yield curve building with futures & swaps

8/7/2019 Yield Curve Building with Futures & Swaps

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Yield Curve Modeling

Constructing the Curve withFutures & Swaps

Copyright 1996-2006 Investment Analytics


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Slide: 2Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps

Ingredients for Building the ZeroCoupon Spot Curve

Cash RatesFRA Rates (T-Bills)Futures Prices

Swap Rates


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Lab: Constructing the Short End

Ask Bid Days3 Month $ LIBOR 4.25% 4.23% 91

$ FRAs 3-6 4.40% 4.35% 91

6-9 4.55% 4.50% 929-12 4.70% 4.65% 91


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Lab: Constructing the Short End

Excel Spreadsheet: Yield CurveModeling.xlsWorksheet: Short End

Use discount factor methodSee cell notes for hints


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Solution: Short End

Period DF Spot Rate

3 months 0.9894 4.2500%6 months 0.9785 4.3486%9 months 0.9672 4.4498%

12 months 0.9559 4.5519%Notes:DF6-3 = (1 + Ask x Days / 360)DF

6= DF

6-3x DF

3R6 = (-1 + 1/DF 6) x 360 / (91 + 91)


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Futures & Forwards

Assumption is often that 100-F = forward rateNot exact for several reasons:

Interest differentials on margin surplus & funding

Futures are marked to marketConvexity - stochastic interest rates give rise todifferences (Cox, Ingersoll, Ross JFE)


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Convexity

Definition: Positive ConvexityPresent value increases with rate decline

exceeds

Present value decline with rate increaseWhat is the convexity of a Euro $ future andan interest rate swap?Are convexity differences priced?


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Pricing Convexity Differences

If not pricedShort swap/short futures buys positive convexityfor free Significant for longer tenor securities 5+ years Arbitrage gains with rate increases/declines

If priced

Forward rates implied by FRAs or swaps differfrom forward rates implied by futures


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Lab: FRA-Futures ConvexitySell $100 81 v 84M IMM dated FRA @ 5.00%Hedge by Selling Futures @ 95.00Yield curve is flat at 5%Work out:

Equivalent futures positionGain or loss on FRA and equivalent Futures position forparallel shifts +/- 4%

Worksheet: FRA-Futures

See cell notes for help


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Solution: FRA - Futures

D81 = 0.7150FRA Value for 1bp change in YC = $1,788Therefore equivalent position is:

$1,788/ $25 = Long 72 Futures(or if hedging, sell 72 futures contracts)

Examine changes in position value due toshifts in spot & forward rates


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Chart: FRA-Futures Convexity

-800,000

-600,000

-400,000

-200,000

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1% 2% 3% 4% 5% 6% 7% 8% 9%Spot & Forward Rates

Short FRA

Long Futures

Difference


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ConvexityShort FRA has positive convexityFutures have zero convexityDifference must be paid for:

Forward rate is lower than implied by futures price

(100 - futures price) is greater than forward rateNeed adjustment factor to take account of thevolatility of the two rates and their correlation


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Convexity Adjustment Factor

Depends on term structure dynamicsRule of thumb (Burghardt & Hoskins)Change in spread between forward rate andfutures:

S = f x zcb x f,zcb f = standard deviation of change in forward rate zcb = standard deviation of zero coupon bond return

f,zcb = correlation of forward rate change and zerocoupon bond returns


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Convexity Adjustment FactorFutures and forwards are the same at expiryRule gives change in difference over timeCalculate the change for each three month periodStandard deviation of returns on ZCB:

Duration (Maturity) * SD of Yield (Spot Rate) changes Standard deviations and correlations will be slightly different for each

period Derived from historical data or option prices


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Adjustment Factors: Typical ValuesSix Months: 0.25bpOne Year: 0.5bpTwo Years: 1.0bpThree Years: 3.5 bpFive Years: 17bpTen Years: 63bp

Significant 3 years and beyond


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Building the Curve Using Futures

Find spot rate to expiry of first futures contractInterpolate from cash rates

Calculate 90-day forward rate from expiry100-futures price less adjustment factor

Combine to give spot rate to 90 days from expiryExtrapolate to expiry of next futures contractRepeat steps above for successive futurescontracts


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Extending the Curve Using Swaps

A swap is an exchange of an FRN for a bondThe FRN trades at par on coupon payment dates

FRN Price = 100 = D 1C1 + D2C2 + D3C3 + 100D 3

Cash flows are known from the swap coupon

The curve has already been built out to 2 years usingfuturesD1 and D 2 are known, calculate D 3 by bootstrapping

C1 C2 C3

Year 1 Year 2 Year 3Today

Bond Price


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Bootstrap Method with Swaps

Example:Three year swap rate = 5%D1 = 0.9655 D 2 = 0.9259

100 = 0.9655 x 5 + 0.9259 x 5 + 105 x D 3D3 = (100 - 4.8275 - 4.6295) / 105 = 0.8623

Repeat with 4, 5, . . . year swaps to

complete the curve


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Lab: Delmar Capital

Excel Workbook: Yield Curve Modeling .xlsWorksheet: Delmar CapitalBuild yield curve using:

Cash & FuturesCash , Futures & SwapsCash & Adjusted Futures

See Notes & Solution


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Delmar Capital: Yield Curves

4.5%

4.6%

4.7%

4.8%

4.9%

5.0%

5.1%

5.2%

5.3%

5.4%

0 500 1000 1500 2000

Futures Sw aps Adjusted Futures

yield curve building with futures & swaps

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