Derivatives 2010-2011 Swap Pro – Wrap up

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

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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

May 5, 2011 |9

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.