Application chapters 10 - 15 Flashcards
solving equations
1
Q
Using the same values except that the
construction costs are fixed at $86,667 (the original expected value), find the
value of the land. Value of land if economy improves $160,000. Then with the value of land if economy falters $70,000. Then the risk-neutral probability of the economy improvement 33.33%. The riskless interest rate is zero.
A
Current Value = Expected Value = (UpValue x UpProb)
+ [DownValue x (1− UpProb)]
(value of land if economy improves - construction cost if economy improves) x risk-neutral probability of economy improvement + (value of land if economy falters - construction cost if economy falters) x 1 -risk-neutral probability of economy faltering
($160,000.00 - $86,667.00) x 2/3 + ($70,000 - $86,667) x 1 -2/3
= 24,441.89
riskless interest rate meaning that when the value goes below zero, it then is equal to zero.
(-$11,111.89)
2
Q
Using the same values except that the
construction costs are $100,000 if the economy improves, $80,000 if the economy falters, find the
value of the land. Value of land if economy improves $160,000. Then with the value of land if economy falters $70,000. Then the risk-neutral probability of the economy improvement 33.33%. The riskless interest rate is zero.
A
Current value = expected value = (UpValue x UpProb) + (DownValue x (1 - UpProb))
(160,000 - 100,000)(1/3) + (70,000 - 80,000)(1 - 1/3) = $20.000
3
Q
Using the same values except that the
construction costs are $75,000 if the economy improves, $37,500 if the economy falters, find the
value of the land. Value of land if economy improves $150,000. Then with the value of land if economy falters $75,000. Then the risk-neutral probability of the economy improvement 25.00%. The riskless interest rate is zero.
A
?
4
Q
Using the same values except that the
construction costs are $125,000 if the economy improves, $62,500 if the economy falters, find the
value of the land. Value of land if economy improves $250,000. Then with the value of land if economy falters $125,000. Then the risk-neutral probability of the economy improvement 50%. The riskless interest rate is zero.
A
?
5
Q
Using the same values except that the
construction costs are $100,000 if the economy improves, $50,000 if the economy falters, find the
value of the land. Value of land if economy improves $200,000. Then with the value of land if economy falters $100,000. Then the risk-neutral probability of the economy improvement 75%. The riskless interest rate is zero.
A
?
6
Q
Using the same values except that the
construction costs are $50,000 if the economy improves, $25,000 if the economy falters, find the
value of the land. Value of land if economy improves $100,000. Then with the value of land if economy falters $50,000. Then the risk-neutral probability of the economy improvement 10%. The riskless interest rate is zero.
A
?
7
Q
Using the same values except that the
construction costs are $47,500 if the economy improves, $23,750 if the economy falters, find the
value of the land. Value of land if economy improves $100,000. Then with the value of land if economy falters $47,500. Then the risk-neutral probability of the economy improvement 20%. The riskless interest rate is zero.
A
?
8
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $180,000 or fall to $60,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 1/3). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
Current value = expected value = (UpValue x UpProb) + (DownValue x (1 - UpProb)
The option price rises to $26,667 ($80,000 x 1/3). This value is higher
than in the original example and demonstrates that volatility favors the option
holder. Higher volatility increases the upside profit potential without increasing
the loss potential due to the limited downside risk afforded by long option
positions
9
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $200,000 or fall to $190,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 1/3). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
10
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $250,000 or fall to $100,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 25%). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
11
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $150,000 or fall to $50,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 50%). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
12
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $200,000 or fall to $100,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 75%). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
13
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $350,000 or fall to $200,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 10%). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
14
Q
Return to the original values and find the value
of the land, assuming that economic uncertainty increases such that improved
properties either rise to $100,000 or fall to $50,000, with all other values
remaining the same. Following the same math, the risk-neutral probabilities are
the same (the up probability is 20%). The value to developing is $80,000 in the up
state and $0 in the down state (the construction costs exceed the developed
value). Construction cost if the economy improves $100,000.
A
?
15
Q
Land that remains undeveloped is estimated to
generate an expected return of 5%, and land that is developed is estimated to
generate an expected single-period return of 25%. If the probability that a parcel
of land will be developed is 10% over the next period, what is its expected
return?
A
Current value = expected value = (UpValue x UpProb) + (DownValue (1 - UpProb)
(0.25% x 0.10%) + (0.05% (1 - 0.10)
= 0.07 (7%)
16
Q
Land that remains undeveloped is estimated to
generate an expected return of 5.50%, and land that is developed is estimated to
generate an expected single-period return of 20%. If the probability that a parcel
of land will be developed is 15% over the next period, what is its expected
return?
A
?
17
Q
Land that remains undeveloped is estimated to
generate an expected return of 10%, and land that is developed is estimated to
generate an expected single-period return of 30%. If the probability that a parcel
of land will be developed is 5% over the next period, what is its expected
return?
A
?
18
Q
Land that remains undeveloped is estimated to
generate an expected return of 2.50%, and land that is developed is estimated to
generate an expected single-period return of 35%. If the probability that a parcel
of land will be developed is 20% over the next period, what is its expected
return?
A
?
19
Q
Land that remains undeveloped is estimated to
generate an expected return of 4%, and land that is developed is estimated to
generate an expected single-period return of 27%. If the probability that a parcel
of land will be developed is 9% over the next period, what is its expected
return?
A
?
20
Q
Land that remains undeveloped is estimated to
generate an expected return of 6%, and land that is developed is estimated to
generate an expected single-period return of 21%. If the probability that a parcel
of land will be developed is 7.5% over the next period, what is its expected
return?
A
?
21
Q
Land that remains undeveloped is estimated to
generate an expected return of 8%, and land that is developed is estimated to
generate an expected single-period return of 33%. If the probability that a parcel
of land will be developed is 6% over the next period, what is its expected
return?
A
?
22
Q
Land that remains undeveloped is estimated to
generate an expected return of 5%, and land that is developed is estimated to
generate an expected single-period return of 25%. If 20% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
A
Current value = expected value = (UpValue x UpProb) + (DownValue (1 - UpProb)
(0.25% x 0.20%) + (0.05% (1 - 0.20%) = 0.09 (9%)
23
Q
Land that remains undeveloped is estimated to
generate an expected return of 5.5%, and land that is developed is estimated to
generate an expected single-period return of 20%. If 15% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
A
?
24
Q
Land that remains undeveloped is estimated to
generate an expected return of 6.5%, and land that is developed is estimated to
generate an expected single-period return of 30%. If 25% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
A
?
25
Land that remains undeveloped is estimated to
generate an expected return of 6%, and land that is developed is estimated to
generate an expected single-period return of 35%. If 30% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
?
26
Land that remains undeveloped is estimated to
generate an expected return of 7%, and land that is developed is estimated to
generate an expected single-period return of 40%. If 35% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
?
27
Land that remains undeveloped is estimated to
generate an expected return of 4.5%, and land that is developed is estimated to
generate an expected single-period return of 45%. If 5% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
?
28
Land that remains undeveloped is estimated to
generate an expected return of 4%, and land that is developed is estimated to
generate an expected single-period return of 27%. If 10% of land in a database is
developed in a particular year, by how much will an index based on land that
remains undeveloped understate the average return on all land?
?
29
If the annual revenue in Exhibit 10.4 is
expected to rise to $40,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($40,000 – $6,000 – $2,000)/0.08]. With a price of $360,000 and an annual
operating income of $40,000, what would the cap rate be?
