FORWARD AND FUTURES CONTRACTS
1. There are many different reasons some futures contracts succeed and some fail, but the most important is demand. If people need a particular contract to expose themselves to or hedge a price risk, then the contract will succeed. Most people use Treasury bond futures to gain exposure to or hedge general long term interest rate risk. The only additional advantage of futures on corporate bonds would be that the investors could gain exposure to changes in the credit spread. Apparently there is little demand for this, either because investors do not want to hedge or gain exposure to this risk or because the underlying market is not liquid enough to support futures. Either way, the lack of futures is motivated by a lack of demand in the asset or futures contract. The lack of chicken contracts most likely derives from a similar lack of demand. It could be that chicken prices are highly correlated with other existing contract prices, so investors do not need the additional chicken contract. Perhaps there are too many different types of chickens to have a single contract that would attract enough trading volume.
2. Before entering into a futures or forward contract, hedgers have exposure to price changes in the underlying asset. To hedge this risk, hedgers enter into contracts that most closely offset this price risk. The problem is that for most hedgers there is not a contract that exactly matches their exposure. Perhaps, the commodity they use is a different grade or needed in a different location than specified in the contract, so differences in prices between the actual asset the company is exposed to and the asset in the contract may exist and change over time. Likewise, a portfolio manager hedging a stock portfolio may hold a portfolio that is not perfectly correlated with the index future he/she is using to hedge with. To minimize basis risk it is necessary to find the contract whose price is most highly correlated with the price of the asset to be hedged.
3(a). To hedge price risk, you could enter into a long position in 100,000 gallons worth of gasoline futures.
3(b). Since you will have to post margin on the futures contract, the price swings in the futures contract will effect how much you earn on the capital posted as margin. If gas prices go up, your margin account will be credited and you will earn more interest. If gas prices go down, your account will be debited and you will earn less interest. If prices go down substantially, you will be required to post additional margin, and therefore tie up additional capital. How this effects your pricing of the forward contract you have sold depends on your opportunity cost of capital.
3(c). In this case, by using futures you will not be able to match the quantity or time of delivery in the forward contract you sold. This gives rise to two types of risk. Since you will be forced to over or under hedge, you will be exposed to the general price movements in gas one way or the other. Next, if you synthetically create a three month expiration using half two-month and half four-month contracts, then you will be exposed to relative price changes in the two futures contracts in the near term. Later, after the two-month contract expires, you will be forced to hedge a shorter term position with a longer term contract, so again relative near-term and longer-term price differentials will lead to basis risk.
4. There are two types of basis risk that this hedge is exposed to. The first is from changes in the shape of the yield curve. Since the company wishes to hedge a seven year issue’s cost with a ten year contract, the hedge is exposed to changes in the relative level of interest rates between the seven and ten year maturities. Specifically, if the seven-year treasury rate rises more relative to the ten-year rate then the hedge will not completely neutralize the position and lose money for the firm. The second source of basis risk is from the quality spread over treasuries in the Eurobond market. Since the company will have to sell its bonds with a spread over the Treasury rate, changes in this spread will also effect the quality of the hedge. Specifically, increases in the spread will lead to losses for the firm.
5. CFA Examination II (1997)
5(a). What Lane Should Do
To protect his investment from declining interest rates, Mike Lane should purchase, or “go long,” $5 million worth of U.S. Treasury five-year-note futures contracts. Lane can actually purchase any of the traded five-year-note futures contracts, depending on what date he actually plans to invest his $5 million.
5(b). Effect of Higher Interest Rates on Lane’s Position
If rates increase by 100 basis points in three months, the price of these futures contracts will decline and Lane will have a loss in his long futures position. The loss will show up in his mark-to-market position over time and will require him to post additional margin money.
5(c). Return from Lane’s Hedged Position vs. Unhedged Return
Two methods can be used to answer Part C.
One answer is that the return from Lane’s hedged position will be lower than the return if he had not hedged. Because of the futures contract’s loss in Part B, the higher yield Lane can earn when he purchases a now lower priced (higher yield) U.S. Treasury five-year note in the cash market is “offset” by the loss from the futures contract. The loss is actually added to the now lower price of the U.S. Treasury five-year notes, thus decreasing Lane’s realized yield.
