1. Options and Speculative Markets 2004-2005 Swapnote – Wrap up Professor André Farber Solvay Business School Université Libre de Bruxelles

2. August 23, 2004 Swapnote 2004 |2 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)

3. August 23, 2004 Swapnote 2004 |3 Summary of results (1)Value of swap for Piggibank: V Swap = €325,337 (2)Duration of Swap: D Swap = 116 Basis Point Value of Swap BVP Swap = - €3,782 (3) Swapnote = futures on 6% notional bond Tick (Value of ∆F = 0.01) = €10 BVP Swapnote = - €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

4. August 23, 2004 Swapnote 2004 |4 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 d 1 + 400,000 d 2 +...+ 10,400,000 d 5 = 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

5. August 23, 2004 Swapnote 2004 |5 Calculation of discount factors Bootstrap method. Solve the following equations: 100 = 102.30 d 1 100 = 2.56 d 1 + 102.56 d 2 100 = 2.83 d 1 + 2.83 d 2 + 102.83 d 3 100 = 3.07 d 1 + 3.07 d 2 + 3.07 d 3 + 103.07 d 4 100 = 3.29 d 1 + 3.29 d 2 + 3.29 d 3 + 3.29 d 4 + 103.29 d 5 Use eq.1 to obtain d 1 Replace d 1 in eq.2 and solve for d 2 Replace d 1 and d 2 in eq.3 and solve for d 3..... or use matrix algebra: d = C -1 P

6. August 23, 2004 Swapnote 2004 |6 2. Duration of swap As:

7. August 23, 2004 Swapnote 2004 |7 Using duration Suppose the interest rate change ∆r = 0.01% (= + 1bp)

8. August 23, 2004 Swapnote 2004 |8 Swapnote A futures contract on a 6% notional coupon bond. Face value = €100,000 To calculate the futures price, use general approach: S 0 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) Today Maturity of futures Coupon 0 2 m 1yr 2 m2 yr 2 m5 yr 2 m Coupon + Principal 0.1671.1672.1675.167

9. August 23, 2004 Swapnote 2004 |9 Spot price calculation 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) = a 0 + a 1 t + a 2 t² + a 3 t 3 where r(t) is the spot rate with continuous compounding for maturity t 3.The discount factor is d(t) = exp(-r(t)t)

10. August 23, 2004 Swapnote 2004 |10 Swapnote quotation S 0 = 111.71 F 0 = 111.71 / 0.99653 = 112.10 The duration of the underlying bond is 4.66. If the interest rate change ∆r = 0.01% (= + 1bp) ∆F 0 = -0.05 (= - 5 bp) (see next slide for details) As the size of the contract is €100,000: ∆r = 0.01% → ∆F 0 = -0.05 → BVP Swapnote = €100,000  (-0.05) / 100 = - €50

11. August 23, 2004 Swapnote 2004 |11 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?

12. August 23, 2004 Swapnote 2004 |12 Setting up the hedge What do we know? If ∆r = 0.01% (= + 1 bp) BVP Swap = - € 3,782 BVP Swapnote = - €50/contract To hedge its swap position, Piggibank should short n futures swapnotes contract so that: