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					Interest Rates

1.    Types of rates
2.    Measuring interest rates
3.    Zero rates
4.    Bond pricing
5.    Determining Treasury zero rates
6.    Forward rates
7.    Forward rate agreements
8.    Duration
9.    Convexity
10.   Theories of the term structure of interest rates
Types of Rates(1/2)

(1) Treasury Rates
       A government to borrow in its own currency.
       An investor earns on Treasury bills and Treasury bonds.
       It is usually assumed “risk-free rates”.
       LIBOR is short for London Interbank Offered Rate.
       Derivatives traders regard LIBOR rates as a better
        indication of the “true” risk-free rate than Treasury rates.
       Large banks also quote LIBID rates ( London Interbank Bid
        Rate ) and is the rate at which they will accept deposits
        from other banks.
Types of Rates(2/2)

(3) Repo Rates
      An investment dealer who owns securities agrees to
       sell them to another company now and buy them
       back later at a slightly higher price.
      The difference between the price is the interest it
       earns. We called the interest rate is repo rate.
      The most common type of repo is an overnight repo.
Measuring interest rates(1/3)
   Suppose that an amount A is invested for n years at
    an interest rate of R per annum.
    If the rate is compounded m times per annum,
    the terminal value of the investment is
                    A (1+ R/m)                       (1)
• With continuous compounding, it can be shown
  that an amount A invested for n years at rate R
  grows to       Ae
    Compounding a sum of money at a continuously compounded rate R for
    n years involves multiplying it by e . Discounting it at a continuously
    compounded rate R for n years involves multiplying e-Rn.
Measuring interest rates(2/3)

• Suppose that Rc is a rate of interest with
 continuous compounding and Rm is the
 equivalent rate with compounding m times per
 annum. From the results in equations (1) and
 (2), we have
         Rc              m
        e = ( 1+ Rm/m )

           Rc = m ln(1+ Rm/m)                  (3)
                Rm=m (e             -1)              (4)
 These equations can be used to convert a rate with a compounding frequency
 of m times per annum to a continuously compounded rate and vice versa.
Measuring interest rates(3/3)

       Rm mn
A(1  )  Ae Rc n                       Rm
       m               Rc n  mn ln(1     )
           Rm mn
e  (1  )
 Rc n
                       Rc n
                             ln(1 
           m           mn            m
                Rm         Rc                Rm
Rc n  mn ln(1  )     e   m

                 m         Rc
                 Rm                  Rm
                       e   m
                                 1
 Rc  m ln(1  )(3)                 m
                 m                       Rc
                       Rm  m(e          m
                                                1)(4)
Zero rates

1. The n-year zero-coupon interest rate is the
   rate of interest earned on an investment
   that starts today and lasts for n years. All
   the interest and principal is realized at the
   end of n years.

2. Sometimes also referred to as the n-year
   spot rate.
Bond Pricing(1/4)
Example     (Table 2)
Suppose that a 2-year Treasury bond with a
principal of $100 provides coupons at the rate of 6%
per annum semiannually.
         Maturity         Zero Rate
         (years)        (% cont comp)
           0.5               5.0
            1.0               5.8
            1.5               6.4
            2.0               6.8
Bond Pricing(2/4)
(1)Theoretical price of the bond
 3e-0.05*0.5 + 3e-0.058*1.0 + 3e-0.064*1.5 + 103e-0.068*2.0=98.39

    Maturity          Zero Rate
    (years)         (% cont comp)
      0.5                5.0
        1.0                  5.8
        1.5                  6.4                Compounding
        2.0                  6.8                     Ae-Rn
Bond Pricing(3/4)
(2) Bond Yield
   A bond’s yield is the single discount rate that,
 when applied to all cash flows, gives a bond price
 equal to its market price. Assumed: y is the yield
 on the bond.

     -y*0.5          -y*1.0         -y*1.5            -y*2.0
3e            + 3e            +3e            + 103e            =98.39

                               y=6.76% (trial and error )
 Bond Pricing(4/4)
 (3) Par Yield
     The par yield for a certain bond maturity is the
   coupon rate that causes the bond price to equal its
   par value. Suppose that the coupon on a 2-year
   bond in our example is c per annum.


  The 2-year par yield is 6.87% per annum with
  semiannual compounding.
Determining Treasury zero rates(1/5)

    The most popular approach is known as the
     bootstrap method.
     Table 3 Date for bootstrap method
           Bond        Time to         Annual     Bond Cash
        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

       *Half the stated coupon is assumed to be paid every 6 months.
Determining Treasury zero rates(2/5)

    The bootstrap method
   The 3-month bond provides a return of 2.5 in 3 months on an
    initial investment of 97.5.

    With quarterly compounding, the 3-month zero rate is
    (4×2.5)/97.5=10.256%(per annum).

