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Just Enough Interest RatesAnd Foreign Exchange

Joe Troccolo, Managing DirectorUBS Financial Markets Education


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Preface

Everyone involved in the financial markets needs to know something about interest ratesand foreign exchange. Even people who deal entirely with equities are affected through

the influence of interest rates on equity prices and foreign exchange fluctuations on

corporate balance sheets and income statements. Unfortunately most books on these

subjects are written for specialists in the rates and foreign exchange market and tend to

be too complete. The effect of this is that someone who wants to know "just enough" has

to sift through a lot of information to find the truly important parts. Its probably true that

with most forms of expression books, movies, music, paintings and even everyday

conversation, knowing what to leave out is at least as important as knowing what to

include. This "short" book is our answer to the question of what everyone needs to know

about interest rates and foreign exchange. If, having read through the book, you feel we

included something unnecessary, feel free to forget it! If you find there is something you

really needed to know but we left it out, first ask yourself if everyone needs to know it. If

so, let us know and well include it next time.

I would like to thank my colleagues in UBS Financial Markets Education for their help and

encouragement in the course of writing this book. Particularly I would like to thank

Spencer Morris who proof read the text exhaustively. Naturally any errors are the

responsibility of the author. Should any errors be found please send us an email or write to

us at the following address and if your discovery is verified, we will send you a unit of yourcountrys currency whose value most closely approximates one pound sterling.

Joe Troccolo

Managing Director

UBS Financial Markets Education

1 Finsbury Avenue

London EC2M 2PP

United Kingdom

Email: [email protected]


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Contents

Section 1 Interest Rates on Deposits and Discount Securities 1The W orld of Interest Rates 1A nsw ers to Exercises 11

Section 2 Bonds 12Yield To M aturity 16

The Price-Yield Relationship 19

Bond Risk M easures 22

D uration 25

W orksheet Bond Pricing, D uration, C onvexity 33

Section 3 Forward Rates and Eurodollar Futures 35FRA s 35

Futures 36Forw ard Rates 39

A nsw ers to Exercises 41

Bond Futures 42

Section 4 Interest Rate Swaps 45A nsw ers to Exercises 52

Section 5 Building a Yield Curve 53The U S Treasury M arket 53

Bond Yields 59

Form ulas and Bootstrapping the Yield C urve 61

The Relationship Betw een Forw ard, Zero and Par Rates 64Practical C onstruction of the Treasury Yield C urve 67

LIBO R C urve 68

W orksheet Yield C urve Relationships 69

W orksheet -Stripping and Recom bining 72

Exercise Sw aps and Forw ard Start Sw aps 73

A nsw ers to Exercises 74

Section 6 Just Enough Foreign Exchange 79Bids and O ffers 81

Spot and Forw ard Transactions 83

Forw ard Points 85

Interest Rate Parity 89

C urrency Futures 93

C urrency Bond Sw aps 94

Appendix 1 Interest Rate and Bond Taxonomy 102

Appendix 2 Day Basis 106

Global Disclaimer 112


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Section 1Interest Rates on Deposits andDiscount Securities

The world of interest ratesM an (hum ankind, if you like) m ay be the m easure of all things, but m oney

is the m easure of all investm ents. M oney, though, is the one good that you enjoy m ost

w hen you get rid of it, presum ably by spending it on som ething you w ant. M isers m ay

get enjoym ent from just looking at a stack of bills, or running their hands over a pile of

Kruggerands, or m aybe just looking at a large bank balance, but, given our preferences,

m ost of us w ould just as soon buy things. O ne reason w e dont spend all our m oney on

im m ediate consum ption (or m aybe you do), is that w e are w illing to w ait until a future

tim e w hen w e m ight have a greater need to buy, but a lesser ability to do so i.e. w e save

for a rainy day. A nother reason for not spending m oney now is that there m ay not beso m uch you need, or w ant, to buy. If you dont have m uch m oney you are likely to spend

all of it; but if you have a lot of it you m ight get to the point w here there isnt really

anything you w ant! Youve bought the Porsche and the Rolex w atch, you have your

clothes tailor-m ade, youre renting the $50,000 a m onth penthouse and you get take-out

from Le C irque 2000 m ost nights. There just isnt anything left! In the m eantim e there

are other people w ho need to spend m oney now but dont have it. M aybe som eone

w ants to m ake a m ajor purchase like a car or a house, or a college education. O r m aybe

the governm ent w ants to build a new highw ay or a business w ants to expand. These

potential borrow ers are w illing to pay you m ore m oney in the future if you w ill let them

use your m oney from now until then. M aybe som e of them w ill approach you directly,

probably relatives and old college friends. But you dont often find the C FO of a m ajor

corporation in your lobby w aiting for you to com e out in the m orning. W ell, not m ost

m ornings anyw ay. So to m ake the picture com plete w e need som eone to channel the

m oney from you, the investor, to the borrow er. This is usually done by a bank or som e

other type of financial institution. In this sense the interest rate w orld is really a pretty

sim ple place. So w hy do you need a book on interest rates?

Like other m arkets, the w orld of interest rates is populated w ith individuals and institutions

w ith w idely varying m otives for being there. But there are three prim ary them es am ong the

participants they need to borrow m oney, or they have m oney to invest or they w ant to

interm ediate betw een the first tw o. The m ost im portant of these are the investors. The

astonishing variety of types of investm ents available in the interest rate m arkets is due to

the differing appetites of the investors. The borrow ers corporations, governm ents, banks

and other financial institutions, along w ith their advisers the investm ent banks, w ill tailor

their borrow ing to suit the investor w ho is lending the m oney. It is the w ide range of

instrum ents, pricing conventions and uses for the instrum ents that has lead to this book.

W e have tried to keep it as short as possible w hile still covering the m ost interesting and

im portant aspects of rates. For no extra charge w e have included a section on foreignexchange, as the interest rate and foreign exchange m arkets are so closely related.


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Risk

The defining characteristics of interest rate investm ents are m aturity and risk. Investors

choose the form of their investm ent prim arily based on these tw o dim ensions. M aturity

sim ply refers to how long the investor is lending the m oney for a day, a w eek, a year or

som e longer period of tim e. Risk refers to both the uncertainty of the return the investor

w ill get on the investm ent, w hich can range from no uncertainty to nearly total andw hether the borrow er w ill fail to live up to one or m ore of the obligations represented by

the investm ent. The first type of risk w e callmarket riskand the second is credit risk.

A ctually an investor m ight not even think that he/she is lending m oney. For exam ple, if you

buy a bond in the secondary m arket, you are not lending any m oney to the issuer of the

bond. Rather it is the original buyer of the bond w ho did that. W hat you are doing is taking

that investm ent over from som eone w ho no longer w ants it. Still, once you have bought

the bond you have assum ed all the risk the original investor did. So from a practical point

of view you are the lender now .

Basic principles

Lets begin w ith som e obvious principles. The am ount of interest received m ust becom e

larger as the tim e lapse betw een w hen it is lent and w hen it is returned grow s larger. That

is, if you borrow for a year, you w ill have to pay m ore than if you borrow for just 6 m onths.

A second principle, so obvious it seem s unnecessary to state, is that the m ore you borrow ,

the m ore you pay. That is, the am ount of interest you pay on 100,000 is m ore than the

am ount you w ould pay on 50,000.

Cash flows

C ash flow s involve am ounts of cash and tim ing. If you receive 500 today and repay

510.35 in 3 m onths, then the first cash flow is 500, w hich you take in and the second

is the 510.35 w hich you pay out. The 500 is referred to as a Present Value (PV) because

it is valued as of today. The 510.35 is called a Future Value (FV) because it is deferred until

a future date.

Calculating interest

Suppose you w ent to a bank and asked to borrow 500 for 3 m onths and the banker toldyou that you w ould need to repay 510.35 at the term ination of the loan. W ouldnt you

w onder w here the num ber 510.35 cam e from ? O r suppose you w anted to m ake an

investm ent rather than a loan. You w ish to invest 750 for 9 m onths. The banker says you

can receive 771.25 in 9 m onths. But he gives you an alternative. Instead he says, you can

receive 750 in 9 m onths if you w ill give him 731 today? W hich is the better deal? H ow

w ould you figure it out? The answ er is by using an interest rate.

PV = 500

FV = 510.35

Figure 1


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Add-on interest rates

W hen w e know the present value, w e usually arrive at the future value by adding the

interest to the initial am ount. The com putation of the am ount of interest is done using an

interest rate. The w ord rateim plies that the result w ill be proportional to the am ount

borrow ed or invested, just like the distance you travel in a car is proportional to the rate

of speed at w hich you travel. C ontinuing that analogy, if you w ere driving at a rate of100kph, w e could not say how far you have travelled until w e knew how long i.e. for how

m uch tim e, you had travelled at that rate. It is the sam e w ith interest. W e need to know

both the speedi.e. the rate and the tim e as w ell as the present value, in order to know

how m uch interest there w ill be.

Example

W e lend 500 for 3 m onths at a rate of 8.28% per annum . The 8.28% per annum is the

interest rate. It says that for every 1 invested, you w ould receive 1.0828 in a years tim e.

But w e are not investing 1 for 1 year; w e are investing 500 for 3 m onths. Thus the

8.28% m ust be adjusted for the am ount - 500 - and the tim e - 3 m onths - of the loan.

O ne of our principles said that the am ount of interest should be proportional to the

am ount of the loan. This m eans w e need to m ultiply the am ount - 500 - by the rate -

8.28% - (expressed as a decim al: 0.0828). But the other principle said that the interest

m ust also be proportional to the am ount of tim e of the loan - 3 m onths -. So w e m ust also

m ultiply by the tim e (expressed as fraction of a year:3/12).

