Jarrow and Turnbull Interest Rate Swap Valuation

Derivative Securities by Robert Jarrow and Stuart Turnbull (South-Western College Publishing, Year 2000, ISBN 0-538-87740-5).  This example appears on Pages 435-437.

EXAMPLE

SWAP VALUATION

This example illustrates the valuation of a two-year interest rate swap.  Consider a financial institution that is receiving fixed payments at the rate of 7.15 percent per annum and paying floating payments in a two-year swap.  Payments are made every six months.  Details are shown in Table 14.2 about the payment dates, the term structure of Treasury rates, and the term structure of LIBOR rates.

Consider first the fixed side of the swap.  At the first payment date, t1, the dollar value of the payment is
                                            NP X 0.0715 X (182/365),

 

Payment
Dates		 Days Between
Payment Dates		 Treasury Bill Prices
B(0,T)		 Eurodollar Deposit
L(0,T)
t1 = 182
t2 = 365
t3 = 548
t4 = 730
 182
183
183
182

 0.9679
0.9362
0.9052
0.8749

 0.9669
0.9338
0.9010
0.8684
B(0.T) denotes the present value of receiving for sure one dollar at date T.
L(0,T) denotes the present value of receiving one Eurodollar at date T.
These prices are respectively derived from the Treasury and Eurodollar terms structures.

 

where NP denotes the notional principal.   The present value today of receiving one dollar for sure at date t1 is B(0, t1) = 0.9679.  Therefore, the present value of the first fixed swap payment is
                                            NP X 0.9679 X 0.0715 X (182/365).

By repeating this analysis, the present value of all fixed payments is

 

                                            VR(0) = NP{0.9679 X 0.0715 X (182/365)
                                                         + 0.9362 X 0.0715 X (183/365)
                                                         + 0.9052 X 0.0715 X (183/365)
                                                         + 0.8749 X 0.0715 X (182/365)}
                                                     = NP X 0.1317.

Now let us consider the floating side of the swap.  The pattern of payments is similar to that of a floating rate bond, with the important proviso that there is no principal payment in a swap.  From the last chapter we know that on the date when the interest rate is reset, the bond sells at par value.  Hence at time 0, the present value of the sequence of floating rate payments is the notional principal, NP.   But, given that there is no principal payment in a swap, we must subtract the present value of principal repayment that would normally occur at time t4.   The present value of the floating rate payments is thus
                                            VF(0) = NP X L(0, t4)
                                                      = NP[1 - 0.8684]
                                                      = NP X 0.1316,

where L(0, t4) is the present value of receiving one Eurodollar at date t4.

The value of the swap to the financial institution receiving fixed and paying floating is
                                            Value of Swap = VR(0) - VF(0)
                                                                   = NP X [0.1317 - 0.1316]
                                                                   = NP X 0.0001.

If the notional principal is 25 million dollars, the value of the swap is $2,500.

In this example, the Treasury bond prices are used to discount the cash flows based on the Treasury note rate, and the Eurodollar discount factors are used to measure the present value of the LIBOR cash flows.  This incorporates the different risks implicit in these different cash flow streams.