Chapter 2

Bond Prices and Yields

FIXED-INCOME SECURITIES

Outline

• Bond Pricing• Time-Value of Money• Present Value Formula• Interest Rates• Frequency• Continuous Compounding• Coupon Rate • Current Yield• Yield-to-Maturity• Bank Discount Rate• Forward Rates

Bond Pricing

• Bond pricing is a 2 steps process– Step 1: find the cash-flows the bondholder is entitled to– Step 2: find the bond price as the discounted value of the cash-flows

• Step 1 - Example– Government of Canada bond issued in the domestic market pays one-half

of its coupon rate times its principal value every six months up to and including the maturity date

– Thus, a bond with an 8% coupon and $5,000 face value maturing on December 1, 2005 will make future coupon payments of 4% of principal value every 6 months

– That is $200 on each June 1 and December 1 between the purchase date and the maturity date

Bond Pricing

• Step 2 is discounting

T

tt

t

r

FP

10 )1(

• Does it make sense to discount all cash-flows with same discount rate?

• Notion of the term structure of interest rates – see next chapter

• Rationale behind discounting: time value of money

Time-Value of Money

• Would you prefer to receive $1 now or $1 in a year from now?

• Chances are that you would go for money now• First, you might have a consumption need sooner

rather than later– That shouldn’t matter: that’s what fixed-income markets are for– You may as well borrow today against this future income, and consume

now

• In the presence of money market, the only reason why one would prefer receiving $1 as opposed to $1 in a year from now is because of time-value of money

Present Value Formula

• If you receive $1 today– Invest it in the money market (say buy a one-year T-Bill) – Obtain some interest r on it– Better off as long as r strictly positive: 1+r>1 iff r>0

• How much is worth a piece of paper (contract, bond) promising $1 in 1 year?

– Since you are not willing to exchange $1 now for $1 in a year from now, it must be that the present value of $1 in a year from now is less than $1

– Now, how much exactly is worth this $1 received in a year from now?– Would you be willing to pay 90, 80, 20, 10 cents to acquire this dollar paid

in a year from now?

• Answer is 1/(1+r) : the exact amount of money that allows you to get $1 in 1 year

Interest Rates

• Specifying the rate is not enough• One should also specify

– Maturity– Frequency of interest payments– Date of interest rates payment (beginning or end of periods)

• Basic formula– After 1 period, capital is C1= C0 (1+ r )– After n period, capital is Cn = C0(1+ r )n

– Interests : I = Cn - C0

• Example– Invest $10,000 for 3 years at 6% with annual compounding– Obtain $11,910 = 10,000 x (1+ .06)3 at the end of the 3 years– Interests: $1,910

Frequency

• Watch out for – Time-basis (rates are usually expressed on an annual basis) – Compounding frequency

• Examples– Invest $100 at a 6% two-year annual rate with semi-annual compounding

• 100 x (1+ 3%) after 6 months• 100 x (1+ 3%)2 after 1 year• 100 x (1+ 3%)3 after 1.5 year• 100 x (1+ 3%)4 after 2 years

– Invest $100 at a 6% one-year annual rate with monthly compounding• 100 x (1+ 6/12%) after 1 month• 100 x (1+ 6/12%)2 after 2 months• …. 100 x (1+ 6/12%)12 = $106.1678 after 1 year• Equivalent to 6.1678% annual rate with annual compounding

• The effective equivalent annual (i.e., compounded once a year) rate ra is defined as the solution to

• More generally– Amount x invested at the interest rate r– Expressed in an annual basis – Compounded n times per year – For T years– Grows to the amount

Frequency

1nT

rx

n

1 (1 )nT

a Trx x r

n

11

na

n

rr

or

• The equivalent annual rate of a 6% continuously compounded interest rate is e6% –1 = 6.1837%

• Very convenient: present value of X is Xe-rT

• One may of course easily obtain the effective equivalent annual ra

Continuous Compounding

• What happens if we get continuous compounding• The amount of money obtained per dollar invested

after T years islim 1

nTrT

n

rx xe

n

(1 ) 1rT a T a re r r e

Bond Prices

• Bond price

T

tTt rr

F

r

FP

1 1

11

)1(

T

tt

t

r

FP

1 )1(

TT

tTTt r

N

rr

cN

r

N

r

cNP

11

11

1)1(1

• Shortcut when cash-flows are all identical (can you prove it?)

