问题描述：
Homework Assignment 2
Instructions
This assignment is due to be turned in at the start of class on Wednesday February 11. You can work on this assignment by yourself or in a team of two. If you choose to work in a team of two, you should turn in only one assignment per team (with both member’s student IDs on the assignment).The assignment consists of two parts. Part A contains discussion questions. These are NOT to be turned in, they are intended to solidify your understanding of the theoretical concepts that we have discussed. I suggest writing down your answers and comparing them with the solutions that I will upload once the assignments have been turned in. Part B contains assignment problems that must be turned in. Remember that these assignments are graded on a √- ,√, √+ system as detailed in the syllabus. You are free to turn in printouts or a clear, legible, handwritten assignment.
Part A: Discussion questions (not to be turned in).
1. Explain carefully why liquidity preference theory is consistent with the observation that the term structure of interest rates tends to be upward sloping more often than it is downward sloping.
If long-term rates were simply a reflection of expected future short-term rates, we would expect the term structure to be downward sloping as often as it is upward sloping. (This is based on the assumption that half of the time investors expect rates to increase and half of the time investors expect rates to decrease). Liquidity preference theory argues that long term rates are high relative to expected future short-term rates. This means that the term structure should be upward sloping more often than it is downward sloping.
2. What is meant by (a) an investment asset and (b) a consumption asset? Why is the distinction between investment and consumption assets important in the determination of forward and futures prices?
An investment asset is an asset held for investment by a significant number of people or companies. A consumption asset is an asset that is nearly always held to be consumed (either directly or in some sort of manufacturing process). The forward/futures price can be determined from the spot price for an investment asset. In the case of a consumption asset all that can be determined is an upper bound for the forward/futures price.
3. What is the cost of carry for (a) a non-dividend-paying stock, (b) a stock index, (c) a commodity with storage costs, and (d) a foreign currency?
a) the risk-free rate, b) the excess of the risk-free rate over the dividend yield c) the risk-free rate plus the storage cost, d) the excess of the domestic risk-free rate over the foreign risk-freerate.
4. Do we expect the futures price of a stock index to be greater than or less than the expected future value of the index? Explain your answer.
The futures price of a stock index is always less than the expected future value of the index. This follows from Section 5.14 and the fact that the index has positive systematic risk. For an alternative argument, let μ be the expected return required by investors on the index so that () 0() q T T E S S e -=μ. Because r >μ and() 00r q T F S e -=, it follows that 0() T E S F >.
Part B: Homework problems (to be turned in)
1. The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bond that will pay coupons of $4 every six months currently sells for $94.84. A two-year bond that will pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month, one-year, 1.5-year, and two-year zero rates.
The 6-month Treasury bill provides a return of 6946383%/=. in six months. This is 2638312766%?.=. per annum with semiannual compounding or
2ln(106383) 1238%.=. per annum with continuous compounding. The 12-month rate is 118912360%/=. with annual compounding or ln(11236) 1165%.=. with continuous compounding. For the 11 year bond we must have 012380501165115441049484R e e e -.?.-.?-.++=.
where R is the 12
year zero rate. It follows that 15153763561049484
084150115
R R e e R -.-..+.+=.=.=.
or 11.5%. For the 2-year bond we must have
012380501165101151525551059712R e e e e -.?.-.?-.?.-+++=. where R is the 2-year zero rate. It follows that 207977
0113R e R -=.=.
or 11.3%.
2. Problem 4.9. What rate of interest with continuous compounding is equivalent to 15% per annum with monthly compounding?
The rate of interest is R where: 12015112R e .??=+ ???
i.e., 01512ln 112R .??=+ ??
?
01491=.
The rate of interest is therefore 14.91% per annum.
3. Problem 4.12. A three-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond’s yield?
Ans: The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond
yield is the value of y that solves
05101520253044444104104y y y y y y e e e e e e -.-.-.-.-.-.+++++= Using theSolver orGoal Seek tool in Excel 006407y =. or 6.407%.
