ate, with limited riskand limited return, in the rise of a currency.Butterﬂy spread A combination of a bull spread and a bear spread; the strategy normallygives a maximum return and maximum loss.Calendar spread A strategy involving the buying and selling of options with differentexpiration dates.Call option An option which gives the holder the right to buy, and the writer the obligationto sell, a predetermined amount of a currency to a predetermined date at a predeterminedexchange rate.Clearing corporation An organization which matches and guarantees option trades on anexchange.Combination A strategy involving the buying of call and put options with different strikesbut with the same expiration dates.Condor spread A variation on a butterﬂy spread but with strikes further apart.Conversion arbitrage A riskless strategy involving the buying of a currency and the simultaneous buying of a put and writing of a call option, both normally European style and ofthe same strikes and expiration.Covered write A strategy involving the buying of a currency and the writing of a call option,or the selling of a currency and the writing of a put option.Credit premium The premium received when an option is written.Debit premium The premium paid when an option is purchased.Delta The ratio by which the price of the option moves relative to the underlying spot orfutures contract.
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Delta spread (trade or hedge) A trade involving the adjustment of the long or short optionspositions by the ratio of the delta.Discount Term used to describe an option trading for less than its intrinsic value.Downside protection For covered calls, the ‘cushion’ against loss provided by the optionpremium received.Early exercise The exercise of an option before its expiration date.European option An option which may only be exercised on the expiration day.Exchange-traded market The organized marketplace for option trading purposes.Exercise Process by which the holder of an option elects to take delivery of (call) or deliver(put) a currency according to the contract terms.Exercise price The price at which the option holder has the right to buy or sell the underlyingcurrency or currency futures contract.Expiration cycle In the exchange-traded options market, the time frame in which listedoptions run.Expiration date The last day on which a holder can exercise his or her option.Expiration time In the over-the-counter market the latest time an option may be exercisedis usually 3 p.m. London time, 10 a.m. New York time, or 3 p.m. Tokyo time, on thatparticular day.Fair value Usually refers to the value of an option premium according to a mathematicalmodel.Gamma The change in the delta for a unit change in the spot price.Hedge ratio The ratio of options to buy or sell against a spot position in order to create ariskless hedge.Implied volatility The expected standard deviation of percentage price changes.In-the-money An option that has intrinsic value. For a call, the strike is below the spot rate.For a put, the strike is above the spot rate.Intrinsic value The value of an option, were it to be exercised immediately.Leg One component of a multiple option strategy.Margin Initial margin is the amount required to be put up as collateral by the option writer tothe clearing corporation. It is equivalent to a performance bond. Variation or maintenancemargin is further cover required, should the option position move against the writer.Mark to market Daily adjustment of an account to reﬂect accrued proﬁts and losses.Money spread Strategy involving the buying and writing of options with different strikesbut with the same expiration dates. It can be put on for a credit or debit to take advantage ofa directional market move.Naked position A short option position which is not covered by the underlying currency orwith another option.Out-of-the-money An option with no intrinsic value. For a call, the strike is above the spotrate and for a put the strike is below the spot rate.Over-the-counter market Customized option market usually traded directly between banksand their customers or with other banks.Premium The amount of money paid by a buyer and received by a seller for an option.Put option An option giving the holder the right to sell and the writer the obligation to buy, apredetermined amount of currency to a predetermined date at a predetermined exchange rate.Ratio spread Strategy involving the sale of an amount of call options in excess of the amountof a long call option position held, or the sale of put options in excess of the amount of along put option position held.
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Ratio write Strategy involving the sale of call options in excess of the amount of a longcurrency position held, or the sale of put options in excess of the amount of a short currencyposition held.Reversal arbitrage Riskless trade involving the selling of a currency and the simultaneousbuying of a currency call and writing of a currency put option, both normally European styleand of the same strikes and expiration.Settlement date Two business days following exercise. It is the day on which the currenciesinvolved in the option transaction are exchanged.Spread Strategy involving the simultaneous buying and selling of options on the samecurrency.Straddle Strategy involving the buying of call and put options with the same strikes andmaturity.Strangle Strategy involving the buying of call and put options with different strikes but withthe same expiration dates.Strike The price at which the option holder has the right to buy or sell the underlying currencyor currency futures contract.Theta The change in the premium for a unit change in time.Time value The amount by which an option premium exceeds its intrinsic or in-the-moneyvalue.Vega The change in the premium for a unit change in implied volatility.Volatility The standard deviation of percentage price changes.Writer One who sells an option.
