Chapter 26: Derivatives and Hedging Risk
26.11. Futures contracts are standardized and traded on exchanges, while forward contracts are tailor–made to suit the specific needs of two counterparties. The standardization of contracts increases the liquidity of futures markets in comparison to forward markets and also allows traders to enter into their positions with a certain degree of anonymity.
2. The holder of a futures contract is insulated from default risk due to clearing corporations and margin requirements. The owner of a forward contract has no guarantee that his counterparty will not default, and therefore forward holders must carefully evaluate each others’ credit risk before entering into a contract.
3. Since futures positions are marked–to–market at the close of trading, gains and losses on futures positions are realized daily, while gains or losses on a forward contract are not realized until the delivery of the asset.
When you need tailor-made contracts, you may prefer forwards over futures.
If the firm is selling futures contracts on lumber to hedge, it must have lumber to sell in the future and worry about future price drop.
26.2a.1. Since the futures price of wheat is $3.09 per bushel at the end of trading on March 18, the delivery price on that date is $3.09 per bushel.
2. On the delivery date, the long and short positions in a futures contract transact with the clearing corporation at the current futures price. Therefore, you will pay the current futures price of $3.09 per barrel in order to receive the wheat. The difference between the price that you pay at delivery and the price at which you entered into the contract is reconciled by daily marked–to–market gains and losses.
3. On March 15, you entered into a long futures position in wheat at a price of $3.00 per bushel. Since the closing futures price is $3.06 per bushel, your account receives a cash inflow of $0.06 at the end of the day. Your position in wheat futures increases to $3.06 per bushel (= $3.00 + $0.06).
On March 16, your opening long position in wheat futures is $3.06 per bushel. Since the closing futures price is $3.11 per bushel, your account receives a cash inflow of $0.05 at the end of the day. Your position in wheat futures increases to $3.11 per bushel (= $3.00 + $0.06+ $0.05).
On March 17, your opening long position in wheat futures is $3.11 per bushel. Since the closing futures price is $3.16 per bushel, your account receives a cash inflow of $0.05 at the end of the day. Your position in wheat futures increases to $3.16 per bushel (= $3.00 + $0.06 + $0.05 + $0.05).
On March 18, your opening long position in wheat futures is $3.16 per bushel. Since the closing futures price is $3.09 per bushel, your account experiences a cash outflow of $0.07 at the end of the day. Your position in wheat futures decreases to $3.09 per bushel (=$3.00 + $0.06 + $0.05 + $0.05– $0.07). Since you receive a notice of delivery on this date, you will pay the $3.09 futures price and receive 1 bushel of wheat.
4.The following is a summary of your futures position:
Therefore, the net amount that you pay for one bushel of wheat is $3.00 per bushel.
b.1.Since the futures price wheat is $3.07 per bushel at the end of trading on March 19, the delivery price on that date is $3.07 per bushel.
2.On the delivery date, the long and short positions in a futures contract transact with the clearing corporation at the current futures price. Therefore, you will pay the current futures price of $3.07 per barrel in order to receive the wheat. The difference between the price that you pay at delivery and the price at which you entered into the contract is reconciled by daily marked–to–market gains and losses.
3.On March 15, you entered into a long futures position in wheat at a price of $3.00 per bushel. Since the closing futures price is $3.06 per bushel, your account receives a cash inflow of $0.06 at the end of the day. Your position in wheat futures increases to $3.06 per bushel (= $3.00 + $0.06).
On March 16, your opening long position in wheat futures is $3.06 per bushel. Since the closing futures price is $3.11 per bushel, your account receives a cash inflow of $0.05 at the end of the day. Your position in wheat futures increases to $3.11 per bushel (= $3.00 + $0.06+ $0.05).
On March 17, your opening long position in wheat futures is $3.11 per bushel. Since the closing futures price is $3.16 per bushel, your account receives a cash inflow of $0.05 at the end of the day. Your position in wheat futures increases to $3.16 per bushel (= $3.00 + $0.06 + $0.05 + $0.05).
On March 18, your opening long position in wheat futures is $3.16 per bushel. Since the closing futures price is $3.09 per bushel, your account experiences a cash outflow of $0.07 at the end of the day. Your position in wheat futures decreases to $3.09 per bushel (=$3.00 + $0.06 + $0.05 + $0.05– $0.07).
