a swap difference of 7 points.Therefore the bank buys dollars outright one week after spotat 1.7505 − 0.0007 = 1.7498. The other side of the one weekoutright price is 1.7510 − 0.0005 = 1.7505.(ii) Suppose the customer wishes to sell dollars for outrightvalue tomorrow. This is equivalent to selling dollars for valuespot and, at the same time, undertaking a swap to sell dollarsfor value tomorrow and buy dollars back for value spot.Again, the bank spot dealer buys dollars for value spot onthe left at 1.7505. However, the bank forward dealer ‘buys andsells’ dollars tomorrow against spot on the right at a swapdifference of 14 point. Furthermore (because Swiss interest ratesare lower than dollar rates), the dollar is at a discount to theSwiss franc: the ‘bigger number’ 12 is on the left. The dollar istherefore worth less on the far date (i.e. spot) and more on thenear date (i.e. tomorrow). The swap difference is therefore addedto the spot rate, rather than subtracted, to give an outrightvalue tomorrow price of 1.7505 + 14 = 1.750525. The other sideof the value tomorrow outright price is 1.7510 + 12 = 1.75105.
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A simple rule to remember for the calculation of dates earlier thanspot is ‘reverse the swap points and proceed exactly as for a forwardlater than spot’. In the above example, this would mean reversing‘ 21 / 14 ’ to ‘ 41 / 12 ’. The outright value tomorrow price is then (1.7505 +1)/(1.7510 + 12 ), obtained by adding the swap points to the spot rate4because the ‘bigger’ swap number is now on the right. However, itis important always to remember to make this reversal in yourhead only! Never actually quote the price in reverse!O/N prices are the only regular swap prices not involving the spotvalue date – i.e., the O/N swap is from today until tomorrow, notfrom today until spot. To calculate an outright value today price, it istherefore necessary to combine the spot with both the overnight priceand the tom-next price:
(iii) Suppose the customer wishes to sell dollars for outrightvalue today. This is equivalent to three separate transactions:selling dollars for value spot, undertaking a swap to sell dollarsfor value tomorrow and buy dollars back for value spot (tomnext swap) and undertaking another swap to sell dollars forvalue today and buy dollars back for value tomorrow (overnightswap). The price is therefore 1.7505 + 14 + 34 = 1.7506.The rules can be thought of in terms of premiums and discounts,which apply in the same way as with forwards after spot. The swapsin the previous example show a dollar discount because Swiss francinterest rates are lower than dollar interest rates. Consequently, ifthe customer sells dollars value today and not value spot, he willreceive the currency with the lower interest rate two days early. Theextra point which he receives from the bank reﬂects this.Example 15The following are rates for EUR/USD:Spot:O/N:T/N:S/N:
0.8763/581.30/1.801.60/2.101.70/2.20
spot:S/N
0.8763 /0.87680.00017/0.00022
outright day after spot:
0.87647/0.87702
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spot:T/N
0.8763 /0.87680.00016/0.00021
outright tomorrow:
0.87609/0.87664
spot:T/NO/N
0.8763 /0.87680.00016/0.000210.00013/0.00018
outright today:
0.87591/0.87651
Forward-forwardsA forward-forward swap is a swap deal between two forward datesrather than from spot to a forward date – for example, to sell USdollars 1 month forward and buy them back 3 months forward. Inthis case, the swap is for the 2-month period between the 1-monthdate and the 3-month date. A company might undertake such a swapbecause it has previously bought dollars forward but wishes now todefer the transaction by a further two months, as it will not need thedollars as soon as it thought.As with forward-forward interest rates, FX forward-forwards arereferred to by the beginning and end dates of the forward period,compared with the spot value date. Thus the forward-forwarddescribed here is a ‘1v3’, meaning that the forward period starts1 month from now and ends 3 months from now.From the bank’s point of view, a forward-forward swap can be seenas constructed from two separate swaps, each based on spot.Example 16USD/CHF spot rate:1-month swap:3-month swap:
1.7325/ 3565/ 61160/155
If our bank’s counterparty wishes to sell dollars 1 month forward,and buy them back 3 months forward, this is the same as undertaking one swap to buy dollars spot and sell dollars 1 monthforward, and another swap to sell dollars spot and buy dollars3 months forward.As swaps are always quoted as how the quoting bank ‘sells andbuys’ the base currency on the left, and ‘buys and sells’ the basecurrency on the right, the counterparty can ‘buy and sell’ dollars
