The ** Section 871(m) regulations** create a distinction between so-called “simple contracts” and “complex contracts”. A simple contract is a derivative that references a single, fixed number of shares of an issuer(s) in order to determine a payout. In addition, the contract must have a single maturity or exercise date on which all amounts (other than upfront or periodic payments) are required to be calculated; however, the fact that a contract has more than one expiry (or continuous expiry) does not preclude it from being a simple contact (preserving simple contract status of American-style options). Simple contracts are basically the derivatives that you are most likely familiar with–call and put options, forward purchase contracts, swaps, etc.

The necessity for the distinction between simple and complex contracts is due to Treasury’s use of a delta threshold of 0.80 in determining whether a derivative will be subject to the new withholding requirements. In short, the delta threshold requirement of 0.80 is not amenable to more complex derivatives. My understanding of the delta calculation is that it indicates how much the derivative price will change given the change in the value of the reference share. It is not a “per share” concept. Despite that, the delta threshold does work better for simple contracts since the amount of shares referenced doesn’t change. That is, delta per share can be easily calculated because there is only one choice for the denominator (i.e., a fixed number of reference shares). However, this becomes more difficult, for example, when the calculation of the economics of a derivative reference a different number of shares depending on the value of the underlying reference. Which shares would you use in such cases?

Because of these complications, the regulations apply a far more complicated (more like dizzying) test to a complex contract. The core of the test is to first calculate how much the value of the reference shares (they use “initial hedge) will move as compared to how much the value complex contract will move at different testing prices of the reference shares. These results are compared to the value movement produced by a simple contract that would have tripped the delta threshold. The upshot is determining whether the complex contract is “substantially equivalent” to a simple contract that would itself have been subject to the regulations.

Now, the first thing I’ll do is run through the test step-by-step using an example provided in the regulations and I’ll make some technical comments along the way. In Part 2, I will make some broader policy points about this test.

The guts of the test are contained in Temp. Reg. 1.871-15T(h)(4) and I’ll walk through the test using the example in the regulations. The example provides that, for a $10,000 initial investment, the Investor is entitled to a return equal to their initial investment (x) plus 200% of any increase in the value of 100 shares of stock capped at 110x, (y) minus 100% of any decrease in the value of 100 shares of stock below 90x. The Issuer hedges the issuance by acquiring 64 shares of the reference stock.

The first thing to note is that this is a complex contract because the factor of the payout over 100 shares varies depending on the price of the stock (e.g., 200% in the case of an increase up to 110x versus 100% in the case of a decrease below 90x). As indicated above, applying the simple delta threshold of 0.80 is difficult here because it is not clear what the “per share” delta could possibly be. Is it 200 shares or 100 shares, or something in-between? Therefore, the regulations require resort to the substantial equivalence test.

**Step 1: ** *(A) Determining the change in value (as described in paragraph (h)(4)(ii) of this section) of the complex contract with respect to the underlying security at each testing price (as described in paragraph (h)(4)(iii) of this section);*

Here, you must determine the change in the value of complex contract itself based on two different prices for the reference. That is, if your contract references, say IBM, and IBM increases in value by $10, how much does the contract value change.

The determination of two testing prices to be used requires some knowledge of statistics and particularly how it applies to financial concepts because the regulations use the concept of standard deviation to determine the testing prices.

Here we encounter our first technical issues with the temporary regulations. I preface this by stating I am no expert in statistics. I took a statistics course in college and have a passing familiarity with the concepts over the course of my career, but my knowledge is limited. I’ll try to describe this to my best ability but caveat that I may screw up the concepts a bit (which is going to be a later point as well!).

Standard deviation is a concept used in statistics to describe the dispersion of a data set. That is, if the average (or mean) for a given data set is, say 100, the standard deviation tries to tell you how closely bunched that data is to 100. For a “normal distribution” of numbers one standard deviation tells you that about 68% of all the data in the set are within that standard deviation. For example, if one standard deviation is +/- 5 where the average is 100, then one would expect about 68% of all data to be within the range of 95 to 105.

Therefore, I suspect that the regulations are simply trying to choose testing points that include a large portion of the possible outcomes (e.g., 68%) and exclude the less probable outcomes. Whether this is the correct testing point or not I have no idea. Moreover, I don’t really know if testing only two data points is sufficient, but I suspect in some cases it might be and in some cases it might not. How’s that for definitive?!?

In any event, while the question of data points used to conduct tests is beyond my reach (and likely beyond Treasury’s), I will confine myself to a couple of easier technical issues that relate to Treasury’s use of standard deviation. Recall that standard deviation tells how a set of data is dispersed about the “mean” of that data. However, other than the current price of the stock, Treasury does not give any indication of how to plot that data. Treasury presumably wants to determine the data set that includes all potential future prices of the stock but they have to include certain assumptions that the financial engineers will use to create the data set on which the calculation is based (at least implicitly).

For example, Treasury leaves out the time period over which that change in the price of stock is expected to occur. Obviously, smaller changes in the price of the stock would be expected to occur over shorter time frames. Consequently, one standard deviation will necessarily lead to a much smaller band of price movement and defining the time frame is very important for the calculation. I imagine Treasury will use the maturity date of the complex contract as the date but this can lead to complications where a single instrument has multiple dates that can be considered a maturity date.

In addition to this, the stock’s volatility is needed as another input in order to determine the probability of potential future prices of stock. Stocks with high volatility will obviously have a greater price variability over one standard deviation. However, the amount of dispersion may vary depending on whether the volatility used to measure the dispersion is implied volatility or historical volatility and, in the case of the latter, the time frame that is used to measure historical volatility . I imagine the intent is to use the implied volatility but would think that needs to be spelled out. Ambiguity in the regulations could lead to differing interpretations in order to take advantage of different results.

