as the present value ofUSD 1,018,550.00 because it is the value now of that cashﬂow inthe future. Given a choice between receiving USD 1,018,550.00 after126 days, or USD 1,000,000 now, I would be equally happy witheither, because if I start with the USD 1,000,000 now and put it ondeposit, I can accumulate USD 1,018,550.00 by the end of 126 days.The present value so calculated depends on the interest rate usedand the time until the future cashﬂow.We can do the same thing for any amount, any period, any interestrate and any day/year convention. For example, if we need to knowhow much we need to put on deposit now so that, after 57 days at6.1%, we will receive back GBP 2,483,892.71, the answer is:
GBP 2,483,892.71 = GBP 2,460,454.36571 + 0.061 ×365
amount of future cashﬂowPresent value = days1 + interest rate ×year
14
Foreign exchange and money markets
In general, this calculation demonstrates the fundamental principlebehind market calculations – the time value of money. As long asinterest rates are not negative, it is better to have any given amountof money sooner rather than later because, if you have it sooner, youcan place it on deposit to earn interest on it. The extent to which itis worthwhile having the money sooner depends on the interest rateand the time period involved.Present values are fundamental to pricing money market instruments, because the present value of a future cashﬂow is its value, orworth, now. It is therefore the amount of money which one is preparedto pay now for any instrument which generates that cashﬂow in thefuture.The calculation of a present value is sometimes known as discounting a future value to a present value and the interest rate used issometimes known as the rate of discount.
Example 5What is the present value at 6.5% of a cashﬂow of USD 1 millionwhich will arise after 90 days?amount of future cashﬂowPresent value = days1 + interest rate ×yearUSD 1,000,000 = USD 984,009.84= 901 + 0.065 ×360
Calculating the yield on aninvestment or the cost of aborrowingAnother useful related calculation is the answer to the question: ‘‘If weknow how much money we invest at the beginning of an investmentand we know the total amount achieved at the end, including anyinterest, what is our rate of return, or yield, on the investment overthe period we have held it?’’ The same question can be asked aboutthe cost of a borrowing, if we know the amount borrowed at thestart and the total amount repaid at the end, including interest. Theanswer is found by turning round the formula above, to give:
Essential ﬁnancial arithmetic
Yield =
15
yearcashﬂow at the end−1 ×cashﬂow at the startdays
Note that this formula is correct only where there are no cashﬂowsin the middle of the investment – only at the start and the end. Ifthere were more cashﬂows, we would need to compound, because wewould be able to earn interest on interest.Exactly the same idea works for the cost of a borrowing.Example 6I buy a painting for EUR 24,890. I keep the painting for 198 days.I then sell it for EUR 26,525. What yield have I made on thisinvestment?cashﬂow at the endyear−1 ×Yield =cashﬂow at the startdays36026,525−1 ×= 0.1194 = 11.94%=24,890198
Example 7I buy a painting for GBP 24,890. I keep the painting for 198 days.I then sell it for GBP 23,000. What yield have I made on thisinvestment?cashﬂow at the endyear−1 ×Yield =cashﬂow at the startdays36523,000−1 ×= −0.1400 = −14.00%=24,890198In this case, I have a negative yield – i.e., I have made a loss.Example 8I borrow USD 356,897.21 for 213 days. At the end, I repay a totalof USD 368,363.43. What is the total cost of this borrowing?cashﬂow at the endyear−1 ×Rate =cashﬂow at the startdays360368,363.43−1 ×= 0.0543 = 5.43%=356,897.21213
16
Foreign exchange and money markets
Compounding two or moreinterest rates together (a strip)Suppose that I borrow EUR 1 million for 73 days at 4.7%. At maturity,I borrow the whole amount again – principal and the interest – foranother 124 days at 4.9%. What is my simple interest rate over thecombined period of 197 days?At the end of 73 days, the interest cost is EUR 9,530.56:EUR 1,000,000 × 0.047 ×
73= EUR 9,530.56360
The whole amount to be reﬁnanced is therefore EUR 1,009,530.56.The interest cost on this at 4.9% for a further 124 days is EUR17,038.63:EUR 1,009,530.56 × 0.049 ×
