Deri DFBVSDFHSDTH DRTRJDT TFJ

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Transcript of Deri DFBVSDFHSDTH DRTRJDT TFJ

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DescriptionDescription

A swap is an agreement between two parties to exchange (swap) payments at certain dates in the future.

Counterparty A Counterparty B

A’s payments to B

B’s payments to A

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DescriptionDescription

The payments can beDifferent currencies (currency swap)Different interest payments (coupon swap)Different commodities (e.g. oil swap)

Payment dates are fixed at settlement and extend until a fixed expiration date.

Payments are determined at least one payment period in advance, (payment in arrears).

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Plain vanilla swapPlain vanilla swap

A standard swap is an interest rate swap in whichInterest payments based on a fixed rate are exchanged forInterest payments based on a floating rate

Interest is calculated on an amount called the notional principal (or just the notional) of the swap.

The notional is not exchanged.

For example, if payments are made annually, the fixed leg payment is rfixed times Bnotional and the floating leg payment is rfloating(t) times Bnotional.

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Plain vanilla swapPlain vanilla swap

Counterparty A is called the fixed rate payer or swap buyer

Counterparty B is called the floating rate payer or swap seller

Counterparty A Counterparty B

Fixed rate payments

Floating rate payments

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ExampleExample

In this five-year swap, 12-month LIBOR is swapped for 2.67% fixed, on $100 million.

At initiation, the planned payments are:

Floating Leg Fixed LegYear 1-yr LIBOR Payment Fixed rate Payment

0 1.52% 2.67%1 2.00% 1,520,000$ 2.67% 2,670,000$ 2 2.60% 2,000,000$ 2.67% 2,670,000$ 3 3.30% 2,600,000$ 2.67% 2,670,000$ 4 4.12% 3,300,000$ 2.67% 2,670,000$ 5 4,120,000$ 2,670,000$

Hypothetical 5-year Swap

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A par bond and a floaterA par bond and a floater

Except for the receipt and payment of principal,The fixed leg of this swap is the set of payments the swap

buyer would make if she issued a five-year 2.67% par bond on $100 million, and

The floating leg is the set of payments she would receive if she invested $100 million in a five-year annual “floater” bond paying 12-month LIBOR.

That is, the fixed-rate payer’s position is equivalent to being short the 2.67% par bond, and long the floater, except that principal payments are not exchanged

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A par bond and a floaterA par bond and a floater

If Vfixed is the value of the par bond, and Vfloating is the value of the floater, then

To the fixed-rate payer, the value of the swap is Vfloating - Vfixed

And to the floating-rate payer, the value of the swap isVfixed - Vfloating

At initiation, the value must be the same to both counterparties, which can only be when

V fixed =V floating

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Valuing the floaterValuing the floater

The floater is valued by working backward from the final maturity.

In the example, 12-month LIBOR at the 4th set date (beginning of year 5) is 4.12%.

One year later, at maturity, the floater would pay (1+.0412)xN where N is the notional.

At the set date, the value of this future payment (and the value of the floater) is

1+ .0412( )N

1+ .0412( )= N

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Valuing the floaterValuing the floater

In fact, at any set date t = 0, … , T-1

1+ LIBOR(t, t +1)( )N

1+ LIBOR(t, t +1)( )= N

That is, at all set dates, (including in particular the initial one, t = 0), the value of the floater is par.

At t = 0, the value of the fixed leg must therefore also be par. This means the fixed rate (swap rate) must be the yield on the T-year LIBOR par bond.

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Finding the swap rate given the Finding the swap rate given the LIBOR curveLIBOR curve

The LIBOR curve is a set of LIBOR forward rates.

In the example above, the LIBOR curve was given by 12-month LIBOR rates extending 5 years into the future.

Year 1-yr LIBOR0 1.52%1 2.00%2 2.60%3 3.30%4 4.12%5

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Finding the swap rate given the Finding the swap rate given the LIBOR curveLIBOR curve

From the LIBOR curve, the discount function, d(s), which gives the present value of $1 to be delivered at date s, can be computed as

Year 1-yr LIBOR Discount0 1.52%1 2.00% 0.9850275812 2.60% 0.9657133143 3.30% 0.9412410474 4.12% 0.9111723595 0.875090854

d s( ) =1

1+ r t, t +1( )( )t= 0

s−1

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Finding the swap rate from the Finding the swap rate from the LIBOR curveLIBOR curve

The LIBOR spot (zero rate) curve is computed from the discount function.