```
exhibit 10.4:
Purchase price
$300,000
Financing
$150,000
Equity investment
$150,000
Annual revenues
$30,000
Less real estate taxes
$6,000
Less insurance
$2,000
Operating income
$22,000
Less interest
$12,000
Net income
$10,000
ROE = $10,000/$150,000 = 6.67%
```
Value of Real Estate = Annual Operating Income / Cap Rate
$360,000 = $40,000 / cap rate
Cap rate = 40,000/360,000 = 11.11%
30
If the annual revenue in Exhibit 10.4 is
expected to rise to $45,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($45,000 – $6,000 – $2,000)/0.08]. With a price of $365,000 and an annual
operating income of $45,000, what would the cap rate be?
value of real estate = annual operating income / cap rate
365,000 = 45,000 / cap rate
cap rate = 45,000/365,000 = 0.123287671233 (12.33%)
31
If the annual revenue in Exhibit 10.4 is
expected to rise to $55,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($55,000 – $6,000 – $2,000)/0.08]. With a price of $375,000 and an annual
operating income of $55,000, what would the cap rate be?
?
32
If the annual revenue in Exhibit 10.4 is
expected to rise to $60,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($60,000 – $6,000 – $2,000)/0.08]. With a price of $380,000 and an annual
operating income of $60,000, what would the cap rate be?
?
33
If the annual revenue in Exhibit 10.4 is
expected to rise to $65,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($65,000 – $6,000 – $2,000)/0.08]. With a price of $385,000 and an annual
operating income of $65,000, what would the cap rate be?
?
34
If the annual revenue in Exhibit 10.4 is
expected to rise to $35,000 and the market cap rate rises to 8%, then with all
other values remaining constant, the farmland's price would rise to $400,000
[($35,000 – $6,000 – $2,000)/0.08]. With a price of $400,000 and an annual
operating income of $35,000, what would the cap rate be?
?
35
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of five contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $99. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
Market price - Future market price
$100 - $99 = $1
Each contract 1,000 barrels x number of contracts
1000 x 5 = 5,000
The long position lost 5,000 multiplied by $1 = $5,000
The short position is exactly the opposite so it gained $5,000.
The futures margin is $10 per barrel. So, both long and short, posted $10 x 5000 = $50,000 margin.
Long position = $50,000 - $5,000 = $45,000
Short position = $50,000 + $5,000 = $55,000
36
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of five contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $100. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
contracts x barrel size
5 x 1000 = 5,000
x margin
5,000 x 10 = 50,000
both long and short are $50,000 as there are not gains or losses in the next day market price.
37
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of ten contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $100. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
?
38
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of five contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $99. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
?
39
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of ten contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $102. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
?
40
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of five contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $95. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
?
41
Futures contracts on crude oil are often
denominated in 1,000-barrel sizes. In other words, each contract calls for the
holder of a short position at the delivery date of the futures contract to deliver to
the long side 1,000 barrels of the specified grade of oil using stated delivery
methods. Assume that a trader establishes a long position of ten contracts in
crude oil futures at the then-current futures market price of $100 per barrel. Both
the trader on the long side of the contract and the trader on the short side of the
contract post collateral (margin) of, say, $10 per barrel. At the end of the day, the
market price of the futures contract falls to $106. How much money will each side
of the contract have (assuming that the required collateral was the only cash and
that there were no other positions)?
?
42
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in five
silver futures contracts, settling in each quarter for the next four quarters (20
contracts total). If the initial margin requirement is $11,000 per contract, what is
the firm’s total initial margin requirement?
number of contracts x the initial margin per contract = total margin (available collateral to establish position)
11000 x 20 = $220,000
43
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in 8.75
silver futures contracts, settling in each quarter for the next four quarters (35
contracts total). If the initial margin requirement is $10,000 per contract, what is
the firm’s total initial margin requirement?
?
44
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in 11.25
silver futures contracts, settling in each quarter for the next four quarters (45
contracts total). If the initial margin requirement is $25,000 per contract, what is
the firm’s total initial margin requirement?
?