124= EUR 17,038.63360
The total repayment of principal plus interest after 197 days is therefore EUR 1,026,569.19 (=EUR 1,009,530.56 + EUR 17,038.63). Theinterest cost is therefore 4.855%:yearcashﬂow at the end−1 ×Interest rate =cashﬂow at the startdays3601,026,569.19−1 ×= 0.04855 = 4.855%=1,000,000197The same idea can be used if the borrowing is rolled over more thanonce, as long as the total length of time is not longer than a year.This process of compounding together more than one rate is calledstripping. The overall result can be expressed as a formula:Creating a stripThe simple interest rate for a period up to one year⎛⎞days in ﬁrst period1 + ﬁrst interest rate ×⎟⎜year ⎟⎜⎟⎜days in second period⎜ × 1 + second interest rate ×⎟⎟⎜=⎜year⎟⎟⎜ ×·····················⎟⎜⎝⎠days in last period−1× 1 + last interest rate ×yearyear×total days
Essential ﬁnancial arithmetic
17
This can be applied to the example above, to give 4.855% as expected: 12436073× 1 + 0.049 ×−1 ×1 + 0.047 ×360360197= 0.04855 = 4.855%This idea works in exactly the same way for rolling over an investmentas for rolling over a borrowing, as in the following example.
Example 9An investor deposits USD 3 million at 5.3% for 92 days. At maturity, he rolls over the whole principal and interest at 5.4% for30 days. At maturity, he again rolls over the accumulated principal and interest at 5.6% for 62 days. What is his overall simplerate of interest earned over the whole six months?  ⎞⎛9230× 1 + 0.054 ×⎟ 360⎜ 1 + 0.053 × 360360⎜⎟×⎝⎠ 18462× 1 + 0.056 ×−1360= 0.05463 = 5.463%
InterpolationIn the money market and foreign exchange market, prices are generally quoted for standard periods such as 1 month, 2 months etc.If a dealer needs to quote a price for an odd date (or broken date)between these periods, he needs to interpolate. In the money markets,straight-line interpolation is generally used, as follows.Suppose, for example, that the 1-month rate (30 days) is 8.0% andthat the 2-month rate (61 days) is 8.5%. The rate for 39 days assumesthat interest rates increase steadily (i.e. in a straight line) from the 1month rate to the 2-month rate. The increase from 30 days to 39 dayswill therefore be a 9/31 proportion of the increase from 30 days to61 days. The 39-day rate is therefore:9= 8.1452%8.0% + (8.5% − 8.0%) ×31The same process can be used for interpolating exchange rates.
18
Foreign exchange and money markets
Straight-line interpolationInterpolated rate = the ﬁrst rate⎞⎛(days between the ﬁrst dateand the required date)⎜ (the second rate⎟×+⎝⎠(days between the ﬁrst date− the ﬁrst rate)and the second date)
Example 10The interest rate for 3 months (92 days) is 15.2%. The interestrate for 4 months (123 days) is 15.0%. What is the rate for104 days, using straight-line interpolation?(104 − 92)= 14.9226%15.0% + (15.0% − 15.2%) ×(123 − 92)
Paying interest with differentfrequenciesIf a dealer lends GBP 1 million to a customer at 5.3% for 182 days, hewill normally receive total proceeds at the end of GBP 1,013,427.40:182= GBP 26,427.40365GBP 1,000,000 + GBP 26,427.40 = GBP 1,026,427.40GBP 1,000,000 × 0.053 ×
If, however, the customer agrees to pay the same interest rate but topay the interest in two instalments – after 91 days and 182 days – thedealer will be better off. This is because, when he receives the ﬁrstinterest instalment after 91 days, he can reinvest it until the end of182 days. This will earn him extra money, as interest on interest. Howmuch extra will depend on the interest rate at which he can reinvestthe instalment. If we assume that the rate is still 5.3%, the totalproceeds achieved by the end of 182 days will be GBP 1,026,602.00:• After 91 days, receive ﬁrst interest instalment of GBP 13,213.70:GBP 1,000,000 × 0.053 ×
91= GBP 13,213.70365
Essential ﬁnancial arithmetic
19