16Foreign Exchange FuturesAlthough futures contracts on commodities have been traded on organized exchanges sincethe 1860s, ﬁnancial futures are relatively new, dating from the introduction of foreign currencyfutures in 1972. The basic form of the futures contract is identical to that of the forwardcontract, whereby a futures contract obligates its owner to purchase a speciﬁed asset at aspeciﬁed exercise price on the contract maturity date. Likewise, currency futures are deﬁnedas a standardized contract/agreement to sell or buy a speciﬁc amount of a currency at a particularprice on a stipulated future date.A future is a contract to buy and sell a standard quantity of a given instrument, at anagreed price, on a given date.Futures developed in response to the substantial volatility for currency trading that occurredfollowing the 1971 shift from ﬁxed to ﬂexible currency exchange rates.
16.1 TWO-SIDED RISKLike a forward contract, the futures contract has a two-sided risk. However, in marked contrastto forwards, credit or default risk can be virtually eliminated in a futures market. Firstly,instead of conveying the value of a contract through a single payment at maturity, any changein the value of a futures contract is conveyed at the end of the day in which it is realized. Forexample, suppose that, on the day after origination, the ﬁnancial price rises and, consequently,the ﬁnancial instrument has a positive value. In the case of a forward foreign exchange contract,this value change would not be received until maturity. With a futures contract, this changein value is received at the end of the day. In the language of the futures markets, the futurescontract is cash-settled or marked-to-market daily.Since the value of the futures contract is paid or received at the end of each day, a futurescontract can be likened to a series of forward contracts. That is, a futures contract is like asequence of forwards in which the ‘forward’ contract written on day 0 is settled on day 1 andis replaced, in effect, with a new ‘forward’ contract reﬂecting the new day 1 expectations. Thisnew contract is itself settled on day 2 and replaced, and so on until the day the contract ends.In other words, a futures contract can be thought of as ‘rolling over’ forward contracts on adaily basis. Strictly speaking, the futures price and the forward price are not quite the same butas a practical matter they are so close that little accuracy is lost in viewing them as identical.Therefore, analogous to forward contracts, the futures price is that contract price which resultsin the futures contract having zero value to both the buyer and the seller each day the contractis settled and restruck.Secondly, all market participants, sellers and buyers alike, post a performance bond, i.e.margin. If a futures contract increases in value during the trading day, this gain is added to
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the margin account at the end of the day. Conversely, if the contract loses value, this loss isdeducted from the margin account. If the margin account balance falls below some agreed-uponminimum, the holder will be required to post an additional bond. Hence, the margin accountmust be replenished or the holder’s position will be closed out.
16.2 EXCHANGE MEMBERSThere are two types of exchange members who can trade any futures contract. First, thereare commission brokers, or ﬂoor brokers, who execute orders for non-members. These ordersfrom non-members will originate through futures commission merchants – which are usuallyorganizations, for example, brokers and commercial banks. These types of organizations willsolicit orders for futures trading. Futures commission merchants also hold their clients’ marginmonies and handle all margin accounting. The ﬂoor broker executing the order may or maynot be afﬁliated with the futures commission merchant that originated the order.The second type of exchange member is called a ‘local’, and is simply an individual tradingfor his or her own account. Essentially, locals are willing to hold positions, inter- or intra-day,acting much like a market maker who hopes to proﬁt from the bid–offer spread or marketmoves.
16.3 CLEARING CORPORATIONAn important feature of an organized futures exchange is the Clearing Corporation. Essentially,the Clearing Corporation interposes itself as the seller to every buyer and the buyer to everyseller. In other words, the Clearing Corporation becomes the counterparty to every trade,guaranteeing the opposite side of every transaction. This has several attractive features:
r Firstly, the buyer of a futures contract need not be concerned with the creditworthiness ofthe seller. If the buyer’s position is doing well, i.e. the futures price is rising, then the buyer isguaranteed the daily receipt of the variation margin by the Clearing Corporation, independentof whether the original seller was able to pay that same variation margin. In this example,the Clearing Corporation looks to the member ﬁrm, who originated the futures sale, for thetimely payment of the daily variation margin independent of whether the original seller haspaid sufﬁcient margin into the account.r Secondly, the margin accounting problem is signiﬁcantly simpliﬁed. There is now only oneentity with which the member ﬁrm must deal in settling margin calls, as opposed to havingto exchange monies with all other member ﬁrms.r In addition, the Clearing Corporation will net out all margin calls and receipts for a singlemember ﬁrm across all of that ﬁrm’s positions, such that only one net amount of funds mustbe transferred at the end of each trading day.
