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## Fixed Income

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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
- Mortgage Pass-Through Securities
- CMO and Stripped MBS
- ABS
- Bonds with Embedded Options
- Analysis of MBS
- Analysis of Convertible Bonds

Plan

- 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

- Issuer
- Time to maturity
- Coupon rate, type and frequency
- Linkage
- Embedded options
- Indentures
- Guarantees or collateral

Sources

- Fabozzi, “Bond Markets, Analysis and Strategies”, Prentice Hall.
- P. Wilmott, Derivatives, Wiley.
- Hull, White, Manuscript.

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

Basic terms

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

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 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

- Interest rate risk
- Default risk
- Reinvestment risk
- Currency risk
- Liquidity risk

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

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

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

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.

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

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

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 assumes that all coupon (and amortizing) payments will be invested at the same yield.

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

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.

The Yield to Maturity

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

Macaulay Duration

Definition of duration, assuming t=0.

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

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

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

ALM Duration

A similar problem with measuring yield

Key rate duration

- Principal component duration
- Partial duration

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

- 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

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

Lending

Repos and reverse repos

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).

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

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

Expectation theory

Market segmentation theory

Liquidity theory

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

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?

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