combined total is then discounted back again tothe previous coupon payment date, also using the current yield asthe rate of discount.
Deposits and coupon-bearing instruments
47
This new result is then again added to the coupon payment actuallymade on that date and the combined total again discounted back.This process continues until the nearest coupon payment outstanding. The value thus reached is ﬁnally discounted back to the transaction settlement date.Throughout the process, both the calculation of each couponpayment and each discounting are performed using the exact numberof days in the relevant coupon period.To value a CD with more than one coupon payment outstanding:• Calculate each future cashﬂow• Discount the ﬁnal cashﬂow back to the previous coupondate, using the current yield as the rate of discount• Add the coupon on that date and repeat the process until thenearest outstanding coupon• Add that coupon and discount back to the settlement dateExample 8A CD with face value USD 10 million and semi-annual couponsof 5.5% per annum is issued on 16 May 2000 with a maturity of5 years. You purchase the CD on 20 August 2003 at a yield of4.5%. What amount do you pay?The previous coupon date was 16 May 2003. As 16 November2003 and 16 May 2004 are both Sundays, the remaining coupondates, lengths of coupon period and coupon amounts are asfollows:Coupon date
Days sincepreviouscoupon
17 November 2003 185
Coupon payment
USD 10 million × 0.056 ×= USD 287,777.78
17 May 2004
182
USD 10 million × 0.056 ×= USD 283,111.11
16 November 2004 183
USD 10 million × 0.056 ×= USD 284,666.67
16 May 2005
181
USD 10 million × 0.056 ×= USD 281,555.56
185360182360183360181360
48
Foreign exchange and money markets
The ﬁnal cashﬂow is therefore USD 10,281,555.56. Discountingthis back 6 months at 4.5% gives a value of:USD 10,281,555.56 = USD 10,054,081.961811 + 0.045 ×360Adding this to the 16 November 2004 coupon of USD 284,666.67gives a total of USD 10,338,748.63. Discounting this back6 months at 4.5% gives a value of:USD 10,338,748.63 = USD 10,107,538.681831 + 0.045 ×360Adding this to the 17 May 2004 coupon of USD 283,111.11 givesa total of USD 10,390,649.79. Discounting this back 6 monthsat 4.5% gives a value of:USD 10,390,649.79 = USD 10,159,520.691821 + 0.045 ×360Adding this to the 17 November 2003 coupon of USD 287,777.78gives a total of USD 10,447,298.47. Discounting this back afurther 89 days from 17 November 2003 to 20 August 2003 at4.5% gives a value of:USD 10,447,298.47 = USD 10,332,351.06891 + 0.045 ×360This is therefore the amount for which the CD can be purchased.
Exercises1
You are quoted 4.50% for a 273-day deposit of GBP 2 million.What amount do you expect to receive back at the end of 9months?2 You lend EUR 10 million for 3 months at 4.28%, you lend EUR 3million for 3 months at 4.23% and you borrow EUR 9 million for3 months at 4.25%. What is your net position and your averagerate?3 A bank issues a CD with a face value of USD 5 million, a term of181 days and a coupon of 4.7%.
Deposits and coupon-bearing instruments
49
(a) What are the total maturity proceeds of the CD?(b) You buy this CD when it has only 129 days left until maturity,on a quote of 4.80%/4.85%. What amount do you pay for it?(c) You then sell this same CD when it has only 122 days leftuntil maturity, on a quote of 5.05/5.10%. What amount doyou receive for it?(d) Over the 7 days that you have held the CD, what yield haveyou earned on your investment?(e) At what different yield would you need to have sold in (c) inorder to have achieved an overall yield on the investment toyou of 4.80%?4 You are a dealer. A customer places a deposit of EUR 5 millionwith you at 4.64% for 92 days. After 61 days, the customer asksto break the deposit. Interest rates at the time are as follows. Yourbank normally funds itself at around LIBOR plus 5 basis points. Ifyou agree to break the deposit, how much cash might you returnto your customer?1 month2 months3 months
4.70/4.80%4.65/4.75%4.60/4.70%
5
DiscountinstrumentsDiscount instrumentsInstruments which pay no coupon are known as discount instruments, because the amount paid for them is always less than theface value. This is so because, otherwise, the investor would earna negative return. Commercial paper, for example, is a discountinstrument.Consider, for example, commercial paper issued for 91 days with aface value of EUR 10 million. On maturity, the investor receives onlythe face value of EUR 10 million. If the yield on the paper is 4.7%,the amount the investor will be willing to pay now for the paper is itspresent value calculated at 4.7%:EUR 10,000,000 = EUR 9,882,589.34Price paid = 911 + 0.047 ×360The idea that the amount paid is the present value is exactly the samehere as in the previous chapter for coupon-bearing instruments. Thedifference in calculation arises only in that the coupon is zero here.
