Chapter 5Chapter 5Valuing BondsValuing Bonds

Chapter 5 Topic Overview

Bond Characteristics Reading Bond Quotes Annual and Semi-Annual Bond

Valuation Finding Returns on Bonds Bond Risk and Other Important Bond

Valuation Relationships

Bond Characteristics Face (or Par) Value = stated face value

that is the amount the issuer must repay, usually $1,000

Coupon Interest Rate Coupon (cpn) = Coupon Rate x Face Value Maturity Date = when the face value is

repaid. This makes a bond’s cash flows look like this:

Characteristics of Bonds

Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the face value at maturity.

00 1 1 2 . . .2 . . . nn

$I $I $I $I $I $I+$M$I $I $I $I $I $I+$M

The Financial Pages: Treasury Bonds

Maturity AskRate Mo/Yr Bid Asked Chg Yld6.5 Oct 06n 112:17 112:18 -2 2.23• Most values expressed as a %age of par ($1000).• xxx:## = xxx and ##/32nd % of parAsked = investor purchase price = 112 18/32% of

$1000 = $1,125.625Bid = investor selling price = $1,125.3125Rate = Annual coupon rate = 6.5% of par $65/year:

$32.50 semiannually Chg = change in price from previous day in 32nds

of % of parAsk Yld = 2.23% annual rate of return if purchased

and held until maturity in Oct 2006

BondsWARNINGWARNING

The coupon rate IS NOT the discount rate used in the Present Value calculations.

The coupon rate merely tells us what cash flow the bond will produce.

Since the coupon rate is listed as a %, this misconception is quite common.

Bond PricingThe price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.PV cpn

rcpnr

cpn parr t

( ) ( )

.... ( )( )1 1 11 2

Bond Valuation• Discount the bond’s cash flows at the

investor’s required rate of return.– the coupon payment stream (an annuity).– the face (par) value payment (a single sum).– PV = cpn (PVAF r, t) + par /(1+r)t

00 1 1 2 . . . 2 . . . nn

cpn cpn cpncpn cpn+parcpn+par

Bond Valuation Example #1

Duff’s Beer has $1,000 par value bonds outstanding that make annual coupon payments. These bonds have an 8% annual coupon rate and 12 years left to maturity. Bonds with similar risk have a required return of 10%, and Moe Szyslak thinks this required return is reasonable.

What’s the most that Moe is willing to pay for a Duff’s Beer bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

10 = I/Y 1,000 = FV80 = PMT

CPT PV = -$863.73

Note: If the coupon rate < discount rate, the bond will sell for less than the par value: a discount.

Let’s Play with Example #1

Homer Simpson is interested in buying a Duff Beer bond but demands an 8 percent required return.

What is the most Homer would pay for this bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

8 = I/Y 1,000 = FV80 = PMT

CPT PV = -$1,000

Note: If the coupon rate = discount rate, the bond will sell for its par value.

Let’s Play with Example #1 some more.

Barney (belch!) Barstool is interested in buying a Duff Beer bond and demands on a 6 percent required return.

What is the most Barney (belch!) would pay for this bond?

0 1 2 3 . . . 12

1000 80 80 80 . . . 80

P/Y = 1 12 = N

6 = I/Y 1,000 = FV80 = PMT

CPT PV = -$1,167.68

Note: If the coupon rate > discount rate, the bond will sell for more than the par value: a premium.

Bond Prices and Interest Rates have an inverse relationship!

Bond Values for 8% Annual Coupon Bonds

0200400600800

100012001400

0% 2% 4% 6% 8% 10% 12%

Required Return

($)M

ark

et V

alue

12-yr Bond

Bonds with Semiannual Coupons

Double the number of years, and divide required return and annual coupon by 2.

VVBB = = annual cpnannual cpn/2/2(PVAF(PVAFr/2,2tr/2,2t) + par ) + par /(1+r/2)2t

Semiannual Example A $1000 par value bond with an annual

coupon rate of 9% pays coupons semiannually with 15 years left to maturity. What is the most you would be willing to pay for this bond if your required return is 8% APR?

Semiannual coupon = 9%/2($1000) = Semiannual coupon = 9%/2($1000) = $45$45

15x2 = 30 remaining coupons15x2 = 30 remaining coupons

0 1 2 3 . . . 30

1000 45 45 45 . . . 45

P/Y = 1 15x2 =30 = N

8/2 = 4 = I/Y 1,000 = FV

90/2 = 45 = PMTCPT PV = -$1,086.46

Bond Yields• Current Yield - Annual coupon payments

divided by bond price.• Yield To Maturity - Interest rate for

which the present value of the bond’s payments equal the price.