45
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in twelve and a half
silver futures contracts, settling in each quarter for the next four quarters (50
contracts total). If the initial margin requirement is $13,000 per contract, what is
the firm’s total initial margin requirement?
?
46
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in 16.25
silver futures contracts, settling in each quarter for the next four quarters (65
contracts total). If the initial margin requirement is $15,000 per contract, what is
the firm’s total initial margin requirement?
?
47
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in twenty five
silver futures contracts, settling in each quarter for the next four quarters (100
contracts total). If the initial margin requirement is $900 per contract, what is
the firm’s total initial margin requirement?
?
48
To lock in sales prices for its anticipated
production, HiHo Silver Mining Company wishes to take short positions in 26.25
silver futures contracts, settling in each quarter for the next four quarters (105
contracts total). If the initial margin requirement is $20,000 per contract, what is
the firm’s total initial margin requirement?
?
49
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$100 per barrel, the five contracts call for delivery of 5,000 barrels (five contracts
x 1,000 barrels). The trader posts exactly the required initial margin of $50,000
($10,000 per contract). Suppose that the maintenance margin requirement is
$25,000 ($5,000 per contract) and that the price of oil drops $6 per barrel. What
is the trader’s margin balance after the price decline?
Required initial margin:
$50,000
Maintenance margin requirement is $25,000 (5000 per contract)
oil drop $6 per barrel (5000 barrels)
6 x 5000 = $30,000
Required initial margin - Oil drop = $20,000
As the margin balance is below the maintenance margin requirement of $25,000, the trader would receive a MARGIN CALL, which would require an additional $30,000 to bring the margin back the initial margin requirement.
50
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$95 per barrel, the ten contracts call for delivery of 10,000 barrels (ten contracts
x 1,000 barrels). The trader posts exactly the required initial margin of? Suppose that the maintenance margin requirement is
$50,000 ($5,000 per contract) and that the price of oil drops $5 per barrel. What
is the trader’s margin balance after the price decline?
?
51
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$99 per barrel, the five contracts call for delivery of 15,000 barrels (fifteen contracts
x 1,000 barrels). The trader posts exactly the required initial margin?. Suppose that the maintenance margin requirement is
$75,000 ($5,000 per contract) and that the price of oil drops $14 per barrel. What
is the trader’s margin balance after the price decline?
?
52
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$85 per barrel, the twenty contracts call for delivery of 20,000 barrels (twenty contracts
x 1,000 barrels). The trader posts exactly the required initial margin of $50,000
($10,000 per contract). Suppose that the maintenance margin requirement is
$100,000 ($5,000 per contract) and that the price of oil drops $5 per barrel. What
is the trader’s margin balance after the price decline?
?
53
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$64 per barrel, the five contracts call for delivery of 5,000 barrels (five contracts
x 1,000 barrels). The trader posts exactly the required initial margin of $50,000
($10,000 per contract). Suppose that the maintenance margin requirement is
$25,000 ($5,000 per contract) and that the price of oil rises $4 per barrel. What
is the trader’s margin balance after the price decline?
?
54
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$45 per barrel, the ten contracts call for delivery of 5,000 barrels (ten contracts
x 1,000 barrels). The trader posts exactly the required initial margin of $50,000
($10,000 per contract). Suppose that the maintenance margin requirement is
$50,000 ($5,000 per contract) and that the price of oil drops $35 per barrel. What
is the trader’s margin balance after the price decline?
?
55
Returning to the previous example of an oil
trader with a long position of five contracts established at an initial futures price of
$40 per barrel, the fifteen contracts call for delivery of 5,000 barrels (fifteen contracts
x 1,000 barrels). The trader posts exactly the required initial margin of $50,000
($10,000 per contract). Suppose that the maintenance margin requirement is
$75,000 ($5,000 per contract) and that the price of oil drops $1 per barrel. What
is the trader’s margin balance after the price decline?
?
56
A six-month forward contract on a commodity
trades at a spot price of $50. The commodity has marketwide convenience yields
of 3%, storage costs of 2%, and financing costs (interest rates) of 7%. What is the
price of the six-month forward contract on the commodity?
F(T ) = e(r+c− y)T S
F(T) = e^(x) (Interest Rate + Costs - Yield) (Time) (Spot price)
e^(x)(0.07% + 0.02% - 0.03%)(0.5 six months)(50)
calculator:
0. 07 + 0.02 - 0.03
x 0.5
2nd e^(x)
x 50
= $51.52
57
A 1.10 year forward contract on a commodity
trades at a spot price of $38. The commodity has marketwide convenience yields
of 2%, storage costs of 5%, and financing costs (interest rates) of 3%. What is the
price of the 1.10 year forward contract on the commodity?
F(T) = e^(x) (r + c - y) (T) (S)
0. 03 + 0.05 - 0.02
0. 06 x 1.1 = 0.066
0. 066 2nd e^(x) = 1.068227
1. 068227 x 38 = $40.59
58
A 1.30 year forward contract on a commodity
trades at a spot price of $26. The commodity has marketwide convenience yields
of 1%, storage costs of 2%, and financing costs (interest rates) of 2%. What is the
price of the 1.30 year forward contract on the commodity?
?
59
A 4.8 years forward contract on a commodity
trades at a spot price of $14. The commodity has marketwide convenience yields
of 4%, storage costs of 2%, and financing costs (interest rates) of 1%. What is the
price of the 4.8 year forward contract on the commodity?
?
60
A 0.37 year forward contract on a commodity
trades at a spot price of $2. The commodity has marketwide convenience yields
of 3%, storage costs of 5%, and financing costs (interest rates) of 3%. What is the
price of the 0.37 year forward contract on the commodity?
?
61
A seven year forward contract on a commodity
trades at a spot price of $10. The commodity has marketwide convenience yields
of 4%, storage costs of 2%, and financing costs (interest rates) of 2%. What is the
price of the seven year forward contract on the commodity?
?
62
A 0.15 year forward contract on a commodity
trades at a spot price of $22. The commodity has marketwide convenience yields
of 5%, storage costs of 2%, and financing costs (interest rates) of 4%. What is the
price of the 0.15 year forward contract on the commodity?
?
63
Suppose that an important grain, such as corn,
is trading in the spot or cash market at $8 per bushel because bad weather
caused a decrease in supply during the previous harvest. Market participants
expect a bountiful harvest in about six months, which is expected to drive market
prices down to $5 per bushel. Forward prices with delivery dates after the next
harvest are trading in the range of $5 per bushel. How could arbitrageurs attempt
to profit from these prices?
Arbitrageurs might theorize that they can (1) borrow corn, (2) sell the corn for $8
per bushel in the cash market, (3) take a long position in a forward contract with
a delivery in six months at a forward price of roughly $5, and (4) take delivery of
the corn in six months at a price of $5. They would use the delivered corn from
the forward contract to return the corn previously borrowed and pocket a riskless
$3 profit (ignoring financing costs). But they would find that nobody with an
available corn supply would be willing to lend the corn at little or no cost. The
reason is that entities holding inventories of corn might need the corn in the next
six months. If these entities had a surplus of corn, they could sell it for $8 in the
spot market and use a long position in a forward contract to lock in a $5 cost to
replenish their inventory when needed (in six months).
Essentially, the arbitrageur is selling the more expensive corn right now, in order
to buy it back for $3 cheaper in the future. The idea does not take into
consideration convenience yield. If the entities holding the corn did not have an
immediate use for the corn, it would be likely that the suppliers themselves would
sell the corn immediately in order to eliminate the storage costs, but also to profit
from the spread between the spot and 6 month futures price. But because the
entities holding the corn have tremendously high convenience yields (e.g.,
keeping their livestock alive), they are willing to hold corn inventories at market
prices that greatly exceed the forward price.
