A Probabilistic Analysis of Autocallable Optimization Securities
AA Probabilistic Analysis of Autocallable Optimization Securities
Gilna K. Samuel ∗ and Donald St. P. Richards † September 6, 2016
Abstract
We consider in this paper some structured financial products, known as reverse convertiblenotes, that resulted in substantial losses to certain buyers of these notes in recent years. We shallfocus on specific reverse convertible notes known as “Autocallable Optimization Securities withContingent Protection Linked to the S&P 500 Financial Index,” because these notes are repre-sentative of the broad spectrum of reverse convertibles notes. Therefore, the analysis provided inthis paper is applicable to many other reverse convertible notes.We begin by describing the notes in detail and identifying potential areas of confusion inthe pricing supplement to the prospectus for the notes. We deduce two possible interpretationsof the payment procedure for the notes and apply the Law of Total Expectation to develop aprobabilistic analysis for each interpretation. We also determine the corresponding expected netpayments to note-holders under various scenarios for the financial markets and show that, undera broad range of scenarios, note-holders were likely to suffer substantial losses.As a consequence, we infer that the prospectus is sufficiently complex that financial advisersgenerally lacked the mathematical knowledge and expertise to understand the prospectus com-pletely. Therefore, financial advisers who recommended purchases of the notes did not havethe knowledge and expertise that is required by a fiduciary relationship, hence were unable toexercise fiduciary duty, and ultimately misguided their clients. We conclude that these reverseconvertibles notes were designed by financial institutions to insure themselves, against significantdeclines in the equities markets, at the expense of note-holders.
In July, 2011 the U.S. Financial Industry Regulatory Authority (FINRA) issued an alert [2]concerning the sale of structured financial products known as “reverse convertible notes.”These notes were issued by many financial firms and sold widely to clients who had an op-timistic view of future market conditions and who were seeking to diversify their financialportfolios. Clients were promised high yields, and phrases such as “autocallable optimiza-tion” or “contingent protection” in the titles of the notes imbued clients with a high sense ofconfidence in the future of these notes. However, the notes turned out not to be as simple ∗ Department of Finance, Rensselaer Polytechnic Institute, Troy, NY 12180. † Department of Statistics, Pennsylvania State University, University Park, PA 16802. : Primary 60E99.
Key words and phrases . Financial derivatives; Law of Total Expectation; Return optimization securities;Contingent Protection; S&P 500 Financials Index. a r X i v : . [ q -f i n . S T ] A p r r as safe as customers had thought initially, and buyers of these notes eventually sufferedenormous losses when the financial markets experienced a significant downturn in 2008.The U.S. Securities Exchange and Commission (SEC) defines a reverse convertible noteto be a financial product whose return is linked to the performance of a “reference asset” or a“basket of reference assets,” usually consisting of the stock price of an unrelated company orthe level of a stock market index. Some common reference assets for such notes are the S&P500 Financials Index, the EURO STOXX 50, and the common stock of JPMorgan Chase &Co. or other prominent corporations. In this paper, we will investigate an autocallable note, acommon type of reverse convertible note. Autocallable notes have one or more “call dates,”which are dates on which the note can be redeemed prior to the maturity date and which aredetermined by specific market conditions. A wide variety of reverse convertible notes havebeen sold to the public since 2006, with sales continuing even today, and these notes canbe examined in a manner similar to the analysis given in this paper. Therefore, the analysisprovided in this paper serves as a template for the study of many reverse convertibles notes.In this paper, we investigate a particular reverse convertible note known as “AutocallableOptimization Securities with Contingent Protection Linked to the S&P 500 Financials Index.”This financial product was issued by Lehman Brothers Holdings Inc. in early 2008, before the2007-2008 financial crisis reached its nadir, and the pricing supplement to the prospectus forthis note can be obtained from the SEC’s website at [8]. We will study the pricing supplementto the prospectus, and provide a description of the note and its key features. This pricingsupplement is similar to the pricing supplements for many other reverse convertible notesissued by other financial companies.At first glance, the pricing supplement to the prospectus appears to provide a promisingdescription of the note; however, on examining the pricing supplement in detail, we foundsome disturbing attributes. First, the term “contingent protection” in the title causes thisfinancial product to appear more secure than it really is. The phrase “contingent protection”may suggest that there is some legal or financial protection on the notes, thereby causingsome customers to become more confident about the returns on their purchases. Therefore,we will first assess the weaknesses of the contingent-protection feature of these notes.Another concern is that the details of the pricing supplement could have been confusingto a financially unsophisticated client and as a result, many clients would have found it dif-ficult to understand complete details of the note. Our second task, therefore, is to describesome vaguenesses that we have found in the prospectus regarding the payment procedureand redemption rules. In order to assess these details we will analyze the method used todetermine payment and describe two interpretations of the payment procedure.Next, we provide a probabilistic analysis of the payment procedure and use the Law ofTotal Expectation to calculate the expected net returns to clients. We will show, based on theresults derived from our analysis, that customers were at a substantial disadvantage from thevery moment they purchased this note. We deduce from our analysis that the average clientwould have lost a significant proportion of their principal; in some situations, those lossescould be as high as 50%.At this point, we provide some remarks on the concept of fiduciary duty . Many state lawsand the U.S. Investment Advisers Act of 1940 require that certain financial advisers act as2 duciaries in their business transactions with clients. West’s Encyclopedia of American Law [3] defines a fiduciary relationship as “one which encompasses the idea of faith and confi-dence and is generally established only when the confidence is given by one and accepted byanother.” “A fiduciary has greater knowledge and expertise of the matters being handled andis held to a standard of conduct and trust above that of a casual business person” (Hill [5]).Fiduciary duty is “the obligation to act in the best interest of the beneficiary of the fiduciaryrelationship” (Rahaim [9]).As a consequence, we will conclude that financial advisers who recommended purchasesof reverse convertible notes in early 2008 did not exercise fiduciary duty to their clients.Indeed, the prospectus is sufficiently complex that financial advisers themselves could nothave understood every detail of the prospectus. Further, financial advisers lacked the math-ematical knowledge and expertise necessary to determine the consequences of details in theprospectus. Therefore, financial advisers simply were unable to exercise fiduciary duty totheir clients, and ultimately the advisers misguided their clients.We remark that related probabilistic analyses of other types of financial derivatives weregiven by Richards and Hundal [10] and Richards [11]; cf. Samuel [13]. Those papers an-alyzed structured products, such as, return optimization securities, yield magnet notes, re-verse exchangeable securities, and principal-protected notes. In each case, it was shown thatpurchasing these structured products during the mid-2000’s would likely lead to substantiallosses. In this paper, a similar probabilistic analysis will be done on reverse convertible notesand our analysis will show that the purchase of these notes were likely to result in significantlosses also.Moreover, Richards [11] observed that certain structured financial products were de-signed “to insure the banks against substantial declines in the markets; such an arrangementallowed the banks to avoid direct stock sales on the open market, which could have triggeredwidespread market declines.” Likewise, the reverse convertible notes analyzed in this paperprovided a similar safety net to the financial institutions which issued them, but at the expenseof note-holders.We conclude the introduction with a summary of the results to follow in the remainingsections of this paper. In Section 2, we provide an assessment of the pricing supplement tothe prospectus. We will point out that seemingly positive features of the payment procedurefor the notes appear to provide overly optimistic views of future market conditions seem tosuggest that clients’ principal are protected against adverse market conditions, and that thechances of a loss of principal are low. Moreover, the pricing supplement presents positivefeatures of the notes in the earlier part of the document, and in greater detail. On the otherhand, negative aspects of the notes are described in a less prominent manner; those aspectsare relegated to later sections which are less likely to be read by a typical client; and thesupplement does not provide a comprehensive list of possible payout scenarios for the notes.We shall also identify problems with the payment procedure for the notes. Indeed, thedescription of the payment procedure provided in the payment supplement seems incompleteand vague. Moreover, the conditions under which the note will be called is unclear, leadingto at least two plausible interpretations of the payment procedure.In Section 3, we describe and analyze a first interpretation of the payment procedure. We3hall use the Law of Total Expectation to obtain mathematical formulas for the expected, oraverage, payout to a randomly chosen note-holder under a variety of market-outlook scenar-ios ranging from pessimistic to optimistic. By analyzing graphs of the expected net paymentfunctions, we shall prove that for the expected net payment to be positive, the market outlookmust necessarily be highly optimistic, and even in such a best-case scenario the note-holderis likely to be at a disadvantage to the issuing bank because of the call provisions of the note.As regards pessimistic scenarios, we show that the average return to note-holders can be aslow as -50% in some instances.In Section 4, we describe a second plausible interpretation of the payment procedure.We shall show that the expected net payment under this interpretation can never be greaterthan the expected net payment under the interpretation analyzed in Section 3. Therefore,the outcomes for note-holders can be expected generally to be worse than those described inSection 3. In Section 5, we present some closing remarks on our analysis. We will concludethat these reverse convertible notes were simply clever marketing scheme used by financialinstitutions wanting to insure themselves against future market declines.
Reverse convertibles notes known as “Autocallable Optimization Securities with ContingentProtection Linked to the S&P 500 Financials Index” were sold to the public in multiples of$10. The pricing supplement [8] to the prospectus states that the trade date for this financialproduct was February 5, 2008, with a settlement date three days later on February 8, 2008.These notes had an 18-month “observation period,” from May 5, 2008 to August 5, 2009,with a total of six “observation dates” every three months. The final valuation date was onAugust 5, 2009, and the maturity date was five days later on August 10, 2009.The pricing supplement also mentions information such as the advantages and disadvan-tages of the note and possible returns that clients might receive from their purchases. How-ever, the information provided seem biased toward positive aspects of the note. For instance,the positive aspects are emphasized in a “Features Section” on the first page of the pricingsupplement, and are highlighted in boldfaced font while the explanation of each feature is ina smaller font. The first feature listed states that the notes provide “positive call return in flator bullish scenarios,” implying that clients will receive positive returns as long as the S&P500 Financials Index remains above a certain level.Another positive attribute listed in the “Features Section” is that of “contingent principalprotection.” This feature suggests that clients have some protection against loss of principal;however, the specified market conditions under which the contingent protection applies isprinted in a smaller font on the same page.The last feature states that the notes “express a bullish view of the U.S. Financial ServicesSector.” Given that the notes are linked to the S&P 500 Financials Index which comprises of93 companies in the financial services sector of the S&P 500 Index, financial advisers recom-mending purchases of these notes should have provided their clients with an estimate of thelikelihood that the S&P 500 Financials Index would continue on an upward trend. However,4either advisers nor the pricing supplement appear to have provided any such information.Indeed, the features listed in the “Features Section” seem to promote optimistic views offuture market conditions, leading clients to believe that their future gains will be high. Thesedetails imply that the clients’ principals are protected against adverse market conditions andtheir chances of a loss of principal are low. Moreover, buyers of these notes were morelikely to read the first few sections in greater detail compared to the later sections. Thus, thepositive aspects of these notes may have played a more influential role in clients’ decisionsto purchase the notes.On the other hand, negative aspects of the note are portrayed in a less prominent mannerand are postponed to later sections of the pricing supplement. For instance, the fact that thenote is not insured by the Federal Deposit Insurance Corporation (FDIC) is stated in smallerfont. The section “Key Risks,” which lists numerous risk factors, appears as one of thelast sections in the pricing supplement. Also, the pricing supplement provides no mentionof the likelihood that downward trends in the market can occur, leading to a total loss ofcapital. Moreover, the prospectus implies that future market conditions will be positive butthen warns later against the expectation of a “positive-return environment.” Further, based onthe pricing supplement’s guidelines for net payment, the greatest possible return on the noteis only 31.26%, whereas the greatest possible loss is 100% of capital. That is, profits on thenote are limited to a specific percentage while there is no similar limit on the percentage ofcapital that clients might lose.Another major concern is the description of the payment method given in the pricingsupplement; in our view the description is incomplete. The pricing supplement provides onlyfour examples of possible scenarios of the future market trends. In our view, the supplementshould have provided a more comprehensive list of such scenarios. In two of the scenariospresented the client receives a positive return; in one scenario the client breaks even, and inonly one scenario does the client lose part of their principal. Also, no example is given todemonstrate how a client could have suffered a loss of their entire capital. Therefore, theseexamples may have led some clients to believe that they were more likely to receive positivereturns rather than losses.Moreover, the details in the scenario analysis are vague. Specifically, the determinationof the conditions under which the note will be called is unclear, and therefore we will nowinvestigate the payment procedure in great detail.In order to describe the payment method, we need to define several terms. The
IndexStarting Level is the closing level of the S&P 500 Financials Index on February 5, 2008. Thepricing supplement states the Index Starting Level to be . . The
Trigger Level is . , which is 50% of the Index Starting Level. The Index Ending Level is the closing level of theS&P 500 Financials Index on the corresponding observation or trade date. Having definedthese terms, we can define the
Index Return , denoted by I , to be I = Index Ending Level − Index Starting LevelIndex Starting Level . By means of this formula, we see that if the Index Ending Level on any trade date is at orabove the Index Starting Level then the Index Return will be positive, i.e. I ≥ , while if5n any trade date the Index Ending Level is less than the Index Starting Level then the IndexReturn will be negative, i.e. I < . Also, if the Index Ending Level is less than the TriggerLevel then I < − . and conversely.The section of the pricing supplement entitled “Payment at Maturity” contains a diagramwhich is used to illustrate the rules for calculating net payments to clients. The diagram pre-sented here is typical, and hence representative of related diagrams appearing in the pricingsupplements of many other autocallable notes found via the SEC’s website. This diagram isthe primary source in the pricing supplement devoted to determining the payment procedureand it is reproduced in Figure 1, as follows: Was the closinglevel of the Indexon any ObservationDate at or abovethe Index StartingLevel? You will receive the applicable CallPrice as described under “IndicativeTerms—Return on Call Date” if theNotes have not been previously called.Was the closinglevel of the Indexbelow the TriggerLevel on anytrading day duringthe ObservationPeriod? At maturity, you will receive yourprincipal of $10 per Note.Determine theIndex Return At maturity, you will receive yourprincipal reduced by an amount basedon the percentage decrease in theIndex from the Index Starting Level tothe Index Ending Level, calculated asfollows:$10 × (1 + Index Return) In this scenario, you could losesome or all of your principaldepending on how much theIndex decreases.