• Reinvest this amount at 5.3% for the next 91 days, to earn anextra GBP 170.79:91= GBP 174.60GBP 13,213.70 × 0.053 ×365• At the end of 182 days, receive the second interest instalment ofGBP 13,213.70 plus the principal back of GBP 1,000,000.• The total proceeds are GBP 1,026,306.99:GBP 13,213.70 + GBP 174.60 + GBP 1,013,213.70= GBP 1,026,602.00The total interest which the dealer has achieved is thereforeGBP 26,602.00, on a principal of GBP 1 million. This is the sameresult the dealer would have achieved if he had quoted a rate of5.335% with all the interest paid at the end (the difference of GBP 0.08is due to rounding):182= GBP 26,601.923655.3% is said to be the nominal rate quoted to the customer with3-monthly or quarterly interest payments and 5.335% is said to bethe 6-monthly or semi-annual equivalent rate.These calculations assume that all interim cashﬂows can be reinvested at the same original interest rate. Although this is unrealistic,it is a useful simplifying assumption to make, in order to be able tocompare interest rates with different interest payment frequencies. Itis possible to use a formula to convert the 5.3% quoted with quarterlyinterest payments to the semi-annual equivalent rate of 5.335%:GBP 1,000,000 × 0.05335 ×
Equivalent rate with interest paid M times per year⎞⎛N⎞⎞ M⎛⎛ratewithinterest⎟⎜⎟⎜⎜⎜ paid N times per year ⎟⎟⎟⎜⎜⎟⎟− 1⎟ × M= ⎜⎝1 + ⎜⎠⎠⎝⎟⎜N⎠⎝
In the example we have been considering, the customer has agreedto pay the 5.3% in quarterly instalments – that is, effectively fourtimes per year (even though the loan only lasts for 6 months). Wewant to calculate what this is worth to the dealer if interest were paid6-monthly – that is, effectively twice per year (again, even though theloan only lasts for 6 months). Using the formula above agrees withour answer:
20
Foreign exchange and money markets
Equivalent rate with interest paid twice per year⎞⎛ 42rate with interest paid four times per year⎟⎜− 1⎠ × 2=⎝ 1+4⎛⎞ 420.053⎜⎟=⎝ 1+− 1⎠ × 2 = 0.05335 = 5.335%4Note that the calculation method we have given here is almostalways satisfactory but is not quite correct mathematically. This isbecause the 6-month period of 182 days used in our example is notexactly half a year (that would be 182 12 days, which is not realistic).Similarly, if interest were paid quarterly, the payment periods wouldnot be exactly equal periods of 91 14 days but slightly different periods,depending on weekends etc.Example 11A dealer quotes 5.3% for a 5-year deposit with interest paid onceper year. What equivalent rate should he quote if the customerprefers to receive the interest quarterly?Equivalent rate with interest paid four times per year⎞⎛ 14rate with interest paid once per year⎟⎜− 1⎠ × 4=⎝ 1+1⎛
⎞ 140.053⎜⎟=⎝ 1+− 1⎠ × 4 = 0.05198 = 5.198%1
Effective ratesThe equivalent yield with interest paid only once per year is knownas the annual equivalent rate, or the effective rate. Consider, forexample, a deposit for 3 months. The effective yield is the interestwhich would be accumulated by the end of the year if the depositwere rolled over three times – i.e. the principal and interest at the endof 3 months were all reinvested at the same rate, and then all againat the end of 6 months and all again at the end of 9 months.The same idea can be extended to a deposit for an odd period, suchas, say, 59 days. If the deposit is rolled over repeatedly, its maturitywill never quite coincide with exactly one year, because 365 is not an
Essential ﬁnancial arithmetic
21
exact multiple of 59. Nevertheless, it is still possible to consider theeffective rate as the annual rate which achieves the same compoundresult, as follows: 365daysdaysEffective rate = 1 + nominal rate quoted ×−1year
Example 12A dealer is trying to compare two different US dollar investments.The ﬁrst is for 153 days at a yield of 9.1% and the second is for87 days at a yield of 9.05%. Assuming that there are no othersigniﬁcant differences between the two investments, which isbetter? It may be, for example, that the dealer believes thatinterest rates will be higher after 87 days than now – in whichcase he would probably prefer the 87-day investment becausethen he could roll it over at a higher yield when it matures.Ignoring any such expectations, however, it is useful to be ableto compare the two investments on a like-for-like basis.One approach therefore is to compare the effective rate of eachinvestment. The ﬁrst has an effective rate of 9.475% and thesecond 9.501%, so on this basis, the shorter investment has alower nominal rate but a higher effective rate: 365153153− 1 = 0.09475 = 9.475%1 + 0.091 ×360 36587871 + 0.0905 ×− 1 = 0.09501 = 9.501%360
