Derivatives2010-2011Swap Pro – Wrap up
André Farber & Stéphanie ColletSolvay Brussels School of Economics and ManagementUniversité Libre de Bruxelles
May 5, 2011 |2
Outline
(1) interest rate swap - valueSwap Pro is short (receives fixed rate and pays floating rate) on:
– A 4% 7-year swap– notional principal of $1 million.The current 7-yr swap rate is 2.99% (Exhibit 1) the value of the swap is positiveStep 1 of the analysis is to calculate this value.
(2) interest rate swap - durationInterest 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) interest rate swap futuresSwap Pro considers using the CBOT Swap 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.
May 5, 2011 |3
Outline
(4) interest rate swap futures – hedgingSwap Pro considers hedging its swap position using the CBOT Swap futures.
Step 4 of the analysis is to calculate the number of Swapnote needed to hedge the swap position using the CBOT 7-Year Swap futures
- short - number of Swapnote equal to the ratio: BVP(Swap)/BVP(Swapnote)
(5) Rollercoast IR swapSwap Pro is long (receives floating rate and pays fixed rate) on:
– A 2.8% 6-year swap– Notional principal progressively from $10 million to $50 million the next 2 years
and then progressively declines to zero– The current 6-yr swap rate is 2.74% (Exhibit 1)
Step 5 of the analysis is to value the Rollercoast IR swap
May 5, 2011 |4
Swap Pro « results interest rate swap »
May 5, 2011 |5
Short solution (bypass term structure problem)
( , )(1 )n
FP C a r nr
1 ( , ) (1 )y yD n a r n rr r
Bond price formula Duration Babcock’s formula
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1. Current value of the interest rate swap
• Swap Pro is short on a 4% 7 yr swap with a notional principal of $1million.• 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 = 20,000 d0.5 + 20,000 d1 + ...+ 1,020,000 d7 = 1,064,294
3- Subtract the value of the floating rate bond (equal to the principal)• Vfloat = 1,000,000
Vswap = 1,064,294 – 1,000,000 = 64,294
May 5, 2011 |7
Calculation of discount factors
Bootstrap method - Solve the following equations:100 = 100.43 d1
100 = 0.89 d1 + 100.89 d2
100 = 1.43 d1 + 1.43 d2 + 101.43 d3
100 = 1.91 d1 + 1.91 d2 + 1.91 d3 + 101.91 d4…
100 = 2.99 d1 + 2.99 d2 + 2.99 d3 + 2.99 d4 + 2.99 d5 + 2.99 d6+102.99 d7
• 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
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Calculation of discount factors
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Spot price calculation
Some sort of interpollation is required to find the proper discount factor atall times.
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 maturityt
3. The discount factor is d(t) = exp(-r(t)t)
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Calculation of discount factors
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
3.00%
3.50%
4.00%
0 2 4 6 8 10
Spot
rate
(con
t.com
p.)
Maturity
May 5, 2011 |11
2. Duration of swap
101,323,663 100,000,0004.67 0.51,323,663 1,323,663
319.7
fix floatSwap fix float
Swap Swap
V VD D D
V V
= 6.18 * (1,064,294 / 64,294)
- 0.5 * (1,000,000/ 64,294)
= 94 . 5
May 5, 2011 |12
Using duration
• As: floatfixswap VVV
rVD
rVVV
DVV
D
rVDVD
VVV
SwapSwap
SwapSwap
floatfloat
Swap
fixfix
floatfloatfixfix
floatfixSwap
)(
)(
Question: by how much the value of the swap will change if interest rates change by 0.01%?
May 5, 2011 |13
Using duration
• Suppose the interest rate change ∆r = 0.01% (= + 1bp)
4.67 101,323,663 0.01% $47,391
0.5 100,000,000 0.01% $5,000
$42,391
fix
float
Swap Swap
VVV BVP
- 6.18 * 1,064,294 * 0.01% = -$658
= -$608
- 0.5 * 1,000,000 * 0.01% = -$50
May 5, 2011 |14
3. Swap Futures
• A futures contract on a 4% 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 4% coupon bond)• T is the maturity of the futures contract (3 month = 0.25 yr)• r is the 3-month interest rate (with continuous compounding)
rTeSF 00
Today
Maturity of futures Coupon Coupon
03 m 9 m 1 yr 3 m 7 yr 3 m
Coupon + Principal
0.25 0.75 1.25 7.25
May 5, 2011 |15
Swapnote quotation
• S0 =• F0 = 102.782 / 0.99954= 102.830
• The duration of the underlying bond is 6.19.
• If the interest rate change ∆r = 0.01% (= + 1bp)• ∆F0 = -0.0630 (= - 6.30 bp) (see next slide for details)
• As the size of the contract is $100,000:• ∆r = 0.01% → ∆F0 = -0.0630• → BVPSwapnote = $ 100,000 (-0.0630) / 100 = - $63• Tick (Value of ∆F = 0.01) = $ 10
0 0.75 1.25 1.75 10.252 2 2 ... 102S d d d d
May 5, 2011 |16
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?
0 4% 0
8.57 102.97 0.01%0.0882
BondS D S r
0 0 0( )0.0882 0.25 103.04 0.01%0.0857
rT rTF S e TS e r
=-0.0657
=-0.0630
May 5, 2011
(100,000)100
42,391 49585,7
SwapFV n
n
|17
4. Setting up the hedge
• What do we know?
• If ∆r = 0.01% (= + 1 bp)• BVPSwap = - $ 608• BVPSwapnote = - $63/contract
• If interest rates ↑→Futures price ↓ short swapnote
• To hedge its swap position, Swap Pro should short n futures with:
608 / 63 = 9.6
May 5, 2011 |18
5. RollerCoaster IR Swap
Swap Pro is long (receives floating rate and pays fixed rate) on:
• 2.8% 6-year swap•Notional principal progressively from $10 million to $50 million the next 2 years and then progressively declines to zero•The current 6-yr swap rate is 2.74% (Exhibit 1)
May 5, 2011 |19
5. RollerCoaster IR Swap
May 5, 2011 |20
SwapPro Summary of results
(1) Value of swap for CPE: VSwap
(2) Duration of Swap: DSwap
Basis Point Value of Swap BVPSwap
(3) Swapnote = futures on 4% notional bondTick (Value of ∆F = 0.01) = €10BVPSwapnote
Note: if interest rates ↑→Futures price ↓ short swapnote
(4) Number of swapnotes to short to hedge position:
(5) RollerCoaster IR Swap: brain-teaser through FRA’s decompositionor valuation based on forward rate.