fixed income

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Zvi Wiener

Fixed Income

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Plan

• Pricing of Bonds

• Measuring yield

• Bond Price Volatility

• Factors Affecting Yields and the Term Structure of IR

• Treasury and Agency Securities Markets

• Corporate Debt Instruments

• Municipals

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Plan

• Non-US Bonds

• Mortgage Loans

• Mortgage Pass-Through Securities

• CMO and Stripped MBS

• ABS

• Bonds with Embedded Options

• Analysis of MBS

• Analysis of Convertible Bonds

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Plan

• Active Bond Portfolio Management

• Indexing

• Liability Funding Strategies

• Bond Performance Measurement

• Interest Rate Futures

• Interest Rate Options

• Interest Rate Swaps, Caps, Floors

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Characteristics of a Bond• Issuer

• Time to maturity

• Coupon rate, type and frequency

• Linkage

• Embedded options

• Indentures

• Guarantees or collateral

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Sources

• Fabozzi, “Bond Markets, Analysis and

Strategies”, Prentice Hall.

• P. Wilmott, Derivatives, Wiley.

• Hull, White, Manuscript.

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Sectors• Treasury sector: bills, notes, bonds

• Agency sector: debentures (no collateral)

• Municipal sector: tax exempt

• Corporate sector: US and Yankee issues– bonds, notes, structured notes, CP– investment grade and noninvestment grade

• Asset-backed securities sector

• MBS sector

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Basic terms• Principal• Coupon, discount and premium bonds• Zero coupon bonds• Floating rate bonds• Inverse floaters• Deferred coupon bonds• Amortization schedule• Convertible bonds

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Basic Terms

• The Money Market Account

• LIBOR = London Interbank Offer Rate, see BBA Internet site

• FRA = Forward Rate Agreement

• Repos, reverse repos

• Strips = Separate Trading of Registeres Interest and Principal of Securities

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Basic Terms

• gilts (bonds issued by the UK government)

• JGB = Japanese Government Bonds

• Yen denominated issued by non-Japanese institutions are called Samurai bonds

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Major risks

• Interest rate risk

• Default risk

• Reinvestment risk

• Currency risk

• Liquidity risk

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Time Value of Money

• present value PV = CFt/(1+r)t

• Future value FV = CFt(1+r)t

• Net present value NPV = sum of all PV-PV 5 5 5 5 105

5

4

1 )1(

105

)1(

5

rrPV

tt

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TT

T

tt

t

r

C

r

CPV

)1()1(1

Term structure of interest rates

TT

TT

tt

t

t

r

C

r

CPV

)1()1(1

Yield = IRR

TT

T

tt

t

y

C

y

Cice

)1()1(Pr

1

How do we know that there is a solution?

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Price-Yield Relationship

• Price and yield (of a straight bond) move in opposite directions.

yield

price

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General pricing formula

11

1 )1()1()1()1(

nv

nn

ttv

t

rr

C

rr

CP

periodmonthssixindays

couponnextandsettlementbetweendaysv

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Accrued Interest

Accrued interest = interest due in full period*

(number of days since last coupon)/

(number of days in period between coupon payments)

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Day Count Convention

Actual/Actual - true number of days

30/360 - assume that there are 30 days in each month and 360 days in a year.

Actual/360

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Floater

The coupon rate of a floater is equal to a

reference rate plus a spread.

For example LIBOR + 50 bp.

Sometimes it has a cap or a floor.

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Inverse Floater

Is usually created from a fixed rate security.

Floater coupon = LIBOR + 1%

Inverse Floater coupon = 10% - LIBOR

Note that the sum is a fixed rate security.

If LIBOR>10% there is typically a floor.

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Price Quotes and Accrued Interest

Assume that the par value of a bond is $1,000.

Price quote is in % of par + accrued interest

the accrued interest must compensate the

seller for the next coupon.

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Annualizing Yield

Effective annual yield = (1+periodic rate)m-1 examples

Effective annual yield = 1.042-1=8.16%

Effective annual yield = 1.024-1=8.24%

price

ratecouponannualyieldcurrent

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Bond selling at Relationship

Par Coupon rate=current yield=YTM

Discount Coupon rate<current yield<YTM

Premium Coupon rate>current yield>YTM

Yield to call uses the first call as cashflow.

