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					Swaps
                               Swaps
   A swap is a contract calling for an exchange of
    payments over time.
   Almost anything can be swapped these days:
           Interest rate payments
           Asset cash flows
           Credit risk
           Commodity
   Obviously our primary concern will be with interest rate
    swaps, but in the beginning we will follow McDonald
    rather closely and discuss commodity swaps. The reason
    for doing so it to get a sense of the economics of swaps.
                     Forward Prices
   Swaps are intrinsically linked to forward prices, which we
    covered extensively in FINN 6211. We still need a little
    bit of a review, but basically we have the following facts
    about forward contracts:
       A forward contract is an agreement signed today to purchase a
        good at a price that is determined today.
       Both the long party (the party that will take delivery of the
        good) and the short party (the party that will sell the good) are
        obligated under the contract to perform their side of the deal.
        This is not an options contract.
       In essence the short part “holds” the asset on behalf of the long
        party during the period of the forward contract.
                 Forward Prices
   Obviously the short party is tying up capital “holding” the asset
    on behalf of the long party. Since there is no uncertainty in the
    price of the good or in the delivery date, the short party will
    earn the risk-free rate on the tied-up capital.
   Thus, if the asset pays no cash flows between time 0 and the
    maturity date of the forward contract, the forward price is given
    by:


                         F0  S0e        rT
                 Forward Prices
   If the asset pays out discrete payments Dt during the life of the
    forward contract, the forward price is given by:

                     F  S  I e
                       0          0
                                            rT

   Where I is the present value of the future payments discounted
    at the risk-free rate:
                              n
                         I   D e r
                              1
                     Forward Prices
       Finally, if the asset pays a continuous dividend yield (as is the
        case with say a stock index) y, the forward price is given by:

                                          ( r  y )T
                         F0  S0e
   Example: Let’s say that the spot price for oil is $18.835
    per barrel and that the one-year risk-free rate is 6%.
    What would be the forward price of a barrel of oil if the
    maturity date of the contract were 1 year?

                   F0  18.835e        .06(1)
                                                 $20.00
                     Forward Prices
   Let’s say that you entered into a two year forward
    contract to buy a stock. The stock is currently priced at
    $50. The stock will pay a dividend to $2 in six months,
    and a dividend of $3 in eighteen months. The six month,
    eighteen month and twenty-four month risk free rates
    are 4%, 5% and 6% respectively. What is the current
    forward price of this stock?
       First, determine the present value of the two future cash flows:

                I  2e.04(.5)  3e.05(1.5)  4.7436
                     Forward Prices
       Then, determine the forward price itself
                F0  (S  I )erT  (50  4.7436)e.06(2)  51.026

   Finally, assume that a stock index is currently priced at
    10,000 and that it pays a dividend yield of 3%. If the
    current 2 year risk-free rate is 6%, what is the 2 year
    forward price on this index?

            F0  S0e( r  y )T  10,000e(.06.03)(2)  10,618.37
       Obviously there is a lot more that we are not covering in here,
        and I will put my FINN 6211 forwards notes on my web-site for
        those of you that have not had that class yet.
                   Commodity Swaps
   McDonald begins with an example of a commodity swap,
    and I think it bears examining in detail.
       The company PI is going to by 100,000 barrels of oil in exactly
        one year and another 100,000 barrels in exactly two years. The
        one and two year risk free rates are 6% and 6.5% respectively.
        Suppose that the forward prices for oil in one and two years are
        20 and 21 dollars respectively (note that these are not
        consistent, but we will use them anyway!)
            The company could enter into two forward contracts to lock in the
             price of this oil today. The present value of the two contracts would
             be:
                               20    21
                                       2
                                           $37.373
                              1.06 1.065
            Assuming that the law of one price holds, IP should be able to pay
             $37.383 today and lock in delivery of the oil. This would be called a
             prepaid forward contract or a prepaid swap.
            Commodity Swaps
   IP could enter into a prepaid swap today, but they would then
    be taking on credit risk, in that there is a chance the person
    that IP has paid will default on the contract and simply fail to
    deliver the oil. This is also sometimes called counterparty risk.
   For this reason, IP would probably prefer to structure the deal
    so that they would pay for the oil only when the actually
    received it. For this reason, commodity swaps typically are
    structured so that payments are made when delivery is made.
   The present value of the total scheduled payments must be
    equal to $37.383, and any cash flow stream meeting that criteria
    would, in theory, be acceptable. Typically to keep things simple,
    however, the prices are set to be equal each period.
                 Commodity Swaps
       This means that the following must be true:

                            x     x
                                      37.385
                          1.06 1.0652

       And solving for this we find that the price would be $20.483.


