fnce 30001 week 8 coupon bonds

FNCE 30001 Investments 8.0

FNCE 30001 InvestmentsSemester 2, 2011

14 & 16 September 2011Week 8: Coupon Bonds

Professor Rob Brown


Week 8: Coupon Bonds

Overview of Lecture1. Review: Zeros and Fixed-coupon Bonds2. Yield-to-maturity (ytm)3. Bond Pricing in Practice (A): Australian Conventions4. Bond Pricing in Practice (B): US Conventions5. The Yield Curve6. The Par Yield Curve7. Finding the Zero Rate Curve: Bootstrapping8. The Holding Period Return


Week 8: Coupon Bonds

Readings

Bodie et al, Chapters 14 and 15.Reserve Bank of Australia, Pricing Formulae for Commonwealth

Government Securities, available from the RBA website at: www.rba.gov.au/mkt-operations/tech-notes/pricing_formulae.html.


1. Review: Zeros and Fixed-coupon Bonds


Review: Zeros and Fixed-coupon Bonds

Notation from last lecture Par: Par value or face value or principal. P0 : Current (time 0) price z0T : Zero rate from time 0 to time T. d0T : Discount factor from time 0 to time T. HPR0T : Holding period rate of return from time 0 to time T.



Zero-coupon bonds (zeros) A single future cash flow (par, or face value). Pricing formula:

The price (P0) is related to: The par value (Par): positively The zero rate (z 0T): negatively The term to maturity (T): negatively

For a zero, P0 is always less than Par. This is not necessarily true of fixed-coupon bonds.

0 001 TTTParP d Parz



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 at the end of the bonds life ie when the bond matures.

Interest (coupon payment): Additional pre-specified payments made before (and on) the maturity date at pre-specified intervals (eg yearly, half-yearly, quarterly). Usually expressed as an annual simple interest rate (%).


Example: Commonwealth Govt bond 6.50% May 2013Par 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 x 6.50% x $100 = $3.25If you bought this bond on, say, 31 August 2011, you would get these cash flows:

On 15 November 2011: $3.25On 15 May 2012: $3.25On 15 November 2012: $3.25On 15 May 2013: $3.25Also on 15 May 2013: $100.00



Example: Commonwealth Govt bond 6.50% May 2013 (contd.)

Of course, in practice you cant 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: $325,000On 15 May 2012: $325,000On 15 November 2012: $325,000On 15 May 2013: $325,000Also on 15 May 2013: $10,000,000



A fixed-coupon bond is like a portfolio of constituent zeros. If we know the prices (and hence the zero rates) of the

constituent zeros, then pricing a coupon bond is simple:

This is called pricing off the zero curve. Practical problem: few zeros exist

Solutions to this problem are covered later in the lecture.

0 2 3 101 02 03 00, 1

...1 1 1 11

T TTT

C C C C C ParPz z z zz




Notation new this lecture

c : Coupon rate (%) per period C: Coupon amount ($) per period = c Par ytm :Yield-to-maturity (or just yield for short). rr : Reinvestment rate


2. Yield-to-Maturity (ytm)


Yield-to-Maturity (ytm) The yield-to-maturity (or just yield for short) is the single

discount rate that equates the bond price to the present value of all future cash flows of the bond. That is, the yield is the bonds internal rate of return.

The yield is a handy way to summarise a bond trade in a way that everyone can immediately relate to.

In practice many bond dealers think in terms of yields rather than the constituent zero rates.


Yield-to-Maturity (ytm) When a bond is traded in the market, it is rare for the dollar

price of a trade to be reported. For example, The Australian Financial Review reports on the

bond market but it doesnt provide the bond price it provides the yield.

The next slide shows the yields for bond trading on 8 June 2011, as reported in The Australian Financial Review on 9 June 2011.


Bond series Yield (% pa) Bond series Yield (% pa)

5.75% Jun 2011 4.685 4.75% Jun 2016 5.040

5.75% Apr 2012 4.825 6.00% Feb 2017 5.115

4.75% Nov 2012 4.845 5.50% Jan 2018 5.175

6.50% May 2013 4.855 5.25% Mar 2019 5.215

5.50% Dec 2013 4.870 4.50% Apr 2020 5.230

6.25% Jun 2014 4.895 5.75% May 2021 5.255

4.50% Oct 2014 4.925 5.75% Jul 2022 5.290

6.25% Apr 2015 4.950 5.50% Apr 2023 5.360

Yield-to-Maturity (ytm)



Consider a bond that pays annual coupons.Consider its price on a coupon date.By definition, ytm is:

For a fixed-coupon bond, C is constant.Using the formula for the present value of an ordinary annuity,

this formula can be simplified to:

011

1 1T TC ParPytm ytm ytm

0 2 ...1 1 1 T

C C C ParPytm ytm ytm



ExampleConsider a 3-year bond with these features:

Par value: $10,000,000Coupon rate: 8% paCoupon frequency: 1 per year

The current zero rates are: 6.250% pa (1 year); 6.725% pa (2 years) and 6.875% (3 years).