On March 19, your opening long position in wheat futures was $3.09 per bushel. Since the closing futures price is $3.07 per bushel, you will experience a cash outflow of $0.02 at the end of the day. Your position in wheat futures decreases to $3.07 per bushel (= $3.00 + $0.06 + $0.05 + $0.05– $0.07- $0.02). Since you will receive a notice of delivery on this date, you will pay the $3.07 futures price and receive 1 bushel of wheat. Notice that even though you only paid $3.07 for the delivery of wheat, the net amount that you paid for it out of your pocket is $3.00 per bushel, the futures price at which you originally entered into the position.
4.The following is a summary of your futures position:
Therefore, the net amount that you pay for one bushel of wheat is $3.00 per bushel.
26.3 a.
Since you will receive the bond’s face value of $1,000 in 16 years and the 16 year spot interest rate is currently 8.5 percent, the current price of the bond is:
Current bond price = $1,000 / (1.085)16
Current bond price = $271.10
Since the forward contract defers delivery of the bond for one year, the appropriate interest rate to use in the forward pricing equation is the one–year spot interest rate of 4.2 percent:
Forward price = $271.10 (1.042)
Forward price = $282.48
b. If both the 1–year and 16–year spot interest rates unexpectedly shift downward by 2 percent, the appropriate interest rates to use when pricing the bond is 6.5 percent, and the appropriate interest rate to use in the forward pricing equation is 2.2 percent. Given these changes, the new price of the bond will be:
New bond price = $1,000 / (1.065)16
New bond price = $365.09
And the new forward price of the contract is:
Forward price = 365.09 (1.022)
Forward price = $373.13
26.4The coupon payment is not specified in the question. Hence, the most reasonable assumption is to use a zero–coupon bond.
- You’ll receive $1,000 in 18 months. Use 18-month spot rate to discount the cash flow back to time 0 and then arrive at the future value of it at the end of month 6. The forward price is therefore:
Forward Price = [$1,000 /1.0623] (1.055) = $880.80
b.It is important to remember that 100 basis points equals 1% and one basis point equals 0.01%. Therefore, if all rates increase by 15 basis points, each rate increases by 0.0015. The new 18–month spot rate is 0.0605 (= 0.0620 – 0.0015), and the new 6–month spot rate is 0.0535 (= 0.055 – 0.0015).
Bond price = [$1,000 /1.06053](1.0535) = $883.28
26.5If Jonathon Simpleton believes that the futures price of silver will fall over the next month, he should take on a short position in silver futures contracts with approximately one month until expiration. By selling futures contracts now, he will be locking in a sales price that is higher than what he believes he will be able to purchase silver futures for in one month’s time.
26.8When you purchase the contracts, the initial value is:
Initial value = 10(100)($680)
Initial value = $680,000
At the end of the first day, the value of you account is:
Day 1 account value = 10(100)($673)
Day 1 account value = $673,000
So, your cash flow is:
Day 1 cash flow = $673,000 – $680,000
Day 1 cash flow = –$7,000
The day 2 account value is:
Day 2 account value = 10(100)($679)
Day 2 account value = $679,000
So, your cash flow is:
Day 2 cash flow = $679,000 – $673,000
Day 2 cash flow = $6,000
The day 3 account value is:
Day 3 account value = 10(100)($682)
Day 3 account value = $682,000
So, your cash flow is:
Day 3 cash flow = $682,000 – $679,000
Day 3 cash flow = $3,000
The day 4 account value is:
Day 4 account value = 10(100)($686)
Day 4 account value = $686,000
So, your cash flow is:
Day 4 cash flow = $686,000 – $682,000
Day 4 cash flow = $4,000
You total profit for the transaction is:
Profit = $686,000 – 680,000
Profit = $6,000
Note that: (1) $6,000 is the last contract price – initial contract price
(2) $6,000 is also the sum of all of the above daily gains/losses.
26.9When you purchase the contracts, your cash outflow is:
Cash outflow = 25(42,000)($1.52)
Cash outflow = $1,596,000
At the end of the first day, the value of you account is:
Day 1 account value = 25(42,000)($1.46)
Day 1 account value = $1,533,000
Remember, on a short position you gain when the price declines, and lose when the price increase.