Finally then, the am ount of interest is:

500 x 0.0828 x 3/12 = 10.35

So the future value, w hich w e receive in 3 m onths tim e is:

500 + 10.35 = 510.35.

Interest rate problems

Several types of problem s com m only occur w hen w orking w ith interest rates:

1. Solving for Future Value, given Present Value, Interest Rate and Time.

H ow m uch w ill you have to repay if you borrow 150,000 for 8 m onths at an interest

rate of 9% per annum ?

Exercise 1

Pre se nt Va lue (Inve st me nt ) Int ere st Ra te Tim e Fut ure Va lue1 ,00 0 8 .5 0% 1 year$2 ,60 0 6 .2 5% 6 m onthse3 0 ,0 00 5 .1 0% 1 m onth


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2. Solving for Present Value, given Future Value, Interest Rate and Time.

You need to have $10,000 in 1 years tim e in order to m ake a dow n paym ent on a new

car. If you can earn interest at 6.50% per annum , how m uch w ould you have to

deposit today in order to have $10,000 in one year?

3. Solving for the Interest Rate, given Present Value, Future Value and Time.A U K corporate is selling com m ercial paper (a security) that w ill pay the ow ner 1,000

6 m onths from now . If the paper costs 980 to purchase today, w hat interest rate is

the investor earning?

4. Solving for Time, given Present Value, Future Value and the Interest Rate.

H ow long w ill the m oney have to be on deposit if w e w ant to invest 6,000 and w ant

to earn 337.50 in interest, if the bank is offering an interest rate of 7.50% per annum ?

1. Solve for Future Value

The first of these is the type of problem w e started w ith. W e solved it by first com puting

the interest then adding it on to the Present Value, i.e.

Future Value = Present Value + Interest

If you didnt faint, you have just survived an encounter w ith an equation! The equation

represents a relationship betw een the three quantities FV, PV and I. You have now

survived an encounter w ith m athem atical shorthand. Its quicker to w rite PV than Present

Value. C ontinuing to push you, recall that w e calculated the Interest by m ultiplying the

present value by the rate and the tim e, i.e.

Interest = Present Value x Interest Rate x Tim e

C om bining the tw o equations (w ill this never end!), w e get

FV = PV + PV x r x t

For our next trick, w e can sim plifythe right hand side of the equation by w riting:

FV = PV(1 + rt)

w hich is very succinct*. Since you saw how w e built up this equation, w e hope it is not

m ysterious to you. Lets see how w e w ould use it on one of our previous problem s.

If w e deposit 500 for 3 m onths at a rate of 8.28% , then the am ount w e receive back (FV) is:

FV = 500(1 + 0.0828 x 3/12)

*Notice how w e dropped the xfrom r x t; w hen tw o quantities (r,t) are w ritten together, they are assum ed to be m ultiplied.


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Parentheses () indicate that the quantity inside m ust be calculated first before the final

result can be obtained:

FV = 500 x 1.0207 = 510.35

w here the 1.0207 = 1 + 0.0828 x 3/12.

The reason for w riting the relationship betw een FV, PV, r and t in sym bolic form is that the

rest of the problem s can be solved using (gasp) algebra.

2. Solve for PV

FV = PV(1 + rt)

PV = FV1 + rt

You need to have 10,000 in 1 years tim e in order to m ake a dow n paym ent on a new car.

If you can earn interest at 6.50% per annum , how m uch w ould you have to deposit today

in order to have 10,000 in one year?

PV = 10,000 = 9,389.671 + 0.065

3. Solve for r

r =1

(FV

1)t PV

A U K corporate is selling com m ercial paper (a security) that w ill pay the ow ner 1,000 six

m onths from now . If the paper costs 980 to purchase today, w hat interest rate is the

investor earning?

r = 1 (1,000 1) = 0.0408 = 4.08%1/2 980

4. Solve for t

t = 1 (FV 1)r PV

H ow long w ill the m oney have to be on deposit if w e w ant to invest 6,000 and w ant to

earn 337.50 in interest, if the bank is offering an interest rate of 7.50% per annum ?

t = 1 (6,337.50 1) = 0.75 = 9 m onths

0.0750 6,000


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Summary of Formulas:

FV = PV(1 + rt)

PV = FV

1 + rt

r = 1 (FV 1)t PV

t = 1 (FV 1)r PV

Day basis

W e have presented a sim plified view of interest rate calculations. M ost calculations are m ore

com plicated because of m arket conventions w hich regulate exactly how the calculations are

to be done. This involves how the num ber of days that earn interest are calculated and

w hether the year is on a 360 or 365 day count. These topics are taken up in the appendix.

If you are interested, you can look there for m ore details. In this section w e w ill keep itsim pleand just use fractions of a year.

Discount rates

Som e securities are sold at a discount to their face value. This m eans that the security pays

a specified am ount at the m aturity of the instrum ent and the buyer pays som ething less

than that so-called face am ount today. In our previous term inology, the security has a

specific Future Value and the Present Value is todays price. This w ould all be very sim ple

except that the convention in the m arket is to quote the price today in term s of an annual

discount rate. H ere is w hat this m eans. If the security pays 100 in 1 year and the discount

rate is given as 8.00% , it m eans that todays price (the PV ) is found by sim ply subtracting

the discount from the face value i.e. the price = 100 100 x 0.08 = 92. This is not the sam e

thing as using the 8.00% as if it w ere an add-on interest rate. If it had been an add-on rate

w e w ould have found the present value using our previous form ula: PV = 100/1.08 = 92.59.

This w ould be w rong! In the w orld of discount securities interest rates are given as discounts

not add-ons. So w e m ust use them as they are intended. Lets do another exam ple.

Example

A 3 m onth Treasury bill is being sold at a 6.00% annual discount. H ow m uch w ould you

pay for $25,000 face value of this bill?

Exercise 2 Fill in the m issing item s

Present Va lue Fut ure Va lue Int erest Rat e Inte rest Tim e5 ,000 6 .4 5% 6 m onths

4 50 5 3 m onths5 50 575 8 m onths2 ,40 0 2 ,505 .60 5 .5 0%


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Before w e solve this problem , lets be sure w e know w hat is m eant. The security in question

is a Treasury bill. The U S Treasury borrow s m oney by issuing i.e. selling these bills. The bill

pays a specific am ount ($25,000 in this case) at a specific tim e (3 m onths). The investor

buys the bill at less than its face am ount ($25,000 is the face am ount). To find the price w e

subtract the am ount of the discount from the face value of the bill. A s usual w e need a

rate to w ork w ith, 6.00% in this case and a tim e period, 3 m onths. Then:

Price = PV = Face Discount

Price = 25,000 25,000 x 0.0600 x 3/12 = 24,625

N otice that the am ount of the discount w as determ ined by both the rate (6.00% ) and the

tim e (1/4 of a year). The greater the period of tim e, the greater the discount w ould be. The

tim e period of the bill w ould have had to have been an entire year for the discount to be

a full 6.00% (w hich w ould have been $1,500 instead of just $375).

Since you are so used to form ulas by now , w e can determ ine the correct expression for discount

securities:

Price = PV = Face Discount

Price = Face Face x r x t

Price = Face x (1 rt)

W e have a com m on backw ardsproblem too. N ote that Face is the FV here.

Solve for r, given PV, FV and t

r = 1 (1 PV )t FV

Example

A piece of com m ercial paper w ill pay off $5,000 in 4 m onths. If its m arket price is $4,850,w hat is the discount rate?

r = 1 (14,850 ) = 0.0900 = 9.00%4/12 5,000

Exercise 3: Fill in the m issing item s

Face Va lue Discount Rat e Time Price5 ,00 0 8 .5 0% 8 m onthse1 ,00 0 ,0 00 6 .2 0% 3 m onths

2 ,00 0 ,0 00 ,000 6 m onths 1 ,910 ,00 0 ,0 00


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Comparing rates

Rem em ber this problem w e posed early on?

You w ish to invest 750 for 9 m onths. The banker says you can receive 771.25 in

9 m onths. But he gives you an alternative. Instead he says, you can receive 750 in

9 m onths if you w ill give him 731 today? W hich is the better deal?

Since you are investing 750 for 9 m onths and receiving 771.25, w e know that w e can

solve for the interest rate:

r = 1 (771.25 1) = 0.0378 = 3.78%9/12 750

The second alternative is to invest 731 and get 750 in 9 m onths. W e can solve for the

im plied interest rate:

r = 1 (750 1)= 0.0347 = 3.47%9/12 731

So the first alternative is better. But lets go one step further. The second alternative is really

a discount security. You are told that its face value is 750 and its price is 731, since w e

know the tim e to w hen the security pays off is 9 m onths, w e can solve for the discount rate:

r = 1 (1731 ) = 0.0338 = 3.38%9/12 750

Thats probably one too m any rates for you! W hat is the relationship betw een the answ ers

to the last tw o problem s? In the last problem w e treated the investm ent as a discount

security and found that the discount rate w as 3.38% . That is, if w e started w ith the face

value of the security (750) and discounted it by that rate (3.38% ) w e w ould get its price

(731). A lternatively, if w e thought of the investm ent as being like a deposit, then if w e

started w ith the deposit am ount (731) and used an add-on rate of 3.47% , w e w ould

arrive at its future value (750). The tw o rates, 3.47% and 3.38% , are obviously tw o w ays

to get the sam e result. H ow ever one is used as an add-on and the other is used as adiscount. W e can see, though, that they are equivalent.

