Free Reply to Response to Motion - District Court of Federal Claims - federal

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IN THE UNITED STATES COURT OF FEDERAL CLAIMS No. 05-231 T (Chief Judge Damich) JZ Buckingham Investments, LLC as Tax Matter Partner of JBJZ Partners, a South Carolina general partnership, Plaintiff, v. United States of America, Defendant. _________

REBUTTAL REPORT OF A. LAWRENCE KOLBE ON EXPERT REPORTS OF DON M. CHANCE AND ALAN C. HESS Prepared for the U.S. Department of Justice

The Brattle Group 44 Brattle Street Cambridge, MA 02138-3736 617.864.7900 voice 617.864.1576 fax July 2, 2007

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Table of Contents INTRODUCTION AND SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Purpose of Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Organization of Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 OPTION VALUATION METHODS AND RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Long Option Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problems with Chance and Hess Reports' Option Methodologies . . . . . . . . . . . . . . . . . . 4 Reality Check on Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Forward Rate/Interest Rate No-Arbitrage Conditions . . . . . . . . . . . . . . . . . . . . . . 6 Hess Report Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHANCE REPORT'S CONSIDERATION OF WHETHER A REASONABLE PROBABILITY OF A PAYOFF EXISTED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 CHANCE REPORT'S DIVERSIFICATION COMMENTS AND CALCULATIONS . . . . . . . . . . . . . . . . . . . 12

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I.

INTRODUCTION AND SUMMARY

My name is A. Lawrence Kolbe. My business address is The Brattle Group, 44 Brattle Street, Cambridge, Massachusetts 02138. My Expert Report ("Kolbe Report") in this matter was filed on June 1, 2007. That report describes my professional qualifications and discloses various matters, including the information I have considered (in Appendix B to that report). Appendix B-R to this report updates that disclosure. A. Purpose of Report

I have been asked by counsel for the U.S. Department of Justice ("DOJ") to review the Expert Report of Don M. Chance, Ph.D., CFA ("Chance Report") and the Expert Report of Alan C. Hess, Ph.D. ("Hess Report") in this matter, both filed contemporaneously with my own, and to comment on the economic analyses therein as warranted. This Rebuttal Report does so.1 B. Summary of Findings

This report addresses (1) aspects of the option valuation methods and results in the Chance and Hess Reports, (2) the Chance Report's conclusions on whether a reasonable possibility of profit existed for these options, and (3) the Chance Report's comments on diversification. My findings in these areas are as follows.2 Option Valuation Methods and Results: The Chance and Hess Reports both calculate values for the short options (i.e., those sold by the participants to Deutsche Bank, AG -- "DB") as of the date they were contributed to the JBJZ Partnership.3 The same techniques can be applied to the long options. When this is done, the results confirm my finding that the long options' stated premiums -the "desired loss" in the COBRA Presentation -- materially overstate the actual values of the long options. The methods used by the Chance and Hess Reports contain various infirmities. This is immediately apparent from the fact that some of their results imply that DB voluntarily priced the options in a way that produced a negative fee to DB, i.e., that were irrationally generous to the participants. In point of fact, DB priced the options in a way that produced material fees for itself.

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I would note for completeness that my failure to discuss here a particular point made in either the Chance Report or the Hess Report does not necessarily indicate my agreement with that point. The statements in this section are summaries of detailed discussions in the body of this report. It is prepared only for the convenience of the reader, and this section cannot and is not intended to replace the subsequent, more detailed discussions. For this reason, this summary generally does not contain references to documents or data sources. Footnotes that provide the sources of the information on which I rely appear in the body of the report. I use the same general set of abbreviations here as in my original report. Thus, in addition to "DB," the "Partnership" is JBJZ Partners, and the "COBRA Presentation" is 2003EY001565-83, undated ("longer version") and DOJ 049106-21, dated 11/11/1999 ("shorter version").
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An overriding infirmity is the reports' failure to address whether a bid-ask spread might eliminate the possibility of the long option's paying off when the short option does not. Another example is that both reports fail to consider the forward foreign exchange rate in some of their calculations, which violates no-arbitrage conditions that are routinely satisfied in actual financial markets (since there is so much money to be made when arbitrage opportunities exist). Additionally, the Hess Report's simulation method assumes that (except for one "detrending" adjustment) the processes generating changes in exchange rates were stable over more than a quarter-century and were independent of the level of exchange rates. This is unlikely on its face, and it is not supported by the actual foreign exchange rate change data. These and other infirmities can, and sometimes do, have a material effect on the Chance and Hess Reports' results. Chance Report's Conclusion on Reasonable Probability of Profit: Profit is revenue minus cost. The Chance Report's methodology for addressing whether there was a reasonable probability of profit ignores the stated or actual cost of these particular options. It therefore cannot even determine if the expected profit is positive, let alone "reasonable." The probabilities of exchange-traded options the Chance Report examines are reflected in the market prices of those options, but the COBRA options did not trade at market value. The Chance and Hess Reports' option valuation methodologies themselves confirm this. The failure to address this fact in any way in its profit reasonableness methodology renders the Chance Report's conclusions on this topic meaningless. Moreover, the Chance Report ignores the fees to Ernst & Young and the attorneys. Even if it had not made the fundamental error just described, its conclusions would have no relevance to the issue of profitability after fees. Chance Report's Conclusion on the Value of Diversification: The Chance Report ignores the fact that the expected returns on these options were negative, and were very materially negative after fees. It therefore does not recognize that the best chance for a positive profit from these "investments" was to diversify as little as possible. This is simply an application of the widespread recognition that the casino always wins if you gamble long enough. C. Organization of Report

