fnce 30001 week 6 fixed income fundamentals(1)

7/14/2019 FNCE 30001 Week 6 Fixed Income Fundamentals(1)

http://slidepdf.com/reader/full/fnce-30001-week-6-fixed-income-fundamentals1 1/70

Semester 2, 2011

Week 6: Fixed Income FundamentalsProfessor Rob Brown

FNCE 30001 Investments: 6.0



Overview of Lecture1. Fixed Income Securities: What and who?

2. Bond Types

3. Pricin Zero-cou on Bonds

4. Pricing Money Market Securities

6. Using the Zero Rate Curve to Price Securities

.

Readings : Bodie et al , Chapter 14.




1. Fixed Income Securities: What and who?




What is a Fixed Income Security?

Four characteristics of a fixed income security (note, bill,bond, debenture) are:

1. the issuer (debtor, borrower) promises to repay the

nvestor en er, on o er2. the amount borrowed (principal or price)

3. p us nterest

4. at a specific point or points in time.




Comparing some US and Australian Terminology

US AUSTRALIA

Treasur bills 28, 91, 182 da s Treasur notes u to one ear

Treasury notes (up to 10 years) Government bonds

overnment on s years overnment on s

Commercial paper (up to 270 days) Bills of exchange (30, 90, 180 days)Corporate bonds, Debentures Corporate bonds, Debentures




o s nvo ve

• Issuers : Commonwealth government, state governments,, .

– Note: “semi-government” bonds are issued by theassociated state overnment.

• Investors : Financial intermediaries, investment funds,superannuation funds, corporations, individuals.

• Others:

– Credit Rating Agencies: S&P, Moody’s, Fitch. – Regulators: ASIC, APRA, ATO, RBA

– Industry bodies: AFMA, ABA




600.0

Debt Securities on Issue in Australia (A$b)

400.0

500.0

300.0 Short-term non-govt

Long-term non-govt

Short-term govt

100.0

.Long-term govt

0.0

S e p - 1 9 9 2

A p r - 1 9 9 3

N o v - 1 9 9 3

J u n - 1 9 9 4

J a n - 1 9 9 5

A u g - 1 9 9 5

M a r - 1 9 9 6

O c t - 1 9 9 6

M a y - 1 9 9 7

D e c - 1 9 9 7

J u l - 1 9 9 8

F e b - 1 9 9 9

S e p - 1 9 9 9

A p r - 2 0 0 0

N o v - 2 0 0 0

J u n - 2 0 0 1

J a n - 2 0 0 2

A u g - 2 0 0 2

M a r - 2 0 0 3

O c t - 2 0 0 3

M a y - 2 0 0 4

D e c - 2 0 0 4

J u l - 2 0 0 5

F e b - 2 0 0 6

S e p - 2 0 0 6

A p r - 2 0 0 7

N o v - 2 0 0 7

J u n - 2 0 0 8

J a n - 2 0 0 9

A u g - 2 0 0 9

M a r - 2 0 1 0

O c t - 2 0 1 0




2. Bond Types




1. Fixed-coupon Bonds

• The “fixed-coupon bond” is the classic bond type.• A fixed-coupon bond makes two kinds of payments:

– Par value (face value): The payment the bond holder

receives when the bond matures. – Interest (coupon payment): Additional pre-specified

payments ma e e ore an on t e matur ty ate at pre-specified intervals ( eg yearly, half-yearly, quarterly).

sua y expresse as a s mp e annua ra e.

• Why the term “coupon payment”?




Exam le: Commonwealth Gov’t bond 6.50% Ma 2013

Par value is taken to be $100.

Coupon interest is paid twice per annum on 15 May and 15 November each year.Each half-yearly coupon is ½ x 6.50% x $100 = $3.25

If you bought this bond on, say, 31 August 2011, you would get these cash flows:

.

On 15 May 2012: $3.25

On 15 November 2012: $3.25

On 15 May 2013: $3.25

Also on 15 May 2013: $100.00




Exam le: Commonwealth Gov’t bond 6.50% Ma 2013

(contd.)

