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FX Derivatives
1. FX Futures and Forwards
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FX RISK
Example: ABYZ, a U.S. company, imports wine from France. ABYZ has to pay EUR 5,000,000 on May 2. Today, February 2, the exchange rate is 1.15 USD/EUR.
Situation: Payment due on Mayy 2: EUR 5,000,000.
SSep 2 = 1.15 USD/EUR.
Problem: St is difficult to forecast Uncertainty.
Uncertainty Risk.
Example: on January 2, St > or < 1.15 USD/EUR.
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On SFeb 2, ABYZ total payment would be:
EUR 5,000,000 x 1.15 USD/EUR = USD 5,750,000.
On May 2 we have two potential scenarios:
If the SMay 2 (USD appreciates) ABYZ will pay less USD.
If the SMay 2 (USD depreciates) ABYZ will pay more USD.
→ The second scenario introduces Currency Risk.
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FX Risk: Hedging Tools
• Hedging Tools:
- Market-tools:
Futures/Forward
Money Market (IRPT strategy)
Options
- Firm-tools:
Pricing in domestic currency
Risk-sharing
Matching Outflows and Inflows
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Futures or Forward FX Contracts
Forward markets:Tailor-made contracts.
Location: none.
Reputation/collateral guarantees the contract.
Futures markets: Standardized contracts.
Location: organized exchanges
Clearinghouse guarantees the contract.
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Comparison of Futures and Forward Contracts
Futures Forward
Amount Standardized Negotiated
Delivery Date Standardized Negotiated
Counter-party Clearinghouse Bank
Collateral Margin account Negotiated
Market Auction market Dealer market
Costs Brokerage and exchange fees Bid-ask spread
Secondary market Very liquid Highly illiquid
Regulation Government Self-regulated
Location Central exchange floor Worldwide
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FX Futures/Forwards: Basic Terminology
Short: Agreement to Sell.
Long: Agreement to Buy.
Contract size: number of units of foreign currency in each contract.
CME expiration dates: Mar, June, Sep, and Dec + Two nearby months
Margin account: Amount of money you deposit with a broker to cover your possible losses involved in a futures/forward contract.
Initial Margin: Initial level of margin account.
Maintenance Margin: Lower bound allowed for margin account.
A margin account is like a checking account you have with your broker, but it is marked to market. If your contracts make (lose) money, money is added to (subtracted from) your account.
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If margin account goes below maintenance level, a margin call is issue:
you have to add funds to restore the account to the initial level.
Example: GBP/USD CME futures
initial margin: USD 2,800
maintenance margin: USD 2,100
If losses do not exceed USD 700, no margin call will be issued.
If losses accumulate to USD 850, USD 850 will be added to account. ¶
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Using FX Futures/Forwards
• Iris Oil Inc., a Houston-based energy company, will transfer CAD 300 million to its USD account in 90 days. To avoid FX risk, Iris Oil decides to short a USD/CAD Forward contract.
Data:
St = .8451 USD/CAD
Ft,90-day = .8493 USD/CAD
In 90-days, Iris Oil will receive with certainty:
(CAD 300M) x (.8493 USD/CAD) = USD 254,790,000.
Note: The exchange rate at t+90 (St+90) is, now, irrelevant.
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The payoff diagram makes it clear: Using futures/forwards can isolate a company from uncertainty.
St+90
USD 254.79MForward
Amount Received in t+90
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Hedging with FX Futures Contracts
• FX Hedger
FX Hedger reduces the exposure of an underlying position to currency risk using (at least) another position (hedging position).
Basic Idea of a Hedger
A change in value of an underlying position is compensated with the change in value of a hedging position.
Goal: Make the overall position insensitive to changes in FX rates.
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Hedger has an overall portfolio (OP) composed of (at least) 2 positions:
(1) Underlying position (UP)
(2) Hedging position (HP) with negative correlation with UP
Value of OP = Value of UP + Value of HP.
Perfect hedge: The Value of the OP is insensitive to FX changes.
• Types of FX hedgers using futures:
i. Long hedger: Underlying position: short in the foreign currency. Hedging position: long in currency futures.
ii. Short hedger: Underlying position: long in the foreign currency.
Hedging position: short in currency futures.
Note: Hedging with futures is very simple: Take an opposite position!
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Q: What is the proper size of the Hedging position?A:
Basic (Naive) Approach: Equal hedge
Modern Approach: Optimal hedge
• The Basic Approach: Equal hedgeEqual hedge:
Size of Underlying position = Size of Hedging position.
Example: Long Hedge and Short Hedge(A) Long hedge.A U.S. investor has to pay in 90 days 2.5 million Norwegian kroners.
UP: Short NOK 2.5 M.
HP: Long 90 days futures for NOK 2.5 M.
(B) Short hedge.A U.S. investor has GBP 1 million invested in British gilts.
UP: Long GBP 1 M.
HP: Short futures for GBP 1 M.
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Define:Vt: value of the portfolio of foreign assets measured in GBP at time t.
Vt*: value of the portfolio of foreign assets measured in USD at time t.