Section II addresses the option valuation methods and results in the Chance and Hess Reports. Section III considers the Chance Report's conclusions on whether a reasonable possibility of profit existed for these options. Section IV discusses the Chance Report's comments on diversification. This report is supported by two appendices. Appendix B-R updates the list of materials I have considered. Appendix E-R contains workpapers that present various calculations related to the Chance and Hess Reports. (For completeness, Appendix E-R shows the results of such calculations for all three of the COBRA cases in which both Drs. Chance and Hess and I have filed reports.)

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II.

OPTION VALUATION METHODS AND RESULTS

This section addresses first the values of the long options according to the methods used in the Chance and Hess Reports. It then addresses certain problems with their methods and results. A. Long Option Values

The Chance and Hess Reports both calculate values for the short options on the date of their contribution to the Partnership, evidently at the request of counsel to the Plaintiffs. However, they do not calculate the values of the long options. The result is necessarily an incomplete picture. The Chance and Hess Reports' methods and inputs can be used to calculate these values, however, which I have done for both the options' creation date and their contribution date. In particular, Figure R1 compares the stated value of the JBJZ long options at the time they were written to the high and low valuations implied by the Chance and Hess Reports' methods.4 (This is comparable to Figure 5 in the Kolbe Report, which compares the stated long option premiums with DB's and my own valuations of the long options at the time they were written.) The basic message of Figure R1 is that the Chance and Hess Reports' valuation methods produce estimates of the values of the long options that also are well below the DB stated premiums for these options. This is consistent with my own findings. That is, as discussed in Sections III.B and III.C.1 in my own report, the long option values are well below the stated long option premiums.5

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See Appendix E-R, Workpapers E-R-5A to E-R-5C and Workpaper E-R-8. Equivalent valuations as of the date of contribution are in Appendix E-R, Workpapers E-R-2A to E-R-2C and Workpaper E-R-8. These workpapers report the long option values for the other cases we have in common as well. The workpapers also value the short options using the Chance and Hess Reports' approaches, but with the long option's strike price. (See the Kolbe Report, Section III.C.1.) Also, the next part of this section discusses certain problems with the valuation methods used in the Chance and Hess Reports. Figure R1 takes the Chance and Hess Reports' methods as stated. Moreover, the net values of the long minus short options are materially below the actual net premiums charged. This is how DB received a fee, and DB's fees were economically material. See Section III.E of the Kolbe Report.

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Figure R1

B.

Problems with Chance and Hess Reports' Option Methodologies

This section first addresses the need for a "reality check" in the Chance and Hess Reports' option valuations. It then considers certain problems with their methodologies. A major problem that I do not address subsequently is the Chance and Hess Reports' complete acceptance of the possibility of a long option's paying off when the short option does not. The two-pip-wide strike price difference is so narrow that this issue would seem to be something that any recognition of a bid-ask spread should flag automatically as something to consider, even if Drs. Chance and Hess did not have access to the parts of the record my original report discusses in its Section III.C.1. (Recall that that section explained the reasons that the single-option payoff was not a possible outcome.) However, the Chance and Hess Reports do not address the issue.

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1.

Reality Check on Results

As discussed in my expert report, the COBRA options have an odd payoff pattern.6 The long option is priced as though its payoff were a 2 to 1, "coin toss" probability.7 The short option is priced to offer even better odds.8 Yet the net position has worse odds, a 2½ to 1 payoff. As also noted in my report, this implies that DB had to structure the actual options to offer no better than 2½ to 1 odds on the component options, too, if it wished to avoid a loss on the transaction. It had to offer the participants worse odds still if it wanted to get a fee.9 Therefore, any option payoff probability that exceeds 40 percent (which is the breakeven level for DB with a 2½ to one payoff)10 implies that DB is pricing the options in a way that does not let it expect to break even on average, let alone to earn a fee. It would be economically irrational for DB to price the COBRA options in such a fashion. Yet the Hess Report states the payoff probabilities for the Japanese Yen to U.S. dollar ("JPY/USD") options in the present case are over 40 percent.11 Also, while the Chance Report does not contain probabilities over 40 percent in the present case, Dr.

