7-1By Donglin Li

CHAPTER 7 & 6Bonds and Their Valuation

Key features of bonds Bond valuation Measuring yield, yield curve Assessing default risk

7-2By Donglin Li

What is a bond?

A long-term debt instrument in which a borrower agrees to make payments of principal and interest, on specific dates, to the holders of the bond.

7-3By Donglin Li

Bond markets Primarily traded in the over-the-counter (OTC)

market with dealers connected electronically. Most bonds are owned by and traded among

large financial institutions. Full information on bond trades in the OTC

market is not published, but a representative group of bonds is listed and traded on the bond division of the NYSE.

Information on small company or municipal issues is difficult to get, treasury securities are an exception.

7-4By Donglin Li

Key Features of a Bond Par value – face amount of the bond, which

is paid at maturity (assume $1,000). Coupon rate – stated interest rate

(generally fixed) paid by the issuer. Multiply by par to get dollar payment of interest.

Maturity date – years until the bond must be repaid.

Current Yield - Annual coupon payments divided by bond price.

7-5By Donglin Li

WARNINGWARNING

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.

7-6By Donglin Li

YTM

Yield to maturity (YTM) - rate of return earned on a bond held until maturity.

It is the interest rate (discount rate) for which the present value of the bond’s payments equal the bond price.

ty

parcpn

y

cpn

y

cpnP

)1(

)(....

)1()1( 21

7-7By Donglin Li

Wait a minute, isn’t yield to maturity just the discount rate?

If the discount rates (interest rates) remain same from year to year, YTM=discount rate (interest rate).

Discount rates (interest rates) may be different in different periods. In that case, YTM is some average of the discount rates.

7-8By Donglin Li

The value of financial assets

NN

22

11

r)(1CF

... r)(1

CF

r)(1CF

Value

0 1 2 Nr%

CF1 CFNCF2Value

...

7-9By Donglin Li

Bond price changes as Rates Change

Bond Value = PV of coupons + PV of par= PV annuity + PV of lump sum

As interest rate (discount rate, yields to maturity) increases, the PV’s decrease, so bond prices decrease and vice versa.

7-10By Donglin Li

What is the value of a 10-year, 10% annual coupon bond, if rd = 10%?

$1,000 V$385.54 $38.55 ... $90.91 V

(1.10)$1,000

(1.10)$100

... (1.10)$100

V

B

B

10101B

0 1 2 nr

100 100 + 1,000100VB = ?

...

7-11By Donglin Li

Using a financial calculator to value a bond

This bond has a $1,000 lump sum due at t = 10, and annual $100 coupon payments beginning at t = 1 and continuing through t = 10, the price of the bond can be found by solving for the PV of these cash flows. (PMT and FV need to have the same sign, PV the opposite sign)

INPUTS

OUTPUT

N I/YR PMTPV FV

10 10 100 1000

-1000

7-12By Donglin Li

The same company also has 10-year bonds outstanding with the same risk but a 13% annual coupon rate

This bond has an annual coupon payment of $130. Since the risk is the same the bond has the same yield to maturity as the previous bond (10%). In this case the bond sells at a premium because the coupon rate exceeds the yield to maturity.

INPUTS

OUTPUT

N I/YR PMTPV FV

10 10 130 1000

-1184.34

7-13By Donglin Li

The same company also has 10-year bonds outstanding with the same risk but a 7% annual coupon rate

This bond has an annual coupon payment of $70. Since the risk is the same the bond has the same yield to maturity as the previous bonds (10%). In this case, the bond sells at a discount because the coupon rate is less than the yield to maturity.

INPUTS

OUTPUT

N I/YR PMTPV FV

10 10 70 1000

-815.66

7-14By Donglin Li

Changes in Bond Value over Time

What would happen to the value of these three bonds if its required rate of return remained at 10%:

Years to Maturity

1,184

1,000

816

10 5 0

13% coupon rate

7% coupon rate

10% coupon rate.

VB

7-15By Donglin Li

Bond values over time At maturity, the value of any bond

must equal its par value. If rd remains constant:

The value of a premium bond would decrease over time, until it reached $1,000.

The value of a discount bond would increase over time, until it reached $1,000.

7-16By Donglin Li

What is the YTM on a 10-year, 9% annual coupon, $1,000 par value bond, selling for $887?

Must find the rd that solves this model.

10d

10d

1d

Nd

Nd

1d

B

)r(11,000

)r(1

90 ...

