Homework Soursehere are some points from a yield curve yrs to maturity yiel
here are some points from a yield curve:
yrs to maturity yield rate
1 0.0500
2 0.0600
3 0.0650
4 0.0700
5 0.0725
#1 Determine the present value, to nearest penny, of:
a. $1,000 payable at time 2
b. $1,000 payable at time 4
#2 Determine the following forward rates to 6 decimal places eg .080011:
a. the forward rate that applies from year 1 to year 2
b. the forward rate that applies from year 2 to year 3
c. the forward rate that applies from year 4 to year 5
Solution
July 6
A
B
5
$294.50
July 6
C
B
10
$294.00
July 6
Settlement Price
$294.00
July 7
D
A
10
$293.50
July 7
B
D
5
$293.80
July 7 Settlement Price
$293.80
July 8
B
A
7
$293.70
July 8
Settlement Price
$299.50
When trading commences, each trader has a zero net position. The open interest, which is the sum of the trader\'s net long positions, is also zero. Looking at each trade sequentially, we can track the traders\' net positions and open interest as follows:
Trade
Trader A
Trader B
Trader C
Trader D
Open Interest
1
+ 5
05
0
0
5
2
+ 5
-15
+ 10
0
15
3
-5
-15
+ 10
+ 10
20
4
-5
-10
+ 10
+ 5
15
5
-12
-3
+ 10
+ 5
15
B. Calculate the gains and losses for Trader A. Assume that at the time of each change in position, Trader A must bring the margin back to the initial margin amount. Compute the amount in Trader A\'s margin account at the end of each trading day. Will Trader A get a margin call? If so, when and how much addi-tional margin must be posted?
Date
Buy Seil
Position
Price
(loss)
Variation
Margin
Variation
July 6
5
+ 5
$294.50
_
$0
$5,065
$5,065
July 6
settlement
+ 5
$294.00
($250)
$4,815
$0
$4,815
July 7
10
,5
$293.50
($250)
$4,565
$500
$5,065
July 7
settlement
15
$293.80
($150)
$4,915
$0
$4,915
July 8
7
$12
$293.70
$50
$4,965
$7,191
$12,156
July 8
settlement
$12
$299.50
($6,960)
$5,196
$6,960
$12,156
On July 6, Trader A goes long 5 contracts at a price of $294.50. At this time, she is required to post $1,013 margin per contract for a total of $5,065. The July 6 settlement price of $294.00 results in a $50 per contract loss (100 X —$0.50). Trader A\'s 5 contracts lose a total of $250 which is taken from the margin account. Trader A\'s margin account still has $963 per contract which is in excess of the minimum maintenance margin of $750 per contract. Therefore, no margin call occurs. On July 7, Trader A reverses her long position and goes short 5 contracts. A $250 loss occurs on the reversed long position which is assessed against the margin account, bringing it down to $4,565. Since Trader A is now short 5 contracts, she adds $500 to her margin account to bring it back to the initial margin requirement of $5,065. At settlement on July 7, Trader A\'s position lost $150 which is taken from her margin account bringing it to $4,915. No margin call occurs. On July 8, Trader A sells an additional 7 contracts, bringing her total to short 12 contracts. At the time of the transaction, her original position of 5 short contracts has gained $50 which is credited to her margin account, bringing it to $4,965. The required initial margin for 12 contracts is $12,156, so Trader A adds $7,191 to her margin account. July 8 is not a good day for Trader A. The futures price of gold increases by $5.80 per ounce. Trader A\'s 12-contract short position loses $6,960, bringing her margin account to $5,196 or $433 per contract. Since this is below the maintenance level, Trader A gets a margin call. She must add $6,960 to bring her margin back to the initial margin level. Alternatively, she can close her position.
20. Today is June 30. You have an anticipated liability of $10 million due on December 31. To fund this liability, you plan to sell part of your store of gold. Looking in The Wall Street Journal, you note the follow-ing futures prices and bond equivalent yields for the T-bills maturing at or near the expiration of the futures contracts:
Cash price 293.00
AUG 297.40 4.86%
DEC 302.00 5.35%
A. What is the basis for the December futures contract?
The basis is computed as the cash price minus the futures price.
Basis = $293.00 - $302.00 = -$9
B. Is the market normal or inverted? Explain.
In a normal market, the prices for the more distant contracts are higher than the prices for the nearby contracts. In an inverted market, futures prices decline as we go from the nearby contracts to the more distant contracts.
In this question, prices increase the more distant the delivery date, so the market is normal.
C. If you wanted to eliminate the risk of gold price variation, explain your alternatives. There are several alternatives for eliminating price risk.
Alternative 1: Sell gold via the December futures contract. This effectively locks in a December selling price of $302.Alternative 2: Sell gold today in the spot market and invest the proceeds in the 6-month T-bill. Alternative 3: Sell gold today in the spot market and invest the proceeds in the 2-month T-bill. At the same time, buy gold by using the August futures contract and sell gold using the December futures contract. The most attractive alternative is the one which allows us to meet the $10 million liability while selling the least amount of gold.
D. Given the above prices, justify which of the alternatives you would prefer. Alternative 1: Sell gold using the December futures contract.
r> + t> c \\a $10 million
Ounces to Be Sold =----- -----= 33,113 ounces
$302
Alternative 2: Sell gold today in the spot market and invest the proceeds in the 6-month T-bill.
Cash Needed Today = ,$1° million = $9.739 million
r> D c UT a $9.739million -,-,-., Ounces to Be Sold T oday =------ämi------= 33,241 ounces
Alternative 3: Sell gold today in the spot market and invest the proceeds in the 2-month T-bill. At the same time, buy gold using the August futures contract and sell gold using the December futures contract. We already know from alternative 1 that 33,113 ounces must be sold in December. Then 33,113 ounces must be bought in August.
Cash Needed in August = 33,113 x $297.40 = $9.848 million
r um a a t a $9.848million acn-n -ir
Cash Needed Today =----------------;—— = $9.769 million
1 + 0.0486
r> D c UT a $9.769million -,-,-,.,, Ounces to Be Sold T oday =------~i^>------= 33,340 ounces
$293
Conclusion: Selling gold using the December futures contract requires the fewest ounces of gold to be sold.
E. At what August and December futures prices would you be indifferent between the alternatives?
Assuming negligible costs other than the time value of money, the following relationship must hold for you to be indifferent between alternatives:
In this relationship the cost of carry, C, is the borrowing and lending rate. For the December contract, the futures price at which you are indifferent between alternatives 1 and 2 is:
F0,dec = 293 I 1 + ^y^ I = $300.84 The August futures price at which you are indifferent between alternatives 1 and 3
| July 6 | A | B | 5 | $294.50 |
| July 6 | C | B | 10 | $294.00 |
| July 6 | Settlement Price | $294.00 | ||
| July 7 | D | A | 10 | $293.50 |
| July 7 | B | D | 5 | $293.80 |
| July 7 Settlement Price | $293.80 | |||
| July 8 | B | A | 7 | $293.70 |
| July 8 | Settlement Price | $299.50 |