Of course, in practice you can’t buy as little as a $100 bond.Suppose you bought bonds with a par value of $10 million.

If you bought this bond on 31 August 2011, you would get these

cash flows:On 15 November 2011: $325,000

On 15 May 2012: $325,000

On 15 November 2012: $325,000On 15 May 2013: $325,000

Also on 15 May 2013: $10,000,000




2. Ca ital Indexed Bonds

• A capital indexed bond is the same as a fixed-coupon bond

exce t that the ar value and cou on a ments are stated inreal ( ie after-inflation) terms.

• That is, a CPI adjustment is made at each coupon date so that

the investor earns the stated real interest rate.• Example : The Australian government has issued capital indexed

bonds.




3. Floatin Rate Bonds Notes

• Like the fixed-coupon bond, a floating rate bond pays regular

cou ons and the ar value at maturit .

• But the coupon rate is not fixed.

• T icall each cou on a ment is linked to a short-term

interest rate current at the beginning of the coupon period.• Therefore:

– If interest rates rise during the life of the bond, the coupon

payments also rise. – But if interest rates fall during the life of the bond then the

coupon payments also fall.




4. Convertible Bonds Notes

• Like a standard bond ( eg a fixed-coupon bond) plus the

investor has an option to convert to shares at maturity.• For example, a convertible note may have a par value of $100

and at maturity the investor can choose to get:

– e n cas or

– 20 shares in the borrower

• ,maturity date is less than $5 per share.

• A convertible bond must be worth more than an otherwiseequivalent straight bond.

• This may show up as a lower coupon interest rate.




5. Callable Bonds

• Like a standard bond ( eg a fixed-coupon bond) plus the

borrower has an o tion to re a the bonds earl .

• This option may not apply until ( eg ) the last 2 years of thebond’s life.

• The borrower may choose to repay early if interest rates havefallen since the money was borrowed.

• A callable bond must be worth less than an otherwise

equivalent straight bond.• This may show up as a higher coupon interest rate.




6. Domestic vs International Bonds

• Domestic – eg a US company issues a USD bond in the US.

• International

– Foreign bond eg an Australian company issues a USD bond in the US

(known as a “Yankee bond”)

– Eurobond eg a company issues a USD bond in the UK.

• These bonds may be coupon-paying, convertible, callable etc .




7. ero-coupon on s nstruments

a.k.a. Pure Discount Securities• Only one cash flow (the par value), which occurs on the

maturity date.

xam es : any s or - erm e secur es suc as:

– Treasury Notes (Australia)

–

– commercial bills.

• - - but in practice there are few.

• But note: the zero coupon bond is an important building


block for theory and practice.



3. Pricing Zero-coupon Bonds




-

n eterm n ng t e pr ce o a zero-coupon on a zero orshort), the market takes into account:

e ea ures o e on :

o Time to maturity (–)

o e e au r s o e orrower –

• Tax (–)

• qu ty n t e secon ary mar et

• Expected inflation (–)




-

• Time value of money : $1 to be received in the future has a lower

value than $1 to be received today.

• ere are many ways to represent t e t me va ue o money:

– Prices of zero-coupon bonds.

– ero-coupon rates.

– Yields-to-maturity.

– Forward rates.

– Discount factors.




-

Notation

• Par : Par value or face value or principal.• P 0 : Current (time 0) bond price.

• z 0T : Zero-coupon interest rate (pa) from time 0 to time T .

• d 0T : Discount factor from time 0 to time T .• HPR 0T : Holding period rate of return (pa) from time 0 to

time T .




-

A: Zero rates where there is no default risk

Consider a default-free zero that pays $Par T years from now me .

• Its price is:

001 T T

P z

where z 0T is the interest rate that applies per year from time 0 totime T .

“ ”0T -

– Note that zero rates are on a compound interest basis,re ardless of their term and are uoted er annum.




-

For a given par value, if we know z 0T we can calculate P 0.

