Chapter 10

Bond Prices and Yields

10-2

Straight Bond • Obligates the issuer of the bond to pay the

holder of the bond:• A fixed sum of money (principal, par value, or face

value) at the bond’s maturity• Constant, periodic interest payments (coupons)

during the life of the bond (Sometimes)

• Special features may be attached• Convertible bonds• Callable bonds• Putable bonds

10-3

Straight Bond Basics

• $1,000 face value• Semiannual coupon payments

Price Bond

Coupon Annual YieldCurrent (10.2)

ValuePar

Coupon Annual Rate Coupon (10.1)

10-4

Straight Bonds• Suppose a straight bond pays a semiannual

coupon of $45 and is currently priced at $960.• What is the coupon rate?• What is the current yield?

%.

%.

3759$960

2 $45 YieldCurrent (10.2)

009$1,000

2 $45 Rate Coupon (10.1)

10-5

Straight Bond Prices & Yield to Maturity• Bond Price:

• Present value of the bond’s coupon payments

• + Present value of the bond’s face value

• Yield to maturity (YTM):• The discount rate that equates today’s

bond price with the present value of the future cash flows of the bond

10-6

Bond Pricing Formula

MM YTM

FV

YTMYTM

C22

2121

11 Price Bond (10.3)

Where:

C = Annual coupon payment

FV = Face value

M = Maturity in years

YTM = Yield to maturity

PV of coupons PV of FV

10-7

Straight Bond Prices Calculator Solution

MM YTM

FV

YTMYTM

C22

2121

11 Price Bond (10.3)

Where:

C = Annual coupon payment

FV = Face value

M = Maturity in years

YTM = Yield to maturity

N = 2M

I/Y = YTM/2

PMT = C/2

FV = 1000

CPT PV

10-8

Straight Bond Prices

41457

2081

11

08

60 Coupons ofPV

24.$

..

MM YTM

FV

YTMYTM

C22

2121

11 Price Bond (10.3)

PV of coupons PV of FV

12390

2081

1000 FV ofPV

24.$

.

For a straight bond with 12 years to maturity, a coupon rate of 6% and a YTM of 8%, what is the current price?

Price = $457.41 + $390.12 = $847.53

10-9

Calculating a Straight Bond Price Using Excel

• Excel function to price straight bonds:

=PRICE(“Today”,“Maturity”,Coupon Rate,YTM,100,2,3)

• Enter “Today” and “Maturity” in quotes, using mm/dd/yyyy format.• Enter the Coupon Rate and the YTM as a decimal.• The "100" tells Excel to us $100 as the par value.• The "2" tells Excel to use semi-annual coupons.• The "3" tells Excel to use an actual day count with 365 days per year.

Note: Excel returns a price per $100 face.

10-10

Spreadsheet Analysis

10-11

Par, Premium and Discount Bonds

Par bonds: Price = par valueYTM = coupon rate

Premium bonds: Price > par value

YTM < coupon rate

The longer the term to maturity, the greater the premium over par

Discount bonds: Price < par value

YTM > coupon rate

The longer the term to maturity, the greater the discount from

par

10-12

Premium Bond Price

42677

2061

11

06

80 Coupons ofPV

24.$

..

93491

2031

1000 FV ofPV

24.$

.

12 years to maturity

8% coupon rate, paid semiannually

YTM = 6%

Price = $457.41 + $390.12 = $1,169.36

10-13

Discount Bonds

15906

2081

1000

2081

11

08

60 yr)(6 Price Bond

1212.$

...

Consider two straight bonds with a coupon rate of 6% and a YTM of 8%.

If one bond matures in 6 years and one in 12, what are their current prices?

53847

2081

1000

2081

11

08

60 yr)(12Price Bond

2442.$

...

10-14

Premium Bonds

361691

2061

1000

2061

11

06

80 yr)(12 Price Bond

2424.,$

...

Consider two straight bonds with a coupon rate of 8% and a YTM of 6%.

If one bond matures in 6 years and one in 12, what are their current prices?

540991

2061

1000

2061

11

06

80 yr)(6 Price Bond

1212.,$

...

10-15

M

Premium

1,000

Discount

12 6 0

CR>YTM 8%>6%

CR<YTM 6%<8%

YTM = CR

Bond Value ($) vs Years to Maturity

10-16

Premium and Discount Bonds• In general, when the coupon rate and YTM are

held constant:

For premium bonds: the longer the term to maturity, the greater the premium over par value.