A second answer draws from the Clarke reading. The reading expresses the combined futures contract’s price and cash market price in terms of “net price.” Net price equals the new cash market price plus the original futures price minus the new futures price. Subtracting net price from 100 gives the investor the realized yield and thus the investor’s return from the combined futures and cash markets. Because rates have increased, the new lower cash market price (higher yield) is increased by the loss from the futures contract’s position, which reduces realized yield. The reading expresses this net price in several forms, but all are the same formula with rearranged terms.
6. Both Eurodollar and Treasury Bill futures are designed so that the long position benefits from a decline in the respective reference rates (relative to the contract yield) while the short position benefits from a rate increase. In both cases, the purpose of the hypothetical price index is to translate rate declines into price increases—and vice versa—which is a more natural way to think of the contract holder’s position. Despite the fact that the underlying rate for these contracts are quoted on different conventions, they can both use the same price index calculation because all that really matters at maturity (or contract unwind) is the number of basis points the settlement price differs from the original contract price, with each basis point being worth $25 (= $1,000,000) x .0001 x 90/360).
7. It is most likely that a single position in an index futures market would be the best hedge. There are several reasons for this. The most important is cost. Since there are no exchange traded futures for individual stocks, entering 50 different positions would have to be done through an over-the-counter derivative dealer. This typically would mean higher transaction costs to cover the fees from setting up the one-off deal, the lower liquidity of individual stocks, and the increased commissions for the dealer. Another disadvantage is the liquidity of the position. Index options are very liquid and can be closed out quickly with little trading cost. Closing the 50 different positions would entail paying many of the start-up costs twice. Finally, it is easy to short an index future but rather hard and more expensive to short the underlying stocks which the OTC dealer would have to do to hedge the position in the 50 different stocks. The only advantage to the 50 different positions is that they would provide a nearly perfect hedge, whereas there would be some basis risk in the index futures position. Since the portfolio is well diversified, this should not be a major problem.
8. CFA Examination II (1991)
The fourth factor affecting the price of a stock index futures contract is the risk-free interest rate, usually measured by the Treasury bill rate. Futures prices increase with increases in the risk-free interest rate. Investors can create portfolios having identical levels of risk by either investing directly in a diversified equity portfolio or purchasing an equivalent position in stock index futures and placing the remainder in risk-free assets. The stock portfolio earns the price appreciation of the stocks plus their dividend yield; the futures portfolio earns the price appreciation of the futures plus the risk-free interest rate. Since futures are marked to the market, the futures price will equal the spot price of the stocks at the futures contract’s expiration date. Market forces (arbitrage activity) results in stock index futures being priced such that their price is equal to the future value of the current spot price, using the “cost of carry” as the discount rate. The cost of carry is the risk free interest rate minus the dividend yield on the stock portfolio.
If the risk-free interest rate subsequently increases, it becomes more profitable to purchase the futures/Treasury bill combination than to invest directly in the stocks themselves, because of the higher return on the Treasury bills now available. As a result, the price of the futures contract will be bid up until it is again equal to the future value of the current spot price of the equivalent stock portfolio. There is, thus, a direct and positive relationship between the risk-free interest rate and futures prices.
9. When the index futures price is below its theoretical level, the arbitrage involves buying the futures contract and selling the underlying index of stocks short. When the index futures price is above its theoretical level, the arbitrage involves selling the futures contract and buying the underlying index of stocks. The practicalities of selling the index short make the first type of arbitrage more difficult. First, there is an up-tick rule for short sales that prevents short sales from being possible for all of the index’s stocks at the current market price. This is because, on average, about half of the market prices will have just moved down a tick and the up-tick rule prevents short sales until they trade up a tick. Second, margin requirements and limitations on the use of proceeds from short sales demand the use of more capital than long positions in the underlying stocks. Finally, for some stocks there may not be shares available to borrow and then short, or shares borrowed may be recalled by the original owner prior to the arbitrageur’s preferred time to close the position.
10. CFA Examination III (1989)
If WEC invests in Bunds, they can use the current spot rate to calculate how many bonds they will receive today for $30 million. We can assume WEC is satisfied with the Bund interest rate and motivated primarily by a desire to diversify and reduce interest rate risk. Even if they could guarantee the Deutschemark value of the bonds they will hold, however, they would still face exchange rate risk. A currency futures contract sets the rate for future exchange of marks for dollars. With a prior agreement, like a futures contract, WEC can guarantee their exchange rate six months hence.
Short futures positions will incur losses as the exchange rate rises and gains as the exchange rate falls. The dollar value of the Bunds will change in the opposite direction. A perfect hedge would have WEC buy just enough futures (face value in marks) to cover the marks they will repatriate.
