Financial Markets Interest Rate Swaps Session #17 Alejandro Reynoso

The Bank for International Settlements reports that interest rate swaps are the largest component of the global OTC derivative market. The notional amount outstanding as of June 2009 in OTC interest rate swaps was $342 trillion, up from $310 trillion in Dec 2007. The gross market value was $13.9 trillion in June 2009, up from $6.2 trillion in Dec 2007. Interest rate swaps can now be traded as an Index through the FTSE MTIRS Index.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH In an interest rate swap, each counterparty agrees to pay either a fixed or floating rate denominated in a particular currency to the other counterparty. The fixed or floating rate is multiplied by a notional principal amount (say, USD 1 million) and an accrual factor given by the appropriate day count convention. When both legs are in the same currency, this notional amount is typically not exchanged between counterparties, but is used only for calculating the size of cashflows to be exchanged. When the legs are in different currencies, the respective notional amounts are typically exchanged at the start and the end of the swap. The most common interest rate swap is one where one counterparty A pays a fixed rate (the swap rate) to counterparty B, while receiving a floating rate indexed to a reference rate (such as LIBOR or EURIBOR). By market convention, the counterparty paying the fixed rate is called the "payer" (while receiving the floating rate), and the counterparty receiving the fixed rate is called the "receiver" (while paying the floating rate). A pays fixed rate to B (A receives floating rate) B pays floating rate to A (B receives fixed rate) Currently, A borrows from Market @ LIBOR +1.5%. B borrows from Market @ 8.5%. Consider the following swap in which Party A agrees to pay Party B periodic fixed interest rate payments of 8.65%, in exchange for periodic variable interest rate payments of LIBOR + 70 bps (0.70%) in the same currency. Note that there is no exchange of the principal amounts and that the interest rates are on a "notional" (i.e. imaginary) principal amount. Also note that the interest payments are settled in net (e.g. Party A pays (LIBOR + 1.50%)+8.65% - (LIBOR+0.70%) = 9.45% net). The fixed rate (8.65% in this example) is referred to as the swap rate. At the point of initiation of the swap, the swap is priced so that it has a net present value of zero. If one party wants to pay 50 bps above the par swap rate, the other party has to pay approximately 50bps over LIBOR to compensate for this.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH Being OTC instruments, interest rate swaps can come in a huge number of varieties and can be structured to meet the specific needs of the counterparties. For example, the legs of the swap can be in the same currency or in different currencies. The notional of the swap could be amortized over time. The reset dates of the floating rate could be non-regular, etc. However, in the interbank market, just a few, standardized types are traded. They are listed below. Each currency has its own standard market conventions regarding the frequency of payments, the day count conventions and the end-of-month rule. Fixed-for-floating rate swap, different currencies. Party P pays/receives fixed interest in currency A to receive/pay floating rate in currency B indexed to X on a notional N at an initial exchange rate of FX for a tenure of T years. For example, you pay fixed 5.32% on the USD notional 10 million quarterly to receive JPY 3M (TIBOR) monthly on a JPY notional 1.2 billion (at an initial exchange rate of USD/JPY 120) for 3 years. For nondeliverable swaps, the USD equivalent of JPY interest will be paid/received (according to the FX rate on the FX fixing date for the interest payment day). No initial exchange of the notional amount occurs unless the Fx fixing date and the swap start date fall in the future. Fixed-for-floating swaps in different currencies are used to convert a fixed rate asset/liability in one currency to a floating rate asset/liability in a different currency, or vice versa. For example, if a company has a fixed rate USD 10 million loan at 5.3% paid monthly and a floating rate investment of JPY 1.2 billion that returns JPY 1M Libor +50bps monthly, and wants to lock in the profit in USD as they expect the JPY 1M Libor to go down or USDJPY to go up (JPY depreciate against USD), then they may enter into a Fixed-Floating swap in different currency where the company pays floating JPY 1M Libor+50bps and receives 5.6% fixed rate, locking in 30bps profit against the interest rate and the fx exposure. Floating-for-floating rate swap, same currency. Party P pays/receives floating interest in currency A Indexed to X to receive/pay floating rate in currency A indexed to Y on a notional N for a tenure of T years. For example, you pay JPY 1M LIBOR monthly to receive JPY 1M TIBOR monthly on a notional JPY 1 billion for 3 years or you pay EUR 3M EURIBOR quarterly to receive EUR 6M EURIBOR semi-annually. The second type of example, where the indexes are of the same type but with different tenors, are the most liquid and most traded same currency floating-for-floating swaps. Floating-for-floating rate swaps are used to hedge against or speculate on the spread between the two indexes widening or narrowing. For example, if a company has a floating rate loan at JPY 1M LIBOR and the company has an investment that returns JPY 1M TIBOR + 30bps and currently the JPY 1M TIBOR = JPY 1M LIBOR + 10bps. At the moment, this company has a net profit of 40bps. If the company thinks JPY 1M TIBOR is going to come down (relative to the LIBOR) or JPY 1M LIBOR is going to increase in the future (relative to the TIBOR) and wants to insulate from this risk, they can enter into a float-float swap in same currency where they pay, say, JPY TIBOR + 30bps and receive JPY LIBOR + 35bps. With this, they have effectively locked in a 35bit/s profit instead of running with a current 40bps gain and index risk. The 5bps difference (w.r.t. the current rate difference) comes from the swap cost which includes the market expectations of the future rate difference between these two indices and the bid/offer spread which is the swap commission for the swap dealer. Floating-for-floating rate swaps are also seen where both sides reference the same index, but on different payment dates, or use different business day conventions. This can be vital for asset-liability management. An example would be swapping 3M LIBOR being paid with prior non-business day convention, quarterly on JAJO (i.e. Jan, Apr, Jul, Oct) 30, into FMAN (i.e. Feb, May, Aug, Nov) 28 modified following. Floating-for-floating rate swap, different currencies Party P pays/receives floating interest in currency A indexed to X to receive/pay floating rate in currency B indexed to Y on a notional N at an initial exchange rate of FX for a tenure of T years. The notional is usually exchanged at the start and at the end of the swap. This is the most liquid type of swap with different currencies. For example, you pay floating USD 3M LIBOR on the USD notional 10 million quarterly to receive JPY 3M TIBOR quarterly on a JPY notional 1.2 billion (at an initial exchange rate of USDJPY 120) for 4 years; at the start you receive the notional in USD and pay the notional in JPY and at the end you pay back the same USD notional (10 million) and receive back the same JPY notional (1.2 billion). To explain the use of this type of swap, consider a US company operating in Japan. To fund their Japanese growth, they need JPY 10 billion. The easiest option for the company is to issue debt in Japan. As the company might be new in the Japanese market without a well known reputation among the Japanese investors, this can be an expensive option. Added on top of this, the company might not have appropriate debt issuance program in Japan and they might lack sophisticated treasury operation in Japan. To overcome the above problems, it can issue USD debt and convert to JPY in the FX market. Although this option solves the first problem, it introduces two new risks to the company: FX risk. If this USDJPY spot goes up at the maturity of the debt, then when the company converts the JPY to USD to pay back its matured debt, it receives less USD and suffers a loss. USD and JPY interest rate risk. If the JPY rates come down, the return on the investment in Japan might go down and this introduces an interest rate risk component. The first exposure in the above can be hedged using long dated FX forward contracts but this introduces a new risk where the implied rate from the FX spot and the FX forward is a fixed rate but the JPY investment returns a floating rate. Although there are several alternatives to hedge both the exposures effectively without introducing new risks, the easiest and the most cost effective alternative would be to use a floating-for-floating swap in different currencies. In this, Fixed-for-fixed rate swap, different currencies Party P pays/receives fixed interest in currency A to receive/pay fixed rate in currency B for a term of T years. For example, you pay JPY 1.6% on a JPY notional of 1.2 billion and receive USD 5.36% on the USD equivalent notional of 10 million at an initial exchange rate of USDJPY 120. JSFH

