Chapter 8. Valuation of Known Cash Flows: Bonds

1

FinanceFinance School of Management School of Management

ObjectiveExplain the principles of bond pricingUnderstand the features that affect

bond prices

Chapter 8. Valuation of Known Chapter 8. Valuation of Known Cash Flows: BondsCash Flows: Bonds

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Chapter 8 ContentsChapter 8 Contents

Using Present Value Formulas to Value Known Flows The Basic Building Blocks: Pure Discount Bonds Coupon Bonds, Current Yield, and Yield-To-Maturity Reading Bond Listings Why Yields for the Same Maturity Differ The Behavior of Bond Prices over Time

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Valuation and Fixed-Income SecuritiesValuation and Fixed-Income Securities

Essence of valuation process– To estimate an asset’s market value using information about

the prices of comparable assets. Valuation models

– A quantitative method used to infer an asset’s value from market information about the prices of other assets and market interest rates.

Fixed-income securities and other contracts promising a stream of known future cash payments – Bonds– Mortgages – Pension annuities

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Reasons for Valuing Fixed-Income Reasons for Valuing Fixed-Income SecuritiesSecurities

To have an agreed-upon valuation procedure in setting the terms of the contracts at the outset.

To revaluate the securities when they are sold before maturity.

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Using Present Value Formulas to Value Using Present Value Formulas to Value Known Cash FlowsKnown Cash Flows

A fixed-income security that promises to pay $100 each year for the next three years.

The appropriate discount rate is 6% per year.

An hour after you buy the security, the risk-free interest rate rises from 6% to 7% per year.

i

iPMTPV

n

)1(1

n i PV FV PMT Result 3 6% ? 0 100 PV=267.30

n i PV FV PMT Result 3 7% ? 0 100 PV=262.43

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Bond Prices Fall as the Interest Bond Prices Fall as the Interest Rates RiseRates Rise

Write the PV of the fixed-income security as the sum terms

n

n

n

n

n

j

j

j

ipmt

ipmt

ipmt

ipmt

ipmtPV

1

1*

1

1*...

1

1*

1

1*

1

1*

1

1

2

2

1

1

1

7


The Difficulty of Valuation of The Difficulty of Valuation of Known Cash FlowsKnown Cash Flows

We do not know usually which discount rate to use in the present value formula.

Is it correct to use the interest rate corresponding to a three-year maturity in valuing the three-year annuity in the previous example?

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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

(%

)

9


The Basic Building Blocks: Pure The Basic Building Blocks: Pure Discount BondsDiscount Bonds

The difficulties of finding equivalent fixed-income securities, or comparables and making adjustments for differences.

Any fixed-income security can be decomposed into a series of known payments at different time points in the future.

Pure discount bonds (zero-coupon bonds): Promising a single payment of cash at the maturity date (in the future).

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Pure Discount BondsPure Discount Bonds

The pure discount bond is an example of the present value of a lump sum equation we analyzed in Chapter 4.

Solving this, the yield-to-maturity on a pure discount bond is given by the relationship:

11

1

nn

P

FiiPF

11



11

1

nn

P

FiiPF

In this equation,– P is the present value or price of the bond– F is the face or future value – n is the investment period– i is the yield-to-maturity

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%60.61880

10001

2

11

n

P

Fi

n i PV FV PMT Result 2 ? 880 1000 0 i=6.60%

A two-year pure discount bond with a face value of $1,000 and a price of $880

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Pricing a Coupon BondPricing a Coupon Bond

A 3-year bond with a face value of $1,000 that makes annual coupon payments at a coupon rate 10%

Prices of pure discount bonds

Maturity Prices per $1 ofFace Value

1 year 0.962 years 0.893 years 0.81

Prices of Pure Discount Bonds

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First Solution MethodFirst Solution Method

00.1076$

100100081.010089.010096.0

P

P

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Second Solution MethodSecond Solution Method

91.075,1$0728.1

1001000

0600.1

100

0417.1

100

%28.71810

000,1

%00.61890

000,1

%17.41960

000,1

32

3

1

3,0

2

1

2,0

1

1

1,0

P

P

i

i

i

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The YTM of the Coupon BondThe YTM of the Coupon Bond

It would be a mistake to discount all three cash flows using the same three-year yield of 7.28%.

The single discount rate that we can use to discount all three cash flows is the yield-to-maturity (YTM).