Bond YieldsCalculating Yield to Maturity

(YTM=r)If you are given the price of a bond (PV) and the coupon rate, the yield to maturity can be found by solving for r.PV cpn

rcpnr

cpn parr t

( ) ( )

.... ( )( )1 1 11 2

Yield to Maturity Example

$1000 face value bond with a 10% coupon rate paid annually with 20 years left to maturity sells for $1091.29.

What is this bond’s yield to maturity?

0 1 2 3 . . . 20

1000-1091.29 100 100 100 . . . 100

P/Y = 1 -1091.29 = PV

20 = N1,000 = FV100 = PMT

CPT I/Y = 9% = YTM

Bond YieldsRate of Return - Earnings per period

per dollar invested.Rate of return = total income

investment

Rate of return = Coupon income + price changeinvestment

Let’s try this together.• Imagine a year later, the discount

(required) rate for the bond from the YTM example fell to 8%.

• What is the bond’s expected price?• What is the rate of return, if we sell the

bond at this time assuming we bought the bond a year earlier at 1091.29?

• PMT =100, FV = 1000

YTM for semiannual coupon bonds: back to

our T-bond Maturity Ask

Rate Mo/Yr Bid Asked Chg Yld6.5 Oct 06n 112:17 112:18 -2 2.23• $1000 par value, today’s price = $1125.625 = PV• Semiannual coupon = $1000(6.5%/2) = $32.50• Assume 2006-2003 = 3 years to maturity x 2 = 6

semiannual payments left.• -1,125.625 = PV, 32.50 = PMT, $1000 = FV, 6 = N, CPT I/Y

= 1.1% semiannually• Annual YTM = 2(1.1%) = 2.2% APR

Bond Value Changes Over Time

Returning to the original example #1, where k = 10%, N = 12, cpn (PMT) = $80, par (FV) = $1000, & PV = $863.73.

What is bond value one year later when N = 11 and r is still = 10%?

80 = PMT, 1000 = FV, 11 = N, 10 = I/Y, CPT PV = 870.10

PV = $80(PVAF10%,11) + $1000/(1.10)11 = $870.10

What is the bond’s return over this year? Rate of Return = (Annual Coupon + Price

Change)/Beg. Price Annual Coupon = $80 Beg. Price = $863.73, End Price =

$870.10 Price Change = $870.10 - $863.73 =

$6.37 Rate of Return = ($80 +

$6.37)/$863.73 = 10%

Bond Prices over time approach par value as maturity date approaches assuming same YTM

Bond Values Over Tim e

$870.10$ 8 6 3 .7 3

$ 1 ,1 6 7 .6 8

$800.00

$900.00

$1,000.00

$1,100.00

$1,200.00

12 10 8 6 4 2 0

Tim e to Maturity

Bo

nd

Val

ue

k = 1 0 %k = 8 %k = 6 %

Interest Rate Risk• Measures Bond Price Sensitivity to

changes in interest rates.• In general, long-term bonds have

more interest rate risk than short-term bonds.

Interest Rate Risk Example

• Recall from our earlier example (#1), the 12-year, 8% annual coupon bond has the following values at kd = 6%, 8%, & 10%. Let’s compare with a 2-yr, 8% annual coupon bond.

• 12-year bond 2-year bondr=6%: PV = $1,167.68 PV = $1,036.67r=8%: PV = $1,000 PV = $1,000r=10%: PV = $863.73 PV = $965.29

Bond Price Sensitivity Graph

Bond Values for 8% Annual Coupon Bonds

400650900

1150140016501900215024002650

2% 4% 6% 8% 10% 12% 14%

2-yr Bond12-yr Bond30-yr Bond

Default Risk• Credit risk• Default premium• Investment grade• Junk bonds

Default RiskStandard

Moody' s & Poor's Safety

Aaa AAA The strongest rating; ability to repay interest and principalis very strong.

Aa AA Very strong likelihood that interest and principal will berepaid

A A Strong ability to repay, but some vulnerability to changes incircumstances

Baa BBB Adequate capacity to repay; more vulnerability to changesin economic circumstances

Ba BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over

sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C C-rated bonds offer little prospect for interest or principal

on the debt ever to be repaid.

Other Types of Bonds Zero Coupon Bonds: no coupon payments, just

par value. Convertible Bonds: can be converted into (fixed

# of) shares of stock. Floating Rate (Indexed) Bonds: coupon

payments and/or par value indexed to inflation. TIPs: Indexed US Treasury coupon bond, fixed coupon rate,

face value indexed. Callable Bonds: Company can buy back the

bonds before maturity for a call price. More likely as interest rates fall. Yield to Call: calculate like yield to maturity but use time to

earliest call date as N, and call price as FV.