   Now, so far we have assumed that IP would be taking
    physical delivery of the oil, but there is really no reason
    this needs to be the case. The two parties could simply
    agree to cash settlement or, as it is sometimes known,
    financial settlement.
                   Commodity Swaps
   How would cash settlement work?
       Well, basically what would happen is that on a payment date IP
        would pay to the counterparty an amount equal to:
                            Swap Price – Spot price
       What does this do? Let’s look at two examples:
            Ex. 1: Spot price = $15. IP agreed to buy the oil at $20.483, so in
             essence they have overpaid. If the counterparty bought the oil in
             the market at $15 and then sold it to IP for $20.483 they earn
             $5.483. It is exactly the same as if IP simply paid the counterparty
             $5.483 and then bought the oil on their own in the spot market.
            Ex. 2: Spot price = $30. The counterparty would pay $9.517 to IP.
             The counterparty could go buy the oil for $30 and sell it to IP for
             20.483, and they would lose $9.517. Its simply easier for them to
             pay the $9.517 than got through the physical delivery hassles.
                 Commodity Swaps
   So is the only reason for cash settlement ease of
    delivery? No. It actually expands the sphere of investors
    that can take on this type of risk, i.e. the number of
    market participants!
       If you had to make physical delivery, only firms that had the
        infrastructure to make physical delivery, i.e. oil firms, could enter
        into this type of contract.
       With cash settlement, any firm that was willing to bear oil price
        risk can enter the market. This could be investment banks,
        commercial banks, corporations, even, in theory, individuals.
                 Commodity Swaps
   Another way of viewing this transaction, by the way, is
    to simply say that it is an agreement between IP and the
    counterparty such that on each payment date IP will pay
    $20.483 to the counterparty, and the counterparty will
    pay the market price to IP.
       In the parlance of swaps, IP is paying the “fixed” price, and the
        counterparty is paying the “floating” price.
   In our discussion of cash settlement, we have dealt with
    a per barrel price of oil. If the swap were on 100,000
    barrels, we would just multiply the cash flows by
    100,000 to get the actual dollar amounts. In this case
    100,000 would be the notional amount of the swap.
       The notional amount is the amount of the underlying
        instrument that is used to determine the actual cash flows.
                Commodity Swaps
   So why is the swap price ($20.483) not the same as the
    average of the two forward prices (i.e.
    (20+21)/2=20.50?)
       Well, consider that if the price were 20.50, IP would be
        committing to pay $0.50 more in year 1 than the forward price,
        and to pay $0.50 less than the forward price in year 2.
       In essence IP would be making an interest free loan of $0.50 to
        the counterparty for the year.
       With a price of $20.483, IP overpays by $0.483 in year 1, but
        pays $0.517 less in year 2. In essence they lend the
        counterparty $0.483 in year 1, and are paid back $0.517 in year
        2.
             Commodity Swaps
   What is the return on this loan?
         0.483(1  y )  0.517
         or
            0.517
         y=         1  0.07039
            0.483
         or
         y=7.03%.