For a discount instrumentMaturity proceeds = face valueAmount paidface value= days from settlement to maturity1 + yield ×year
Discount instruments
51
Example 1A dealer buys US dollar Euro-CP with face value USD 3 million,at a yield of 5.17%, when it is issued with 30 days’ maturity. Theamount he pays for it is:USD 3,000,000 = USD 2,987,130.45301 + 0.0517 ×3607 days later, he sells it at a yield of 5.17%. It now has only23 days to maturity, so the price he sells it for is:USD 3,000,000 = USD 2,990,123.46231 + 0.0517 ×360The return he has made on his investment over the 7 days isgiven by the same formula as before:Yield over the holding periodcashﬂow at the endyear=−1 ×cashﬂow at the startdays held2,990,123.46360=−1 ×= 0.0515 = 5.15%2,987,130.457In the last example, the yield earned over the holding period of7 days was calculated as shown in the section on ‘Calculating theyield on an investment or the cost of a borrowing’ in Chapter 2. Analternative, as in the previous chapter, is to use the following formula:For instruments quoted on a yieldYield over holding period⎛⎛⎛
⎞⎞⎞days from purchaseto maturity⎜⎟⎟⎜⎜⎟⎠⎠⎜ ⎝1 + ⎝purchase yield ×⎟year⎜⎟⎜⎟⎟=⎜−1⎜⎟⎜ 1 + sale yield × days from sale to maturity⎟⎟⎜year⎠⎝×
yeardays held
52
Foreign exchange and money markets
In the previous example, this gives as before:⎛⎞301 + 0.0517 ×⎜⎟ 360360 − 1⎟Yield over holding period = ⎜⎝⎠× 7231 + 0.0517 ×360= 0.0515 = 5.15%
Discount/true yieldFor certain discount instruments in the USA and the UK, a furthercomplication arises in the way the interest rate is generally quoted –as a discount rate instead of a yield.With a yield, the investor in a security asks the question: ‘‘If I investa certain amount at the beginning, how much extra will I receiveby the end?’’ The answer is his yield on the investment, expressedas a proportion of the amount invested. For example, if he investsGBP 100 for one year and receives back a total of GBP 110 at the endof a year, he has earned GBP 10 extra. This represents 10% of theGBP 100 originally invested.With a discount rate, however, the investor asks instead: ‘‘If Ireceive a certain amount back at the end, how much less will I needto invest at the beginning?’’ The answer is the discount amount onthe investment. With the same investment as above, if he will receiveback GBP 110 at the end of the year but only invests GBP 100 atthe beginning, there is a discount amount of GBP 10. This represents9.091% of the GBP 110 received at the end.In this example, therefore, the yield is 10% but the discount rateis 9.091%. In general, the discount rate is the amount of discountexpressed as a percentage per year of the face value, rather than asa percentage per year of the original amount paid.The discount rate is always less than the corresponding yield. Anyinvestment could be expressed either in terms of yield or in terms ofdiscount rate. In practice, however, the market convention is to useyield in almost all ﬁnancial markets worldwide, but to use discountrate in just ﬁve instruments. These are listed below.Terminology can be confusing. What we are calling ‘discountrate’ is also known in the USA as ‘discount yield’ or even just‘yield’. When we use the term ‘yield’ in this book, we alwaysmean yield as we have described – also known as ‘true yield’, or‘investment yield’; we never mean discount rate.