Exercises1 An amount of 5 million is placed on deposit from 17 August to 31December at 6.7%. What would be the interest earned if the ratehad been quoted on an ACT/365, ACT/360 or 30/360 basis?2 A dealer quotes a rate of 6.7% on a 30/360 basis for an investmentfrom 17 August to 31 December. What would be the equivalentrate if the customer asked him to quote it instead on an ACT/365basis?
22
Foreign exchange and money markets
3
What is the present value of a cashﬂow of EUR 5,327.21 arisingin 276 days, using 4.2% as the rate of discount?4 I buy an investment for GBP 2,345,678.91 and sell it one weeklater for GBP 2,350,000.00. What is my yield?5 The interest rate for 6 months (183 days) is 9.00% and the ratefor 7 months (214 days) is 9.15%. What is the rate for 193 days?6 A dealer quotes a customer 7.35% to borrow for one year withinterest paid monthly. The customer prefers to pay interest quarterly. What rate should the dealer quote instead?7 Which of the following is the cheapest for a borrower?••••
6.7% annual money market basis6.7% semi-annual money market basis6.7% annual bond basis6.7% semi-annual bond basis.
8 What is the effective yield of a 45-day deposit in euros, with anominal rate quoted of 4.6%?9 A company borrows US dollars at 5.1% for 183 days and thenat maturity reﬁnances the principal and interest at 5.3% for afurther 92 days. What is the simple cost of borrowing over the 9months?
This Page Intentionally Left Blank
Part 2Money Markets
3
Overview ofmoney marketinstrumentsIntroductionThe term money market is used to include all short-term ﬁnancialinstruments which are based on an interest rate, whether the interestrate is actually paid (as in a cash deposit) or only implied in the waythe instrument is priced and settled (as in an FRA). Longer-terminstruments are covered by the term capital market.The underlying instruments are essentially those used by one party(the borrower, seller or issuer) to borrow money and by the otherparty (the lender, buyer or investor) to lend money. The main suchinstruments are:
Fixed depositCertiﬁcate of deposit (CD)Commercial paper (CP)Bill of exchangeTreasury bill
borrowingby banksborrowing by companies(or in some cases, banks)borrowing by companiesborrowing by a government
These instruments have different names in different markets – forexample, each country’s government issues something equivalentto a Treasury bill in its own market – but the characteristics areessentially the same as described here.Each of the above instruments represents an obligation on theborrower to repay the amount borrowed at maturity, plus interest
26
Foreign exchange and money markets
if appropriate. As well as these underlying borrowing instruments,there are other money market instruments which are linked to these,or at least to the same interest rate structure, but which are notdirect obligations on the issuer in the same way:
Repurchase agreement
Forward-forwardborrowingFutures contractForward rate agreement
effectively cash borrowing usinganother instrument (such as abond) as collateral
⎫⎪⎪⎬ used to trade or hedgeshort-term interest rates for⎪⎪⎭ future periods
The money market is linked to other markets though arbitragemechanisms. Arbitrage occurs when it is possible to achieve thesame result in terms of position and risk though two alternativemechanisms which have a slightly different price. The arbitrageinvolves achieving the result via the cheaper method and simultaneously reversing the position via the more expensive method – therebylocking in a proﬁt which is free from market risk (although still probably subject to credit risk). For example, if I can buy one instrumentcheaply and simultaneously sell at a higher price another instrumentor combination of instruments which has identical characteristics,I am arbitraging. In a completely free market with no other priceconsiderations involved, the supply and demand effect of arbitragetends to drive the two prices together until the arbitrage opportunityceases to exist.For example, the money market is linked in this way to the forwardforeign exchange market, through the ability to create syntheticdeposits in one currency, by using foreign exchange deals combinedwith money market instruments. Similarly it is linked to the capitalmarkets (long-term ﬁnancial instruments) through the ability tocreate longer-term instruments from a series of short-term instruments (such as a 2-year interest rate swap from a series of 3-monthFRAs).
