Options and Speculative Markets2004-2005Swapnote – Wrap up
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
Swapnote 2004 |2August 23, 2004
Outline
(1) Piggibank is short (receives fixed rate and pays floating rate) on:
– a 4% 5-year swap
– notional principal of €10 million.
The current 5-yr swap rate is 3.29% (Exhibit 1). So the value of this swap is positive for Piggibank.
Step 1 of the analysis is to calculate this value.
(2) Interest rates might change. This would modify the value of the swap.
Step 2 of the analysis is to calculate by how much the value of the swap will change if interest rates change by 0.01% (1 basis point – bp) – the Basis Point Value (BVP) of the swap.
(3) Piggibank considers hedging its swap position using Swapnote futures.
Step 3 of the analysis is to understand by the payoff on one futures contract if interest rates change by 0.01% - the Basis Point Value of one Swapnote.
(4) The number of Swapnote to short is equal to the ratio:
BVP(Swap)/BVP(Swapnote)
Swapnote 2004 |3August 23, 2004
Summary of results
(1) Value of swap for Piggibank: VSwap = €325,337
(2) Duration of Swap: DSwap = 116
Basis Point Value of Swap BVPSwap = - €3,782
(3) Swapnote = futures on 6% notional bond
Tick (Value of ∆F = 0.01) = €10
BVPSwapnote = - €50.35
Note: if interest rates ↑→Futures price ↓ short swapnote
(4) Number of swapnotes to short to hedge position:
n = (- 3,782) / (- 50.35) = 75
Swapnote 2004 |4August 23, 2004
1. Current value of the swap of Piggibank
• Piggibank is short on a 4% 5 yr swap with a notional principal of €10 million.
• To value this swap:
• 1- Calculate the discount factors from the current swap rates.• See next slide for details
• 2- Calculate the value of the fixed rate bond• Vfix = 400,000 d1 + 400,000 d2 + ...+ 10,400,000 d5
• = 10,325,337
• 3- Subtract the value of the floating rate bond (equal to the principal)• Vfloat = 10,000,000
• Vswap = 10,325,337 – 10,000,000
• = 325,337
Swapnote 2004 |5August 23, 2004
Calculation of discount factors
• Bootstrap method. Solve the following equations:
100 = 102.30 d1
100 = 2.56 d1 + 102.56 d2
100 = 2.83 d1 + 2.83 d2 + 102.83 d3
100 = 3.07 d1 + 3.07 d2 + 3.07 d3 + 103.07 d4
100 = 3.29 d1 + 3.29 d2 + 3.29 d3 + 3.29 d4 + 103.29 d5
• Use eq.1 to obtain d1
• Replace d1 in eq.2 and solve for d2
• Replace d1 and d2 in eq.3 and solve for d3
• .....
• or use matrix algebra: d = C-1 P
Swapnote 2004 |6August 23, 2004
2. Duration of swap
• As: floatfixswap VVV
rVD
rVV
VD
V
VD
rVDVD
VVV
SwapSwap
SwapSwap
floatfloat
Swap
fixfix
floatfloatfixfix
floatfixSwap
)(
)(
3.116
337,325
000,000,101
337,325
337,325,1063.4
Swap
floatfloat
Swap
fixfixSwap V
VD
V
VDD
Swapnote 2004 |7August 23, 2004
Using duration
• Suppose the interest rate change ∆r = 0.01% (= + 1bp)
782,3€
000,1€%01.0000,000,101
782,4€%01.0337,325,1063.4
SwapSwap
float
fix
BVPV
V
V
Swapnote 2004 |8August 23, 2004
Swapnote
• A futures contract on a 6% notional coupon bond.
• Face value = €100,000
• To calculate the futures price, use general approach:
• S0 is the spot price of the underlying asset (a 6% coupon bond)
• T is the maturity of the futures contract (2 month = 0.167 yr)
• r is the 2-month interest rate (with continuous compounding)
rTeSF 00
Today
Maturity of futures Coupon Coupon
02 m 1yr 2 m 2 yr 2 m 5 yr 2 m
Coupon + Principal
0.167 1.167 2.167 5.167
Swapnote 2004 |9August 23, 2004
Spot price calculation
167.5167.2167.1167.00 106...666 ddddS
Some sort of interpollation is required to find the proper discount factor.
In the Excel spreadsheet, I proceed as follow:
1. I compute the spot interest rates (with continuous compounding) for various maturities
2. I fit a polynomial function:
r(t) = a0 + a1 t + a2 t² + a3 t3
where r(t) is the spot rate with continuous compounding for maturity t
3. The discount factor is d(t) = exp(-r(t)t)
Swapnote 2004 |10August 23, 2004
Swapnote quotation
• S0 = 111.71
• F0 = 111.71 / 0.99653 = 112.10
• The duration of the underlying bond is 4.66.
• If the interest rate change ∆r = 0.01% (= + 1bp)
• ∆F0 = -0.05 (= - 5 bp) (see next slide for details)
• As the size of the contract is €100,000:
• ∆r = 0.01% → ∆F0 = -0.05
• → BVPSwapnote = €100,000 (-0.05) / 100 = - €50
Swapnote 2004 |11August 23, 2004
Duration of swapnote (details)
• Suppose the interest rate change ∆r = 0.01% (= + 1bp)
• By how much will the price of the swapnote change?
• What about the futures price?
0521.0
%01.071.11166.40%60
rSDS Bond
05.0
%01.071.111167.0052.0
)( 000
00
reTSeSF
eSFrTrT
rT
Swapnote 2004 |12August 23, 2004
Setting up the hedge
• What do we know?
• If ∆r = 0.01% (= + 1 bp)
• BVPSwap = - € 3,782
• BVPSwapnote = - €50/contract
• To hedge its swap position, Piggibank should short n futures swapnotes contract so that:
7550
782,3100
)000,100(
n
FnVSwap