16.3.1 Major ExchangesThe major exchanges for ﬁnancial futures include the Chicago Board of Trade (CBT), theChicago Mercantile Exchange (CME), the International Monetary Market (IMM), the LondonInternational Financial Futures Exchange (LIFFE), the New York Futures Exchange (NYFE)and the Kansas City Board of Trade (KC).
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16.4 QUOTING CURRENCY FUTURESHow are currency futures quoted? Generally, in the foreign exchange market, currencies arequoted against the American dollar. For example, a rate of 1.67 Swiss francs per dollar meansthat it takes 1.67 Swiss francs to buy/sell 1 dollar. Of course, there are the exceptions to thisrule, for example sterling. However, currency futures are priced in American terms, in that itquotes how many dollars it takes to buy one unit of foreign currency. They are the reciprocal ofthose used in the cash market. Thus a rate of 1.67 Swiss francs per dollar would be quoted in thefutures market as 0.5988 dollars per Swiss franc (1 divided by 1.67), which means that it costs60 cents to buy one Swiss franc. For each contract, there is a speciﬁc contract size, for exampleone Swiss franc contract is worth 125,000 francs, the Japanese yen is worth 12,500,000 yen,sterling is worth 62,500 pounds, while the euro is worth 125,000 euros.
16.5 TICKS AND DELIVERY MONTHSThe minimum price movement of a currency futures contract is called a tick. The value of a tickis determined by multiplying the minimum tick size by the size of the contract. For example,using the Swiss franc against the dollar, 1-point is $.0001 per Swiss franc, which equals $12.50per contract, while 1-point sterling is worth $.0001 per pound, which equals $6.25 per contract.The contract trading months are on the same quarterly cycle as other ﬁnancial instruments:March, June, September and December. They are also known as the delivery months, becausethe seller of a contract must be prepared to deliver the speciﬁed amount of foreign currency tothe buyer if the seller has not cancelled the obligation with an offsetting purchase. It must besaid that the vast majority of market participants close out their positions before delivery.
16.6 CONTRACT SPECIFICATIONSThe contract speciﬁcations for currencies against the dollar futures are listed below. There arealso, of course, contracts for crosses.ProductAustralian dollarBrazilian real
Trading unit and point description100 000 dollars – physically delivered1 point = $0.0001 per dollar = $10.00 per contract100 000 real – cash settled/2 point = $0.0005 per real = $5.00 per contract
1
British pounds
62 500 pounds – physically delivered1 point = $0.0001 per pound = $6.25 per contract
Canadian dollars
100 000 dollars – physically delivered1 point = $0.0001 per dollar – $10.00 per contract
Euro
125 000 euros – physically delivered1 point = $0.0001 per euro = $12.50 per contract
Japanese yen
12 500 000 yen – physically delivered1 point = $0.000001 per yen = $12.50 per contract
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Mexican peso
500 000 peso – physically delivered1 point = $0.00001 per peso = $5.00 per contract
New Zealand dollars
100 000 dollars – physically delivered1 point = $0.0001 per dollar = $10.00 per contract
‘New’ Russian ruble
2 500 000 ruble – cash settled1 point = $0.00001 per ruble = $25.00 per contract
South African rand
500 000 rand – physically settled1 point = $0.00001 per rand = $5.00 per contract
Swiss franc
125 000 franc – physically delivered1 point = $0.0001 per franc = $12.50 per contract
Swedish krona
2 000 000 krona – physically delivered1 point = 0.00001 dollar/krona = $20.00 per contract
Norwegian krone
2 000 000 krone – physically delivered1 point = 0.00001 dollar/krone = $20.00 per contract
CONCLUDING REMARKSIn summary, a foreign exchange futures contract is a forward contract for standardized currencyamounts and for standard value dates. Buyers and sellers of foreign exchange futures arerequired to post initial margin or security deposits for each contract. Participants also haveto pay brokerage commissions that can be ﬁxed or negotiated depending on the size of thetrade. Foreign exchange futures are only traded on regulated exchanges. In general, futures areused by banks, commodity trading advisers and arbitrage houses, i.e. by ‘professional’ tradersrather than by corporations.
17Exchange for PhysicalAn exchange for physical (EFP) refers to exchanging a physical (cash) position for a futuresposition. This is where a spot interbank transaction can be converted into a futures positionvia an exchange for physical. Consequently, when a cash position is exchanged for a futureposition, the EFP is simply a mechanism by which the cash position is converted to its IMMor Finex equivalent. The EFP represents the current spot (cash) price plus or minus the interestrate differential (cost of carry) between the two currencies, expressed in the futures price, andit is essentially an ex-pit transaction.
17.1 EXAMPLES17.1.1 Example 1In this example, the mechanics of an EFP transaction would be where on, say, 13 December theinterbank spot price for francs for value 15 December is 1.6700 bid–offer 1.6705. The marketuser sells 25 million Swiss francs at 1.6705 for value 15 December and then decides to convertthe short cash position of 25 million Swiss francs to the IMM equivalent for short contractsfor March. Assume the forward pips for 3-month $/Sfr is 1.7/1.9. The resulting transactioncan be viewed as:buys Sfr 25 million at 1.6705 value 15 Decembersells 200 March IMM contracts at 0.5986(1.6705 + 1.9 = 1.67069 and then 1 divided by 1.67069, giving 0.5986)The result for the cash position is:short Sfr 25 million at 1.6705 value 15 Decemberlong Sfr 25 million at 1.6705 value 15 Decemberso that the net cash position is ﬂat and the market user is left with an open IMM futures positionof being short 200 March IMM futures contracts at 0.5986.17.1.2 Example 21. On 13 December, the interbank spot price for British pounds for value 15 December was1.5000 bid–offer 1.5005.2. The trader sells £10 million (equivalent of 160 IMM contracts) at 1.5000 value 15 December.3. The trader decides to convert the short cash position of £10 million to the IMM equivalentof short 160 contracts for March.In the resulting transaction the dealer interbank swap price for 16 March is 70–67, thus thetrader executes a simultaneous swap transaction:
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IMM futures
Single counterparty riskUnregulated marketTailored maturity datesTailored currency amountsGreater liquiditySingle average priceNo exchange feesNo reporting levelsUnrealized gains can only bewithdrawn upon maturity date
Counterparty risk with ExchangeRegulated marketLimited delivery monthsSpeciﬁc contract speciﬁcationsLower average volumeMultiple price ﬁlls
buys £10 million at 1.5000 value 15 December andsells 160 March IMM contracts at 1.4930 (1.5000 − 0.0070)Hence, the result is:short £10 million at 1.5000 value 15 December andlong £10 million at 1.5000 value 15 December.This results in a net cash position which is ﬂat (square) and the trader is short 160 March IMMfutures contracts at 1.4930.
17.2 POINT OF THE EXERCISEThe whole point of the exercise is that the client’s spot cash position is ﬂat and no proﬁt orloss will be generated. The client, however, will have a futures position at only one averageprice for the full amount traded. The EFP transaction is ‘posted’ on the required exchange.Also, there are no commissions or fees charged on the cash side of the transaction, as shownin Table 17.1.
CONCLUDING REMARKSIn summary, the main points are that with the EFP execution, the client’s spot cash positionis ﬂat. No proﬁt or loss will be generated. The client will have a futures position at only oneaverage price for the full amount traded. With the execution of the EFP, the futures price willbe posted (reported) to the exchange and the client will receive a conﬁrmation, as he or shewould for an IMM or Finex transaction.Additionally, fees and commissions will be recorded in exactly the same manner as if thetransaction was executed on any of the exchanges.
Part IIIEssential Knowledge
18Foreign Exchange Dealing RoomsEver since the market experienced problems of unauthorized dealing, fraud and collusion inseveral institutions in the early 1970s, banks and institutions have been encouraged by theirregulatory authorities to impose strict rules and regulations on the type of business transacted,its controls and the manner in which it is monitored. One of the most important aspects inthe control of a dealing operation is the division of duties between dealing and back-ofﬁceprocessing. Dealers should (a) effect the deals in the interbank market, or with clients, (b)maintain sufﬁcient informal records as required to monitor their positions, (c) complete asource document (deal ticket) and, from that moment on, have no further part in or controlover the processing, conﬁrmation or settlement procedures involved.