YesYesNoNo NoNoYesYes
Figure 1: Payment at Maturity, as Depicted in the Pricing SupplementFrom Figure 1, we deduce that if
I > on an observation date, then the note might becalled and redeemed for an amount stated in the pricing supplement. This redemption value6s based on a rate of 20.84% per annum, with actual amounts stated in the “Final Terms”section of the supplement. This “Final Terms” section states each observation date and thecorresponding percentage return if the note is called on that date.However, if I < on a particular observation date then the note is not called. Also, if I < − . on any trading date during the observation period then there is a possibility thatthe client will receive a negative return at maturity. In particular, the diagram stated that theclient can lose up to a 100% of their capital but provided no indication of the probabilityof this occurrence. Also, the diagram itself does not make it explicit when the note will becalled, but as long as the notes are not called then the calculation of the Index Return will bedone on every subsequent trading date until the final valuation date.To determine the actual return that note-holders will receive on their notes is particularlyvague. It appears that the notes could have been called before the final valuation date, andat the complete discretion of the sellers. Suppose also that the Index Return, when measuredon the final valuation date, is substantially greater than zero, i.e., the Index Ending Levelis higher than the Index Starting Level; then the language used in the pricing supplementappears to give the seller of the note the power to have called the note in a way so as tominimize the return to note holders. Therefore, it seems that the manner in which paymentsto clients are determined can be interpreted in different ways. In the following two sectionswe will analyze two likely interpretations of the payment procedure that could have beenused and we will estimate the average return to clients under each interpretation. In this section, we describe and analyze one of the possible interpretations of the paymentprocedure for calculating net payment. For this interpretation, we will show that even underoptimistic scenarios, an average buyer will lose a substantial amount of their principal.For each observation date within the 18-month period from February 5, 2008 to August5, 2009, we define the corresponding index return as follows: I : Index return on May 5, 2008 I : Index return on August 5, 2008 I : Index return on November 5, 2008 I : Index return on February 5, 2009 I : Index return on May 5, 2009 I : Index return on August 5, 2009Let n denote the total number of trading days during the 18-month observation period.We have verified through Google Finance’s website on historical closing prices of the S&P500 Financials Index, that for this note n = 381 trading days. For , . . . , n , define d i to bethe cumulative index return for Day i ; thus d i represents the total index return from Day 1,February 5, 2008, to Day i . 7et d min = min ( d , d , ..., d n ) be the smallest daily cumulative return over the entireobservation period.Using this notation, the steps for this interpretation of the payment procedure can now bedescribed as follows:Step 1: Calculate the Index Return, I r on the observation date for the r thquarter.Step 2: If the Index Return on the observation date is non-negative, i.e. if I r ≥ , then the note is called and the client receives an amount asstated in the pricing supplement on the Final Valuation date.Step 3: If the Index Return on the observation date is negative, i.e. if I r < then we calculate the cumulative returns, d i , for each trading dateduring the quarter corresponding to the given observation date. Ifany of those cumulative returns are less than − then we recordthis occurrence for future reference.Step 4: Return to Step 1 for the next observation date and repeat the process.Steps 1-4 are carried out on every observation date until the final valuation date or until thenote is called.At the Final Valuation date, one of the following three cases can occur:Case 1: If the Index Return is non-negative on the Final Valuation date i.e.if I ≥ , then the client receives the amount stated in the pricingsupplement.Case 2: If the Index Return on the Final Valuation date is negative and everydaily Index Ending Level remained at or above the Trigger Level oneach trading date during the entire observation period, i.e. if I < and d min ≥ − , then the client receives $10 at maturity.Case 3: If on any trading date prior to the Final Valuation Date at least onedaily Index Ending Level breached the Trigger Level, i.e. if d min < − , then the client receives a reduced payment equal to I ) ,representing a negative return of I to note-holders.The following tree diagram describes this interpretation of the payment procedure:8 rade Date (Feb. 5, 2008)Go to Observation Date I ≥ on May 5, 2008?No: Then was any d i < − %during Feb. 5-May 5, 2008? Yes: Then call the notes;pay note-holders $10.52Yes: Notes are not called No: Then go to Observation Date I ≥ on Aug. 5, 2008No: Then was any d i < − %during May 6-Aug. 5, 2008? Yes: Then call the notes;pay note-holders $11.04Yes: Notes are not called No: Then go to Observation Date I ≥ on Nov. 5, 2008?No: Then was any d i < − %during Aug. 6-Nov. 5, 2008? Yes: Then call the notes;pay note-holders $11.56Yes: Notes are not called No: Then go to Observation Date I ≥ on Feb. 5, 2009No: Then was any d i < − %during Nov. 6, 2008-Feb. 5, 2009? Yes: Then call the notes;pay note-holders $12.08Yes: Notes are not called No: Then go to Observation Date I ≥ on May 5, 2009?No: Then was any d i < − %during Feb. 6-May 5, 2009? Yes: Then call the notes;pay note-holders $12.61Yes: Notes are not called No: Then go to the Final ValuationDate: Is I ≥ on Aug. 5, 2009?No: Then was any d i < − %during the observation period? Yes: Then call the notes;pay note-holders $13.13Yes: Notes mature;pay note-holders $ I ) No: Notes mature;pay note-holders $10
