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Lecture 9: Derivatives and Hedging Drag picture to placeholder or click icon to add
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Lecture 9: Derivatives and Hedging
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Futures and forwards
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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 controversialOrange County, California Bankers TrustAllfirst Bank (Allied Irish)
As of 2000, FASB requires that derivatives be marked to market Transparency of losses and gains on financial
statements
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Spot and Forward Contracts
Spot ContractAgreement at t=0 for immediate delivery and
immediate payment.
Forward ContractAgreement to exchange an asset at a
specified future date for a price which is set at t=0.Counterparty risk
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Futures Contracts
Futures Contract similar to a forward contract except:Marked to marketExchange traded Rapid growth of off market trading systems
Standardized contractsSmaller denomination than forward
Lower default risk than forward contracts.
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Hedging Interest Rate Risk
Example: 20-year $1 million face value bond. Duration of the bonds is 9 years. 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/(1+R)
P/ $970,000 = -9 [.02/1.08]
P = -$161,666.67
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Example continued: Naive hedge
Hedged by selling 3 months forward at forward price of $970,000.
Suppose interest rate rises from 8%to 10%.
$970,000 - $808,333 = $161,667
(forward (spot price price) at t=3 months)Exactly offsets the on-balance-sheet
loss.Immunized.
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Hedging with futures
Futures more commonly used than forwards.Microhedging Individual assets.
MacrohedgingHedging entire duration gapFound more effective and generally lower
cost.Basis riskExact matching is uncommonStandardized delivery dates of futures
reduces likelihood of exact matching.
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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.Partially hedge duration gap or
individual assets or liabilities
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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/(1+R)]
Suppose: DA = 5 years, DL = 3 years, Assets = $100 millions, L = $90 millions and interest rate expected to rise from 10% to 11%. A = $100 million.
DE = -(5 - (.9)(3)) $100 (.01/1.1) = -$2.091 million.
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Risk-Minimizing Futures Position
Sensitivity of the futures contract:
DF/F = -DF [DR/(1+R)]
Or,
DF = -DF × [DR/(1+R)] × F and
F = NF × PF
Where
NF is the number of contracts bought or sold, and
PF is the price of each contract
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Risk-Minimizing Futures Position
Fully hedged requires DF = DEDF(NF × PF) = (DA - kDL) × A
Number of futures to sell:
NF = (DA- kDL)A/(DF × PF)
Perfect hedge may be impossible since number of contracts must be rounded down.
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Basis Risk
Spot and futures prices are not perfectly correlated.
We assumed in our example that
DR/(1+R) = DRF/(1+RF)
Basis risk remains when this condition does not hold. Adjusting for basis risk,
NF = (DA- kDL)A/(DF × PF × br) where
br = [DRF/(1+RF)]/ [DR/(1+R)]
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Hedging FX Risk
Hedging of FX exposure parallels hedging of interest rate risk.
If spot and futures prices are not perfectly correlated, then basis risk remains.
Tailing the hedge Interest income effects of marking to market
allows hedger to reduce number of futures contracts that must be sold to hedge
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Basis Risk
In order to adjust for basis risk, we require the hedge ratio,
h = DSt/Dft
Where:Nf = (Long asset position × estimate of h)/(size of one contract).
ft Futures price ($/£) for the contract
St Spot exchange rate ($/£)
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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
CSF defines agreed forward credit spread at time contract written
CST = actual credit spread at maturity of forward
Credit Spread Credit Spread Credit Spreadat End Seller BuyerCST> CSF Receives Pays
(CST - CSF)MD(A) (CST -C SF)MD(A)
CSF>CST Pays Receives
(CSF - CST)MD(A) (CSF - CST)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 (insured losses to premiums)
Example: Payoff = contract size × realized loss ratio – contract size × contracted futures loss ratio. $25,000 × 1.5 - $25,000 – 0.8 = $17,500 per contract.
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Regulatory Policy
Three levels of regulation:Permissible activitiesSupervisory oversight of permissible
activitiesOverall integrity and compliance
Functional regulatorsSEC and CFTC
As of 2000, derivative positions must be marked-to-market.
Exchange traded futures not subject to capital requirements: OTC forwards potentially subject to capital requirements
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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
Allfirst/Allied Irish: Existing (and apparently inadequate) policies were circumvented via fraud and deceit.
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Websites
Federal Reserve www.federalreserve.gov
Chicago Board of Trade www.cbot.org
Chicago Mercantile Exchange www.cme.com
CFTC www.cftc.gov
FDIC www.fdic.gov
FASB www.fasb.org
OCC www.ustreas.gov
SEC www.sec.gov
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Options, Caps, Floors and
Collars
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Overview
Derivative securities as a whole have become increasingly important in the management of risk and this chapter details the use of options in that vein. A review of basic options –puts and calls– is followed by a discussion of fixed-income, or interest rate options. The chapter also explains options that address foreign exchange risk, credit risks, and catastrophe risk. Caps, floors, and collars are also discussed.
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Option Terms
Long position in an option is synonymous with: Holder, buyer, purchaser, the longHolder of an option has the right, but not the
obligation to exercise the option
Short position in an option is synonymous with: Writer, seller, the shortObliged to fulfill terms of the option if the
option holder chooses to exercise.