So, your cash flow is:
Day 1 cash flow = $1,596,000 – $1,533,000
Day 1 cash flow = $63,000
The day 2 account value is:
Day 2 account value = 25(42,000)($1.55)
Day 2 account value = $1,627,500
So, your cash flow is:
Day 2 cash flow = $1,533,000 – $1,627,500
Day 2 cash flow = –$94,500
The day 3 account value is:
Day 3 account value = 25(42,000)($1.59)
Day 3 account value = $1,669,500
So, your cash flow is:
Day 3 cash flow = $1,627,500 – $1,669,500
Day 3 cash flow = –$42,000
The day 4 account value is:
Day 4 account value = 25(42,000)($1.62)
Day 4 account value = $1,701,000
So, your cash flow is:
Day 4 cash flow = $1,669,500 – $1,701,000
Day 4 cash flow = –$31,500
You total profit for the transaction is:
Profit = $1,596,000 – $1,701,000
Profit = –$105,000
26.12The duration of a bond is the average time to payment of the bond’s cash flows, weighted by the
ratio of the present value of each payment to the price of the bond. Since the bond is selling at par,
the market interest rate must equal 10 percent, the annual coupon rate on the bond. The price of a
bond selling at par is equal to its face value. Therefore, the price of this bond is $1,000. The relative
value of each payment is the present value of the payment divided by the price of the bond. The
contribution of each payment to the duration of the bond is the relative value of the payment
multiplied by the amount of time (in years) until the payment occurs. So, the duration of the bond is:
Payment / PV of Payment / Relative Value / Time to Payment (in years) / Duration$100 / $90.91 / 0.0909 / 1 / 0.0909
$100 / $82.64 / 0.0826 / 2 / 0.1653
$1,100 / $826.45 / 0.8264 / 3 / 2.4793
Duration = / 2.7355 years
26.13The duration of a portfolio of assets or liabilities is the weighted average of the duration of the
portfolio’s individual items, weighted by their relative market values. Fed. fund deposits, receivables, loans (i.e., loans lent out) and mortgages (i.e., mortgage held) are assets, and deposits (deposits from outside) and long-term financing are liabilities.
a. The total market value of assets in millions is:
Market value of assets = $28 + $580 + $390 + $84 + $315
Market value of assets = $1,397
So, the market value weight of each asset is:
Federal funds deposits = $28 / $1,397 = 0.020
Accounts receivable = $580 / $1,397 = 0.415
Short–term loans = $390 / $1,397 = 0.279
Long–term loans = $84 / $1,397 = 0.060
Mortgages = $315 / $1,397 = 0.225
Since the duration of a group of assets is the weighted average of the durations of each
individual asset in the group, the duration of assets is:
Duration of assets = 0.020(0) + 0.415(0.20) + 0.279(0.65) + 0.060(5.25) + 0.225(14.25)
Duration of assets = 3.79 years
b. The total market value of liabilities in millions is:
Market value of liabilities = $520 + $340 + $260
Market value of liabilities = $1,120
Note that equity is not included in this calculation since it is not a liability. So, the market value weight of each asset is:
Checking and savings deposits = $520 / $1,120 = 0.464
Certificates of deposit = $340 / $1,120 = 0.304
Long–term financing = $260 / $1,120 = 0.232
Since the duration of a group of liabilities is the weighted average of the durations of each
individual asset in the group, the duration of liabilities is:
Duration of liabilities = 0.464(0) + 0.304(1.60) + 0.232(9.80)
Duration of liabilities = 2.76 years
c. Since the duration of assets does not equal the duration of its liabilities, the bank is not immunefrom interest rate risk.
26.14Since the cost is $30,000 at the beginning of
each year for four years, we can find the present value of each payment using the PV equation (you can, of course, use the annuity formula. Note that if you use it, make sure you understand the formula is for the regular annuity only):
PV = FV / (1 + r)t
So, the PV each year of college is:
Year 1 PV = $30,000 / (1.10)3 = $22,539.44
Year 2 PV = $30,000 / (1.10)4 = $20,490.40
Year 3 PV = $30,000 / (1.10)5 = $18,627.64
Year 4 PV = $30,000 / (1.10)6 = $16,934.22
So, the total PV of the college cost is:
PV of college = $22,539.44+ $20,490.40+ $18,627.64 + $16,934.22
PV of college = $78,591.71
Now, we can set up the following table to calculate the liability’s duration. The relative value of each payment is the present value of the payment divided by the present value of the entire liability. The contribution of each payment to the duration of the entire liability is the relative value of the
payment multiplied by the amount of time (in years) until the payment occurs.
Year PV of payment Relative value Payment weight
3$22,539.440.28679 0.86037
4$20,490.400.26072 1.04288
5$18,627.640.23702 1.18509
6$16,934.220.21547 1.29282
PV of college $78,591.71 Duration = 4.38117 years
26.17a.The price of a bond equals the present value of its cash flows.
Since Bond One pays an annual coupon of 5%, the bond’s owner will receive $50 (= 0.05 * $1,000) at the end of each year in addition to the bond’s $1,000 face value when the bond matures at the end of year 7..
Price of Bond One = $50+1000/(1+0.0456)7
= $1025.87
The price of Bond One is $1025.87.