Figure 2: Schematic Description of the Payment Procedure9hen, according to the pricing supplement [8] and this payment procedure, the net pay-ment in U.S. dollars, to note-holders is given by the function,Net Payment = . , if I ≥ . , if I < , and I ≥ . , if I j < , j = 1 , and I ≥ . , if I j < , j = 1 , , and I ≥ . , if I j < , ≤ j ≤ and I ≥ . , if I j < , ≤ j ≤ and I ≥ , if I j < , ≤ j ≤ and d min ≥ − I , if I j < , ≤ j ≤ and d min < − (3.1) • The first six cases occur if any of the index returns, I , . . . , I are non-negative, inwhich case, the note is called on the corresponding observation date, and the clientreceives a positive return on the Final Valuation date in those instances. • The seventh case occurs if all of the index returns, I , . . . , I are negative and theminimum cumulative daily index return during the observation period, d min is at least − . In this case, the note matures and the client receives a net payment of zero onthe Final Valuation date, i.e. the client breaks even. • The eighth and last case occurs if the note is not called previously and d min , the mini-mum cumulative daily index return during the observation period , is less than − ;i.e. the smallest Index Ending Level falls below the Trigger Level. In this case, thenote matures and on the Final Valuation date the client receives a negative net paymentof I .In the net payment function in (3.1), the index returns I , . . . , I are continuous randomvariables whose joint probability density function is strictly positive for I , . . . , I in theinterval ( − , ; i.e. there is a non-zero probability of a downward market trend in the future.Hence, P ( d min < − | I j < , ≤ j ≤ (cid:54) = 0 (3.2)since there always exists the possibility of decreasing market trends.We now apply the Law of Total Expectation (Ross [12], page 333, Equation (5.1b)) to the10et payment function in (3.1); then we obtain E ( Net Payment ) = 0 . × P ( I ≥ . × P ( I < , and I ≥ . × P ( I j < , j = 1 , and I ≥ . × P ( I j < , j = 1 , , and I ≥ . × P ( I j < , ≤ j ≤ and I ≥ . × P ( I j < , ≤ j ≤ and I ≥ × P ( I j < , ≤ j ≤ and d min ≥ − )+ 10 × E ( I | d min < − and I j < , ≤ j ≤ × P ( d min < − | I j < , ≤ j ≤ × P ( I j < , ≤ j ≤ . (3.3)In a real-world setting, the index return random variables I , . . . , I are unlikely to beindependent. In fact, there are numerous research articles which have found evidence thatmomentum-trading “strategies which buy stocks that have performed well in the past andsell stocks that have performed poorly in the past generate significant positive returns over3- to 12-month holding periods” (Jegadeesh and Titman [7]; Conrad and Kaul [1]); thisphenomenon has been observed to hold not only for U.S. financial markets but also inter-nationally (Hurn [6]). Therefore, if the financial index has recorded a negative return in athree-month period then it is natural to expect traders to be pessimistic about the ensuingthree-month period. Therefore, we would expect to find that, for example, P ( I ≥ | I < ≤ P ( I < | I < (3.4)in a real-world setting.We recognize that there can be exceptions to this inequality; if, for instance, the marketsare in a euphoric state of mind, where any downturn is regarded optimistically as a sign ofpositive future developments then the above inequality might even be reversed at times. Sinceit is unrealistic and unwise to treat the markets as being in a permanent state of euphoria, wewill treat this inequality as being valid generally. Therefore for the purposes of analyzing theexpected net payment to note-holders, we will assume that the inequality (3.4) holds.We remark that because P ( I ≥ | I <
0) + P ( I < | I <
0) = 1 , (3.5)then the inequality (3.4) implies that P ( I ≥ | I < ≤ ≤ P ( I < | I < . (3.6)Also, (3.5) implies that if P ( I ≥ | I < is large, i.e. close to one, then, P ( I < | I < is small, i.e. close to zero. 11igure 3: Closing Levels of the S&P 500 Financials Index for September 20, 2002 to De-cember 28, 2007At this point, it is worthwhile to provide historical context relating to the state of thefinancial markets in early 2008, at the same time when these reverse convertible notes werebeing sold to the public. Figure 3 was obtained from Google Finance’s website and is basedon historical closing prices from September 20, 2002 to December 28, 2007 of the S&P 500Financials Index.As Figure 3 shows, the S&P 500 Financial Index had undergone a sustained increaseduring the period 2002 to late 2007. By early 2008, many investors were concerned thatthe bull market had gone too far and that stock prices might be due for substantial declines.Therefore, it was reasonable to expect that P ( I r < was large for the r th quarter, hence, P ( I r ≥ was small for that quarter.Given the statistical evidence that momentum-trading strategies are successful over 3- to12-month periods, we also would have expected P ( I < | I < to be high in late 2007,hence that P ( I ≥ | I < was small. Then it follows that P ( I < , and I ≥ ≡ P ( I ≥ | I < P ( I < < P ( I ≥ | I < , which proves that P ( I < , and I ≥ also is small. By repeating this argument, wededuce that, in early 2008, each of the probabilities in the first six terms in (3.3) are verysmall. Therefore, E ( Net Payment ) (cid:39) E ( I | d min < − and I j < , ≤ j ≤ × P ( d min < − | I j < , ≤ j ≤ × P ( I j < , ≤ j ≤ (3.7)which, clearly, is negative. Consequently, we deduce that clients who purchased these reverseconvertible notes in early 2008 likely were destined to obtain negative net returns, on average.Even more can be deduced from (3.7). By late 2007, P ( I j < , ≤ j ≤ and P ( d min < − | I j < , ≤ j ≤ both were very large. Hence, by (3.7), E ( Net Payment ) (cid:39) × E ( I | d min < − and I j < , ≤ j ≤ . (3.8)12f it was believed in early 2008 that the financial markets were likely to have a precipitous,quarter-over-quarter, fall over the next 18 months, then it would mean that I (cid:39) d min , inwhich case it follows from (3.8) that E ( Net Payment ) (cid:39) × E ( d min | d min < − and I j < , ≤ j ≤ < × − . − . Consequently, the outcome of purchasing these reverse convertible notes in early 2008 wasto undertake a substantial risk of least a 50% loss of capital, on average.Admittedly, the analysis provided above is based on a pessimistic view in 2007-2008 ofcoming trends in the financial markets. To provide an analysis which is based on an unbi-ased approach toward future market trends, we will therefore treat the market as a “randomwalk” in which I , . . . , I are mutually independent and identically distributed random vari-ables. These assumptions imply that medium-term momentum-based strategies will providea trader