JSFH Interest rate swaps expose users to interest rate risk and credit risk. Market Risk: A typical swap consists of two legs, one fixed, the other floating. The risks of these two component will naturally differ. Newcomers to market finance may think that the risky component is the floating leg, since the underlying interest rate floats, and hence, is unknown. This first impression is wrong. The risky component is in fact the fixed leg and it is very easy to see why this is so. The discussion of pricing interest rate swaps illustrated an important point. Regardless of what happens to future Libor rates, the value of a rolling deposit or FRN always equals the notional amount N at the reset dates. Between the reset dates this value may be different than N, but the discrepancy cannot be very large since the \316\264 will be 3 or 6 months. Interest rate fluctuations have minimal effect on the values of fixed instruments with such maturities. In other words, the value of the floating leg changes very little during the life of a swap. On the other hand the fixed leg of a swap is equivalent to a coupon bond and fluctuations of the swap rate may have major effects on the value of the future fixed payments. Credit Risk. Credit risk on the swap comes into play if the swap is in the money or not. If one of the parties is in the money, then that party faces credit risk of possible default by another party. However, when the swap is negotiated through an intermediary financial institution, usually the intermediary assumes the default risk in exchange for a fixed percentage of the transaction (the bid-ask spread). In an intermediated swap, the two parties are not typically even aware of the identity of the second party to the transaction, making a quantification of the other party's credit risk not only irrelevant, but impossible. JSFH