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The usefulness of this equivalence is in com paring investm ents. If one investm ent is on an

add-on basis and the other is priced at a discount, w e m ust convert one rate to the other

type in order to com pare them and see w hich is better. That is, w e m ust find the add-onrate that gives the sam e answ er as the discount rate or the other w ay around. This is all

w e have to say on this topic for now , but if you w ere to look in a financial new spaper like

the W all Street Journal at the prices of U S Treasury bills, you w ill see that the discount rate

is given and along w ith it, the equivalent add-on interest rate. This is for the convenience

of the new spapers readers since they have the opportunity of either investing their m oney

at their local savings institution, w hich w ould use an add-on rate, or buying U S Treasury

bills, w hich are priced at a discount. By looking in the W all Street Journal and com paring

to the local savings rate, they can see w hich gives a better return.

Example

A bank is advertising a rate of 7.50% for 3 m onth deposits. A highly rated U S C orporation

is selling its 3 m onth com m ercial paper at a discount of 7.40% . W hich investm ent gives a

better return?

Suppose you w ere going to buy $1,000 face value of the com m ercial paper. You w ould

have to pay

1,000 1,000 x 0.0740 x 3/12 = $981.50

today.

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Figure 2 Add-On, Discount and Equivalent Rates

6 month Add-On rate of 8%

PV IN T FV100 + 100 x 0.0800 x 1/2 = 104

(a)

6 month Discount rate of 8%

PV D iscount FV

96 + 100 x 0.0800 x 1/2 = 100

(b)

6 month Add-On Rate equivalent to 6 month Discount Rate of 8%

96 + 96 x 0.0833 x 1/2 = 100

(c)


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The im plied add-on rate is:

r = 1 (1,000 1) = 0.0754 = 7.54%3/12 981.50

so the return on the com m ercial paper is better than the return on the bank deposit.

A ssum ing the corporation has a credit rating at least as good as the bank, w e w ould be

better off investing in the C P (C om m ercial Paper).

In each case assum e you have a discount security w ith a face value of 1000. Find its price,

then determ ine w hich add-on rate w ould give you the sam e return on your initial

investm ent. The first problem is solved for you:

Price = 1,000 1,000 x 0.0800 x 4/12 = 973.33

r = 1 (1,000 1) = 0.0822 = 8.22%4/12 973.33

Market interest rates

LIBO R

The m arket for deposits is an inter-bank m arket. Each day banks borrow from and lend to

each other and their custom ers. The benchm ark rates for deposits and loans are referred to

as LIBO R, w hich stands for the London Inter-Bank O ffered Rate, the rate at w hich large banks

w ill lend (offer funds) to one another. The rate at w hich the bank w ill borrow is called LIBID

and the average of the tw o is called LIM EA N . The significance of these rates, w hich are

quoted for term s of overnight to one year, is that they are visible to all m arket participants

and so form a reference point for all other cash rates. This rate exists in all the m ajor

currencies such as U SD , G BP, JPY and EU R.

Table 1Euro Deposit Rates on 19 August 1999

Euro Yen Sw iss St erling USD HKD M XN

1 w e e k 2 .10 0 .0 275 0 .8 8 4 .84 5 .26 5 .97 19 .00

1 month 2 .57 0 .0 33 0 .9 11 4 .98 5 .30 6 .12 20 .25

3 m onths 2 .65 0 .0 575 1 .0 63 5 .08 5 .43 6 .37 20 .70

6 m onths 2 .86 0 .2 7 1 .4 27 5 .39 5 .81 7 .03 22 .75

1 year 3 .16 0 .2 862 1 .6 51 5 .73 5 .92 7 .50 24 .40

Exercise 4:

Tim e Discount Rate Price (per 1 ,0 00 ) Add-On Rate4 months 8 .0 0 %1 year 5 .2 5 %6 months 4 .8 0 %9 months 6 .0 0 %


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Answers to Exercises

Exercise 4

Tim e Discount Rate Price (per 1 000 ) Add-On Rate4 m onths 8 .0 0 % 973 .33 8 .2 2%1 year 5 .2 5 % 947 .50 5 .5 4%6 m onths 4 .8 0 % 976 4 .9 2%9 m onths 6 .0 0 % 955 6 .2 8%

Exercise 3

Face Value Discount Rate Tim e Price5 0 00 8 .5 0 % 8 m onths 4 ,716 .67

e1 00 0 0 00 6 .2 0 % 3 m onths e984 ,500

2,000,000,000 9 .00% 6 months 1,910,000,000

Exercise 2:

Present Value Future Value Interest Rate Interest Tim e4843.79 5 ,0 0 0 6.4 5% 156 .21 6 months45 0 4 55 4 .44 % 5 3 m onths55 0 5 7 5 6 .82 % 25 8 months24 0 0 2 ,5 0 5.6 0 5.5 0% 10 5 .60 0 .8 years

(9.6 mont hs)

Exercise 1:

Pre se nt Va lu e (I nve st m en t) I nt e re st Ra te Ti me Fu tu re Va lu e1 0 00 8 .5 0 % 1 year 1 ,085$2 6 00 6 .2 5 % 6 m onths $2 ,681 .25

e3 00 0 0 5 .1 0 % 1 m onth e30 ,127 .50


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Bonds are issued by corporations, banks, financial institutions - such as insurancecom panies and brokerages, governm ents, governm ent agencies and so-called

supranationalssuch as the W orld Bank. A bond is just a m arket-based m ethod for these

entities to borrow m oney. A bond really represents a loan, but w hereas loans are taken

outfrom a bank, bonds are securities and so are soldin a m arket. In this section w e are

going to, as usual, ignore the fine details of how bonds are originated and sold. For now

w e w ant to go over som e of the basic term inology of the bond m arket.

Principal Amount

W hen a bond is sold the principal am ount is the total am ount of m oney

being borrow ed. G enerally the bond m arket only deals in significant sizes of m any m illions

of dollars. A typical size for a bond offering by a corporation w ould be 100 to 200 m illion

dollars, but there have been m any bonds of over 1 billion dollars. Bonds are used by

corporations to fund their capital investm ents, to expand their operations and frequently

to m ake acquisitions of other corporations. The W orld Bank raises m oney to fund loans to

underdeveloped countries, w hile the IM F, w hich is part of the W orld Bank, uses funds to

help stabilise the econom ic system s in developing countries. The m ost significant feature

of the principal am ount is that it is used as the basis for com puting the interest paym ents

to be m ade.

Interest

The m ost com m on type of bond pays a fixed interest rate. The interest rate is also called

the coupon rate, w hich refers back to the days w hen bonds w ere physical pieces of paper.

In those days, the interest paym ents w ere sym bolised by coupons attached to the physical

security. W hen it w as tim e for one of the paym ents to be m ade, the bond holder w ould

clipthe coupon off and bring or send it to the issuers bank for paym ent.

W hen the interest rate is of the fixed type, interest is calculated by m ultiplying the coupon

rate tim es the principal am ount and the tim e period covered by the paym ent. The usual

day basis* used is either 30/360 or actual/365, but in the U S Treasury m arket, w hich is the

largest single bond m arket in the w orld, the convention is actual/actual. A s usual these are

described later in m ore detail.

Floating Rate

Som e bonds have an interest rate that is reset every period (just like the interest rate sw aps

discussed in Section 4). These bonds are usually called Floating Rate N otes (FRN s). The m ost

com m on reference rate for resetting the interest rate is LIBO R, the London Interbank

*See the Appendix Day Basis

12

Section 2Bonds


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O ffered Rate, or one of its relatives in other currencies such as EU RIBO R (LIBO R for

paym ents in Euro).

Maturity Date

O n the m aturity date, the issuer of the bond w ill repay the principal am ount of the bond

as w ell as any coupon that m ay be due on that date.

Example

Issuer XYZ C orporation

Principal A m ount U SD250,000,000

Paym ent Date 15 July 1999

C oupon Rate 6.50%

First Coupon 15 January 2000

Frequency sem i-annual

D ay Basis 30/360M aturity Date 15 July 2004

This is an exam ple of a five year, fixed coupon bond issued by a corporation. The principal

am ount w ould be received by the com pany on 15 July 1999 and the full am ount w ould be

repaid on 15 July 2004. The coupon paym ents are m ade every 6 m onths on 15 July and

15 January. For the detail m inded, w hen the day on w hich a paym ent is m eant to be m ade

falls on a non-business day, w hich generally m eans the banks are not open, the paym ent

rolls forw ard to the next business day. In the event the next business day w ould be in the

next m onth, how ever, the paym ent is usually rolled back to the previous business day. Thism ethod of determ ining w hen a paym ent happens is an exam ple of a Business D ay

C onvention. Because this bond uses the 30/360 m ethod of counting days, each coupon

period w ill contain 180 days (ignoring the non-business day problem ) so one-half of the

coupon w ill be paid out at each coupon date.

The principal am ount of the bond is also referred to as the param ount. W hen the price

of the bond, w hich fluctuates, happens to equal the par value of the bond w e say the bond

is trading at par. The coupon on the bond is then called the par coupon. Prior to a bond

being issued, w e m ight ask w hat w ould the par coupon be?W hat this question asks is

w hat coupon w ould the bond need to have in order for buyers to be w illing to pay the full

am ount for the bond? The higher the coupon, the m ore attractive the bond w ill be to

investors but the w orse for the borrow er. C onversely, a low coupon w ould be good for the

borrow er but unattractive to an investor. Because the borrow er usually needs the m oney

and the investor usually has lots of choices for investing, the requirem ents of the investors

usually predom inate in determ ining w hat the interest paym ent w ill be. The rate the investor

w ill dem and in turn depends on the general level of rates and the specific credit quality of

the borrow er. The general level of rates refers to the rate being paid by the highest quality

issuers such as governm ents and the credit quality refers to the perceived ability of the

borrow er to m ake all the bond paym ents as scheduled.

