BUS FINANCE 826

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BUS FINANCE 826
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Overview

• Derivative securities have become increasingly
  important as FIs seek methods to hedge risk
  exposures. The growth of derivative usage is not
  without controversy since misuse can increase
  risk. This chapter explores the role of futures
  and forwards in risk management.

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Futures and Forwards

• Second largest group of interest rate derivatives
  in terms of notional value and largest group of
  FX derivatives.
• Swaps are the largest.

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Derivatives

• Rapid growth of derivatives use has been
  controversial
• Orange County, California
• Bankers Trust
• As of 2000, FASB requires that derivatives be
  marked to market

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Web Resources

• For further information on the web, visit
• FASB www.fasb.org

Web Surf
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Spot and Forward Contracts

• Spot Contract
• Agreement at t0 for immediate delivery and
  immediate payment.
• Forward Contract
• Agreement to exchange an asset at a specified
  future date for a price which is set at t0.

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Futures Contracts

• Futures Contract
• Similar to a forward contract except
• Marked to market
• Exchange traded (standardized contracts)
• Lower default risk than forward contracts.

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Hedging Interest Rate Risk

• Example 20-year 1 million face value bond.
  Current price 970,000. Interest rates expected
  to increase from 8 to 10 over next 3 months.
• From duration model, change in bond value
• ?P/P -D ? ?R/(1R)
• ?P/ 970,000 -9 ? .02/1.08
• ?P -161,666.67

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Example continuedNaive hedge

• Hedged by selling 3 months forward at forward
  price of 970,000.
• Suppose interest rate rises from 8to 10.
• 970,000 - 808,333 161,667
• (forward (spot price
• price) at t3 months)
• Exactly offsets the on-balance-sheet loss.
• Immunized.

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Hedging with futures

• Futures used more commonly used than forwards.
• Microhedging
• Individual assets.
• Macrohedging
• Hedging entire duration gap.
• Basis risk
• Exact matching is uncommon.

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Routine versus Selective Hedging

• Routine hedging reduces interest rate risk to
  lowest possible level.
• Low risk - low return.
• Selective hedging manager may selectively hedge
  based on expectations of future interest rates
  and risk preferences.

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Macrohedging with Futures

• Number of futures contracts depends on interest
  rate exposure and risk-return tradeoff.
• DE -DA - kDL A DR/(1R)
• Suppose DA 5 years, DL 3 years and interest
  rate expected to rise from 10 to 11. A 100
  million.
• DE -(5 - (.9)(3)) 100 (.01/1.1) -2.09
  million.

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Risk-Minimizing Futures Position

• Sensitivity of the futures contract
• DF/F -DF DR/(1R)
• Or,
• DF -DF DR/(1R) F and
• F NF PF

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Risk-Minimizing Futures Position

• Fully hedged requires
• DF DE
• DF(NF PF) (DA - kDL) A
• Number of futures to sell
• NF (DA- kDL)A/(DF PF)
• Perfect hedge may be impossible since number of
  contracts must be rounded down.

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Payoff profiles
Long Position
Short Position
Futures Price
Futures Price
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Futures Price Quotes

• T-bond futures contract 100,000 face value
• T-bill futures contract 1,000,000 face value
• quote is price per 100 of face value
• Example 103 14/32 for T-bond indicates purchase
  price of 103,437.50 per contract
• Delivery options
• Conversion factors used to compute invoice price
  if bond other than the benchmark bond delivered

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Basis Risk

• Spot and futures prices are not perfectly
  correlated.
• We assumed in our example that
• DR/(1R) DRF/(1RF)
• Basis risk remains when this condition does not
  hold. Adjusting for basis risk,
• NF (DA- kDL)A/(DF PF br) where
• br DRF/(1RF)/ DR/(1R)

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Hedging FX Risk

• Hedging of FX exposure parallels hedging of
  interest rate risk.
• If spot and futures prices are not perfectly
  correlated, then basis risk remains.
• Tailing the hedge
• Interest income effects of marking to market
  allows hedger to reduce number of futures
  contracts that must be sold to hedge

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Basis Risk

• In order to adjust for basis risk, we require the
  hedge ratio,
• h DSt/Dft
• Nf (Long asset position h)/(size of one
  contract).

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Estimating the Hedge Ratio

• The hedge ratio may be estimated using ordinary
  least squares regression
• DSt a bDft ut
• The hedge ratio, h will be equal to the
  coefficient b. The R2 from the regression reveals
  the effectiveness of the hedge.

