Tomas Vitvar, Marco Brambilla tomas.vitvar@deri, [email protected]
Deri DFBVSDFHSDTH DRTRJDT TFJ
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Transcript of Deri DFBVSDFHSDTH DRTRJDT TFJ
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
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).
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.
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
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
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
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
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
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.
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
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
∏
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
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
∑
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.
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
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
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
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
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
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
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
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( )
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