   Notice how this relates to our two spot interest rates. Recall that
    r1= 0.06, and r2 =0.065. Recall that we said the forward rate is
    given by:

                   1  r2                     1.065
                            2                             2

            f1,2                1 so f1,2                   1  7%
                    1  r1                     1.06
                Commodity Swaps
   In our discussion of forward rates, we used (in keeping
    with the book) discrete compounding, such as semi-
    annual compounding.
       Determining forward rates in continuous time is actually much
        easier to do.
       Assume that you have two continuously compounded forward
        rates, r1 and r2, what is the continuously compounded forward
        rate between them?
                    Commodity Swaps
   Thus, we can set up the problem and solve for the
    forward rate:
                                              1e r2 (2)
                                  f    (1)
               1* e r1 (1) * e 1,2
               or, simplifying and following the rules of exponents:
                 r  f1,2
               e1            e 2 r2
               taking logs:
               r1  f1,2  2* r2
               or
                f1,2  2r2  r1

       So if in this case r1=.06 and r2=0.065, we can see that f1,2 is
                     f1,2=2*0.065-.06=0.13-.06=0.07, or 7%!
                 Commodity Swaps
   What about the counterparty, who are they?
       There is a fairly large swaps market, and there are plenty of
        swaps dealers.
       Normally they seek to create a “matched book”, that is, they will
        be paying fixed/receiving float on one transaction and paying
        float, receiving fixed on another (at the same prices, of course).
                   Commodity Swaps
   What about the counterparty, who are they?
       There is a fairly large swaps market, and there are plenty of
        swaps dealers.
       Normally they seek to create a “matched book”, that is, they will
        be paying fixed/receiving float on one transaction and paying
        float, receiving fixed on another (at the same prices, of course).
            Begin by entering into the first trade…

                     Receive
                     Fixed
         Trade                     Swaps
             A                     Dealer

                      Pay
                      Float
                   Commodity Swaps
   What about the counterparty, who are they?
       There is a fairly large swaps market, and there are plenty of
        swaps dealers.
       Normally they seek to create a “matched book”, that is, they will
        be paying fixed/receiving float on one transaction and paying
        float, receiving fixed on another (at the same prices, of course).
            And then put on an offsetting trade to close out the position.

                     Receive
                     Fixed                    Pay Fixed
         Trade                     Swaps                    Trade

             A                     Dealer                     B
                                                Receive
                      Pay
                                                Float
                      Float
                 Commodity Swaps
   Of course in reality the dealer would set it up to make a
    profit.
       They would agree to pay and receive float (so that it was a net
        wash), but they would try to set the fixed that they received at a
        slightly higher rate than the fixed that they would pay.
       The swap dealer has credit risk, but not price risk in this
        transaction.
                   Receive
                   Fixed                  Pay Fixed
         Trade                  Swaps                 Trade

           A                    Dealer                  B
                                           Receive
                   Pay
                                           Float
                   Float
                 Commodity Swaps
   What if the dealer could not find the second party, i.e.
    Trade B?
       Well, they could not have a “matched book” at this point, so
        they would have to hedge their price risk in some way.
       The risk they face is that the floating price that they have to pay
        will increase above the fixed price they are going to get.
       They could hedge this through a forward contract…
                   Receive               At t1 pay f1, get oil and sell at spot price.
                   Fixed
                                                                Foward0,1
         Trade                  Swaps
           A                    Dealer                          Foward0,2

                    Pay                  At t2 pay f2, get oil and sell at spot price.
                    Float
                  Commodity Swaps
   Unfortunately, there is still risk for the dealer.
           Recall that f1 is 20 and f2 is 21. The fixed price is 20.483.
           So at time 1 the dealer pay 20, but receives 20.483 from the swap.
            They must invest the $0.483
           At time 2, the dealer pays 21, but only receives 20.483. They have
            to earn at least 7% on the 0.483 in order for it to grow to be 0.517
            so that they will have (20.483+0.517=21) to pay for the oil under
            the forward at time 2.
                    Receive 20.483          At t1 pay 20 and sell oil at spot.
                                                                  Foward0,1
        Trade                     Swaps
            A                     Dealer                          Foward0,2

