illustrates these effects by starting with a graph of equation 6 that has NT_N = 500 and S_T = 0. Next, it varies the values of S_T while keeping NT_N fixed. As S_T is successively increased to 0.6 and 4.0, the reader can see that the transition period becomes successively shorter/steeper. Then, as S_T is successively decreased to –0.55 and –0.75, it is clear that the transition period become successively longer/flatter. In essence, the parameter S_T represents a second “degree-of-freedom” that is available to the user in equation 6.

Mission Accomplished

The Schleifer-Lipp Modulated Dollar Offset represents the culmination of the present research effort to enhance the simple dollar-offset algorithm. Although it is not as simple as dollar offset, it should be considerably easier to understand than most statistical methods.

Moreover, as noted in the introduction, it relies upon the fair values and change-in-fair-values that must already be captured for performing FAS 133 earnings calculations. Perhaps best of all, though, is the ease with which one can duplicate equation 6 in a spreadsheet and test it out on real financial instruments, to get a feel for what it would predict in real-life situations.

With equation 6 and the two user-definable parameters—NT_N and S_T —a hedging corporation should be able to get just about any type of transition-behavior desired. Of course, just because one can physically set the noise threshold--NT_N--at astronomical levels and thereby guarantee perpetual effectiveness doesn't entitle anyone to actually get away with it!

Moreover, it is quite clear that doing so would never even be in the corporation's best interest, as it is tantamount to denying that any economic ineffectiveness exists--a serious impediment to dealing with such ineffectiveness when it actually occurs (and it will occur!).

The benefit provided by the algorithm is simply this: Instead of looking far-and-wide at the myriad approaches to measuring effectiveness—each with its advantages and disadvantages, and many of which are computationally intensive and mathematically esoteric—the hedging corporation should be able to convince both its management and auditors of the reasonableness and practicality of the approach discussed herein.

This means that the only remaining issue is to hammer out—with the approval of both management and auditors—the actual values of the parameters NT_N and S_T that will be used in equation 6. Once this is done, these values can be hardwired in the system. And at that point, the system will be relying upon an effectiveness algorithm that the hedger, his management, and his auditors can all understand and defend.