JSFH The valuation of vanilla swaps was often done using the so-called textbook formulas using a unique curve in each currency. Some early literature described some incoherence introduced by that approach and multiple banks were using different techniques to reduce them. It became even more apparent with the 2007\342\200\2232012 global financial crisis that the approach was not appropriate. The now standard pricing framework is the multi-curves framework. The present value of a plain vanilla (i.e. fixed rate for floating rate) swap can be computed by determining the present value (PV) of the fixed leg and the floating leg. The value of the fixed leg is given by the present value of the fixed coupon payments known at the start of the swap, i.e. 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the swap rate, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the number of fixed payments, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the notional amount, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYyis the accrual factor according to the day count convention for the fixed rate period and 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 is the discount factor for the payment time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiVRImlGJ0YvRjIvRjNRJ25vcm1hbEYnRj0vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGPQ== . The value of the floating leg is given by the present value of the floating coupon payments determined at the agreed dates of each payment. However, at the start of the swap, only the actual payment rates of the fixed leg are known in the future, whereas the forward rates are unknown. The forward rate for each floating payment date is calculated using the forward curves. The forward rate for the period 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 with accrual factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSgmZGVsdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUYsNiVRImlGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0Y6LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOkYrRjo= is given by 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 where I is the market index, such as USD LIBOR, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnUEluZGV4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEifkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMRjk=is the discount factor associated to the relevant forward curve. The value of the floating leg is given by the following: 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the number of floating payments, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYkLUYsNiVRImlGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjIvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVuRjI= is the accrual factor according to the floating leg day count convention. The fixed rate offered in the swap is the rate which values the fixed rates payments at the same PV as the variable rate payments using today's forward rates, i.e.: Therefore, at the time the contract is entered into, there is no advantage to either party, i.e., 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 Thus, the swap requires no upfront payment from either party. During the life of the swap the same valuation technique is used, but since, over time, both the discounting factors and the forward rates change, the PV of the swap will deviate from its initial value. Therefore, the swap will be an asset to one party and a liability to the other. The way these changes in value are reported is the subject of IAS 39 for jurisdictions following IFRS, and FAS 133 for U.S. GAAP. Swaps are marked to market by debt security traders to visualize their inventory at a certain time. JSFH