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Kolbe Report, Sections III.B and III.E. This statement ignores the fact that the payoff is not immediate, which makes the payoff slightly less valuable. I do not repeat this caveat, but it applies generally to this discussion. It does not affect the issues I address here. That is, for the 1999 COBRA parameters, $93.75 million to $47.5 million, in the COBRA Presentation example, or 1.97 to one (which implies better than a 50-50 chance of winning). COBRA Presentation, 2003EY001579-80, 82-83 (longer version) and DOJ 0491117-18, 20-21 (shorter version). Kolbe Report, Section III.E. That section found that DB appeared to price the COBRA options in a way that produced an average payoff probability in the 28 to 29 percent range over the entire set of 1999 COBRA transactions. This produced an average fee for DB on the order of 1.5 percent of the stated long option premium -- the COBRA Presentation's "desired loss". ( COBRA Presentation, 2003EY001572, longer version.) When combined with fees of 1.5 percent of the stated long option premium to Ernst & Young and 3 percent to attorneys, the total fees on the 1999 COBRA transactions averaged about 6 percent of the "desired loss", a figure also cited in an Ernst & Young email (a document with two Bates numbers, 2003EY020377 and PSI-EY020472). Additionally, handwritten notes that I understand were taken during discussions to develop the COBRA product (then apparently designated as "BEST" -- JGNC087849-58) discuss a 1 percent fee to Deutsche Bank in a context in which six percent is said to be the "max. to be competitive", and the listed fees sum to 5.5 percent when Deutsche Bank was shown at 1 percent. (JG-NC087882-83.) For example, if $1.00 is bet for a $2.50 payoff, the expected gross payoff with a 40 percent win probability is $2.50× 0.40 = $1.00. The expected net payoff = the $1.00 in expected winnings ! the $1.00 bet = $0.00, which is breakeven. Higher odds than 40 percent would imply a positive expected return for the bettor, and therefore a negative expected return for the counterparty (DB, in this case). Hess Report, Table 1, p. 19.

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Chance's report in another case (in which he relies on a similar methodology) does contain probabilities that are well over 40 percent.12 Accordingly, something in the methodologies used in the Chance and Hess Reports is producing results that fail a basic reality check, i.e., that imply DB would price the options in a way that was irrationally generous to the participants, resulting in an expected loss for DB. For reasons discussed in my own report, there is ample evidence that this did not happen. 2. Forward Rate/Interest Rate No-Arbitrage Conditions

The Hess Report at p. 31 states that: I do not include the U.S. and foreign interest rates in the computations for two reasons. First, the time intervals between the trade date and termination date are sufficiently short that compounding is numerically unimportant. Second, there are no data that accurately represent the risk free rates that appear in the theory. Using available data, in my opinion, introduces noise into the calculations, which makes the calculations imprecise with no offsetting benefit. Interest rates have two uses in valuing digital cash-or-nothing foreign exchange rate options. One is to calculate the present value of the payoff, and the one-month duration of these options does make that use of relatively little import. On the other hand, when formally valuing the options, why omit a known adjustment factor? However, the other use is to forecast the current estimate of the exchange rate on the termination date, which is an input to the probability-of-payoff calculation. That is, to avoid arbitrage opportunities, investors must be indifferent between two strategies: · Invest a quantity of dollars at the U.S. risk-free rate for the specified period, which produces at the end of the period an amount of dollars that is known with certainty today, and Follow a two-step strategy: (a) use the initial amount of dollars instead to buy foreign currency and invest that currency for the specified period at the risk-free foreign currency
See the EXPERT REPORT (McCombs), Don M. Chance, Ph.D., CFA, in Gary Woods, as Tax Matters Partner of Tesoro Drive Partners, a Texas general partnership, Plaintiff, v. United States of America, Defendant, (Civil No. SA-05-CA-0216-XR) consolidated with Gary Woods, as Tax Matters Partner of SA Tesoro Investment Partners, a Texas general partnership, Plaintiff, v. United States of America, Defendant (Civil No. SA-05-CA-0217-XR) in the United States District Court for the Western District of Texas, San Antonio Division, at Table 5, p. 33. The specific problem appears to be that the Chance Report uses noon exchange rates for the Swiss franc, while DB used rates that were closer to the 10am rates, and there was a material change in the level of the exchange rates between 10am and noon on the day these options were written. Appendix E-R, Workpapers E-R-3A, E-R-7A, and the DB Deal Sheets (DB COBRA 04211-28, 04133-66; and DB-DOJ-COBRA 00240679 - 799). Note also that whether or not Drs. Chance and Hess were familiar with the COBRA Presentation, the COBRA payoff ratios for the long and short options and for the net position are implied by the terms of the actual options in the present case.

