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Fixed Income. Zvi Wiener. 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. Plan. Non-US Bonds Mortgage Loans

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fixed income

Fixed Income

Zvi Wiener

  • 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
  • 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
  • Active Bond Portfolio Management
  • Indexing
  • Liability Funding Strategies
  • Bond Performance Measurement
  • Interest Rate Futures
  • Interest Rate Options
  • Interest Rate Swaps, Caps, Floors
characteristics of a bond
Characteristics of a Bond
  • Issuer
  • Time to maturity
  • Coupon rate, type and frequency
  • Linkage
  • Embedded options
  • Indentures
  • Guarantees or collateral
  • Fabozzi, “Bond Markets, Analysis and Strategies”, Prentice Hall.
  • P. Wilmott, Derivatives, Wiley.
  • Hull, White, Manuscript.
  • 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
basic terms
Basic terms
  • Principal
  • Coupon, discount and premium bonds
  • Zero coupon bonds
  • Floating rate bonds
  • Inverse floaters
  • Deferred coupon bonds
  • Amortization schedule
  • Convertible bonds
basic terms1
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
basic terms2
Basic Terms
  • gilts (bonds issued by the UK government)
  • JGB = Japanese Government Bonds
  • Yen denominated issued by non-Japanese institutions are called Samurai bonds
major risks
Major risks
  • Interest rate risk
  • Default risk
  • Reinvestment risk
  • Currency risk
  • Liquidity risk
time value of money

-PV 5 5 5 5 105

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

Term structure of interest rates

Yield = IRR

How do we know that there is a solution?

price yield relationship
Price-Yield Relationship
  • Price and yield (of a straight bond) move in opposite directions.



accrued interest
Accrued Interest

Accrued interest = interest due in full period*

(number of days since last coupon)/

(number of days in period between coupon payments)

day count convention
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.



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.

inverse floater
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.

price quotes and accrued interest
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.

annualizing yield
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%

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.

ytm and reinvestment risk
YTM and Reinvestment Risk
  • YTM assumes that all coupon (and amortizing) payments will be invested at the same yield.
ytm and reinvestment risk1
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?

bond price volatility
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.


Bond duration price impact of +1% YTM

A 3 yr

B 1 yr

C 10 yr

D 20 yr





the yield to maturity
The Yield to Maturity

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

macaulay duration
Macaulay Duration

Definition of duration, assuming t=0.

macaulay duration1
Macaulay Duration

What is the duration of a zero coupon bond?

A weighted sum of times to maturities of each coupon.

fra forward rate agreement
FRA Forward Rate Agreement

A 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=0

hint: it is equal to forward rate

alm duration
ALM Duration
  • Does NOT work!
  • Wrong units of measurement
  • Division by a small number
alm duration1
ALM Duration

A similar problem with measuring yield

Key rate duration
  • Principal component duration
  • Partial duration
factors affecting bond yields and ts
Factors affecting Bond yields and TS
  • Base interest rate - benchmark interest rate
  • Risk Premium - spread
  • Expected liquidity
  • Market forces - Demand and supply
taxability of interest
Taxability of interest
  • qualified municipal bonds are exempts from federal taxes.

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

do not use yield curve to price bonds
Do not use yield curve to price bonds

Period A B

1-9 $6 $1

10 $106 $101

They 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!

On-the-run Treasury issues

Off-the-run Treasury issues

Special securities


Repos and reverse repos

forward rates
Forward Rates

Buy a two years bond

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


forward rates1
Forward Rates

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

Term structure of instantaneous forward rates.

determinants of the term structure
Determinants of the Term Structure

Expectation theory

Market segmentation theory

Liquidity theory

Mathematical models: Ho-Lee, Vasichek, Hull-White, HJM, etc.

home assignment
Home Assignment
  • 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?