For discount bonds: the longer the term to maturity, the greater the discount from par value.

10-17

Relationships among Yield Measures

For premium bonds: coupon rate > current yield > YTM

For discount bonds:coupon rate < current yield < YTM

For par value bonds:coupon rate = current yield = YTM

10-18

A Note on Bond Quotations

• If you buy a bond between coupon dates:• You will receive the next coupon payment • You might have to pay taxes on it• You must compensate the seller for any

accrued interest.

10-19

A Note on Bond Quotations• Clean Price = Flat Price

• Bond quoting convention ignores accrued interest.

• Clean price = a quoted price net of accrued interest

• Dirty Price = Full Price = Invoice Price• The price the buyer actually pays • Includes accrued interest added to the clean

price.

10-20

Clean vs. Dirty PricesExample

• Today is April 1. Suppose you want to buy a bond with a 8% annual coupon payable on January 1 and July 1.

• The bond is currently quoted at $1,020

• The Clean price = the quoted price = $1,020• The Dirty or Invoice price = $1,020 plus

(3mo/6mo)*$40 = $1,040

10-21

Calculating Yields

MM YTM

FV

YTMYTM

C22

2121

11 Price Bond

• Trial and error

• Calculator

• Spreadsheet

10-22

Calculating YieldsTrial & Error

A 5% bond with 12 years to maturity is priced at 90% of par ($900).

Selling at a discount YTM > 5%

Try 6% --- price = $915.32 too high

Try 6.5% --- price = $876.34 too low

Try 6.25% --- price = $895.56 a little low

Actual = 6.1933%

10-23

Calculating YieldsCalculatorA 5% bond with 12 years to maturity is priced at

90% of par ($900).

N = 24

PV = -900

PMT = 25

FV = 1000

CPT I/Y = 3.0966 x 2 = 6.1933%

10-24

Calculating YieldsSpreadsheet5% bond with 12 years to maturity,priced at 90% of par

=YIELD(“Now”,”Maturity”,Coupon, Price,100,2,3)

“Now” = “06/01/2008”

“Maturity” = “06/01/2020”

Coupon = .05

Price = 90 (entered as a % of par)

100 redemption value as a % of face value

“2” semiannual coupon payments

“3” actual day count (365)

=YIELD(“06/01/2008”,”06/01/2020”,0.05,90,100,2,3) = 0.06193276

10-25

Spreadsheet Analysis

10-26

Callable Bonds• Gives the issuer the option to:

• Buy back the bond • At a specified call price • Anytime after an initial call protection

period.

• Most bonds are callable Yield-to-call may be more relevant

10-27

Yield to Call

TT YTC

CP

YTCYTC

C22

2121

11 Price Bond Callable

Where:

C = constant annual coupon

CP = Call price of bond

T = Time in years to earliest call date

YTC = Yield to call

10-28

Yield to Call

• Suppose a 5% bond, priced at 104% of par with 12 years to maturity is callable in 2 years with a $20 call premium. What is its yield to call?

N = 4 # periods to first call date

PV = -1040

PMT = 25

FV = 1,020 Face value + call premium

CPT I/Y = 1.9368 x 2 = 3.874%

Remember: resulting rate = 6 month rate

10-29

Interest Rate Risk

• Interest Rate Risk = possibility that changes in interest rates will result in losses in the bond’s value

• Realized Yield = yield actually earned or “realized” on a bond

• Realized yield is almost never exactly equal to the yield to maturity, or promised yield

10-30

Interest Rate Risk and Maturity

10-31

Malkiel’s Theorems1. Bond prices and bond yields move in

opposite directions.

2. For a given change in a bond’s YTM, the longer the term to maturity, the greater the magnitude of the change in the bond’s price.

3. For a given change in a bond’s YTM, the size of the change in the bond’s price increases at a diminishing rate as the bond’s term ot maturity lengthens.

10-32

Malkiel’s Theorems

4. For a given change in a bond’s YTM, the absolute magnitude of the resulting change in the bond’s price is inversely related to the bond’s coupon rate.

5. For a given absolute change in a bond’s YTM, the magnitude of the price increase caused by a decrease in yield is greater than the price decrease caused by an increase in yield.

10-33

Bond Prices and Yields

10-34

Duration• Duration measure the sensitivity of a bond

price to changes in bond yields.