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rate, to produce at the end of the period an amount of foreign currency that is known with certainty today, and (b) sell, today, that end-of-period known amount of foreign currency in exchange for dollars at the end of the period, at today's forward rate between dollars and that currency (which also produces at the end of the period an amount of dollars that is known with certainty today). If both these strategies do not yield the same amount of dollars at the end of the period, an arbitrage opportunity -- a way to make money using no net wealth -- exists. Financial markets are replete with people (and computer programs, in the modern world) that look to exploit such opportunities, and by doing so, to eliminate them.13 There are therefore two ways to specify the foreign exchange rate as of the expiration date of an option: (a) adjust today's exchange rate for the difference between the foreign and dollar interest rates, and (b) use the current value of forward exchange rate for the expiration date. Doing neither is simply a methodological error. Nor does the absence of an ideal measure of the interest rate justify the exclusion. First, use of no interest rates will virtually always lead to a bigger mistake than use of imperfect interest rates. Second, market participants solve this problem routinely (else the arbitrage opportunity elimination process would not work). DB, for example, used the difference between the current spot rate and the forward rate (the "forward points to expiry" field) in its ODETS system.14 Forward rates are widely available. Alternatively, one could use current London Inter-Bank Offer Rates ("LIBORs") for the various currencies, which are close to risk-free and also are widely available, for a variety of maturities.15 The Chance Report recognizes this no-arbitrage condition in its Technical Appendix A, pp. 48-49, and it appears to make the necessary adjustments when valuing the short options as of the contribution date (see pp. 39-40). However, it also appears to reject the point at pp. 30-31, when calculating the profit probabilities for the options in this case. I am unable to explain this apparent inconsistency, but it (and other details of the calculations) can make a material difference.16

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Of course, the opposite transaction is possible, too. If you want to start and end with foreign currency, you can either (a) invest an amount of that currency at its own risk-free rate, or (b) follow the equivalent two-step strategy, i.e., (i) buy dollars with the foreign currency and invest at the dollar risk-free rate, and (ii) lock in the final amount of foreign currency by selling today, at today's forward rate, that amount of dollars in exchange for the foreign currency, for delivery at the end of the period. If both approaches do not give you the same quantity of foreign currency at the end of the period, a great deal of money can be made by the party or parties fast enough to be first to exploit the opportunity. DB COBRA 04214. See http://www.bba.org.uk/bba/jsp/polopoly.jsp?d=141. It seems quite odd that the report would accept the point in one area and reject it in another, but that interpretation is the only way I have been able to replicate the values reported in the Chance Report's Tables 5 and 12. (See Appendix E-R, Workpapers E-R-1A to E-R-1C and Workpapers E-R-3A to E-R(continued...)

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In particular, consider the Chance Report's Swiss franc probabilities in the Tesoro case, the ones that flunk the above reality check because they materially exceed 40 percent (and thus imply DB was being irrationally generous to the participants when writing the net option position). Row [1] of Table R1 reports those probabilities for the three Chance Report volatility measures. Row [2] shows what those probabilities would have been had the Chance Report considered the forward points in its calculations. All three probabilities drop below 40 percent, although not by enough to give DB a material fee. Table R1. Corrections for Two Problems with Chance Report Probability Methodology

Chance Volatility Measure 30 Day 60 Day 90 Day [1] Chance Reported Trade Date Probability for CHF/USD Options [2] Chance Probability Adjusted to Reflect Forward Points [3] Chance Probability Adjusted to Include Forward Points & Fed 10AM Spot Rates [4] Total Impact 43.94% 38.81% 29.32% 14.62% 44.30% 39.52% 30.63% 13.67% 44.34% 39.61% 30.79% 13.55%

Sources and Notes: [1]: Values from Chance Tesoro expert report dated May 8, 2007. [2]: 30 Day value from Appendix E-R, Workpaper E-R-5A; 60 Day value from Appendix E-R, Workpaper E-R-5B; 90 Day value from Appendix E-R, Workpaper E-R-5C. [3]: 30 Day value from Appendix E-R, Workpaper E-R-7A; 60 Day value from Appendix E-R, Workpaper E-R-7B; 90 Day value from Appendix E-R, Workpaper E-R-7C. [4]: Calculated as [1] - [3].

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(...continued) 3C; these calculations again show the results for all of the common cases Dr. Chance and I analyze.) In particular, I can make no sense in the present context of the argument on p. 30 that use of the interest rate adjustment somehow assumes that one currency will strengthen relative to another indefinitely. These are not infinite-lived options, they only last one month, which is all the time the interest rate adjustment needs to consider. The interest rate adjustment is simply a way to recognize the current forward price for that date.

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To further explore this issue, note that the Chance Report uses noon spot rates in its probability calculations,17 and the CHF/USD spot price changed noticeably between 10 a.m. and noon on the trade date in question.18 Row [3] shows that the Chance Report's payoff probabilities drop by an even larger amount if the 10 a.m. spot price instead of the noon spot price is used.19 After this additional adjustment, the probabilities are in fact sufficiently low to imply a material fee for DB.20 3. Hess Report Simulation Method

The Hess Report's simulation method for valuing the options relies on individual daily changes in exchange rate data stretching back to the early 1970s, if available.21 This implicitly assumes that the process generating daily exchange rate movements has been unchanged over that period, except as accounted for in the Hess Report's "detrending" adjustment. On its face, this seems quite unlikely. For example, suppose that (a) the process generating relative (percentage) day-to-day changes were in fact stable, but (b) the average exchange rate were different in different historical periods. In this case, the same average percentage change in an exchange rate would produce larger or smaller average absolute changes in different periods. That is, a 1 percentage point change in a JPY/USD rate of 300 is 3 JPY/USD; a 1 percentage point change in a JPY/USD rate of 100 is 1 JPY/USD. The Hess Report's process would treat the 3 JPY/USD change as exactly as likely at the 1 JPY/USD change, even if the exchange rate at the time the options were written were 100 JPY/USD exactly. This potentially matters because the actual exchange rates on which the Hess Report relies do vary materially over the historical period. As it turns out, neither the absolute nor the relative average changes are constant in the actual historical data.22 Since the Hess Report's simulation method is
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The Chance Report takes its values from the Federal reserve statistical release at http://federalreserve.gov/releases/h10/Hist/default1999.htm. This database contains only Noon buy rates in New York. The Federal Reserve Bank of New York publishes both 10AM and Noon rates. http://www.ny.frb.org/markets/fxrates/historical/home.cfm. See

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The effects of these two corrections on all the other Chance Report probabilities are contained in Appendix E-R, Workpaper E-R-9. DB itself used spot prices more consistent with the 10 a.m. series on the day it wrote the options. See Appendix E-R, Workpapers E-R-3A to E-R-3C, Workpapers E-R-7A to E-R-7C, and the DB Deal Sheets (DB COBRA 04211-28, 04133-66; and DB-DOJ-COBRA 00240679 - 799). Please note that the problems with the Chance and Hess Reports' methodologies addressed in Table 1 do not always lead to lower probabilities, as they do in that case. See Appendix E-R, Workpaper E-R-9. However, they are problems regardless of the direction of their impact. Hess Report, p. 17. See Appendix E-R, Workpaper E-R-11. The Hess Report also appears always to start its simulations using the exchange rate from the day before the trade date (as opposed to the day of the trade date as his (continued...)

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just that, the result of a particular set of simulations, it is impossible to be sure exactly what the results would have been if it had used relative instead of absolute changes. However, simulations that otherwise replicate the Hess Report's methods suggest the differences were likely to have been material in some cases, and that the estimated probabilities with the Hess Report's absolute changes would tend to be higher than those with relative changes.23 4. Conclusion

The methodologies used by the Chance and Hess Reports contain a number of infirmities, which is apparent from the fact that some of their results imply DB was writing options that would have expected positive payoffs to the participants and expected negative payoffs to DB (i.e., that would amount to DB's paying rather than receiving a fee). These infirmities can, and in some cases do, affect the Chance and Hess Reports' estimated probabilities of success and their option valuations materially.

III.

CHANCE REPORT'S CONSIDERATION OF WHETHER A REASONABLE PROBABILITY OF A PAYOFF EXISTED

The Chance Report states at p. 3 that one of its purposes is to discover "whether the foreign exchange digital option spread strategies include a reasonable probability of profit". It purports to do that, but "profit" is simply revenue minus cost. The Chance Report's methodology for addressing this question completely ignores the cost of the options, and therefore cannot address even what the expected profit was, let alone whether it was "reasonable." Specifically, the Chance Report describes its consideration of whether the payoff probability was reasonable as follows: I found that the transactions did have a reasonable probability of profit. I drew this conclusion on the basis of a comparison of the probability of earning a profit on the transactions in this case to the probability of earning a profit on standard option transactions that are conducted on U. S. options exchanges. For example, I estimated the probabilities of profit of a large number of options on Google and the S&P 500 and found those probabilities to be comparable if not much lower than those of the transactions in this case.24 ...

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(...continued) report asserts), which also can make a material difference in the results in some cases. Additionally, when valuing the options on the contribution date, it uses one too many days in its simulations. See Appendix E-R, Workpaper E-R-10. See Appendix E-R, Workpaper E-R-10. Chance Report, pp. 3-4.

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The results [of analyses of the payoff probabilities on exchange-traded options on Google and the S&P 500 stock index] are shown in Table 6. Note how the probabilities of profit are inversely related to the exercise price. All probabilities are less than 50%. Note that the probabilities for the three deepest out-of-the-money options for Google are less than 3%. For the three deepest out-of-the-money S&P 500 options, the probabilities are less than 6% and yet over 100,000 of those contracts traded. These comparative results using the options of two widely known underlying stocks, Google and the S&P 500, show that the probabilities of the digital currency options in this case are not at all out of line with commonly traded options and what option investors expect when buying and selling options.25 This approach uses an economically unreasonable definition of "reasonable probability of profit." Recall Section III.A of my original report, on risk-return principles, and particularly Equations [1] and [2], which I reproduce here for convenience: E[$Return] = [(Probability of Payoff) × ($Payoff)] !$Cost of Option [1]

where E[...] indicates the "expected value" (in the statistical sense) of the quantity in the brackets.26 The expected rate of return is the expected dollar return divided by the cost of the option: E[%Return] = E[$Return] / $Cost of Option [2]

The probability of payoff is a key input to both the dollar and the percentage return, but so too is the dollar cost of the option. "Profit", the quantity the Chance Report purports to analyze, is just the dollar return. One cannot even decide if the profit is positive, let alone "reasonable," without consideration of the cost of the investment. The Chance Report simply ignores the second half of the equation. Put differently, the Chance Report's Table 6 describes a wide range of probabilities, but it does not report the market prices of the options that underlie those probabilities. The market prices of the lower-probability options will be lower, all else equal, so that Equation [1] applied to those options would still yield a positive value for the expected dollar return.27 Similarly, the market prices of the higher-probability options will be higher, so that the expected dollar and percentage returns offer
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Chance Report, p. 34. For completeness, I will note that Equation [1] is only exact for cash-or-nothing digital options. For other options, you would need to replace "$Payoff" with "E[$Payoff]". I will also note that the pricing formulas for other types of options are more complicated than those for cash-or-nothing digital options, which would make "E[$Payoff]" (and possibly "Probability of Payoff") more complicated to calculate. But none of this affects the conclusions in this section of the present report. Another way to see this is to note that if the prices were not lower (and if other features of the Chance Report's options were the same except for strike price), no one would buy the lower-probability options. Why pay the same amount for a lower expected payoff? Thus, the very existence of different payoff probabilities for these options is evidence that their prices (or some other important feature) must vary.

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just-fair compensation for the risks borne. One of the most basic findings of financial economics is that asset prices in well-functioning capital markets equilibrate to offer just-fair rates of return for the risks investors must bear.28 But the options in this case did not trade at market prices. As discussed at length in Section III of my original report, the stated prices of the components and the actual price for the net position were well in excess of the actual values of these options. Since the prices actually stated or charged were well above market value, the expected "profits" were negative, not positive. Moreover, the Chance (and the Hess) Report also ignores the fees paid to Ernst & Young and attorneys.29 Section III.C of my original report shows that the expected rates of return on these options were materially negative even without those fees, but that they were much more negative with the fees. Therefore, the profit probabilities for the options in this case were unreasonably low because they produced materially negative expected rates of return. By failing to consider the price paid (and the fees), the Chance Report's methodology overlooks this vital fact. In short, it is economically impossible to assess whether the profit probability on these options was "reasonable" without taking account of the prices paid for them. The Chance Report omits this key step and so reaches an unwarranted and economically insupportable conclusion.

IV.

CHANCE REPORT'S DIVERSIFICATION COMMENTS AND CALCULATIONS

The Chance Report calculates profit probabilities for the partnership. In the course of those calculations, it states:30 Whether these numbers [the probabilities of a positive return for the partnership] are higher or lower than the corresponding figures for the individual transactions depends on two factors: the correlations between the currencies and whether the transactions are calls or puts. The correlation effect is similar to what is commonly referred to as diversification in investment terminology. An improvement in diversification will increase the probability of profit. An investor who holds only two stocks that are highly correlated gets little benefit from diversification. Likewise, calls on two highly positively correlated currencies would have little benefit from diversification. A call and a put on two highly positively correlated currencies would provide a diversification benefit. Calls on two highly negatively correlated currencies would provide diversification, while a call and a put on two

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Note that the Chance Report itself relies on this principle at p. 33, when it uses the "Capital Asset Pricing Model," the earliest and simplest of the models used in this area, to estimate expected rates of return. In fairness, the Chance and Hess Reports do not contain any indication of awareness of these fees. At the same time, it would be logical to expect that transaction costs of some sort would be associated with these transactions, and to consider the possibility that such costs could be substantial. Chance Report, pp. 37-38, emphasis in the original.

