# 1 finc3131 business finance chapter 7: bonds and their valuation

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FINC3131Business Finance

Chapter 7: Bonds and Their Valuation

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Learning objectives1. Compute the price of a consol2. Compute the price of a zero-coupon bond.3. Compute the price of a fixed-coupon bond.

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.

Key Features of a Bond

1. Par value – face amount of the bond, which is paid at maturity (assume $1,000).

2. Coupon interest rate – stated interest rate (generally fixed) paid by the issuer. Multiply by par value to get dollar payment of interest.

3. Maturity date – the date when the par value must be repaid.

4. Issue date – when the bond was issued.

5. Yield to maturity - rate of return earned on a bond held until maturity.

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Bond Types

1. Consols,

2. Zero-coupon bonds

3. Fixed-coupon bonds

Other types (features) of bonds

1. Convertible bond – may be exchanged for common stock of the firm, at the holder’s option.

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

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

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

The value of financial assets

NN

22

11

r)(1CF

... r)(1

CF

r)(1CF

Value

0 1 2 Nr%

CF1 CFNCF2Value

...

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Consols 11. Pays a fixed coupon every period forever.

2. Has no maturity.

3. Investor who buys a consol is buying the perpetuity of the fixed coupon.

So, use PV formula of a perpetuity to find the

present value/price of the consol

Price of consol

=consolon return of rate required sinvestor'

dollarsin coupon fixed

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Consols 2 Remember earlier that cost of capital =

investor’s required rate of return. So, we can re-arrange the equation to find the firm’s cost of consol capital.

Cost of consol capital consol of price

dollarsin coupon fixed

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Consol problem 1 ABC Corp. wants to issue perpetual debt in order to

raise capital. It plans to pay a coupon of $90 per year on each bond with face value $1,000. Consols of a comparable firm with a coupon of $100 per year are selling at $1,050. What is the cost of debt capital for ABC? What will be the price at which it will issue its consols?

1. Verify that cost of debt = 0.0952 or 9.52%

2. Use cost of debt from above to find price of consol.Verify that price of consol = 90/0.0952 = 945.38

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Consol problem 2 If ABC (from the problem above) wanted to raise $100

million dollars in debt, how many such consols would it have to issue (to nearest whole number)?

No. of consols = 100 million/ consol price = 105,778

If ABC wanted to issue it’s consols at par, that is, at a price of $1,000, what coupon must it pay?Use coupon = price x required rate of returnVerify that coupon = $95.20 per year

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Zero-coupon bond (ZCB) 11. Call this ZCB for short.

2. Zero coupon rate, no coupon paid during bond’s life.

3. Bond holder receives one payment at maturity, the face value (usually $1000).

4. Price of a ZCB, PZCB

NZCBd

ZCBr

FP

1

F = face value of the bond

cost of ZCB debt capital (in decimals)

N = number of years to maturity

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Zero-coupon bond (ZCB) 2

1. As long as interest rates are positive, the price of a ZCB must be less than its face value.

2. Why? With positive interest rates, the present value of the face value (i.e., the price) has to be less than the face value.

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ZCB Problems1) Find the price of a ZCB with 20 years to

maturity, par value of $1000 and a required rate of return of 15% p.a.

N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10

2) XYZ Corp.’s ZCB has a market price of $ 354. The bond has 16 years to maturity and its face value is $1000. What is the cost of debt for the ZCB (i.e., the required rate of return).

PV=-354, FV=1000, N=16, PMT=0.

Required rate of return/ Cost of debt =6.71% p.a.

These problems are just basic TVM problems where you receive a single cash flow in the future.

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Fixed-coupon bond (FCB) 11. Call this FCB for short. 2. Firm pays a fixed amount (‘coupon’) to the

investor every period until bond matures.3. At maturity, firm pays face value of the bond to

investor. 4. Face value also called par value. Unless

otherwise stated, always assume face value to be $1000.

5. Period: can be year, half-year (6 months), quarter (3 months).

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Fixed-coupon bond (FCB) 21. FCB gives you a stream of fixed payments

plus a single payment (face value) at maturity.

