Assignment_2_Solutions

201Tutorial8/Marked Assignment22010 topic:Static gameshand in date:day21Monday24Mayreading:Perloffchapters13,14Q1Chewing guma)Twofirms,A and B,compete in the chewing gum industry.Theproducts of the twofirms are differentiated.The demand functionsfor the products of thesefirms are Q A=120−2P A+P B and Q B=120−2P B+P A,where the subscript denotes thefirm.Eachfirm haszero production costs.(a)Suppose that the twofirms play a Bertrand game.Write downthe objective function forfirm A.The profit offirm A isπA=P A Q A=(120−2P A+P B)P A.(b)Assume that prices can only be set in tens,with a maximum of50.The following matrix gives the payoffs possible values of A’sprice(rows)and B’s price(columns).Firm A’s payoffs are givenfirst.Identify(i)the best responses with the circling method and(ii)the Nash equilibrium.150015002500330035003500 402400240032003200210013002700270033002500 2020002000240024001100110013002100150015002040Firm A’s best response is indicated with bold payoffs in thefollowing table.1Moreover,the game is symmetric,so we mightexpect the equilibrium to feature both players choosing the sameprice.Cells in which both players choose the same price areindicated with payoffs in italics.Consequently,the symmetricNash equilibrium has a bold and italicised payoff,ie.with bothfirms choosing a price of$40.1500250035002000280036002100270033001800220026001100130015002040(c)Now assume that prices can be any nonnegative real number.Take afirst-order condition and use it tofind the best-response function of eachfirm.Firm A’s optimal price satisfies thefirst-order condition∂πA.Similarly,firm B’s reaction4function is P B=120+P Ax 2be the location of Firm 2.Assume that the location decisions are made simultaneously.The payoffs to the two firms are:π1(x 1,x 2)=12−x 21−(x 2−x 1)2π2(x 1,x 2)=12−(1−x 2)2−(x 2−x 1)2a)Derive best response functions for the two firms.Sketch them in(x 1,x 2)space.MB 1=−2x 1+2(x 2−x 1)=0and MB 2=2(1−x 2)−2(x 2−x 1)=0.So the two best response functions are:B 1(x 2)=x 22.b)Identify the Nash equilibrium.State it and also show it in your dia-gram. x 1=1300.20.40.60.81.000.20.40.60.81.0x 2x 1(x 1)3Q3WarrantiesAcura and Volvo offer warranties on their cars,where w A is the number of years of an Acura warranty and w V is the number of years of a Volvo warranty.The revenue forfirm i,i=A for Acura and V for Volvo,is R i= 27,000w i/(w A+w V).The cost of providing the warranty is C i=2,000w i.Acura and Volvo play a warranty-setting game in which they simultaneously set the length of warranties.2Volvo’s indifference curves are drawn in(w V,w A)space below.a)Sketch Volvo’s best response function into the indifference curve dia-gram.The trick is tofind the point of zero slope of each indifference curveand then connect the dots.You should get a backward-bending best-response for Volvo,as illustrated by the blue curve.Note that thetwo diagrams don’t have the same scale-the second one starts from(1,1)rather than(0,0).In fact the best response goes to(0,0)in thelimit,but it is next to impossible to determine that from looking atthe indifference curve diagram.Unfortunately,I did give them the payofffunctions,so although I amnot expecting or requiring it,some people may attempt a mathemat-ical derivation.If they do,Volvo’s FOC is27w A/(w A+w V)2−2=0.Solving for w V gives the best response function w V=B V(w A)=2Technically,payoffs are undefined when wA=w V=0,so let’s ignore warranties of zero length.4The students haven’t been taught about contract curves,as general equilibrium is no longer in the course.So don’t be distraught if they mostly don’t get this right.1234561 2 3 4 5 627*y/(y+x)2-227*x/(y+x)2-25。

Course Meeting Time and Location:Mondays and Wednesdays, 11:15am - 12:40pmClass meets in room 220 of the Stuart Building (220 SB)Textbook:Chi-Tsong Chen, Linear System Theory and Design, 3rd ed., Oxford University Press, 1999. ISBN: 0-19-511777-8Course Objectives:After completing this course, the student should be able to do the following things correctly: ∙formulate state space descriptions of linear dynamical systems, in both time-invariant and time-varying cases, and for both continuous-time and discrete-time systems;∙find the analytic solution of state equations and give a geometric interpretation of the state space in terms of the system dynamics;∙apply the concepts of stability, controllability and observability in interpreting and analyzing system behavior;∙formulate input-output descriptions of linear dynamical systems, in both time-invariant and time-varying cases, and for both continuous-time and discrete-time systems;∙construct realizations of input-output system descriptions via state space system dynamics;∙use state feedback to reshape system dynamics;∙use observers to infer knowledge of the system states given input and output measurements;∙use computer-based analysis and design tools (such as {\sc Matlab} software) in the analysis of linear, time-invariant systems.Course Documents:These documents are available in pdf or html format.