with no advantages over the general market. Also, such an assumption will be moreprofitable for a client who purchased reverse convertible notes in 2008, for it means that thevarious probabilities appearing in the first six cases in the expected net-payment function arelikely to be more favorable to the clients than would have been the case under the real-worldassumption underlying the inequality (3.4).Nevertheless, under the assumptions that I , . . . , I are mutually independent and identi-cally distributed, we shall show that the expected net-payment to note-holders still remainssignificantly negative. Hence, we will deduce that even in a “random walk” scenario anaverage note-holder would have faced losses except under highly optimistic environments.Using the assumption that I , . . . , I are independent, we see that equation (3.3) becomes E ( Net Payment ) = 0 . × P ( I ≥ . × P ( I < × P ( I ≥ . × (cid:34) (cid:89) j =1 P ( I j < (cid:35) × P ( I ≥ . × (cid:34) (cid:89) j =1 P ( I j < (cid:35) × P ( I ≥ . × (cid:34) (cid:89) j =1 P ( I j < (cid:35) × P ( I ≥ . × (cid:34) (cid:89) j =1 P ( I j < (cid:35) × P ( I ≥ × E ( I | d min < − and I j < , ≤ j ≤ × P ( d min < − | I j < , ≤ j ≤ × (cid:89) j =1 P ( I j < . (3.9)13enote P ( I ≥ by p , because of the assumption that I , . . . , I are identically dis-tributed, it follows that p ≡ P ( I j ≥ for all j = 1 , . . . , . Note that, p is the probability thatthe note is called, i.e. that the Index Return on any given observation date was non-negative.Then, we obtain E ( Net Payment ) = 0 . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p + 10 × E ( I | d min < − and I j < , ≤ j ≤ × P ( d min < − | I j < , ≤ j ≤ × (1 − p ) . (3.10)In order to derive explicit values for the terms E ( I | d min < − and I j < , ≤ j ≤ and P ( d min < − | I j < , ≤ j ≤ , we would need to make strong assumptions about the random variables I , . . . , I , which weprefer to avoid doing. Therefore, we will assign values to these terms to reflect a variety ofmarket conditions, and then we will study the resulting behavior of the function (3.10) foreach set of values.Define B ≡ − E ( I | d min < − and I j < , ≤ j ≤ . (3.11)Note that the expectation E ( I | d min < − and I j < , ≤ j ≤ clearly is negative; hence, B < . Also, because I ≥ − , then B ≤ . Therefore, we have < B ≤ .Also, define B ≡ P ( d min < − | I j < , ≤ j ≤ (3.12)clearly, ≤ B ≤ because it is a probability. Moreover, by (3.2), B (cid:54) = 0 ; therefore < B ≤ .By (3.10), we obtain E ( Net Payment ) = 0 . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p − B B (1 − p ) . (3.13)For p = 0 , we obtain E ( Net Payment ) = − B B < . Therefore, there will always be asmall interval around p = 0 where the expected net payment is negative.As regards an interpretation for values of B , we offer the following. Suppose that B is small; then it follows from (3.11) that E ( I | d min < − and I j < , ≤ j ≤ is closeto zero and is negative. Hence, for small values of B , a note-holder will receive a small,but negative, overall expected return despite the conditioning event that I , . . . , I all arenegative. That is to say, although the index returns on all six observation dates were negative,and the minimum cumulative daily return is less than -50%, the final outcome was that theaverage loss to note-holders nevertheless turned out to be small.14ence, small values of B indicate that the financial markets are likely to have maintainedan optimistic outlook throughout the 18-month observation period despite consecutive quar-terly losses. This leads us to associate small values of B with general optimism amongstmarket participants, and by a similar argument, we can associate large values of B withbroad market pessimism.In the case of B , it follows from (3.12) that if B is small then there is a small probabilityof any daily cumulative index return breaching the Trigger Level, conditional on six consec-utive quarterly losses. In a real-world situation, consecutive quarterly losses usually inducesgeneral gloom on market participants; so it would indeed be an optimistic assumption to be-lieve that there remains a small probability of avoiding the Trigger Level. Therefore, we seethat small values of B can be associated with a generally optimistic market outlook; and bythe same argument, large values of B can be associated with broad market pessimism.Consider the case in which B = B = 0 . . Using these values, we plot (3.13) withrespect to p , ≤ p ≤ , to obtain in Figure 4 a graph of the expected net payment function:Figure 4: A Graph of Expected Net Payments, where B = 0 . and B = 0 . In this case, the maximum expected net payment is 1.15, approximately, i.e., an averagereturn of about 11.5%. Also, the minimum expected net payment is − . , approximately,i.e., an average loss of about 10%. Thus, even under this optimistic scenario, the averageclient could lose as much as 10% of their capital.We remark also that the expected net payment is maximized at p = 0 . , approximately.Although it might have been presumed that consecutive quarterly increases in the financialmarkets, i.e., p = 1 , would be more beneficial to note-holders, such a scenario would havecaused the note to be called early due to the sustained market increases. In such a case,paradoxically, average returns to note-holders would have been diminished.Under a less-optimistic scenario for the financial markets, in which B = B = 0 . , thegraph of expected net payment function in (3.13) is given in Figure 5:15igure 5: A Graph of Expected Net Payments, where B = 0 . and B = 0 . For this situation, the maximum expected net payment is 0.97, approximately, i.e., anaverage return of about 9.7%. Also, the minimum expected net payment is − . , approxi-mately, representing an average loss of about 25% of principal. As in the previous scenario,moderate values of p turn out to be more beneficial to note-holders than high values of p .If financial markets are expected to undergo significant downturns where the values of B and B are relatively high, say B = 0 . and B = 0 . , then we obtain in Figure 6 the graphof the expected net payment function in (3.13):Figure 6: A Graph of Expected Net Payments, where B = 0 . and B = 0 . Under this pessimistic scenario, the highest expected net payment is 0.89, approximately,16hich means that the maximum percentage