ExampleCalculate the price of a zero with 10 year maturity and par value$100. The current 10-year zero coupon rate is 7.5% pa; that is,

0,10 . . .

Answer

0

01T

T

P z

101.075


.



-

If we know P 0 we can calculate z 0T . Rearranging the equation:

1 T

Par

Example

00

T P

A 5-year zero with a par value of $1,000,000 is sold for $650,000. What is the implied 5-year zero rate?

Answer

1

0

0

1

T

T

Par z

P

1 5

0,5

$1,000,0001

$650,000z


8.9977% pa.



-

Comparative Statics

• Recall that the pricing formula is:

0

01T

T

Par P

z

• The price ( P 0 ) is related to: – The par value ( Par ): positively

– Today’s zero-coupon rate ( z 0T ): negatively

– The term to maturity ( T ): negatively

, 0 .

This is not necessarily true of coupon-paying bonds.




-

• P 0 is negatively related to z 0T

Examples : from previous example:if z 0,10 = 10% pa, then P 0 = $38.55

if z 0 10 = 8% pa, then P 0 = $46.32

• P 0 is negatively related to T

if T = 1, then P 0 = $93.02

= 1 , t en 0 = 33.80




-• Less obviously, when the term to maturity ( T ) is larger, the bond price is more

sensitive to chan es in the interest rate:

Term

T

Price if

=7.5 a

Price if Change in

rice

Change in

rice0T

100

1 year – $0.4307 – 0.463%

0 1.075

$93.0233

P

0 1.08

$92.5926

P

$100 $100 – . – .

0 101.075

0 101.08


. .



-

B: Zero rates where there is default risk

• When there is default risk we need to distinguish between the promised interest rate and the expected interest rate.

• Consider a T -year zero with a face value of Par.

– The probability of default is b . – If default occurs, the probability of recovering some of the

amount owe s r .

– The proportion recovered is π and will be received at time

.




-

ar

π x Par

1 – b

r

Defaultb

Zero1 – r




-

0

Promised Cash Flow T P

0T

T

Par

0

1

1

T

T Par

Of course this is the same formula as in the default-free case.

0




-

- *

0

Expected Cash Flow T P

01

1

T z

b Par b r π Par

01

1 1

T z

Par b r π

*0

1

*

1

1 1

T

T

z

Par b r π

00

*0 0Of course, if 0 then expected promised .

T

T T

z P

b z z




-

Comparative Statics

The expected 1-year zero rate is related to: *

0T z – The probability of default ( b ): negatively

– The probability of recovery ( r ): positively

– The proportion expected to be recovered ( π

: positively




-

Suppose Par = $1,000,000; P 0 = $740,000; T = 3; b = 0.005;

. . .

Then the promised zero rate is:

0

0

1T Par z P

1 3$1,000,000

1

$740,000 10.558% pa.




-

.

The expected zero rate is:1 T

*0

0

1T

ar r π

z P

$1,000,000 1 0.005 1 0.6 0.4 1$740,000

1 3$996,200

1$740,000

10.418% pa.




4. Pricing Money Market Securities




A: Australian Money Market Securities

• In Australia, these securities are priced using simple interest.

• Suppose the interest rate for a commercial bill for the next90 ays s 4 per annum. e par va ue s 100,000.

– What is the price?

• The interest rate for the 90-day period is calculated as:

90

0.04 0.0098630137

• So the price is:

100 000

01.0098630137

$99,023.33

P




Putting this into a single formula, money market

securities in Australia are priced using:

0 ,

1365

ar P

n s

w ere n s t e num er o ays unt matur ty.

s is the simple annual interest rate (yield)

Rearranging the equation, if we are given the price P 0,the implied yield s is:

0

1ar

s P n




B: US Money Market Securities

• US money market dealers trade in terms of “bank discountrates”, which are defined this way:

n

0

360 where means the rate quoted.q

• Note there are two differences here:

e pr ce s expresse as a su rac on rom no aproportion of) the par value.


- - .