LIBOR/SwapYear 1-yr LIBOR Discount Zero Rate

0 1.52%1 2.00% 0.985027581 1.52%2 2.60% 0.965713314 1.76%3 3.30% 0.941241047 2.04%4 4.12% 0.911172359 2.35%5 0.875090854 2.70%

i(0,s) =1

d s( )

⎣ ⎢

⎦ ⎥

12

−1

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Finding the swap rate given the Finding the swap rate given the LIBOR curveLIBOR curve

The par yield (swap rate) curve can be determined from the discount function:

LIBOR/Swap LIBOR/SwapYear 1-yr LIBOR Discount Zero Rate Par Rate

0 1.52%1 2.00% 0.985027581 1.52% 1.52%2 2.60% 0.965713314 1.76% 1.76%3 3.30% 0.941241047 2.04% 2.03%4 4.12% 0.911172359 2.35% 2.34%5 0.875090854 2.70% 2.67%

Hypothetical 5-year Swap

y par s( ) =1− d s( )

d t( )t=1

s

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Pricing a seasoned swapPricing a seasoned swap

Swaps have zero value at initiation, but after will change in value as interest rates change.The duration of the fixed leg is longer than the

duration of the floating leg, soThe value of the fixed leg will decrease (increase)

more than the floating leg when interest rates rise (decline).

Thus, the buyer of a swap will benefit when rate rise and the seller of a swap will benefit when rates fall.

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Pricing a seasoned swapPricing a seasoned swap

To the fixed rate payer, the value of a seasoned swap (i.e. at 0<t<T) is the time t value of the remaining floating rate payments minus the time t value of the remaining fixed rate payments.

tt+s-1 t+s T

Transaction date Swap maturity date

Previous set date Next set date

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Pricing a seasoned swapPricing a seasoned swap

The floating rate leg can be valued by Taking the present (time t) value of a floater paying LIBOR

plus notional at t+s, andRealizing that at each set date between t+s and T, the floater is

valued at par.

tt+s-1 t+s T

Transaction date Swap maturity date

Previous set date Next set date

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Pricing a seasoned swapPricing a seasoned swap

The fixed rate leg can be valued by discounting the fixed rate payments between t+s and T (and notional at T) using the LIBOR/swap zero curve observed at t.

tt+s-1 t+s T

Transaction date Swap maturity date

Previous set date Next set date

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ExampleExample

Mid-way through the 5-year swap described above, there are three payments remaining.12-month LIBOR has remained low (rates have

fallen compared to previous expectations).

Time from Discount toYear Today 1-yr LIBOR Today

2 -0.5 1.50%3 0.5 2.00% 0.9925833344 1.5 2.50% 0.9731209165 2.5 0.949386259

2 1/2 year Residual Swap

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ExampleExample

The anticipated cash flows of the residual swap are

Time from Floating Leg Fixed LegYear Today 1-yr LIBOR Payment Payment

2 -0.5 1.50%3 0.5 2.00% 1,500,000$ 2,670,000$ 4 1.5 2.50% 2,000,000$ 2,670,000$ 5 2.5 2,500,000$ 2,670,000$

2 1/2 year Residual Swap

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ExampleExample

To get the value of the floating leg, discount the $1.5 million payment due 6 months from now plus the $100 million notional using the discount function for s = 1/2

d 12( ) =

1

1+ .0150( )1

2

V floating = $101,500,000 × d 12( )

= $100,747,208

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ExampleExample

To get the value of the fixed leg, discount the future payments of $2.67 million on each of the remaining settlement dates plus the $100 million notional at time T using the discount functions for s = 1/2, 1 1/2, and 2 1/2.

V fixed = $2,670,000 × d 12( ) + d 1 1

2( )( ) + $102,670,000 × d 2 12( )

= $102,721,918

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ExampleExample

The value of the swap to the fixed rate payer is

Vswap =V floating −V fixed= $100,747,208 − $102,721,918

= $ 161,396( )

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A more realistic exampleA more realistic example

In practice, the standard swap is a semi-annual pay fixed rate for 3-month LIBOR

Term in years 5Fixed rate 4.97%Notional principal $10,000,000

Set 3-month Pay Act/360 Floating Pay 30/360 Fixed PayDate LIBOR Date Days Amount Days Amount

3/2/06 4.83% 3/4/066/2/06 5.26% 6/5/06 93 $124,775 919/5/06 5.39% 9/7/06 94 $137,344 92 $252,642

12/4/06 5.35% 12/6/06 90 $134,750 893/2/07 5.34% 3/5/07 89 $132,264 89 $245,7396/4/07 5.36% 6/6/07 93 $137,950 919/4/07 5.85% 9/6/07 92 $136,978 90 $249,881

12/3/07 5.30% 12/5/07 90 $146,250 893/3/08 3.07% 3/5/08 91 $133,972 90 $247,1196/3/08 2.98% 6/5/08 92 $78,456 909/2/08 3.00% 9/4/08 91 $75,328 89 $247,119

12/2/08 3.00% 12/4/08 91 $75,833 903/2/09 1.65% 3/4/09 90 $75,000 90 $248,5006/2/09 1.20% 6/4/09 92 $42,167 909/2/09 9/4/09 92 $30,667 90 $248,500

Three and a half years into a 5-year swap

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