Chapter 6-bonds (1)

Valuing Bonds

Chapter Six

Bond Terminology Bond

• A security that obligates the issuer to make fixed payments to the bondholder until the maturity of the bond (fixed income security)

Face Value (par, principal or maturity value)• Payment made at the maturity of the bond. In the US it is

usually $1,000 Coupon

• The interest payments made to the bondholder. In the USA coupons are paid twice a year

Coupon Rate• Total annual interest payment as percent of bond face

value. Only used to calculate coupon. Not used as discount rate, but upon issuance the coupon rate is usually close or equal to the discount or required rate of return.

Bond Terminology and Pricing Bond Price

• Equals the present value of all future coupon payments and the future par value payment.

Bond Yield• The annualized (APR) discount rate used to calculate the

bond’s present value. Also known as the yield to maturity or required rate of return.

Current Yield• Annual coupon payments divided by bond price.

t21 )r1(

)parcoupon(....

)r1(

coupon

)r1(

couponPV

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%.

Calculator: N=3, I/Y=2.15, PMT=.05x1,000=50, FV=1,000, PV=?

95.081,1$

)0215.1(

050,1

)0215.1(

50

)0215.1(

50321

PV

PV

Bond Cash Flows

Bond Pricing Example

What is the price of the bond if the required rate of return is 2.15% and the coupons are paid semi-annually?

Calculator: N=3 years x 2 payments/year=6, I/Y=2.15/2 payments/year = 1.075, PMT=.05x1,000/2 payments/year=25, FV=1,000, PV=?

37.082,1$

)01075.1(

025,1

)01075.1(

25...

)01075.1(

25

)01075.1(

256521

PV

PV

Semi-Annual Coupons

How did the calculation change, given semi-annual coupons versus annual coupon payments?

Time Periods

Paying coupons twice a year, instead of once doubles the total number of cash flows to

be discounted in the PV formula.

Discount Rate

Since the time periods are now half years, the discount rate is also

changed from the annual rate to the half year rate.

Bond Pricing Example

What is the price of the annual bond if the required rate of return is 5.0 %?

000,1$

)050.1(

050,1

)050.1(

50

)050.1(

50321

PV

PV

Bond Pricing Example

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

69.922$

)08.1(

050,1

)08.1(

50

)08.1(

50321

PV

PV

Interest Rate Risk

The value of an existing bond falls as interest (discount) rates rise

700

800

900

1,000

1,100

1,200

0 2 4 6 8 10 12 14 16

Bo

nd

pri

ce ($

)

Interest rate (%)

Interest Rate Risk

The value of an existing bond falls as interest (discount) rates rise.

The intuition is that, when buying a bond, we agree on a fixed interest payment.

Afterwards, when current interest rates go up, our bond pays less interest than newer bonds, making our bond worth less.

Treasury Yields

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

0

2

4

6

8

10

12

14

1619

00

1905

1910

1915

1920

1925

1930

1935

1940

1945

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

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Year

Yie

ld %

Pricing of Zero-Coupon Bonds

Price a 10 year zero-coupon bond at a 12% discount rate.

PV = CFt / (1 + r)t = 1000 / (1 + .12)10

Price = 1000 / (1+0.12)10 = $321.97

Bond Yield orYield to Maturity (YTM)

When a normal coupon bond is initially issued its coupon rate should be close to or equal to its yield to maturity (discount rate)

But market conditions like changing interest rates can cause the bond prices and yields to change over time while the coupon rate stays constant

If given the bond price, you can calculate its YTM by solving for r below.

t21 )r1(

)parcoupon(....

)r1(

coupon

)r1(

couponPV

Bond Yields Example

What is the YTM of a 5.0 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,081.95.

95.081,1$

)1(

050,1

)1(

50

)1(

50321

PV

rrrPV

YTM = 2.15%

The Yield Curve

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

Yield Curve - Graph of the term structure.

The Yield Curve

0

1

2

3

4

5

6

1 3 5 7 9

11

13

15

17

19

21

23

25

27

29

Maturity (years)

Yie

ld %

Treasury zero coupon bonds (strips) are bonds that make a single payment. The yields on Treasury strips in February 2008 show that investors received a higher yield on longer term bonds.

Bond Rates of Return

Rate of Return – actual earnings of an investor during a certain period of time per dollar invested in a bond.

Rate of return =

Rate of return =

investmentprice) in hangec (coupons

0

01

Price

)Price – Price (coupons

Bonds and Credit RisksCredit risk is the risk of default by the issuer, which is the inability to pay coupons or face value at maturity. US Government debt has no credit risk by convention.

Investors require higher yields on riskier bonds.

The default premium is the difference between the yields on government bonds and corporate bonds with the same maturity, but more default risk.

Default Risk and Ratings

• Rating companies– Moody’s Investor Service– Standard & Poor’s– Duff and Phelps

• Rating Categories– Investment grade = BBB, Baa and above– Speculative grade or Junk bonds = BB and

below

Bond RatingsStandard

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.

Default Risk

0

2

4

6

8

10

12

1980

1983

1986

1989

1992

1995

1998

2001

2004

2007

Yie

ld s

pre

ad %

Junk bonds

Baa-rated bonds

Aaa-rated bonds

Yield spreads between corporate and 10-year Treasury bonds