Valuing Bonds

Professor: Burcu Esmer

Valuing Bonds

A bond is a debt instrument issued by governments or corporations to raise money

The successful investor must be able to:• Understand bond structure

• Calculate bond rates of return

• Understand interest rate risk

• Differentiate between real and nominal returns

2

Bond Basics

Bond: long-term debt security usually issued by a corporation or governmentbody

• Notes

Notes are issued in two-, three-, five- and 10-year terms

• Bonds

Bonds are long-term investments with terms of more than 10 years

• Mortgage Bonds

These bonds are typically backed by real estate holdings and/or realproperty such as equipment.

• Collateral Trust Bonds

A bond that is secured by a financial asset - such as stock or otherbonds - that is deposited and held by a trustee for the holders of the bond.

• Debentures

A type of debt instrument that is not secured by physical asset orcollateral.

3

Bond Basics (cont.)

Bond indenture: contract between issuer and investor thatspecifies terms of agreement

• face (par) value: principal to be repaid at end of loan

• coupon rate: (CR) the amount of the coupon payment (C) as a %of the face value of the bond

• coupons: (coupon payment) periodic interest payments madeover the life of the bond

• maturity date: when bond’s face value is paid

• frequency of payments: usually semiannually for U.S. corporatebonds

http://0.tqn.com/d/beginnersinvest/1/0/W/H/investing_in_bonds_bond_certificate.jpg 4

4

Straight BondsAn annual bond pays the holder a coupon payment, C, each year and returns the “face” or “par” value, FV, at maturity. C=(Coupon rate)x(Face Value)

C C C C+FV

0 1 2 3 Nr

Decompose into a $C, N-period annuity + a lump sum of $FV received in N-

periods.

Coupon rate is a stated rate written

onto the bond. It does not change!

Pricing Bonds

• value of any financial asset: depends on amount, timing, and riskiness of cash flows• => use Discounted Cash Flow (DCF) valuation:

Find PV of cash flows!

6

Bond Pricing: Example

What is the price of a 9% annual coupon bond with a par value of $1,000 that matures in 3 years? Assume a required rate of return of 4%.

7

Bond Pricing

A bond is a package of two investments: an annuity and a final repayment.

( ) ( )

1 (1 )where

1and

(1 )

Bond Coupons ParValue

Bond

t

t

PV PV PV

PV coupon Annuity Factor par value Discount Factor

rAnnuity Factor

r

Discount Factorr

Bond price= 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡

𝑟x 1 −

1

(1+𝑟)𝑛+ 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒

(1+𝑟)𝑛

8

Bond Pricing: Example

What is the value of a 3-year annuity that pays $90 each year and an additional $1,000 at the date of the final repayment? Assume a discount rate of 4%.

3

3

1 (1 .04) 1$90 $1,000

.04 (1 .04)

$1,138.75

BondPV

9

Semiannual Coupons

• Most bonds in the U.S. pay interest twice a year (1/2 of the annual coupon).

• coupon rates and yield (YTM)s quoted on annual basis

• Adjustment needed:

• divide coupon payment and yield (YTM) by 2

• multiply n by 2.

10

Example• PK Inc. issues 10% bonds with 20 years to maturity. Similar

bonds have a YTM of 11%. What is the price of the PK Inc.bond if coupons payments occur annually? What is the priceof the PK Inc. bond if coupons payments occur semi-annually

11

• Annual coupon pmt: $100, N=20. FV=1000, YTM=11%

PV of annual coupon payments: 796.33Annual pmt: . FV=1000, N=20, YTM=11%PV of face value : 124.03Total price= 796.33 + 124.03 =920.36

• Semi-Annual coupon pmt: $50, N=40. FV=1000, YTM=5.5%

PV of semi-coupon payments: 802.31PV of face value : 117.46Total price= 919.77

Sample Treasury bond quotes for May 14, 2010

12

Bid-ask spread

Bond Yields

To calculate how much we earn on a bond investment, we can calculate two types of 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 equals the price

13

Current Yield: Example

Suppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3 years.

What is the bond’s current yield?

14Your income as a proportion of the initial outlay.

How about capital gain return? What will happen to the price of the bond after 3 years?

Yield to Maturity

Yield to Maturity:

tr

parcoupon

r

coupon

r

couponPV

)1(

)(....

)1()1( 21

15

Yield to Maturity: Example

Suppose you spend $1,150 for a $1,000 face value bond that pays a $60 annual coupon payment for 3 years.

What is the bond’s yield to maturity?

321 )1(

)000,1$60($

)1(

60$

)1(

60$150,1$

rrr

16

Pricing Bonds

• To price a bond: discount the coupon payments and face value at appropriate market rate

• Yield to Maturity (YTM): the required market interest ratethat makes the discounted cash flows of the bond equal to thebond’s price

17

WARNING

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.