Discount instruments
53
Using a discount rate, the amount to be paid for a particulardiscount instrument is as follows:daysAmount paid = face value × 1 − discount rate ×yearSince the difference between the start amount and the end amountcan in theory be expressed as either a yield or a discount rate, it ispossible to express one in terms of the other. This is useful whenit is necessary to compare an instrument quoted with a yield (forexample, US dollar commercial paper issued internationally) withan instrument quoted with a discount rate (for example, US dollarcommercial paper issued domestically in the USA).For instruments quoted on a discount rateAmount paid = face valuedays from settlement to maturity× 1 − discount rate ×yeardiscount rateYield = days to maturity1 − discount rate ×yearyieldDiscount rate = days to maturity1 + yield ×yearInstruments quoted on a discount rate include Treasury bills,domestic CP and trade bills/BAs in the USA, and sterling Treasurybills and trade bills/BAs in the UK. However, all loans/ﬁxed depositsand CDs in dollars and sterling are quoted on a yield. Note thatdomestic CP in the UK, and US dollar CP issued in the Euromarket(i.e. dollar Euro-commercial paper) are all quoted on a yield. Notealso that while the Treasury-bills issued by the UK government insterling are quoted on a discount rate, the Treasury-bills it issues ineuros are quoted on a yield, as are other international instrumentsin euros.Instruments quoted on a discount rateUSA:
T-billBA
UK:
T-bill (sterling only)BACP
54
Foreign exchange and money markets
Example 2A dealer buys US domestic CP with face value USD 3 million at adiscount rate of 5.17%, when it is issued with 30 days’ maturity.What is the yield?discount rateYield = days1 − discount rate ×year0.0517=301 − 0.0517 ×360= 0.05192 = 5.192%The amount the dealer pays for it is:30USD 3,000,000 × 1 − 0.0517 ×360= USD 2,987,075.00Seven days later, he sells it, again at a discount rate of 5.17%. Itnow has only 23 days to maturity, so the price he sells it for is:23USD 3,000,000 × 1 − 0.0517 ×360= USD 2,990,090.83The return he has made on his investment over the 7 days isgiven by the same formula as before:Yield over the holding periodyearcashﬂow at the end−1 ×=cashﬂow at the startdays held3602,990,090.83−1 ×=2,987,075.007= 0.0519 = 5.19%In the previous example, the yield earned over the holding periodof 7 days was calculated as shown in the section on ‘Calculating theyield on an investment or the cost of a borrowing’ in Chapter 2. Inthis case of discount rates rather than yields, an alternative is to useinstead the following formula:
Discount instruments
55
For instruments quoted on a discount rateYield over holding period⎛
⎞
⎜ ⎟days from sale to maturity⎜⎟discount⎜ 1− rate on sale ×⎟⎜⎟year⎟=⎜−1⎞⎞⎛⎛⎜⎟days from purchase⎜⎟⎜ ⎜ ⎜ discount rate⎟to maturity⎟⎟⎝ ⎝1− ⎝⎠×⎠⎠on purchaseyear×
yeardays held
In the previous example, this gives as before:⎛⎞231 − 0.0517 ×⎜⎟ 360360 − 1⎟Yield over holding period = ⎜⎝⎠× 7301 − 0.0517 ×360= 0.0519 = 5.19%In Example 1, the dealer originally purchased at a yield of 5.17%.However, when he sold the investment at the same yield of 5.17%, itturned out that the yield he earned over the holding period was infact 5.15%. It is always true that if you buy at a particular yield andthen sell later at the same yield, then the yield earned is less.In Example 2, however, the dealer originally purchased at a discount rate of 5.17%, which was equivalent to a yield of 5.19%. Inthis case, when he sold the investment at the same discount rate of5.17%, it turned out that the yield he earned over the holding periodwas the same 5.19% at which he had originally invested. It is alwaystrue that if you buy at a particular discount rate and then sell laterat the same discount rate, then the yield earned is the same as theoriginal equivalent yield.
If you buy at particular discount rate and later sell at the samediscount rate, the yield earned over the holding period is thesame as the purchase yield.
56
Foreign exchange and money markets
Bond-equivalent yields for USTreasury billsFor trading purposes, a government Treasury bond which has lessthan a year left to maturity may be just as acceptable to a moneymarket trader as a Treasury bill with the same remaining maturity.As the method used for quoting yields generally differs between thetwo instruments, the rate quoted for Treasury bills in the USA is oftenconverted to a bond-equivalent yield for comparison.The bond-equivalent yield of a Treasury bill is deﬁned as the couponof a theoretical US Treasury bond, trading at par, with the samematurity date, which would give the same return as the bill.We show below the formulas for calculating this. If the bill has182 days or less until maturity, the calculation is the same conversionfrom discount rate to yield as we have already seen above, except thatit is also then converted to a 365-day year.If the bill has more than 182 days until maturity, however, thecalculation must take account of the fact that the equivalent bondwould pay a coupon during the period as well as at the end. In eithercase, if 29 February falls in the 12-month period starting on thepurchase date, 365 is conventionally replaced by 366.Bond-equivalent yield for US Treasury billIf there are 182 days or less to maturity:⎛
⎞
⎟ 365⎜discount rate⎟×Bond-equivalent yield = ⎜⎝days ⎠ 3601 − discount rate×360If there are more than 182 days to maturity:Bond-equivalent yield⎛⎜ days⎜ 365⎜ ⎛⎜⎜⎝× ⎝=
2
+ 2×
⎞0.5days− 0.5365
11 − discount rate × days360
⎟⎟⎞⎟⎟⎟⎠−1 ⎠
−
days365
days365 − 0.5
If 29 February falls in the 12-month period starting on thepurchase date, replace 365 by 366.
