Chapter 12 Forwards, Futures, Futures options, and Swaps Contents ...
Yield Curve Building with Futures & Swaps
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Transcript of 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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Slide: 3Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 4Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
Lab: Constructing the Short End
Excel Spreadsheet: Yield CurveModeling.xlsWorksheet: Short End
Use discount factor methodSee cell notes for hints
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Slide: 5Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 7Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 8Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 10Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 11Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 12Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 13Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 14Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 15Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 16Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 18Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 20Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 21Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 22Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 23Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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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Slide: 24Copyright 1996-2006 Investment Analytics Yield Curve Building with Futures & Swaps
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