Even a perfect fixed futures hedge does not preserve the entire $30 million. The futures settlement rate for six months hence is almost surely less favorable than the exchange rate today. WEC can hedge but probably will lock in a loss, even without transaction costs. This loss is part of the opportunity cost of hedging forgoing the chance of exchange rate gains in return for preventing exchange rate losses.
Fixed currency hedges are rarely perfect because German mark futures contracts are for a fixed amount (currently 125,000 DM) and may not be an integer multiple of the number of marks purchased today. A typical fixed futures hedge is either slightly over or under hedged.
Currency futures positions require margin. Losses must be paid daily. If WEC does not liquidate gains on the Bunds to fund losses on the futures, they may need extra cash some time during the six months.
Fixed currency hedging also presents problems when the Bunds mature beyond the six month holding period. WEC will not know exactly how many marks it will need to repatriate six months hence. Hedging by shorting a fixed number of contracts is rarely done unless the foreign investment is a pure discount security.
11. First, enter into a forward position agreeing to exchange pesos for dollars in two months. Then, enter into another forward contract agreeing to exchange those dollars for Swiss francs, also in two months.
12. Because the funds from one country could be converted to another currency and then through the use of forward contracts converted back to the original currency at the same rate, an arbitrage opportunity is available. Investors from the Country B could borrow at the lower rate, convert into the currency of Country A, earn a higher rate of return on the money, and then pay back their loan pocketing the difference in interest payments. The market forces from these transactions would tend to either equalize interest rates or change the forward exchange rates so that the currency of Country A trades at a forward discount.
Answers to Problems
Price Adjustment Margin Maintenance
March 9 $173.00 0 3000 0
April 9 $179.75 -675 2325 0
May 9 $189.00 -925 3000 1600
June 9 $182.50 650 3650 0
July 9 $174.25 825 4475 0
Net loss of $4475 – ($3000 + $1600) = -$125
Total Return = -125/3000 = – 4.17%
1(b). Cost of Carry = 1.5%+8%=9.5%
Theoretical spot price on March 9
S = PV(F)
S = 173*exp(1.095*.3333)
S = $167.61
Implied May 9 price
S = $189*exp(-.095*.16667)
S = $186.03
1(c). Futures (Forwards) unwind without (with) discounting net differential, so
Short Futures Long Forward Net
May 9 -(189 – 173) x100= ($1,600) (189-173) x 100 x exp(-0.8x .1667) = $1,578.81 -21.19
June 9 -(182.5 -173)x100=($950.00) (182.5 – 173) x 100 x exp(-0.8 x 0.0833)=$943.69 -6.31
This implies that the forwards underhedge with equal notional amounts of forwards and futures.
2(a). Client A would need to go short 10,000 units of the June 1 contract at $24.95 and long 15,000 units of the September 1 contract at $25.85 The change in value of the two contracts to Client A is
= (24.95 25.85) x 10,000 + (25.85 25.65) x 15,000 = ($2,500.00)
So, you would receive this amount since this is the amount of their losses.
2(b). You would need to add in the interest received on the margin balance.
2(c). If Client B called to default, it would not be to your advantage since they had a long position at $26.40 and the price has declined to $25.85. So you would lose 25,000 x ($26.40-$25.65) = $18,750.00 at expiration, which in present value terms is $18,610.42.
2(d). Yes, you were short the equivalent of 25,000-15,000=10,000 contracts so you would be harmed by a price increase.
3(a). You buy the coffee at 58.56 cents per pound. This will cost 75,000 x ($.5856) = $43,920. Your futures profit will be 75,000 x ($ .592 – $ .5595) = $2,437.50. This reduces the effective price at which you buy the coffee to $43,920-$2,437.50 = $41,482.50. This is an effective price per pound of $41,482.50/75,000 = $ .5531. So you paid 55.31 cents per pound.
Buying two contracts for 75,000 pounds at 55.95 cents/pound leaves 7,000 pounds unhedged and therefore purchased in the spot market for 58.56 cent/lb. The effective price per pound is therefore = (75,000 x 55.95 + 7,000 x 58.56)/82,000 = 56.17 cents/pound.
The difference in price between spot and future is probably due to delivery costs.