   The rate is expressed with continuous compounding, it becomes
    4 ln (1+0.10256/4)=0.10127

   Similarly the 6 month and 1 year rates are 10.469% and 10.536
    % with continuous compounding.
Determining Treasury zero rates(3/5)

The bootstrap method
   To calculate the 1.5 year zero rate, the
    payments are as follows:
    6 months : $4 ; 1 year:$4 ; 1.5 years :$104
   Suppose the 1.5-year zero rate is denoted by R.
         0.104690.5          0.105361.0             R1.5
    4e                   4e                   104e              96
   Similarly the two-year rate is 10.808%
Determining Treasury zero rates(4/5)

 Zero Curve Calculated from the Data
                                                   (Figure 1)
            Rate (%)

                                       10.68       10.808
                      10.469   10.53     1
   10                            6

                                               Maturity (yrs)
        0             0.5       1       1.5            2        2.5
Determining Treasury zero rates(5/5)
   A common assumption is that the zero curve is
    linear between the points determined using the
    bootstrap method.

   The zero curve is horizontal prior to the first point
    and horizontal beyond the last point.

   By using longer maturity bonds, the zero curve
    would be more accurately determined beyond 2
Forward rates(1/5)

   Definition: the rates of interest implied by
    current zero rates for periods of time in the
Forward rates(2/5)
Calculation of Forward Rates (Table 5)

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

        1            3.0
        2            4.0              5.0
        3            4.6              5.8
        4            5.0              6.2
        5            5.3              6.5
 Forward rates(3/5)
Calculation of Forward Rates (Table 5)
• The 4% per annum rate for 2 years mean that, in return
for an investment of $100 today, the receive
            100e0.04*2=$108.33 .
• Suppose that $100 is invested. A rate of 3% for the first
year and 5% for the second year gives at the end of the
second year.
 When interest rates are continuously compounded and in
 successive time periods are combined, the overall
 equivalent rate is simply the average rate during the
 whole period.
 Forward rates(4/5)
Formula of Forward Rates
   The forward rate for year 3 is the rate of interest that
    is implied by a 4% per annum 2-year zero rate and
    a 4.6% per annum 3-year zero rate.
   If R1and R2 are the zero rates for maturities T1 and
    T2 ,respectively, and RF is the forward interest rate for
    the period of time between T1 and T2 ,then
                         R2T2  R1T1
                  RF 
                           T2  T1                (5)

             RF    R2  ( R 2  R1 )
                                      T2  T1      (6)
Forward rates(5/5)
Instantaneous Forward Rate

   The instantaneous forward rate for a maturity
    T is the forward rate that applies for a very
    short time period starting at T. It is
             RF  R  T
    where R is the T-year rate
Forward rate agreements(1/3)

    Definition: FRA is an over-the-counter
     agreement that a certain interest rate will
     apply to either borrowing or lending a certain
     principal during a specified future period of
Forward rate agreements(2/3)

Valuation Formulas Define

 Rk: The rate of interest agreed to in the FRA
 RF: The forward LIBOR interest rate for the
     period between times T1 and T2
     calculated today.
 RM: The actual LIBOR interest rate observed
    in the market at times T1 for the period
    between times T1 and T2
 L: The principal underlying the contract.
Forward rate agreements(3/3)

Valuation Formulas
     Value of FRA where a fixed rate RK will be
      received on a principal L between times T1
      and T2 is          V  L( RK  RF )(T2  T1 )e R T
                                                   2 2

     Value of FRA where a fixed rate is paid is
         V  L( RF  RK )(T2  T1 )e R2T2
     RF is the forward rate for the period and R2
      is the zero rate for maturity T2

   Definition: A measure of how long on average
    the holder of the bond has to wait before
    receiving cash payment.
   A zero-coupon bond that lasts n years has a
    duration of n years.
   A coupon-bearing bond lasting n years has a
    duration of less than n years.

• Duration of a bond that provides cash flow c i at time t i is
              ci e      yti
    D   ti                                      n

        i 1  B                 (12)        B   Ci e  yti
                                                   i 1

where B is its price and y is its yield (continuously
• This leads to
                            Dy

Modified Duration
 • The preceding analysis is based on the
   assumption that y is expressed with
   continuous compounding. If y is expressed
   with a compounding frequency of m times
   per year, then              BDy
                  B  
                             1 y / m
 A variable D*, defined by
       D*  
              1 y / m

   The duration relationship applies only to
    small changes in yields.
   It shows the relationship between the
    percentage change in value and change in
    yield for two bond portfolios having the same
    duration. ( Figure 2)
Convexity(2/2)               (Figure 2)



               Y2       Y0         Y1
For large yield changes, the portfolio X has more
curvature in its relationship with yields than portfolio Y.
 Theories of the term structure of
 interest rates
•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 rates are the rates paid by a
    government on borrowings in its own currency.

   LIBOR rates are short-term lending rates
    offered by banks in the interbank market.

   The n-year zero or spot rate is the applicable to
    an investment lasting for n years when all of the
    return is realized at the end.

   Forward rates are the rates applicable to future
    periods of time implied by today’s zero rates.

   FRA is an over-the-counter agreement that a
    certain interest rate will apply to either borrowing
    or lending a certain principal at LIBOR during a
    specified future period of time.

   Duration measures the sensitivity of the value of
    a bond portfolio to a small parallel shift in the
    zero-coupon yield curve.

   Liquidity preference theory can be used to
    explain the interest rate term structures that are
    observed in practice.

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