18.1 COMPOSITION OF A DEALING ROOMThe composition of an institution’s dealing room very much depends on that institution’s size,its spread of market interests and its client base (see Figure 18.1). However, in general, themajority of institutions actively involved in the foreign exchange markets will have some orall of the personnel listed below.18.1.1 Spot dealersSpot dealers are the dealing section the media love the most. Whenever there is pressure on anycurrency, that pressure will be most noticeable on the currency’s spot value. They are marketmakers.18.1.2 Forward dealersForward dealers will normally be active market makers in the same major currencies as tradedby the spot dealers. Here, rather than dealing on the minute-by-minute value of the currency,forward dealers are looking at the relative currency interest rates over the standard forwardperiod dates quoted by the market.18.1.3 Money-market dealersMoney-market dealers control the interbank deposit and loan positions of the bank. The interbank market deals in all periods from overnight (depending on currency) to one-year fundsand above. They are also responsible for funding the bank’s commercial asset portfolio.18.1.4 Treasury product dealersTreasury product dealers are involved in numerous other activities, involving other ﬁnancial instruments and newer off-balance-sheet products. Some of the treasury products lend themselves
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Chiefdealer
Seniortrader
high-levelmanagement
Corporatedesk
currencyorders
FOREIGN EXCHANGETRADER
assistance inperformingdeals
Positionkeeper
monitoring position
errors to reconcile
marketanalysis
Economist+ analysts
completeddealtickets
Backoffice
Figure 18.1 Graphic example of the relationship between all the different personnel in a dealing room
to trading activities (futures, options, etc.), while others are offered by the treasury productsdealers as a customer service in-house, to providing assistance wherever it is needed in themanagement of all types of ﬁnancial risk.18.1.5 Corporate dealersCorporate dealers undertake a totally different role to that of the interbank dealers. In mostsituations, they will not make a market, but will instead obtain rates from the market makers in
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response to their client’s dealing requests and quote for those accordingly. They are, in effect,the ‘middle men’ between the market makers and the clients. In their own right, they have atremendous marketing role to play as in many respects they are the front-line representativesof the institution.18.1.6 Research personnelResearch personnel, generally, have a presence on the trading ﬂoor because they assist thetraders in their task with technical and statistical research, market commentaries and graphicalanalyses, among other things.18.1.7 Position clerksPosition clerks are the workhorses of the dealing room, as they tend to carry out all of theclerical functions that require immediate execution.
18.2 BACK OFFICEIn most institutional organizations, the term ‘back ofﬁce’ is generally accepted as a descriptionof all processing functions connected with a dealing operation outside the actual dealing itself.The back ofﬁce starts, therefore, at the dealing room door and is a vital part of any dealingroom function. It is concerned with settlement of trades, conﬁrmations, maturity ﬁles andreconciliations. Other departments are ﬁnancial control and information technology.
19Managing the Relationship withan InstitutionIn today’s volatile markets, it is becoming increasingly important that salespeople, corporateforeign exchange dealers, corporate foreign exchange advisers or international treasury advisers (various titles depending on the organization) keep themselves abreast of developments inthe markets. More often than not, corporate treasurers, in particular, but also speculators, willuse a small number of people who are constantly in touch with them, to advise accordinglyon market developments. For such a relationship to work smoothly, it is advisable that bothparties establish at the onset some ground rules – namely, what each party expects from theanother. In general, the following should be expected from an adviser:
r Regular market updatesr Prompt and timely response to enquires, questions and issues relating to treasury risk management
r Sound advice on hedging strategies based on fundamental and technical analysis.In return, the adviser needs the following to enhance the quality of service:
r Knowledge of the individual’s philosophy to risk managementr The client’s exposure situation – inﬂows/outﬂows by currency and timingr Short-term or long-term trading strategyr Legal/tax constraints, if any.19.1 ROLE OF THE ADVISEROnce the prerequisites of the relationship have been established, the adviser will have todemonstrate an overall understanding of the foreign exchange market to earn the trust of theclient. However, clients must recognize that they and only they are responsible for any tradingdecision. The adviser is only an adviser, for good or bad.One role of the adviser that is often under-utilized is that of market watcher. By physicallybeing located in the institution’s dealing room, the adviser can follow the market very closely.The client, on the other hand, on many occasions, has to leave and will not be able to watch themarkets. Thus, one of the adviser’s functions is to ‘watch’ a price level. In fact, the client caneven leave an order with the adviser for execution once the necessary conditions are met. Asa client, you can be assured that the adviser will work for your interest. You are the client andthe adviser wants to get your business, but as the institution pays the adviser, he or she needsto make money for the institution. You can assist advisers by dealing with them if their pricesare market prices, and, even at times, by dealing when their prices are slightly off. There is apremium for quality service. The better the service, the more the client beneﬁts.
