APPENDIX 4A: CONTINUOUS COMPOUNDING
5.4 FORWARD VERSUS FUTURES MARKET HEDGES
Forward and futures contracts are equivalent once they are adjusted for differ- ences in contract terms and liquidity. Indeed, the difference between a futures and a forward contract is operational rather than valuational, in that it depends on the contracts themselves (the deliverable asset, settlement procedures, maturity dates, and amounts) and not directly on prices.1As with forward contracts, the price Futtd/f of a futures contract is determined by relative interest rates and the current spot rate of exchange according to interest rate parity.
Futtd/f=Ftd/f=s0d/f[(1+id)/(1+if)]t (5.1) As with forwards, futures contracts allow you to hedge against nominal, but not real, changes in currency values. If inflation in the foreign currency is more than expected, then the forward rate won’t buy as much purchasing power as you expected. Currency forward and futures contracts can eliminate currency risk, but not inflation or interest rate risk within any single currency.
December January February March +€250,000
−¥37,500,000 today
Watanabe is short yen and long euros three months forward. Watanabe’s yen and euro cash flow exposures andrisk(payoff)profilesare as follows:
+€250,000
−¥37,500,000
∆V$/€
∆S$/€
∆V$/¥
∆S$/¥
Depending on the exchange rates St$/¥and St$/ , Watanabe might be spending some sleepless nights between now and March.
Forward Market Hedges
Watanabe can hedge these exposures by buying ¥37,500,000 forward and selling 250,000 forward. Suppose forward rates are equal to current spot rates such that S0$/¥ =F0,T$/¥ =$0.00800/¥ and S0$/ =F0,T$/ =$1.2000/ . Buying yen forward is equivalent to selling (¥37,500,000)($0.00800/¥)=$300,000 forward. Selling euros forward is equivalent to buying ( 250,000)($1.2000/ )=$300,000 forward.
These forward contracts lock in the following cash flows and payoff profiles:
+$300,000
−$300,000 +¥37,500,000
−€250,000
Buy yen forward Sell euros forward
∆V$/€
∆S$/€
∆V$/¥
∆S$/¥
In this example, the $300,000 cash outflow of the long yen position exactly offsets the $300,000 inflow of the short euro position. When combined with Watanabe’s underlying short yen and long euro positions, these transactions exactly neutralize Watanabe’s exposures to the yen and euro.
+$300,000
−$300,000 Net yen position
Net euro position
∆V$/€
∆S$/€
∆V$/¥
∆S$/¥
The net position has no exposure to currency risk, and Watanabe can now sleep soundly at night.
Futures Market Hedges
These cash flows are an integer multiple of the CME futures contract and occur on a CME expiration date, so they can be hedged exactly. Watanabe needs to buy three CME 3-month yen futures contracts worth ¥37,500,000 and sell two CME 3-month euro futures contracts worth 250,000. Cash inflows in each currency will be exactly offset by outflows, and Watanabe has no net exposure to currency risk.
Forwards versus Futures: Viv ´ela Diff ´erence
The biggest difference between futures and forwards is that changes in the underlying spot rate are settled daily in futures, whereas they are settled at maturity in a forward.
Suppose the yen spot rate is S0$/¥ =$0.010000/¥ and that 180-day interest rates are i$=4.03 and i¥=1.00 percent. Today’s futures and forward prices for exchange in six months are given by interest rate parity.
Fut0,1$/¥=F0,1$/¥=S0$/¥[(1+i$)/(1+i¥)]1−0
=($0.010000/¥)[(1.0403)/(1.0100)]1
=$0.010300/¥
The yen is selling at a 3 percent forward premium because the ratio of Eurodollar and Euroyen interest rates is 3 percent.
Futures are marked-to-market daily.
Suppose actual spot rates rise by $0.000005/¥ per day over each of the next 180 days to S1$/¥=($0.010000/¥)+($0.000005/¥)(180)=$0.010900/¥. This is a 9 percent increase over the current rate of $0.010000/¥. The purchaser of a yen forward would pay F1$/¥=$0.010300/¥ at expiration for yen worth $0.010900/¥
in the spot market, for a gain of $0.000600/¥ at expiration.
day 1 day 178 day 179 day 180
+$0.010900/¥
−$0.010300/¥
+$0.000600/¥
Settlement of a forward contract at expiration
This is a profit of ($0.000600/¥)/($0.010000/¥)=0.06, or 6 percent on each yen purchased.
In contrast, the futures contract is settled one day at a time. According to interest rate parity, the spot price is expected to rise by ($0.0003/¥)/(180 days)
= $0.0000016/¥ per day. If in fact the yen rises by ($0.0009/¥)/(180 days)=
$0.000005/¥ per day, there is a net gain at each daily settlement of ($0.0006/¥)/(180
days)=$0.000003/¥. Accumulated over 180 days, this equals a 6 percent gain. At expiration, the accumulated gain on the futures contract is the same as the gain on the forward contract. The difference is that the futures gain is received one day at a time.
day 1 day 178 day 179 day 180
Daily settlement of a futures contract (sum of all 180 days = $0.0006/¥)
+$0.000003/¥ +$0.000003/¥ +$0.000003/¥ +$0.000003/¥
The net gain or loss on futures is the same as on a forward.
In the more general case in which exchange rates fluctuate randomly over time, the net gain at the expiration of the forward contract still equals the sum of the daily settlements on a comparable futures contract. Figure 5.4 shows spot and futures prices that begin at S0$/¥=$0.010000/¥ and Fut0,1$/¥=$0.0103000/¥ and then fluctuate randomly toward a spot price at expiration of ST$/¥=$0.010900/¥. As in the previous example, day-to-day changes in the futures price are settled daily through the maintenance margin account as the contract is marked-to-market at each day’s close. At the end of the contract, the futures price will have converged to the spot exchange rate. Since the beginning and ending points are the same as in the previous example, the sum of the payments to or from each customer’s margin account over the life of the futures contract must equal the gain or loss at expiration on a comparable forward contract. The size and timing of the cash flows from the futures contract depend on the time path of the futures price, but the net gain or loss is the same as on the forward contract. This is the reason futures and forwards are near substitutes for hedging purposes and share the same risk profiles.
Standardized or Customized: Which Do You Choose?
A perfect hedge exactly offsets the underlying exposure.
The size, timing, and currency underlying a forward contract are negotiated between the bank and its client, so the transaction exposure of a foreign currency cash inflow or outflow can be exactly matched with a forward contract. If the size and timing of the foreign currency cash flow are exactly offset by a forward contract, the forward provides aperfect hedgeagainst currency risk.
Futures provide a perfect hedge against currency risk only when the underlying transaction falls on the same day and is in an integer multiple of a futures contract.
To the extent that the amount or timing of cash flows does not match an exchange- traded contract, futures provide only an imperfect hedge. The size mismatch is a problem only for small transactions. The maturity mismatch can be important, because exchange-traded contracts cannot be tailored to the maturity date of the
−3%
−2%
−1%
0%
1%
2%
3% Daily gain or loss
FutTd/f = STd/f
T (expiration) 0
Gain or loss on long futures
= (Futtd/f – Fut0d/f) Std/f
Fut0d/f S0d/f
Forward premium at t = 0
Gain or loss on long spot
= (Std/f – S0d/f)
FIGURE 5.4 Futures and Spot Price Convergence.
exposure. For the same reason, forward and futures contracts cannot be compared on cost alone unless the size and maturity of the forward and futures positions are identical.