Yield of a portfolio is calculated with the total cashflow.

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YTM and Reinvestment Risk

• YTM assumes that all coupon (and

amortizing) payments will be invested at the

same yield.

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YTM and Reinvestment Risk• An investor has a 5 years horizon

Bond Coupon Maturity YTM

A 5% 3 9.0%

B 6% 20 8.6%

C 11% 15 9.2%

D 8% 5 8.0%

What is the best choice?

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Bond Price Volatility

Consider only IR as a risk factor

Longer TTM means higher volatility

Lower coupons means higher volatility

Floaters have a very low price volatility

Price is also affected by coupon payments

Price value of a Basis Point = price change resulting from a change of 0.01% in the yield.

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Duration and IR sensitivity

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Duration

nn y

M

y

C

y

C

y

CP

)1()1()1()1( 2

nn y

nM

y

nC

y

C

y

C

P

DurationMacaulay

)1()1()1(

2

)1(

112

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Duration

y

DurationMacaulayDurationModified

1

DurationModifiedPdy

dP 1

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DurationBond duration price impact of +1%

YTM

A 3 yr

B 1 yr

C 10 yr

D 20 yr

-3%-1%-10%-20%

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Measuring Price Change

errordydy

Pddy

dy

dPdP 2

2

2

)(2

1

P

errordy

ConvDdy

P

dP 2)(

2

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The Yield to Maturity

The yield to maturity of a fixed coupon bond y is given by

n

i

ytTi

iectp1

)()(

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Macaulay Duration

Definition of duration, assuming t=0.

p

ecTD

n

i

yTii

i

1

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Macaulay Duration

What is the duration of a zero coupon bond?

T

tt

tT

tt y

CFt

iceBondwtD

11 )1(Pr

1

A weighted sum of times to maturities of each coupon.

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Meaning of Duration

Dpecdy

d

dy

dp n

i

yTi

i

1

r

$

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Convexity

r

$

2

2

y

pC

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FRA Forward Rate AgreementA contract entered at t=0, where the parties (a lender

and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T].

Assuming continuous compounding we have

at time S: -K

at time T: KeR*(T-S)

Calculate the FRA rate R* which makes PV=0hint: it is equal to forward rate

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ALM Duration

• Does NOT work!• Wrong units of measurement• Division by a small number

r

A

ADA

1r

L

LDL

1

r

LA

LAD LA

)(1

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ALM Duration

A similar problem with measuring yield

r

P

VaR P 1

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Do not think of duration as a measure of time!

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• Key rate duration

• Principal component duration

• Partial duration

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Factors affecting Bond yields and TS

• Base interest rate - benchmark interest rate

• Risk Premium - spread

• Expected liquidity

• Market forces - Demand and supply

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Taxability of interest• qualified municipal bonds are exempts from

federal taxes.

After tax yield = pretax yield (1- marginal tax rate)

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Do not use yield curve to price bondsPeriod A B1-9 $6 $110 $106 $101They can not be priced by discounting

cashflow with the same yield because of different structure of CF.

Use spot rates (yield on zero-coupon Treasuries) instead!

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On-the-run Treasury issues

Off-the-run Treasury issues

Special securities

Lending

Repos and reverse repos

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Forward RatesBuy a two years bond

Buy a one year bond and then use the money to buy another bond (the price can be fixed today).

(1+r2)=(1+r1)(1+f12)

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Forward Rates

(1+r3)=(1+r1)(1+f13)= (1+r1)(1+f12)(1+f13)

Term structure of instantaneous forward rates.

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Determinants of the Term Structure

Expectation theory

Market segmentation theory

Liquidity theory

Mathematical models: Ho-Lee, Vasichek,

Hull-White, HJM, etc.

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• What is the duration of a floater?

• What is the duration of an inverse floater?

• How coupon payments affect duration?

• Why modified duration is better than

Macaulay duration?

• How duration can be used for hedging?

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