What is the market price of the bond?What is its yield-to-maturity?



Answer to ExampleThe market price of the bond is:

0 2 3$800, 000 $800, 000 $10, 800, 000

1.0625 1.06725 1.06875$752, 941.18 $702, 356.59 $8, 846, 986.66$10, 302, 284.43

P



Answer to Example (contd.)To find the yield-to-maturity we need to know what

value of ytm will solve this equation:

There is no analytic solution use numerical methods.In practice, use (eg) Excels IRR function.We find that ytm is approximately 6.85127% pa.

2 3$800, 000 $800, 000 $10,800, 000 $10, 302, 284.431 1 1ytm ytm ytm



Answer to Example (contd.)To show this is right, we calculate the price:

0 2 3

2 3

1 1 1$800, 000 $800, 000 $10,800, 0001.0685127 1.0685127 1.0685127$748,704.25 $700,697.57 $8,852,882.30$10, 302, 284.12

(rounding error of 31 cents)

C C C ParPytm ytm ytm



We can think of the yield as a complex kind of average of the interest rates of the constituent zeros:

BUT the yield is also a function of the coupon rate.

DateCash

Flow ($)PV (zero) PV (ytm)

PV(ytm) lessPV(zero)

1 $800,000 $752,941.18 $748,704.25 $4236.932 $800,000 $702,356.59 $700,697.57 $1659.023 $10,800,000 $8,846,986.66 $8,852,882.30 +$5895.64

SUM $10,302,284.43 $10,302,284.12$0.31

(rounding)



Bonds that pay different coupons, but are otherwise equivalent, have different prices and hence must have different yields.

ExampleWe will redo the example but with a coupon rate of 5% instead of 8%.Recall: Par = $10,000,000; T = 3 years.

1-year zero rate = 6.250% pa2-year zero rate = 6.725% pa3-year zero rate = 6.875% pa



Calculating the price:

Solving for the yield gives ytm = 6.85932% pa

0 2 3$500, 000 $500, 000 $10,500, 000

1.0625 1.06725 1.06875$470,588.24 $438,972.87 $8,601, 237.03$9,510,798.14

P



Summarising, for these 3-year bonds:8% coupon yield = 6.85127% pa5% coupon yield = 6.85932% pa

Difference = 0.00805% pa (ie 0.8 of a basis point) So this must be trivial, right? Who would worry about 0.8

of a basis point? A bond dealer might.

Remember, in the bond markets deals may often be, say, $200m at a time. If you could make 0.8 of a basis point (pa) on a $200m deal, thats $16,000 (pa).


Yield-to-Maturity (ytm)Comparative statics If ytm increases, then P0 decreases.

If ytm decreases, then P0 increases. A higher C produces a higher P0.

A lower C produces a lower P0. When the price is calculated an instant after a coupon

payment: if ytm < c, then P0 > Par if ytm > c, then P0 < Par if ytm = c, then P0 = Par

Note: These relationships are not necessarily true if the price is calculated at a time between coupon dates.


3. Bond Pricing in Practice (A): Australian Conventions


Bond Pricing in Practice (A): Australian Conventions

The bond pricing formula we have been using is a textbook version.

It is a simplification of reality in three main respects:1. It assumes annual coupons.

But Australian government bonds pay half-yearly coupons

2. It assumes that the first coupon will be paid exactly one year from the pricing date. So it can be used only twice a year for Australian

government bonds.


3. It assumes that bonds are paid for (settled) on the same day that the transaction is done (the trade date). But for most Australian government bonds the

settlement date is a few days after the trade date.