                     Pay                    At t2 pay 21 and sell at spot
                     Float
                Commodity Swaps
   So the dealer has hedged their price risk but not their
    interest rate risk!
       The dealer would have to go to the interest rate forwards
        markets to hedge the interest rate risk.
                     Commodity Swaps
   Indeed, we can see that from IP’s perspective, the swap is
    equivalent to:
          1.   A forward contract to buy oil at time 1 for $20.00
          2.   A forward contract to buy oil at time 2 for $21.00
          3.   An agreement to lend the swap dealer $0.483 between time 1 and 2 at a
               7% interest rate.
        Note that at time zero these all have zero value, meaning that they you
         do not have to pay anything to enter into the contracts.
        You could unwind immediately by taking an offsetting position in the
         swap and you would not owe anything but commissions and bid/ask
         spread.
        Thus, we say that the market value (or price) of the swap at time 0 is
         0.
                   Commodity Swaps
   That price will not stay 0, however, for a variety of
    reasons.
       Consider that at time 0 the buyer “overpays” for the oil by
        0.483: in essence they have leant the money to the dealer.
       If the dealer (counterparty) wanted to get out of the transaction
        at time 1 – even if the spot and forward oil prices had not
        changed – they would have to pay IP to get out of it. Why?
            Well, essentially they would have to repay the loan.
       Of course in oil prices change, the value of the swap would no
        longer be 0.
            To see this, let’s assume that the forward price of oil rises by
             $2/barrel immediately after the parties enter into the contract.
               Commodity Swaps
   The new one year forward price would be 22, and the new two
    year forward price would be 23. The new swap price would be
    $22.483.
        Intuitively you should be able to realize who this benefits. The swap
         allows IP to purchase oil at times 1 and 2 for 20.483, but if the
         swap were signed now they would pay 22.483. Clearly being able to
         buy at 20.483 is valuable – they have gained in value.
        To put it another way, if the counterparty said to IP, we would like
         to get out of this swap, IP would demand that they be paid enough
         so that they could put on a new swap (at 22.483 dollars) without
         losing wealth.
   So what is the new value of the swap?
        Ultimately, its just the present value of the difference in the new
         and old swaps.
             Commodity Swaps
   In this case, the floating prices are just “washes”, so we really
    only have to focus on the difference in the fixed prices.
   IP would have to pay $2 more in each period. Assuming that
    interest rates are unchanged, the differences in the swaps are
    just $2 at time 1 and $2 at time 2. This present value is:
                           2     2
                                  2
                                      $3.65
                         1.06 1.065

   Thus, we would say that $3.65 is the market value of the swap.
                Interest Rate Swaps
   The commodity swap is helpful for understanding the
    basic mechanics of a swap, and to see why we might
    chose to “cash settle” the swap.
   Of course, since this is a fixed-income class, we are
    primarily concerned with the role of swaps in the interest
    rate markets.
       Let’s motivate this discussion with a relatively simple example.
            XYZ corporate has $200 million of floating rate debt at LIBOR.
            This means that they at the beginning of the year they reset the
             interest rate on the loan to LIBOR, and pay that rate for the rest of
             the year.
            They would prefer to have a fixed rate.
               Interest Rate Swaps
   Here is what the current situation is:
                     Pays
                     LIBOR
                               XYZ
                               Corp



   How could XYZ switch to a fixed rate loan?
       They could retire the current loan and issue a fixed-rate loan.
       They could enter into a forward rate agreement (FRA).
       They could enter into an interest rate swap.
              Interest Rate Swaps
   We will come back to Forward Rate Agreements in a
    little while, but for now let’s see what would happen if
    they entered into a swap.
   What they could do would be to enter into a swap
    agreement in which they agreed to pay a fixed rate of
    interest (say 6.9548%) on a notional amount, and then
    to receive the LIBOR rate on that amount as well.
       Once again, we will cash settle, meaning that we never really
        trade the notional amount back and forth.
       Every XYZ pays to the counterparty an amount equal to
        6.9548-LIBOR. If LIBOR>6.9548, then the payment is negative,
        meaning that the counterparty pays XYZ corporation.
               Interest Rate Swaps
   This can be illustrated as:

            Pays
            LIBOR              Receives LIBOR
                      XYZ                        Swap
                      Corp      Pays 6.9548%     Dealer