Example (continuation): Calculating the short hedger's profits.It’s September 12. An investor decides to hedge using Dec futures.Situation:
UP: British bonds worth GBP 1,000,000.
FSep 12,Dec = 1.55 USD/GBP
Futures contract size: GBP 62,500.
SSep 12: 1.60 USD/GBP.
number of contracts = ?
HP: Investor sells
GBP 1,000,000 / (62,500 GBP/contract) = 16 contracts.
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Example (continuation): Calculating the short hedger's profits.• On October 29, we have:
Sep 12
Oct 29
ChangeVt (GBP)
1,000,000
1,000,000
0Vt* (USD)
1,600,000
1,500,000
-100,000St
1.60
1.50
0.10Ft,T=Dec
1.55
1.45
0.10
The USD change in UP ("long GBP bond position") is given by:
Vt* - V0* = VtSt - V0S0 = V0 x (St - S0)
= USD 1.5M - USD 1.6M = USD -0.1M.
The USD change in HP ("short GBP futures position") is given by:
-V0 x (Ft,T – F0,T) = Realized gain
(GBP -1M) x USD/GBP (1.45 - 1.55) = USD 0.1M.
USD Change in OP = USD Change of UP + USD Change of HP = 0
=> This is a perfect hedge! ¶
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Note: In the previous example, we had a perfect hedge. We were lucky!VSep 12 = VOct 29 = GBP 1,000,000
(FSep 12,Dec - SSep 12 ) = (FOct 29,Dec – SOct 29 ) = USD .05
An equal position hedge is not a perfect hedge if:(1) Vt changes.
(2) The basis (Ft-St) changes.
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• Changes in Vt
If Vt changes, the value of OP will also change.
Example: Reconsider previous example. On October 29: FOct 29,Dec = 1.45 USD/GBP.
SOct 29= 1.50 USD/GBP.
VOct 29 increases 2%.
Size of UP (long): GBP 1,020,000. Size of HP (short): GBP 1,000,000.
USD change in UP (long GBP bond) is
VOct 29* (long) = GBP 1.02M x 1.50 USD/GBP = USD 1,530,000
VSep 12* (long) = GBP 1.0M x 1.60 USD/GBP = USD 1,600,000
USD Change in Vt* (long) = USD -70,000
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USD Change in HP (short GBP futures) = USD 100,000.
Net change on the overall portfolio = USD -70,000 + USD 100,000= USD 30,000. ¶
=> Not a perfect hedge: only the principal (GPB 1 million) was hedged!
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• Basis ChangeDefinition: Basis = Futures price - Spot Price = Ft,T - St.
Basis risk arises if Ft,T - St deviates from a constant basis per period.
• If there is no basis risk completely hedge the underlying position (including changes in Vt).
• If the basis changes "equal" hedge is not perfect.
• In general, if (Ft,T - St) (or "weakens"), the short hedger loses.
if (Ft,T - St) (or "strengthens"), the short hedger wins.
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Example (continuation): Now, on October 29, the market data is: FOct 29,Dec = 1.50 USD/GBP.
SOct 29 = 1.50 USD/GBP.
BasisSep 12 = FSep 12,Dec - SSep 12 = 1.55 -1.60 = -.05 (5 points)
BasisOct 29 = FOct 29,Dec - SOct 29 = 1.50 -1.50 = 0.
Compared to previous Example, basis increased from -5 points to 0 points. (The basis has weakened from USD .05 to USD 0.)
Date Long Position ("Buy") Dec Futures ("Sell")September 12 1,600,000 1,550,000October 29 1,500,000 1,500,000Gain -100,000 50,000
Note: (Ft-St) Short hedger's profits loses (USD -50,000).
Equal hedge is not perfect! ¶
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• Basis risk• Recall IRPT. For the USD/GBP exchange rate, we have:
Assume T =360. After some algebra, we have:
The basis is proportional to the interest rate differential. As interest rates change, the basis changes too.
GBP
GBPUSDttTt i
iiSSF
1,
)360
1(
)360
1(
, Txi
Txi
SF
GBP
USD
tTt
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• Derivation of the Optimal hedge ratio
Additional notation:ns: number of units of foreign currency held.
nf: number of futures foreign exchange units held.
(Number of contracts = nf/size of the contract)
πh,t: uncertain profit of the hedger at time t.
h = hedge ratio =(nf/ns) = number of futures per spot in UP.
We want to calculate h* (optimal h). For that, we minimize the variability of πh,t.
πh,T = ΔST ns + ΔFT,T nf (Or, πh,T / ns = ΔST + h ΔFT,T.)
We want to select h to minimize: Var(πh,T/ns) = Var(ΔST) + h2 Var(ΔFT,T) + 2 h Covar(ΔST,ΔFT,T)
h*= -σSF/σF2.
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• Optimal hedge ratio: h* = -σSF/σF2.
(A covariance over a variance => A (OLS) regression estimate!)
Remarks:(1) h* is the (negative) slope of an OLS regression of ΔST against ΔFT:
ΔSt = μ + θ ΔFt,T + εt => h* = - b (OLS estimate of θ)
(2) Recall IRPT: Ft,T = St (1 + id)/(1 + if) = δ St.