We require a formula that will take account of these three features (and some others too).The formula we will develop is the one used by the Reserve Bank of Australia and the Australian Financial Markets Association.



f

h

0 1 2 n-1 n

h = number of days in the half-year ending on the next coupon paymentf = number of days from the settlement date to the next coupon payment date

-1

$C $C $C $C +Par

Time Line for Coupon Bond with n Coupons Remaining



Viewed from the payment date of the first coupon: this is an annuity-due of n cash flows of $C, plus Par. equivalently, it is an immediate cash flow of $C plus an

ordinary annuity of n 1 cash flows of $C, plus Par. We will use the second of these descriptions.



Using the second description, given a yield of ytm per half-year, the value as at time 1 is:

We now need to discount V1 back to time 0. The length of this period is f / h of a half-year. Therefore:

1 1 111

1 1n nC ParV Cytm ytm ytm

0 1/1

1 f hP V

ytm



Putting both parts together, we get the RBA pricing formula for coupon bonds:

where:ytm is the yield to maturity per half-yearn is the number of half-yearly coupons to be receivedC is the half-yearly couponPar is the par (face) value of the bondf is the number of days from the settlement date to the next coupon

paymenth is the number of days in the half-year ending on the next coupon

payment date

0 / 1 11 11

1 1 1f h n nC ParP Cytmytm ytm ytm



Seven further technicalities:1. In trading and reporting, the yield will be quoted per year (not

per half-year): the quoted yield (pa) will be 2 ytm.

2. The coupon rate (c) is also stated on an annual basis. Therefore:

3. For bonds with more than 6 months to run to maturity, the settlement date is 3 business days after the trade date. Non-business days are Saturdays, Sundays and days that are public holidays in both Sydney and Melbourne. All other days are business days.

4. The price (per $100 par value) is taken to 3 decimal places.

12

C c Par



Seven further technicalities (contd.):

5. There is an ex-interest period beginning a week (7 days) before every coupon date. If a bond is bought during that week (that is, if the trade

date falls in that week), the buyer is not entitled to receive the next coupon.

So, delete the first C from the pricing equation.

6. If the first coupon payment date falls on a non-business day (egon a Saturday or a Sunday), then: f is calculated as the number of days from the settlement

date to the first business day after the next coupon date but h is calculated as the number of days in the half-year

ending on the next coupon date.



Seven further technicalities (contd.):7. When a bond goes ex-interest for the second-last time (ie 6 months plus

7 days before maturity), the cash flows consist of a single payment at maturity, which is equal to the par value plus one half-yearly coupon. To maintain consistency with the procedures for Treasury notes and

other short-term securities, simple interest is used. Also, for these bonds, the settlement date is:

for trades completed before 12.00 noon, the same day; for trades completed after 12.00 noon, the next business day.



Seven further technicalities (contd.)Example of the 7th Technicality

At 2.00 pm on Friday 2 September 2011, price a 6.5% February 2012 bond if the yield is 4.850% pa.

AnswerSettlement date = next business day = Monday 5 September 2011.Maturity date = Wednesday 15 February 2012.Term = 25 + 31 + 30 + 31 + 31 + 15 = 163 days. Coupon = $3.25.

0$103.251631 0.0485365

$101.061(to 3 decimal places)

P



Example of the Main Formula (Slide 8.31)On Thursday 9 June 2011, The Australian Financial Review reported that on Wednesday 8 June 2011 the market yield on the 6.00% February 2017 Commonwealth Government bond was 5.115% pa. If the par value was $10,000,000 what was the bond price?



Answer:The settlement date is Wednesday 8 June 2011 + 3 business days.Monday 13 June 2011 was a public holiday in Sydney and Melbourne.So the settlement date is Tuesday 14 June 2011.The previous coupon date

was 15 February 2011.The next coupon date is Monday 15 August 2011 (which is a business day).

16 31 15 62 days13 31 30 31 30 31 15 181 days1 6.00% $100 $3.00 phy2

1 5.115% pa 2.5575% phy2

12

fh

C

ytm

n coupon payments



0 / 1 1

62/181 11 11

1 111 1 1

1 $3.00 1 $100$3.00 10.0255751.025575 1.025575 1.025575

0.991386960 $3.00 $28.45072348 $75.745758240.991386960 $10

f h n nC ParP Cytmytm ytm ytm

7.1964817$106.273 (to 3 decimal places)

$10, 000, 000$106.273$100

$10,627, 300

P



Most government bond trading in Australia is wholesale over-the-counter (eg interbank) rather than retail. Although the Reserve Bank operates a trading service for

retail clients (face values between $1000 and $250,000). See www.rba.gov.au/fin-services/bond-facility/

The wholesale market for Australian Government bonds is very liquid.

