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Solutions to the Exercises on Caps Floors and Swaps

VIEWS: 152 PAGES: 3

									MF860, Derivatives and Risk Management                                             Prof. Marcus
Problem Set 4, Solutions

1. The rate on the reverse floater falls 2.5% for every percentage point increase in
   LIBOR. So to offset the LIBOR exposure the firm (which wants to create synthetic
   fixed-rate debt) would need to enter swaps with two and one-half times the notional
   principal, or £250 million. In the swap, the firm will receive the fixed rate of 8.10%
   and pay LIBOR.

   In addition, the payments on the geared inverse floater stay at zero and stop falling
   once LIBOR rises above 12%. But the firm's payments on the swap will continue to
   rise as LIBOR increases beyond 12%. Therefore the firm also must buy a cap with a
   strike rate of 12% and notional principal of £250 million to offset this exposure.

   The firm's net cash flows (in £ millions) are as follows:

                      LIBOR ≤ 12%                                    LIBOR ≥ 12%

   Inverse floater    –100  [.30 – 2.5  LIBOR]                     0
   Swap                250  [.081 – LIBOR]                          250  [.081 – LIBOR]
   Cap                 0                                             250  [LIBOR – .12]
   Total CF           –9.75                                          –9.75

   The CF of the total portfolio is therefore equivalent to that of a bond paying a 9.75%
   fixed rate coupon on par value of £100 million. The firm has successfully hedged its
   exposure to LIBOR.

   While the effective coupon rate of the bond is 9.75%, and the inverse floater is issued
   at par, you must remember that the firm has to pay for the caps. The cost of the caps
   raises the fixed rate cost of funds above 9.75%, since net of that cost, the firm realizes
   less than par value for the geared inverse floater:

   The all-in fixed cost of funds is 10.74% (bond-equivalent yield), calculated on a
   financial calculator as follows:

   PV = Proceeds of reverse floater net of cost of cap: 100 – (250  .015) = 96.25
   n = semiannual periods, = 10
   pmt = semiannual payment = 9.75/2
   FV = 100
   Compute i = 5.37% semiannually = 10.74% BEY.

   Note that 9.75% is the synthetic fixed coupon rate; the all-in cost (i.e., the effective
   yield to maturity) exceeds that in the same manner that the yield on a bond selling at a
   discount from par always exceeds its coupon rate.




                                            -1-
2. If the firm issues the PLB, buys a floor and enters a swap (to buy LIBOR for a fixed
   rate), it will convert the debt to synthetic fixed rate debt:

   Initial CF per $100 of par value:

                      CF at t = 0      LIBOR < 6              LIBOR > 6
   Sell PLB           $102             6                     L
   Buy floor            .35           6L                     0
   Buy swap               0            L  8.5                 L  8.5
   TOTAL              $101.65          8.5                    8.5

   Buy floor at ask price; floor converts PLB into effective LIBOR note
   Sell swap for ask; for converts effective floating rate to fixed rate of 8.5%, at ask price

   To find YTM:

       Net price = 101.65              PV
       Final payment = 100             FV
       Maturity = 5                    n
       Payment = 8.5                   PMT

       Compute i = 8.086%

Alternative:   Issue PLB and buy a cap [cap coverts PLB into fixed rate debt at 6% coupon]

                      CF at t = 0      LIBOR < 6              LIBOR > 6
   Sell PLB            $102            6                     L
   Buy cap              9.3            0                      L6
   TOTAL                $ 92.7         6                      6

       PV = Net price = 92.7 [i.e., net cash flow realized at t = 0]
       PMT = 6
       n=5
       FV = 100

       Compute i = 7.82%




                                              -2-
3. Fixed payments for FRAs:

       0  6 FRA:       .04  .5 = .02
       6  12 FRA: .05  .5 = .025

   Discount factors:

   First six-month period: 4%  .5 = 2%
   Forward rate for next six-month period: 5%  .5 = 2.5%
   Discount factor for a payment that is paid in month 12: 1.02  1.025 = 1.0455

   Therefore, using the weighted-average of FRA rate formula:

             1                                         1
       B6 = 1.02 = .9804                      B12 = 1.0455 = .9565

               .9804                                    9565
       w6 = .9804 + 9565 = .50617             w12 = .9804 + 9565 = .49383

   
       SFR = .50617  4% + .49383  5% = 4.4938%

   Alternative approach: equate the PV of the FRA fixed-rate payments to those of the swap:



                .02    .025             (SFR  .5)  (SFR  .5)
               1.02 + 1.0455 = .04352 =    1.02    + 1.0455    .


   Solve to find that SFR = .04494 = 4.4938%



Problem 4
              A              B           C               D               E

   1         Maturity   Price of ZCB         weight   forward rate   col C x col D
   2                0           1000
   3                1         952.38      .2734              0.050        .01367
   4                2         898.47      .2580              0.060        .01548
   5                3         843.64      .2422              0.065        .01574
   6                4         788.45      .2264              0.070        .01585
   7        sum(3:6)         3482.94     1.0000                           .06074 = SFR = 6.074%




                                             -3-

								
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