Since Bond Two pays an annual coupon of 3.5%, the bond’s owner will receive $35 (= 0.035 * $1,000) twice a year in addition to the bond’s $1,000 face value when the bond matures at the end of year 6. The effective discount rate for the 6-month period is:
Price of Bond Two = $35 +1000/(1+0.0225)12
= $1,130.19
The price of Bond Two is $1,130.19.
Since Bond Three pays an annual coupon of 10%, the bond’s owner will receive $100 (= 0.1 * $1,000) at the end of each year in addition to the bond’s $1,000 face value when the bond matures at the end of year 9.
Price of Bond Three = $100+1000/(1+0.0456)9
= $1394.35
The price of Bond Three is $1394.35.
The duration of a bond is the average time to payment of the bond’s cash flows, weighted by the ratio of the present value of each payment to the price of the bond.
The relative value of each payment is the present value of the payment divided by the price of the bond. The contribution of each payment to the duration of the bond is the relative value of the payment multiplied by the amount of time (in years) until the payment occurs.
Bond One
Payment / PV of Payment / Relative Value / Time to Payment (in years) / Duration$50 / $47.82 / 0.0466 / 1 / 0.0466
$50 / $45.73 / 0.0446 / 2 / 0.0892
$50 / $43.74 / 0.0426 / 3 / 0.1279
$50 / $41.83 / 0.0408 / 4 / 0.1631
$50 / $40.01 / 0.0390 / 5 / 0.1950
$50 / $38.26 / 0.0373 / 6 / 0.2238
$1,050 / $768.48 / 0.7491 / 7 / 5.2437
6.0892
The duration of Bond One is 6.0892 years.
Bond Two
Payment / PV of Payment / Relative Value / Time to Payment (in years) / Duration35.00 / 34.23 / 0.0303 / 1 / 0.0303
35.00 / 33.48 / 0.0296 / 2 / 0.0592
35.00 / 32.74 / 0.0290 / 3 / 0.0869
35.00 / 32.02 / 0.0283 / 4 / 0.1133
35.00 / 31.31 / 0.0277 / 5 / 0.1385
35.00 / 30.63 / 0.0271 / 6 / 0.1626
35.00 / 29.95 / 0.0265 / 7 / 0.1855
35.00 / 29.29 / 0.0259 / 8 / 0.2073
35.00 / 28.65 / 0.0253 / 9 / 0.2281
35.00 / 28.02 / 0.0248 / 10 / 0.2479
35.00 / 27.40 / 0.0242 / 11 / 0.2667
1035.00 / 792.47 / 0.7012 / 12 / 8.4142
10.14
The duration of Bond Two is 10.1406 6-month periods or 5.0703 years.
Bond Three
Payment / PV of Payment / Relative Value / Time to Payment (in years) / Duration$100 / $95.64 / 0.0686 / 1 / 0.0686
$100 / $91.47 / 0.0656 / 2 / 0.1312
$100 / $87.48 / 0.0627 / 3 / 0.1882
$100 / $83.66 / 0.0600 / 4 / 0.2400
$100 / $80.02 / 0.0574 / 5 / 0.2869
$100 / $76.53 / 0.0549 / 6 / 0.3293
$100 / $73.19 / 0.0525 / 7 / 0.3674
$100 / $70.00 / 0.0502 / 8 / 0.4016
$1,100 / $736.38 / 0.5281 / 9 / 4.7530
6.7663
The duration of Bond B is 6.7663 years.
b.If the market interest rate increases to 6% per annum:
Price of Bond One = $50+1000/(1+0.06)7
= $944.17
The price of Bond One is $944.17.
The effective discount rate for the 6-month period is:
Price of Bond Two = $35+1000/(1+0.0296)12
= $1053.88
The price of Bond Two is $1053.88.
Price of Bond Three = $100+1000/(1+0.06)9
= $1,272.06
The price of Bond Three is $1,272.06.
Bond Three should experience a greater percentage change in its price. Bond Three has a higher duration than Bond One and Two. Bonds with higher durations will experience greater percentage changes in price for a given movement in the interest rate.
d. The percentage change in the price of each bond is:
Percentage Change in Bond Price = (New Price / Old Price) – 1
Percentage Change in Bond One = ($944.17 / $1025.87) – 1
= –0.0796 or –7.96%
Percentage Change in Bond Two = ($1053.88/ $1,130.19) – 1
= –0.0675 or – 6.75%
Percentage Change in Bond Three = ($1,272.06 / $1,394.35) – 1
= –0.0877or – 8.77%
e.Hubcap should issue Bond Three since it has the biggest percentage change.
1
Answers to End–of–Chapter ProblemsB–