∙Course syllabuso Syllabus as an html documento Syllabus as an Excel document∙Suggested additional references∙The Kalman Decomposition: a document describing how to find the coordinate transformation leading to the Kalman decomposition and giving an example of theapproach.Homework Policy and Assignments:Please follow these rules when submitting homework papers. Any exceptions to these basic policies should be confirmed with the instructor before submitted your paper.∙Use 8 1/2 x 11 inch or A4 paper (other sizes are not accepted), with multiple pages stapled together in the upper left corner. Do not use any means other than stapling tohold pages together (in that event, the grader has full authority to discard all but the first page).∙Your homework paper should present answers to the questions in the same order that they appear in the assignment. At the grader's discretion, zero credit may be given forwork the appears out of order.∙No email submissions are accepted unless prior arrangements have been made due to special circumstances, or unless the assignment specifically requests email submission.∙No late homeworks will be accepted without prior approval by the instructor. (Generally, approval of late homework submission requires there to be very extenuatingcircumstances. I drop the lowest homework score in part to accommodate the commonsituations that prevent persons from completing the assignments on time.)Homework assignments, when available, will be posted here in pdf format.∙Assignment #1 [posted 26 Jan 2007] (due 7 February 2007) [Note the new (later) due date!]∙Assignment #2 [posted 7 Feb 2007] (due 14 February 2007)∙Assignment #3 [posted 14 Feb 2007] (due 28 February 2007)∙Assignment #4 [posted 13 Mar 2007] (due 28 March 2007) [Problem #1 typo corrected15 Mar 2007]o Here are a couple of supplementary problems for controllable and Kalman decompositions that are non-degenerative (unlike problems 4 and 5 on HomeworkAssignment \#4). For your convenience the various system matrices are stored inthe file hw4_supp_data.mat. To access the variables, save this file to yourcomputer, and change the working directory of Matlab to the directory where yousaved the file (or alternatively add that directory to Matlab's path). You then loadin the system matrices by typing "load hw4_supp_data" at the Matlab prompt.▪Assignment #4 supplement▪hw4_supp_data.mat∙Assignment #5 [posted 26 Mar 2007] (due 13 April 2007, on-campus students please submit homework by putting it in my mailbox in the ECE Department; off campusstudents may submit homework in the usual way)∙Assignment #6 [posted 12 Apr 2007] (due 25 April 2007)Homework Solutions:Homework solutions will be posted either here or at the Galvin Library's Electronic Reserves.∙Solutions to Assignment #1 [submitted for posting at the Galvin Library's Electronic Reserves on 7 Feb 2007 and posted there on 8 Feb 2007]∙Solutions to Assignment #2 [submitted for posting at the Galvin Library's Electronic Reserves on 14 Feb 2007 and posted there on 15 Feb 2007]∙Solutions to Assignment #3 [submitted for posting at the Galvin Library's Electronic Reserves on 28 Feb 2007 and posted there on 1 Mar 2007]o ECE531_07S_hw3_p3.m: Matlab m-file for problem 3 of Assignment #3 (save to your own computer and run from there)∙Solutions to Assignment #4 [submitted for posting at the Galvin Library's Electronic Reserves on 28 Mar 2007 and posted there on 28 Mar 2007]∙Solutions to Assignment #5 [submitted for posting at the Galvin Library's Electronic Reserves on 16 Apr 2007 and posted there on 17 Apr 2007]∙Solutions to Assignment #6 [submitted for posting at the Galvin Library's Electronic Reserves on 25 Apr 2007 and posted there on 26 Apr 2007]Examinations:There will be two take-home examinations during the course of the term, and one in-class final examination. The take-home examinations, when available, will be posted here in pdf format.