Discount instruments
57
Example 3What is the bond-equivalent yield for a US Treasury bill with48 days left to maturity and trading at a discount rate of 5.24%on 15 April?3650.0524 ×Bond-equivalent yield = 483601 − 0.0524 × 360= 0.05350 = 5.35%What is the bond-equivalent yield for a US Treasury bill with248 days left to maturity and trading at a discount rate of 5.24%on 13 October 2003?Bond-equivalent yield 248 248 2 + 2 × 366 − 0.5 × 366
0.51 − 1− 2483662481 − 0.0524 × 360 248− 0.5366
== 0.05487 = 5.487%
Effective rates and NorwegianTreasury billsIn Chapter 2 we introduced the idea of an effective interest rate – i.e.an annual equivalent interest rate. In Norway, Treasury bills arequoted on the basis of an effective yield. This gives the followingformula for the price of a Norwegian Treasury bill:
Amount paid = ⎛⎜Effective yield = ⎝
face value1 + effective yield

face valueamount paid

days to maturity365
365days to maturity

⎞⎟⎠−1
58
Foreign exchange and money markets
The following formula converts the effective yield to a simple yield ona 365-day basis as quoted in other markets, in order to facilitate acomparison with other instruments. This can then be converted to a360-day basis in the usual way if necessary.
Simple yield on 365-day basis days to maturity365= 1 + effective yield−1 ×
365days to maturity
Example 4You purchase a Norwegian Treasury bill, face value NOK10 million, with 87 days left to maturity, at an effective yieldof 6.73%. How much do you pay? What is the simple yield ofthis bill?NOK 10,000,000Price paid =  87   = NOK 9,845, 952.491.0673 365Simple yield (365-day basis) 87 365365= 0.06564 = 6.564%−1 ×= 1.067387
Exercises1
2
You own ECP with face value USD 10 million. The paper wasissued at a yield of 4.50% on 15 March for 59 days, maturing on13 May.It is now 17 April and you wish to sell the paper for value19 April (24 days left until maturity). You are quoted 4.30/4.35%for the paper. How much money will you receive on the sale?You own US domestic CP with face value USD 10 million. Thepaper was issued at a discount rate of 4.50% on 15 March for59 days, maturing on 13 May.It is now 19 April and you wish to sell the paper for value19 April (24 days left until maturity). You are quoted 4.30/4.35%for the paper. How much money will you receive on the sale?
Discount instruments
345
6
7
8
9
10
59
A US Treasury bill with 147 days left to maturity is quoted in themarket at 4.97%. What is the bill’s yield on a 365-day basis?A UK Treasury bill with 83 days left to maturity has a yield of4.5567%. At what discount rate would it be quoted?You buy sterling commercial paper with 163 days left to maturity,face value GBP 10 million, at a yield of 5.1%. You sell it one weeklater at a yield of 5.15%. What yield per annum have you madeover the week?You buy a US Treasury bill with 163 days left to maturity, facevalue USD 10 million, at a discount rate of 5.1%. You sell it oneweek later at a discount rate of 5.15%. What yield per annumhave you made over the week?What is the bond-equivalent yield for a US Treasury bill with190 days left to maturity, trading at a discount rate of 4.99% on10 July 2003?You sell a Norwegian Treasury bill, with 103 days left to maturity,at a price of NOK 98.08 per NOK 100 face value. At what effectiveyield have you sold?Which has the higher yield – a Norwegian Treasury bill with aneffective yield of 5.7%, or a US Treasury bill quoted at a discountrate of 5.5%? Both have 119 days left to maturity.Place the following instruments in ascending order of yield, giventhe rate quoted for each. All the instruments have exactly 91 daysleft to maturity.UK sterling Treasury billUK euro Treasury billSterling domestic CPSterling ECPUS dollar Treasury billUS dollar domestic CPUS dollar ECP
4.98%4.98%5.10%5.10%4.81%4.93%4.93%
6
Forward interestrates, FRAs andintroduction tofuturesOverviewForward-forwards, forward rate agreements (FRAs) and futures arevery closely linked instruments, all relating to an interest rate appliedto some period which starts in the future. We shall outline them ﬁrstand then examine each more closely.• A forward-forward is a cash borrowing or deposit which starts onone forward date and ends on another forward date. The term,amount and interest rate are all ﬁxed in advance. Someone whoexpects to borrow or deposit cash in the future can use this toremove any uncertainty relating to what interest rates will be whenthe time arrives.• An FRA is an off-balance sheet instrument which can be used toachieve the same economic effect as a forward-forward. Someonewho expects to borrow cash in the future can buy an FRA toﬁx in advance the interest rate on the borrowing. When the timeto borrow arrives, he borrows the cash in the usual way. Underthe FRA, which remains quite separate, he receives or pays thedifference between the cash borrowing rate and the FRA rate, so
Forward interest rates, FRAs and introduction to futures
61
that he achieves the same net effect as with a forward-forwardborrowing.Similarly, an investor or other lender can use an FRA to protectthe return on a future investment.There is of course no need to have any underlying borrowing orinvestment when trading an FRA. In this way, an FRA an be usedspeculatively, to take a view on whether interest rates will rise orfall, and also in arbitrage strategies.• A futures contract is similar to an FRA. Futures are, however,traded only on an exchange, and differ from FRAs in a variety oftechnical ways.