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Hedging Credit Risk

• More FIs fail due to credit-risk exposures than
  to either interest-rate or FX exposures.
• In recent years, development of derivatives for
  hedging credit risk has accelerated.
• Credit forwards, credit options and credit swaps.

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Credit Forwards

• Credit forwards hedge against decline in credit
  quality of borrower.
• Common buyers are insurance companies.
• Common sellers are banks.
• Specifies a credit spread on a benchmark bond
  issued by a borrower.
• Example BBB bond at time of origination may have
  2 spread over U.S. Treasury of same maturity.

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Credit Forwards

• SF defines credit spread at time contract written
• ST actual credit spread at maturity of forward
• Credit Spread Credit Spread Credit Spread
• at End Seller Buyer
• STgt SF Receives Pays
• (ST - SF)MD(A) (ST - SF)MD(A)
• SFgtST Pays Receives
• (SF - ST)MD(A) (SF - ST)MD(A)

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Futures and Catastrophe Risk

• CBOT introduced futures and options for
  catastrophe insurance.
• Contract volume is rising.
• Catastrophe futures to allow PC insurers to hedge
  against extreme losses such as hurricanes.
• Payoff linked to loss ratio

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Regulatory Policy

• Three levels of regulation
• Permissible activities
• Supervisory oversight of permissible activities
• Overall integrity and compliance
• Functional regulators
• SEC and CFTC
• Beginning in 2000, derivative positions must be
  marked-to-market.

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Regulatory Policy for Banks

• Federal Reserve, FDIC and OCC require banks
• Establish internal guidelines regarding hedging.
• Establish trading limits.
• Disclose large contract positions that materially
  affect bank risk to shareholders and outside
  investors.
• Discourage speculation and encourage hedging

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Pertinent websites

• Federal Reserve www.federalreserve.gov
• Chicago Board of Trade www.cbot.org
• CFTC www.cftc.gov
• FDIC www.fdic.gov
• FASB www.fasb.org
• OCC www.occ.ustreas.gov
• SEC www.sec.gov

Web Surf
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Call option

• A call provides the holder (or long position)
  with the right, but not the obligation, to
  purchase an underlying security at a prespecified
  exercise or strike price.
• Expiration date American and European options
• The purchaser of a call pays the writer of the
  call (or the short position) a fee, or call
  premium in exchange.

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Payoff to Buyer of a Call Option

• If the price of the bond underlying the call
  option rises above the exercise price, by more
  than the amount of the premium, then exercising
  the call generates a profit for the holder of the
  call.
• Since bond prices and interest rates move in
  opposite directions, the purchaser of a call
  profits if interest rates fall.

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The Short Call Position

• Zero-sum game
• The writer of a call (short call position)
  profits when the call is not exercised (or if the
  bond price is not far enough above the exercise
  price to erode the entire call premium).
• Gains for the short position are losses for the
  long position.
• Gains for the long position are losses for the
  short position.

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Writing a Call

• Since there is no theoretical limit to upward
  movements in the bond price, the writer of a call
  is exposed to the risk of very large losses.
• Recall that losses to the writer are gains to the
  purchaser of the call. Therefore, potential
  profit to call purchaser is theoretically
  unlimited.
• Maximum gain for the writer occurs if bond price
  falls below exercise price.

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Call Options on Bonds

• Buy a call Write a call

X
X
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Put Option

• A put provides the holder (or long position) with
  the right, but not the obligation, to sell an
  underlying security at a prespecified exercise or
  strike price.
• Expiration date American and European options
• The purchaser of a put pays the writer of the put
  (or the short position) a fee, or put premium in
  exchange.

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Payoff to Buyer of a Put Option

• If the price of the bond underlying the put
  option falls below the exercise price, by more
  than the amount of the premium, then exercising
  the put generates a profit for the holder of the
  put.
• Since bond prices and interest rates move in
  opposite directions, the purchaser of a put
  profits if interest rates rise.

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The Short Put Position

• Zero-sum game
• The writer of a put (short put position) profits
  when the put is not exercised (or if the bond
  price is not far enough below the exercise price
  to erode the entire put premium).
• Gains for the short position are losses for the
  long position. Gains for the long position are
  losses for the short position.

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Writing a Put

• Since the bond price cannot be negative, the
  maximum loss for the writer of a put occurs when
  the bond price falls to zero.
• Maximum loss exercise price minus the premium

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Put Options on Bonds

• Buy a Put Write a Put

X
X
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Writing versus Buying Options

• Many smaller FIs constrained to buying rather
  than writing options.
• Economic reasons
• Potentially unlimited downside losses.
• Regulatory reasons
• Risk associated with writing naked options.