However, can we get it?

n i PV FV PMT Result

3 7.28% ? 1000 100 PV=$1071


3 ? 1076 1000 100 i=7.10%

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You would like to create a 2-year synthetic zero-coupon bond.

Assume you are aware of the following information: – 1-year zero-coupon bonds are trading for $0.93 per

dollar of face value, and– 2-year 7% coupon bonds (annual payments) are selling

at $985.30 (Face value = $1,000). Assume you can purchase the 2-year coupon bond and

unbundle the two cash flows and sell them. – You would receive .93×$70 = $65.10 from the sale of

the first payment.– To break even, you would need to receive $985.30-

$65.10 = $920.20 from the sale of the 2-year strip.

Coupon StrippingCoupon Stripping

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TThe Principle of STRIPshe Principle of STRIPs

Compensation for reinvestment risk

Single term

Low valueNo compensation for

reinvestment risk

Multiple terms

Secondary market

High value

Output Security

n-year coupon treasure bond

6-month zeros

1-year zeros

n-year zeros

Input Security

Decompose the CFs

Investment Bank

Term Intermediation

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In 1982, Merrill Lynch: TIGRs—Treasury Investment Growth Receipts.

Follow up: Salomon Brother’s Certificates of Accrual on Treasury Security (CATs) 、 Lehman Investment Opportunity Notes (LIONs) —‘Animals’.

In 1984, American government: STRIPS—Separate Trading of Registered Interest and Principal of Securities.

In 1985, the outstanding face value is over 100 billion dollars.

The Development of STRIPsThe Development of STRIPs

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Coupon RateCoupon Rate

Coupon rate is the interest rate applied to the face value to compute the coupon payment.

– A bond with a face value of $1,000 and a coupon rate of 10%

– An annuity component of $100 per year and a “balloon” or “bullet” payment at maturity

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Current Yield and Yield-to-maturityCurrent Yield and Yield-to-maturity

Current yield is the annual coupon divided by the bond’s price.

Yield-to-maturity is the discount rate that makes the present value of a bond’s stream of promised cash payments equal to its price.

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Example 1Example 1 A 20-year-maturity

bond with a face value of $1,000 and a coupon rate of 10% was originally issued 19 years ago.

At that time, the yield curve was flat at 10% per year.

Now the interest rate on one-year bonds is 5% per year.

Its market price will now be

Its current yield is

Its yield-to-maturity is

62.1047$05.1

1100P

%55.962.1047

100yieldCurrent

%5162.1047

1100 maturitytoYield

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Example 2Example 2

A bond with a face value of $1,000 and a coupon rate of 4% will mature in two years.

Its market price is $950.

Its current yield is

Its yield-to-maturity

%21.4950

40yieldCurrent


2 ? 950 1000 40 i=6.76%

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Bonds Trading at ParBonds Trading at Par

Bond Pricing Principle #1: (Par Bonds)– If a bond’s price equals its face value, then its

yield-to-maturity = current yield = coupon rate.

Proof:

Fi

pmtP

ii

pmt

iP

FPi

Fii

pmtP

nn

nn

1

11

1

11

&1

1

1

11

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Bonds Trading at Premium or DiscountBonds Trading at Premium or Discount

Bond Pricing Principle #2: (Premium Bonds)– If a bond has a price higher than its face value, then

its yield-to-maturity < current yield < coupon rate.

Bond Pricing Principle #3: (Discount Bonds)– If a bond has a price lower than its face value, then

its yield-to-maturity > current yield > coupon rate.

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Proof:Proof:

11

111

11

0&0&0&1

1

1

11

n

n

n

nn

iP

PFP

pmt

FiP

ipmti

pmtnii

Fii

pmtP

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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_yy_t_m

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Beware of “High-Yield” US Treasure Beware of “High-Yield” US Treasure Bond FundsBond Funds

You have $10,000 to invest for one year. You are deciding between: – Putting your money in a one-year government-insured

bank CD offering an interest rate of 5%;

– Investing in the shares of a U.S. Treasure bond fund that holds one-year bonds with a coupon rate of 8%.

The bonds are selling at a premium: you must pay $10,285.71 for $10,000 of face value.

The fund advertises a yield of 7.78%. The fund charges a 1% annual fee for their services.