13


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15

Bond Price

A s noted above, a bond has a principal am ount, w hich is the am ount of m oney being

borrow ed. In the m arket w e do not usually have m uch reason to refer to the principal

am ount. W e are m ore concerned w ith the m arket price of the bonds. If the principal am ount

is USD 250 m illion, it is unlikely that any one person or institution w ill buy all of the bonds.

Each buyer of a bond is a lender to the issuer. The bonds them selves w ill be bought inam ounts that are sm aller than the entire size of the issue. U sually there w ill be som e sm allest

am ount of the bond that w ill be offered for sale to investors. Because the bond m arket is

m ostly a professional and institutional m arket, the sm allest size is usually 1 m illion dollars. It

m ay be possible for individual investors to buy sm aller am ounts of the bond through their

broker but w e w ill ignore this retailm arket. A lso m ost bonds are no longer physical pieces

of paper, rather they are kept in book-entry form and bought and sold through banks or

brokers. The price of the bond in the m arket is quoted as a percentage. A price of 105, for

exam ple, w ould m ean that an investor w ould need to pay 105% of the principal am ount of

the bond in order to buy it. If the investor w anted to buy 5 m illion dollars of principal, or face

value of the bond, the price w ould be 5.25 m illion dollars. If, instead, the investor w anted to

buy 10 m illion dollars of face value, the price w ould be 10.5 m illion dollars. This w ay of

quoting the bond price is convenient because it can be used to calculate the actual price to

be paid for any face value of bonds. This is analogous to quoting stock on a per share basis.

If the price of IBM is USD 200 then the investor pays 200 m ultiplied by the num ber of shares

bought. Just think of percent of par as being the fundam ental unit of price.

Accrued Interest

By its nature a bond is a lot like a savings account. W hen you put m oney in the bank you

expect to be paid interest. Even if the bank credits your account just once a year, if you w ere

to w ithdraw your m oney part w ay through the year, you w ould expect to receive the interest

already earned. This is the concept of accrued interest. If a bond has an annual 6% coupon,

the issuer is prom ising to pay 6% interest every year, on the coupon paym ent date. If the

entire bond issue w as ow ned by just one person or institution and if the bonds w ere never

traded in the secondary m arket, there w ould be no problem w ith this. But as a m atter of fact,

after the bonds are issued there w ill be secondary trading i.e. trading by people w ho did not

buy the bond originally from the issuer. Each tim e the bond is traded, the buyer of the bond

is required to pay the am ount of interest that has accrued from the last coupon paym ent to

the trade date. This is because w hen the issuer pays the coupon, the full am ount of the

coupon w ill go to the person w ho ow ns the bond on the coupon paym ent date, no m atter

how long they have actually ow ned the bond, even if it is as short as one day. This idea of

accruing a benefit like interest is different from the case of dividends on stock. If you ow n a

stock for years and then sell it the day before it goes ex dividendyou w ill not receive any

part of that dividend. Basically the reason for this is that investors realise that corporations are

not required to pay dividends and even if they do, the am ount of the dividend is uncertain.

So there w ould be no basis for determ ining how m uch had accrued. Rather than try to

m ake som ething up, it is custom ary for w hoever ow ns the share on the record dateto

receive the entire dividend. But as noted this is not the case in the bond m arket.


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18

change? Since w e w ould be using a higher interest rate w ith w hich to discount the cash

flow s, the price w ould certainly be less. The calculation below verifies this.

Tim e present 1 year 2 years 3 years 4 years

Cash Flow 10 10 1 0 11 0

Present Value 103 .24 9 .1 7 8 .4 2 7 .72 77 .93

Exercise 1

Fill in the table below if the yield to m aturity of the bond is 7% rather than 8% or 9% .

Tim e present 1 year 2 years 3 years 4 years

Cash Flow 1 0 1 0 10 110

Present Value

Example

A 5 year bond w ith a 6% , sem i-annual coupon has a yield to m aturity of 4% . W hat is its price?

This exercise involves doing 10 calculations. C learly it w ould be better done by a calculator.

W e can use the H P12C to get the answ er.

Coupon Bond Pricing Formula

For bonds which pay an annual coupon:

F = Face Valuey = y ield to matur i tyc = coupon raten = years to matu r i ty

I f the bond p ays a coupon more than once in a year, then the formula is :

w here interest is paid m t imes per year.

Price = cF + F

(1 + y)k (1 + y)n

Price =

cFm

+(1+ )y

m

F

(1+ m x n)y

m

k

k=n

k=1

k=m x n

k=1

Figure 3

Keyst roke Display Notes

f FIN 0 clear the f inancial registers10 n 1 0 1 0 paym ents

3 PM T 3 each one is 3

10 0 FV 1 0 0 principal repaym ent

2 i 2 yield /periods per year

PV -1 0 8 .9 8 price is show n as negat ive indicat ing you w ou ld

pay this to receive the cash flows

FV

PM T PM T PM T PM T PM T PM T PM T PM T PM T PM T

PV


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The H P12C does financial calculations using a cash-flow approach. The diagram illustrates

how the calculator view s the problem and the term s it uses for its calculations. The

calculator distinguishes betw een the last coupon paym ent, w hich is part of the PM T

sequence and the repaym ent of the bond principal w hich is FV, even though it assum es the

coupon and principal paym ents occur at the sam e tim e.

Price of 10% 20 year semi-annual bond

The Price-Yield Relationship

A s the graph in Figure 4 show s, the price of a fixed coupon bond decreases as the yield

increases. This m akes sense because w e are present valuing each cash flow w ith a higher

interest rate, w hich m eans the denom inator of each term in the expression for the bond

price is larger, w hich m akes the value of the fraction sm aller. N otice though that the

relationship betw een yield and price is not linear.

Table 1

Yie ld (% ) 1 3 5 7 9 1 1 1 3 1 5

Price 262 .78 204 .71 162 .76 1 32 .03 10 9 .20 91 .98 78 .78 68 .51

W hen the yield is increased from 1% to 3% , the bond price drops by about 58 points or

22% , w hile if w e increase the yield from 13% to 15% , the price decreases by about 10

points or 13% . N otice also that w hen the yield is 10% , the bond price is 100 (this can be

seen from the graph). Recall that w e talked earlier about par, prem ium and discount bonds.A nother w ay to think about this is that a bond w ill be priced at par if its yield is equal to

its coupon. Prem ium bonds have coupons greater than their yield, w hile the yield on a

discount bond is greater than the coupon.

Yield From Price

If the bond in the previous exam ple, w hich w as a 20 year bond w ith a sem i-annual coupon

of 10% , is trading for 88 in the m arket, w hat is its yield? From the graph and the chart

below it, w e can guess that the yield m ust be m ore than 11% but less than 13% . But how

could w e find its exact value?

Figure 4

0.00

50 .00

10 0.00

15 0.00

20 0.00

25 0.00

30 0.00

1 4 710

13

16

19

22

25

28

Yield

Price

19


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D eterm ining the yield of a bond from its price is not an easy m athem atical problem to

solve. W e could use various approxim ations w hich give good results or w e could use a

process of trial and error, trying different yields until w e got one that resulted in a price

close to 88. For exam ple, w e could try 12% . U sing the H P12C :

Figure 540 n 40 n

100 FV 100 FV

5 PM T 5 PM T

6 i 5.75 i

PV -84.95 PV -88.35

Since 12% w as too large, w e then tried 11.5% . N ote that since the coupon is sem i-annual

w e had to use 40 periods (2 x 20) , half the coupon (5) and half the proposed yield (6 or

5.75). W e could go on like this, by next trying 11.75, but fortunately the H P12C can solve

this problem for us. A ll w e have to do is ask! This tim e put in the price (as a negative

num ber) for the PV.

Figure 6

40 n

100 FV

5 PM T

-88 PV

i > 5.775

2 x > 11.55

This show s the yield to m aturity is 11.55% . O nce again, w e had to m ultiply the answ er

(5.775) by 2 since the coupon w as sem i-annual and the value that the H P12C gave us w as

the one-period interest rate.

Understanding Yield To Maturity

Yield to m aturity is useful because it is so w ell accepted in the m arket. It is w orth analysing

m ore carefully though to see w hat insight it can give us about bond value. Lets use our

first exam ple, w hich w as a 4 year 10% annual coupon bond that w as priced at 106.623,

thus yielding 8% :

106.623* =10

+10

+1.08 1.08

2

A criticism of yield to m aturity is that it seem s to im ply that the yield curve is flati.e.

interest rates are the sam e for investm ents of any m aturity. This does seem to be im plied

by the coupon bond pricing form ula. H ow ever this criticism is not truly correct as w e w ill

see later w hen w e discuss the yield curve in m ore detail. Still, it is true that the form ula uses

*The exact value is 106.6243 if you use a bond calculator

10+

110

1.083

1.084

20


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21

the sam e rate for each paym ent. W e can gain som e insight into the equation by looking

at it in a different w ay:

106.623(1.084

) = 10(1.083

) + 10(1.082

) + 10 (1.08) + 110

Lets look at the tw o sides of the equation separately. The left hand side is the am ount w e

w ould have in four years if w e put 106.623 into a bank deposit that paid 8% interest on

an annual-com pounded basis. The right hand side has four term s. The first one, 10(1.083

),

is the am ount w e w ould have four years from now if w e w ere able to invest the first

coupon, w hich w e w ill receive in one year, for three additional years in an account paying

that sam e 8% interest. Sim ilarly the next tw o term s represent the am ount in an account

four years from now if w e are able to reinvest the coupons w e receive at 8% from the tim e

w e receive them to the end of the four year period. The last term , 110, is sim ply the final

paym ent from the bond w ith its last coupon. A s these are received in four years tim e, no

reinvestm ent of those cash flow s is necessary.

