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Chapter 8: Valuation of Chapter 8: Valuation of Known Cash Flows: Known Cash Flows: BondsBonds

Copyright © Prentice Hall Inc. 1999. Author: Nick Bagley

ObjectiveValuation of fixed income securities

Explain why bond prices change

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Chapter 8 ContentsChapter 8 Contents

• 1 Using Present Value Formulas to Value Known 1 Using Present Value Formulas to Value Known FlowsFlows

• 2 The Basic Building Blocks: Pure Discount Bonds2 The Basic Building Blocks: Pure Discount Bonds

• 3 Coupon Bonds, Current Yield, and Yield-to-3 Coupon Bonds, Current Yield, and Yield-to-MaturityMaturity

• 4 Reading Bond Listings4 Reading Bond Listings

• 5 Why Yields for the same Maturity Differ5 Why Yields for the same Maturity Differ

• 6 The Behavior of Bond Prices Over Time6 The Behavior of Bond Prices Over Time

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Yield CurveYield Curve

• A typical yield curve:A typical yield curve:

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US Treasury Yiled Curve, Jan 97

4.50

5.00

5.50

6.00

6.50

7.00

7.50

0 5 10 15 20 25 30

Years to Maturity

An

nu

aliz

ed Y

ield

(%

)

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8.2 and 8.3 - Pure 8.2 and 8.3 - Pure Discount Bonds and Discount Bonds and Coupon BondsCoupon Bonds• A pure discount bond is a security that A pure discount bond is a security that

pays a specified single cash payment pays a specified single cash payment ((face valueface value or or par valuepar value) at a specified ) at a specified date called its maturity datedate called its maturity date

• We can always analyze any fixed We can always analyze any fixed income contract with known cash flows income contract with known cash flows into a sum of into a sum of $1-face value$1-face value pure pure discount bonds of different maturitiesdiscount bonds of different maturities

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Example (based on Prob. Example (based on Prob. 3)3)

• Let P1 and P2 be the prices of Let P1 and P2 be the prices of $1 face value $1 face value pure discount bondspure discount bonds maturing at time 1 and maturing at time 1 and 2 respectively. 2 respectively.

• Hence, P1 = $0.97 = $1/(1+iHence, P1 = $0.97 = $1/(1+i11))1 1 and and P2 = $0.90 P2 = $0.90 =$1/(1+i=$1/(1+i22))22

• where iwhere i11 and i and i22 are 1-year and 2-year yields are 1-year and 2-year yields on pure discount bondson pure discount bonds

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Example Continued - Example Continued -

• The price of a 2-year, 6% coupon The price of a 2-year, 6% coupon bond with a face value of $100 will bond with a face value of $100 will equal: equal:

• $6/(1+i$6/(1+i11))1 1 + $106/(1+i+ $106/(1+i22))2, 2, or or = 6 * P1 = 6 * P1 + 106 *P2 + 106 *P2 = 6 * $0.97 = 6 * $0.97 + 106 * $0.90 = + 106 * $0.90 = $101.22$101.22

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Example Continued -Example Continued -

• Hence, a 2-year, 6% coupon bond Hence, a 2-year, 6% coupon bond with a face value of $100 is with a face value of $100 is equivalent to the sum of six $1-face equivalent to the sum of six $1-face value pure discount bonds maturing value pure discount bonds maturing in one year and one hundred and six in one year and one hundred and six $1-face value pure discount bonds $1-face value pure discount bonds maturing in two yearsmaturing in two years

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Example ContinuedExample Continued

• What is the relationship between iWhat is the relationship between i11, , ii22, and the two-year yield to , and the two-year yield to maturity, imaturity, icc on the 6% coupon bond? on the 6% coupon bond?

• $101.22 = $6/(1+i$101.22 = $6/(1+i11))1 1 + $106/(1+i+ $106/(1+i22))2, 2,

= $6/(1+i = $6/(1+icc))1 1 + + $106/(1+i$106/(1+icc))2, 2,

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Example ContinuedExample Continued

• Recall that P1 = $0.97 = $1/(1+iRecall that P1 = $0.97 = $1/(1+i11))11, which , which implies iimplies i11 = 3.093% = 3.093%

• Also, P2 = $0.90 =$1 /(1+iAlso, P2 = $0.90 =$1 /(1+i22))2 2 which which implies iimplies i22 = 5.41% = 5.41%

• Should the yield to maturity iShould the yield to maturity icc on a 2- on a 2-year 6% coupon bond be some complicated year 6% coupon bond be some complicated average of 3.093% and 5.41%?average of 3.093% and 5.41%?

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Example CompletedExample Completed

• The yield to maturity on the 2-year The yield to maturity on the 2-year 6% coupon bond with face value of 6% coupon bond with face value of $100 can be calculated as:$100 can be calculated as:

• $101.22 = $6/(1+i$101.22 = $6/(1+icc))1 1 + $106/(1+i+ $106/(1+icc))2, 2,

• iicc = 5.34% = 5.34%

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Coupon Stripping Example Coupon Stripping Example (based on Prob. 4)(based on Prob. 4)

• Let P1 and P2 be the prices of Let P1 and P2 be the prices of $1 face value $1 face value pure discount bondspure discount bonds maturing at time 1 and maturing at time 1 and 2 respectively. 2 respectively.