There may be additional inputs that need to be spelled out in the regulations and those that actually do these types of calculations are probably better suited to comment on those may be. While future comments may sharpen these regulations, it does indicate that Treasury may be venturing too far into difficult concepts that it may not have a firm grasp on which I discuss more in Part 2.

So, with that, back to the example (that didn’t take long!).

Fortunately (unfortunately?) for us, the example in the regulations does not actually calculate the testing prices. It simply stipulates that the two testing prices for the reference shares, conveniently sidestepping the issues raised above. In the example, the current stock price was $100, and a one standard deviation increase in the price of stock was $120 and a one standard deviation decrease was $79.

Based on these testing prices, the change in the value of the complex contract was +$2,000 for the $120 price (200% x 100 shares x $10 cap) and -$1,100 for the $79 price (100% x 100 shares x -$11).

**Step 2: ***(B) Determining the change in value of the initial hedge for the complex contract at each testing price;*

Because the Issuer hedged with 64 shares, the testing prices would result in a change in value of the initial hedge of $1,280 (64 shares x +$20 per share) and -$1,344 (64 shares x -$21 per share).

**Step 3: ***(C) Determining the absolute value of the difference between the change in value of the complex contract determined in paragraph (h)(4)(i)(A) of this section and the change in value of the initial hedge determined in paragraph (h)(4)(i)(B) of this section at each testing price;*

Basically subtract things and get rid of the “negatives”.

For the testing price of $120, the complex contract had change in value of $2,000 and the initial hedge had a change in value of $1,280. The absolute value of the difference is $720.

For the testing price of $79, the complex contract had change in value of $1,100 and the initial hedge had change in value of $1,344. The absolute value of the difference is $244.

**Step 4:** *(D) Determining the probability (as described in paragraph (h)(4)(iv) of this section) associated with each testing price;*

“Probability” is based on the percentage chance that the stock price will increase or decrease. Here, it appears that Treasury wants to “weight” the values determined in Step 4. In other words, if there is a 60% chance that the price will go up (and conversely a 40% that the price will go down), they want you to weight the results of the $120 testing price at 60% and the results of the $79 testing price at 40%.

I must admit that I’m not a finance expert but this seems a little goofy to me. If financial engineers could actually come up with a probability of a particular stock increasing or decreasing, I’d imagine one of two things. First, if it was accurate, someone’s going to make a lot of money. I sure as heck would like to know the probability that stock price will increase or decrease, wouldn’t you? Second, for everybody else in the world, I would think that the probability would revert to 50/50. The financial engineer who accurately came up with that percentage would keep buying up the stock until the price point is reached such that the next movement will be 50/50.

Of course, I admit that I could be wrong and naïve about this as perhaps those kinds of calculations are in fact done. If my instincts are correct, however, this is a needlessly complicated addition to the test.

In any event, similar to the prior steps of the test, the Treasury example simply stipulates the percentage chance of an increase in the value of the stock at 52% (and conversely a 48% chance of a decrease) and does not actually derive the probability.

**Step 5: ***(E) Multiplying the absolute value for each testing price determined in paragraph (h)(4)(i)(C) of this section by the corresponding probability for that testing price determined in paragraph (h)(4)(i)(D) of this section;*

Well, this one is fairly mechanical and easy to do.

Since the absolute value of the difference at the $120 testing price is $720, we multiply $720 by 52% and get $374.40. Since the absolute value of the difference at the $79 testing price is $244, we multiply $244 by 48% and get $117.12.

**Step 6: ***(F) Adding the product of each calculation determined in paragraph (h)(4)(i)(E) of this section; and*

Yay, another spreadsheet step that’s easy to accomplish. This is summing $374.40 and $117.12 to get $491.52.

**Step 7: ***(G) Dividing the sum determined in paragraph (h)(4)(i)(F) of this section by the initial hedge for the complex contract.*

Now we’re flying!!! Since the initial hedge used 64 shares, this is simply a matter of dividing $491.52 by 64 to get $7.68.

**Step 8: **Do everything in Steps 1-7 all over again applying it to a simple contract benchmark. This is a simple contract that has a delta of 0.80 and the same maturity as the complex contract. Apparently you can use a call or put option (or any other simple contract) depending on which one better matches the complex contract (whatever that means).

I won’t go through the entire calculation again with a simple contract. Suffice it to say that the regulations perform these calculations with a one-year call option that settles on the same date as the complex contract and has a delta of 0.8. The value derived with respect to the call option after going through steps 1-7 was $4.473.

**Step 9: ** Compare the values obtained for the complex contract ($7.68) and the simple contract ($4.473). If the value of the complex contract is larger than the value obtained for the simple contract, it is not “substantially equivalent”. If the value obtained for the complex contract is less than or equal to the value obtained for the simple contract, it is “substantially equivalent”.

Consequently, the conclusion in the example is that the complex contract is not substantially equivalent.

As to this final test, I can’t really tell you if it makes sense or not.

Very generally, there is some intuitive appeal as it does appear to be testing how close the result of the complex contract will be to the result obtained from the simple contract benchmark. But beyond intuition, I have no clue. And this makes me wonder how many people at Treasury really know anything beyond an intuitive level as well. I’ve met a lot of tax lawyers and the Venn diagram that represents the intersection of people who are tax lawyers and those that are statistics/financial experts is fairly small.

Anyway, in my next post on complex contracts, I’ll talk about some additional complexities that result from this concept; in particular, complexities that result from the concept of combined contracts. Moreover, I’ll talk about some policy issues and whether tax law should really be going into these sorts of complexities that are beyond any a layman’s ability to grasp.

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