   The net effect is, of course, that XYZ corporation is now
    paying a fixed rate of 6.9548% on $200 million.
       The term of the swap – that is, how long it lasts, is called the
        swap tenor.
                Interest Rate Swaps
   We can see that XYZ corporation has an incentive to
    enter into this contract, they are able to convert a
    floating rate commitment into a fixed rate one, but why
    would the dealer enter into this arrangement?
       One potential reason is that the dealer has a fixed rate
        commitment that they would like to convert into a floating rate
        instrument.
       More likely, however, they are doing this simply to earn a fee,
        meaning they will have to hedge their (newly acquired) interest
        rate exposure.
            They can do this by entering into forward rate agreements, or
            They can enter into a nearly-offsetting agreement with a second
             counterparty.
               Interest Rate Swaps
   Let us assume that there is now a second corporation,
    ABC Corp, that is currently paying 7.00% on a fixed rate
    bond, and they wish to convert that into a floating rate
    instrument.

Pays
LIBOR           Receives LIBOR                         Pays 7.00%
        XYZ                      Swap           ABC
        Corp    Pays 6.9548%     Dealer         Corp
               Interest Rate Swaps
   If the swap dealer agreed to a second swap with ABC,
    one where ABC paid LIBOR to the dealer and received
    6.85% fixed, the net effect is that the dealer earns
    .1048% on the notional.

Pays
LIBOR           Receives LIBOR          Pays LIBOR            Pays 7.00%
        XYZ                      Swap                  ABC
        Corp    Pays 6.9548%     Dealer Receives 6.85% Corp
                 Interest Rate Swaps
   If the swap dealer agreed to a second swap with ABC,
    one where ABC paid LIBOR to the dealer and received
    6.85% fixed, the net effect is that the dealer earns
    .1048% on the notional.

Pays
LIBOR              Receives LIBOR             Pays LIBOR            Pays 7.00%
          XYZ                          Swap                  ABC
          Corp     Pays 6.9548%        Dealer Receives 6.85% Corp

    XYZ Net:                        Dealer Net:              ABC Net:
    Pays 6.9548%                    Receives 0.1048%         Pays LIBOR+.15
          Comparative Advantage
   In one sense interest rate swaps are kind of strange
    creatures. Why do them at all? Why not simply issue the
    type of debt you want to have in the first place?
       One argument that is frequently raised is that firms may have a
        comparative advantage in either the fixed or floating rate
        markets, and so it may be advantageous to issue debt in one
        market over the other, and then swap to get into the other type
        of debt.
       Let’s work an extended example to see this.
          Comparative Advantage
   Let’s say that there are two companies, AAA and BBB,
    who can borrow in either the fixed or floating rate
    markets at the following rates:

    Company              Fixed          Floating
    AAA                  10.0%          6-Month LIBOR + 0.3%
    BBB                  11.2%          6-Month LIBOR + 1.0%

       Clearly AAA has an absolute advantage in both markets, but
        since BBB pays only 0.7% more in the floating markets (as
        opposed to the 1.2% more they pay in the fixed markets), they
        have a comparative advantage in the floating market.
          Comparative Advantage
   In this case we can structure a swap transaction that will
    be beneficial to both AAA and BBB (and the dealer!).
       First, both AAA and BBB issue debt in the markets in which they
        both have comparative advantages (say $100m).


10.0%
                                                                LIBOR+1.0%
         AAA                      Swap                   BBB
         Corp                     Dealer                 Corp
            Comparative Advantage
   In this case we can structure a swap transaction that will
    be beneficial to both AAA and BBB (and the dealer!).
           Next, they enter into swaps with the dealer. AAA agrees to pay
            LIBOR and receive 9.90% fixed. Their net position is that they now
            pay LIBOR + 0.10%, which is better than the LIBOR +0.3% they
            would pay in the floating rate market!
10.0%
                  9.90%                                              LIBOR+1.0%
        AAA                           Swap                    BBB
        Corp       LIBOR              Dealer                  Corp

 AAA Net:
 Pays LIBOR+0.10%
 Net Gain: .20%
            Comparative Advantage
   In this case we can structure a swap transaction that will
    be beneficial to both AAA and BBB (and the dealer!).
           Then BBB enters into a swap where they receive LIBOR and pay a
            fixed rate of 10%. Their net is to pay fixed 11%, which is better
            than the 11.2% they could get by issuing debt in the fixed market.