Client
Commercial banker
Analyst + economist
Salesperson
Back office
Head of sales
Market maker/trader
Figure 19.1 Client–adviser relationshipTarget market
Identify client
THE MARKETINGPROCESS
The sales pitch
Close the sale
THE SALESPROCESS
Deliver the productTHE DEALINGPROCESS
After sales service
Steady earnings stream
Figure 19.2 The marketing process
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19.2 CLIENT–ADVISER RELATIONSHIPAs the salesperson or adviser or corporate trader, etc., will be the primary contact between amarket maker and the client, this relationship is illustrated in Figure 19.1.
19.3 MARKETING PROCESSInstitutions with established client relationships will continue to expand their client lists bycarefully selecting suitable targets. The marketing process for ﬁnancial services is summarizedgraphically in Figure 19.2.
20Foreign Exchange DealingsToday, dealings in the foreign exchange market are usually done over the telephone or via theInternet. The Internet is a very quick and easy way to make a foreign exchange transaction,with ‘click and deal’ or ‘request for price’ systems dominating. However, the majority of alltrading is still transacted over the telephone. Use of the telephone allows for prompt and timelyexecution, but also leaves room for errors in communication.
20.1 ASKING FOR A QUOTEWhen asking for a quote, the following basic information needs to be conveyed to the marketmaker directly or to a corporate foreign exchange dealer: type of transaction, currency pair,and quantity. For example, a client would say:‘I want to buy Sfr 5 million value spot – what is your quote please?’Today, a market maker normally quotes a two-way price, where the trader stands ready to bidfor, or offer up to, some standard amount. The difference between the two prices is the spread.Market convention, where trading is between market ‘professionals’, is not to quote the bigﬁgures. Instead, a trader tends to quote only the last two ﬁgures of the price, the pips. Forexample, if the rate of dollars against the yen were 123.35/40, then the trader would only quote35/40. That is, the trader bids for dollars at 123.35 and offers dollars at 123.40. If a client ischecking prices with other market makers, then the client should inform the trader by sayingthat it is at his risk, i.e. the quote can be changed by the trader. The client should then askwhen a new price is needed. If the client wishes to deal, then the trader’s price would be hit,that is, where one side of the price or the other is accepted. Written conﬁrmation of this deal,whether oral or electronic, will be exchanged and instructions taken as to whether this trade isto be settled or not and the currency amounts are transferred into the designated accounts onthe value date.
20.2 EXAMPLES20.2.1 Example 1XYZ Corporation, based in Japan, needs to raise dollars to pay for a delivery of machine partsfrom America. The treasurer gets in contact with his bank dealer to arrange to buy the dollars inthe spot market. The treasurer’s screens display indicative prices contributed by certain majorbanks. This gives the treasurer a good idea of the current exchange rate. However, this is onlyan indicative rate and it is not a dealable price, i.e. not a price on which to deal. Therefore,the following steps are taken:Treasurer: Please quote me dollar/yen in 10 million dollars. (At this stage there is no mentionof whether the client wants to buy or sell dollars.)
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Dealer: 52/57.Treasurer: I buy 10 million dollars.Dealer: To conﬁrm, you buy 10 million dollars against yen at 123.57 value spot.At this stage, the dealer ﬁlls in a deal ticket with the details of the trade including currencies,amount, which currency is bought and which one is sold, value date, exchange rate, counterpartyand settlement details if known.The treasurer could also have said ‘at 57’ or ‘mine’. All three ways would be correct andwithin market practice. Also, the dealer knew that the treasurer was used to market conventionsand, hence, did not ﬁrst quote the big ﬁgure of 123.
20.