APPENDIX 4A: CONTINUOUS COMPOUNDING
5.6 HEDGING WITH CURRENCY FUTURES
Forward contracts hedge foreign currency cash flows one-to-one when the forward contract matches the size, timing, and currency of the underlying exposure. Futures hedges also provide a perfect hedge against currency risk when the amount of a transaction that is exposed to currency risk is an even multiple of a futures contract and matures on the same date as a futures contract in the same currency.
Exchange-traded futures contracts cannot be tailored to meet the unique needs of each customer because they come in only a limited number of contract sizes, maturities, and currencies. Figure 5.5 presents a classification of futures hedges as a function of the maturity and currency of the underlying exposure. The rest of this section discusses these futures hedges.
Maturity Mismatches and Delta Hedges
A futures hedge is called a delta hedge when there is a mismatch between the maturity — but not the currency — of a futures contract and the underlying exposure.
When there is a maturity mismatch, a futures hedge cannot provide a perfect hedge against currency risk.
Hedge (hedge ratio estimation)
Cross hedge std/f = α + β std/f2 + et
Mismatch Exact match
Currency
Exact match
Mismatch Maturity
Perfect hedge: std/f = α + β std/f + et (such that α = 0, β = 1, and r2 = 1)
Delta hedge std/f = α + β futtd/f + et
Delta hedge std/f1 = α + β futtd/f2 + et
FIGURE 5.5 A Classification of Future Hedges.
A delta hedge has a maturity mismatch.
Suppose that today is Friday, March 13 (time 0), and that Chen Machinery Company has a S$10 million (Singapore dollar) obligation due on Friday, October 26. There are 227 days between March 13 and October 26, so with annual compounding this is t=(227/365) of one year. The nearest CME Singapore dollar futures contracts mature on Friday, September 11, and on Friday, December 16.
This maturity mismatch is shown here.
time t Oct 26
time T Dec 16 Sep 11
time 0 Mar 13
−S$10 million
A hedge with the futures contract that expires on September 11 hedges only against currency risk through that date. Chen remains exposed to changes in currency values from the end of the contract through October 26. The December futures contract is a better choice because it can hedge currency risk through October 26 and can then be canceled. December 16 is 278 days after March 13, so the time until expiration of the December contract is T=(278/365) of one year.
Suppose the spot rate is S0$/S$=$0.6010/S$ on March 13. Annual interest rates in the United States and Singapore are i$=6.24% and iS$=4.04%, respectively.
According to interest rate parity, the forward price for exchange on October 26 is F0,t$/S$=S0$/S$[(1+i$)/(1+iS$)]t
=($0.6010/S$)[(1.0624)/(1.0404)](227/365)
≈$0.6089/S$ (5.3)
Chen can form a perfect hedge with a long forward for delivery of S$10 million on October 26 in exchange for ($0.6089/S$)(S$10,000,000)=$6,089,000. As we shall see, a futures hedge using the December 16 futures contract can eliminate most — but not all — of Chen’s S$ exposure.
The Basis Risk of a Delta Hedge
Basis is the difference between nominal interest rates.
In a futures hedge, the underlying position is settled in the spot market and the futures position is settled at the futures price. Although futures converge to spot prices at expiration, prior to expiration there is a risk that interest rates will change in one or both currencies. If interest rates change, the forward premium or discount also will change through interest rate parity.
The interest rate differential often is approximated by the simple difference in nominal interest rates (id−if). This difference is called thebasis.The basis changes as interest rates rise and fall. The risk of unexpected change in the relation between the futures price and the spot price is called basis risk. When there is a maturity mismatch between a futures contract and the underlying exposure, basis risk makes a futures hedge slightly riskier than a forward hedge.
Using the Chen Machinery Company example, here is how basis is determined and how it can change prior to expiration. As with a forward contract, the price of the March 13 S$ futures contract for December delivery (i.e., at time T in 278 days) is determined by interest rate parity.
Fut0,T$/S$=S0$/S$[(1+i$)/(1+iS$)]T
=($0.6010/S$)[(1.0624)/(1.0404)](278/365)
≈$0.6107/S$ (5.4)
When this price is set on March 13, the expectation is that on October 26 the spot price will not have risen by the full amount. The expectation of the October 26 spot price is the same as the price for forward delivery on that date.
Basis risk: unexpected change in the relation between spot and futures prices.
F0,t$/S$=E[S0,t$/S$]=S0$/S$[(1+i$)/(1+iS$)]t
=($0.6010/S$)[(1.0624)/(1.0404)](227/365)
≈$0.6089/S$ (5.5)
This expectation will hold over the life of the exposure only if the interest rate ratio (1+i$)/(1+iS$)=1.0624/1.0404=1.0211 remains constant. This ratio is the ‘‘basis’’ for changes in futures prices.
The convergence of futures prices to the spot price at expiration is almost linear over time, so the basis (i$−iS$)=(6.24%−4.04%)=2.20 percent often is used in lieu of the ratio of interest rates in the interest rate parity relation.
Using the basis approximation, the spot price on October 26 is predicted to be
time 0 Mar 13
time t Oct 26
time T Dec 16 –S$10 million Actual profit (loss) on long S$ futures position:
Unexpected profit (loss) on short S$ spot position: Equation Time zero:
⇒ (5.4)
⇒ (5.5)
Scenario #1:
and
Futt,T$/S$ = ($0.6089/S$) [(1.0624)/(1.0404)](51/365)≈ $0.6107/S$ (5.4) Profit on long futures: +($0.6107/S$ – $0.6107/S$) +$0.0000/S$
Profit on short spot: –($0.6089/S$ – $0.6089/S$) –$0.0000/S$
Net gain $0.0000/S$
(5.4)
⇒ Futt,T$/S$ = ($0.6255/S$) [(1.0624)/(1.0454)](51/365)≈ $0.6269/S$
Profit on long futures: +($0.6269/S$ – $0.6107/S$) +$0.0162/S$
Profit on short spot: –($0.6255/S$ – $0.6089/S$) –$0.0166/S$
Net gain –$0.0004/S$
(5.4)
⇒
Profit on long futures: +($0.5795/S$ – $0.6107/S$) –$0.0312/S$
Profit on short spot: –($0.5774/S$ – $0.6089/S$) +$0.0315/S$
Net gain $0.0003/S$
Scenario #2:
Scenario #3:
(Futt,T$/S$ – Fut0,T$/S$) –(St$/S$ – E[St$/S$]) S0$/S$ = $0.6010/S$ with i$ = 6.24% and iS$ = 4.04%
Fut0,T$/S$ = S0$/S$[(1 + i$)/(1 + iS$)]T
= ($0.6010/S$) [(1.0624)/(1.0404)](278/365)≈ $0.6107/S$
E[St$/S$] = S0$/S$[(1 + i$)/(1 + iS$)]t [(1 + i$)/(1 + iS$)t]
= ($0.6010/S$) [(1.0624)/(1.0404)](277/365)≈ $0.6089/S$
St$/S$ = $0.6089/S$ with i$ = 6.24% and iS$ = 4.04%
St$/S$ = $0.6255/S$ with i$ = 6.24% and iS$ = 4.54%
St$/S$ = $0.5774/S$ with i$ = 6.74% and iS$ = 4.04%
Futt,T$/S$ = ($0.5774/S$) [(1.0624)/(1.0404)](51/365)≈ $0.5795/S$
FIGURE 5.6 An Example of a Delta Hedge.
(0.0220)(227/365)=0.0137, or 1.37 percent above the March spot price. This suggests an October spot price of ($0.6010/S$)(1.0137)=$0.6092/S$, which is fairly close to the forward price of $0.6089/S$ from Equation 5.5.
On October 26, there are 51 days remaining on the contract. This contract provides a perfect hedge of Chen’s exposure so long as the basis does not change. If the basis changes, then the hedge is imperfect and there will be some variability in the hedged payoffs. Figure 5.6 provides an example using three scenarios.
Scenario #1 Scenario #1 reflects the market’s expectation. In this scenario, the basis (i$−iS$) has not changed and the spot rate on October 26 turns out to be the
$0.6089/S$ rate predicted by Equation 5.5. On October 26, the futures price for December delivery is based on the prevailing spot exchange rate of $0.6089/S$, the basis of 2.20 percent per year, and the (T−t)=(278−227)=51 days remaining on the futures contract according to Equation 5.4.
Futt,T$/S$=St$/S$[(1+i$)/(1+iS$)]T−t
=($0.6089/S$)[(1.0624)/(1.0404)](51/365)
≈$0.6107/S$
This is the same time T price expected at time t=0. In this scenario, there are no gains or losses on the long futures position or on the underlying short position in spot currency.
Profit on long futures: (Futt,T$/S$−Fut0,T$/S$)=($0.6107/S$−$0.6107/S$)
=$0.00/S$
Profit on short spot: −(St$/S$−E[St$/S$])= −($0.6089/S$−$0.6089/S$)
=$0.00/S$
Consequently, in this scenario there is no gain or loss on the combined position.
Scenario #2 In this scenario, the S$ interest rate rose to iS$=4.54% and the Singapore dollar rose to St$/S$=$0.6255/S$ on October 26. With these new rates, the October futures price for December delivery is
Futt,T$/S$=St$/S$[(1+i$)/(1+iS$)]T−t
=($0.6255/S$)[(1.0624)/(1.0454)](51/365)
≈$0.6269/S$
The gains (losses) on the futures and spot positions are now as follows:
Profit on long futures: (Futt,T$/S$−Fut0,T$/S$)=($0.6269/S$−$0.6107/S$)
= +$0.0162/S$
Profit on short spot: −(St$/S$−E[St$/S$])= −($0.6255/S$−$0.6089/S$)
= −$0.0166/S$
The net position is then +($0.0162/S$)−($0.0166/S$)= −$0.0004/S$, or
−$4,000 based on the S$10 million underlying positions. This loss arises because of a change in the Singapore dollar interest rate and not because of change in the spot exchange rate.2
Scenario #3 In this scenario, dollar interest rates rose to i$=6.74 percent and the spot rate fell to St$/S$=$0.5774/S$. Singaporean interest rates remain unchanged at iS$=4.04 percent. The October futures price for December delivery is
Futt,$/S$=St$/S$[(1+i$)/(1+iS$)]T−t
=($0.5774/S$)[(1.0674)/(1.0404)](51/365)
≈$0.5795/S$
In this instance, the gains (losses) on the two positions are
Profit on long futures: (Futt,T$/S$−Fut0,T$/S$)=($0.5795/S$−$0.6107/S$)
= −$0.0312/S$
Profit on short spot: −(St$/S$−E[St$/S$])= −($0.5774/S$−$0.6089/S$)
= +$0.0315/S$
The net gain is (−$0.0312/S$+$0.0315/S$)= +$0.0003/S$, or $3,000 based on the S$10 million short and long positions. Again, it is basis risk that spoils the futures hedge.
Chen’s underlying short position in Singapore dollars is exposed to considerable currency risk. If the range of spot rates is from $0.5774/S$ to $0.6255/S$, as in Scenarios #2 and #3, then the range of dollar obligations is $481,000 (from
−$5,774,000 to−$6,255,000) on the underlying exposure in the spot market. This risk arises from variability in the level of the exchange rate. A forward contract can reduce the variability of the hedged position to zero. The futures hedge does almost as well, producing a $7,000 range of outcomes (from−$4,000 to+$3,000).
The remaining risk in the futures hedge arises from variability in the basis— the risk that interest rates in one or both currencies will change unexpectedly. The futures hedge transforms the nature of Chen’s currency risk exposure from a bet on exchange rates to a bet on the difference between domestic and foreign interest rates.
Futures Hedging Using the Hedge Ratio
The optimalhedge ratioNF∗of a forward position is defined as
NF∗=Amount in forward position/Amount exposed to currency risk (5.6) In a perfect forward hedge, the forward contract is the same size as the underlying exposure, and the optimal hedge ratio is NF∗= −1. The minus sign indicates that the forward position is opposite (short) the underlying exposure. A forward contract provides a perfect hedge because gains (losses) on the underlying position are exactly offset by losses (gains) on the forward position.