So a w ay to understand yield to m aturity is that buying a bond is equivalent to investing its

price in a deposit that pays interest equal to the bonds yield to m aturity, com pounded w ith

the sam e frequency as the bonds coupon is paid. In other w ords, w e w ould be indifferent

betw een buying the bond and investing its price in such an account. N ote that this analysis

indicates that the bonds yield to m aturity is an im plied reinvestm ent rate for its coupons. This

m akes clear that reinvestm ent risk is a significant feature of coupon bonds.

But w ere not done yet! Everyone does talk about reinvestm ent of the coupons as part of

the risk of the bond and w e are not going to disagree w ith that here. But lets have onem ore look at that bank account the one that pays 8% interest on an annual com pounded

basis. Suppose w e do put the bonds price into that account and then each year w ithdraw

10 from it and reinvest w hat rem ains. H ere are the cash flow s*:

Table 2

Tim e 0 1 2 3 4

Amount 10 6 .6 243 11 5 .1 542 1 13 .566 6 11 1 .8 519 110

Withdraw 0 10 1 0 10 110

Remains 10 5 .1 542 1 03 .566 6 10 1 .8 579 0

W hat this says is that the bonds price placed in the account at 8% interest can exactly

duplicate the bonds cash flow s. Replicating one instrum ent w ith another is the foundation

of arbitrage, a them e w e w ill return to later.

Summary

W e have seen three w ays to understand yield to m aturity:

1. The single interest rate that equates the bonds price to the sum of the present

values of its cash flow s.

*W e used values correct to four decim al places to avoid rounding errors.


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22

2. The assum ed reinvestm ent rate of the bonds coupons.

3. The equivalent rate of interest on a bank deposit of the bonds price.

A ll of these different w ays to understand the yield to m aturity concept help to clarify the

strengths of this m easure as w ell as pointing to som e of its w eaknesses.

Bond Risk Measures

Return versus Yield

Som etim es these tw o term s are used interchangeably, but they are quite different. Yield,

as w e m entioned above, is a forw ard looking concept. You can see this from our discussion

of yield to m aturity. The yield to m aturity of a bond can be view ed as the im plied

reinvestm ent rate of the bonds coupons. Return is retrospective i.e. a backw ard-looking

concept. It applies to any type of investm ent not just bonds. The idea is sim ple. W hen you

enter into an investm ent w hat really m atters are the cash flow s you get from it and w hat

you do w ith them . For exam ple if you buy a stock, you m ight w ant to look at it on som e

tim e horizon such as 5 years. O ver that 5 year period as you hold the stock, you m ay receive

cash flow s in the form of dividends. You can either put them in a deposit or you could use

them to purchase m ore shares. The decision you m ake w ill affect the perform ance of your

investm ent. A t the end of the 5 years you could look at the cash value of your position at

that tim e and ask, w hat annual-com pounded rate of interest w ould I need to have been

paid on m y original investm ent to have this final am ount of cash? That interest rate w ould

be your realised annualised rate of return.

Example

You buy a share for 100, w hich pays a dividend at the end of each year. W hen you receive

a dividend you reinvest it in the shares. Your investm ent experience over the 5 year period

is show n below :

Table 3

Tim e 0 1 2 3 4 5

Share price 100 1 10 1 05 9 5 12 5 13 5

Dividend 4 4 5 5 6

Number of Shares 1 1 .0 36 36 1 .0 74 46 1 .1 270 9 1 .1 670 9 1 .21 15 3

Cash Value 100 11 4 .0 00 00 11 2 .8 18 18 10 7 .0 735 9 14 5 .8 86 31 16 3 .55 72 1Return* 14 % 6.2 % 2 .3 % 9 .9 % 10 .3 %

A s the table show s, after 5 years the cash value of your position is 163.56, w hich is the

cash value of the approxim ately 1.2 shares you ow n. You have earned an annual

com pounded 10.3% rate of return. N otice that the annual rate of return over the 5 year

period w as quite volatile. You earned 14% in your first year, but lost m oney in the second

and third. Fortunately the stock price cam e back in years 4 and 5.

*A nnual-com pounded return


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Exercise 2

Fill in the table again but assum e that instead of reinvesting the dividend into the share,

you deposit the cash am ount of the dividend in an account that pays 5% annual interest.Time 0 1 2 3 4 5

Share price 1 00 11 0 105 9 5 12 5 135

Dividend 4 4 5 5 6

Number of Shares 1 1 1 1 1 1

Cash Value 1 00

Return

Returning to bonds, suppose that the current yield for 20 year bonds is 9% so that a 20

year bond w ith a 9% annual coupon w ould be priced at par.

Calculator Note

Before w e get into our m ain discussion lets explain the use of the H P12C in doing return

calculations. W e already know that 9% is the im plied reinvestm ent rate for the coupons

on our bond, but if w e are able to reinvest those coupons at 9% , how m uch cash w ill

w e have in 20 years? The H P12C can easily handle this calculation show n in Figure (7a):

N ow if w e add in the repaym ent of the bond principal, w e w ill have 560.44 in total at the

end of 20 years. So w hat rate of return did w e earn? Figure (7b) show s the calculation.

Rem em ber w e are asking, if w e put the bonds price into a deposit that pays an annual

com pounded rate, w hat rate w ould w e need to be paid to have the sam e am ount in cash

after 20 years? The unsurprising answ er is 9% .

W hat if yields w ere suddenly to go up to 10% im m ediately after w e bought the bond?

Then the bonds price w ould drop to 91.49 (check this on your calculator). C onversely, if

the 20 year rate w ere to go dow n to 8% , the bonds price w ould go up to 109.82. This

illustrates the price risk of ow ning the bond, due to the change in yield. W e can see this

price risk in the pricing form ula for the bond:

Figure (7b)

20 n

-100 PV

0 PM T

560.44 FVi > 9%

Figure (7a)

20 n

0 PV

9 PM T

9 iFV > -460.44

23

Figure 8

Price = cF + F

(1 + y)k

(1 + y)n

k=n

k=1


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24

A larger yield (y) w ill give a low er price, w hile a sm aller yield w ill give a higher price. Recall

once again that the yield is the im plied reinvestm ent rate for the coupons. W e verified

above that if w e are able to reinvest the coupons at 9% for our sam ple bond, then the

realised return does equal the yield.

Suppose now that rates go to 8% and that w e are only able to reinvest the bonds coupons

at this reduced rate. H ow m uch cash w ill w e have in 20 years and w hat w ill our realised rate

of return be? C learly the return should be less than 9% and the calculation show n below

confirm s this.

Sim ilarly, if rates go to 10% and w e are able to reinvest the coupons at that rate, w e w ould

expect that in 20 years, our realised return w ould be m ore than 9% . You should verify as

an exercise that the return w ould, in fact, be 9.51% .

These calculations illustrate the reinvestm ent risk of a bond. N ote that the tw o risks, price

risk and reinvestm ent risk w ork in opposite directions. W hen yields go up, it is bad for the

bond price but good for the reinvestm ent of the coupons. C onversely w hen rates go dow n,

the bond price w ill increase but the total coupon reinvestm ent w ill be less.

Holding Period Return

In the previous section w e considered w hat w ould happen to a bond and our rate of return

under tw o interest rate and tim e scenarios. If rates change and w e im m ediately sell the

bond w e profit if rates have gone dow n and lose if rates have gone up; if instead w e hold

the bond to m aturity, the opposite is true. Think of these as tw o different holding periods

for our investm ent, either 1 day or to m aturity. W hat if w e hold the bond for som e

interm ediate am ount of tim e? W e w ill explore this idea in this section.

The am ount of tim e from w hen w e buy the bond to w hen w e term inate the investm ent,

either by selling it or because it m atures, is called the holding period. W e assum e that w hile

w e hold the bond w e w ill reinvest any coupons w e get at w hatever the current rate of

interest happens to be. In order to conduct our investigation w e have to m ake som e

sim plifying assum ptions. Lets not w orry right now about how realistic these are. W e w ill

discuss that later. If w e can understand how things w ork in a sim ple environm ent w e can

use it to extend our understanding to m ore realistic situations. A t the very least w e should

understand the lim itations of our analysis.

W e w ill assum e that the yield curve is flat. This m eans that all bonds have the sam e yield,

irrespective of their m aturity. W e also assum e that w hen the yield curve changes, it does

Figure 9

20 n 20 n

0 PV -100 PV

9 PM T 0 PM T

8 i FV 511.86

FV > -411.86 i > 8.51%


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25

so instantaneously and in a w ay that affects all m aturities in the sam e w ay i.e., a parallel

shift. Finally w e assum e that w e buy the 20 year 9% bond for par and that instantly the

yield curve m oves to a new level and stays there for the entire holding period.

Table 4 below show s the realised rates of return for different holding periods for three

interest rate levels.