Pricing a forward-forwardExample 1I will need to borrow dollars for a 3-month period, starting in2 months’ time. I believe that interest rates will rise over the next2 months, so that if I wait, I will have a higher borrowing costthan if I could borrow now.The dollar interest rate now for 2 months (61 days) is 5.6/5.7%and the rate for 5 months (153 days) is 5.3/5.4%.If I borrow now for 5 months at 5.4% and simultaneouslydeposit the cash borrowed at 5.6% for 2 months, the positionover the ﬁrst 2 months is a net zero one. I have therefore synthetically created a net borrowing which begins in 2 months andends in 5 months – that is, a forward-forward borrowing. All thecashﬂows are known in advance, so it must be possible to calculate the cost of borrowing over the 3-month forward period. Thisrate is in fact given by the formula below.If I am a bank customer, I could more simply ask the bank toprovide the forward-forward loan from 2 months to 5 months at anagreed rate, rather than construct it myself as in Example 1. However,the bank would be able to construct it in exactly the same way andthen present it to me as a forward-forward loan: the bank borrows for5 months, lends the cash in the interbank market for 2 months andthen has the cash available to lend to me for the forward period from2 months to 5 months.
62
Foreign exchange and money markets
Forward-forward rate⎛⎞interest rate for days in longer period1+×⎜⎟yearlonger period − 1⎟=⎜⎝⎠interest rate for days in shorter period1+×shorter periodyearyear×days difference
In the example above, this formula gives a forward-forward rate of:⎛⎞1531 + 0.054 ×⎜⎟360360⎜ − 1⎟⎝⎠ × (153 − 61) = 0.05218 = 5.218%611 + 0.056 ×360Note that this formula for a theoretical forward-forward rate appliesonly for periods up to one year. A money market deposit for longerthan one year typically pays interim interest after one year (or eachsix months). This extra cashﬂow must be taken into account in calculating a forward-forward rate for any period ending after one year.Forward-forwards and FRAs are referred to by the beginning andend dates of the forward period, compared with the dealing date orspot value date. Thus the forward-forward described here is a ‘2v5’(or ‘2x5’, or 2/5, or ’twos against ﬁves’), meaning that the forwardperiod starts 2 months from now and ends 5 months from now.
Break-even on a forward positionIf a dealer has borrowed for a longer period than he has lent – forexample, he has borrowed USD 10 million for 8 months and lentUSD 10 million for two months – he is said to be over-borrowed,because at a future point (in 2 months’ time in this case), hisborrowing will be greater than his lending. He will therefore needto lend again in the future to square his book. Similarly, a dealerwho has lent for a longer period than he has borrowed is said to beover-lent.The forward-forward formula is useful for calculating the breakeven rate which a dealer needs to achieve to cover such an unmatchedposition, because the forward-forward rate is exactly the rate whichis consistent with the rate for the shorter period and the rate for thelonger period.
Forward interest rates, FRAs and introduction to futures
63
Example 2A dealer borrows USD 10 million for 8 months (244 days) at4.95% and lends USD 10 million for 2 months (92 days) at4.91%. At what 6-month rate does the dealer need to lendagain in the future in order just to break even?Break-even rate⎞⎛interest rate for days in longer period×1+⎟⎜yearlonger period − 1⎟=⎜⎠⎝interest rate for days in shorter period1+×shorter periodyearyear×days difference⎛⎞2441 + 0.0495 ×⎜⎟ 360360 − 1⎟=⎜⎝⎠ × 152 = 0.04913 =