JSFH Objective We said beforethat we can use yield curves to calculatethe forward rates that will help us in pricing a swap. Another option is to use some sort of forecast model. Maple provide us with a way of forecasting the rates udinghistorical data. To build the corresponding structures we need to start seting up a date structure and a rate structure. this is how this looks like. Maple has thre following references: AUDLIBOR, CADLIBOR, CDOR, CHFLIBOR, DKKLIBOR, EURIBOR, GBPLIBOR, JIBAR, JPYLIBOR, NZDLIBOR, TIBOR, TRLIBOR, USDLIBOR, ZIBOR. Maple has historical data. Before the evaluation date it will use historic data. After the evaluation date it will use the forecasts. Flat Forward Curve and Variable Swap Spread The building blocks of the calculator. For the variable leg of the swap we will nedd to have a view on the model that generates such rates. In structuring such process with Maple we ned some minimal elements: a.) a "Starting Date" for the forecasting model; b.) a "Calendar Convention", which in general can be the AUDLIBOR, CADLIBOR, CDOR, CHFLIBOR, DKKLIBOR, EURIBOR, GBPLIBOR, JIBAR, JPYLIBOR, NZDLIBOR, TIBOR, TRLIBOR, USDLIBOR, ZIBOR; c.) We need a curve and a specification of such curve to produce the corresponding forward rates. Such curve can be a Zero, Par or Forward depending on how we want to interpret the future. We can have a more sophisticated approach and model it with a Montecarlo approach, for which we will need a specification of short rates. The simplest example is one where we have a swap that starts with a Flat Forward Curve at a given level. Ahother important parameter is the Swap Spread. The calculator allows for a variable swap spread, however the default is set to zero. This concept is defined as the spread that the paying fixed leg of the swap has above the running Treasury Rate. Here would be defined above a generic concept of a "Running Rate". In comparing both flows we calculate the fair spread, that is, the spread for which the net present value of the swap is zero. The discount rate for that process will be the "Running Rate". CalculatorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= INTEREST RATE SWAP PAYS FIXED RATE AND RECEIVES VARIABLE RATE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Reference Date JSFH SCHEDULE OF THE VARIABLE (RECEIVING) LEG Starting Date Ending Date Frequency INTEREST RATE MODEL OF THE VARIABLE (RECEIVING) LEG Periods Units Forward (Flat) Curve Calendar Convention SCHEDULE OF THE FIXED (PAYING) LEG Starting Date Ending Date Frequency INTEREST RATE MODEL OF THE PAYING (FIXED) LEG Running Rate Swap SpreadLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= restart; use DocumentTools, Finance, plots in #######################################################################3 # LOAD THE INFORMATION Do(a[1]=DocumentTools[GetProperty]('TextArea0','value')): Do(a[2]=DocumentTools[GetProperty]('TextArea1','value')): Do(a[3]=DocumentTools[GetProperty]('MathContainer11','expression')): Do(a[4]=DocumentTools[GetProperty]('TextArea3','value')): Do(a[5]=DocumentTools[GetProperty]('MathContainer9','expression')): Do(a[6]=DocumentTools[GetProperty]('MathContainer10','expression')): Do(a[7]=DocumentTools[GetProperty]('TextArea2','value')): Do(a[8]=DocumentTools[GetProperty]('TextArea4','value')): Do(a[9]=DocumentTools[GetProperty]('MathContainer12','expression')): Do(a[101]=DocumentTools[GetProperty]('RadioButton0','value')): Do(a[102]=DocumentTools[GetProperty]('RadioButton1','value')): Do(a[103]=DocumentTools[GetProperty]('RadioButton2','value')): Do(a[104]=DocumentTools[GetProperty]('RadioButton3','value')): Do(a[105]=DocumentTools[GetProperty]('RadioButton4','value')): if a[101]="true" then a[10]:=USDLIBOR: elif a[102]="true" then a[10]:=GBPLIBOR: elif a[103]="true" then a[10]:=JPYLIBOR: elif a[104]="true" then a[10]:=CADLIBOR: elif a[105]="true" then a[10]:=EURIBOR: end if: Do(a[12]=DocumentTools[GetProperty]('MathContainer9','expression')): Do(a[13]=DocumentTools[GetProperty]('MathContainer10','expression')): Do(a[14]=DocumentTools[GetProperty]('MathContainer15','expression')): Do(a[15]=DocumentTools[GetProperty]('MathContainer14','expression')):#Running Rate Do(a[16]=DocumentTools[GetProperty]('MathContainer36','expression')):#Swap Spread #######################################################################3 # CREATE THE SERIES SCHEDULES SetEvaluationDate(a[4]); Schedule1 := Schedule(a[1], a[2], a[3]): Schedule2 := Schedule(a[7], a[8], a[9]); Rate_1 := BenchmarkRate(a[12], a[14], a[10], ForwardCurve(a[13], compounding = Continuous)); Rate_2:=a[15]: #######################################################################3 # PRODUCE THE SWAPS Swap1 := InterestRateSwap(1000, Rate_2, Schedule2, Rate_1, Schedule1, a[16]); PayingLeg1 := CashFlows(Swap1, paying); ReceivingLeg1 := CashFlows(Swap1, receiving): b := nops(ReceivingLeg1); c:=[seq(ReceivingLeg1[j], j = 1 .. b)]: b:=nops(PayingLeg1); g:=[seq(PayingLeg1[j], j = 1 .. b)]: f:=evalf((1/1000)*NetPresentValue(Swap1, a[15]), 10): Do(%MathContainer16=f): h:=evalf(FairSpread(Swap1, a[15]), 10): Do(%MathContainer17=h): end use: # CLICK THE ICON TO DISPLAY RECEIVING END c;