∙Take-home Exam #1 (due 7 March 2007 at 11:25am) [Problem 8 corrected on 2 March 2007 to include a missing transpose.]o Solutions to Take-home Exam #1 [submitted for posting at the Galvin Library's Electronic Reserves on 22 Mar 2007 and posted there on 23 Mar 2007] ∙Take-home Exam #2 (due 2 May 2007 at 11:25am)o Solutions to Take-home Exam #2 [submitted for posting at the Galvin Library's Electronic Reserves on 7 May 2007 and posted there on 7 May 2007]. Note: theproblem 2 solutions shown here are for a problem not on the exam. The problem 3solutions solve problem 2 on the exam. The problem 4 solutions solve problem 3.Problem 4's solutions are missing, but are included in the supplement.o Supplement to solutions to Take-home Exam #2 [submitted for posting at the Galvin Library's Electronic Reserves on 7 May 2007 and posted there on 7 May2007]. These are the missing problem 4 solutions.∙In-class final examination: Wednesday, 9 May 2007, 2:00 - 4:00pm∙Sample in-class examinations are available as follows:o Final Exam, Spring 2001o Final Exam, Spring 2002o Final Exam, Spring 2004o Final Exam, Spring 2005o Final Exam, Spring 2006Grading:∙Homework: 20% (best 5 of 6 assignments)∙Take-home Exam #1: 30%∙Take-home Exam #2: 30%In-class Final: 20% (the final is comprehensive)Academic Honesty:It is your responsibility to be familiar with IIT's Code of Academic Honesty. (Consult the IIT Student Handbook for this code.)In particular, the work that you submit for homework and individual project assignments and your work on examination papers must be your own. You may consult with other students about homework and project assignments. In fact, discussion of the assignments is encouraged. However, the written material that is submitted must be your own. Such written material includes computer programs and the results of using computer programs (computed values, plots, etc.). If the above policy or any part of IIT's Code of Academic Honesty is violated in regard to a submitted homework assignment, a grade of zero will be assigned to the work. If the above policy or any part of IIT's Code of Academic Honesty is violated in regard to a submitted project assignment or an examination paper, a punitive failing grade will be given in the course. In both cases, the matter will be reported to the appropriate university officials and offices. In the case of a second offense (with the first offense in this or any other course), expulsion from the university will be advocated.。

Speaking of Statistics(12) - Assignment 1 – Suggested Solution1. Below we list several variables. Which of these variables are quantitative and which arequalitative? Explain.a. The dollar amount on an accounts receivable invoice. → Quantitativeb. The net profit for a company in 2005. → Quantitativec. The stock exchange on which a company’s stock is traded. → Qualitative d. The national debt of the United States in 2005. → Quantitativee. The advertising medium (radio, television, or print) used to promote a product. → Qualitative2. Classify each of the following qualitative variables as ordinal or nominative. Explain youranswers.Qualitative Variables The Corresponding CategoriesStatistics course letter grade A B C D F →Ordinal Personal computer ownership Yes No → Nominal Restaurant rating ***** **** *** ** * → Ordinal Incoming tax filing status Married filing jointly; Married filing separately; Single; Headof household; Qualifying widow(er) → Nominal3. Forbes magazine publishes the Forbes Platinum 400 –a list of the ―Best Big Companies inAmerica‖ as selected by the magazine’s writers and editors. Table above gives the best companies in the retailing industry as given in this list on the Forbes website on March 16, 2005. (Data in the last page ).b. Organize data of the profit margin percentages into a frequency distribution.Number of class K=6 (Since n K≥2 where n is the total number of data = 35.) The class length is calculated as 9.381.361.10.24≈=-=K rangea. Construct a stem-and leaf display of the return on capital percentages for the retailers in Table.Find the median and then describe the distribution of the return.The median is the 18th measurement, 15.4.The data are widely spread out, with majority falling in the range [8.3, 22.8]. The distribution is somewhat symmetric, with extreme values at both ends.c.Construct a histogram of the profit margin percentages.4. In order to control costs, a company wishes to study the amount of money its sales forcespends entertaining clients. The following is a random sample of six entertaining expenses (dinner costs for four people) from expense reports submitted by members of the sales force $157 $132 $109 $145 $125 $139a. Show the equivalence of the following two formulas:,.Solutions:()⎥⎦⎤⎢⎣⎡--=⎥⎥⎦⎤⎢⎢⎣⎡⎪⎪⎭⎫ ⎝⎛+--=++++++-+++-=+-+++-++--=-++-+--=--=--=∑∑∑∑∑∑∑∑22222221222212222222121222212221 1 1)(2(11)]...