Therefore equation 11.3 indicates that the forward price must not exceed the
value e(r+c-y)T (because arbitrageurs could buy the spot and sell the forward at a
profit) but the forward price can be less than e(r+c-y)T because arbitrageur will likely
be unable to short the spot price (without paying the lender a fee equal to the
lender’s convenience yield).
64
Consider a calendar spread that is long the
two-year forward contract and short the one-year forward contract on a physical
commodity with a spot price of $100. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs rose one percentage point (from 3% to 4%)?
Changes in the spot price will not affect calendar spreads as long as none of the
carrying costs change from r + c – y = 0. All forward prices will continue to match
the spot price, and the basis of all contracts will remain zero. The trader is hedged
against changes in the spot price by holding an equal number of long and
short contracts. In the second scenario, when storage costs rise from 3% to
4%, r + c – y will no longer equal 0, and forward prices will rise relative to spot
prices. In this example, the longer delivery date of the long position (two years)
will cause the forward price of the two-year forward to rise in price by more than
the one-year forward, netting the trader a profit from correctly speculating that the
storage costs would rise. Specifically, the two-year forward rises from $100 to
$102.020, and the one-year forward rises from $100 to $101.005, netting the
trader a profit of $1.015 from being long the two-year forward and short the oneyear
forward. Note that the values are based on continuous compounding.
F(T) = e^(x) (r + c - y)(T)(S)
Two year long position:
0. 02 + 0.04 - 0.05
0. 01 x 2 = 0.02
0. 02 2nd e^(x) = 1.020201
1. 020201(100) = 102.020134 ($102.02)
one year short position:
0.02 + 0.04 - 0.05
0. 01 x 1 = 0.01
0. 02 2nd e^(x) = 1.010050
1. 010050 x 100 = $101.01
65
Consider a calendar spread that is long the
one-year forward contract and short the two-year forward contract on a physical
commodity with a spot price of $50. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs rose two percentage point (from 3% to 5%)?
?
66
Consider a calendar spread that is long the
two-year forward contract and short the one-year forward contract on a physical
commodity with a spot price of $50. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs rose two percentage point (from 3% to 5%)?
?
67
Consider a calendar spread that is long the
one-year forward contract and short the three-year forward contract on a physical
commodity with a spot price of $50. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs fell one percentage point (from 3% to 2%)?
?
68
Consider a calendar spread that is long the
four-year forward contract and short the two-year forward contract on a physical
commodity with a spot price of $40. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs fell three percentage point (from 3% to 0%)?
?
69
Consider a calendar spread that is long the
five-year forward contract and short the three-year forward contract on a physical
commodity with a spot price of $40. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs rose three percentage point (from 3% to 6%)?
?
70
Consider a calendar spread that is long the
three-year forward contract and short the five-year forward contract on a physical
commodity with a spot price of $40. Assume that the number of contracts in the
long position equals the number of contracts in the short position. The trader put
the spread on in anticipation that storage costs, c, will rise. Assume that the
forward prices adhere to Equation 11.2 and that r = 2%, c = 3%, and y = 5%.
Note that these values were chosen for the simplicity that r + c – y = 0% so that
the forward prices equal the spot prices. What would the profit or loss be to the
trader if spot prices rose $1? What would the profit or loss be to the trader if the
storage costs did not change?
?
71
Consider a calendar spread that is long the
two-year forward contract and short the one-year forward contract on a physical
commodity with a spot price of $100. Each contract calls for delivery of one unit
of the spot asset (currently trading at $100). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the oneyear
forward contract would hedge the $1,000,000 notional value position in the
two-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs fell one percentage point (from 3% to 2%)?