Calculating changes: ΔFt,T = δ ΔSt => ΔSt = (1/δ) ΔFt,T
Then, h*=-b estimates (1/δ) in the IRPT equation: δ=1/h*.
(3) Two cases regarding hedge ratios:- When Ft,T is denominated in the same currency as the asset being
hedged, we can use IRPT to get the hedge ratio, h*.
- When Ft,T is denominated in a different currency than the asset being
hedged (cross-hedging), OLS provides an estimate of h*.
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• OLS Estimation of the Optimal Hedge RatioConsider the following regression equation:
ΔSt = μ + θ ΔFt,T + εt.
=> OLS produces b = σSF/σF2 = - h*
• A hedge is perfect only if ΔSt and ΔFt,T are perfectly correlated:
to have a perfect hedge we need εt to be always zero.
• The R² of regression measures the efficiency of a hedge.
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Example: We estimate a hedge ratio using OLS:We use five years of monthly data for a total of 60 observations.
ΔSt = .001 + .92 ΔFt,T, R2 = .95.
h = -.92. nf/size of the contract = h ns / size of the contract =
= -.92 x 1,000,000 / 62,500 = -14.7 15 contracts sold! ¶
• The high R2 points out the efficiency of the hedge:
Changes in futures USD/GBP prices are highly correlated with changes in USD/GBP spot prices.
Note: A different interpretation of the R2: Hedging reduces the variance of the CF by an estimated 95%.
Remark: OLS estimates of the hedge ratio are based on historical data. The hedge we construct is for a future period. Problem!
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• Time-varying hedge ratios: ht* = -σSF,t /σF,t2
GARCH models: The variance changes with the arrival of news.
A GARCH(1,1) specification is a good approximation: 2
S,t = S0 + S1 2S,t-1 + ßS1 2
S,t-1
S,t-1 = forecasting error at t-1. (Under RWM, S,t-1 is the change in St-1.)
Recall that GARCH models accommodate two features of financial data:- Large changes tend to be followed by large changes of either sign.- Distribution is leptokurtic –i.e., fatter tails than a normal.
• Statistical packages that estimate GARCH models: SAS, E-views.
To estimate a time-varying hedge ratio, you need a model for the bivariate distribution of St and Ft,T (to get covariances)..
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Hedging Strategies
• Three problems associated with hedging in the futures market:
- Contract size is fixed.
- Expiration dates are also fixed.
- Choice of underlying assets in the futures market is limited.
• Imperfect hedges:
- Delta-hedge when the maturities do not match
- Cross-hedge when the currencies do not match.
• Another important consideration: liquidity.
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• Contract Terms (Delta Hedging)Major decision: choice of contract terms.
• Advantages of a short-term hedging:
- Short-term Ft,T closely follows St.
Recall linearized IRPT : Ft,T ≈ St [1+ (id - if)xT/360]
As T → 0, Ft,T → St (UP and HP will move closely)
- Short-term Ft,T has greater trading volume (more liquid).
• Disadvantages of a short-term hedging:- Short-term hedges need to be rolled over: cost!
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• Contract Terms (Delta Hedging)
• Short-term hedges are usually done with short-term contracts.
• Longer-term hedges are done using three basic contract terms:- Short-term contracts, which must be rolled over at maturity;- Contracts with a matching maturity (usually done with a
forward);- Longer-term contracts with a maturity beyond the hedging
period.
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Three Hedging Strategies for an Expected Hedge Period of 6 Months
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• Different Currencies (Cross-Hedging)Q: Under what circumstances do investors use cross-hedging?
• An investor may prefer a cross-hedge if:
(1) There is no available contract for the currency she wishes to hedge.Futures contracts are actively traded for the major currencies (at the CME: GBP, JPY, EUR, CHF, MXN, CAD, BRR).Example: Want to hedge a NOK position using CME futures:
you must cross-hedge.
(2) Cheaper and easier to use a different contract.Banks offer forward contracts for many currencies. These contracts might not be liquid (and expensive!).
• Empirical results:(i) Optimal same-currency-hedge ratios are very effective. (ii) Optimal cross-hedge ratios are quite unstable.
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Example: Calculation of Cross-hedge ratios.Situation:- Veron SA, a U.S. firm, has to pay HUF 10 million in 180 days. - No futures contract on the HUF.- Liquid contracts on currencies highly correlated to the HUF.
Solution: Cross-hedge using the EUR and the GBP. • Calculation of the appropriate OLS hedge ratios.Dependent variable: USD/HUF changes Independent variables: USD/EUR 6-mo. futures changes
USD/GBP 6-mo. futures changes Exchange rates: .0043 EUR/HUF
.0020 GBP/HUFSUSD/HUF = α + .84 FUSD/EUR + 0.76 FUSD/GBP, R2 = 0.81.
The number of contracts bought by Veron SA is given by:EUR: (-10,000,000 x .0043/125,000) x -0.84 = 2.89 3 contracts.GBP: (-10,000,000 x .0020/62,500) x -0.76 = 3.20 3 contracts. ¶