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Hedging

• Payoffs to Bond Put

Bond
X
Put
Net
X
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Tips for plotting payoffs

• Students often find it helpful to tabulate the
  payoffs at critical values of the underlying
  security
• Value of the position when bond price equals zero
• Value of the position when bond price equals X
• Value of position when bond price exceeds X
• Value of net position equals sum of individual
  payoffs

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Tips for plotting payoffs
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Futures versus Options Hedging

• Hedging with futures eliminates both upside and
  downside
• Hedging with options eliminates risk in one
  direction only

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Hedging with Futures
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Hedging Bonds

• Weaknesses of Black-Scholes model.
• Assumes short-term interest rate constant
• Assumes constant variance of returns on
  underlying asset.
• Behavior of bond prices between issuance and
  maturity
• Pull-to-par.

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Hedging With Bond Options Using Binomial Model

• Example FI purchases zero-coupon bond with 2
  years to maturity, at P0 80.45. This means YTM
  11.5.
• Assume FI may have to sell at t1. Current yield
  on 1-year bonds is 10 and forecast for next
  years 1-year rate is that rates will rise to
  either 13.82 or 12.18.
• If r113.82, P1 100/1.1382 87.86
• If r112.18, P1 100/1.1218 89.14

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Example (continued)

• If the 1-year rates of 13.82 and 12.18 are
  equally likely, expected 1-year rate 13 and
  E(P1) 100/1.13 88.50.
• To ensure that the FI receives at least 88.50 at
  end of 1 year, buy put with X 88.50.

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Value of the Put

• At t 1, equally likely outcomes that bond with
  1 year to maturity trading at 87.86 or 89.14.
• Value of put at t1
• Max88.5-87.86, 0 .64
• Or, Max88.5-89.14, 0 0.
• Value at t0
• P .5(.64) .5(0)/1.10 0.29.

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Actual Bond Options

• Most pure bond options trade over-the-counter.
• Open interest on CBOE relatively small
• Preferred method of hedging is an option on an
  interest rate futures contract.
• Combines best features of futures contracts with
  asymmetric payoff features of options.

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Web Resources

• Visit
• Chicago Board Options Exchange www.cboe.com

Web Surf
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Hedging with Put Options

• To hedge net worth exposure,
• ? P - ?E
• Np (DA-kDL)?A ? ? ? D ? B
• Adjustment for basis risk
• Np (DA-kDL)?A ? ? ? D ? B ?br

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Using Options to Hedge FX Risk

• Example FI is long in 1-month T-bill paying 100
  million. FIs liabilities are in dollars. Suppose
  they hedge with put options, with X1.60 /1.
  Contract size 31,250.
• FI needs to buy 100,000,000/31,250 3,200
  contracts. If cost of put 0.20 cents per ,
  then each contract costs 62.50. Total cost
  200,000 (62.50 3,200).

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Hedging Credit Risk With Options

• Credit spread call option
• Payoff increases as (default) yield spread on a
  specified benchmark bond on the borrower
  increases above some exercise spread S.
• Digital default option
• Pays a stated amount in the event of a loan
  default.

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Hedging Catastrophe Risk

• Catastrophe (CAT) call spread options to hedge
  unexpectedly high loss events such as hurricanes,
  faced by PC insurers.
• Provides coverage within a bracket of
  loss-ratios. Example Increasing payoff if
  loss-ratio between 50 and 80. No payoff if
  below 50. Capped at 80.

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Caps, Floors, Collars

• Cap buy call (or succession of calls) on
  interest rates.
• Floor buy a put on interest rates.
• Collar Cap Floor.
• Caps, Floors and Collars create exposure to
  counterparty credit risk since they involve
  multiple exercise over-the-counter contracts.

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Fair Cap Premium

• Two period cap
• Fair premium P
• PV of year 1 option PV of year 2 option
• Cost of a cap (C)
• Cost Notional Value of cap fair cap premium
  (as percent of notional face value)
• C NVc ? pc

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Buy a Cap and Sell a Floor

• Net cost of long cap and short floor
• Cost (NVc pc) - (NVf pf )
• Cost of cap - Revenue from floor

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Pertinent websites

• Chicago Board of Trade www.cbot.org
• CBOE www.cboe.com
• Chicago Mercantile Exchange www.cme.com
• Philadelphia Options Exchange www.phlx.com

Web Surf
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Overview

• The market for swaps has grown enormously and
  this has raised serious regulatory concerns
  regarding credit risk exposures. Such concerns
  motivated the BIS risk-based capital reforms. At
  the same time, the growth in exotic swaps such as
  inverse floater have also generated controversy
  (e.g., Orange County, CA). Generic swaps in order
  of quantitative importance interest rate,
  currency, credit, commodity and equity swaps.