Two bonds with the same duration, but not necessarily the same maturity, will have approximately the same price sensitivity to a (small) change in bond yields.

2YTM1

YTM in ChangeDurationPrice Bond %

10-35

Macaulay Duration

% Bond Price ≈ -Duration x 2YTM1

ΔYTM

A bond has a Macaulay Duration = 10 years, its yield increases from 7% to 7.5%.

How much will its price change?

Duration = 10

Change in YTM = .075-.070 = .005

YTM/2 = .035

%Price ≈ -10 x (.005/1.035) = -4.83%

(10.5)

10-36

Modified Duration• Some analysts prefer a variation of Macaulay’s

Duration, known as Modified Duration.

• The relationship between percentage changes in bond prices and changes in bond yields is approximately:

2YTM

1

DurationMacaulayDurationModified

YTM in Change Duration Modified- Price Bond in Change Pct.

10-37

Modified Duration

2YTM1

Duration Macaulay Duration Modified

(10.7) %Bond Price ≈ -Modified Duration x YTM

(10.6)

A bond has a Macaulay duration of 9.2 years and a YTM of 7%. What is it’s modified duration?

Modified duration = 9.2/1.035 = 8.89 years

10-38

Modified Duration

2YTM1

Duration Macaulay Duration Modified

(10.7) %Bond Price ≈ -Modified Duration x YTM

(10.6)

A bond has a Modified duration of 7.2 years and a YTM of 7%. If the yield increases to 7.5%, what happens to the price?

%Price ≈ -7.2 x .005 = -3.6%

10-39

Calculating Macaulay’s Duration

• Macaulay’s duration values stated in years• Often called a bond’s effective maturity

• For a zero-coupon bond:Duration = maturity

• For a coupon bond:Duration = a weighted average of individual maturities of all the bond’s separate cash flows, where the weights are proportionate to the present values of each cash flow.

10-40

Calculating Macaulay Duration

M22YTM1

11

YTM

2YTM1 duration bond ValuePar(10.8)

Suppose a par value bond has 12 years to maturity and an 8% coupon. What is its duration?

years10.472.081

11

.08

2.081 duration bond ValuePar

24

10-41

General Macaulay Duration Formula

12YTM1CPRYTM

YTM)M(CPR2YTM1

YTM

2YTM1Duration 2M

(10.9)

Where:

CPR = Constant annual coupon rate

M = Bond maturity in years

YTM = Yield to maturity assuming semiannual coupons

10-42

Calculating Macaulay’s Duration

• In general, for a bond paying constant semiannual coupons, the formula for Macaulay’s Duration is:

• In the formula, C is the annual coupon rate, M is the bond maturity (in years), and YTM is the yield to maturity, assuming semiannual coupons.

12YTM1CYTM

YTMCM2YTM1

YTM2

YTM1Duration

2M

10-43

Using the General Macaulay Duration Formula

• What is the modified duration for a bond that matures in 15 years, has a coupon rate of 5% and a yield to maturity of 6.5%?

• Steps:1. Calculate Macaulay duration using 10.9

2. Convert to Modified duration using 10.6

10-44

Using the General Macaulay Duration FormulaBond matures in 15 yearsCoupon rate = 5% Yield to maturity = 6.5%

1. Calculate Macaulay duration using 10.9

years10.34 Duration

12.0651.05.065

.065)15(.052.0651

.065

2.0651Duration 30

12YTM1CPRYTM

YTM)M(CPR2YTM1

YTM

2YTM1Duration 2M

10-45

Using the General Macaulay Duration Formula

Bond matures in 15 yearsCoupon rate = 5% Yield to maturity = 6.5%Macaulay Duration = 10.34 years

2. Convert to Modified duration using 10.6

2YTM1

Duration Macaulay Duration Modified

years01102.0651

10.34 Duration Modified .

10-46

Calculating Duration Using Excel

• Macaulay Duration -- DURATION function• Modified Duration -- MDURATION function

=DURATION(“Today”,“Maturity”,Coupon Rate,YTM,2,3)

10-47

Calculating Macaulay and Modified Duration

10-48

Duration Properties: All Else Equal

1. The longer a bond’s maturity, the longer its duration.

2. A bond’s duration increases at a decreasing rate as maturity lengthens.

3. The higher a bond’s coupon, the shorter is its duration.

4. A higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.

10-49

Properties of Duration

10-50

Bond Risk Measures based on Duration

• Dollar Value of an 01: (10.10)

≈ Modified Duration x Bond Price x 0.0001

= Value of a basis point change

• Yield Value of a 32nd: (10.11)

≈ 01 an of ValueDollar32

1

In both cases, the bond price is per $100 face value.