The yield curveIf I wish to borrow money for 3 months, the interest rate is likely tobe different – possibly higher and possibly lower – from what the rate
Overview of money market instruments
27
would be if I wished instead to borrow for 6 months. The yield curveshows how interest rates vary with term to maturity. For example, aReuters screen might show the following rates:1 month2 months3 months6 months12 months2 years
9.5%9.7%10.0%10.0%10.2%10.5%
In a free market, these rates show where the market on averagebelieves rates ‘should’ be, as supply and demand would otherwisetend to move the rates up or down. A yield curve where longer-termrates are higher than shorter-term rates is known as a normal orpositive yield curve; a curve where longer-term rates are lower thanshorter-term rates is known as negative or inverted; a curve whererates are approximately the same across the range of maturities isknown as ﬂat.There are in fact an inﬁnite number of yield curves, becauseyield depends on credit risk as well as maturity. For example, if Iam choosing between investing for 6 months in a security issuedby the government of a particular country, and a security issuedby a medium-risk company in that same country, I would expectthe latter to have a signiﬁcantly higher yield to compensate for thehigher risk. A security issued by a government in its own currency isgenerally considered to represent virtually zero credit risk, becausethe government can always print its own money to redeem thesecurity if necessary, regardless of the state of the economy. Thisdoes not apply to a security issued by a government in a foreigncurrency.There are therefore as many different yield curves as there are levelsof creditworthiness. In practice, one generally considers a yield curveas referring to either government risk or ﬁrst-class bank risk.The yield on any security also depends on liquidity – how easy itis to liquidate a position in that security by selling it if necessary. Ifthere are two securities issued by exactly the same organization butone is more liquid than the other, it should be traded at a slightlylower yield because investors will be more willing to own it.Liquidity in turn depends partly on creditworthiness. In any domestic money market, a Treasury bill (see below) should be the most liquidinstrument, since, even in dire circumstances, the government canin theory always print money to redeem its own securities. In the UK
28
Foreign exchange and money markets
market, eligible bills (see below) are also very liquid, as they can besold to the Bank of England.There are various factors which inﬂuence the shape of the yieldcurve:• Economic performance Expectations of future rates, and hencecurrent longer-term rates also, are affected by market expectationsof future inﬂation.• Central bank operations Intervention in the market by thecentral bank, involving buying and selling securities, can changethe availability of securities of different maturities. If this causesa shortage of supply for investors at a particular maturity, yieldswill rise at that maturity. If the market believes that the centralbank is about to change ofﬁcial 3-month rates, for example, thisexpectation will already have been factored into the market 3month rate.• Borrowing demand An increase or decrease in borrowing inany particular maturity will tend to increase or decrease yieldsat that maturity. For example, if the government either increasesor decreases its borrowing requirement, this will affect the yieldcurve at the maturities sought by the government. Similarly if thegovernment shortens or lengthens the maturity of its debt whenit rolls over maturing debt, this tends to make longer-term yieldslower than shorter-term yields or vice versa.• Liquidity preference The typical shape for the yield curve, inthe absence of other factors, is positive because, if there is noadvantage in investing for a longer term, investors would preferto invest short-term since their investments are then more liquid.Also, the risk of losing money is greater with a longer-term security than with a shorter-term security, because its price is moresensitive to yield changes. Therefore, an incentive is necessary toattract longer-term investors. This is a supply and demand effect:borrowers prefer long-term and investors prefer short-term, so theprice of money is driven up for the long term.• Market segmentation Bank investors tend to invest short-termand institutional investors like pension funds tend to invest longterm, with fewer investors in the medium-term. In the absence ofany other factors, this would tend to encourage a humped yieldcurve shape.