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Interest Rate Swaps

• Interest rate swap as succession of forwards.
• Swap buyer agrees to pay fixed-rate
• Swap seller agrees to pay floating-rate.
• Purpose of swap
• Allows FIs to economically convert variable-rate
  instruments into fixed-rate (or vice versa) in
  order to better match the duration of assets and
  liabilities.
• Off-balance-sheet transaction.

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Plain Vanilla Interest Rate Swap Example

• Consider money center bank that has raised 100
  million by issuing 4-year notes with 10 fixed
  coupons. On asset side CI loans linked to
  LIBOR. Duration gap is negative.
• DA - kDL lt 0
• Second party is savings bank with 100 million in
  fixed-rate mortgages of long duration funded with
  CDs having duration of 1 year.
• DA - kDL gt 0

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Example (continued)

• Savings bank can reduce duration gap by buying a
  swap (taking fixed-payment side).
• Notional value of the swap is 100 million.
• Maturity is 4 years with 10 fixed-payments.
• Suppose that LIBOR currently equals 8 and bank
  agrees to pay LIBOR 2.

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Realized Cash Flows on Swap

• Suppose realized rates are as follows
• End of Year LIBOR
• 1 9
• 2 9
• 3 7
• 4 6

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Swap Payments

• End of LIBOR MCB Savings
• Year 2 Payment Bank Net
• 1 11 11 10 1
• 2 11 11 10 1
• 3 9 9 10 - 1
• 4 8 8 10 - 2
• Total 39 40 - 1

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Off-market Swaps

• Swaps can be molded to suit needs
• Special interest terms
• Varying notional value
• Increasing or decreasing over life of swap.
• Structured-note inverse floater
• Example Government agency issues note with
  coupon equal to 7 percent minus LIBOR and
  converts it into a LIBOR liability through a swap.

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Macrohedging with Swaps

• Assume a thrift has positive gap such that
• DE -(DA - kDL)A DR/(1R) gt0 if rates rise.
• Suppose choose to hedge with 10-year swaps.
  Fixed-rate payments are equivalent to payments on
  a 10-year T-bond. Floating-rate payments repriced
  to LIBOR every year. Changes in swap value DS,
  depend on duration difference (D10 - D1).
• DS -(DFixed - DFloat) NS DR/(1R)

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Macrohedging (continued)

• Optimal notional value requires
• DS DE
• -(DFixed - DFloat) NS DR/(1R)
• -(DA - kDL) A DR/(1R)
• NS (DA - kDL) A/(DFixed - DFloat)

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Pricing an Interest Rate Swap

• Example
• Assume 4-year swap with fixed payments at end of
  year.
• We derive expected one-year rates from the
  on-the-run Treasury yield curve treating the
  individual payments as separate zero-coupon bonds
  and iterating forward.

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Solving the Discount Yield Curve

• P1 108/(1R1) 100 gt R1 8 gt d1 8
• P2 9/(1R2) 109/(1R2)2 100 gt R2 9
• 9/(1d1) 109/(1d2)2 100 gt d2 9.045
• Similarly, d3 9.58 and d4 10.147

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Solving Implied Forward Rates

• d1 8 gt E(r1) 8
• 1 E(r2) (1d2)2/(1d1) gt E(r2) 10.1
• 1 E(r3) (1d3)3/(1d2)2 gt E(r3) 10.658
• 1 E(r4) (1d4)4/(1d3)3 gt E(r4) 11.866

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Currency Swaps

• Fixed-Fixed
• Example U.S. bank with fixed-rate assets
  denominated in dollars, partly financed with 50
  million in 4-year 10 percent (fixed) notes. By
  comparison, U.K. bank has assets partly funded by
  100 million 4-year 10 percent notes.
• Solution Enter into currency swap.

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Cash Flows from Swap
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Fixed-Floating Currency

• Fixed-Floating currency swaps.
• Allows hedging of interest rate and currency
  exposures simultaneously

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Credit Swaps

• Credit swaps designed to hedge credit risk.
• Total return swap
• Pure credit swap
• Interest-rate sensitive element stripped out
  leaving only the credit risk.

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Credit Risk Concerns

• Credit risk concerns partly mitigated by netting
  of swap payments.
• Netting by novation
• When there are many contracts between parties.
• Payment flows are interest and not principal.
• Standby letters of credit may be required.

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Pertinent Websites

• Visit
• American Banker www.americanbanker.com
• BIS www.bis.org

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