()2...22()...[(11)]2(...)2()2[(11])(...)()[(11)(111)(i i i i i i n n n n n i ix n x n n x n x n x x n x x x x x x x x x x x n x x x x x x x x x x x x n x x x x x x n x x n n x x sb. Calculate x , 2s , and s for the expense data, use the two different formulas in (a) to calculate 2s .Solutions 5.1346139125145109132157=+++++=x7.27616)5.134 139()5.134 132()5.134 157(2222=+++= s , 63.167.276==sOr using another formula:925,1091391251451091321572222222=+++++=∑ix()249,651)807()139125145109132157(222==+++++=∑ix()⎥⎦⎤⎢⎣⎡--=∑∑2221 1 1i i x n x n s 7.276)249,651(61–925,1091–61=⎥⎦⎤⎢⎣⎡=5. On February 15, 2005, the Gallup Organization released the results of a Gallup UK poll regarding Internet usage in Great Britain. Each of 1,009 randomly selected UK adults was asked to respond to the following question:How much time, if at all, do you personally spend using the Internet-more than an hour a day, up to one hour a day, a few times a week, a few times a month or less, or never?The poll’s results were as follows: More than an hour a day (22%); Up to an hour a day (14%);A few times a week (15%); A few times a month or less (10%); Never (39%). Use these datato construct a bar chart and a pie chart.Bar Chart:Pie Chart:6. Thirteen internists in the Midwest are randomly selected, and each internist is asked to report last year’s income. The incomes obtained (in thousands of dollars) are127 132 138 141 144 146 152 154 162 171 177 192 241 Find:There are 13 measurements in total (n=13). They are already in ascending order. a. The 90th percentile.Calculate the index of the 90thpercentile: 7.111310090=⨯=i Round 11.7 up to 12, hence the 90th percentile is the 90th measurement 192.b. The median.The median is equivalent to the 50thpercentile: 5.61310050=⨯=i Round 6.5 up to 7, hence the median is the 7th measurement 152.c. The first quartile.We know the index of the first quartile can be found as: 25.31310025=⨯=i Round 3.25 up to 4, hence the first quartile is the 4th measurement 141.d. The third quartileWe know the index of the third quartile can be found as: 75.91310075=⨯=i Round 9.75 up to 10, hence the third quartile is the 10th measurement 171.e. The 10th percentile.Calculate the index of the 10thpercentile: 3.11310010=⨯=i Round 1.3 up to 2, hence the 10th percentile is the 2nd measurement 132.f. The interquartile range.Interquartile range is the difference between first and third quartile3014117113=-=-=Q Q IQRg. Develop a five-number summary and a box-and-whiskers display.7. According to the website, the 2004 total return percentages for several popular funds were as follow:Fund 2004 Total Return% Vanguard 500 Index 10.7Wasatch Core Growth 21.7 Fidelity Stock Selector 9.9 Fidelity Dividend Growth6.0Janus Worldwide 5.2Suppose that an investor had $100,000 invested in the Vanguard 500 Index fund, $500,000 invested in the Wasatch Core Growth fund, $500,000 invested in the Fidelity Stock Selector fund, $200,000 invested in the Fidelity Dividend Growth fund, and $50,000 invested in the Janus Worldwide fund.a. Compute a weighted mean that measures the 2004 average total return for the investor ’s portfolio.Compare your weighted mean with the un-weighted mean of the five total return percentages. Explain why they differ.The weighted mean is%56.13000,350,1000,305,18000,50000,200000,500000,500000,100)5.5(000,50)8.5(000,200)9.9(000,500)7.21(000,500)7.10(000,100==++++++++=μAnd the un-weighted mean is%72.1055.58.59.97.217.10=++++The average total return calculated using weighted mean is higher because more money was invested in funds with larger gains, such as Wasatch Core Growth.8. As given on the Morningstar website, the mean return for Fidelity Leveraged Company Stock is 17.98 percent with a standard deviation of 10.02 percent, while the mean return for Baron Opportunity Retail Stock is 16.43 percent with a standard deviation of 12.23 percent. Which stock do you buy? (Which is riskier than the other one?)The mean return for Fidelity Leveraged Company Stock is higher than of Baron Opportunity Retail Stock.When we compute and compare the coefficient of variation for these two stocks using 100mean deviationstandard ⨯ , we