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= # CLICK THE ICON TO DISPLAY PAYING END g;

NypgNiVJLE1vZHVsZVByaW50RzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0ksX24xNTAxMDgxNTJHNiJJJWRhdGVHNiMvRihJLF9uMTUwMTk2NjA4R0YrSSdhbW91bnRHNiMvRihJLF9uMTUwMTk2NTc2R0YrSSxfbTE1MDE5NjY0MEdGK0YrYEYkSSxfbTE1MDIwODEyOEdGK0YrYEYkSSxfbTE1MTA1NjE2MEdGK0YrYEYkSSxfbTE1MTA2MjM2OEdGK0YrYEYkSSxfbTE1MTA2NjQ2NEdGK0YrYEYkSSxfbTE1MTA3MjEyOEdGK0YrYEYkSSxfbTE1MTA5MTkwNEdGK0YrYEYkSSxfbTE1MTA5NDk0NEdGK0Yr

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Net Present Value of the Swap Fair Spread of the Swap LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 1. Create a time schedule for each one of the legs of the swap 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 2. Define the interest rates to be exchanged We will define the two legs using the following syntaxis. We will se the impact of each one of the commands as we solve an example. When defining rates, Maple will use historical data when available and the given forecast when we give a reference. We can provide a curve or additional information to create the data structure. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 3. Calculate the Swap payments. We will consider a simple case where we have a notional amount of $1,000; we have Rate1 as the fixed leg and refered to the Schedule1, while the variable is rate is Rate2 with the corresponding Schedule2. The swap spread is zero in this case. This is a simple example where one exchanges two flat curves. 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print(); # input placeholder

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print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 4. Calculate the fair Swap Spread 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print(); # input placeholder

JCEiI0Yj

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print(); # input placeholder

JCIiJyEiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH

JSFH Objective We add now some more realism to the exercise. Here we consider the possibility of starting from a Forward Curve. From there we generate the schedule of the variable leg. The rest of the exercise, proceed exactly the same as the previous section. CalculatorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= INTEREST RATE SWAP PAYS FIXED RATE AND RECEIVES VARIABLE RATE LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Reference Date JSFH SCHEDULE OF THE VARIABLE (RECEIVING) LEG Starting Date Ending Date Frequency INTEREST RATE MODEL OF THE VARIABLE (RECEIVING) LEG Forward Curve Times Rates SCHEDULE OF THE FIXED (PAYING) LEG Starting Date Ending Date Frequency INTEREST RATE MODEL OF THE FIXED (PAYING) LEG Running Rate Swap Spread LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= restart; use DocumentTools, Finance, plots in #################################################################################### # LOAD THE INFORMATION Do(a[1]=DocumentTools[GetProperty]('TextArea11','value')):# Initial Date Variable Leg Do(a[2]=DocumentTools[GetProperty]('TextArea12','value')):# Final Date Variable Leg Do(a[3]=DocumentTools[GetProperty]('MathContainer20','expression')): # Frequency Variable Leg Do(a[4]=DocumentTools[GetProperty]('TextArea10','value')):#Reference Date Do(a[7]=DocumentTools[GetProperty]('TextArea13','value')):# Initial Date Fixed Leg Do(a[8]=DocumentTools[GetProperty]('TextArea14','value')):# Final Date Variable Leg Do(a[9]=DocumentTools[GetProperty]('MathContainer27','expression')): #Frequency Fixed Leg Do(a[12]=DocumentTools[GetProperty]('MathContainer25','expression')):#Times Do(a[14]=DocumentTools[GetProperty]('MathContainer26','expression')):#Rates Do(a[15]=DocumentTools[GetProperty]('MathContainer28','expression')):#Running Rate Do(a[20]=DocumentTools[GetProperty]('MathContainer34','expression')):#SwapSpread ############################################################################################## # CREATE THE SERIES SCHEDULES SetEvaluationDate(a[4]); Schedule2 := Schedule(a[1], a[2], a[3]): Schedule1 := Schedule(a[7], a[8], a[9]); ############################################################################################## #MODELING THE YIELD CURVE times := a[12]; rates := a[14]; forwardcurve := ForwardCurve(times, rates, referencedate = a[4]): Rate_2:=BenchmarkRate(ForwardCurve(times, rates, referencedate = a[4])): Rate_1:= a[15]: ################################################################################################ # PRODUCE THE SWAPS Swap1 := InterestRateSwap(1000, Rate_2, Schedule2, Rate_1, Schedule1, a[20]); PayingLeg1 := CashFlows(Swap1, paying); ReceivingLeg1 := CashFlows(Swap1, receiving): b := nops(ReceivingLeg1); c:=[seq(ReceivingLeg1[j], j = 1 .. b)]: b:=nops(PayingLeg1); g:=[seq(PayingLeg1[j], j = 1 .. b)]: f:=evalf((1/1000)*NetPresentValue(Swap1, a[15]+a[20]), 3): Do(%MathContainer29=f): h:='h': q:=evalf(Finance:-FairSpread(Swap1, a[15]+a[20]), 10): Do(%MathContainer30=q): end use: # DISPLAY RECEIVING END c;