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highly negatively correlated currencies would provide little benefit from diversification. In contrast, my own report noted that diversification was harmful for "investments" with negative expected rates of return.31 The Chance Report does not calculate expected rates of return, only payoff probabilities, which (as noted in the previous section of this report) ignores the cost of the investment. As a consequence, the Chance Report does not calculate and fails to take into account in any way whether the expected net payoff is positive or negative. It also ignores the fees to Ernst & Young and the attorneys. For these reasons, it reaches incorrect conclusions on the value of diversification in the present circumstances.32 I explain why with a series of figures. First, Figures R2 and R3, on the next page, illustrate the general point about diversification with positive versus negative expected net payoffs.33 Figure R2 shows how diversification helps assure a positive actual net payoff in cases in which the expected net payoff is positive, and Figure R3 shows how diversification works equally well to help assure a negative actual net payoff in cases in which the expected payoff is negative.

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Kolbe Report, Section III.F. The Hess Report similarly does not calculate expected rates of return or consider the fees to Ernst & Young and attorneys. However, neither does it claim that diversification is valuable in the present circumstances. See Appendix E-R, Workpaper E-R-15A for the data that underlie these figures. The phrase "probability density" is standard; it recognizes that with a continuous (infinite) set of possible outcomes, the probability of any one outcome is infinitesimal. The area under either curve between any two points on the horizontal axis represents the probability that the outcome will fall between those two points. The total area under each of the curves is 1.0 in conventional probability terms, i.e., 100 percent. That is, the outcome will definitely fall somewhere on the horizontal axis.

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Figure R2

Figure R3 14

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In Figure R2, diversification narrows the distribution of possible returns. Since the expected return is positive, this reduces the part of the curve that produces negative outcomes (i.e., Area A of the figure). Diversification equally narrows the outcomes in Figure R3, but since the expected payoff is negative, this tends to eliminate the chance of positive outcomes (i.e., Area B of that figure). Diversification works the same way in both figures, but whether this process is beneficial or harmful depends on whether narrowing the distribution helps lock in a positive or a negative outcome. Figures R2 and R3 are drawn to illustrate a continuous range of possible outcomes. However, the digital options in the present case have a discrete (countable) range of possible outcomes. With one option, there are two possible outcomes, win or lose. With two options, there are four: win both, win the first and lose the second, win the second and lose the first, and lose both. To illustrate the basic principles, it is useful first to consider the simplifying assumption of identical, independently distributed bets. In that case, since both options are identical, the middle two of the four outcomes with two options are the same, and we need focus on only three states of the world: win both, win one (which can happen in two ways), and lose both. Under the assumption of identical bets, the number of possible outcomes with N bets is always N+1, and we need only keep track of how many ways the intermediate outcomes may happen. Figure R4, below, takes this approach for two quite different cases, one bet or nine bets.34 It considers bets like the example in the COBRA Presentation: the probability of winning any individual bet is 38 percent, the total amount bet is $2.5 million, and the total potential payoff if all bets are won is $6.25 million, 2½ times the amount bet. The single bet wagers the whole $2.5 million at once, while the nine independent bets just divide the $2.5 million amount bet into nine equal pieces.

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See Appendix E-R, Workpaper E-R-16A for the data that underlie this figure.

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Figure R4

The vertical axis depicts the probability of each event, while the horizontal axis depicts the net payoff if that event occurs. The expected (i.e., the statistical mean) net payoff -- the vertical dashed line in the figures -- is the same regardless of the number of bets, and it is slightly negative. The expected net payoff is mildly negative because a 40 percent chance of winning is necessary to expect break even on such bets, as discussed above.35 The single bet has two possible outcomes, depicted by the purple diamonds. The nine bets have ten possible outcomes, depicted by the brown circles, from losing all nine (the far left circle) to winning all nine (the far right circle). Note that the possible outcomes with nine bets, while discrete, start to sketch out the sort of "bell shaped curve" depicted in Figures R2 and R3. Of course, there were more than one but many fewer than nine bets in the actual COBRA transactions. Figure R5 below adds two intermediate cases to Figure R4, two bets and three bets.36

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Of course, the actual probabilities of winning were well below 38 percent and the actual expected net payoffs materially more negative, but these assumptions are enough to illustrate the issues involved. See Appendix E-R, Workpaper E-R-16A for the data that underlie this figure.