)r(190

$887

)r(1M

)r(1

INT ...

)r(1INT

V

7-17By Donglin Li

Using a financial calculator to find YTM

Solving for I/YR, the YTM of this bond is 10.91%. This bond sells at a discount, because YTM > coupon rate.

INPUTS

OUTPUT

N I/YR PMTPV FV

10

10.91

90 1000- 887

7-18By Donglin Li

Find YTM, if the bond price was $1,134.20.

Solving for I/YR, the YTM of this bond is 7.08%. This bond sells at a premium, because YTM < coupon rate (9%).

INPUTS

OUTPUT

N I/YR PMTPV FV

10

7.08

90 1000-1134.2

7-19By Donglin Li

Bond Prices: Relationship Between Coupon and Yield If YTM = coupon rate, then par value

= bond price If YTM > coupon rate, then par value

> bond price Selling at a discount, called a discount

bond If YTM < coupon rate, then par value

< bond price Selling at a premium, called a premium

bond

7-20By Donglin Li

Definitions

Annual coupon paymentCurrent yield (CY)

Current price

Change in priceCapital gains yield (CGY)

Beginning price

Total return YTM CY CGY

7-21By Donglin Li

An example: Current and capital gains yield

Find the current yield and the capital gains yield for a 10-year, 9% annual coupon bond that sells for $887, and has a face value of $1,000.

Current yield = $90 / $887

= 0.1015 = 10.15%

7-22By Donglin Li

Calculating capital gains yield

YTM = Current yield + Capital gains yield

CGY = YTM – CY= 10.91% - 10.15%= 0.76%

Could also find the expected price one year from now and divide the change in price by the beginning price, which gives the same answer.

7-23By Donglin Li

What is interest rate (or price, maturity) risk?

Interest rate risk is the concern that rising rd will cause the value of a bond to fall.

rd 1-year Change 10-year Change5% $1,048 $1,38610% 1,000 1,00015% 956 749

The 10-year bond is more sensitive to interest rate changes, and hence has more interest rate risk.

+ 4.8%

– 4.4%

+38.6%

–25.1%

7-24By Donglin Li

Interest Rate Risk Change in price due to changes in

interest rates Long-term bonds have more interest

risk than short-term bonds. Also called maturity risk.

7-26By Donglin Li

Semiannual bonds

1. Multiply years by 2 : number of periods = 2N.2. Divide nominal rate by 2 : periodic rate (I/YR) =

rd / 2.

3. Divide annual coupon by 2 : PMT = ann cpn / 2.

INPUTS

OUTPUT

N I/YR PMTPV FV

2N rd / 2 cpn / 2 OKOK

7-27By Donglin Li

What is the value of a 10-year, 10% semiannual coupon bond, if rd = 13%?

1. Multiply years by 2 : N = 2 * 10 = 20.2. Divide nominal rate by 2 : I/YR = 13 / 2 = 6.5.3. Divide annual coupon by 2 : PMT = 100 / 2 =

50.

INPUTS

OUTPUT

N I/YR PMTPV FV

20 6.5 50 1000

- 834.72

7-28By Donglin Li

Would you prefer to buy a 10-year, 10% annual coupon bond or a 10-year, 10% semiannual coupon bond, all else equal?

The semiannual bond’s effective rate is:

10.25% > 10% (the annual bond’s effective rate), so you would prefer the semiannual bond.

10.25%12

0.1011

mi

1EFF%2m

Nom

7-29By Donglin Li

If the proper price for this semiannual bond is $1,000, what would be the proper price for the annual coupon bond?

The semiannual coupon bond has an effective rate of 10.25%, and the annual coupon bond should earn the same EAR. At these prices, the annual and semiannual coupon bonds earn the same effective return.

INPUTS

OUTPUT

N I/YR PMTPV FV

10 10.25 100 1000

- 984.80

7-30By Donglin Li

Zero Coupon Bonds (also called zeros.) Make no periodic interest payments

(coupon rate = 0%) The entire yield-to-maturity comes from

the difference between the purchase price and the par value

Question: is YTM=CGY for zeros? Will never sell for more than par value Treasury Bills is an example of zeroes.

7-31By Donglin Li

Government Bonds Treasury Securities

T-bills – pure discount bonds with original maturity of one year or less

T-notes – coupon debt with original maturity between one and ten years

T-bonds coupon debt with original maturity greater than ten years

Municipal Securities Debt of state and local governments There is default risk. Interest received is tax-exempt at the federal tax level

7-32By Donglin Li

Other types (features) of bonds Convertible bond – may be exchanged for

common stock of the firm, at the holder’s option.

