bonds guide and formulation

7/31/2019 Bonds Guide and Formulation

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1Copyright Michael R. Roberts

Bonds

Finance 100

Prof. Michael R. Roberts


Topic Overview

z Introduction to bonds and bond markets

z Zero coupon bonds Valuation

Yield-to-Maturity & Yield Curve

Spot Rates

Interest rate sensitivity DVO1

z Coupon bonds Valuation

Arbitrage

Bond Prices Over Time Yield Curve Revisited

Interest rate sensitivity Duration & Immunization

z Forward Rates


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2


What is a Bond and What are its Features?

z A bond is a security that obligates the issuer to make interest and principalpayments to the holder on specified dates.

Maturity (or term)

Face value (or par): Notional amount used to compute interest payments

Coupon rate: Determines the amount of each coupon payment, expressed as anAPR

z Bonds differ in several respects:

Repayment type

Issuer

Maturity

Security

Priority in case of default

Coupon Rate Face Value

Number of Coupon Payments per YearCoupon

=


Repayment Schemes

z Bonds with a balloon (or bullet) payment Pure discount or zero-coupon bonds

Pay no coupons prior to maturity.

Coupon bonds

Pay a stated coupon at periodic intervals prior to maturity.

Floating-rate bonds

Pay a variable coupon, reset periodically to a reference rate.

z Bonds without a balloon payment Perpetual bonds

Pay a stated coupon at periodic intervals.

Annuity or self-amortizing bonds

Pay a regular fixed amount each payment period.

Principal repaid over time rather than at maturity.


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Who Issues Bonds?

z US Government (Treasuries)

T-bills: 4,13,16-week maturity, zero coupon bonds

T-notes: 2,3,5,10 year, semi-annual coupon bonds

T-bonds: 20 & 30-year, semi-annual coupon bonds

TIPS: 5,10,20-year, semi-annual coupon bond, principal -adjusted

Strips: Wide-ranging maturity, zero-coupon bond, IB-structured

z Foreign Governments

z Municipalities

Maturities from one month to 40 years, semiannual coupons

Exempt from federal taxes (sometimes state and local as well).

Generally two types: Revenue bonds vs General Obligation bonds

Riskier than government bonds (e.g., Orange County)


Who Issues Bonds? (Cont.)

z Agencies: E.g. Government National Mortgage Association (Ginnie Mae),

Student Loan Marketing Association (Sallie Mae)

Most issues are mortgage-backed, pass-through securities.

Typically 30-year, monthly paying annuities mirroring underlyingsecurities

Prepayment risk.

z Corporations 4 types: notes, debentures, mortgage, asset-backed

~30 year maturity, semi-annual coupon set to price at par

Additional features/provisions: Callable: right to retire all bonds on (or after) call date, for call price

convertible bonds

putable bonds


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Bond Ratings

Moodys S&P Quality of Issue

Aaa AAA Highest quality. Very small risk of default.

Aa AA High quality. Small risk of default.

A A High-Medium quality. Strong attributes, but potentiallyvulnerable.

Baa BBB Medium quality. Currently adequate, but potentiallyunreliable.

Ba BB Some speculative element. Long-run prospectsquestionable.

B B Able to pay currently, but at risk of default in the future.

Caa CCC Poor quality. Clear danger of default.

Ca CC High speculative quality. May be in default.

C C Lowest rated. Poor prospects of repayment.

D - In default.


The US Bond Market FlowsAmount ($bil.). Source: Flow of Funds Data 2005-2007

132.3104.494.5Consumer Credit

1417.5

53.6

195

307.3

2005

1397.1

213.4

177.3

183.7

2006

1053.2

314.1

214.6

237.5

2007

Mortgages

Corporate

Municipal

U.S. Gov.

Debt

Instrument

Dollar volume of bonds traded daily is 10 times that of equity markets!

Outstanding investment-grade dollar denominated debt is about $8.3 trillion (e.g.,treasuries, agencies, corporate and MBSs


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Zero Coupon Bonds(a.k.a. Pure Discount Bonds)

z Notation Reminder: Vn=Bn = Market price of the bond in period n F= Face value R= Annual percentage rate m = compounding periods (annual m = 1, semiannualm = 2,) i = Effective periodic interest rate; i=R/m T= Maturity (in years) N= Number of compounding periods;N = T*m r= discount rate

z Two cash flows to buyer of a zero coupon bond (a.k.a. zero): -V0 at time 0 F at time T

z What is the price of a bond?

