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CHAPTER 5

Bonds, Bond Valuation, and

Interest Rates

Omar Al Nasser, Ph.D.

FIN 6352

1

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Key Concepts and Skills• Know the important bond features and

bond types

• Understand bond values and why they fluctuate

• Understand bond ratings and what they mean

• Understand the impact of inflation on interest rates

• Understand the term structure of interest rates and the determinants of bond yields

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Chapter Outline

• Bonds and Bond Valuation

• More on Bond Features

• Bond Ratings

• Some Different Types of Bonds

• Bond Markets

• Inflation and Interest Rates

• Determinants of Bond Yields

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Value = + + +FCF1 FCF2 FCF∞

(1 + WACC)1 (1 + WACC)∞

(1 + WACC)2

Free cash flow(FCF)

Market interest rates

Firm’s business riskMarket risk aversion

Firm’s debt/equity mixCost of debt

Cost of equity

Cost of debt

Cost of equity

Weighted averagecost of capital

(WACC)

Net operatingprofit after taxes

Required investmentsin operating capital−

=

Determinants of Intrinsic Value: The Cost of Debt

...

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Key Features of a Bond• A bond is along-term contract under which a

borrower agrees to make payments of interest and principal on specific date to the holder of the bond. – For example, if a company borrowed $50 million

dollars by issuing $50 millions of bonds. Lets assume that the company sold 50,000 for 1,000 each.

• Par value: – The face value of the bond.– The par value represents the amount of money the

firm borrow and promises to repay on the maturity date.

– We generally assume a face value of $1000, although any multiple of $1000 ($5000 or $5m) can be used.

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Key Features of a Bond• Coupon interest rate:

– The coupon payments represent the periodic interest payments from the bond issuer to the bondholder. The annual coupon payment is calculated by multiplying the coupon rate by the bond's face value.

• Maturity:– A specified date at which the par value of a bond

must be repaid.

• Yield or Yield to maturity:– The rate required in the market on a bond.

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Zero Coupon Bonds• 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

• A bond bought at a price lower than its face value, with the face value repaid at the time of maturity.

• These bonds pays no annual interest. Thus, these bonds provide capital appreciation rather than interest income.

• Examples of zero-coupon bonds include U.S. Treasury bills, U.S. savings bonds.

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Bond Valuation

• Bonds are valued using time value of money concepts (PV of cash flows).

• Their coupon, or interest, payments are treated like an equal cash flow stream.

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PV =

CF

1+ r ... +

CF

1+r

1 n

1

2

21

CF

rn

.

0 1 2 n

r

CF1 CFnCF2Value

...

+ ++

Financial Assets Valuation

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PV of Cash Flows as Rates Change

• Bond Value = PV of coupons + PV of par

• Remember, as interest rates increase, the PVs decrease

• So, as interest rates increase, bond prices decrease and vice versa

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Valuing a Discount Bond with Annual Coupons

• Consider a bond with a coupon rate of 10% and coupons paid annually. The par value is $1,000 and the bond has 15 years to maturity. The yield to maturity is (rd) 10%. What is the value of the bond?

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15 10 100 1000N I/YR PV PMT FV

-1,000

INPUTS

OUTPUT

A bond with a coupon rate of 10% and coupons paid annually. The yield to maturity is (rd) 10%, the par value is $1,000 and the bond has 15 years to maturity. What is the value of the bond?

• The bond is selling at a price equal to its par value because rd equal the coupon rate.

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When rd rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount. Discount Bond is a bond that sells below its par value which occurs whenever the required interest is above the coupon rate.

15 13 100 1000N I/YR PV PMT FV

-806.13

INPUTS

OUTPUT

What would happen if expected inflation rose by 3%, causing r = 13%?

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What would happen if inflation fell, and rd declined to 7%?

If coupon rate > rd, price rises above par, and bond sells at a premium. Premium Bond is a bond that sells above its par value which occurs whenever the required interest is below the coupon rate.

