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Interest Rate

   Chapter 5

            Types of Rates
• Treasury rates
• LIBOR rates
• Repo rates
              Zero Rates
A zero rate (or spot rate), for maturity T, is
the rate of interest earned on an investment
that provides a payoff only at time T

               Bond Pricing
• To calculate the cash price of a bond we discount
  each cash flow at the appropriate zero rate
• In our example, the theoretical price of a two-year
  bond providing a 6% coupon semiannually is
                 Bond Yield
• The bond yield is the discount rate that makes the
  present value of the cash flows on the bond equal
  to the market price of the bond
• Suppose that the market price of the bond in our
  example equals its theoretical price of 98.39
• The bond yield is given by solving

  to get y=0.0676 or 6.76%.
                  Par Yield
• The par yield for a certain maturity is the coupon
  rate that causes the bond price to equal its face
• In our example we solve
The Bootstrap Method                                5.7
How do we calculate the zero rates?

       Bond      Time to     Annual       Bond
     Principal   Maturity   Coupon        Price
     (dollars)   (years)    (dollars)   (dollars)

       100         0.25         0         97.5

       100         0.50         0         94.9

       100         1.00         0         90.0

       100         1.50         8         96.0

       100         2.00        12        101.6
Zero Curve Calculated from the
  Rate (%)

                               10.681       10.808
             10.469   10.536

                                        Maturity (yrs)
Day Count Conventions in the US

Day count conventions matter so that you can correctly
calculate the interest payments over a period of time.

When you buy or sell interest-bearing securities, you need
to adjust the price for accrued interest.

          Forward Rates

The forward rate is the future zero rate
implied by today’s term structure of interest
Calculation of Forward Rates

                 Zero Rate for     Forward Rate
             an n -year Investment for n th Year
 Year (n )      (% per annum)     (% per annum)

    1                10.0
    2                10.5              ?
    3                10.8              ?
    4                11.0              ?
    5                11.1              ?
                              Forward Rates
                          (Comment on Textbook)
•   For continuously compounded interest rates:

•   For discrete compounding (such as money market rates):

    – where L (S) denotes longer-term (shorter-term) rate, TL (TS ) denotes the number of days
      in the longer-term (shorter-term) period, and TF denotes the number of days in the
      forward period
    – You will use the discrete compounding method when you calculate
      forward rates for money market instruments like LIBOR term deposits

     Forward Rate Agreement
• A forward rate agreement (FRA) is an agreement
  that a certain rate will apply to a certain principal
  during a certain future time period
• An FRA is equivalent to an agreement where
  interest at a predetermined rate, RK is exchanged
  for interest at the market rate
• An FRA can be valued by assuming that the
  forward interest rate is certain to be realized

Theories of the Term Structure
• Expectations Theory: forward rates equal
  expected future zero rates
• Market Segmentation: short, medium and
  long rates determined independently of each
• Liquidity Preference Theory: forward rates
  higher than expected future zero rates
               Treasury Bill Futures
          (Chicago Mercantile Exchange)
• The Contract
  –Physical delivery of $1 million in T-bills maturing in 90, 91, or 92 days
  from the first contract delivery date
  –Tick size = $25 per basis point on a discounted basis
• Quotation and Pricing in the Cash Market
  –Cash T-bills are discount securities
  –The annualized discount yield, DY, on a T-bill maturing in n days with
  the current cash price, Y
                           DY = [(360/n)]x(100 – Y)
  The cash (or invoice) price for a T-bill with n days to maturity (per
  $100) is:
                            Y = 100 – DY*(n/360)
                    where DY is quoted as 5.0%, for example
  US Treasury Bills: Futures Pricing
• T-Bill Futures quoted price, Z:
           Z = (100 – corresponding Treasury bill quoted price )
                        Z = 100 - (360/n)*(100 - Y)

• The futures cash invoice price =
               10,000[100 - (n/360)*(100 - Z)]
   – the futures price is a function of the spot cash price of
     Treasury Bills.