10.0%                                             10%
                  9.90%                                              LIBOR+1.0%
        AAA                           Swap                    BBB
        Corp       LIBOR              Dealer   LIBOR          Corp

 AAA Net:                                                      ABC Net:
 Pays LIBOR+0.10%                                              Pays 11%
 Net Gain: .20%                                                Net gain: 0.2%
            Comparative Advantage
   In this case we can structure a swap transaction that will
    be beneficial to both AAA and BBB (and the dealer!).
           Note that the dealer now has no interest rate risk, but earns 0.1%
            on the deal – essentially for brokering the deal.


10.0%                                             10%
                  9.90%                                               LIBOR+1.0%
        AAA                           Swap                     BBB
        Corp       LIBOR              Dealer    LIBOR          Corp

 AAA Net:                         Dealer Net:                  ABC Net:
 Pays LIBOR+0.10%                 Receives 0.10%               Pays 11%
 Net Gain: .20%                                                Net gain: 0.2%
          Comparative Advantage
   Now we should point out that the swaps dealer does
    bear credit risk: AAA or BBB could default, but the dealer
    would have to honor the swap to the other party.
   Notice something interesting in that swap: AAA and BBB
    each improved their net position by 20 basis points, and
    the dealer earned 10 basis points, meaning that there
    was 50 basis points of “value” created by the swap.
    Where did it come from?
       It is the difference in the spread between AAA and BBB in the
        fixed and floating rate market.
         Comparative Advantage
   To see this, go back to the original situation facing AAA
    and BBB:
    Company           Fixed         Floating
    AAA               10.0%         6-Month LIBOR + 0.3%
    BBB               11.2%         6-Month LIBOR + 1.0%

    Spread             1.2%         0.7%

   The difference in the spreads between the two firms is
    50 basis points, and a swap can allocate those 50 basis
    points between the two companies and the swap dealer.
             Comparative Advantage
   There is something troubling about the comparative
    advantage argument, why is it that the floating rate
    market charges a lower spread on the weaker company
    (BBB) than does the fixed rate market?
       The reason is because in the floating rate market if the credit
        quality of BBB deteriorates, then the lender can, at the next
        payment resetting date, raise the spread they charge on the
        loan. They cannot easily do this on the fixed rate bond.
       Note that if BBB were to lose its ability to borrow at LIBOR
        +1.0%, they would actually pay more on the swap than 11%.
            For example, if their borrowing cost went to LIBOR + 2.0%, they
             would wind up paying 12% on their debt.
            In essence they get the better rate in the swaps market than they
             could in the fixed rate market because they bear credit risk that
             normally the fixed rate lender would bear!
        Forward Rate Agreements
   Before we can understand interest rate swap pricing and
    analysis, we need to understand how to price forward
    rate agreements.
       McDonald discusses forward rate agreements (FRA) in section
        7.2 of his book (starting on page 208.)
   A forward rate agreement (FRA) is an over-the-
    counter contract that guarantees a borrowing or lending
    rate on a given notional principal amount.
       No principal changes hands and the contract is essentially cash-
        settled.
       The FRA can be settled either at the initiation of or the end of
        the period. If done at the end it is said to be settled in
        arrears.
        Forward Rate Agreements
       The reference rate is the market interest rate against which the
        forward rate is compared. The long party is paid if the FRA rate is
        above the reference rate, and pays if the FRA is below the reference
        rate.


   It may be easiest to understand FRAs by using the example that
    McDonald creates.
       Consider a firm that expects to borrow $100m for 91 days beginning
        120 days from today, in June.
       The loan will be repaid in September.
       Assume that the effective quarterly rate at that time can be either 1.5%
        or 2.0%.
       The implied June 91 day forward rate is 1.8%.
          Forward Rate Agreements
   We can draw this as a time line…



                                       rquarterly=either 1.5% or 2.0%
      120 days                                  91 days
t=0                                 June                      September
(Today)
                   211 days total
          Forward Rate Agreements
   We can draw this as a time line…



                                       rquarterly=either 1.5% or 2.0%
      120 days                                  91 days
t=0                                 June                      September
(Today)
                   211 days total