• Coupon bond

– Note that when r=c, P=N (see next example)

Bond Prices - Example

• Example– Consider a bond with 5% coupon rate– 10 year maturity – $1,000 face value– All discount rates equal to 6%

• Present value

10

10 10 101

50 1,000 50 1 1,0001 $926.3991

(1 6%) 6%1 6% 1 6% 1 6%i

i

P

• We could have guessed that price was below par– You do not want to pay the full price for a bond paying 5% when interest rates are at

6%

• What happens if rates decrease to 5%?– Price = $1,000

Perpetuity

• When the bond has infinite maturity (consol bond)

r

cN

r

N

rr

cNP

TTT

11

11

000,2$05.

100P

• Example– How much money should you be willing to pay to buy a contract offering

$100 per year for perpetuity? – Assume the discount rate is 5% – The answer is

– Perpetuities are issued by the British government (consol bonds)

Coupon Rate and Current Yield

• Coupon rate is the stated interest rate on a security– It is referred to as an annual percentage of face value– It is usually paid twice a year – It is called the coupon rate because bearer bonds carry coupons for

interest payments – It is only used to obtain the cash-flows

• Current yield gives you a first idea of the return on a bond

P

cNyc

%78.7900

70cy

• Example– A $1,000 bond has a coupon rate of 7 percent– If you buy the bond for $900, your actual current yield is

Yield to Maturity (YTM)

• It is the interest rate that makes the present value of the bond’s payments equal to its price

• It is the solution to (T is number of periods)

T

tt

t

YTM

FP

1 )1(

• YTM is the IRR of cash-flows delivered by bonds– YTM may easily be computed by trial-and-error– YTM is typically a semi-annual rate because coupons usually paid semi-

annually– Each cash-flow is discounted using the same rate– Implicitly assume that the yield curve is flat at a point in time– It is a complex average of pure discount rates (see below)

BEY versus EAY

• Bond equivalent yield (BEY): obtained using simple interest to annualize the semi-annual YTM (street convention): y = 2 YTM

• One can always turn a bond yield into an effective annual yield (EAY), i.e., an interest rate expressed on a yearly basis with annual compounding

• Example – What is the effective annual yield of a bond with a 5.5% annual YTM– Answer is

%5756.512

%5.51

2

ar

One Last Complication

• What happens if we don’t have integer # of periods?• Example

– Consider the US T-Bond with coupon 4.625% and maturity date 05/15/2006, quoted price is 101.739641 on 01/07/2002

– What is the YTM and EAY?

• Solution (street convention)– There are 128 calendar days between 01/07/2002 and the next coupon

date (05/15/2002)

%353.4)21(

3125.102

)21(

2%625.4

739641.1018181

128

8

0 181128

YTMYTMYTMt

t

• EAY is %40.41

2

%353.41

2

– Fed convention: =1+YTM/2*128/181

Quoted Bond Prices - Screen

Source: Wall Street Journal

Government Bonds & Notes Friday, October 16, 1998

Rate Maturity Mo/Yr

Bid Asked Chg Ask Yield

7 1/2 Feb 05n 116:30 117:02 -5 4.38

6 1/2 May 05n 112:02 112:06 -3 4.35

8 1/4 May 00-05 105:22 105:24 -1 4.42

12 May 05 142:10 142:16 -5 4.47

5 1/2 Aug 28 108:10 108:11 -8 4.96

Quoted Bond Prices - Paper

Quoted Bond Prices

• Bonds are– Sold in denominations of $1,000 par value

– Quoted as a percentage of par value

• Prices– Integer number + n/32ths (Treasury bonds) or + n/8ths (corporate bonds)

– Example: 112:06 = 112 6/32 = 112.1875%

– Change -5: closing bid price went down by 5/32%

• Ask yield– YTM based on ask price (APR basis:1/2 year x 2)

– Not compounded (Bond Equivalent Yield as opposed to Effective Annual Yield)

Examples

• Example– Consider a $1,000 face value 2-year bond with 8% coupon

– Current price is 103:23

– What is the yield to maturity of this bond?