3(b). There are a couple of types of basis risk. First the anticipated amount is not exactly hedgable because of the contract size. This means you will either have to over-hedge or under-hedge. Also you may not know the exact amount that you will really need at the future date. If you are really going to purchase the coffee somewhere else and were only using the futures to hedge (i.e. close your position before delivery) then you will be exposed to changes in the relative prices between the market you purchase in and the futures market.
4. CFA Examination III (1999)
4(a). Futures are an efficient, low-cost tool that can be used to alter the risk and return characteristics of an entire portfolio with less disruption than using conventional methods there may also be both institutional constraints and unfavorable tax consequences that prevent a portfolio manager such as Klein from liquidating the entire portfolio. Because Treasury bonds and Treasury bond futures have a very high correlation, the futures approach allows one to effectively create a temporary fully liquidated position without disturbing the portfolio. Futures can be sold against the portfolio to replicate the price response of then portfolio with the desired duration. In addition, there are cost advantages of using futures contracts including lower execution costs (bid-ask spread), speed and ease of executions (time required), and the higher marketability and/liquidity of futures contracts. The bond sale strategy may well be disadvantageous on all counts. Shortening the duration by liquidating the bond portfolio would be more costly, time consuming, and disruptive to the portfolio, with possible adverse tax implications as well. In Klein’s case, there may be more bonds to sell than futures contracts, because many bonds in the portfolio could be in denominations as low as $1,000. Also, the bond sales would invoke liquidity problems not encountered by the bond futures strategy.
4(b). The value of the futures contract is 94-05 (i.e., 94 5/32% of $100,000), which translates into 0.9415625 x $100,000 = $94,156.25.
Using the information given, there are at least two ways, modified duration (MD) or basis point value (BPV), to calculate the number of contracts.
Using modified duration,
Target Change in Value using MD
= Change in Value using MDhedge + Change in Value using MDportfolio
= (MDhedge x change in yield x Valuehedge) +
(MDportfolio x change in yield x Valueportfolio)
= (MD per futures x change in yield x N x contract value) +
(MDportfolio x change in yield x Valueportfolio)
where N = the number of futures contracts.
Because the target MD is zero, then:
N = -(MDportfolio x Valueportfolio)/(MD per futures x contract value)
= -(10 x $100,000,000)/(8 x $94,156.25)
= – 1328 (exact answer – 1327.58) or short 1328 contracts.
Using basis point value,
BPVtarget = BPVportfolio – BPVhedge
BPVtarget = BPVportfolio – (N x BPV per futures)
And N = (BPVtarget – BPVportfolio)/BPV per futures
Because the target BPV is zero, then:
N = ($0 – $100,000)/$75.32
= -1328 (exact answer –1327.67) or short 1328 contracts.
Klein is selling the contracts as indicated by the negative value of the contracts. The difference in the two exact answers is due to rounding the BPV number to the nearest cent.
4(c). Because the newly modified portfolio has approximately a zero modified duration and basis point value, the value of this portfolio would remain relatively constant for small parallel changes in rates. With an interest rate increase, the bond portfolio’s immediate market value would decline, but the positive cash flow from the Treasury bond futures contracts would offset this loss. As shown in part B, either modified duration or basis point value can be used to compute the change in value.
Change in value using MD = MD x change in yield x value, or
Change in Value using BPV = BPV x BP change
4(c)i. The $100,000 BPV for the portfolio means that the portfolio value will decrease (increase) by $100,000 for each basis point increase (decrease). A 10 basis point increase in interest rates would mean a $1,000,000 decline (or loss in the market value of the original portfolio.
Change in Value = MD x change in yield x change in value
= 10 x .001 x $100,000,000
Change in Value = BPV x BP change
= 10 x $100,000
4(c)ii. A $75.32 BPV for the futures contract represents a $75.32 change in value per basis point per contract. When rates increase by 1 basis point, each futures contract will decrease by $75.32. However, because Klein is short contracts, she will receive a cash flow of $75.32 from each short contract for each basis point increase.
Using MD, the total cash inflow from the futures position is:
$94,156 x 8 x .0001 x 1,328 = $1,000,316
Using BPV, the total cash inflow from the futures position is:
10 x $75.32 x 1,328 = $1,000,249
Differences from exactly $1,000,000 are due to rounding the number of contracts.
4(c)iii. The change in the value of the hedged portfolio is the sum of the change in value of the original portfolio and the cash flow from the hedge (futures) position, or:
Newly-hedged portfolio change = -$1,000,000 + $1,000,316 ~ $0 (using MD).