10-51

Calculating Bond Risk MesuresBond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065

Macaulay Duration = 10.34 yearsModified Duration = 10.01 years

• First we have to find the price of the bond:

64857

20651

1000

20651

11

065

50 Price Bond

2121

11 Price Bond (10.3)

3030

22

.$...

YTM

FV

YTMYTM

CMM

10-52

Bond Risk Measures based on Duration

• Dollar Value of an 01: (10.10)

≈ Modified Duration x Bond Price x 0.0001

≈ 10.01 x 85.764 x 0.0001 = 0.0859

Bond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065Macaulay Duration = 10.34 yearsModified Duration = 10.01 yearsPrice = 85.764

10-53

Bond Risk Measures based on Duration

• Yield Value of a 32nd: (10.11)

≈ 364.00859.032

1

01 an of ValueDollar32

1

Bond matures in 15 years M = 15Coupon rate = 5% C = $50Yield to maturity = 6.5% YTM = .065Macaulay Duration = 10.34 yearsModified Duration = 10.01 yearsPrice = 85.764

10-54

Dedicated Portfolios

• Bond portfolio created to prepare for a future cash payment, e.g. pension funds

• Target Date = date the payment is due

10-55

Reinvestment Risk & Price Risk

• Reinvestment Rate Risk:• Uncertainty about the value of the portfolio on the

target date• Stems from the need to reinvest bond coupons at

yields not known in advance

• Price Risk:• Risk that bond prices will decrease• Arises in dedicated portfolios when the target date

value of a bond is not known with certainty

10-56

Price Risk vs. Reinvestment Rate Risk For a Dedicated Portfolio• Interest rate increases have two effects:

in interest rates decrease bond prices, but

in interest rates increase the future value of reinvested coupons

• Interest rate decreases have two effects: in interest rates increase bond prices, but

Decreases in interest rates decrease the future value of reinvested coupons

10-57

Immunization

• Immunization = constructing a dedicated portfolio that minimizes uncertainty surrounding the target date value • Engineer a portfolio so that price risk and

reinvestment rate risk offset each other (just about entirely).

• Duration matching = matching the duration of the portfolio to its target date

10-58

Immunization by Duration Matching

10-59

Dynamic Immunization

• Periodic rebalancing of a dedicated bond portfolio for the purpose of maintaining a duration that matches the target maturity date• Advantage = reinvestment risk greatly

reduced• Drawback = each rebalancing incurs

management and transaction costs

10-60

Constructing a Dedicated Portfolio

• Suppose a Pension Fund estimates it will need to pay out about $50 million in 4 years. The fund decides to buy coupon bonds paying 6%, maturing in 4 years and selling at par.

• Assuming interest rates do not change over the next four years, how much should the fund invest to have $50 million in 4 years?

10-61

Constructing a Dedicated Portfolio

2M

M2M2

2)YTM(1

Required ValueFuture ValueFace

value Face12YTM1YTM

C)2YTM1(P

10.12

10.13

Future Value required = $50 million

Time = 4 years (M=8)

Par value bonds paying 6%

462,470,39$)03.(1

0$50,000,00 ValueFace

8

10-62

Constructing a Dedicated Portfolio6-month Value at End

Year Period Payment of Year 51 1 $1,184,113.86 1,456,310.69$

2 $1,184,113.86 1,413,893.87$ 2 3 $1,184,113.86 1,372,712.50$

4 $1,184,113.86 1,332,730.58$ 3 5 $1,184,113.86 1,293,913.19$

6 $1,184,113.86 1,256,226.39$ 4 7 $1,184,113.86 1,219,637.28$

8 $1,184,113.86 1,184,113.86$

Future Value of Coupons 10,529,538.36$ Face Value 39,470,462.00$

50,000,000.36$

10-63

Useful Internet Sites

• www.sifma.org (check out the bonds section)• www.jamesbaker.com (a practical view of bond

portfolio management)• www.bondsonline.com (bond basics and current

market data)• www.investinginbonds.com (bond basics and

current market data) • www.bloomberg.com (for information on

government bonds)