• To answer that question– First note that 103:23 means 103 + (23/32)%=103.72%

– And obtain the following equation

432 )21(

040,1

)21(

40

)21(

40

)21(

402.037,1

yyyy

– With solution y/2 = 3% or y = 6%

Accrued Interest

• The quoted price (or market price) of a bond is usually its clean price, that is its gross price (or dirty or full price) minus the accrued interest

• Example– An investor buys on 12/10/01 a given amount of the US Treasury bond with

coupon 3.5% and maturity 11/15/2006– The current market price is 96.15625– The accrued interest period is equal to 26 days; this is the number of calendar

days between the settlement date (12/11/2001) and the last coupon payment date (11/15/2001)

– Hence the accrued interest is equal to the coupon payment (1.75) times 26 divided by the number of calendar days between the next coupon payment date (05/15/2002) and the last coupon payment date (11/15/2001)

– In this case, the accrued interest is equal to $1.75x(26/181) = $0.25138– The investor will pay 96.40763 = 96.15625 + 0.25138 for this bond

– where P is price of T-Bill– n is # of days until maturity

• Example: 90 days T-Bill, P = $9,800

Bank Discount Rate (T-Bills)

• Bank discount rate is the quoted rate on T-Bills

%890

360

000,10

800,9000,10

BDr

• Can’t compare T-bill directly to bond– 360 vs 365 days – Return is figured on par vs. price paid

10,000

10,000 / 360 10,000

10,000 360

10,000

BD

BD

P Discount

P r n

Pr

n

Bond Equivalent Yield

• Adjust the bank discounted rate to make it comparable 10,000 10,000 365

1365

BEY

BEY

PP r

n P nr

%28.890

365

800,9

800,9000,10

BEYr

• Example: same as before

BDBEY rP

r 365

360

000,10

• BDR versus BEY

• (exercise: Show it!)

Spot Zero-Coupon (or Discount) Rate

• Spot Zero-Coupon (or Discount) Rate is the annualized rate on a pure discount bond

– where B(0,t) is the market price at date 0 of a bond paying off $1 at date t– See Chapter 4 for how to extract implicit spot rates from bond prices

• General pricing formula

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t

,0)1(

1

,0

01 10,

0,(1 )

T Tt

ttt tt

FP F B t

R

Bond Par Yield

• Recall that a par bond is a bond with a coupon identical to its yield to maturity

• The bond's price is therefore equal to its principal• Then we define the par yield c(n) so that a n-year

maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes at par

• Typically, the par yield curve is used to determine the coupon level of a bond issued at par

n

ii

i

nn

nn

n

ii

i

R

Rnc

RR

nc

1 ,0

,0

,01 ,0

)1(1

)1(1

1

)()1(

100

)1(

)(100100

Forward Rates

• One may represent the term structure of interest rates as set of implicit forward rates

• Consider two choices for a 2-year horizon:– Choice A: Buy 2-year zero

– Choice B: Buy 1-year zero and rollover for 1 year

• What yield from year 1 to year 2 will make you indifferent between the two choices?

)1(

)1(1

1,0

22,0

1,1 R

RF

Forward Rates (continued)

• They are ‘implicit’ in the term structure• Rates that explain the relationship between spot

rates of different maturity• Example:

– Suppose the one year spot rate is 4% and the eighteen month spot rate is 4.5%

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)1)(1()1(

6,122/1

6,122/3

2/16,121,0

2/318,0

mmmm

mmm

FF

FRR

Recap: Taxonomy of Rates

• Coupon Rate • Current Yield• Yield to Maturity • Zero-Coupon Rate• Bond Par Yield • Forward Rate