The Futures Hedge As with forward contracts, most of the change in the value of a futures contract is derived from change in the underlying spot rate. However, because futures contracts are exposed to basis risk, there is not a one-to-one relation between spot prices and futures prices. For this reason, futures contracts generally do not provide perfect hedges against currency exposure. However, futures contracts can provide very good hedges, because basis risk is small relative to currency risk.
The relation between spot and futures price changes can be viewed as a regression equation
St$/S$=α+βfutt$/S$+et (5.7)
−1.5
−1
−0.5 0 0.5 1 1.5
−1.5 −1 −0.5 0 0.5 1 1.5
std/f
futtd/f β = ρ
s,fut (σ
s /σ
fut)
Hedge quality is measured by r-square; that is, the percent of the variation in
std/f that is explained by variation in futtd/f
FIGURE 5.7 Linear Regression and the Hedge Ratio.
where st$/S$=(St$/S$−St
−1
$/S$)/St
−1
$/S$and futt$/S$=(Futt$/S$−Futt
−1
$/S$)/Futt
−1
$/S$
are percentage changes in spot and futures prices during period t. In the Chen example, this regression should be estimated using futures contracts that mature in 71/2months (e.g., from March through October). The regression then provides an estimate of how well changes in futures prices predict changes in spot prices over 71/2-month maturities.
The regression in Equation 5.7 is shown graphically in Figure 5.7. Since spot and futures prices are close to a random walk, the expectations of both Futt$/S$and st$/S$are zero and the intercept termαin this regression is usually ignored. As in any regression, the slopeβin Equation 5.7 is equal to
β=(σs,fut)/(σfut2)=ρs,fut(σs/σfut) (5.8) The slope coefficientβmeasures changes in futures prices relative to changes in spot prices. The error term et captures any variation in spot rate changes st$/S$ that is unrelated to futures price changes futt$/S$.
If the historical relation between spot prices and futures prices is a reasonable approximation of the expected future relation, then this regression can be used to estimate the number of futures contracts that will minimize the variance of the hedged position. Let NSbe the size of the underlying exposure to currency risk and NFut the amount of currency to be bought or sold in the futures market to offset the underlying exposure. The optimal amount in futures to minimize the risk of the futures hedge is
NFut∗=Amount in futures contracts/Amount exposed to currency risk (5.9)
= −β
In this context, the hedge ratio provides the optimal amount in the futures hedge per unit of value exposed to currency risk. A futures hedge formed in this fashion is called a delta hedge because it minimizes the variance (the , or delta) of the hedged position.3
MARKET UPDATE Megallgesellschaft’s Oil Futures Hedge
Metallgesellschaft A.G. was a large MNC based in Germany with interests in engineering, metals, and mining. In 1991, Metallgesellschaft’s U.S. subsidiary MG Refining and Marketing (MGRM) nearly drove Metallgesellschaft into bankruptcy through an ill-fated hedging strategy in crude oil futures.* MGRM had arranged long-term contracts to supply U.S. retailers with gasoline, heating oil, and jet fuel. Many of these were fixed rate contracts that guaranteed a set price over the life of the contract.
To hedge the risk of these delivery obligations, MGRM formed a ‘‘rolling hedge’’ of long positions in crude oil futures contracts of the nearest maturity.
Each quarter, the long position was rolled over into the next quarter’s contract.
MGRM used a one-to-one hedging strategy in which long-term obligations were hedged dollar-for-dollar with positions in near-term crude oil futures contracts.
Although the intent of this hedging strategy was well-intentioned, the mismatch between the long-term short positions in delivery contracts and the short-term long positions in oil futures created havoc for MGRM. Futures price fluctuations resulted in wildly fluctuating short-term cash flows in MGRM’s margin account that did not match the maturity of MGRM’s long-term delivery contracts. Metallgesellschaft nearly went bankrupt in 1991 as a result of a
$1.4 billion loss from its hedge. Metallgesellschaft’s experience is a reminder that the exposure (i.e., maturity) of a financial hedge must match the exposure of the underlying transaction.
*Metallgesellschaft’s difficulties are described in the Spring 1995 issue of theJournal of Applied Corporate Finance.Metallgesellschaft is now a part of Germany’s GEA Group AG.
Hedge quality is measured by ther-squareof the regression in Equation 5.7.
