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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Yield = IRR
How do we know that there is a solution?
Accrued interest = interest due in full period*
(number of days since last coupon)/
(number of days in period between coupon payments)
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.
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.
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.
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%
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.
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?
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 of a fixed coupon bond y is given by
Definition of duration, assuming t=0.
What is the duration of a zero coupon bond?
A weighted sum of times to maturities of each coupon.
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
A similar problem with measuring yield
After tax yield = pretax yield (1- marginal tax rate)
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!
Off-the-run Treasury issues
Repos and reverse repos
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).
Term structure of instantaneous forward rates.
Market segmentation theory
Mathematical models: Ho-Lee, Vasichek, Hull-White, HJM, etc.