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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 call position are losses for the long call position.
Gains for the long call position are losses for the short call position.
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Writing a Call Since the price of the bond could rise to equal
the sum of the principal and interest payments (zero rate of interest), 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 could be very large. (Note that call options on stocks have no theoretical payoff limit at all).
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
(Long Put) (Short Put)
X
X
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Writing versus Buying Options
Many smaller FIs constrained to buying rather than writing options.Economic reasonsPotentially large downside losses for calls.Potentially large losses for putsGains can be no greater than the premiums
so less satisfactory as a hedge against losses in bond positions
Regulatory reasonsRisk associated with writing naked options.
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Combining Long and Short Option Positions
The overall cost of hedging can be custom tailored by combining long and short option positions in combination with (or alternative to) adjusting the exercise price. Example: Suppose the necessary hedge requires a
long call option but the hedger wishes to lower the cost. A higher exercise price would lower the premium but provides less protection. Alternatively, the hedger could buy the desired call and simultaneously sell a put (with a lower exercise price). The put premium offsets the call premium. Presumably any losses on the short put would be offset by gains in the bond portfolio being hedged.
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Hedging Downside with Long Put
Payoffs to Bond + Put
X
X
Put
Bond
Net
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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
zeroValue of the position when bond price equals
XValue of position when bond price exceeds XValue of net position equals sum of individual
payoffs
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Tips for plotting payoffs
B=0 B < X B=X B > X
Long call Short call
- C + C
- C + C
-C +C
B-X-C X+C-B
Long put Short put
X - P P - X
X-B-P P+B-X
-P +P
-P +P
Long Bond
B = 0
B
B=X
B
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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
Bond Portfolio
Bond Price
Purchased Futures Contract
X
0
Gain
Loss
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Hedging Bonds
Weaknesses of Black-Scholes model.Assumes short-term interest rate constantAssumes constant variance of returns on
underlying asset.Behavior of bond prices between issuance
and maturityPull-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 BP0 = $80.45. This means YTM = 11.5%.
Assume FI may have to sell at t=1. Current yield on 1-year bonds is 10% and forecast for next year’s 1-year rate is that rates will rise to either 13.82% or 12.18%.
If r1=13.82%, BP1= 100/1.1382 = $87.86
If r1=12.18%, BP1= 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(BP1) = 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 t=1:
Max[88.5-87.86, 0] = .64Or, Max[88.5-89.14, 0] = 0.Value at t=0: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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Hedging with Put Options
To hedge net worth exposure, P = - E
Np = [(DA-kDL)A] [ D B]
Where: is the delta of the option.D is the duration of the bondAdjustment for basis risk:
Np = [(DA-kDL)A] [ D B br]
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Macrohedge of Interest Rate Risk Using a Put Option
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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 X=$1.60 /£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 optionPayoff increases as (default) yield spread on
a specified benchmark bond on the borrower increases above some exercise spread S.
Digital default optionPays 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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Collars: 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
Counterparty credit risk is an issue
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Pertinent websites
Chicago Board of Trade www.cbot.com
CBOE www.cboe.com
Chicago Mercantile Exchange www.cme.com
Wall Street Journal www.wsj.com
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Swaps
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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-rateSwap seller agrees to pay floating-
rate.Purpose of interest rate swapAllows 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: loans linked to LIBOR. Duration gap is negative.
DA - kDL < 0Second party is savings bank with
$100 million in fixed-rate mortgages of long duration funded with CDs having duration of 1 year.
DA - kDL > 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 MCBYear + 2% Payment Bank Net1 11% $11 $10 +12 11 11 10 +13 9 9 10 - 14 8 8 10 - 2Total 39 40 - 1
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Off-market Swaps
Swaps can be molded to suit needsSpecial interest termsVarying notional value Increasing or decreasing over life of swap.
Structured-note inverse floaterExample: 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 thatDE = -(DA - kDL)A [DR/(1+R)] >0 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/(1+R)]
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Macrohedging (continued)
Optimal notional value requiresDS = DE
-(DFixed - DFloat) × NS × [DR/(1+R)]
= -(DA - kDL) × A × [DR/(1+R)]
NS = [(DA - kDL) × A]/(DFixed - DFloat)
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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 Swap73
Fixed-Floating + Currency
Fixed-Floating currency swaps.Allows hedging of interest rate and currency
exposures simultaneously Combined Interest Rate and Currency
(CIRCUS) Swap
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Credit Swaps
Credit swaps designed to hedge credit risk. Involvement of other FIs in the credit risk
shift
Total return swap Hedge possible change in credit risk
exposure
Pure credit swap Interest-rate sensitive element stripped out
leaving only the credit risk.
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Swaps and 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.Greenspan stated that credit swap market has
helped strengthen the banking system’s ability to deal with recession
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Websites
BIS www.bis.orgFederal Reserve www.federalreserve.gov FDIC www.fdic.gov International Swaps and Derivatives
Association www.isda.org Moody’s Investor Services
www.moodys.com
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