find that the coefficient of variation of Fidelity Leverage(73.5510017.9810.02=⨯) is lower than the one of Baron Opportunity (44.7410016.4312.23=⨯), meaning the variation in returns for the Baron Opportunity is higher, so investing in Baron Opportunity is riskier.Therefore investing in Fidelity Leveraged Company Stock is more advisable.9. The card game of Euchre employs a deck that consists of all four of each of the aces, kings, queens, jacks, tens, and nines (one of each suit —clubs, diamonds, spades, and hearts). Find the probability that a randomly selected card from a Euchre deck isThere are a total of 24 cards (4 aces, 4 kings, 4 queens, 4 jacks, 4 tens and 4 nines.a. A jack (J)The probability of selecting a jack is ()61244==J P b. A spade (S)The probability of selecting a spade card is 41246)(==S P c. A jack or an ace (A)The probability of selecting a jack OR an ace is()()()()3106161=-+=⋂-+=⋃A J P A P J P A J P d. A jack or a spade.The probability of selecting a jack OR a spade card is()()()()832414161=-+=⋂-+=⋃S J P S P J P S J P e. Are the events J and A mutually exclusive? J and S? Why or Why not?The events J and A are mutually exclusive because one cannot draw a Jack AND an Ace at the same time in one single draw.The events J and S are NOT mutually exclusive because one can draw a jack of spade under one single draw, hence J and S could occur simultaneously.10. Fifteen percent of the employees in a company have managerial positions, and 25 percent of the employees in the company have MBA degrees. Also, 60 percent of the managers have MBA degree.a. What proportion of employees are managers and have MBA degrees. 9% of employees are managers AND have MBA degrees09.0)6.0)(15.0()|()()(===⋂Manager MBA P Manager P MBA Manager Pb. What proportion of MBAs are managers. 36% of MBAs are managers36.025.009.0)()()|(==⋂=MBA P MBA Manager P MBA Manager Pc. Are the events being a manager and having an MBA independent? Justify your answer.No. Because 15.0)(36.0)|(=≠=ManagerP MBA Manager P11. In a survey of 100 insurance claims, 40 are fire claims (FIRE ), 16 of which are fraudulent (FRAUD ). Also, there are a total of 40 fraudulent claims. a. Construct a contingency table summarizing the claims data. Use the pairs of events FIRE and FIRE , FRAUD and FRAUD .Contingency Table:FireFIRETotal Fraud 16 24 40 Fraud 24 36 60Total4060100b. What proportion of the fire claims are fraudulent? 40% of the fire claims are fraudulent4.04016)()()|(==⋂=Fire P Fire Fraud P Fire Fraud Pc. Are the events a claim is fraudulent and a claim is a fire claim independent? Use your probability of part b to prove your answer. Yes. Because10040)(4.0)|(===Fraud P Fire Fraud P12. Each month a brokerage house studies various companies and rates each company’s stock as being either ―low risk‖ or ―moderate to high risk.‖ In a recent report, the brokerage house summarized its findings about 15a.The probability that the company’s stock is moderate to high risk giventhat the firm is an aerospace company.P(ModToHigh | Areo) = 9/15=0.6b.T he probability that the company’s stock is moderate to high risk giventhat the firm is a food retailer.P(ModToHigh | Food) = 10/25=0.4c.Determine if the company type is independent of the level of risk of thefirm’s stock.If the company type is independent of the level of risk of the firm’s stock, the we should find equalities such asP(Aero | Low) = P(Aero)However, P(Aero | Low) = 2/7 while P(Aero) = 3/8.The two probabilities are not equal. Hence the two events are dependent.Table for question 3:The Forbes platinum list: Best-performing retailers as listed at on March 16, 2005。

(Partial) Solutions to Assignment 2pp.73-761.16In each of the following systems, let or be the input and or be the output. Determine whether each systems is (1) linear, (2) time invariant, (3) causal, (4) BIBO stable(g).