Some more terminology• Negotiable security A security is a ﬁnancial asset sold initiallyfor cash by a borrowing organization (the issuer). A security which
Overview of money market instruments
•
•
•
•
29
can be sold and bought again subsequently – i.e. transferred fromone party to another – is referred to as negotiable.Primary market and secondary market When a security isissued, the original purchaser is said to be buying it in the primarymarket. All subsequent transactions – whether they are on thesame day or much later during the life of the security – are said tobe secondary market transactions.Face value/notional principal For securities, the principalamount is generally known as the face value (which can bethought of as the principal amount written on the face of theinstrument). It is also sometimes called the nominal amount. Withsome instruments such as an FRA (see Chapter 6), the principalamount is never actually transferred to either party, but is usedas a reference amount for calculation purposes. In this case, it isreferred to as a notional principal.Coupon/yield A certiﬁcate of deposit pays interest at maturity – sometimes only at maturity but sometimes also during theCD’s life – as well as repaying the principal. For example, a CDmight be issued with a face value of £1 million, which is repaidon maturity together with interest of say 10% calculated on thenumber of days between issue and maturity. The 10% interest rateis called the coupon. The coupon is ﬁxed once the CD is issued.This should not be confused with the yield, which is the currentrate of return available in the market when investing, or buyingand selling an instrument, and varies continually.Discount1 An instrument which does not carry a coupon is a discountinstrument. Because there is no interest paid on the principal,a buyer will only ever buy it for less than its face value (unlessyields are negative!) – i.e. at a discount. For example, Treasurybills are almost always discount instruments.2 The word discount is also used in the very specialized contextof a discount rate quoted in the US and UK markets on certaininstruments. This is explained in detail below.3 To discount a cashﬂow means to calculate its present value,as discussed in Chapter 2.
• Bearer/registered A bearer security is one where the issuer paysthe principal (and coupon if there is one) to whoever is holdingthe security at maturity. This enables the security to be heldanonymously. A registered security by contrast is one where theowner is considered to be whoever is registered centrally as theowner; this registration is changed each time the security changeshands.
30
Foreign exchange and money markets
• LIBOR LIBOR means London interbank offered rate – theinterest rate at which one London bank offers money as a cashdeposit to another London bank of top creditworthiness. LIBIDmeans London interbank bid rate – the interest rate at whichone London bank of top creditworthiness bids for money as acash deposit from another bank. LIBOR is therefore always thehigher side of a two-sided interest rate quotation (which is quotedhigh–low in some markets and low–high in others). LIMEAN isthe average between the two sides. In practice, the offered rate fora particular currency at any moment is generally no different inLondon from any other major center, although it is possible forreserve costs, credit considerations etc. to make a slight difference.LIBOR is therefore often just shorthand for offered interest rate.Speciﬁcally, however, LIBOR for a particular currency alsomeans the average offered rate for that currency quoted by a groupof banks in London at 11:00 a.m. This is a reference rate, calculated by the BBA (British Bankers’ Association) each day, whichis used as a benchmark for agreeing rates between two partiesfor loans, FRAs and other instruments. Similarly