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= # DISPLAY PAYING END g;

NypgNiVJLE1vZHVsZVByaW50RzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0ksX24xODA3MzM1ODRHNiJJJWRhdGVHNiMvRihJLF9uMTgxMDMxNTUyR0YrSSdhbW91bnRHNiMvRihJLF9uMTgxMDMxNTIwR0YrSSxfbTE4MTAzMTU4NEdGK0YrYEYkSSxfbTE4MTAyNzIwMEdGK0YrYEYkSSxfbTE4MTAzMDI0MEdGK0YrYEYkSSxfbTE4MTAyNTEyMEdGK0YrYEYkSSxfbTE4MTAyMDAzMkdGK0YrYEYkSSxfbTE4MDEyNjA0OEdGK0YrYEYkSSxfbTE4MDEyOTA4OEdGK0YrYEYkSSxfbTE4MDEyMzkzNkdGK0Yr

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Net Present Value of the Swap Fair Spread of the Swap LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 1. Create a time schedule for each one of the legs of the swap 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 2. Define the interest rates to be exchanged 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 3. Define the interest rates to be exchanged 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 print(); # input placeholder 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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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JSFH Our starting point will be to set up the scaffolding for our exercise, by choosing a random process and creating the underlying curve of Zeros. From there, we build the curve of forwards. Finally with that information we construct the swap and determine the fair spread. Step 1. Build the scaffolding of the interest rate model. Remember thet the objective is to create a series of interest rate paths. The underlying sochastic process will produce curves of Zero Coupons. Then, we need to generate the implied Forwar Rates which are the basic input for the interest rate swap. 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 print(); # input placeholder JSFH

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjWkNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSpaZXJvQ3VydmVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSZ0aW1lc0YnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRJnJhdGVzRidGL0YyLyUrZXhlY3V0YWJsZUdGPS8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0Y5RjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2I1EhRidGX29GYW9GOQ== print(); # input placeholder 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 print(); # input placeholder 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 JSFH Step 2. Produce the Forward Curve. From that curve we will apply the command BenchamrkRate which will be used in producing the swap. 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Display the Paying Leg of the Swap 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print(); # input placeholder

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Step 4. Produce the Receiving Leg of the Swap. 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print(); # input placeholder

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JSFH Step 5. Determine the Fair Spread 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print(); # input placeholder