( ) ( )0 0 0 0or

1 1T N

F FV B V B

r i

= = = =

+ +


Zero Coupon BondExamples

z Value a 5 year, U.S. Treasury strip with face value of $1,000.

The APR is 7.5% with quarterly compounding?

Approach 1: UsingR (APR) and i (effective periodic rate)

Approach 2: Using r(EAR)

Approach 3: Using r(periodic discount rate)

?

?

?


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Yield to Maturity

z The Yield to Maturity (YTM) is the one discount rate that

sets the present value of the promised bond payments equal to

the current market price of the bond Doesnt this sound vaguely familiar

z Example: Zero-Coupon Bond

But this is just the IRR since

( )

1/

0

0

11

T

T

F FV r YTM y

Vr

= = = =

+

?

( )

1/

0

0

0 11

T

T

F FV IRR YTM y

VIRR

= = = =

+


Yields for Different Maturities

z Note: bonds of different maturities have different YTMs


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Spot Rates, Term Structure, Yield Curve

z A spot rate is the interest rate on a T-year loan that is to be made today

r1=5% indicates that the current rate for a one-year loan today is 5%.

r2=6% indicates that the current rate for a two-year loan today is 6%.

Etc.

Spot rate = YTM on default-free zero bonds.

z The term structure of interest rates is the series of spot rates r1, r2, r3,relating interest rates to investment term

z The yield curve is just a plot of the term structure: interest rates againstinvestment term (or maturity)

Zero-Coupon Yield Curve:built from zero-coupon bond yields (STRIPS)

Coupon Yield Curve: built from coupon bond yields (Treasuries)

Corporate Yield Curve: built from corporate bond yields of similar risk (i.e.,credit rating)


Term Structure of Risk-Free U.S. InterestRates, January 2004, 2005, and 2006


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Using the Yield Curve

z We should discount each cash flow by its appropriate discount

rate, governed by the timing of the cash flow

z Example: What is the present value of $100, 10 years from

today (Use the term structure from January 2004)

z Generally speaking, we must use the appropriate discount rate

for each cash flow:

1 2

211 2

1 (1 ) (1 ) (1 )== + + + =+ + + +"

NN N

N nnN n

C CC CPV r r r r

?


A Cautionary Note

z All of our valuation formulas (e.g., perpetuity, annuity)assume aflat term structure. I.e., there is only one discount rate for cash flows received at anypoint

in time

z Recall: Growing Annuity:

Growing Perpetuity:

r is implicitly assumed to be the same every period

1 11

( ) (1 )

N

gPV C

r g r

+ = +

( )

CPV

r g=


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Interest Rate SensitivityZero Coupon Bonds

z Why do zero-coupon bond prices change?...Interest rateschange!

z The price of a zero-coupon bond maturing in one year fromtoday with face value $100 and an APR of 10% is:

z Example:Now imagine that immediately after you buy thebond, the interest rate increase to 15%. What is the price of the

bond now

( )0

1N

FV

i=

+

( )0 1

100$90.91

1 0.10V = =

+

?


Characterizing the Price Rate Sensitivityof Zero Coupon Bonds

z Consider the following 1, 2 and 10-year zero-coupon bonds, all with

F=$1,000

APR of R=10%, compounded annually.

$0

$200

$400

$600

$800

$1,000

$1,200

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

1-Year

2-Year

10-Year

Note 4 things:

1. Bond prices areinversely relatedto IR

2. Fix the interestrate: Longer term

bonds are lessexpensive

3. Longer termbonds are moresensitive to IRchanges thanshort term bonds

4. The lower the IR,the more sensitivethe price.


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Quantifying the Interest Rate Sensitivityof Zero Coupon Bonds DV01

z Whats the natural thing to do? Compute the derivative

If we change the interest rate by a little (e.g., 0.0001 or 1 basis point) than

multiplying this number by the derivative should tell me how much the price

will change, all else equal (i.e., DV01 = Dollar Value of 1 Basis Point)

z Alternatively, we can just compute the prices at two different interest ratesand look at the difference:B0(i) B0 (i+0.0001)

( )

( )( )

( ) ( )( )

0

10

220

2

1

1 0 (Negative slope in )

1 1 0 (Convex function of )

N

N

N

FV

i

VFN i i

i

VFN N i i

i

+

+

=+

= +


Valuing Coupon BondsAmortization Bonds

z Consider an amortization bond maturing in two years with

semiannual payments of $1,000. Assume that the APR is 10%

with semiannual compounding

z How can we value this security?