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spot against 1 month at a swap price of −65, with settlementrates of spot and (spot − 0.0065). He can simultaneously ‘sell andbuy’ dollars spot against 3 months at a swap price of −155 withsettlement rates of spot and (spot − 0.0155). He can thereforedo both – ‘sell and buy’ dollars 1 month against 3 months – atsettlement rates of (spot − 0.0065) and (spot − 0.0155), whichimplies a difference between the two forward settlement rates of(−155) − (−65) = −90 points.Conversely, the counterparty can ‘buy and sell’ dollars1 month against 3 months at a swap price of (−160) − (−61) =−99 points. The two-way price is therefore −99/−90, quoted asusual without the ‘−’ signs, as 99/90.As with a swap from spot to a forward date, the two settlement prices in a forward-forward must be based on a currentmarket rate. In the above example, using the middle spot rate, forexample, the settlement rates could be 1.7265 (= 1.7330 − 0.0065)for 1 month forward and 1.7175 (= 1.7330 − 0.0155) for 3 monthsforward.These settlement rates would enable our forward dealer to coverhis position exactly with another bank. We could, for example, askanother bank for a 1-month swap price to cover the ﬁrst leg of theforward-forward. Assuming prices have not moved, we could dealat −65 points with settlement rates of 1.7330 (spot) and 1.7265(1 month). We could then cover the second leg with a 3-month swapat another bank’s price of −155, with settlement rates of 1.7330 (spot)and 1.7175 (3 months). The spot settlements would offset each otherand the forward settlements would exactly offset the settlements withour own counterparty.In practice, however, forward dealers often base the settlement ratefor the ﬁrst leg on a middle rate for spot and also a middle rate for thenear forward date. In the example above, this would give a settlementrate of 1.7330 (middle) − 0.0063 (middle) = 1.7267. The settlementrate for the second leg would then be 1.7267 − 0.0090 = 1.7177. Thedifference between the two settlement rates is still the −90 pointsagreed, but the settlement rates are slightly different.If the yield curves of the two currencies cross over at some pointbetween the two forward dates, the forward points will show apremium for one period but a discount for the other. In this case, it isimportant to make sure that the ‘+’ and ‘−’ signs are correctly shown
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before constructing the forward-forward price, and also to rememberthat two negatives make a positive.
Forward-forward price after spotFirst, insert ‘−’ signs where necessary. Then:left side = (left side of far-date swap)− (right side of near-date swap)right side = (right side of far-date swap)− (left side of near-date swap)The bid–offer spread of the resulting price is the sum of the twoseparate bid–offer spreads.
Example 17Given the following prices, you need to ‘sell and buy’ sterlingagainst euros forward-forward 2v5.EUR/GBP spot:2-month swap:5-month swap:
0.6235/4517/1325/31
The 2-month price is −17/−13 (a euro discount) but the 5-monthprice is +25/+31 (a euro premium). The 2v5 price is therefore:25 − (−13) = 38 on the leftand
31 − (−17) = 48 on the right
On this forward-forward price of 38/48, you deal at 38 (thequoting bank always ‘sells and buys’ the base currency on theleft).The middle rate for 2 months forward is 0.6225 (=0.6240 −0.0015). You therefore agree to settlement rates of 0.6225 for2 months and 0.6263 (=0.6225 + 0.0038) for 5 months.Care needs to be taken with swaps from before spot until after spot.
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Example 18USD/CHF spot rateT/N swap:3-month swap:
1.7325/353/2160/155
If a counterparty requests a price to ‘sell and buy’ dollarstomorrow against 3 months after spot, this can be seen asa price to sell tomorrow and buy spot at (−2) points withsettlement rates of (spot + 0.0002) and spot, and a price to sellspot and buy 3 months later at (−155) points, with settlementrates of spot and (spot − 0.0155). The total price is therefore the difference between (spot − 0.0155) and (spot + 0.0002),which is (−155) − (+2) = −157. The other side of the price is(−160) − (+3) = −163. The 2-sided price is therefore 163/157.
Forward-forward price from before spot until after spotFirst, insert ‘−’ signs where necessary. Then:left side = (left side of far-date swap)+ (left side of T/N or T/N and O/N)right side = (right side of far-date swap)+ (right side of T/N or T/N and O/N)The bid–offer spread of the resulting price is the sum of theseparate bid–offer spreads.