4. Problem 4.14. Suppose that zero interest rates with continuous compounding are as follows:
Calculate forward interest rates for the second, third, fourth, and fifth years.
For year4 2,
$1*exp(2+f)=exp(2*3)
f=6-2 = 4%
Year 2: 4.0%
Similarly, for year 3, 2*3+f = 3.7*3. Therefore f=5.1%
Year 3: 5.1%
Year 4: 5.7%
Year 5: 5.7%
5. A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock.
a. What are the forward price and the initial value of the forward contract?
b. Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract?
a) The present value, I , of the income from the security is given by:
0082120085121119540I e e -.?/-.?/=?+?=. From equation (5.2) the forward price, 0F , is given by:
008050(5019540) 5001F e .?.=-.=.
or $50.01. The initial value of the forward contract is (by design) zero. The fact that the forward price is very close to the spot price should come as no surprise. When the compounding frequency is ignored the dividend yield on the stock equals the risk-free rate of interest.
b) In three months:
00821209868I e -.?/==. The delivery price, K , is 50.01. From equation (5.6) the value of the short forward contract, f , is given by
008312(48098685001) 201f e -.?/=--.-.=.
and the forward price is
008312(4809868) 4796e .?/-.=.
6. The current price of silver is $30 per ounce. The storage costs are $0.48 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in nine months.
The present value of the storage costs for nine months are
0.12 + 0.12e ?0.10×0.25 + 0.12e ?0.10×0.5 = 0.351
or $0.351. The futures price is from equation (5.11) given by0F where
F 0 = (30 + 0.351)e 0.1×0.75= 32.72
i.e., it is $32.72 per ounce.
7. In early 2012, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0404 ($ per franc). Interest rates in the U.S. and Switzerland were 0.25% and 0% per annumrespectively, with continuous compounding. The three-month forward exchange rate was1.0300 ($ per franc). What arbitrage strategy was possible? How does your
answer change if the exchange rate is 1.0500 ($ per franc). The true forward price should be: F =Se(r-rf)T = 1.0404 * exp((0.25% -0)*3/12) = 1.04105 With the three month forward at 1.03 $ per franc , rather than the 1.04105 $ per franc it should be, I want to convert $s to Francs 3 months forward. Therefore, I borrow 1 franc, convert it spot to get 1.0404 dollars and invest them at the risk free rate. I also buy a 3 month swiss francs forward. My US investment yields: 1.0404*exp(0.25%*0.25)=1.04105. I convert that using the forward contract, to yield 1.04105/1.03 = 1.01073 francs. I thus make a risk free profit of 0.01073 per franc invested. . With three month forward rate as 1.05, my strategy is: Borrow 1 dollar, convert it spot to francs, (I get 1/1.0400 francs) in Switzerland, sell francs forward. Use the forward to convert it back to dollars. At the end of three months, I need to pay out exp(0.25%*0.25) = 1.00063 from my US borrowing. I have 1.0500*1/1.0404 =1.00927. I thereby make a profit of 1.00927-1.00063 =0.00864 for every dollar invested. 8. The current spot price of oil is $67.25 per barrel, the 6 month futures price is $69.75/barrel, and the risk-free rate is 4.2% (annualized and continuously compounded). What is the implied storage cost of oil expressed as an annualized continuously compounded rate? Whose storage cost do you think this reflects (Hint: Could you store oil at this rate?)
F 0 = S0e (r+s)T
69.75=67.25e(0.042+s)0.5
s = 2* (Ln(69.75/67.25)-0.021)
= 3.1%
This is the storage cost for the firm with the lowest marginal storage cost in the market. If for example this was not true, and there exists someone with lower marginal storage costs than 3.1% per year,they would execute the strategy of buying spot, and storing and be able to make a risk free arbitrage profit. They would do so until either (1) their marginal storage cost rose, or (2) the forward price fell to reflect their cost.
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