VIEWS: 3 PAGES: 67 POSTED ON: 10/14/2011
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 4e0.1(.25) 4e.105(.75) 104e11(1.25) $98.24 million and B float 5.1e0.1(.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)4e0.1(.25) (.75)4e.105(.75) (1.25)104e.11(1.25) 117.4075 dr And dB float .25 105.1e0.1(.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.