(i).ans: omitted----------------------------------------------------1.17 A linear time invariant system has impulse response Determine the output sequence for each of the followign input signals:(b)(f)(b) ans:h n is given byThe z-transform of []where ROC1:x n is given byz-transform of []where ROC2:h n is given byTherefore, the z-transform of the output []y nPerform inverse z to get [](f) ans: using the same method as in (b) (details omitted )----------------------------------------------------1.18. A linear time invariant system is defined by the difference equationb. Determine the output of the system when the intpu isc. Determine the output of the system when the input isans: omitted----------------------------------------------------1.19 The following expressions define linear time invariant systems. For each one determine the impulse respnose(a)(e)(a) ans: the impulse response is(e) ans: the impulse response is----------------------------------------------------1.20 Each of the following expressions defines a linear time invariant system. For each one determine whether it is BIBO stable or not(g)(k)BIBO: Bounded input and bounded output(g) ans: omitted(k) ans: omitted----------------------------------------------------1.21. Using the geometric series, for each of the following sequence determine the z-transform and its ROC(d)(g)(i)(d) ans:where ROC:(g) ans:The first part is equal towhere ROC1 isThe second part is equal towhere ROC2 isTherefore combining both parts:where ROC={ROC1 and ROC2}:(i) ans:where ROC: whole complex domain----------------------------------------------------1.22. You know what the and are. Using theproperties only (do not reuse the definition of the z-transform.) determine the z-transform of the following signals(c)(g)where ROC1:where ROC2:(c) ans: using z-transform property:We have:where ROC:(g) ans:details omitted. The final answer isTherefore combining both parts:where ROC={ROC1 and ROC2}:----------------------------------------------------1.23 Using partial fraction expansion, determine the inverse z-transform of the following functions:(c) ,(e) ,(c) ans:(e) ans:procedures are the same as above. details omitted.----------------------------------------------------1.24. For each of the followign linear difference equations, determine the impulse response, and indicate whether the system is BIBO stable or not(a)(c)(a) ans:Take z-transform on both sideswhere ROC:Because is finiteTherefore, the system is BIBO stable(c) ans: omitted (the same as (a))----------------------------------------------------1.25. Although most of the time we assume causality, a linear difference equation can be interpreted in a number of ways. Consider the linear difference equation(a) Determine the transfer function and the impulse response. Is the system causal ? BIBO stable ?(a) omitted.----------------------------------------------------1.26. 1.26 Consider the linear difference equation(a) Determine the transfer function . Do you have enough information to determine theregion of convergenceans:Don't have enough information to determine ROC.----------------------------------------------------1.27. Given the system described by the linear difference equationDetermine the output for each of the following input signals(a)(e)(a) ans:Take z-transform on both sides:----------------------------------------------------1.28. Repeat Problem 1.27 when the system is given in terms of the impulse responseBefore you do anything, is the system stable ? Does the frequency responseexist ?ans: omitted.----------------------------------------------------1.29. Repeat Problem 1.27 when the system isgiven by the linear difference equationBefore you do anything, is the system stable ? Does the frequency response exist ?Ans: omitted.----------------------------------------------------。

AK/ADMS 4503 3.0 Derivative SecuritiesFall 2008Assignment #1 SolutionsInstructions:(1) This assignment is to be done individually . You must sign and submit the standard cover page supplied as the last page of this assignment.(2) This assignment is due on October 19, 2008.(3) The work can be typed or handwritten. If it is handwritten and too difficultto read due to messiness and poor handwriting, it will receive zero credit.(4) You must show your work to receive full credit.(5) This assignment contains 5 questions and carries a total of 30 points .Question 1 (6 marks)The NASDAQ-100 futures trade at the CME, and each contract is on $100 times the index. The NASDAQ-100 spot is 1,670 points, and is expected to pay a dividend yield of 1% per annum continuously compounded. The risk-free rate is 2% per annum continuously compounded.