15 7 100 1000N I/YR PV PMT FV

-1,273.24

INPUTS

OUTPUT

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Valuing a Premium Bond with Annual Coupons

• Suppose you are looking at a bond that has a 10% annual coupon and a face value of $1,000. There are 20 years to maturity and the yield to maturity is 8%. What is the price of this bond?– Using the formula:

• B = PV of annuity + PV of lump sum• B = $100[1 – 1/(1.08)20] / .08 + $1,000 / (1.08)20

• B = $981.81 + 214.55 = $1,196.36

– Using the calculator:• N = 20; I/Y = 8; PMT = 100; FV = 1,000• CPT PV = -1,196.36

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M

1,372

1,211

1,000

837

775

0 2 4 6 8 10 12 14

rd = 7%.

rd = 13%.

rd = 10%.

Bond Value ($) vs Years remaining to Maturity

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

• The value of a premium bond would decrease to $1,000.

• The value of a discount bond would increase to $1,000.

• A par bond stays at $1,000 if rd remains constant.

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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– Price below par = “discount” bond

• If YTM < coupon rate, then par value < bond price– Price above par = “premium” bond

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Computing Yield to Maturity • Yield to maturity is the rate implied by the

current bond price• YTM is the rate of return earned on a bond

held to maturity. Also called “promised yield.”

• If you have a financial calculator, enter N, PV, PMT and FV, remembering the sign convention (PMT and FV need to have the same sign; PV the opposite sign)

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Suppose you were offered a 10-years, 9% annual coupon, $1,000 par value bond at a price of $887. What rate of interest would you earn in your investment if you held the bond to maturity?

90 90 90

0 1 9 10rd=?

1,000PV1 . . .PV10

PVM

887 Find rd that “works”!

...

Computing Yield to Maturity

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10 -887 90 1000N I/YR PV PMT FV

10.91

V

INT

r

M

rB

dN

dN

1 11

... +INT

1+ r d

887 90

1

1,000

11 10 10

r rd d

+90

1+r d

+ + + +

++++

INPUTS

OUTPUT

...

Find rd

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Find YTM if price were $1,134.20.

Sells at a premium. Because coupon = 9% > rd = 7.08%, bond’s value > par.

10 -1134.2 90 1000N I/YR PV PMT FV

7.08

INPUTS

OUTPUT

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YTM with Annual Coupons

• Consider a bond with a 10% annual coupon rate, 15 years to maturity, and a par value of $1,000. The current price is $928.09. Will the yield be more or less than 10%? N = 15; PV = -928.09; FV = 1,000; PMT =

100 CPT I/Y = 11%

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Semiannual Bonds

1. Multiply years to maturity, N, by 2 to determine the number of semiannual periods. N = 2n.2. Divide nominal rate, rd, by 2 to determine the semiannual interest rate. I/Y= rd/2.3. Divide the annual coupon interest payment (INT) by 2 to determine the dollars of interest paid each 6 months. PMT = INT/2.

2n rd/2 OK INT/2 OK

N I/YR PV PMT FV

INPUTS

OUTPUT

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2(15) 5/2 100/230 2.5 50 1000N I/YR PV PMT FV

-1523.26

INPUTS

OUTPUT

Value of 15-year, 10% coupon, semiannual bond if rd = 5%.

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YTM with Semiannual Coupons• Suppose a bond with a 10% coupon

rate and semiannual coupons, has a face value of $1,000, 20 years to maturity and is selling for $1,197.93. Is the YTM more or less than 10%? What is the semiannual coupon payment? How many periods are there? N = 40; PV = -1,197.93; PMT = 50; FV =

1,000; CPT I/Y = 4% (Is this the YTM?) YTM = 4%*2 = 8%

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Current Yield

The current yield refers to the yield of the bond at the current moment. It does not reflect the total return over the life of the bond.

Current yield = Annual coupon pmtCurrent price

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Current Yield

Current yield =

= 0.1015 = 10.15%

$90 $887

Suppose you were offered a 10-years, 9% annual coupon, $1,000 par value bond at a price of $887. What is the current yield?

This measure looks at the current price of a bond instead of its face value. It represents the amount of cash income that a bond will generate in a given year.