18

Pricing Bonds

In general,Bond Value= PV of coupons + PV of par

= PVA(r,n,pmt=coupon) + PV(r,n,FV=par)

• r = YTM per coupon period for this type of bond

• n = # of coupon periods until maturity

Bond price= 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡

𝑟x 1 −

1

(1+𝑟)𝑛+ 𝑝𝑎𝑟 𝑣𝑎𝑙𝑢𝑒

(1+𝑟)𝑛

19

Treasury Yields

20

The interest rate on 10-year U.S. Treasury bonds

Bond Prices & Interest RatesAs interest rates change, so do bond prices.

What is the present value of a 4% coupon bond with face value

$1,000 that matures in 3 years? Assume a discount rate of 5%.

What is the present value of this same bond at a discount rate of 2%?

21

Bond Pricing

Example

What is the price of a 5.0 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 2.15%.

95.081,1$

)0215.1(

050,1

)0215.1(

50

)0215.1(

50321

PV

PV

22

Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 5 %?

000,1$

)050.1(

050,1

)050.1(

50

)050.1(

50321

PV

PV

23

Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 8 %?

69.922$

)08.1(

050,1

)08.1(

50

)08.1(

50321

PV

PV

24

Dynamic Behavior of Bond Prices• Discount

• A bond is selling at a discount if the price is less than the facevalue.

• Par

• A bond is selling at par if the price is equal to the face value.

• Premium

• A bond is selling at a premium if the price is greater than the facevalue.

25

Discounts and Premiums

• If a coupon bond trades at a discount, an investor will earn areturn both from receiving the coupons and from receiving aface value that exceeds the price paid for the bond.

• If a bond trades at a discount, its yield to maturity will exceed itscoupon rate.

• What is the relationship between current yield and the return onbonds in this case?

• If a coupon bond trades at a premium it will earn a returnfrom receiving the coupons but this return will be diminishedby receiving a face value less than the price paid for the bond.

• Most coupon bonds have a coupon rate sothat the bonds will initially trade at, or very close to, par.

26

Discounts and Premiums (cont'd)

Bond Prices Immediately After a Coupon Payment

27

Example

28

Example (cont'd)

29

Interest Rate Risk

30

700

800

900

1,000

1,100

1,200

0 2 4 6 8 10 12 14 16

Interest rate (%)

Bo

nd

pri

ce

($

)

Note: The value of the 5% bond falls as interest rates rise

Definition: changes in bond prices arising from fluctuating market interest rates

Fixed vs. variable components of a bond:

•WARNING!!!

• fixed: coupon, face value, maturity date

• variable: time to maturity, YTM

31

32

Interest rate sensitivity

Bond A: 8% Coupon, FV=$1,000, and matures in 5 years

Bond B: 8% Coupon, FV=$1,000, and matures in 10 years

Which bond is more sensitive to interest rates? Why?

Interest Rate Risk

33

-

500

1,000

1,500

2,000

2,500

3,000

0 2 4 6 8 10

YTM

$ B

on

d P

ric

e

30 yr bond

3 yr bond

When the interest rate equals

the 5.0% coupon rate, both

bonds sell at face value

34

Interest rate sensitivity (Cont’d)

Bond A: 0% Coupon, FV=$1,000, and matures in 10 years

Bond B: 8% Coupon, FV=$1,000, and matures in 10 years

Which bond is more sensitive to interest rates? Why?

35

Yield to Maturity - YTMWhat rate of return would you earn if pay $935.82 for a $1000 face value

bond that pays an 8% coupon and that has 10 years to maturity?

80 80 80 80+1,000

0 1 2 3 10r=YTM=?

P0=935.82

SOLVE: 935.82 = 80(PVIFAYTM=?,10) + 1000(PVIFYTM=?,10)

ITERATE (1st try 10%, then 9%!)

( or use your financial calculator…

N=10, PMT=80, FV=1,000, PV=-935.82, I=?=YTM=9% )

36

What happens to the price of the bond if interest rates change causing

your required return to increase to 12%? To decrease to 4%?

P0= 80(PVIFA12%,10) + 1000(PVIF12%,10) =$773.99

P0= 80(PVIFA4%,10) + 1000(PVIF4%,10) =$1,324.44

Bond sells at a “discount”

Bond sells at a “premium”

If you buy this bond today and hold it to maturity your return will be the yield to maturity!

Bond Rate of Return

• Rate of Return - Earnings per period per dollar invested.

37

Rate of return =total income

investment

Rate of return =Coupon income + price change

investment

Do not confuse the bond’s rate of return over a particular investment period with its YTM!

Rate of Return

38

Rate of Return: Example

Suppose you purchase a 5% coupon bond, par value $1,000, with 5 years until maturity, for $975.00 today. After one year you sell the bond for $965.00.

What was the rate of return during the period?

What is the YTM when you bought the bond? Lower or higher than 4.10% ?What happened to YTM after 1 year when you sold the bond?

39

Interest Rate Risk• 30-year maturity, 6% coupon PREMIUM bond with fixed 4% YTM and

• 30-year maturity, 2% coupon DISCOUNT bond with fixed 4% YTM

40

600

700

800

900

1,000

1,100

1,200

1,300

1,400

0 5 10 15 20 25 30

Bo

nd

Pri

ce

Time to Maturity

Price path for

Premium Bond

Price path for

Discount Bond

Today

Maturity

Time and Bond Prices

• Holding all other things constant, the price of discount orpremium bond will move towards par value over time.