Table 4(a)

Rate 8%

Holding period 0 2 4 6 8 10 12 1 4 16 18 20

Bond price 1 09 .82 109 .37 108 .8 5 108 .24 10 7 .54 106 .71 105 .7 5 104 .62 103 .31 101 .78 100 .00

Reinvested coupons 0 .00 18 .72 40 .5 6 66 .02 9 5 .73 130 .38 170 .7 9 217 .93 272 .92 337 .05 411 .86

Total value 1 09 .82 128 .09 149 .4 1 174 .27 20 3 .27 237 .09 276 .5 4 322 .56 376 .23 438 .84 511 .86

Holding period return 1 3 .1 8% 10 .5 6% 9 .7 0% 9 .2 7% 9 .0 2% 8 .8 5% 8 .7 2% 8 .6 3% 8 .5 6% 8 .5 1%

(b)

Rate 9%

Holding period 0 2 4 6 8 10 12 1 4 16 18 20

Bond price 1 00 .00 100 .00 100 .0 0 100 .00 10 0 .00 100 .00 100 .0 0 100 .00 100 .00 100 .00 100 .00


Total value 1 00 .00 118 .81 141 .1 6 167 .71 19 9 .26 236 .74 281 .2 7 334 .17 397 .03 471 .71 560 .44

Holding period return 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0% 9 .0 0%

(c)

Rate 1 0%

Holding period 0 2 4 6 8 10 12 1 4 16 18 20

Bond price 91 .49 91 .80 92 .1 8 92 .63 9 3 .19 93 .86 94 .6 7 95 .64 96 .83 98 .26 100 .00


Total value 91 .49 110 .70 133 .9 5 162 .07 19 6 .11 237 .29 287 .1 2 347 .42 420 .38 508 .66 615 .47

Holding period return 5 .2 1% 7 .5 8% 8 .3 8% 8 .7 8% 9 .0 3% 9 .1 9% 9 .3 0% 9 .3 9% 9 .4 6% 9 .5 1%

A s w e can see in (a), w hen rates m ove dow n to 8% , w e have a high rate of return if w e have

a short holding period. For exam ple, if w e hold the bond for 6 years w e earn 9.70% .

C onversely if rates m ove to 10% as in (c) w e have a low rate of return if the holding period

is short. For that sam e 6 year holding period the return w ould be just 8.38% . The m ost

interesting thing in the table though is that if w e have a ten year holding period, the realised

return is just about the sam e irrespective of w hether rates go dow n to 8% , up to 10% or

stay the sam e at 9% . This observation leads us to the m ain topic of this section duration.

Duration

D uration is a concept that is im portant for both bonds and sw aps. W e w ill see applications

of duration in m any parts of the w orld of fixed incom e and interest rates. W e w ill look at

duration in these w ays:

Price Risk versus Reinvestm ent Risk

G eom etric View

A s a w eighted average of tim e


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26

In the preceding exam ple w e saw that a 10 year holding period im m unisesthe bond ow ner

to a sudden, sustained parallel shift in the yield curve. This is our first view of duration.

Price Risk Versus Reinvestment Risk

In the early years of a bonds life price risk is dom inant, w hile as the tim e the bond is held

approaches m aturity, reinvestm ent risk is dom inant. D uration is the point in tim e w hereprice risk is equal to and opposite in sign to reinvestm ent risk. That is, duration is the

holding period for w hich the bonds return does not depend on the level of interest rates.

N aturally one has to be careful in using this idea. D uration is an instantaneous risk m easure.

A s tim e goes on and interest rates do change so w ill duration, a feature w e discuss below .

Geometric View

A sim ple physical analogy is often used to illustrate another w ay to understand duration.

Im agine that w e put w eights along a plank (like a see-saw ). Each point along the plank

represents a point in tim e w here w e receive a cash flow from the bond, either coupon orprincipal. The w eight is not the cash flow itself, but the present value of that cash flow . For

sim plicity w e w ill assum e that w e discount all the cash flow s using the bonds yield to

m aturity. So at tim e t (expressed in years) w e place a w eight equal to:

C ash flow t

(1 + y)t

w here, as usual, y denotes the yield on the bond*.

Example

C onsider a 4 year 10% annual coupon bond w hich is currently yielding 8% . The

distribution of w eights is show n below .

The point along the plank w here w e could put a fulcrum and have the w hole system in

balance is the bonds duration. This turns out to be about 3.5 years. W e w ill see in the next

section how to calculate this num ber directly. For now w e can verify that the m om ents of

forcearound that point are equal, w hich they m ust be to have the system balance.

*This form is correct for annual coupon bonds. For other bonds w e w ould m ake the appropriate change to the denom inator.

Figure 10

9.38.6

7.9

80.9

1 2 3 4


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W hich w ould have a longer duration, the 4 year 10% bond w e considered earlier or a 4

year 6% bond? Low ering the coupon m akes the initial w eights (present value of coupons)

sm aller. It also m akes the final w eight (present value of coupon plus principal) slightly

sm aller but the effect is dw arfed by the return of principal. In order to keep the system in

balance w e w ould need to m ove the fulcrum to the right (this decreases the m om ent of

force from the final cash flow ). So low ering the coupon lengthens the duration. C onversely

if w e increased the coupon to 12% , the three initial coupons w ould be larger and the

fulcrum (duration) w ould m ove to the left (becom e less). N um erically the durations are

show n below .

Table 5

Coupon 6% 10 % 1 2%Durat ion 3 .6 6 3 .51 3 .44

Duration and Time to Maturity

W hat w ould happen if w e changed our 4 year 10% bond to a 5 year 10% bond? Now there

w ould be one m ore coupon (w eight). A lso the return of face value w ould m ove 1 year to

the right. Since the final paym ent is so large, m oving one year to the right w ill greatly

increase its force (the force w as 0.5 x 80.9, now it is 1.5 x 74.9). This w ill cause the fulcrum

to m ove to the right i.e. as the tim e to m aturity increases, so does the duration. The size of

this effect on duration dim inishes as the tim e to m aturity continues to increase because the

size of the w eight gets sm aller as w e m ove it out in tim e. For exam ple, if the bond has a 20

year m aturity, the final paym ent of 110 w ould have a present value (at 8% ) of only 23.6,

w hich is less than the total present value of the first three coupons (25.77). This suggests

that although the duration increases, the rate of increase lessens as w e extend the m aturity.Figure 11 illustrates this.

The black horizontal line is both a ceiling and a lim it to the duration of a par or prem ium

bond (coupon yield). The level at w hich the line is draw n is (1+y)/y, w here y is the current

yield level. This value is the duration of a so-called perpetual bond one that pays a coupon

indefinitely but never repays its principal. A t 8% this w ould be 13.5 years. O ne useful

observation suggested by the picture is that the difference in duration betw een tw o bonds

w ith tim e to m aturity 20 and 25 years is less than the difference in duration betw een tw o

bonds w ith m aturities of 10 and 15 years, assum ing all the bonds have the sam e coupon

and are yielding the sam e.

Bonds w ith coupons less than the current yield are called discount bonds because they w ill

be priced at less than par. D iscount bonds w ith short m aturities tend to act like zero

coupon bonds, but the resem blance dim inishes as the m aturity increases. Eventually even

discount bonds w ill have durations that approach the theoretical lim it. The full picture is

show n overleaf.

28


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Duration and Yield Level

W hat happens as the general level of yields changes? Recall that the ceilingin our picture

w as at the level (1+y)/y, w here y is the current level of yields.

Yield level 6 % 8% 1 0%Duration limit 1 7 .6 7 years 13 .50 years 1 1 .0 0 years

So as the level of rates goes dow n, the lim it of duration of par and prem ium bonds goes up;

conversely, as the level of yields goes up, the lim it of the duration goes dow n. This m akes sense

because, at a low er level of yields, the cash flow s are being discounted at a low er rate. This

has a greater effect on paym ents that are farther out in tim e, giving them greater w eight. This

causes us to have to m ove the balancing point to the right i.e. to a larger num ber of years.

The table below sum m arises the relationships w e have discussed:

Increase in Coupon Tim e to m aturit y Level o f yieldsEffect on duration decreases increases* decreases

Figure 11

W eve postponed the m athem atics of duration until a good understanding of duration w as

achieved. In order to do quantitative w ork though, w e have to w ork w ith the m ath.

Weighted Average of Time

D uration is a w eighted average of tim e. This is w hat w e learned from the geom etric picture

of duration. The m athem atics that goes along w ith the picture is:

Table 6

*There are som e m inor exceptions - please dont send us any e-m ail about it.

zero

discou nt

par

perpetual

0 10 20 30 40 50 60

time

du

ration

60

50

40

30

20

10

0

Figure 11

29

w h e re F = f a ce am o un tc = coupon raten = t ime to maturi t yy = yield

P = price

D uration = 1 cF x 1 + 1 cF x 2 + ...+ 1 cF + F x nP 1 + y P (1 + y)

2

P (1 + y)n


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The w eight at tim e t is the proportion of the bonds price that is attributable to the present

value of the paym ent at that tim e. By m ultiplying each tim e by the appropriate w eight, w e

get a w eighted average of tim e. This particular form of duration w as proposed by

M acauley and so is called M acauleys duration.

Price SensitivityD uration is often used to determ ine the price sensitivity of a bond that results from a change

in yield. If w e letP stand for a change in price and y stand for a change in yield, w e w ant

a w ay to calculate P/y. This expression is a rate of change. In calculus w e learn that

rates of change are calculated from derivatives i.e. P/y is approxim ately equal to the

derivative of price as a function of yield. Figure 12 illustrates geom etrically the follow ing

m athem atical relationship:

dP = 1 PDdy 1 + y

w here the left hand side is the rate of change of the bonds price as a function of the level

of yield. The reason for the negative sign is that as yield increases, price decreases i.e., there

is an inverse relationship betw een price and yield.