(a) What is the theoretical 1-year futures price? (2 marks)(b) The 1-year futures price is 1,701 points. Show that there is an arbitrage and show how to benefit from it? Show all details. (4 marks)Solution(a) The theoretical 1-year futures price given by:()()78.686,11%)1%2(exp 670,1)(exp 0=×−×=×−×=T T F T q r S F(b) Since the market overprices the contract, there is an arbitrage that consists in shorting the 1-year futures contract, borrow the money at the risk-free rate to buy the index today. By doing this, you lock in a net profit of $1,421.62. Here are the details:Today, you must: - Short one 1-year contract to sell the index at 1,701 in 1 year - Borrow exactly exp(-1%) x 1,670 = 1,653.38 , at the risk-free rate 2% for1 year (to buy 0.99005 units of the index)In 1 year:- The dividend yield paid on your holdings will make you having exactly 1unit of index- You deliver the index for 1,701 according to your short contract- You pay back your loan at 1,653.38 x exp(2%) = 1,686.78 - Your profit is then (1,701 – 1,686.78) = 14.2162Your net dollar profit is $100 x 14.2162 = $1,421.62Question 2 (6 marks)Consider a coupon-bearing bond selling at $950 and paying coupons in 5 months and 11 months from today. The risk-free interest is 2% per annum continuously compounded. The face value of the bond is $1,000.(a) If the 1-year forward contract on this bond is selling at a fair price of $949.05, what is the coupon rate? (2 marks)(b) What is the theoretical 6-month forward price? (1 mark)(c) The 6-month forward contract is selling at $955 in the market. Is there any arbitrage opportunity? If yes, show how to benefit from it. Show all details.(3 marks)Solution(a) We know that %)2exp()950($05.949$×−=I where:())12/11%2exp()12/5%2exp(×−+×−×=Coupon ISolving for the coupon, we find that the coupon is $10, which means that the coupon rate 2% APR semi-annually compounded.(b) Given that the coupon is $10, the theoretical price for the 6-month contract must be:()[]53.949$)5.0%2exp(12/5%2exp 10$950$0=×××−−=F(c) Since the contract is overpriced by the market, an arbitrageur can lock in a profit by borrowing/buying the bond and taking a short 6-month forward.Here are the details of the strategy:Today- Borrow a total amount of $950 to buy the bond:o Borrow $9.92 at 2% today to be reimbursed at $10 in 5 months (thisis exactly the coupon that you would receive in 5 months)o Borrow ($950 - $9.92 = $940.08) at 2% today to be reimbursed at$949.53 in 6 months-Short one forward contract today to sell the bond at $955 in 6 monthsIn 5 months-Receive the coupon of $10 and pay back the small part of the loanIn 6 months-Deliver the bond at $955-Pay back the large part of the loan at $949.53-The net profit is $5.47Question 3 (6 marks)Consider a Canadian company that is planning to buy some equipment from aBritish manufacturer in Oct 1, 2009 (that is in one year). The cost of this machinery is GBP 10 million. You have been asked to analyze the consequencesof entering into a futures contract to reduce the company’s exposure to foreign exchange risk. The current quotes are available from the market:Spot CAD/GBP 1.90Canadian TBill Rate 2%UK TBill Rate 3%(a) What is the theoretical 1-year (October 2009 contract) forward CAD/GBP?(2 marks)(b) Based on the following scenarios for the spot exchange rate one year fromnow, CAD/GBP = 1.7 or 2.1, explain why the company should hedge itscurrency risk exposure. Explain which strategy may be appropriate for thecompany and what will be the total cost (for the equipment) in CAD in oneyear. (2 marks)(c) Assume now that you enter into the strategy proposed in (b) and that aftersix months, i.e. April 1, 2009, the management of the company decides tobuy immediately the equipment from the British manufacturer and to closeout the forward position. Assume that on April 1, 2009, the spot CAD/GBP= 1.95, the forward (with six months remaining to the maturity) rateCAD/GBP = 1.93, what is the effective total cost in CAD for theequipment? (2 marks)Solution(a) The 1-year forward contract is given by:()()8811.11%)3%2(exp 90.1)(exp 000=×−×=×−×=F T r r S F f(b) CAD/GBP 1.70 2.10Cost CAD 17 million CAD 21 millionThis shows that the company has a considerable exchange rate risk exposure. If the exchange rate moves from 1.9 to 1.95 over one year, the company makes a loss of CAD 500,000! The company will face a loss if the exchange rate CAD/GBP increases dramatically since it will have to buy GBP 10 million.To hedge against any dramatic increase, the firm must take a long position in the 1-year forward to buy GBP at CAD 1.8811. The total cost will be CAD 18.811 million (= GBP 10 million x 1.8811) whatever the spot exchange rate will be in one year.