   Note that we will know in June, what the quarterly rate
    is, so we can determine then what the payment under
    the FRA will be.
        Forward Rate Agreements
   As of June, therefore, all uncertainty is resolved. In
    reality this means that the two parties have a choice,
    they can have the FRA payment made at either the
    beginning or the end of the FRA.
       In this case this means the payment could be set up in June or it
        could occur in September, depending upon what the two parties
        wanted.
       If payment is made at the end of the contract, it is said to be
        “settled in arrears.”
       Since the interest on the FRA is earned from June to September,
        the difference is that if paid in June, the payment amount is
        present value (discounted at the risk-free rate) of the amount
        that would be paid in September.
        Forward Rate Agreements
   Payout formulas
       FRA settlement in arrears:
                    Payment   rquarterly  rFRA  * notional amount

       FRA settlement at beginning of period:
                               rquarterly  rFRA  
                   Payment                          * Notional Amount
                                  1  rquarterly 
                                                    
       So if the rquarterly wound up being 1.5%, then payouts would be:
        Forward Rate Agreements
   Payout formulas
       FRA settlement in arrears:
                       Payment   rquarterly  rFRA  * notional amount

       FRA settlement at beginning of period:
                                   rquarterly  rFRA  
                       Payment                          * Notional Amount
                                      1  rquarterly 
                                                        
       So if the rquarterly wound up being 1.5%, then payouts would be:

            Payment Paid in Arrears  .015  .018  *100, 000, 000  $300, 000.00
                                        .015  .018  
            Payment Paid at Begining                   *100, 000, 000  $295,566.50
                                           1.015       
        Forward Rate Agreements
   Payout formulas
       FRA settlement in arrears:
                       Payment   rquarterly  rFRA  * notional amount

       FRA settlement at beginning of period:
                                   rquarterly  rFRA  
                       Payment                          * Notional Amount
                                      1  rquarterly 
                                                        
       Similarly, if rquarterly=2%, then the payouts would be:

             Payment Paid in Arrears  .020  .018  *100, 000, 000  $200, 000.00
                                         .020  .018  
             Payment Paid at Begining                   *100, 000, 000  $196, 078.43
                                             1.015      
                   Swap Pricing
   Clearly a forward rate agreement bears a striking
    resemblance to a swap, and will be a major issue in
    valuing a swap.
   The first thing to realize about swaps is that normally we
    construct them specifically so that at time 0 nobody has
    to pay anything to enter into them (either the swap
    dealer or their counterparty): this means that at time
    0, the value of the swap is 0!
   Over time, however, the value will diverge from 0.
                       Swap Pricing
   McDonald makes a few points about swap pricing that
    are interesting:
       If you enter into a swap where you pay float and you received
        fixed, what you have effectively done is to borrow in the floating
        rate market to buy a fixed rate bond!
       Not surprisingly, then, the fixed rate on a swap should be the
        same as a coupon rate on a bond of the same maturity (and
        credit risk.)
       When it comes to pricing the swap, however, I think Hull’s book
        does the best job, but even he appeals to the notion that we can
        price the swap by treating it as a position that is long (short) a
        floating rate bond and short (long) a fixed rate bond. We are
        going to use an extended example from his book (page 134) to
        illustrate swap pricing.
                     Swap Pricing
   We begin with the notion that the swap can be thought
    of as the difference between being in a long fixed bond
    position and a short floating rate position.
     Since at time zero the swap value (Vswap) is zero, we know that:
                            Vswap,t=0=Bfloat-Bfixed
      it must must also be the case that:
                                Bfloat=Bfixed.
     Now, it must also be the case on a rate-setting date a floating
      rate bond will have value of par, since the discount rate and the
      coupon rate are the same, thus:
                                Bfloat=$100
    And so
                                Bfixed=$100
                   Swap Pricing
   As a result, the fixed side of the swap must have the
    same coupon as a standard coupon bearing bond!

   Of course, as time progresses, the fixed rate bond will
    probably diverge from its par value, and so will the
    floating rate bond (at least between payment dates.)