Turnover is often once in every 2 to 6 weeks. The next two slides gives details of trading for one (randomly

selected) day.



Bond Turnover 7 June 2011

Bond series Turnover $mTotal on issue $m

% Turn-

over

Turnover = once every

5.75% Jun 2011 175 11199 1.56% 64 days

5.75% Apr 2012 474 14055 3.37% 30 days

4.75% Nov 2012 1698 8900 19.08% 5 days

6.50% May 2013 1638 16698 9.81% 10 days

5.50% Dec 2013 605 9300 6.51% 15 days

6.25% Jun 2014 561 11899 4.71% 21 days

4.50% Oct 2014 1507 8450 17.83% 6 days

6.25% Apr 2015 223 13348 1.67% 60 days

4.75% Jun 2016 515 9550 5.39% 19 days



Bond Turnover 7 June 2011 (contd.)

Bond series Turnover $mTotal on issue $m

% Turn-

over

Turnover = once every

6.00% Feb 2017 23 13748 0.17% 598 days

5.50% Jan 2018 230 3500 6.57% 15 days

5.25% Mar 2019 215 13248 1.62% 62 days

4.50% Apr 2020 910 14297 6.36% 16 days

5.75% May 2021 759 12050 6.30% 16 days

5.75% Jul 2022 366 7400 4.95% 20 days

5.50% Apr 2023 53 1000 5.30% 19 days

TOTAL 9952 168642 5.90% 17 days

Source: Reserve Bank of Australia, published in The Australian Financial Review, 9 June 2011, p.40.



4. Bond Pricing in Practice (B):US Conventions


Recall that the textbook formula is:

In US terminology, this is called the flat price. Also known as the clean price when quoted per $100 par

value. Taken literally, it is the price of the bond exactly half a year

before the next coupon. Bond prices reported in newspapers are typically clean

prices.

011

1 1T TC ParPytm ytm ytm

Bond Pricing in Practice (B): US Conventions


As time passes since the last coupon payment date, the amount of accrued interest grows.

The amount of accrued interest (I) is measured by:

The invoice price (ie the price paid by the buyer) is equal to the flat price plus the accrued interest. Also known as the dirty price when quoted per $100 par

value.

Days since last coupon paymentDays between coupon payments

I C



For US Treasury bonds: Coupons are paid every 6 months. Pricing is in 32nds of a dollar. Settlement is one business day after the trade is agreed. Accrued interest is calculated using actual day counts (ie not

assuming a 360-day year).Example

Today is 12 September 2011. The flat price is $108:4. The coupon rate is 6.25% pa. The last coupon was paid on 15 August. What is the invoice price?



Answer

The flat price is $108:4 = $108 4/32 = $108.125.The half-yearly coupon = x $6.25 = $3.125.Days from 15 August to 13 September = 29Days from 15 August to 15 February = 184


Invoice price Flat price Accrued interest29$108.125 $3.125

184$108.617527


5. The Yield Curve


The Yield Curve

The yield curve plots yields-to-maturity (vertical axis) for coupon bonds against their terms to maturity (horizontal axis).

It is frequently referred to by practitioners, in the financial press, in academic research etc.

The yield curve is typically close to, but is rarely coincident with, the zero rate curve.


4.6004.7004.8004.9005.0005.1005.2005.3005.400

0 1000 2000 3000 4000 5000

%

p

a

Days to maturity

Australian Yield Curve 8 June 2011

The Yield Curve


Term (years)

Zero rates

% pa

Coupon bond yields % pa

Zero rates

% pa

Coupon bond yields % pa

1 8.500 8.500 8.500 8.500

2 9.500 9.479 7.500 7.520

3 9.950 9.916 6.950 6.986

4 10.250 10.203 6.650 6.696

5 10.300 10.255 6.600 6.642

Example

Coupon = 4% pa

The Yield Curve

Rising zero rates:

Yields are lessthan zero rates

Falling zero rates:

Yields are greaterthan zero rates


The Yield Curve

If all bonds have the same coupon rate (4%), then:Zero rates rising

Zero rates falling

Yields if coupon = 4%


(not to scale)

All zero rates equalZero rates = Yields

Term to maturity

% pa


Term (years)