2. This cash flow stream is just an annuity plus a single cash flow at maturity.

3. Therefore, we calculate the price of a FCB by finding the PV of the annuity and the single payment.

4. We use the financial calculator to compute the price of the FCB.

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Fixed-coupon bond (FCB) 2

Price of the FCB, PFCB

Nd

N

tt

d

FCBr

F

r

CP

111

Number of periods to maturity

Fixed periodic coupon Face value

Cost of debt capital

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 = ?

...

Using a financial calculator to value a bond

This bond has a $1,000 lump sum (the par value) due at maturity (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.

INPUTS

OUTPUT

N I/YR PMTPV FV

10 10 100 1000

-1000

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

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

Bond values over time

1. At maturity, the value of any bond must equal its par value.

2. If rd remains constant:1. The value of a premium bond would

decrease over time, until it reached $1,000.

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

3. A value of a par bond stays at $1,000.

Interest rate risk

Interest rate risk means rising (decreasing) rd will cause the value of a bond to fall (rise).

(assume Coupon rate =10%, rd=10%, FV=1000 )

rd 1-year Change 10-yearChange5% 1,048 1,38610% 1,000 1,00015% 956 749

* The 10-year bond is more sensitive to interest rate changes because of longer time to maturity, and hence has more interest rate risk.

+ 4.8%

– 4.4%

+38.6%

–25.1%

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

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

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Find FCB price

1. A $1,000 par value bond has coupon rate of 5% and the coupon is paid semi-annually. The bond matures in 20 years and has a required rate of return of 10%. Compute the current price of this bond.

FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV.

PV = -571.02. Thus, price = $571.02 < par value

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Useful property 1Go back to the bond in the last problem. Suppose annual coupon rate = 10%. Verify that price = $1000 = par value

Suppose annual coupon rate = 12% Verify that price = $1,171.59 > par value.

It turns out that the following property is true.

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Useful property 2

Coupon rate < discount rate Price < face value Bond is selling at a discount

Coupon rate = discount rate Price = face value Bond is selling at par

Coupon rate > discount rate Price > face value Bond is selling at a premium

Note: discount rate = cost of debt = required rate of return = yield to maturity

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Apply what we learnt A 10-year annual coupon bond was issued four years

ago at par. Since then the bond’s yield to maturity (YTM) has decreased from 9% to 7%. Which of the following statements is true about the current market price of the bond?

A. The bond is selling at a discountB. The bond is selling at parC. The bond is selling at a premiumD. The bond is selling at book valueE. Insufficient information

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Try one more One year ago Pell Inc. sold 20-year, $1,000 par value,

annual coupon bonds at a price of $931.54 per bond. At that time the market rate (i.e., yield to maturity) was 9 percent. Today the market rate is 9.5 percent; therefore the bonds are currently selling:

A. at a discount.

B. at a premium.

C. at par.

D. above the market price.

E. not enough information.

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Find YTM, Coupon rate1)A $1,000 par value bond sells for $863.05. It matures in

20 years, has a 10 percent coupon rate, and pays interest semi-annually. What is the bond’s yield to maturity on a per annum basis (to 2 decimal places)?

Verify that YTM = 11.80%2) ABC Inc. just issued a twenty-year semi-annual coupon

bond at a price of $787.39. The face value of the bond is $1,000, and the market interest rate is 9%. What is the annual coupon rate (in percent, to 2 decimal places)?

Verify that annual coupon rate = 6.69%

What happens if bond pays coupon annually? Quarterly?

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Long FCB question1. HMV Inc. needs to raise funds for an expansion

project. The company can choose to issue either zero-coupon bonds or semi-annual coupon bonds. In either case the bonds would have the SAME nominal required rate of return, a 20-year maturity and a par value of $1,000. If the company issues the zero-coupon bonds, they would sell for $153.81. If it issues the semi-annual coupon bonds, they would sell for $756.32. What annual coupon rate is HMV Inc. planning to offer on the coupon bonds? State your answer in percentage terms, rounded to 2 decimal places. (assume semi-annual compounding for ZCB)

2. Verify that annual coupon rate = 6.8231%

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Summary1. Find the price/ present value of debt2. Fixed-coupon bonds are valued as an annuity

plus a lump sum (face value at maturity)3. Adjust inputs depending on how often the

interest rate is compounded.4. Assignment:

questions: 7-4 7-7 problems: 7-1 7-2 7-3 7-6 7-9 7-13 7-15 7-16