return for the average note-holder was only about8.9%. On the other hand, the minimum expected net payment is − . , approximately, im-plying that the average note-holder could have suffered percentage losses of as much as 56%of their capital.We have studied up to now the exact values of the expected net payment function which,as we have seen, involves two unknown parameters, B and B . We now derive upper boundsfor the expected net payment function; these bounds will have the advantage of dependingon only one unknown parameter.We know that d min ≤ I r for any r = 1 , . . . , ; therefore, P ( d min < − | I j < , ≤ j ≤ ≥ P ( I r < − | I j < , ≤ j ≤ . (3.14)Also, the expectation E ( I | d min < − and I j < , ≤ j ≤ < because I < in theconditioning event. Therefore, by inequality (3.14), we see that E ( I | d min < − and I j < , ≤ j ≤ P ( d min < − | I j < , ≤ j ≤ ≤ E ( I | d min < − and I j < , ≤ j ≤ P ( I r < − | I j < , ≤ j ≤ , (3.15)and then, by (3.15), we obtain E ( Net Payment ) ≤ . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p + 10(1 − p ) E ( I | d min < − and I j < , ≤ j ≤ × P ( I r < − | I j < , ≤ j ≤ . (3.16)By the definition of conditional probability and the assumptions that I , . . . , I are inde-pendent and identically distributed, P ( I r < − | I j < , ≤ j ≤
6) = P ( I r < − | I r < P ( I r < − and I r < P ( I r < P ( I r < − ) P ( I r < − p ) − P ( I r < − ) . (3.17)Thus, by (3.16) and (3.17), E ( Net Payment ) ≤ . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p + 10(1 − p ) E ( I | d min < − and I j < , ≤ j ≤ P ( I r < − ) . (3.18)We can also obtain a lower bound on E ( I | d min < − and I j < , ≤ j ≤ . To thatend, define A = { I r < − and I j < , ≤ j ≤ } A = { d min < − and I j < , ≤ j ≤ } . Because d min ≤ I r for r = 1 , . . . , then it follows that A ⊆ A .Consider the cumulative index returns d , . . . , d n , viewed as continuous random vari-ables. The sets A and A above clearly are events that depend on d , . . . , d n . Because A ⊆ A then it follows that P ( A ) ≤ P ( A ) . (3.19)We noted earlier in this section in (3.2) that P ( A ) (cid:54) = 0 ; by a similar argument, it follows that P ( A ) (cid:54) = 0 . As mentioned previously, this condition holds because I , . . . , I are continuousrandom variables whose probability density function is strictly positive on the interval (-1,0),a condition which is equivalent to the assumption that there is a non-zero probability of themarkets moving downward in any quarter.Let f ( d , . . . , d n ) denote the joint probability density function of I , . . . , I . Since I ≡ I ( d , . . . , d n ) is a function of the random variables d , . . . , d n then, by the definition ofexpected value, we obtain E ( I | d min < − and I j < , ≤ j ≤ ≡ E ( I | A )= 1 P ( A ) (cid:90) A I ( x , . . . , x n ) · f ( x , . . . , x n ) d x · · · d x n . (3.20)Because I < for all ( x , . . . , x n ) ∈ A and because A ⊆ A then it follows that (cid:90) A I ( x , . . . , x n ) · f ( x , . . . , x n ) d x · · · d x n ≤ (cid:90) A I ( x , . . . , x n ) · f ( x , . . . , x n ) d x · · · d x n . (3.21)It now follows that from (3.19), (3.20) and (3.21) that E ( I | A ) = 1 P ( A ) (cid:90) A I ( x , . . . , x n ) · f ( x , . . . , x n ) d x · · · d x n ≤ P ( A ) (cid:90) A I ( x , . . . , x n ) · f ( x , . . . , x n ) d x · · · d x n ≡ E ( I | A ) . (3.22)This proves that E ( I | d min < − and I j < , ≤ j ≤ ≤ E ( I | I r < − and I j < , ≤ j ≤ . (3.23)Hence for any r = 1 , . . . , , E ( Net Payment ) ≤ . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p + 10(1 − p ) E ( I | I r < − and I j < , ≤ j ≤ P ( I r < − ) . (3.24)18lso, note that the inequality (3.24) involves the random variables I , . . . , I only, and doesnot depend on d min . Thus, we can select r = 6 to obtain E ( Net Payment ) ≤ . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p + 10(1 − p ) E ( I | I < − and I j < , ≤ j ≤ P ( I < − ) . (3.25)It is clear that E ( I | I < − and I j < , ≤ j ≤ < − . (3.26)We now introduce the notation τ = P ( I < − ) ; then it clear that p + τ ≤ . Therefore, itfollows from (3.25) and (3.26) that E ( Net Payment ) ≤ . p + 1 . − p ) p + 1 . − p ) p + 2 . − p ) p + 2 . − p ) p + 3 . − p ) p − − p ) τ. (3.27)Under an optimistic scenario, we expect there is a small probability of I breaching theTrigger Level, i.e., τ is small. For instance, let τ = 0 . , then we obtain in Figure 7 a graphof the right-hand side of (3.27): [p]Figure 7: Upper Bound for Expected Net Payments, where P ( I < − ) = 0 . In this case, an upper bound for the maximum expected net payment is 1.08, approx-imately, resulting in an upper bound for the average return of about . . On the otherhand, an upper bound for the minimum expected net payment is − . , approximately, i.e., anaverage loss of about 5%. 19uppose we let τ = 0 . ; then we obtain in Figure 8 a graph of the upper bound for theexpected net payment function in (3.27):Figure 8: Upper Bound for Expected Net Payments, where P ( I < − ) = 0 . Under this mildly pessimistic scenario, the highest value for an upper bound for expectednet payment is 0.91, approximately; and an upper bound for the lowest expected net paymentis − . , approximately. Equivalently, an upper bound for the average return to note-holdersis about 9.1%, whereas an upper bound for the average loss to note-holders is about 25%.Now consider a highly pessimistic scenario for the financial markets in which τ = 0 . ;then Figure 9 depicts an upper bound for the expected net payment given by (3.27):Figure 9: Upper Bound for Expected Net Payments, where P ( I < − ) = 0 . − ,approximately. These values imply that an upper bound for the the return for the averageclient is about 8.5% while an upper bound for the possible loss is about 40%.In conclusion, Figures 3.3–3.8 depict upper bounds for expected net payments undervarious scenarios. As a general rule, the highest values for expected net payments occurat moderate values of p and not at high values of p , as one would first presume. Under ascenario in which markets are expected to undergo significant upward trends and the valueof p is high, the note will be called early and therefore will limit returns to note-holders.Based on our analysis of the six scenarios depicted in Figures 3.3–3.8, the best-casescenario leads to a possible maximum an average return of 11.5%, approximately, and