A forward-forward from before spot might be for a regular periodfrom the start date, rather than until a regular date after spot. Forexample, a swap for ‘3 months out of today’ would mean that the enddate is not 3 months after the spot date but 3 months after today. Theregular 3-month price is therefore too long a period to use, and thecorrect period must be interpolated (see Chapter 2) from the 2-monthprice and the 3-month price.Example 19Today is 15 April, spot is 17 April, 2 months from spot is 17June and 3 months from spot is 17 July. You need to quote to
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131
customer a swap price for 3 months out of today which is from15 April to 15 July.GBP/USD spot rateO/N swap:T/N swap:2-month swap (61 days):3-month swap (91 days):
1.4325/354/33/2100/95160/155
The period from spot until 17 June is 61 days.The period from spot until 15 July is 89 days.The period from spot until 17 July is 91 days.Interpolating, the left side of the swap price from spot until 15July is:(89 − 61)= − 156−100 + (−160 − −100) ×(91 − 61)Similarly, the right side is:(89 − 61)−95 + (−155 − −95) ×= − 151(91 − 61)To calculate the swap we need, we therefore add the followingprices:O/N swap:T/N swap:swap from spot to 15 July:
−4/ −3−3/ −2−156/−151−163/−156
Calculation summaryIt might be helpful to collect together here the various rules whichapply to calculating forwards:The currency with higher interest rate (=the currency at adiscount) is worth less in the future. If this is the base currency,the points are negative.The currency with lower interest rate (=the currency at apremium) is worth more in the future. If this is the base currency,the points are positive.
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The bank quoting the price always ‘sells and buys’ the basecurrency on the left and ‘buys and sells’ the base currency onthe right.If the swap price appears larger on the right than the left, it ispositive.If the swap price appears larger on the left than the right, it isnegative.For outright forwards later than spot:The left swap price is added to (or subtracted from) the left spotprice.The right swap price is added to (or subtracted from) the rightspot price.For outright deals earlier than spot:Calculate as if the swap price were reversed.Forward-forwards:First, insert ‘−’ signs where necessary. Then:after spot:left side = (left side of far-date swap)− (right side of near-date swap)right side = (right side of far-date swap)− (left side of near-date swap)from before spot until after spot:left side = (left side of far-date swap)+ (left side of T/N or T/N and O/N)right side = (right side of far-date swap)+ (right side of T/N or T/N and O/N)In general:Of the two prices available, the customer gets the worse one.Thus if the swap price is 3/2 and the customer knows that thepoints are in his favour (the outright will be better than the spot),the price will be 2. If he knows that the points are against him(the outright will be worse than the spot), the price will be 3.The bid–offer spread of the resulting price is the sum of theseparate bid–offer spreads.
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Cross-rate calculationsA forward cross-rate is calculated in a similar way to a spot cross-rateas explained in the previous chapter, but using outrights. The sameis true for any outright calculated from two other outrights and istrue for short dates as well as any other outrights:• To calculate an outright from two exchange rates with the samebase currency (for example, the dollar), divide opposite sides of theindividual outrights.• To calculate an outright from two exchange rates with the samevariable currency, again divide opposite sides of the individualoutrights.• To calculate an outright from two rates where the commoncurrency is the base for one but the variable for the other, multiplythe same sides of the individual outrights.To calculate cross-rate forward swaps, the process above must betaken a step further:(i)(ii)(iii)(iv)
calculate the spot cross-rate as usualcalculate the two individual forward outrights as abovefrom (ii) calculate the forward outright cross-ratesubtract (i) from (iii) to give the cross-rate swap
Example 20Based on the following prices, what is the EUR/GBP 6-monthswap?EUR/USD spot:6 months:GBP/USD spot:6 months:EUR/USD spot:6 months:
0.8734/3923/191.4237/4263/670.8734/ 0.8739−0.0023/−0.0019
outright:
0.8711/ 0.8720
GBP/USD spot:6 months:
1.4237/ 1.42420.0063/ 0.0067
outright:
1.4300/ 1.4309
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0.8734 ÷ 1.4242 = 0.6133
0.8739 ÷ 1.4237 = 0.6138
0.8711 ÷ 1.4309 = 0.6088
0.8720 ÷ 1.4300 = 0.6098
EUR/GBP 6-month outright:EUR/GBP spot:EUR/GBP 6-month swap:
0.6088/ 0.60980.6133/ 0.6138−0.0045/−0.0040 i.e. 45/40
The same approach can be used for calculating a USD-based ratefrom two cross-rates, and also for calculating short-date swaps.Example 21Based on the following prices, what is the USD/SEK O/N swap?EUR/USD spot:O/N:T/N:EUR/SEK spot:O/N:T/N:
0.8734/390.6/0.40.7/0.58.3975/952.4/2.62.3/2.5
EUR/USD spot:T/N:
0.8734 / 0.8739−0.00007/−0.00005
outright tomorrow:O/N
0.87345/ 0.87397−0.00006/−0.00004
outright today:
0.87349/ 0.87403
EUR/SEK spot:T/N:
8.3975 / 8.39950.00023/ 0.00025
outright tomorrow:O/N:
8.39725/ 8.399270.00024/ 0.00026
outright today:
8.39699/ 8.39903
8.39699 ÷ 0.87403 = 9.607210 8.39903 ÷ 0.87349 = 9.6154858.39725 ÷ 0.87397 = 9.608167 8.39927 ÷ 0.87345 = 9.616200USD/SEK outright tomorrow: 9.608167/9.616200USD/SEK outright today:9.607210/9.615485USD/SEK O/N swap:
0.000715/0.000957 i.e. 7.1/9.6
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Remember that in calculating the outright today and outrighttomorrow rates above for EUR/USD and EUR/SEK, each spotprice is combined with the opposite side of the swap price andthe sign is ‘reversed’. Similarly, in the ﬁnal calculation of theO/N swap price for USD/SEK, the swap price appears to be‘reversed’ for the same reason.In order to calculate a cross-rate forward-forward, extend theprocess: calculate the cross-rate swap for the near date of theforward-forward and also the cross-rate swap for the far date, andthen combine them as usual.It might be possible to use cross-rates to construct an outright ina more cost-efﬁcient way if prices are out of line, as in the followingexample.Example 22Based on the following prices available to you in the market,what is best price at which you can buy euros against sterling,6 months forward outright?EUR/GBP spot:6 months:EUR/USD spot:6 months:GBP/USD spot:6 months:
0.6132/3744/390.8734/3923/191.4237/4263/67
The outright EUR/GBP must be constructed from a spot and aswap. We are dealing on the right-hand side.For the spot, the straightforward EUR/GBP price of 0.6137 isbetter than the price of 0.6138 which can be constructed via thecross-rate calculation (see Example 20, which is based on thesame market prices).For the swap, the price of −40 points which can be constructedvia the cross-rate calculation (again see Example 20, which isbased on the same market prices) is better than the straightforward EUR/GBP price of −39 points.Therefore an outright price of 0.6097 (=0.6137 − 0.0040) canbe achieved by:
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Foreign exchange and money markets
buying euros against sterling directly at 0.6137selling and buying euros against dollars at −19 pointsbuying and selling sterling against dollars at +63 points
Non-deliverable forwards(NDFs)A non-deliverable forward (NDF) is a forward outright where, insteadof settling the outright amounts at maturity, the two parties agreeat the outset that they will settle only the change in value betweenthe forward rate dealt and the spot rate two working days beforematurity. The economic effect is the same as if a normal forwardoutright had been dealt and then closed out two days before maturityby an offsetting spot deal.One advantage of an NDF is that it can be used to trade nonconvertible currencies. In this case, the settlement at maturity ofthe NDF is necessarily made in a convertible currency – generallyone of the pair of currencies being traded. An NDF also reducesthe counterparty credit risk, as the risk is limited to the settlementamount and does not involve the usual settlement risk of the wholeprincipal amount of the deal.An NDF is a contract for differences, analogous to an FRA forinterest rates.Example 23It is now 15 April. A company needs to buy TWD 100 millionagainst sterling for settlement in 3 months (17 July). It buys thisamount 3 months forward using an NDF, at a rate of 49.06.On 15 July, the company and the bank agree a GBP/TWDspot rate of 48.43 for settlement.100,000,000= 2,038,320.4249.06100,000,000= 2,064,835.8548.43The proﬁt or loss is settled as if the company had bought TWD100 million at 49.06 and then sold again at 48.43. As this isan NDF, there is no need to transact an actual spot deal on15 July.