Zero rates

% pa

Yield % pa if

coupon = 4%

Yield % pa if

coupon = 16%

Zero rates

% pa

Yield % pa if

coupon = 4%

Yield % pa if

coupon = 16%

1 8.500 8.500 8.500 8.500 8.500 8.500

2 9.500 9.479 9.429 7.500 7.520 7.568

3 9.950 9.916 9.840 6.950 6.986 7.067

4 10.250 10.203 10.106 6.650 6.696 6. 793

5 10.300 10.255 10.169 6.600 6.642 6.736

Rising zero rates: Higher coupons decrease yields

Falling zero rates: Higher coupons increase yields

The Yield Curve


Zero rates rising

Zero rates falling





(not to scale)

Zero rates flatZero rates = Yields

Term to maturity

% pa

The Yield Curve


The Yield Curve

Now suppose that the only bonds traded in this market are 1-year 16% coupon bonds 2-year 4% coupon bonds 3-year 16% coupon bonds 4-year 4% coupon bonds and 5-year 4% coupon bonds

What will the yield curve look like?


The Yield Curve

% pa

Zero rates

Yields (4% coupon)

Yields (16% coupon)

Term (years)2 3 4 51

X

XXXX

Is this bond wrongly priced?


The Yield Curve

Even though the zero rate curve is smooth, the observed yield curve has lumps and bumps in it.

The yield curve may, of course, even give two (or more) yields for the same term to maturity in the example, the yield on the 3-year 4% coupon is

9.916% pa while the yield on the 3-year 16% coupon is 9.840% pa.

So, lumps and bumps in a yield curve may indicate: Nothing much its just how it is, due to different bonds

having different coupon rates. Data problems (eg yields are taken from trades that

occurred at slightly different times). Market inefficiency (some yields are wrong).


6. The Par Yield Curve


In some countries (like the US) there are so many different bonds on issue that inevitably some will be selling at par on a coupon date.

Such a bond is called a par bond and its yield is called the par yield.

From these bonds we can estimate the par yield curve. And it turns out that from the par yield curve we can find the

zero rate curve. which, in turn, can then be used to price a wide range of

securities.

The Par Yield Curve


The Par Yield Curve

0 201 02 0

0 201 02 0

0 01 02 0

We can write the price of any bond as:

...1 1 1

Recall that: .

1...1 1 1

It's easier to use discount factors:

... 1

TT

TT

T

C C Par CPz z z

C c Par

c c cP Parz z z

P Par d c d c d c


The Par Yield Curve

0 01 02 0

001 02 0

0

001 02 0

01 02 0 0

0

0

... 1

Therefore: ... 1

For a par yield bond, we know that = and :

... 1 1

Hence: ... 11So:

T

T

T

T T

TT

P Par d c d c d cPd c d c d c

ParP Par ytm c

Pd ytm d ytm d ytmPar

ytm d d d ddytm

d 1 02 0...where means yield to maturity on a -year par yield bond.

T

T

d dytm T


ExampleSuppose we observe bonds selling at par for terms of 1, 2, 3, 4 and 5 years. Their yields to maturity (ytm) are:

1 Year: 7.100% pa2 Years: 7.750% pa3 Years: 8.125% pa4 Years: 8.300% pa5 Years: 8.400% pa

What are the discount factors for terms of 1, 2, 3, 4 and 5 years?Hence, what is the price (per $100 par) of a 5-year 10.5% bond?

The Par Yield Curve


Answer

01 1By definition 0.9337068161.07100d

022

01 02

02

02

02

1When 2, we have .

1Hence 0.07750 .0.933706816

Solving, we find 0.860916679.

dT ytmd d

dd

d

The Par Yield Curve


Answer (contd.)

Similarly, we find:d04 = 0.725278209 andd05 = 0.666022424

The Par Yield Curve

033

01 02 03

03

03

03

1When 3, we have .

1Hence 0.08125 .0.933706816 0.860916679

Solving, we find 0.78999939.

dT ytmd d d

dd

d


Answer (contd.)

Therefore, the price (per $100 par value) of a 5-year 10.5% bond is:

0.933706816 $10.50 0.860916679 $10.50 0.78999939 $10.500.725278209 $10.50 0.666022424 $110.50$108.3494393

The Par Yield Curve


The Par Yield Curve

How is the par yield curve related to the zero rate curve?

If the zero rate curve is flat, then the par yield curve is also flat (coincident).

If the zero rate curve is upward sloping, then the par yield curve is also upward sloping but lies below the zero rate curve.

If the zero rate curve is downward sloping, then the par yield curve is also downward sloping but lies above the zero rate curve.