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Figure R5

The two intermediate cases in Figure R5 are the possible outcomes with two bets (three possible outcomes, shown as green squares) and three bets (four possible outcomes, shown as blue triangles). To help readers distinguish the new cases, Figure R5 increases the size of the points representing the new cases' outcomes and reduces the size of the outcomes of one or nine bets, relative to Figure R4. The two bets are each for one-half of the $2.5 million, and the three bets are each for one-third of this amount. The above quotation from the Chance Report effectively focuses on points like the middle outcome of the two-bet case in Figure R5, which is the circled green box just to the right of the zero-netpayoff line. It has a higher probability, a little under 50 percent, than the single-bet win (the circled purple diamond at the right edge of the figure), which has a probability of 38 percent in the COBRA Presentation example. The quotation notes that if net payoffs of the two bets are negatively correlated (i.e., are more diversified), the probability of the middle outcome will be higher and of the extreme two-bet outcomes lower than depicted in Figure R5.37 This increases the odds of winning the small net payoff associated with the single-win outcome.38

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That is, since the probabilities sum to 100 percent, increasing the probability of the middle outcome for the two-bet case reduces the probabilities of the other two outcomes. With the COBRA Presentation's example, the gross payoff from one of two equal bets would be ($6.25 million / 2) = $3.125 million. The net payoff would subtract the cost of the two bets, $2.5 million, to (continued...)

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That is true as far as it goes, but it does not go far enough. First, note that the expected value of a positive net payoff from the two-bet case is lower than that from the single-bet case. With a single bet, the net expected value of a positive outcome is $1.425 million.39 With two independent bets, the expected net value of a positive outcome is $0.836 million.40 With negatively correlated bets, the odds of the single-win outcome go up, which improves the chance of the $0.625 million outcome, but at the expense of the $3.75 million, two-win outcome. The expected value of the positive net payoff goes down as a result, not up. The way to maximize the expected winnings would instead be to have perfectly positively correlated bets, which effectively restores the single-bet case.41 Second, the Chance Report ignores the fees to Ernst & Young and attorneys. These fees turn even the COBRA Presentation's bet from mildly negative to materially negative. Figures R6 and R7 on the next page redraw Figures R4 and R5 to include the cost of this $2.25 million in fees.42 They depict the same points and the same series of bets as Figures R4 and R5, respectively, but the numbers on the horizontal axis have been shifted to reflect the cost of the fees, which reduces the value of the expected net payoff line (the vertical dashed line) downward by $2.25 million, also. That is, the position of the vertical line relative to the distribution the points representing the various outcomes is exactly the same. But each of the individual points and the vertical line itself correspond to much more negative values when the fees are taken into account.

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(...continued) produce the $0.625 million net payoff value on the horizontal axis that corresponds to the point in question. That is, [0.38 × ($6.25 million ! $2.5 million)] = [0.38 × $3.75 million] = $1.425 million. That is, [0.38×0.38×$3.75 million + 2×0.38×(1!0.38)×$0.625 million] = $0.836 million. See Appendix E-R, Workpaper E-R-12A. Recall Figure R3. In terms of that figure, this paragraph simply points out that diversification through multiple bets, and especially multiple, negatively correlated bets, is just another way to shrink the size of the positive-payoff part of the distribution, Area B. See Appendix E-R, Workpaper E-R-16A for the data that underlie these figures.

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Figure R6

Figure R7 19

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In Figure R6, only the single-bet case has any realistic chance of a positive payoff. You must win at least seven of the nine bets to avoid losing money. In Figure R7, the single-win outcome in the case of two bets has a materially negative expected net payoff, and even the two-win case for three identical bets is negative.43 Note also that if the bets were negatively correlated, the odds of winning them all would go down, reducing the expected positive payoffs with multiple bets even more. Thus, the maximum expected winnings after fees with the COBRA Presentation's structure comes from betting only once. This is illustrated in Figure R8.44 Moreover, with multiple bets, a negative correlation increases the effect of diversification and so reduces the chance of the sort of extreme outcome needed to win. Positive correlations, which make multiple bets more nearly like a single bet, reduce but do not eliminate (absent perfectly positive correlation) the costs of diversification in the present circumstances.

Figure R8

43

With equal, independent bets and the COBRA Presentation's overly generous 38 percent chance of success, you must win at least 2 of 2, 3 of 3, 4 of 4, 4 of 5, 5 of 6, 6 of 7, 7 of 8, or 7 of 9 bets to achieve a positive outcome after fees. See Appendix E-R, Workpaper E-R-14A. See Appendix E-R, Workpaper E-R-17A for the data that underlie this figure, which, like the others, is overly optimistic relative to the actual options.

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