   So we want to think about how we price the swap on
    days other than its origination.
                 Swaps Pricing
   We will use the following notation:
    ti: Time until the ith (1<=i<=n) payments are exhanged
    L : Notional principal in the swap agreement.
    ri : LIBOR zero rate corresponding to maturity ti.
    k : Fixed payment made on each payment date.
    k*: Floating rate payment to be made at time 1.
   We can define the price of the fixed bond as:
                                n
                      B fix   ke  riti  Le  rntn
                               i 1


   At first it would seem that valuing the floating rate side would
    be more difficult, since we only know the current period’s
    coupon rate. Turns, out, that is not a problem.
                  Swaps Pricing
   The reason it is not a problem is because we know that on the
    next payment date, the coupon will be set to the LIBOR rate on
    that day, and thus the value of the bond will return to par. As a
    result, we only have to discount that par amount, and the next
    payment amount, back at r1 to get the current price of the
    floating rate side:

                                        
                       B float  L  k * e r1t1
                     Swaps Pricing
   It is probably easiest to see this through an example.
       Suppose that a financial institution pays six-month LIBOR and
        receives 8%, with semiannual compounding on a swap with a
        notional principal of $100 million and the remaining payment
        dates are in 3 months, 9 months, and 15 months (so the swap
        has 15 months maturity remaining.) The
       The continuously compounded LIBOR rates for 3, 9 and 15
        month zero coupon maturities are 10%, 10.5% and 11%,
        respectively. The 6-month LIBOR rate at the last payment date
        (which was three months ago!) was 10.2%.
       What is the value of the swap?
                            Swap Pricing
         First, let us figure out the timing and size of the we know.
               The fixed side is a constant of .08*100,000,000 =$4,000,000.
               The next floating rate payment is .102*100,000,000 = $5,100,000.
         The time line is:



                      k=$4m              k=$4m             k=$4m
                      k*=$5.1M           k*=??             k*=??

            0         3                  9                 15
r3= 10%

r9= 10.5%

r15=11%
                         Swap Pricing
   So we can now determine the price of the bonds:
          B fix  4e0.1(.25)  4e.105(.75)  104e11(1.25)  $98.24 million

    and
              B float  5.1e0.1(.25)  100e.1(.25)  $102.51 million

    so the net value of the swap to the financial institution,
    (since they are receiving fixed and paying float) is:
         Vswap=Bfix-Bfloat= 98.24-102.51 = -4.27 million.
                       Swap Pricing
   One issue that is still somewhat unsatisfying is that we
    have not given a really strong reason why a firm would
    enter into a swap.
       Comparative advantage argument has some merit, but it seems
        to not justify the size of the market. Plus, why would two firms
        of equal credit quality ever swap?
       One very common reason for using a swap – and one that is not
        mentioned in any of our books – is that it allows one to change
        the aggregate duration of a portfolio very rapidly, and, at least
        initially, at no cost to the firm.
       To see this, consider the duration of the two halves of the swap.
                     Swap Pricing
   The (dollar) duration of the fixed side will be given by:
                  dB fixed       n
                               (k * ti )e riti  ( L * tn )e rntn
                    dr          i 1


    and the (dollar) duration of the floating side will be given
    by:
                             dBFloat
                              dr
                                                       
                                      t1 * L  k * e r1 t1
                                Swap Pricing
     In the previous example, this results in:
    dB fix
              (.25)4e0.1(.25)  (.75)4e.105(.75)  (1.25)104e.11(1.25)  117.4075
     dr

     And
                                                       
                         dB float
                                     .25 105.1e0.1(.25)  25.6262
                           dr

     So the modified durations are:
                           Mod DurFix  117.4075 / 98.24  1.1951

                         Mod Durfloat  25.6262 /102.51  0.24998
                    Swap Pricing
   In some ways, this example does not fully illustrate an
    important point: the floating rate side is always going to
    have a small duration – simply because it resets every
    few months. The fixed side will tend to have a very long
    duration – something approaching the maturity of the
    swap.
   If I am managing a fixed income portfolio and want to
    decrease my exposure to interest rate risk – i.e. I want
    to reduce my aggregate duration – I can agree to “pay
    fixed”, i.e. receive float, and reduce my aggregate
    duration, at very little initial cost to me.

				
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