= -$1,000,000 + $1,000,249 ~ $0 (using BPV).
4(d). Klein’s hedging strategy might not fully protect the portfolio against interest rate risk for several reasons. First, immunization risk would remain even after execution of the strategy, because of the possibility of non-parallel shifts in the yield curve. If the yield curve shifts in a non-parallel fashion, the modified portfolio is not immunized against interest rate risk because the original bond portfolio and T-bond futures exist at different points on the yield curve and hence face different interest rate changes. If the curve became steeper, for example, then the market value loss on the original bond portfolio would be accompanied by a less-than-compensating value gain on the futures position. Second, the volatility of the yield between the T-bond futures and the government bond portfolio may not be one-to-one. Hence a yield beta adjustment may be needed. Third, basis risk also exists between the T-bond futures and spot T-bonds, so that there would still be risk even if the government portfolio held only T-bonds. Fourth, this may still be a cross-hedge, because the government bonds in the portfolio may not be the same as the cheapest-to-deliver bond. Fifth, the duration will change as time passes, so risk will arise unless continual rebalancing takes place. Sixth, because fractional futures contracts cannot be sold, the duration may not be able to be set exactly to zero.
4(e). The correct strategy would be to short (write or sell) call options and go long (buy) put options. The short call position would create a negative cash flow if rates were to decline, but the long put position would create a positive cash flow if rates were to increase. This fully hedges the portfolio. The call and put options should have the same exercise price and expiration date and the appropriate notional amounts. The following diagram illustrates this strategy:
5(a). The price of the bond as of August 1993 will be 97.39193% of par for a total portfolio value of $97,391,930:
1.03435 1.03943 + [40(0.04 – .039435)]
D = .039435 0.04 x [(1.03943540) 1]+.039435
52 (.03625 x 100) 100
PAug93 = (1 + 0.37395)t (1 + 0.37395)52
5(b). The duration portfolio bond:
1.037395 1.037395 + [52(0.03625 – .037395)]
D = .037395 0.03625 x [(1.0373552) 1]+.037395
= 27.741543 – 3.9796244
= 23.761919 six month periods
= 11.88096 years
modified duration = (11.88096/1.037395) = 11.4537
The duration futures bond:
1.039435 1.039435 + [40(0.04 – .039435)]
D = .039435 0.04 x [(1.03843540) 1]+.039435
= 26.358184 – 5.6688275
= 20.689356 six month periods
= 10.3447 years
modified duration = (10.3447/1.039435) = 9.9522
5(c). The calculation of the hedge ratio:
= (11.4527/9.9522) x 1 x (97,391,930/101,125)
= 1.15077 x 963.0846
= 1108.29 contracts
= 1108 contracts
6(a). Both bonds in portfolio 1 are zero coupon bonds so,
D1 = (4/10) x 14+ (6/10) x 3 = 7.4
ModD1 = 7.4/1.0731= 6.896 years
D = 1.0731 _ 1.0731 + 9 x (.046 – .0731) = 7.4
.0731 .046[(1.0731)9 – 1]+.0731
ModD2 = 7.4/1.0731= 6.896 years
For both portfolios:
P/P -6.896 x .006 = -4.137%
6(b). Though the two portfolios have identical durations, the actual price changes will be different because the portfolios have different convexities. In general, a “barbell” portfolio (i.e. portfolio l) will have greater convexity, meaning that its value will fall less relative to portfolio 2.
6(c). To hedge this position
HR1 = (6.896/10.355) x [0.6 x 1.13 + 0.4 x 1.03] x (-10,000,000/109,750) = -66.14
HR2 = (6.896/10.355) x [1.01] x (-11,500,000/109,750) =70.48
Combining these two gives a net hedge ratio of 4.339. Consequently, the hedge would be to enter into a long position in 4 futures contracts.
7. CFA Examination II (1998)
Arbitrage transactions. In a cash-and-carry strategy, which is what this transaction is, the arbitrageur borrows funds at a short-term rate, buys the asset, and sells the futures contract. On the expiration date, the arbitrageur delivers the asset against the futures, repays the loan with interest, and earns a low-risk profit.
• List of transactions
• Borrow funds
• Buy the asset
• Sell futures
• Deliver the asset against the futures
• Repay the loan with interest
Calculation of arbitrage profits. The cash-and-carry model, so called because the trader buys the cash good and carries it to the expiration of the futures contract, may be used to explain the pricing relationship between the futures market and the cash (or spot) market. Simplistically, the expected price (or “fair value”) of a futures contract is given by the formula:
Futures price = Cash price + Finance charges – Income.