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137
buy TWD 100,000,000, sell GBP 2,038,320.42sell TWD 100,000,000, buy GBP 2,064,835.85proﬁt GBP26,515.42The company therefore receives a settlement of GBP 26,515.42from the bank on 17 July.If this were a speculative transaction, nothing further wouldbe required. However, the company physically needs the TWD100 million. Therefore, it buys TWD 100 million spot on 15July for value 17 July at 48.43 in the normal way. Thiswill cost GBP 2,064,835.85. The net effect for the companyis therefore a cost of GBP 2,038,320.42 (=GBP 2,064,835.85 −GBP 26,515.42), leaving the company in the same position as ifit had been able to buy the TWD 100 million 3 months forwardin the ﬁrst place.In practice, the net effect might be slightly worse than thisbecause the NDF would be settled against a middle referencerate but the spot deal would be at bid rate. However, if thecompany had been able to transact a normal forward outright inthe ﬁrst place, that also would have involved the bid side of thespot rate.A source must be agreed for the reference spot rate to be usedfor calculating the settlement amount if the NDF is left open untilmaturity. The conventional source will vary from currency to currencyand could, for example, be a central bank ﬁxing rate or a ratepublished on Reuters or Telerate.If the NDF is used speculatively, an alternative is for the NDF to beclosed out early and for the settlement amount to be based on thedifference between the two rates dealt. In this case, the settlementcan be paid at maturity date of the two NDFs or discounted to apresent value and settled early.Example 24It is now 15 April. A dealer takes a speculative 3-month positionby buying TWD 100 million against sterling for value 17 July,using an NDF, at a rate of 49.06.On 22 April, the dealer closes out the position with anotherNDF for value 17 July at 48.43.As in the last example, the dealer is due a proﬁt ofGBP 26,515.42 for settlement on 17 July:
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buy TWD 100,000,000, sell GBP 2,038,320.42sell TWD 100,000,000, buy GBP 2,064,835.85proﬁt GBP26,515.42There are 84 days from 24 April to 17 July. Assuming thatthe sterling interest rate for the period is say 5%, the presentvalue of this amount could be settled instead on 24 April asGBP 26,213.78:GBP 26,515.42 = GBP 26,213.78841 + 0.05 ×365
Time optionsWhen a bank makes a forward outright deal with a company, it quotesa rate for a ﬁxed date, which means the company must deliver onecurrency and receive another on that date.If the company has a commitment in the future but does notknow the exact delivery date required, it has an alternative meansof covering this exposure in the traditional foreign exchange market,using a time option (also known as an option forward). This allows thecompany to deal now, but to choose the maturity date later withina speciﬁed period. There is, however, no choice about whether todeliver – delivery must take place at some point during the agreedperiod, for the amount and rate agreed.It is important not to confuse time options in this sense withcurrency options. A currency options entails the up-front payment ofan ‘insurance premium’, in return for which the customer has theright to choose whether or not to deal at all.In pricing a time option, the bank will always assume that thecompany will take delivery of the currency at the worst possible timefor the bank. Therefore the company will always be charged the worstpossible forward rate within the period of the time option.Example 25The bank’s customer wishes to transact a forward deal in dollarsagainst Swiss francs, with a ﬂexible delivery date of between 2and 3 months. Given the following market prices, what rate doesthe bank quote?
Forward outrights and swaps
USD/CHF spot rate:2-month swap:3-month swap:
139
1.7325/ 35130/125160/155
The 2-month outright would be 1.7195/1.7210:1.7325/1.7335130/1251.7195/1.7210The 3-month outright would be 1.7165/1.7180:1.7325/1.7335160/1551.7165/1.7180If the customer is selling dollars, the worse price is the 3-monthprice of 1.7165. If the customer is buying dollars, the worse priceis the 2-month price of 1.7210. Therefore the time option price isa swap price of 160/125, or an outright price of 1.7165/1.7210.The advantage to a company of a time option is its ﬂexibility. Thecompany can lock in a ﬁxed exchange rate at which it knows itcan deal. There is no exposure to interest rate changes which wouldaffect it if commitments were covered with a forward outright whichsubsequently needed to be adjusted by means of forward swaps. Thedisadvantage is the cost, given the wide bid/offer spread involved,particularly if the time period of the option is a long one.A time option price is the best for the bank/worst for thecustomer over the time option period
Long-dated forwardsThe formulas we have already seen for a forward outright less than1 year are:Forward outrightdays1 + variable currency interest rate ×variable year= spot ×days1 + base currency interest rate ×base year
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forward swap⎞⎛days⎜ variable currency interest rate × variable year ⎟⎜ ⎟⎠⎝days− base currency interest rate ×base year= spot × days1 + base currency interest rate ×base yearThese were derived from the assumption that the interest on a depositor loan is paid on a simple basis as described in Chapter 2. Fordeposits and loans over 1 year, the interest must be compoundedbecause interest is normally paid at the end of each year. In thiscase, the corresponding theoretical formulas for a forward outrightand a forward swap are as follows:For long-dated forwards:Forward outright(1 + variable interest rate) number of years= spot ×(1 + base interest rate)Forward swap(1 + variable interest rate) number of years= spot ×−1(1 + base interest rate)The interest rates in this case are bond basis rates – quoted on thebasis of a true calendar year rather than a 360-day year. Thesetheoretical formulas are not precise in practice for two reasons.Firstly, this compounding does not take account of reinvestmentrisk. This problem is overcome by using zero-coupon yields for theinterest rates. More importantly, the market in long-dated forwardsis not very liquid and spreads are very wide. The prices available inpractice