• If a bond’s yield to maturity has not changed, then the rate ofreturn of an investment in the bond equals its yield tomaturity even if you sell the bond early.

41

Example

• One bond has a coupon rate of 8%, another a coupon rate of12%. Both bonds have 10-year maturities and sell at a yield tomaturity of 10%. If their yields to maturity next year are still10%, what is the rate of return on each bond? Does the highercoupon bond give a higher rate of return?

42

AnswerBond 1

Year 0:

Year 1:

Rate of return =

Bond 2

Year 0:

Year 1:

• Rate of return = 43

11.877$10.1

000,1$

)10.1(10.0

1

10.0

180$PV

1010

82.884$10.1

000,1$

)10.1(10.0

1

10.0

180$PV

99

%0.10100.011.877$

)11.877$82.884($80$

89.122,1$10.1

000,1$

)10.1(10.0

1

10.0

1120$PV

1010

18.115,1$10.1

000,1$

)10.1(10.0

1

10.0

1120$PV

99

%0.10100.089.1122$

)89.1122$18.1115($120$

The Yield Curve

Term Structure of Interest Rates - A listing of bond maturitydates and the interest rates that correspond with each date.

Yield Curve - Graph of the term structure.

44

The Yield Curve• Treasury strips are bonds that make a single payment. The yields on

Treasury strips in February 2008 show that investors received ahigher yield on longer term bonds.

• Why do some people prefer short-term bonds then?

45

Nominal and Real Rates of Interest

• TIPS (treasury inflation protected securities)

• The real cash flows are fixed but the nominal cash flows (interestand principle) are increased as the CPI increases.

• E.g. In 2008, 10- year TIPS offered a yield of 1.5% (Real interestrate). The yield on nominal 10-year Treasury bonds was 3.8%.

46

Example• The US treasury issues 3% coupon, 2-year TIPS. Assume 5%

inflation in the first year and further 4% in the second year.

47

Year 1 Year 2

Real Cash flows 30 1030

Year 1 Year 2

Nominal Cash flows 30*1.05=31.5 1030*1.05*1.04=1,124.76

Real vs. Nominal Yields

Red line – Real yield on long-term UK indexed bonds

Blue line – Nominal yield on long-term UK bonds

48

Corporate Bonds and Default Risk (a.k.a. Credit Risk)

• Default premium

• The difference between the promised yield on acorporate bond and the yield on a U.S. TreasuryBond with the same coupon and maturity.

• Investment grade vs. Junk bonds

• Investors pay less for bonds with credit risk thanthey would for an otherwise identical default-freebond.

• The yield of bonds with credit risk will be higherthan that of otherwise identical default-free bonds. 49

Bond Ratings

50

Standard

Moody' s & Poor's Safety

Aaa AAA The strongest rating; ability to repay interest and principal

is very strong.

Aa AA Very strong likelihood that interest and principal will be

repaid

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

circumstances

Baa BBB Adequate capacity to repay; more vulnerability to changes

in 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 be

Ca CC in default or in danger of imminent default

C C C-rated bonds offer little prospect for interest or principal

on the debt ever to be repaid.

Investment grade

Junk bonds

Corporate Yield Curves for Various Ratings, February 2009

Source: Reuters

51

Yield Spreads and the Financial Crisis

Source:

Bloomberg.com52

Corporate Bonds

• Zero coupons

• no periodic interest payments

• issued at a substantial discount from par

• Floating rate bonds

• Coupon rate change over time

• E.g. Treasury rate plus 2%

• Convertible bonds

• Can be exchanged for a specified number of common stock shares.

53

Zero-Coupon Bonds

• Zero-Coupon Bond

• Does not make coupon payments

• Always sells at a discount (a price lower than face value), so theyare also called pure discount bonds

• Treasury Bills are U.S. government zero-coupon bonds with amaturity of up to one year.

54

Zero-Coupon Bonds (cont'd)

• Suppose that a one-year, risk-free, zero-coupon bond with a$100,000 face value has an initial price of $96,618.36. Thecash flows would be:

• Although the bond pays no “interest,” your compensation is thedifference between the initial price and the face value.

55

Zero-Coupon Bonds (cont'd)

• Yield to Maturity

• The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond.

• Price of a Zero-Coupon bond

(1 )

n

n

FVP

YTM

56

Zero-Coupon Bonds (cont'd)

• Yield to Maturity

• For the one-year zero coupon bond:

• Thus, the YTM is 3.5%.

1

100,00096,618.36

(1 )

YTM

1

100,0001 1.035

96,618.36 YTM

57

Zero-Coupon Bonds (cont'd)

• Yield to Maturity

• Yield to Maturity of an n-Year Zero-Coupon Bond

1

1

n

n

FVYTM

P

58