1. Brute force discounting

2. Recognize the stream of cash flows as an annuity

( ) ( ) ( ) ( )0 2 3 4

1000 1000 1000 1000$3545.95

1 0.10 / 2 1 0.10 / 2 1 0.10 / 2 1 0.10 / 2V = + + + =

+ + + +

( )401000

1 (1 0.10 / 2) $3,545.950.10 / 2

V = + =

( ) ( ) ( ) ( )0 0.5 1 1.5 2

1000 1000 1000 1000$3545.95

1 0.1025 1 0.1025 1 0.1025 1 0.1025V = + + + =+ + + +

(i):

EAR (r):

or


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Replication

z Can we construct the same cash flows as our amortization

bond using other securities?


A First Look at Arbitrage

z What if the bond is selling for $3,500 in the market?


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Valuation of Straight Coupon BondExample

z What is the market price of a U.S. Treasury bond that has a

coupon rate of 9%, a face value of $1,000 and matures

exactly 10 years from today if the interest rate is 10%

compounded semiannually?

0 6 12 108 120Months

Cash Flows 45 45 45 1045

Timeline:

Present Value = Current Price = ?


Valuation of Straight Coupon BondGeneral Formula

z What is the market price of a bond that has an annual coupon

C, face value Fand matures exactly Tyears from today if the

required rate of return isR, with m-periodic compounding?

Coupon payment is: c = C/m

Effective periodic interest rate is: i = R/m

number of periodsN = Tm

Note the assumption of a flat term structure

[ ] [ ]

( )

++

+=

+=

N

N

iF

iic

ZeroAnnuityV

1)1(1

0


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Relationship Between Coupon Bond Pricesand Interest Rates

z Bond prices are inversely related to interest rates (or yields).

z A bond sells at par only if its interest rate equals the coupon

rate.

Most bonds set the coupon rate at origination to sell at par

z A bond sells at a premium if its coupon rate is above the

interest rate.

z A bond sells at a discount if its coupon rate is below the

interest rate.


The Effect of Time on Bond Prices


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YTM and BondPrice Fluctuations

Over Time


Yield to MaturityCoupon Bonds

z Recall: The Yield to Maturity is the one discount rate that sets the

present value of the promised bond payments equal to the current market

price of the bond

z Prices are usually given from trade prices

need to infer interest rate that has been used

This is not the annualizedyield, which equalsyield* = ( 1 +yield/ m)m-1

z Typically must solve using a computer

E.g., IRR function in excel or your calculator since:

( ) ( )NN myieldF

myieldmyield

cB

/1/1

11

/ ++

+=

( ) ( )NN myieldF

myieldmyield

cB

/1/1

11

/ ++

+=


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The Yield Curve Revisited

z Treasury Coupon-Paying Yield Curve

Often referred to as the yield curve

Same idea as the zero-coupon yield curve except we use the

yields from coupon paying bonds, as opposed to zero-

coupon bonds.

Treasury notes and bonds are semi-annual coupon paying bonds

We often use On-the-Run Bonds to estimate the yields

On-the-Run Bonds are the most recently issued bonds


Interest Rate SensitivityDuration

z The Duration of a security is the percent sensitivity of theprice to a small parallel shift in the level of interest rates.

A small uniform change dy across maturities might by 1 basis point.

