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