Warrant – long-term option to buy a stated number of shares of common stock at a specified price.

Putable bond – allows holder to sell the bond back to the company prior to maturity.

Income bond – pays interest only when interest is earned by the firm.

Indexed bond – interest rate paid is based upon the rate of inflation.

7-33By Donglin Li

So far, we have used one interest rate for all the cash flows of different periods.

In fact, the discount rate for one period cash flows can be different from the discount rate for cash flows in other periods.

Spot interest rate: the actual interest rate between now and a certain future date, as of now. (r1, r2, r3…)

7-34By Donglin Li

Value the bond (revisit)

Here r1, r2, …, rt are spot rates for period 1, 2, …, t, respectively.

The spot rates (r1, r2, …, rt ) for cash flows in different periods are given by the yield curve.

ttr

parcpn

r

cpn

r

cpnPV

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)(....

)1()1( 22

11

7-35By Donglin Li

Yield curve and the term structure of interest rates Term structure –

relationship between interest rates (or yields) and maturities.

The yield curve is a graph of the term structure.

A Treasury yield curve from November 2005:

0

1

2

3

4

5

6

0.25 0.5 2 5 10 30

Maturity (years)

Yield (% )

7-36

Yield Curve Last September

By Donglin Li

Date 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr

09/01/10 0.16 0.13 0.19 0.25 0.50 0.75 1.41 2.02 2.58 3.35 3.65

09/02/10 0.15 0.14 0.18 0.25 0.50 0.76 1.43 2.06 2.63 3.41 3.72

09/03/10 0.15 0.14 0.19 0.25 0.52 0.81 1.49 2.14 2.72 3.49 3.79

09/07/10 0.13 0.14 0.18 0.25 0.49 0.77 1.41 2.04 2.61 3.37 3.67

09/08/10 0.10 0.14 0.19 0.24 0.52 0.80 1.46 2.09 2.66 3.42 3.72

09/09/10 0.10 0.14 0.19 0.26 0.57 0.87 1.57 2.20 2.77 3.54 3.84

09/10/10 0.10 0.14 0.19 0.27 0.58 0.88 1.59 2.24 2.81 3.58 3.88

09/13/10 0.10 0.15 0.19 0.26 0.53 0.82 1.51 2.15 2.74 3.52 3.83

09/14/10 0.11 0.15 0.20 0.26 0.50 0.77 1.43 2.09 2.68 3.48 3.79

09/15/10 0.11 0.15 0.20 0.26 0.50 0.77 1.46 2.13 2.74 3.55 3.87

09/16/10 0.12 0.16 0.20 0.25 0.48 0.77 1.48 2.17 2.77 3.61 3.92

09/17/10 0.12 0.16 0.20 0.26 0.48 0.75 1.46 2.14 2.75 3.60 3.90

09/20/10 0.12 0.17 0.20 0.26 0.47 0.73 1.43 2.10 2.72 3.57 3.87

7-37

Yield Curve in FebruaryDate 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr02/01/11

0.16 0.15 0.18 0.27 0.61 1.04 2.02 2.79 3.48 4.37 4.62

02/02/11

0.15 0.16 0.18 0.28 0.67 1.12 2.10 2.84 3.52 4.39 4.64

02/03/11

0.14 0.14 0.18 0.29 0.71 1.19 2.18 2.92 3.58 4.43 4.67

02/04/11

0.13 0.15 0.18 0.31 0.77 1.25 2.27 3.01 3.68 4.51 4.73

02/07/11

0.13 0.16 0.18 0.31 0.78 1.28 2.29 3.03 3.68 4.50 4.71

02/08/11

0.14 0.15 0.18 0.31 0.86 1.40 2.39 3.12 3.75 4.56 4.76

02/09/11

0.11 0.14 0.18 0.30 0.81 1.34 2.33 3.05 3.65 4.49 4.71

02/10/11

0.08 0.12 0.16 0.30 0.85 1.40 2.40 3.10 3.70 4.55 4.75

02/11/11

0.08 0.12 0.16 0.30 0.85 1.40 2.38 3.05 3.64 4.49 4.71

02/14/11

0.09 0.13 0.17 0.30 0.87 1.41 2.37 3.04 3.62 4.46 4.67

02/15/11

0.11 0.13 0.17 0.30 0.84 1.39 2.35 3.03 3.61 4.45 4.66

02/16/11

0.10 0.12 0.16 0.29 0.86 1.40 2.37 3.04 3.62 4.46 4.67

02/17/11

0.08 0.09 0.15 0.27 0.80 1.33 2.30 2.99 3.58 4.44 4.66

02/18/11

0.08 0.10 0.15 0.28 0.78 1.32 2.30 2.99 3.59 4.46 4.70

By Donglin Li

7-38

Yield Curve This Month

By Donglin Li

09/01/11

0.02 0.02 0.05 0.10 0.19 0.31 0.90 1.49 2.15 3.10 3.51

09/02/11

0.02 0.02 0.05 0.10 0.20 0.33 0.88 1.41 2.02 2.92 3.32

09/06/11

0.02 0.02 0.07 0.13 0.21 0.33 0.88 1.40 1.98 2.86 3.26

09/07/11

0.00 0.02 0.06 0.11 0.21 0.34 0.92 1.45 2.05 2.96 3.36

09/08/11

0.01 0.02 0.07 0.12 0.19 0.33 0.88 1.41 2.00 2.92 3.32

09/09/11

0.00 0.01 0.05 0.11 0.17 0.31 0.81 1.34 1.93 2.86 3.26

09/12/11

0.01 0.01 0.05 0.11 0.21 0.35 0.87 1.38 1.94 2.84 3.24

09/13/11

0.00 0.01 0.05 0.10 0.21 0.35 0.89 1.42 2.00 2.92 3.32

7-39By Donglin Li

Term Structure of Interest Rates

The previous graph is based on treasury securities, so the yields do not reflect the default risk, liquidity effect, etc.

Yield curve: Normal – upward-sloping, long-term

yields are higher than short-term yields Inverted – downward-sloping, long-term

yields are lower than short-term yields

7-40By Donglin Li

Upward-Sloping Yield Curve

7-41By Donglin Li

Downward-Sloping Yield Curve

7-42By Donglin Li

Example

Please use the following information to value a 10%, four years coupon bond, if the term structure is:

Year Spot rate

1 5% 2 5.4%

3 5.7%

4 5.9%

7-43By Donglin Li

Solution

The interest payment is $100 every year.

1 2 3 4

100 100 100 (100 1,000)

(1 .05) (1 .054) (1 .057) (1 .059)

$1,144.5

P

7-44By Donglin Li

What is the opportunity cost of debt capital?

The discount rate (ri ) is the opportunity cost of capital, and is the rate that could be earned on alternative investments of equal risk.

ri = r* + IP + MRP + DRP + LP

7-45By Donglin Li

What else determines interest rates of corporate bonds?

ri = r* + IP + MRP +DRP + LP

ri = required return on a debt security

r* = real risk-free rate of interestIP = inflation premiumMRP= maturity risk premium (also

called interest rate risk premium) DRP= default risk premiumLP = liquidity premium

7-46By Donglin Li

Premiums added to r* for different types of debt

IP MRP DRP LP

S-T Treasury

L-T Treasury

S-T Corporate

L-T Corporate

7-47By Donglin Li

Default risk If an issuer defaults, investors

receive less than the promised return. Therefore, the expected return on corporate and municipal bonds is less than the promised return.

Influenced by the issuer’s financial strength and the terms of the bond contract.

7-48By Donglin Li

Evaluating default risk:Bond ratings

Bond ratings are designed to reflect the probability of a bond issue going into default.

Investment Grade Junk Bonds

Moody’s

Aaa Aa A Baa Ba B Caa C

S & P AAA AA A BBB BB B CCC D

7-49By Donglin Li

Factors affecting default risk and bond ratings Financial performance

Debt ratio Current ratio

Bond contract provisions Secured vs. Unsecured debt Senior vs. subordinated debt Guarantee and sinking fund provisions Debt maturity

7-50By Donglin Li

Bankruptcy

Two main chapters of the Federal Bankruptcy Act: Chapter 11, Reorganization Chapter 7, Liquidation

Typically, a company wants Chapter 11, while creditors may prefer Chapter 7.

7-51By Donglin Li

Priority of claims in liquidation

1. Secured creditors from sales of secured assets.

2. Trustee’s costs3. Wages, subject to limits4. Taxes5. Unfunded pension liabilities6. Unsecured creditors7. Preferred stock8. Common stock