First, determine whether the futures is overpriced or underpriced relative to cash. Calculate the fair value of the futures contract and compare this value to the contract’s actual value. If the actual value is greater than the fair value, the futures contract is overvalued, and a cash-and-carry strategy will result in profits; if the actual value is less than the fair value, the futures contract is undervalued and a reverse cash-and-carry strategy will yield profits.
The theoretical futures price is $101 + $2.50 – $4.50 = $99. Because the actual futures (invoice) price is $ 100, a profit of $1 can be obtained by employing a cash-and-carry strategy.
(180 x .0750) – (90 x .0750) 1
IFDY90,180 = X
180 – 90 1 – [(90 x .0750)/360]
Since the implied forward rate, 7.6433%, is not equal to the explicit futures rate, an arbitrage opportunity is possible, depending on the size of the profit compared to the transaction costs.
• Go long (or lend) at 7.6433% between days 90 and 180 by buying the 180-day paper and selling the 90 day paper in the current spot market.
• Go short (or borrow) at 7.50% between days 90 and 180 in the futures market.
(all transactions are in British pounds)
Date 0 90 180
Buy 180-day paper in spot market
Pay 96,250 Get 100,000
Sell 90-day paper in spot market
Get 98, 125 Pay 100,000
Sell 90-day paper forward
Get 98,125 Pay 100,000
96,250 = 100,000 x [1 – (180 x .0750)/360]
98,125 = 100,000 x [1 – (90 x .0750)/360]
On a net basis, for each set of transactions having a face value of £100,000, the arbitrageur receives £1,875 on date 0 and pays £1,875 on date 90. The profit, therefore, is the interest that can be earned on £1,875 for 90 days. For example, 90-day paper having a face value of £1,910.83 perhaps can be purchased for £1,875 (a discount yield of 7.50). The net profit of £35.83 would have to be sufficient to cover transaction costs.
9(a). Recalling the convention that the interest expense on floating-rate debt is determined in advance and paid in arrears, the relevant quarterly LIBOR expenses (rounded to the basis point) are:
1st Quarter Expense Rate:
4.60% (i.e., current 90-day LIBOR)
2nd Quarter Expense Rate:
(180 x .0475) (90 x .0460) 1
IFR90,180 = X = 4.84%
180 – 90 1 – [(90 x .0460)/360]
3rd Quarter Expense Rate:
(270 x .0500) (180 x .0475) 1
IFR180,270 = X = 5.37%
270 – 180 1 – [(180 x .0460)/360]
4th Quarter Expense Rate:
(360 x .0530) (270 x .0500) 1
IFR270,360 = X = 5.98%
360 – 270 1 – [(270 x .0500)/360]
Based on a non-amortizing loan balance of $1,000,000 and 90-day quarters, these percentages imply the following sequence of quarterly cash payments:
$11,500 = $1,000,000 x (.046) x (90/360); $ 12,100; $ 13,425; and $ 14,950.
9(b). Although there are four interest payments due, the convention of setting LIBOR at the front-end of a borrowing period means that there are only three uncertain cash flows at the time the funding is originated. This means that the customer will have to “lock in” a 90-day LIBOR on settlement dates 90, 180, and 270 days from now.
This can be done by shorting the following strip of Eurodollar futures contracts: short one 90-day contract, short one 180-day contract, and short one 270-day contract. Notice that this problem would require a short hedge because the floating rate borrower would need a hedge to compensate him/her when LIBOR rose (i.e., Eurodollar futures prices fall), thereby raising their underlying funding cost.
9(c). If Eurodollar prices are consistent with the series of implied forward rates, we would observe the following prices: 90-day contract: 95.16 =100 – 4.84; 180-day contract: 94.63; and 270-day contract: 94.02.