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Beware of “High-Yield” US Treasure Beware of “High-Yield” US Treasure Bond FundsBond Funds

%5171.285,10$

000,10$800$

maturitytoYield

%78.771.285,10$

800$yieldCurrent

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The Effect of Coupon RateThe Effect of Coupon Rate

Two different two-year coupon bonds—one with a coupon rate of 5% and the other with a coupon rate of 10%.

The current market prices and yields of one- and two-year pure discount bonds:

Maturity Prices per $1 ofFace Value

Yield(per year )

1 year $0.961538 4%2 years $0.889996 6%

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The Effect of Coupon RateThe Effect of Coupon Rate The market prices of the two coupon bonds should be

– For the 5%-coupon bond:

– For the 10%-coupon bond:

The yields to maturity on the coupon bonds should be– For the 5%-coupon bond, the YTM is 5.9500%– For the 10%-coupon bond, the YTM is 5.9064%

When the yield curve is not flat, bonds of the same maturity with different coupon rates have different yields to maturity.

57.982$050,1$889996.50$961538.

15.075,1$100,1$889996.100$961538.

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Other Effects on Bond YieldsOther Effects on Bond Yields

Default risk Taxes Callability Convertibility

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The Effect of the Passage of TimeThe Effect of the Passage of Time A 20-year pure discount bond with a face value of $1,000

and a constant yield of 6% should be priced at


20 6% ? 1000 0 PV=$311.80 After one year goes by, its price should be


19 6% ? 1000 0 PV=$330.51 If the yield curve were flat and interest rates did not

change, any default-free discount bond’s price would rise with the passage of time, and any premium bond’s price would fall.

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Movement of a Pure discount Bond’s Price over Time

300

400

500

600

700

800

900

1, 000

0 2 4 6 8 10 12 14 16 18 20Years Elapsed since Date of Issue

Pric

e

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Interest-Rate RiskInterest-Rate Risk

The concept The sensitivity of bond prices to interest rates

− The prices of 30-year 8% coupon par bond would fall by roughly 10% if the level of interest rates were to rise from 8% to 9%.

− The prices of 30-year pure discount bond would fall by roughly 23% if the level of interest rates were to rise from 8% to 9%.

Why?

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Sensitivity of Bond Price to Interest Rates

0

0. 5

1

1. 5

2

2. 5

3

3. 5

4

4 5 6 7 8 9 10 11 12

Interest Rate in % per Year

Pric

e R

atio

ZerosPar Bonds

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T

t

twtD1

)(PV

kCF

PV

CFPVtw

ttt

)1()()(

Dkkd

PVdPV

1)1(

T

t

tt kCFPV

1

)1(

)1(* kDD

Duration and Modified DurationDuration and Modified Duration

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The Duration of A Bond PortfolioThe Duration of A Bond Portfolio

1PV 2PV21

11 PVPV

PVw

21

22 PVPV

PVw

1

1

1

)1(

k

kd

2

2

1

)1(

k

kd

T

t

tt DwDwPVPV

FPVFPVtD

12211

21

21 )()(

)1( 1kd )1( 2kd *22

*11

* DwDwD

39


An Illustration An Illustration An pension fund is selling a new insurance policy (pension

annuity), which promises an annual payment of $100 for 15 years.

At the discount rate of 10%, the PV of the liability is $760.61, and the modified duration is 5.708.

Date CF DCF Weight Multiplication 1 100 90.909 0.120 0.120 2 100 82.645 0.109 0.217 3 100 75.131 0.099 0.296 ┇ ┇ ┇ ┇ ┇ 15 100 23.939 0.031 0.472

Total 760.608 1.000 6.279Modified Duration = 6.279/1.1 = 5.708

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The pension fund will invest the $760.61, requiring at least a return of 10%.

There are two instruments: A 30-year treasure bond paying an interest rate of 12% and selling at par, and a 6-month treasure bill offering an interest rate of 8% per year. The duration for the two securities are 8.080 and 0.481 respectively.

Consider investing in a portfolio of the two treasure securities:

121

**22

*11

ww

DDwDw L

)38.237($%21.31

)23.523($%79.68

2

1

w

w

The rate of return on the portfolio = 10.75%. When the interest rate increases by 0.1%, the change of the liability

value = -4.32, and the value of the 30-year bond and 6-month bill will change by -4.2 and -0.12 respectively, and the total of both changes accounts for -4.32.

Continued……

Documents

Chapter 8. Valuation of Known Cash Flows: Bonds