The Par Yield Curve

Zero rates rising

Zero rates falling

Par yield curve

Par yield curve

(not to scale)

All zero rates equalPar yield curve

Term to maturity

% pa


7. Finding the Zero Rate Curve: Bootstrapping


Finding the Zero Rate Curve: Bootstrapping

Recall, in practice, few zero coupon bonds are traded so we cant observe the zero rate curve.

Another way to find the zero rate curve that lies hidden behind the observed market prices of the various coupon bonds is bootstrapping.



Suppose we are able to observe the market prices of three coupon bonds that are exactly 1, 2 and 3 years before their respective maturity dates.

To keep it simple, we will also assume that these bonds pay annual coupons.

To show how this works we will use the following new notation:

0, is the price of a -year bond at time 0. is the coupon paid on a -year bond.

T

T

P TC T



We can solve this sequentially.

10,1

01

2 20,2 2

01 02

3 3 30,3 2 3

01 02 03

1001

1001 1

1001 1 1

CPz

C CPz z

C C CPz z z



10,1 01

01

2 20,2 022

01 02

3 3 30,3 032 3

01 02 03

1001

1001 1

1001 1 1

CP zz

C CP zz z

C C CP zz z z



ExampleWe observe the following bond yields:

1-year bond with 12% coupon: 9.65% pa2-year bond with 9.75% coupon: 10.174% pa3-year bond with 6.50% coupon: 10.351% pa

Use bootstrapping to identify the zero rate curve.



AnswerThe 1-year bond is effectively a zero. So z01 = 9.650% pa.Using the bond yields, we calculate bond prices (per $100 par value) as follows:

0,2 2

0,3 2 3

$9.75 $109.75 $99.2658471.10174 1.10174$6.50 $6.50 $106.50 $90.482004

1.10351 1.10351 1.10351

P

P



Answer

0,2 202

202

0.502

0,3 2 303

03

$9.75 $109.75$99.2658471.0965 1

$109.75 $90.3739181

1.2143990 1 10.200% pa$6.50 $6.50 $106.50$90.482004 ,1.0965 1.102 1

which solves to give 10.375% pa.

Pz

z

z

Pz

z


8. The Holding Period Return


The Holding Period Return

Recall that a zero-coupon bond held until maturity is certain to achieve a rate of return equal to the zero rate.

This is not true of a coupon bond. Consider a 2-year coupon bond:

The rate of return that will be achieved depends on z12, which today (time 0) is unknown.

0 1 2

P0 + C + 100 + C+(1+ z12)C



This is an example of the reinvestment rate problem. The holding period return (ie the return actually achieved) on a

coupon bond held to maturity depends on the future interest rate(s) that will be earned on the coupon payments.

Because reinvestment rates are unknown (ex ante), the holding period return on a coupon bond may be greater than, or less than, its yield.

If a coupon bond is not held to maturity, then of course the holding period return also depends on the price received when the bond is sold in the future, which is also unknown today.



An important special caseIF:

1. A coupon bond is held to maturity and2. Future reinvestment rates are known today and are constant

and3. This reinvestment rate happens to equal the yield to maturity,THEN:

The holding period return is also equal to the yield to maturity.That is, HPR0T = ytm.



Proof

0

1/

00

11 (1)1 1

1 (2)

where is the sum accumulated at time .In the present case, is the future value of an ordinary annuity of where the interest rate i

T T

TT

T

T

T

C ParPytm ytm ytm

SHPRP

S TSC

s , plus the par value.

1 1 (3)TT

rr

Cie S rr Parrr


The Holding Period ReturnProof (contd.)

0

01/

0

If , then (3) becomes:

1 1 (4 )

From (1) it follows that:

1 1 1 (5)

The right-hand sides of (4) and (5) are the same.Therefore:

1

TT

T T

TT

TT

rr ytmCS ytm Parytm

CP ytm ytm Parytm

S P ytm

SytmP

01

see equation (2)THPR



If it turns out that coupons can be reinvested at a higher (lower) rate than the yield, then the holding period return will be higher (lower) than the yield.

Practical implicationIf you expect interest rates to increase (decrease) in the long term, and you hold bonds until maturity, then the rate of return you will achieve will be higher (lower) than the current yield.

fnce 30001 week 8 coupon bonds

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coupon payment

fixedcoupon bondsnotation

fixedcoupon bonds2

bond pricing

par value par

halfyearly coupon