10 year 9% semi annual bond

Because of the presence of the 1/(1+y) factor, w e are lead to m odify the idea of durationand replace it by:

Dm od

=1 D

1 + y

and call it m odified duration. This gives a cleaner equation:

dP = PDm oddy

H ow can w e use this? A s w e m entioned,P/y is approxim ately equal to dP/dy , so

0.00

50 .00

10 0.00

15 0.00

20 0.00

0 5 10 15 20 25yield

Price

Figure 12

30

P

PD m od orP PD m od

yy

~

~~

~


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31

The equation above tells us w hat happens to price on the vertical axis w hen w e change

the yield on the horizontal axis. In Figure 12 the slope of the line tangent to the graph

is PDm od

.

Example

A 20 year annual coupon bond is yielding 8% and so is priced at par. H ow m uch w ill the

bonds price decrease if the yield increases to 8.10% ?

W e used a bond calculator to find the m odified duration of the bond. The result w as 9.9

years. The change in the yield is 0.1% or 0.001 decim al form . So the approxim ate price

change is:

P = 100 x 9.9 x 0.001 = 0.99

and the new price should be about 99.01. You can check this on the H P12C . The actual

new price is 99.025, so the approxim ation is quite good.

PVBP

The m ost com m on m easure used for price sensitivity is the price value of a basis point

(PVBP). This m easures how m uch a bonds price w ill change if its yield changes by 1 basis

point. Since a basis point is 0.0001, it is sim ple to com pute the PVBP:

PVBP = PDm od

x 0.0001

In the exam ple above the PVBP is:

100 x 9.9 x 0.0001 = 0.099

Traders often refer to this as the bonds price delta. It is the rate at w hich the bonds price

changes as yields change.


Convexity

A bond is m ore sensitive to yield changes w hen yields are low than w hen yields are high.

This is illustrated in the next exam ple. This feature is referred to as convexity.

Figure 13

0.00 00

0.02 00

0.04 00

0.06 00

0.08 000.10 00

0.12 00

0.14 00

0.16 00

0 5 10 15 20 25yield

0.16

0.14

0.12

0.100.08

0.06

0.04

0.02

0.00

PVBP


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32

Example

A 10 year sem i-annual bond w ith a 9% coupon is priced at 122.32 to yield 6% . If yields

change to 5.90% , the bonds price w ould becom e 123.17, so that it increases by 0.85. A s a

percentage of the bonds original price this increase is 0.69% . A t high yields the bond is less

sensitive to yield changes. If the sam e bond w ere yielding 10% so that its price w as 93.77

and yields m oved to 9.90% then the new price w ould be 94.37. This is a increase of 0.60 in

absolute term s and 0.64% as a percentage of the price. In this case both the bonds

percentage price sensitivity (duration) and its absolute price sensitivity (pvbp) are different at

10% than they are at 6% . These changes are called convexity and gam m a, respectively:


C onvexity is the change in a bonds duration due to a change in yield.

C onvexity =D

y

G am m a is the change in a bonds price delta (pvbp) due to a change in the bonds yield.

=

y

W hen yields increase a bonds price decreases and its duration also decreases. This decrease

in duration is a good thing for bond holders since they are losing value as yields rise. The

fact that duration decreases as yields increase m eans that further increases in yield w ill

cause less loss than changes at low er levels. Sim ilarly, as yields increase the bonds price

delta decreases (in absolute sense it is a negative num ber). Because the delta is the

product of the price and the m odified duration, both of w hich decrease as yields go up,the bonds price delta is m ore affected than the duration. This is even better for the bond

holder as the delta m easures the absolute loss the investor is incurring w ith yield increases.

The table below show s the prices and sensitivities of the bond in our exam ple at different

levels of yield.

Table 5

Yie ld 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

M acauley Durat ion 7 .6 3 7 .49 7 .34 7 .1 8 7 .00 6 .81 6 .62 6 .4 2 6 .2 1 6 .00

Modif ied Durat ion 7 .4 8 7 .20 6 .92 6 .6 4 6 .36 6 .08 5 .81 5 .5 3 5 .2 6 5 .00pvbp 0 .122 0 0 .1 015 0 .0 847 0 .071 0 0 .0 597 0 .05 04 0 .04 27 0 .036 3 0 .03 10 0 .02 66

Figure 14

4.00

5.00

6.00

7.00

8.00

0 5 10 15 20 25Yield

Dmod


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Worksheet Bond Pricing, Duration, Convexity

The inform ation below is from the Bloom berg Screen (C T G ovt) on 29 July 1993.

Securit y M arke tcoupon m aturit y b id of fer ytm durr risk

2 year 41/4 7 /9 5 10 0-7 10 0 -8 4 .1 2 1 .9 3 1 .9 0

The current 2 year note, the 41/4s of 7/95 is offered at1008/32. A sked yield is4.12%

a. Present value each of the bond's cash flow s at the stated yield. A dd the present values to verify

the bond price

b. The duration of the note is 1.93 years. Verify this.

The w eight for any cash flow is the proportion of the bond's price that is represented by that

cash flow :

1 C F*

P (1+ r/2)t

D uration (M acauley) is the w eighted average of tim e:

M acauley D uration=

M odified D uration is:

Dm od

=1 D

M acauley1 + r/2

M odified D uration =

c. The riskof the bond is 1.90

Risk = am ount the price w ill change if the bond's yield changes by 1 bp

Risk = pvbp

The pvbp can be calculated from Dm od

and the bond's price:

Risk = 0.0001 x P x Dm od

C alculate: Risk =

d. O n 29 July the current 2 year note w as offered at 4.12% (1008/32);

* C F m eans cash flow

33

Tim e (years) 0 .5 1 .0 1 .5 2

Cash Flow

Present Value

Tim e (years) 0 .5 1 .0 1 .5 2

Cash Flow pv

Weight

CF* x Weight


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34

O n 27 July the current 2 year note w as offered at 4.32% ( 9955/64).

Actual price change:

Calculated price change(using pvbp):

Summary:

Duration m easures the percentage price sensitivity of a bond to changes in yield.

Convexity m easures the change in duration as yields change.

Delta m easures the bond'sprice sensitivity to changes in yield.

Gamma m easures the change in delta as yields change.


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35

There are tw o im portant interest rate contracts that relate to deposits or loans that w ill takeplace at a future date. These are forw ard rate agreem ents (FRAs) and their exchange-traded

equivalents, Euro interest rate futures, w hich w e w ill refer to for convenience as Eurodollar

futures even if they happen to be on som e interest rate other than the U S dollar. W e w ill

describe the idea of these instrum ents and then talk about how they w ork in practice.

Facts about Futures

Futures are contracts traded on Exchanges such as:

LIFFE (London International Financial Futures Exchange) C M E (C hicago M ercantile Exchange)

IM M (International M onetary M arket part of C M E)

SIM EX (Singapore M onetary Exchange)

The underlying is a com m odity, currency, interest rate, bond or index.

The contract is for a standardised am ount and quality.

O nly a sm all num ber of delivery (settlem ent) dates are available.

Trades are done by Exchange m em bers but the exchange itself acts as the

counterparty to both sides.A com bination of a long position and a short position in the sam e contract and held

in the sam e account offset one another, resulting in no position.

Traders are required to post (deposit) m argin to guarantee their perform ance on the

contract.

Futures are m arked to m arket daily w ith profits and losses realised im m ediately

through the m argin account.

If the contract is settled into an actual underlying for w hich the standardgrade is

either non-existent or difficult to obtain or deliver (like live hogs or Treasury bonds), the

party w ho is short the contract w ill have som e choice as to w hat to deliver and the

exact tim ing of delivery.

FRAs

W hen a banks custom er w ants to arrange today to agree an interest rate that they can use

(for borrow ing or lending) at som e future date, the custom er can enter into a forw ard rate

agreem ent (FRA ) w ith the bank. The agreem ent w ould allow the custom er to know today

that they could, for exam ple, borrow U SD 10,000,000 at a rate of 5.50% for a 6 m onth

period beginning 3 m onths from today. This w ould be referred to in the m arket as a 3x9

FRA , since the period of the loan begins in 3 m onths and ends in 9 m onths. The custom eris said to have boughtthe FRA for reasons that w ill becom e clear later. The bank in this

Section 3Forward Rates and Eurodollar Futures


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37

rates are available in dollars only), w hile the FRA s can be on 3 m onth, 6 m onth or 1 year

rates and even on periods of tailor-m ade length. A lso the ED futures contracts have only 4

dates during the year on w hich they expire. The dates are in the m onths of M arch, June,

Septem ber and D ecem ber. These dates are m eant to correspond to the beginning of the

loan period that is being hedged. So if a custom er has a loan that begins on a date that is

not the expiry date of a futures contract, the custom er either has to use an FRA or accept

the basis riskof using a contract that does not exactly m atch his exposure. Futures do

enjoy the advantage of greater liquidity, especially in the nearer m aturity contracts and there

is virtually no credit risk associated w ith these futures contracts.

Marking To Market

A fundam ental feature of futures contracts, not just ED futures, is a process know n as

m arking-to-m arket. This is a prudent m easure on the part of the exchange to reduce,

nearly to 0, the credit risk that the exchange takes on from its custom ers. It w orks in the

follow ing w ay. W hen tw o people trade a contract on the futures exchange, the exchange

agrees to guarantee perform ance. C onsequently, each day the exchange forces anyone

w ho has sustained a loss on the futures contract to realise the loss by m aking a paym ent

to the exchange. The exchange then passes this paym ent on to those people w ho have the

corresponding gain on the contract. Because of this, there is no need at the end of the

contract for the kind of settling up that occurs w ith FRA s. O n the last day the exchange

just passes along the final days m ark to m arket. A t that point everyone w ith a position has

realised through the daily accum ulation of the m arks to the m arket, the full value of their

gain or loss on the contract.