Example

,term of 90 days, when the par value is $100,000 has agreed to

a :

0 1

360n P Par q

90$100, 000 1 0.04

360

, .

$99,000




Example (contd.)

calculated the same way as it is in Australia.

, ,

365

1Par

s

0

$100, 000 3651

,

4.0965% pa




5. The Zero Rate Curve




• Definition of the zero rate curve:

Plot of zero coupon rates (vertical axis) against term to maturity

horizontal axis . Also called the “term structure of zero coupon rates”.

• Example:

Suppose on 31 August 2010 you saw the following current interest rates

reported in the Australian financial press:

- .

90-day bank bill yield: 6.347% pa

180-day bank bill yield: 6.687% pa1-year zero-coupon bond: 7.000% pa

2-year zero-coupon bond: 7.100% pa


ot t e zero curve.



• Stop and THINK!

• Zero rates are by definition compound rates.

• The bond rates will be quoted on a compound interest basis

– but the bill rates will be quoted on a simple interest basis.

• So, first we need to convert the bill rates to their compoundinterest equivalents.




• To do this, we equate the bill price calculated using the simple

interest formula with the bill price calculated using the

.

• That is:Par Par

60 365

0, 60 36560 11 0.0595365

z

which solves to give z 0T = 6.100% pa.




• To show this works, suppose you had to price a bill with a par value of

100 000.

• Using the simple interest rate of 5.95% pa, we get:

$100,000

601 0.0595

365

$100,000

• Usin the com ound interest rate of 6.100% a we et:

1.009780822

$99,031.39

0 60/365

$100,000

1.06100P

$100,000

1.009780981

$99,031.38




• Using the same approach we find:

= .

the zero rate for 180 days = 6.800% pa

• The full set of zero rates is therefore:60 days: 6.100% pa

90 days: 6.500% pa

180 days: 6.800% pa

1 year: 7.000% pa

.




7.0

.

Rate

%pa

6.8

6.5

6.1

2 years1 year 180

days

90

days60




• The previous example produced an upward-sloping zero

curve.

• t oug o ten seen, not a zero curves s ope upwar s.

• For example, zero curves can be (and have been) downward

.

• We will study the term structure of zero rates in more detail.




6. Using the Zero Rate Curve to




• Once we have today’s zero rate curve it is very easy to

calculate the present value (price) of any single future cash

– or any set of future cash flows.

Consider the following (rather odd) security.

prom ses o pay e owner:

$300 after 180 days and

a ter year an

$650 after 2 years.




Example (contd.)

The current zero rate curve is:

60 days: 6.100% pa

90 days: 6.500% pa

180 days: 6.800% pa

1 year: 7.000% pa

2 years: 7.100% pa

How much is this security worth today?




Answer to Example

Because we have the zero rate curve, we can value thissecur ty very eas y:

0 180 365 2

$300 $400 $650P

.. .

$290 .4232 $373 .8318 $566 .6755

1230.93

This procedure is called “pricing off the zero curve”.




Valuing a Fixed-Coupon Bond

• A more likely use of the zero rate curve is to value a fixed-coupon on .

• A fixed-coupon bond is like a portfolio of zeros.

• we now t e pr ces an ence t e zero rates o t e

constituent zeros, then pricing a coupon bond is simple:

0 2 3 101 02 03 00, 1

...1 1 1 11

T T

T T

ar P

z z z z z

• Practical problem: few long-term zeros exist.

where is the coupon amount.C


– o ut ons to t s pro em are covere n t e next ecture.



Exam le

Suppose we can observe the current prices of 1-year, 2-year and

3- ear zeros with notional ar values of $100 each:1-year zero: $94.117647

2- ear zero: 87.794573

3-year zero: $81.916543Use this information to rice a 3- ear cou on bond with the following features:

Par value: $10,000,000Coupon rate: 8% pa

Cou on fre uenc : 1 er ear




Answer to Example

Recall that:

Therefore

00

1T

Par z

P

1/1

01$100 1 6.250% pa

$94.117647z

1/2

02

$1001 6.725% pa

$87.794573z

1/3

03

$1001 6.875% pa

$81.916543z




Answer to Example (contd.)

The coupon, paid annually, is 8% × $10,000,000 = $800,000

ence, e on pr ce s:

0

$800, 000 $800, 000 $10,800, 000P

. . .

$752,941.18 $702, 356.59 $8,846,986.66

This is an example where a coupon bond (unlike a zero) is worth more than its

, , .

. When this happens, the bond is said to be trading “at a premium”.

If a bond’s price is less than its par value, the bond is trading “at a discount”.




How o we now t e on ’s price wi e 10,302,284.43?

Answer: Because, if it is anything else, then there is an arbitrage opportunity.

,us for $10,350,000.

Here’s what we could do:

1. Sell the bond for $10,350,000 and then

2. Buy zeros as follows:

- ,

A 2-year zero with a par value of $800,000 and

A 3- ear zero with a ar value of $10,800,000.3. Laugh




Clearly, this set of investments replicates those that we would get from.

The cost of the strategy is:

$800,000

2

- , .1.0625

$800,000Cost of the 2-year zero = $702,356.59

3

.

$10,800,000Cost of the 3-year zero = $8,846,986.661.06875

These numbers should be familiar to you!

The total cost of the strategy is, of course, $10,302,284.43.

ence, we ave cas e t over o , , – , , . = , . . Another way to arbitrage is to invest the whole of the $10.350m in zeros such

that we get a higher cash flow on every future coupon date.




7. Measuring the Return on Zeros




Yield-to-maturity

• Yield-to-maturity (or simply “yield”) is defined as the single

interest rate that equates the price of the bond to the present value of the future cash flows the bond will generate.

• or zeros, y e s t e same as t e zero rate.

• For coupon-paying bonds, yield is usually close to, but not

, .

• The difference between yield and zero rate will be clearer

.




Discount Factors

• The price of a zero with a par value of $1 and a term of T is

called the zero’s discount factor, denoted d 0T .

0

1T T

d

• If we know z 0T we can calculate d 0T , and vice-versa .

0T




Example 1

Calculate the discount factor for z 01 = 5%.

What is the price if Par is $100? Answers:

1

0

011

T T

T z

1.05

0.9524

0 0.9524 $100 $95.24P




Example 2

Suppose today’s zero rates are:

.

for 2 years: 9.5% pa

for 3 years: 9.8% pa

What are today’s discount factors?

Answers:

1for 1 ear: 0.921659d

02 2

1.085

1for 2 years: 0.834011d

03 3

.1

for 3 years: 0.7554281.098

d




The Holding Period Rate of Return (HPR)

• As with any other investment ( eg shares, property, antiquesilverware ,…) if a bond is bought for price P 0 and sold X yearslater for price P X , and there are no other cash flows involved,

time X is simply P X – P 0.•

0 . X P P

0




• orma y we wou annua se t s rate o return.

• The annualised holding period rate of return is:

1

00

0

1 1

X

X X

P P HPR

P

1/

01

X

X P P

• Because (usually) P X is unknown at time 0, (and anyway X itself may be unknown) the holding period rate of return is also notknown at time 0.

• Hence, the investment is risky


– even t ere s no cre t e au t r s .



• ow cons er t e spec a ut very mportant case w ere adefault-free zero is bought for P 0 at time 0 and is held until

maturit at time T .• The rate of return achieved is:

1/T

P

0

0

1/

1T

T

HPR

P

0

1Par

P

• Because Par, P 0 and T are known at time 0, this is a risk-free return.




• n we can say w at t s certa n rate o return w e.

• Recall that:1/T

’

00

1T

ar z

P

, 0T .

• That is, z 0T = HPR 0T .•

– We are certain to achieve an annualised holding period rateof return ( HPR 0T ) that is equal to the current zero rate ( z 0T ).


fnce 30001 week 6 fixed income fundamentals(1)

Documents