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Derivatives

Derivatives. Lecture 12. Hedge Ratio Determination. 1 - The Duration Model 2 - Naive Hedging Model 3 - Conversion Factor Model 4 - Basis Point Model 5 - Regression Model 6 - Yield Forecast Model. Hedge Ratios. Duration Model. Hedge Ratios. Duration Model

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Derivatives

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  1. Derivatives Lecture 12

  2. Hedge Ratio Determination 1 - The Duration Model 2 - Naive Hedging Model 3 - Conversion Factor Model 4 - Basis Point Model 5 - Regression Model 6 - Yield Forecast Model

  3. Hedge Ratios • Duration Model

  4. Hedge Ratios Duration Model • Your cash position is $1,000,000 10% coupon, 26year bonds, with YTM=12.64% and duration of 8.24 years. • The 6%, 20year, TBill has a duration of 10.14 years, YTM=8.5% • The FC on this bond is priced at 96.87 HR = 79.98x8.24 = 659.04 = .671 96.87x10.14 982.26 (1,000,000 / 100,000) x .671 = 6.71 or 7 contracts

  5. Hedge Ratios Duration Example • In 3 months, you will receive $3.3 mil in cash and must invest it for 6 months. The current 6 month rate is 11.20%. You like that rate, and wish to lock it in. • 6 month tbills have a .50 duration, while 3 month bills have a .25 duration. • If the 3 month futures price is 97.36, what number of Ks are required to lock in the rate? HR = 100 x .5 = 2.05 x (3.3 / .1) = 67.8 contracts 97.36 x .25

  6. Hedge Ratios Naive Model HR = 1.0 (all previous examples were naive hedges) Conversion Factor Model HR = conversion factor CF = Price of deliverable bond @ 6% YTM 100

  7. Hedge Ratios Conversion Factor Model Example • You own a $1mil portfolio you wish to hedge. Your are considering a 3 month futures K. The bond that could be delivered against the contract is a 9.5%(semiannual) bond with a 30year maturity. The bond is callable in 15 years. How many Ks should you use to hedge the position? CF = 134.30/100 = 1.34 x (1mil/.1) = 13 contracts

  8. Hedge Ratios Example - Conversion Factor Model • You have a $1mil portfolio, containing 21.5 year 10 3/8 bonds. Price = 100.5363 (YTM = 10 5/16) • CTD 20year, 8% bond has YTM = 10.43 • Create the hedge. • Assume that in 6 months YTM on your portfolio rises to 12 % and YTM on CTD rises to 12.217% • Create a table showing your position/profit/loss

  9. Hedge Ratios Example - Conversion Factor Model CF = PV of 5.1875 @ 3% for 43 periods / 100 = 1.52 1.52 x (1mil/100,000) = 15 Cash Futures Today Own $1mil Short 15 K @ 100.5363 @ 79.718 (given) ($1,005,363) + $1,195,770 6 mths Sell @ 87.63 buy 15 K @ 71.07 (given) + $876,301 ($1,066,050) (129,062) +129,720

  10. Hedge Ratios Basis Point Model BVCcash = $ change in value per basis point of cash position B = Relative yield volatility of cash to CTD = (Vcash / Vctd) BVCctd= $ change in value per basis point of CTD CFctd=conversion factor of CTD

  11. Hedge Ratios Example • YTM = 9% on semi-annual bonds • Your cash portfolio consists $1mil of 26 year 9 7/8 bonds, that have a yield volatility of .60 • Futures CTD is a 7.25% 26.5 year note with a yield volatility of .50 • Use the basis point model to create a hedge and show the position table for a 3month time period and a change in YTM to 10%.

  12. Hedge Ratios Basis Point Model Use Calculator bond functions for calcualtions

  13. Hedge Ratios example - continued Cash value @ 9% = 108.737 BVCcash = $107 (PV @ 9% - PV @ 9.01) BVCctd= $86 B = .6 / .5 = 1.20 CF = .1.16 (PV of CTD @ 6% / 100) HR* = ( 107 ) x 1.20 = 1.73 ( 86 / 1.16) 1 mil / 100,000 x 1.73 = 17 contracts

  14. Hedge Ratios example - continued (10%) Cash Futures Today $1mil @ 108.737 17K @ 82.44 (given) -$1,087,370 +1,401,480 3 months (YTM = 10%) $1 mil @ 98.82 17K @ 76.45 (given) +$ 988,212 - $1,299,650 Net Position $99,158 loss $101,830 gain net gain of $2,672

  15. Hedge Ratios example - continued Assume YTM = 8% Cash Futures Today $1mil @ 108.737 17K @ 82.44 (given) -$1,087,370 + 1,401,480 3 months (YTM = 8%) $1 mil @ 120.30 17K @ 89.56 (given) +$ 1,203,034 - $1,522,520 Net Position $115,664 gain $121,040 loss net loss of $5,376

  16. Hedge Ratios Regression Model HR = Covariance of Cash & Futures Variance of futures best model if HR = .90, then we know that a $1 change in futures prices correlates to a $0.90 change in cash value. requires constant monitoring because HR changes with duration

  17. Hedge Ratios Yield Forecast Model Given various yield forecasts, the HR changes Term Structure can forecast yields HR = CVdiff / FCV diff Example Cash Value = 97.94 & Futures = 72.50 Forecasted YTM YTM CV YTM FC CV FC CVdiff FCdiff HR 12.65 11.25 101.72 75.06 3.77 2.56 1.48 12.85 11.40 100.14 74.14 2.20 1.64 1.34 13.55 12.05 94.99 70.37 -2.95 -2.13 1.36 13.75 12.20 93.62 69.54 -4.33 -2.96 1.47

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