apossible average loss of 10%, approximately. The worst-case scenario is an average return of8%, approximately, and an average loss of 56%, approximately.Bearing in mind that only six scenarios were described above in detail, we also provide amore comprehensive analysis by graphing the upper bound function in (3.27) over all valuesof ( p, τ ) such that ≤ p, τ ≤ and p + τ ≤ . Then we obtain the following three-dimensional plot of expected net payments:Figure 10: Upper Bound for Expected Net Payments as a function of p and τ From this analysis, the best-case scenario is an expected net payment of at most 1.16,approximately, at p = 0 . and τ = 0 . On the other hand, the worst-case scenario results inan expected net payment of at most − , approximately, at p = 0 and τ = 1 . Thus, the netpercentage return for note-holders ranges from − to 11.6%, approximately. This rangeimplies that on average, the possible returns to note-holders are substantially lower than the21otential losses. As a result, an average note-holder would have suffered a significant loss ofcapital. We now provide a second interpretation of the payment procedure and analyze it similarlyto the way in which the previous interpretation was assessed. In the interpretation of thepayment procedure studied in the previous section, if an Index Ending Level breached theTrigger Level during any quarter of the observation period but the Index Return was positiveon a subsequent observation date, then the possibility of a positive return still existed. Be-cause the prospectus was unclear on this issue, we will assume in this section that if an IndexEnding Level breached the Trigger Level on any trading day during a quarter, i.e. if at leastone cumulative daily index return was less than − , then the client is destined to receivea negative return at the Final Valuation date.Throughout this section the notation for the six observation dates and the daily cumu-lative index return will remain the same as before. We also need to define a collection ofminimum daily cumulative index returns as a function of the various observation dates, viz., d min , : Minimum cumulative daily index return on May 5, 2008 d min , : Minimum cumulative daily index return on August 5, 2008 d min , : Minimum cumulative daily index return on November 5, 2008 d min , : Minimum cumulative daily index return on February 5, 2009 d min , : Minimum cumulative daily index return on May 5, 2009 d min , : Minimum cumulative daily index return on August 5, 2009The last minimum cumulative daily index return, d min:6 , is equivalent to d min in the anal-ysis provided in Section 3. Also, from this notation it follows that: d min , ≥ d min , ≥ d min , ≥ d min , ≥ d min , ≥ d min , . (4.1)With this notation, the steps for this interpretation of the payment procedure can now bedescribed as follows:Step 1: Calculate the Index Return, I r on the observation date for the r thquarter.Step 2: If the Index Return on the observation date is non-negative, i.e. if I r ≥ , then the note is called and the client receives an amount asstated in the pricing supplement on the Final Valuation date.Step 3: Calculate the cumulative returns, d i , for each trading date during thequarter corresponding to the given observation date and determine22he minimum cumulative daily index return, d min ,r at this observationdate.Step 4: If the Index Return on the observation date is negative and the min-imum cumulative daily return is greater than − , i.e. if I r < and d min ,r ≥ − , then we return to Step 1 for the next observationdate and repeat the process.Step 5: If the Index Return on the observation date is negative and minimumcumulative daily return is less than − , i.e. if I r < and d min ,t < − , then no further action is taken until the Final Valuation date.Steps 1-4 are carried out on every observation date until the Final Valuation date or untilthe note is called.We remark that on the Final Valuation date if the note has not been called then all pre-vious minimum cumulative daily index returns are greater than − and by equation (4.1) d min , ≥ − .At the Final Valuation date, one of the following three cases can occur:Case 1: If the Index Return on the observation date is non-negative i.e. I ≥ , then the note is called and the client receives an amount as statedin the pricing supplement on the Final Valuation date.Case 2: If the Index Return is negative and the minimum cumulative dailyindex return is greater than − on the Final Valuation date, i.e. if I < and d min , ≥ − , then the client receives $10 at maturity.Case 3: If any minimum cumulative daily index return is less than − ,i.e., if d min ,r < − for at least one r = 1 , . . . , , then the client re-ceives a reduced payment equal to I ) , representing a negativereturn of I to note-holders.Note that on the final valuation date that if at least one d min ,r < − then it followsfrom inequality (4.1) that d min , ≤ . The following tree diagram describes this interpretation of the payment procedure:23 rade Date (Feb. 5, 2008)Go to Observation Date I ≥ on May 5, 2008?No: Then was any d i < − %during Feb. 5-May 5, 2008? Yes: Then call the notes;pay note-holders $10.52No: Then go to Observation Date I ≥ on Aug. 5, 2008No: Then was any d i < − %during May 6-Aug. 5, 2008? Yes: Then call the notes;pay note-holders $11.04No: Then go to Observation Date I ≥ on Nov. 5, 2008?No: Then was any d i < − %during Aug. 6-Nov. 5, 2008? Yes: Then call the notes;pay note-holders $11.56No: Then go to Observation Date I ≥ on Feb. 5, 2009No: Then was any d i < − %during Nov. 6, 2008-Feb. 5, 2009? Yes: Then call the notes;pay note-holders $12.08No: Then go to Observation Date I ≥ on May 5, 2009?No: Then was any d i < − %during Feb. 6-May 5, 2009? Yes: Then call the notes;pay note-holders $12.61Yes: Notes are not called No: Then go to the Final ValuationDate: Is I ≥ on Aug. 5, 2009?No: Then was any d i < − %during the observation period? Yes: Then call the notes;pay note-holders $13.13Yes: Notes mature;pay note-holders $ I ) No: Notes mature;pay note-holders $10