In order to illustrate this w e have to take note of a peculiarity of the ED futures contract.

The price of the contract is quoted as (100 interest rate). So w hen the future is priced at

94.50, the underlying three m onth interest rate is 5.50% . Suppose then that a C orporate

Treasurer sells 10 June Eurodollar futures contracts at a price of 94.50. The corresponding

rate is 5.50% and the notional underlying of each of the 10 contracts is $1,000,000. The

Treasurer has done this because she needs to borrow m oney for 3 m onths at the tim e that

the futures contract expires in June. She hopes to lock in a rate of 5.50% on that loan.

Suppose that on the day the Treasurer sells the contract, the U S Federal Reserve raises short

term interest rates and the futures m arket responds by selling the June ED future dow n to

94.00. The corresponding rate is 6.00% and reflects the m arkets expectation, based on theFed action, of w hat the 3 m onth rate w ill be in June. The Treasurer has a gain of 50 basis

points on the contract because she had a short position. The price decrease is a profit to

her. The next day the exchange w ill credit her m arginaccount w ith the profit she m ade

on the 50 basis point m ark to m arket.

Margin

The m argin account is a deposit w ith the exchange of a sum of m oney, or m arketable

securities, that assures the exchange that the custom er w ill m ake good on any losses from

the m ark to m arket. You can contrast this to w hat happens if you buy shares of equity.If you pay 100 for the shares and the price decreases so that your shares are only w orth


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38

90, your loss of 10 is real but not realised unless you sell the shares. This is an im portant

distinction not just psychologically but also possibly from a tax point of view , since it applies

equally w ell to profits. In the futures m arkets all gains and losses are realised w hen they

occur. The am ount of m argin the exchange requires depends on the size of the contract

and its volatility. The size of the contract determ ines the m onetary value of a price m ove

and the volatility determ ines the likely am ount the price w ill m ove. The exchange reserves

the right to increase or decrease the am ount it requires for m argin.

FRA and Future Settlement

Settlem ent of futures is actually easier to describe. Because the underlying notional is

$1,000,000 and the interest rate is a 3 m onth rate, the exchange has specified that each

basis point change in the ED futures contract is w orth $25. This can be rationalised as the

value of 1 basis point on $1,000,000 for one quarter of a year. N ote that one basis point

is 0.01% , w hich is represented as a decim al by 0.0001.

Basis Point = 0.0001 x 1000000 x 1/4 = 25

Example

The Treasurer sold 10 futures at a price of 94.50 and the price subsequently closed for the

day at 94.00. So the days m ark-to-m arket is:

10 contracts x 50 basis points x $25 per basis point = $12,500

The Treasurer w ould receive this into her m argin account the next day. The paym ent does

not necessarily com e from the party that bought the contract from the Treasurer. That

person m ay w ell have closed out the position by selling the contract later in the day. Instead

the exchange collects the losses from everyone w ho is long the contract at the end of thetrading day and passes out the required am ounts to the people w ho w ere short. If for any

reason som eone w ho w as long the contract, and therefore had a loss, did not have enough

m oney in their m argin account to pay their loss, the exchange w ould issue an im m ediate

call for the loser to pay up and bring their m argin account back to its full initial level. Failure

to do so w ould result in the exchange closing out the contract at the current price and

initiating legal action to recover the loss from the defaulting parties clearing firm . The

clearing firm is an organisation that guarantees the trades of its m em bers. This structure

protects the exchange from losses.

ExerciseIndicate the dai ly mark to m arket for the fol lowing t radeNote: EDM 1 is the exchange symbol for the Euro Dollar futures contract which expires inJune (M ) o f 200(1 )

Customer buys 20 EDM 1 contracts at a t rade price of 94 .00 on day 0

Day Set t lem ent Price M ark to M arke t0 94 .101 94 .152 94 .053 93 .904 94 .00

Suppose the customer closes out the posi t ion by sel l ing the 2 0 contracts on d ay 4 at aprice of 94.00 . Wh at is the customers accumulated m ark-to-m arket?


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39

FRA Settlement

FRA s are settled differently from futures because they do not have a m ark to m arket.

Therefore gains or losses accum ulate over the length of the contract and have to be paid

out w hen the contract is settled. So now suppose that our Treasurer, instead of selling ED

futures, had decided to use FRA s as a hedge. Because the Treasurer is exposed to rates

going up, she needs to have a position that w ill have a profit w hen this happens. Thecorrect action then w ould have been to buy the 3x6 m onth FRA that expired at the tim e

that the ED future expired. The underlying notional w ould be $10,000,000. O n the trade

date, suppose the Treasurer is able to buy the FRA at a rate of 5.50% . Suppose further that

on the day the FRA expires, the 3 m onth rate is 6.50% , 100 basis points higher than w hen

the Treasurer bought it. Then the Treasurer has a profit of 100 basis points on $10,000,000

for a 3 m onth period. The value of a basis point is not so sim ple as for futures. First of all,

FRA s use an actual day count i.e. the actual num ber of days in the 3 m onth period covered

by the FRA . Let us suppose for sim plicity that this is 90 days exactly. Even then there is still

a com plication. A lthough each basis point is w orth $25 just like w ith the future, FRA

settlem ent is discounted by the current interest rate. So the actual am ount the Treasurer

w ould receive is:

FRA paym ent =100 basis points x 10 contracts x $25 per basis point

= $24,600.251 + 0.065/4

O nce again w e can rationalise the result. The Treasurer w ill have to pay the higher rate of

6.50% to borrow the $10,000,000. But the additional interest w ill not have to be paid out

until the loan m atures in 3 m onths tim e. Since she receives the FRA paym ent now rather

than later, she can either reduce the am ount she borrow s or she can place the paym ent in

an account that bears interest at the 6.50% rate. Either w ay she has been correctly

com pensated for the increase in the interest rate. Futures contracts do not use this m ethod

because of the daily m ark to the m arket and because the need for sim plicity, i.e. $25 per

basis point, is param ount on the exchange.

Forward Rates

Because of the actions of the players in the interest rate m arkets w hich includes hedgers

such as com m ercial banks and corporate treasurers as w ell as m oney m arket fund

m anagers and speculators such as hedge funds and investm ent banks the FRA s and

Exercise

In each case indicate the amount of payment due on the FRA and w ho pays. Assume that 3 mo nthFRAs cover exactly 90 days and 6 mon th FRAs cover exactly 18 0 d ays. The fi rst example is do ne foryou. Note that a bas is point on a 3 m onth rate is wo r th $25 per mi l l ion, but a bas is point on a 6month rate is wor th $50.

Typ e o f FRA Pr ice Tr ad ed Cu st om er n ot io na l Ra t e a t Pa ym en t Fro m(m illions) Set t lem ent

2 x5 6 .0 0 % Sells 1 0 6.25 % 6 1 53 .8 5 Cust

3 x6 6 .1 0 % Sells 20 6.1 5%

2 x8 8 .0 0 % Buys 30 7.7 5%

3 x9 7 .5 0 % Buys 25 7.8 0%


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40

futures represent the consensus of the m arket as to w hat the future interest rates w ill be.

So if the M arch 2002 ED futures contract trades at a price of 93.25, it is because the

m arket view is that the 3 m onth interest rate in M arch 2002 w ill be 6.75% . These contracts

are liquid and easy to trade. Their prices quickly adapt to any changes in m arket variables

and m arket sentim ent. So people concerned w ith the future of interest rates are constantly

w atching the forw ard curvew hich consists of the rates im plied by the various ED futures

contracts and the FRA s.

Euro interest rate futures contracts trade on the interest rates of m any different countries.

They are am ong the m ost successful financial futures contracts ever devised. Listed below

are som e of these contracts and the underlying notional am ount for each. A ll are contracts

on 3 m onth interest rates, except as noted, and are settled against the 3 m onth LIBO R rate

prevailing at the tim e the contract expires.

Currency N ot iona l Contract s

USD USD1m 1 0 years o f 3 m onth fu turesUSD3m 2 m onths o f 1 m onth fu tures

EUR EUR1m 5 years i.e. 20 cont ractsGBP GBP500 k 1 8 m ont hsJPY JPY10 0m 1 8 m ont hsM XN * M XN2m 1 2 m ont hs

* U nderlying is a 91 day Treasury Bill

There are futures on a num ber of other currencies also.

Eurodollar Futures

Cont ract Eurodollar Future

Traded: IM M , LIFFE, SIM EX

Underlying : $1 ,00 0 ,0 00 9 0 day t im e deposit

Delivery M on th s: M a rch (H), Ju ne(M ), Sep tem ber(U ), Decem ber(Z)

Set t lem ent : Cash, d eterm ined by 3 -m onth LIBOR on exp iry

Quotat ion: 10 0 -In terest Rate

M i n im um Pr ice Change: 1 basi s po i nt , val ue = $25 , ref er red t o as a t i ck


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42

Bond Futures

The futures contract that trades on the C hicago Board of Trade (C BO T) is another

successful financial futures contract. H aving said that it is also a som ew hat com plicated

contract. The com plication arises because the contract is w ritten on a nom inalbond. The

reason for this is that there are a great m any U S Treasury bonds but they differ w ith respect

to their coupon rate and their tim e to m aturity. The C BO T could have m ade a contract thatw as either cash-settled or w as m arked to a bond index but it did not do so. Rather, a

person w ho is short the contract during the delivery m onth is given the choice of a num ber

of different bonds to deliver to fulfil the contract. The person w ho is short the contract can

also decide on w hich day of the delivery m onth to m ake delivery. A s w ith all futures

contracts, anyone w ho is long or short the contract can trade out of it by reversing their

po

ubs interest rates

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