The annuity that would be equivalent to locking in the preceding series of quarterly cash expenses with the futures strip is calculated as the solution to:
$11,500 $12,100 $13,425 $14,950
+ + +
1+[(.046)(90)/360] 1+[(.0475)(180)/360] 1+[(.050)(270)/360] 1+[(.053)(270)/360]
Annuity Annuity Annuity Annuity
= + + +
1+[(.046)(90)/360] 1+[(.0475)(180)/360] 1+[(.050)(270)/360] 1+[(.053)(270)/360]
or Annuity = (50,325.8353/3.87895) = $12,974.07
Expressing this dollar amount on a percentage basis on terms comparable to LIBOR leaves:
($12,974.07/$1,000,000)(360/90) = 5.19%
10. CFA Examination III (1999)
10(a). The basis point value BPV of a Eurodollar futures contract can be found by substituting the contract specifications into the following mo0ney market relationship:
BPVFUT = Change in Value = (face value) x (days to maturity/360) x (change in yield)
= ($1 million) x (90/360) x (.0001)
The number of contracts, N, can be found by:
N = (BPV spot)/(BPV futures)
N = (value of spot position)/(face value of each futures contract)
= ($100 million)/($1 million)
N = value of spot position)/ value of futures position)
where the value of the futures position = $1,000,000 x [1 – (0.073)/4)]
= 102 contracts (approximately)
Therefore on September 20, Johnson would sell 100 (or 102) December Eurodollar futures contracts at the 7.3 percent yield. The implied LIBOR rate in December is 7.3 percent as indicated by the December Eurofutures discount yield of 7.3 percent. Thus a borrowing rate of 9.3 percent (7.3 percent + 200 basis points) can be locked in if the hedge is correctly implemented.
A rise in the rate to 7.8 percent represents a 50 basis point (bp) increase over the implied LIBOR rate. For a 50 basis point increase in LIBOR, the cash flow on the short futures position is:
= ($25 per basis point per contract) x 50 bp x 100 contracts
However, the cash flow on the floating rate liability is:
= -0.098 x ($100,000,000/4)
Combining the cash flow from the hedge with the cash flow from the loan results in a net
outflow of $2,325,000, which translates into an annual rate of 9.3 percent:
= ($2,325,000 x 4)/$100,000,000 = 0.093
This is precisely the implied borrowing rate that Johnson locked in on September 20. Regardless of the LIBOR rate on December 20, the net cash outflow will be $2,325,000,which translates into an annualized rate of 9.3 percent. Consequently, the floating rate liability has been converted into a fixed rate liability in the sense that the interest rate uncertainty associated with the March 20 payment (using the December 20 contract) has been removed as of September 20.
10(b). In a strip hedge, Johnson would sell 100 December futures (for the March payment), 100 March futures (for the June payment), and 100 June futures (for the September payment). The objective is to hedge each interest rate payment separately using the appropriate number of contracts. The problem is the same as in Part A except here three cash flows are subject to rising rates and a strip of futures is used to hedge this interest rate risk. This problem is simplified somewhat because the cash flow mismatch between the futures and the loan payment is ignored. Therefore, in order to hedge each cash flow, Johnson simply sells 100 contracts for each payment. The strip hedge transforms the floating rate loan into a strip of fixed rate payments. As was done in Part A, the fixed rates are found by adding 200 basis points to the implied forward LIBOR rate indicated by the discount yield of three different Eurodollar futures contracts. The fixed payments will be equal when the LIBOR term structure is flat for the first year.
11. Since you think that interest rates are going to rise, you would want to have a net long position in T-bill or Eurobond futures. But since you also believe that the credit spread between Treassuries and LIBOR will narrow, you would want to be short this spread. Let LIBOR be expressed as the T-bill rate (R) plus a spread (S). Then create a portfolio by:
1. Shorting the T-bill future (payoff = N*R*T)
2. Offsetting long position in the Eurodollar future (payoff = N*(R+S)*T).
Combining these two payoffs gives
Net payoff = N*R*T -N*(R+S)* T = -N*S*T. So the portfolio value is not related to R. Hence, if R increases and S decreases (S<0) then the payoff on the portfolio will be positive. If you strongly believed that interest rates were going to increase you could also profit from an additional short position in T-bill futures.
12. We can calculate the theoretical spot price for the index as S = F*exp[-(r-d)t] = 614.75*exp[-(.08-.03)*.25] = 607.11. Since this is larger than the actual spot price, there is a theoretical arbitrage opportunity, so program trading might take place. This would involve borrowing the money to buy 1 “index share,” taking a short position in the futures, and “delivering” the share at the future date. The cash flows would be as follows:
T= now T=90 days
1. borrow 602.25 +602.25 -614.42
2. buy 1 index -602.25 S
3. short future (K=614.75) 0 614.75 – S
4. receive dividends on index 0 4.53
Total 0 $4.86
So the arbitrage nets $4.86 in 90 days. This may or may not cover transaction costs or overcome the fact that most investors cannot borrow at the risk-free rate.