Figure 11: Schematic Description of a Payment Procedure24or this interpretation of the payment procedure, we apply the information provided inthe pricing supplement to deduce that the net payment in US dollars is given by:Net Payment = . , if I ≥ . , if I < and I ≥ and d min , ≥ − . , if I j < , j = 1 , and I ≥ and d min , ≥ − . , if I j < , j = 1 , , and I ≥ and d min , ≥ − . , if I j < , ≤ j ≤ and I ≥ and d min , ≥ − . , if I j < , ≤ j ≤ and I ≥ and d min , ≥ − , if I j < , ≤ j ≤ and d min , ≥ − I , if I j < , ≤ j ≤ and d min , < − (4.2) • The first six cases occur if any of the index returns, I , . . . , I are non-negative and theminimum cumulative daily index return of the previous quarter is at least − i.e., I r ≥ and d min ,r − ≥ − , in which case, the note is called on the correspondingobservation date, and the client receives a positive return in these instances. • The seventh case occurs if all of the index returns, I , . . . , I are negative and everyminimum cumulative daily index return is at least − . In this case, the note maturesand the client receives a net payment of zero, i.e., the client breaks even. • The eighth and last case occurs if any d min ,r ≤ − , i.e. the minimum cumulativedaily index return during some quarter of the observation period fell below − . Inthis case, the note matures and the client receives a negative net payment of I .Based on our function for net payment above, we obtain: E ( Net Payment ) = 0 . × P ( I ≥ . × P ( I < and I ≥ and d min , ≥ − )+ 1 . × P ( I j < , j = 1 , and I ≥ and d min , ≥ − )+ 2 . × P ( I j < , ≤ j ≤ and I ≥ and d min , ≥ − )+ 2 . × P ( I j < , ≤ j ≤ and I ≥ and d min , ≥ − )+ 3 . × P ( I j < , ≤ j ≤ and I ≥ and d min , ≥ − )+ 0 × P ( I j < , ≤ j ≤ and d min , ≥ − )+ 10 × E ( I | d min , < − and I j < , ≤ j ≤ × P ( d min , < − | I j < , ≤ j ≤ × P ( I j < , ≤ j ≤ . (4.3)Consider the probabilities appearing in the second thru sixth terms in (4.3), It is clear thatfor each r = 1 , . . . , , P ( I j < , j = 1 , . . . , r − , I r ≥ , d min ,r − ≥ ) ≤ P ( I j < , j = 1 , . . . , r − , I r ≥ . (4.4)25lso, as noted earlier, d min , ≡ d min , the minimum cumulative index return encounteredin Section 3. Therefore, it follows from (4.3) and (4.4) that E ( Net Payment ) ≤ . × P ( I ≥ . × P ( I < , and I ≥ . × P ( I j < , j = 1 , and I ≥ . × P ( I j < , j = 1 , , and I ≥ . × P ( I j < , ≤ j ≤ and I ≥ . × P ( I j < , ≤ j ≤ and I ≥ × P ( I j < , ≤ j ≤ and d min ≥ − )+ 10 × E ( I | d min < − and I j < , ≤ j ≤ × P ( d min < − | I j < , ≤ j ≤ × P ( I j < , ≤ j ≤ . (4.5)The expression on the right hand side is precisely the expected net payment functionwhich we derived in Section 3. Therefore, the average return to clients from the interpre-tation of the payment procedure considered here in Section 4 can be no greater than theaverage return from the interpretation in Section 3. Likewise, average losses for this inter-pretation will be greater than losses in the interpretation of the payment procedure in Section3. Consequently, it is clear that clients are likely to suffer even greater losses under the newinterpretation of the payment procedure considered in this latter section. In our analysis, we did not consider the effects of income taxes or commissions related to thesales of these reverse convertible notes. The inclusion of these expenses would have resultedin smaller returns and greater losses for note-holders.Further, any financial adviser who was required to exercise fiduciary duty to their clients,could not have done so by recommending that clients purchase these notes. The SEC hasstated that a financial adviser has a fiduciary duty to make reasonable investment recommen-dations to clients. Graham [4] defines an investment operation as “one which upon thoroughanalysis promises safety of principal and an adequate return.” Had financial advisers carriedout a thorough analysis of these reverse convertible notes then they would have realized thatthe notes promised neither safety of principal nor adequate return to clients. As a result, anyfinancial adviser acting as a fiduciary, should have opposed the purchase of these notes.We have seen recently an increase in the sales of similar notes by many financial firms;the prospectuses for these notes are available on the SEC’s website. This increase in salesperhaps is due to the fact that the financial markets have become increasingly bullish since2009 and as a result these notes now appear more attractive to persons looking to purchasefixed-income securities with yields higher than available through U.S. Treasury securities.However, we predict that if markets undergo substantial downward movements then clientsare again likely to lose substantial capital from purchases of these notes.26urther, these notes were issued during a time when future market conditions were un-certain and direct sales of stock may accelerated further downturns. Thus, these notes werea clever marketing scheme used by financial institutions which wanted to insure themselvesagainst substantial market declines.It is particularly noteworthy to observe the actual outcome for the reverse convertiblenote [8] which was analyzed in the previous sections. In this case, the Index Starting Levelwas . , the Trigger Level was . , and the Index Ending Level and outcome for eachobservation date are provided in Table 1:Table 1: Actual Outcome of a Reverse Convertible NoteObservation Date Index Ending Level OutcomeMay 5, 2008 365.48 Below Index Starting Level and AboveTrigger Level; Securities NOT calledAugust 5, 2008 302.05 Below Index Starting Level and AboveTrigger Level; Securities NOT calledNovember 5, 2008 201.77 Below Index Starting Level AboveTrigger Level; Securities NOT calledFeb 5, 2008 121.51 Below Index Starting Level and BelowTrigger Level; Securities NOT calledMay 5, 2009 155.52 Below Index Starting Level and BelowTrigger Level; Securities NOT calledAugust 5, 2009 189.37 Below Index Starting Level and AboveTrigger Level; Securities NOT called Settlement Amount (per $10) $5.13 (total return of -48.7%, or acompound return of -35.9% perannum)
Note that the index return on the Final Valuation date equalsIndex Starting Return − Index Ending LevelIndex Starting Level = 189 . − . . − . , which represents a total loss to the note-holder of 48.7%. Thus, the settlement amount is $10 × (1 + Index Return on the final valuation date ) = $10 × (1 − . . . eferences [1] Conrad, J. and Kaul, G. (1998). An anatomy of trading strategies. The Review of Fi-nancial Studies , 11, 489–519.[2] Financial Industry Regulatory Authority (FINRA) (2011).
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