       Eurodollar Futures Contracts
• Among the most liquid and actively traded futures contracts in
  the world
• Eurodollar: a dollar deposit in a U.S. or foreign bank outside
  the U.S.
• The contract:
   – Underlying: 90-day hypothetical Eurodollar CD
   – Contract Size: $1 mm face value
   – Minimum Price Move: (Tick): $12.50
       • One basis point is worth $25
       • One tick, worth $12.50, thus is equal to one-half of basis point
   – Contract Months: March, June, Sep, Dec, serial months, spot
   – Maturities: Available through 10 years

       Eurodollar Futures Quotation
• The settlement price when a contract matures is
                          F = 100 - R
                    where R is the 90-day LIBOR
   – R is annualized relative to a 360-day money market year
   – R is expressed as a percentage to two decimals
       • For example: R=8.25% => F = 91.75
• Because Eurodollars CDs are bilateral, the quotes are bank-
   – The CME surveys several London banks each day to determine
     their quoted LIBOR rates
   – The two highest and the two lowest quotes are discarded
   – R is the average of the remaining survey quotes
           Eurodollar Futures Pricing
•   In general, Eurodollar Futures are priced prior to
    expectation based on risk-neutral expectations model
    of interest rates. Thus, the The quoted Eurodollar
    futures price gives us a type of forward rate.
                         Ft,T = 100 – ET[RT]
      where RT is 90-day rate that will be offered on 90-day Eurodollar
                                deposit at time T
•   For short maturities the Eurodollar futures interest rate
    can be assumed to be the same as the corresponding
    forward interest rates. For longer terms, a convexity
    adjustment is often made.
             T-Bonds futures (CBOT)
• Quotation and Pricing in the Cash Market
        Cash price = Quoted price + Accrued Interest
• T-Bond contract allows delivery of any bond with at least 15
  years to maturity or first call. It is based on 8%, 15 year bond.
  $100,000 nominal face value
    – Party with the short position chooses bond
    – About 30 deliverable bonds
    – Beginning with March 2000 contracts based on 6%
• Value of a tick: Minimum price change is 1/32nd or $31.25 per
• Conversion Factors:
    – Quoted price is based on the assumption of a 6% hypothetical coupon
    – Approximate price, in decimal form, at which the bond would (as of the
      first delivery day of the month) yield 6% to maturity (rounded to whole
      quarters) or to first call if callable.
          T-bonds & T-notes futures:
• The party with the short position will choose which of the
  available deliverable bonds is the “cheapest -to-deliver”
   – bond for which the following is the least:
    Quoted spot price - (Quoted futures price x Conversion factor)

• Factors determining the cheapest-to-deliver
   – Yields > 6%, conversion factors favor delivery of low-coupon,
     long-maturity bonds
   – Yields < 6%, conversion factors favor delivery of high-coupon,
     short-maturity bonds
   – Upward (downward) sloping yield curve favors delivery of long
     (short) maturity bonds
                    U.S. Treasury Bonds
                      Futures Pricing
• Treasury Bonds:
                 cash price = quoted price + accrued interest
• Treasury Bond Futures (per $1,000):
  Full Invoice price = (quoted futures price)×(conversion fct.) + accrued int.
• Quoted Futures Price
   – Over the life of the futures contract, the bond that
     is the “cheapest” to deliver changes
   – The quoted futures price is based on the bond that
     is expected to be the cheapest to deliver
   – Imbedded options in the contract
• Duration of an instrument which provides cash flow
  c i at time t i is

• where B is the bond price and y is the continuously
  compounded bond yield
• This leads to
       Duration Continued
• When the yield y is expressed with
  compounding m times per year

• The expression

 is referred to as the “modified duration”

         Duration Matching
• This involves hedging against interest rate
  risk by matching the durations of assets and
• It provides protection against small parallel
  shifts in the zero curve

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