13. CFA Examination III (1999)
13(a). The number of futures contracts required is:
N = (value of the portfolio/value of the index futures) x beta of the portfolio
= [$15,000,000/(1,000 x 250)] x 0.88
= [$15,000,000/250,000] x 0.88
= 60 x 0.88
= 52.8 contracts
Selling (going short) 52 or 53 contract will hedge $15,000,000 of equity exposure.
13(b). Alternative methods that replicate the futures strategy in part A include:
1. Shorting SPDRs. SPDRs would be more expensive than futures to trade in terms of liquidity and transactions costs. Tracking error, in theory, would be higher for futures than for SPDRs, because S&P 500 futures can close under and over fair value. SPDRs do not incur the cost of rolling over, which a position in futures would incur if held longer than one expiration date.
2. Creating a synthetic short futures position using a combination of calls and puts. Either options on the underlying index or options on the futures could be used. Selling an index call option and purchasing an index put option with the same contract specifications would create a synthetic short futures position. This strategy would likely be more costly than futures because two transactions are required. Longer-term options tend to be less liquid than futures and SPDRs. Unlike futures, the option combination can be traded either as a spread or separately. If the call and put are traded separately, bid/ask spreads and market impact may increase the cost of the strategy. Options on some index futures are American style, which may result in the call option being exercised at an inopportune time.
3. Creating a fixed equity swap in which Andrew pays the appreciation and dividends on the portfolio and receives a fixed rate. The price of this transaction is negotiated between the two parties, but in general, the swap would be more costly than the futures hedge. Also, equity swaps are not liquid and may prove difficult to reverse once entered. Unlike futures, which are standardized contracts with no customization possible, this alternative has the advantage of customization; negotiable terms include the length of contract, margin requirements, cost of closing position early, and timing of payments.
4. Shorting a forward contract on the S&P 500 index. The price of this transaction is negotiated between the two parties. Forwards are not liquid, may prove difficult to reverse once entered, and may involve counterparty risk. Unlike futures, which are standardized contracts with no customization possible, this alternative has the advantage of customization; negotiable terms include length of contract, margin requirements, cost of closing the position early, and timing of payments.
14(a). To get a return of 4.25% by converting to CHF it must be the case that a dollar converted today, invested at rate R, and converted back at the end of a year is then worth $1.0425. So,
($1/.6651) x (1 + R) x .6586 = 1.0425
.990 + .990R = 1.0425
R = 5.28%
14(b). If the actual rate is 5.5% for a one-year Swiss government bond then the return on investing in CHF would be greater. This can be seen by plugging in R= .055 into the above left-hand side to get
($1/.6651) x (1+.055) x .6586 = 1.0447
14(c). An arbitrageur could borrow $250,000 domestically at 4.25%, convert it into 375,883.33 CHF, buy Swiss government bonds, and enter into a forward contract to reconvert the proceeds after a year. After a year invested at 5.5% the arbitrageur would have $396,556.91 which could be converted back into dollars at the forward rate of 0.6586 $/CHF. This would result in $261,172.38 of which $260,625 would be needed to repay the loan plus interest. So the arbitrageur would be left with a $547.38 profit, before commissions.
15. CFA Examination II (2001)
15(a). According to the cost-of-carry rule, the futures price must equal the spot price plus the cost of carrying the spot commodity forward to the delivery date of the futures contract.
Value of December contract:
F0,t = S0 x (1 + C)
S0 = spot price at t = 0
C = Risk-free rate
F0,t = 185.00 x (1 + .06/2)
15(b). Assuming the only carrying charge is the financing cost at an interest rate of 6.00 percent, the lower bound imposed by the reverse cash-and-carry strategy including transactions costs is:
Cash Flows today:
• Buy one contract of TOBEC stock index futures (December contract)
• Sell the index spot at 185 x $100 = $18,500
• Invest the proceeds at the risk-free rate for six months (until the expiration of the six month contract)
$18,500 x (1 + 0.06/2) = $19,055
Six months from now:
At expiration the futures price is assumed to converge to the spot price, and
• Sell one contract of TOBEC stock index futures (December contract)
• But the index spot
• Collect on investment = $19,055
• Pay transactions cost = $15.00
Total = ($19,055 – $15.00) = $19,040
Lower Bound = $19,040/$100 = 190.40