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LOS 34.b: Describe the forward pricing and forward
    rate models and calculate forward and spot prices                                         READING 34: THE TERM STRUCTURE AND
    and rates using those models.                                                                              INTEREST RATE DYNAMICS
                                                                                        MODULE 34.1: SPOT AND FORWARD RATES, PART 1
    THE FORWARD RATE MODEL


    The forward rate
    f(2,3) should
    make investors                        Investor A purchases a $1 face value, zero-coupon                        EXAMPLE: Forward
    indifferent                              bond maturing in = 5 years at a price of P (5)                        pricing: Calculate the
    between buying                                                                                                 forward price two years
    a five-year zero-                                                                                              from now for a $1 par,
    coupon bond                                                                                                    zero-coupon, three-
    versus buying a                                                                                                year bond given the
    two-year zero-             Investor B buys a 2 year $1 face value, zero-coupon bond, it matures in 2           following spot rates:
    coupon bond,               years and reinvests the principal for k = 3 additional years, to get F (2,3)        • two-year spot rate,
    and at maturity                                                                                                   S = 4%;
                                                                                                                        2
    reinvesting the                                                                                                • five-year spot rate,
    principal for                                                                                                     S = 6%.
                                                                                                                        5
    three additional
    years: Consider                                                                                                 Now Calculate the
    2 investors!                                                                                                    implied three-year
                                                                                                                    forward rate for a
                                                                                                                    loan starting two
                                                                                   k
                             j
     [1 + S (j+k) ] (j+k) = (1 + S ) [1 + f(j,k)] k                           Or                           [1 + f(j,k)] = [1 + S (j+k) ] (j+k)  years from now
                            j
                                                                                       (1 + S ) j                   [i.e., f(2,3)].
                                                                                              j
                                                                         [1 + f(2,3)] = [1 + 0.06] 5
                                                                                    3
                                                                                        [1 + 0.04] 2      Note that the
                                                                              f(2,3)    = 7.35%           forward rate f(2,3) > S : Why?
                                                                                                                                  5
                                                                        If the yield curve is upward sloping (S (j+k)  > S , then the forward
                                                                                                                       j)
                                                                        rate corresponding to the period from j to k [i.e., f(j,k)] will be
                                                                        greater than the spot rate for maturity j+k [i.e., S (j+k) ]. The
                                                                        opposite is true if the curve is downward sloping.
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