F(T) = e^(x) (r + c - y) (T)(S)
Short hedging position:
0.10% x 1 = 0.10
0.1 2ns e^(x) = 1.105171
x 100 = $110.52
1,000,000/110.52 = 9,048.37
long hedging position:
0.10% x 2 = 0.20
0.20 2nd e^(x) = 1.221403
x 100 = $122.14
1,000,000/122.14 = 8,187.31
one year short position with 2% storage cost:
0.09 x 1 = 0.09
0.09 2nd e^(x) = 1.094174
x 100 = $109.42
two year long position with 2% storage cost:
0.09 x 2 = 0.18
0.18 2nd e^(x) = 1.197217
x 100 = $119.72
Difference in short position:
110.52-109.42 = 1.1
x 9,048.37 = 9,953.21
Difference in long position:
122.14-119.72 = 2.42
x 8,187.31 = -19,813.29 (minus because it is long)
short - long = 9,953.21 - 19,813.29 = -9,860.08
72
Consider a calendar spread that is long the
two-year forward contract and short the one-year forward contract on a physical
commodity with a spot price of $100. Each contract calls for delivery of one unit
of the spot asset (currently trading at $100). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the oneyear
forward contract would hedge the $1,000,000 notional value position in the
two-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs rose two percentage point (from 3% to 5%)?
?
73
Consider a calendar spread that is long the
one-year forward contract and short the three-year forward contract on a physical
commodity with a spot price of $50. Each contract calls for delivery of one unit
of the spot asset (currently trading at $50). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the three year
forward contract would hedge the $1,000,000 notional value position in the
one-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs fell one percentage point (from 3% to 2%)?
?
74
Consider a calendar spread that is long the
four-year forward contract and short the two-year forward contract on a physical
commodity with a spot price of $50 Each contract calls for delivery of one unit
of the spot asset (currently trading at $50). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the two year
forward contract would hedge the $1,000,000 notional value position in the
four-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs fell one percentage point (from 3% to 0%)?
?
75
Consider a calendar spread that is long the
five-year forward contract and short the three-year forward contract on a physical
commodity with a spot price of $40. Each contract calls for delivery of one unit
of the spot asset (currently trading at $40). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the three year
forward contract would hedge the $1,000,000 notional value position in the
five-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs fell one percentage point (from 3% to 6%)?
?
76
Consider a calendar spread that is long the
three-year forward contract and short the five-year forward contract on a physical
commodity with a spot price of $30. Each contract calls for delivery of one unit
of the spot asset (currently trading at $30). The trader put the spread on in
anticipation that storage costs, c, will rise. Assume that the forward prices adhere
to Equation 11.2 and that r = 7%, c = 3%, and y = 0%. Note that r + c – y = 10%.
Assume that the trader hedges $1,000,000 notional value in the long position
with the same notional value in the short position. What short position in the five year
forward contract would hedge the $1,000,000 notional value position in the
three-year contract? What would the profit or loss be to the trader if the spot price
changed or if the storage costs fell one percentage point (from 3% to 3%)?
?
77
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 6%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 5%, and the required rate of return (r) is
12%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
V(IP, 0) = p x CF(1) / (r - g)
0.06 x 1 / (12 -(- 5))
= 0.35
78
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 14%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 5%, and the required rate of return (r) is
12%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
79
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 5%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 2%, and the required rate of return (r) is
12%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
80
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 10%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 3%, and the required rate of return (r) is
9%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
81
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 8%, the rate at which expected cash flows rise
each year after their initial potential (g) is 1%, and the required rate of return (r) is
10%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
82
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 1%, the rate at which expected cash flows rise
each year after their initial potential (g) is 2%, and the required rate of return (r) is
10%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
83
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 1%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 5%, and the required rate of return (r) is
8%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
84
Loosely following some of the values indicated
earlier in this section for films, assume that the probability of substantial success
for an investment in IP (p) is 12%, the rate at which expected cash flows diminish
each year after their initial potential (g) is 1%, and the required rate of return (r) is
1%. How much would this investment in IP be worth per dollar of projected
possible first-year cash flow (CF1)? This example normalizes the analysis to a
value of $1 for CF1.