JCIrXik+I1tdISM2

JSFH JSFH print(); # input placeholder

JSFHJSFH Concept Sometimes we may prefer to start from a rate-generating process instead of from a given yield or forward curve. The sequence moves closer to the fundamentals of the interest rate dynamics. Here we begin at the level of a Vasicek model. From the process we can produce numerous sample paths of the short rates. From each sample path one can construct a sample Forward Curve. With that at hand we can calculate the Fair Spread of the corresponding swap. This exercise is repeated for as many times as we want to end with a distribution of such Fair Spreads. The parameter structure of the exercise is the same as the ones used in the previous sections, that is, the payment schedules, the reference date and the specification of the fixed leg. JSFHCalculator %; INTEREST RATE SWAP PAYS FIXED RATE AND RECEIVES VARIABLE RATELUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= PARAMETERS OF THE VASICEK MODEL FOR THE VARIABLE LEG OF THE SWAP r \316\274 JSFH JSFH JSFH \316\270 \317\203 JSFH Replications JSFH Reference Date JSFH JSFH SCHEDULE OF THE VARIABLE (RECEIVING) LEG Starting Date Ending Date Frequency LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= SCHEDULE OF THE FIXED (PAYING) LEG Starting Date Ending Date Frequency Running Rate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFHrestart; use DocumentTools, Finance, Statistics, plots in #################################################################################### # PARAMETERS AND SPECIFICATION OF THE VASICEK MODEL Do(m[1]=DocumentTools[GetProperty]('Slider0','value')):# Parameter 1 de Vasicek Do(m[2]=DocumentTools[GetProperty]('Slider1','value')):# Parameter 2 de Vasicek Do(m[3]=DocumentTools[GetProperty]('Slider2','value')):# Parameter 3 de Vasicek Do(m[4]=DocumentTools[GetProperty]('Slider3','value')):# Parameter 4 de Vasicek Do(m[5]=DocumentTools[GetProperty]('Slider4','value')): #Number of Replications Do(n[1]=DocumentTools[GetProperty]('TextArea15','value')):#Starting Date 2 Do(n[2]=DocumentTools[GetProperty]('TextArea16','value')):#Ending Date 2 Do(n[3]=DocumentTools[GetProperty]('MathContainer31','expression')):# Frequency Date 2 Do(n[4]=DocumentTools[GetProperty]('TextArea17','value')):# Starting Date 1 Do(n[5]=DocumentTools[GetProperty]('TextArea18','value')):# Ending Date 1 Do(n[6]=DocumentTools[GetProperty]('MathContainer32','expression')):# Frequency Date 1 Do(n[7]=DocumentTools[GetProperty]('TextArea19','value')):# Reference Date SetEvaluationDate(n[7]); Schedule1 := Schedule(n[4],n[5],n[6]); Schedule2 := Schedule(n[1],n[2],n[3]); M := VasicekModel(m[1],m[2],m[3],m[4]); X := ShortRateProcess(M); S := SamplePath(X(t), t = 0 .. 50, timesteps = 5000, replications = 1); FS := []; PVS := []; W := seq(S[1, k], k = 1 .. 5000): Do(a[3]=DocumentTools[GetProperty]('MathContainer35','expression')): #Rate1 for k from 2 to m[5] do T := [seq(.6*i, i = 0 .. 50)]; R := DiscountBondPrice(M, max(W[k],0), T, output = zerorate); times := [seq(T[j], j = 2 .. 51)]; rates := [seq(R[j], j = 2 .. 51)]; ZC := ZeroCurve(times, rates); FR := [seq(ForwardRate(ZC, times[j], times[j+1]), j = 1 .. 49)]; TIMES := [seq(times[j], j = 1 .. 49)]; forwardcurve := ForwardCurve(TIMES, FR, referencedate = n[7]); Rate1 := a[3]; Rate2 := BenchmarkRate(forwardcurve); Swap1 := InterestRateSwap(1000, Rate1, Schedule1, Rate2, Schedule2, 0.7e-1); fs := FairSpread(Swap1, a[3]); FS := [fs, op(FS)]; pvs := evalf((1/1000)*NetPresentValue(Swap1, a[3]), 1); PVS := [pvs, op(PVS)]: end do: H1:=Histogram(FS, gridlines = true, frequencyscale = relative): #H2:=Histogram(PVS, #gridlines = true, #binbounds=proportional, #frequencyscale = relative): M1:=Statistics:-Mean(FS); #M2:=Statistics:-Mean(PVS): Do(%MathContainer33=M1): #Do(%MathContainer34=M2): Do(%Plot0=H1): #Do(%Plot1=H2): end use: JSFH Simulated Fair Spread 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Average Fair Spread LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH Step 1. Build again the framework for the interest rate model. 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 print(); # input placeholder 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 2. Produce 5000 simulations. These simulations consider 5,000 realizations of the Vasicek model. For each one of them we produce a swap, and for each swap we calculate the Fair Spread. 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 print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 3. We produce the Histogram of the distribution of Fair Spreads 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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Ig==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 4. Display the Histogram of the Present Value of the Swap. 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

%;

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NiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Step 5. Calculate the Mean Values of the Fair Spread and the Present Value of the Swap 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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

XTNGQUY1NzM4QTQ5QjQ2Qjg=

XTNGOTBGMzg3MEY1MUExRTU=

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