Duration gives theproportionate decline in value associated with a risein yield

Negative sign is to cancel negative first derivative

z Alternatively, given a durationDB of a security with priceB, a

uniform change in the level of interest rates brings about achange in value of

1B

dBDuration D

B dy= =

Bd B D d y B=


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Duration of a Coupon Bond

z The mathematical expression forDuration is:

which we can rearrange

( )1 1

1

1 1 11 / (1 / )

Nn N

n

n

dBn c y m N F y m

B dy B m

=

= + + +

( )( )

( )N

( ) ( )

1

1

1

1

Time in Years "Weight" onuntil n payment n payment

1 / (1 / )1 /

1 /

th th

n NNn

n

N n

n

c y mn N F y mD y m

m B m B

PV c PV F n Ny mm B m B

=

=

+ += + +

= + +


Duration of a Coupon BondExample

z Compute the duration of a two-year, semi-annual, 10%coupon, par bond, with face value of $100.


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More on Duration

z Duration is a linear operator:D(B1 + B2) = D(B1) + D(B2)

The duration of a portfolio of securities is the value-weighted sum of

the individual security durations

DVO1 is also a linear operator

z Duration is a local measure

Based on slope of price-yield relation at a specific point

Based on a bond offixed maturitybut maturity declines over time

z Duration of a zero is

( )

11 /

ND y m

m

= +


Duration MatchingExample

z Bank of Philadelphia balance sheet (Figures in $billions, D=duration

assuming flat spot rate curve)

z Duration of liabilities =z The problem:

Increases in interest rates will decrease value of liabilities by more than assets

because of duration mismatch.

Liabilities & Shareholders EquityAssets

$25Total Liabilities (D = ?)25Total Assets (D = 1)

$5Shareholder Equity

$102-Year Notes (D = 1.77)

$10Commercial Paper (D = 0.48)

?


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Duration MatchingExample (Cont.)

z What is the change in assets value when interest rates changeuniformly

z What is the change in liability value when interest rateschange uniformly

z We want our assets and liabilities to experience similar valuechanges when interest rates change, so set these twoexpressions to be equal and solve forDL (DA=1.0):

?

?

?


Duration MatchingExample (Cont.)

z What fraction of the banks liabilities should be in CP and

Notes in order to get a liability duration of 1.25

z How much money should the bank hold in CP and Notes in

order to get a liability duration of 1.25

z How should the bank alter their liabilities to achieve thisstructure

?

?

?


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Forward Rates

z A forward rate is a rate agreed upon today, for a loan that isto be made in the future. (Not necessarily equal to the futurespot rate!) f2,1=7% indicates that we could contract today to borrow money at 7%

for one year, starting two years from today.

z Example: Consider the following term structure

r1=5.00%, r2=5.75%, r3=6.00% Consider two investment strategies:

1. Invest $100 for three years how much do we have?

2. Invest $100 for two years, and invest the proceeds at the one-year forwardrate, two periods hence how much do we have?

When are these two payoffs equal? (i.e. what is the implied forwardrate?)


Forward Rates

z Strategy #1: Invest $100 for three years how much do wehave

z Strategy #2: Invest $100 for two years and then reinvest theproceeds for another year at the one year forward rate, twoperiods hence how much do we have

z When are these two payoffs equal? (i.e. what is the impliedforward rate?)

?

?

?


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Arbitraging Forward RatesExample

z What if the prevailing forward rate in the market is 7%, as

opposed to what calculated in the previous slide?

z Step 1: Is there a mispricing and, if so, what is mispriced

z Step 2: Is the forward loan cheap or expensive

z Step 3: Given your answer to Step 2, what is the first step in

taking advantage of the mispricing

?

?

?


Arbitraging Forward RatesExample


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General Forward Rate Relation

z Forward rates are entirely determined by spot rates (and viceversa) by no arbitrage considerations.

z General Forward Rate Relation: (1+rn+t)n+t=(1+rn)

n(1+fn,t)t

z Think of this picture for intuition:

Time 0 1 2

(1+r2)2

(1+r3)3

(1+f2,1)

(1+f2,1)(1+f1,1)(1+r1)

(1+r1) (1+f1,2)2

3


Summary

z Bonds can be valued by discounting their future cash flows

z Bond prices change inversely with yield

z Price response of bond to interest rates depends on term to

maturity.

Works well for zero-coupon bond, but not for coupon bonds

z Measure interest rate sensitivity using duration.

z The term structure implies terms for future borrowing:

Forward rates

Compare with expected future spot rates

bonds guide and formulation

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