?
85
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 3.0, p is 0.06, and g is –0.05. What is the
investment’s annual rate of return?
r = p x (CF(1) / V(p,0)) + g
0.06 x 3 + (-0.05)
= 0.13
86
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 4.0, p is 0.05, and g is –0.04. What is the
investment’s annual rate of return?
?
87
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 5.0, p is 0.10, and g is –0.03. What is the
investment’s annual rate of return?
?
88
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 1.0, p is 0.08, and g is 0.01. What is the
investment’s annual rate of return?
?
89
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 2.0, p is 0.25, and g is 0.02. What is the
investment’s annual rate of return?
?
90
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 2.0, p is 0.01, and g is –0.05. What is the
investment’s annual rate of return?
?
91
Assume that Equation 13.2 is an appropriate
valuation model and that CF1/Vip,0 is 5.0, p is 0.12, and g is –0.01. What is the
investment’s annual rate of return?
?
92
Assume that a borrower takes out a $100,000,
25-year mortgage (300 months), at a 6% annual nominal interest rate (a monthly
interest rate of 6%/12, or 0.5%). What is the mortgage’s monthly payment?
2nd CLR TVM
12 x 25 = N
6 / 12 = I/Y
100,000 +/- PV
0 FV
CPT > PMT
644.30
93
Assume that a borrower takes out a $100,000,
30-year mortgage (360 months), at a 6.5% annual nominal interest rate (a monthly
interest rate of 6.5%/12). What is the mortgage’s monthly payment?
?
94
Assume that a borrower takes out a $100,000,
10-year mortgage (120 months), at a 5.5% annual nominal interest rate (a monthly
interest rate of 5.5%/12). What is the mortgage’s monthly payment?
?
95
Assume that a borrower takes out a $100,000,
7-year mortgage (84 months), at a 7% annual nominal interest rate (a monthly
interest rate of 7%/12). What is the mortgage’s monthly payment?
?
96
Assume that a borrower takes out a $100,000,
15-year mortgage (180 months), at a 4% annual nominal interest rate (a monthly
interest rate of 4%/12). What is the mortgage’s monthly payment?
?
97
Assume that a borrower takes out a $100,000,
20-year mortgage (240 months), at a 4.5% annual nominal interest rate (a monthly
interest rate of 4.5%/12). What is the mortgage’s monthly payment?
?
98
Assume that a borrower takes out a $100,000, 5-year mortgage (60 months), at a 2.5% annual nominal interest rate (a monthly
interest rate of 2.5%/12). What is the mortgage’s monthly payment?
?
99
What would be the outstanding mortgage
balance at the start of month 61 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 6% annual nominal interest rate?
Mortgage payment $644.3014.
20 x 12 = N
6 / 12 = I/Y
0 = FV
644.3014 = PMT
CPT > PV
100
What would be the outstanding mortgage
balance at the start of month 30 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 7% annual nominal interest rate?
Mortgage payment $706.78.
30/12 = 2.5
25 - 2.5 = 22.5
22.5 x 12 = N
7 / 12 = I/Y
0 = FV
706.78 = PMT
CPT > PV
101
What would be the outstanding mortgage
balance at the start of month 40 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 10% annual nominal interest rate?
Mortgage payment $908.70.
?
102
What would be the outstanding mortgage
balance at the start of month 50 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 5% annual nominal interest rate?
Mortgage payment $584.59.
?
103
What would be the outstanding mortgage
balance at the start of month 10 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 3% annual nominal interest rate?
Mortgage payment $474.21
?
104
What would be the outstanding mortgage
balance at the start of month 15 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 8.50% annual nominal interest rate?
Mortgage payment $768.21
?
105
What would be the outstanding mortgage
balance at the start of month 20 in terms of remaining principal of a $100,000,
25-year mortgage (300 months), at a 6.50% annual nominal interest rate?
Mortgage payment $675.21
?