• Hence, P1 = $0.93 = $1/(1+iHence, P1 = $0.93 = $1/(1+i11))1 1 and and P2 = ? P2 = ? =$1/(1+i=$1/(1+i22))22

• where iwhere i11 and i and i22 are 1-year and 2-year yields are 1-year and 2-year yields on pure discount bondson pure discount bonds

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Example Continued - Example Continued -

• The two cash flows of the 2-year, The two cash flows of the 2-year, 7% coupon bond (with a face value 7% coupon bond (with a face value of $1000) are:of $1000) are:

• $70 at the end of the first year, and $70 at the end of the first year, and $1070 at the end of the $1070 at the end of the second yearsecond year

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Example Continued - Example Continued -

• The price of the 2-year, 7% coupon bond with The price of the 2-year, 7% coupon bond with a face value of $1000 is given as $985.30, or a face value of $1000 is given as $985.30, or

• $985.30 = $70/(1+i$985.30 = $70/(1+i11))1 1 + $1070/(1+i+ $1070/(1+i22))2, 2, = 70* P1 = 70* P1 + 1070*P2 + 1070*P2 = 70* $0.93 = 70* $0.93 + 1070*P2 + 1070*P2 = $65.10 + 1070*P2 = $65.10 + 1070*P2

•

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Example Completed - Example Completed -

• If you unbundle the $70 cash flow If you unbundle the $70 cash flow and $1070 cash flow from the coupon and $1070 cash flow from the coupon bond and sell them separately, then:bond and sell them separately, then:

• Sale of first payment = 70* P1Sale of first payment = 70* P1 = = $65.10$65.10

• Sale of second payment = $1070*P2 Sale of second payment = $1070*P2 = $985.30 - $65.10 = $920.20 = $985.30 - $65.10 = $920.20

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Par, premium, and Par, premium, and Discount BondsDiscount Bonds

• A coupon bond with its current price A coupon bond with its current price equal to its face value (face value is equal to its face value (face value is the same as par value) is a par bondthe same as par value) is a par bond

• If it is trading below face value, it is a If it is trading below face value, it is a discount bonddiscount bond

• If it is trading above face value, it is a If it is trading above face value, it is a premium bondpremium bond

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The Relationship between The Relationship between Coupon Rate, Current Coupon Rate, Current Yield, and Yield-to-Yield, and Yield-to-MaturityMaturity• The The yield-to-maturityyield-to-maturity is the discount is the discount

rate that makes the present value of rate that makes the present value of the cash flows from the bond equal the cash flows from the bond equal to the current price of the bondto the current price of the bond

• Coupon Rate = Coupon/Face ValueCoupon Rate = Coupon/Face Value

• Current Yield = Coupon/Price Current Yield = Coupon/Price

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Bond Pricing PrinciplesBond Pricing Principles

• For Par Bonds, bond price = face value For Par Bonds, bond price = face value ytm = current yield = coupon rate ytm = current yield = coupon rate

• For Prem. Bonds, bond price > face value For Prem. Bonds, bond price > face value ytm < current yield < coupon rate ytm < current yield < coupon rate

• For Disc. Bonds, bond price < face value For Disc. Bonds, bond price < face value ytm > current yield > coupon rate ytm > current yield > coupon rate

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How to Remember How to Remember PrinciplesPrinciples

• Imagine that the bond was issued at Imagine that the bond was issued at parpar– the yield-to-maturity and current yield the yield-to-maturity and current yield

move in the opposite direction to price move in the opposite direction to price

– the coupon rate is unchangingthe coupon rate is unchanging

• This diagram may help:This diagram may help:

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•Yield Relationships

•0

•0.02

•0.04

•0.06

•0.08

•0.1

•0.12

•0.14

•0.16

•0.18

•0.2

•600.00 •800.00 •1000.00 •1200.00 •1400.00 •1600.00 •1800.00

•Price

• Yie

ld

•coupon_y•current_y•y_t_m

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8.4 Reading Bond Listings8.4 Reading Bond Listings

• There are traditions for reporting yields There are traditions for reporting yields and computing earned interest that and computing earned interest that need to be understood before tradingneed to be understood before trading– Coupon bonds are often quoted in terms of Coupon bonds are often quoted in terms of

the annual rate compounded semi-annuallythe annual rate compounded semi-annually

– T-bills are often quoted on a discount basisT-bills are often quoted on a discount basis• e.g., a 1 year T-bill has 365 days e.g., a 1 year T-bill has 365 days

outstanding, but a year has only 360 days…outstanding, but a year has only 360 days…(it gets nasty)(it gets nasty)

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Reading Bond ListingsReading Bond Listings

• Take care that the fractional part of Take care that the fractional part of a number is understooda number is understood– Is it 16ths, 32nds, 64ths, 100ths or Is it 16ths, 32nds, 64ths, 100ths or

some other convention?some other convention?

• Ask price: dealer’s selling priceAsk price: dealer’s selling price

• Bid price: dealer’s buying priceBid price: dealer’s buying price

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8.5 Why Yields for the 8.5 Why Yields for the same Maturity Differsame Maturity Differ

– The fundamental building block of bonds is The fundamental building block of bonds is the pure discount bond: Coupon bonds may the pure discount bond: Coupon bonds may be viewed as a portfolio of discount bondsbe viewed as a portfolio of discount bonds

– The rule of one price applies to bonds The rule of one price applies to bonds through pure discount bondsthrough pure discount bonds

– It is a mistake to assume that coupon bonds It is a mistake to assume that coupon bonds with the same life have the same yield--with the same life have the same yield--their coupon rates differ, leading to a their coupon rates differ, leading to a different % mix of discount bondsdifferent % mix of discount bonds

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8.6 The Behavior of Bond 8.6 The Behavior of Bond Prices Over TimePrices Over Time

• The The expectedexpected price of pure discount price of pure discount bonds rises exponentially to the face bonds rises exponentially to the face value with time, and the actual price value with time, and the actual price never exceeds parnever exceeds par

• Coupon bonds are more complex, and Coupon bonds are more complex, and their price may exceed their par value, their price may exceed their par value, but at maturity they reach their par but at maturity they reach their par valuevalue