R-square is the square of the correlation coefficient (i.e.,ρs,fut2) and also is called the
‘‘coefficient of determination’’ or ‘‘r2.’’ It is bounded by zero and one, and measures the percentage of the variation in st$/S$that is explained by variation in Futt$/S$. A high r-square indicates low basis risk and a high-quality delta hedge. A low r-square means that basis risk is high relative to the underlying currency risk.
R-square measures hedge quality.
The regression in Equation 5.7 is designed to estimate basis risk over the maturity of a proposed hedge. Unfortunately, it is difficult to construct a sample of futures prices of constant maturity t because exchange-traded futures come in only a limited assortment of maturities. In the Chen example, this would be a 71/2-month maturity.
Exchange-traded futures expire only every three months, and the futures prices on
any single contract converge to the spot rate at maturity. Fortunately, interest rate parity determines both the forward price and the futures price for a given maturity.
It is much easier to construct a sample of forward prices of constant maturity than a sample of futures prices of constant maturity, so the hedge ratio conventionally is estimated from the relation of forward price changes to spot changes over the desired maturity.
An Example of a Delta Hedge Suppose the regression in Equation 5.7 yields a regres- sion coefficient ofβ=1.025. The futures hedge should then consist of
NFut
∗=(Amount in futures contracts)/(Amount exposed)= −β
⇒(Amount in futures contracts)=(−β)(Amount exposed)
For Chen’s underlying S$10 million short exposure, this requires a long position of Amount in futures contract=(−1.025)(−S$10,000,000)
=S$10,250,000
Variability in the hedged position can be minimized with S$10,250,000 of December futures. On the CME, this would be worth (S$10,250,000)/(S$125,000/contract)= 82 futures contracts.
Cross Hedges and Delta-Cross Hedges
Adelta-cross hedgeis used when there are both maturity and currency mismatches between the underlying exposure and the futures hedge. The regression in Equation 5.7 must be modified for a delta-cross hedge to include both basis risk from the maturity mismatch as well as currency cross-rate risk from the currency mismatch.
The general form of the regression equation for estimating the optimal hedge ratio of a delta-cross hedge is
Delta-cross hedges have FX and maturity mismatches.
std/f1 =α+βfuttd/f2+et (5.10) for an underlying transaction exposure in currency f1and a futures hedge in currency f2. The interpretation of the slope coefficient as the optimal hedge ratio is the same as in Equation 5.9; that is, buy futures contracts according to the ratio Nfut∗= −β.
Across hedgeis a special case of the delta-cross hedge. As discussed earlier, in a cross hedge there is a currency mismatch but not a maturity mismatch. The optimal hedge ratio of a cross hedge is estimated from
std/f1 =α+βstd/f2+et (5.11)
This is identical to Equation 5.10 except that Futtd/f2 is replaced by std/f2. Spot rate changes std/f2 can be substituted for futtd/f2 because futures prices converge to spot prices at maturity, and the maturity of the futures contract is the same as that of the underlying transaction exposure in the spot market.
If futures are not available in the currency that you wish to hedge, a cross hedge using a futures contract on a currency that is closely related to the desired currency can at least partially hedge against currency risk. As an example, a U.K.-based corporation can hedge a Canadian dollar (C$) obligation with a long U.S. dollar futures contract because the pound values of the U.S. dollar and the Canadian dollar are highly correlated. For a U.S. dollar hedge of a Canadian dollar obligation, the spot exposure is in Canadian dollars and the futures exposure is in U.S. dollars as in the following regression:
A cross hedge has a currency mismatch.
st£/C$=α+βfutt£/$+et (5.12) The quality of this cross-rate futures hedge is only as good as the correlation between the pound sterling values of the U.S. and Canadian dollars.
When both the maturity and the currency match that of the underlying obliga- tion, Equation 5.10 reduces to
std/f=α+βstd/f+et (5.13) Since the correlation of std/f with itself is+1, this is a perfect hedge (r-square=1) and the optimal hedge ratio is NFut∗= −β= −1. In this circumstance, the futures hedge is equivalent to a forward market hedge. There is no basis risk and currency risk can be completely eliminated.