(c) The company decides to buy the machinery after six months for GBP 10 million at CAD/GBP = 1.95, that is a cost of CAD 19.5 million. On the other hand, the company makes a net profit of CAD 0.4841 million on the forward position, (1.93 – 1.8811) x exp(-2% x 0.5) x 10 = 0.4841. Therefore, the effective total cost is 19.5 – 0.4841 = CAD 19.0159 million, or equivalently a CAD/GBP of 1.9016.Question 4 (6 marks)The Aluminum sells at $1.20 per pound and it has a convenience yield y = 2%, and a storage cost of 1% (per annum continuously compounded). The risk-free interest rate is 4% per annum continuously compounded. An investor takes a short position in a 1-year futures contract on Aluminum today. Assume that each contract is on 44,000 pounds.(a) What is the 1-year futures price per pound? (2 marks)(b) Suppose that the investor closes out her position 9 months from now and makes a total profit of $2,200. What is the spot price of Aluminum 9 months from now if there is no arbitrage? (4 marks)Solution(a) The theoretical 1-year futures price is $1.20 x exp(4% + 1% - 2%) = $1.2365(b) If the investor makes a $2,200 total profit on her short position in 9 months, or a $0.05 (= 2,200 / 44,000) profit per pound, this means that the futures price 9 months from now is $1.1865 and that the contract has 3 months remaining. The spot price in 9 months is then given by:S = 1.1865 x exp(-3% x 0.25) = $1.1777Question 5 (6 marks)A portfolio manager has sold short a portfolio of stocks worth $100 million for three months. The beta of the portfolio is 2. The manager would like to use the CME futures contract on the S&P 500 index to hedge the portfolio over the next three months. The index is currently 1,200 points, and each contract is on $250 times the index. The S&P 500 dividend yield is 1% and the risk-free rate is 2%.(a) How many long or short positions should the manager take? (4 marks)(b) What would be the net gain/loss in three months in the following scenarios? (2 marks)Scenario Portfolio value In 3 months S&P 500 Futures In 3 months 1 $110 million 12602 $90 million 1140Solution(a) First, we need to calculate the theoretical S&P 500 futures price today:1200 x exp[(2%-1%) x .25] = 1203.Since the manager is short selling the portfolio, any rise in the stock market would cause big losses. The manager should then take long S&P 500 futures contracts, exactly 665 contracts:6651203250$000,000,100$2*=××==F P N β(b) The net gain/loss for each scenario is given by the gain/loss on the shortposition on the portfolio added to the gain/loss on the futures position.Scenario 1: (100 – 110) million + 665 x 250 x (1260 – 1203) = –$523,750 Scenario 2: (100 – 90) million + 665 x 250 x (1140 – 1203) = –$473,750Atkinson Faculty of Liberal and Professional StudiesYORK UNIVERSITYToronto, OntarioADMS 4503 3.0Derivative SecuritiesTahaniProfessor NabilSections A and BAssignment #1Due Date: October 19, 2008Personal Work StatementI, the undersigned:•warrant that the work submitted herein is my work and not the work of others •acknowledge that I have read and understood the Senate Policy on Academic Honesty•acknowledge that it is a breach of the University Regulations to give and receive unauthorized assistance on a graded piece of workName (typed or printed) York Student # Signature。

Chapter 5+6+8 AssignmentsP5.Consider the 5-bit generator, G= 10011, and suppose that D has the value 1010101010. What is the value of CRC?Solution:If we divide 10011 into 1010101010 0000, we get 1011011100, with a remainder of R=0100.P21. Suppose nodes A and B are on the same 10Mbps Ethernet bus, and the propagation delaybetween the two nodes is 245 bit times. Suppose A and B send frames at the same time, the frames collide, and then A and B choose different values of K in the CSMA/CD algorithm. Assuming no other nodes are active, can the retransmissions from A and B collide? For our purposes, it suffices to work out the following example. Suppose A and B begin transmission at t =0 bit times. They both detect collisions at t =245 bit times. They finish transmitting a jam signal at t = 245 + 48 = 293 bit times. Suppose K A =0 and K B = 1. At what time does B schedule its retransmission? At what time does A begin transmission? (Note: The nodes must wait for an idle channel after returning to Step 2 -- see protocol.) At what time does A's signal reach B? Does B refrain from transmitting at its scheduled time?Solution:Because A's retransmission reaches B before B's scheduled retransmission time (805+96), B refrains from transmitting while A retransmits. Thus A and B donot collide. Thus the factor 512 appearing in the exponential backoff algorithm is sufficiently large.P23. Suppose four nodes, A, B, C, and D, are all connected to a hub via 10Mbps Ethernet cables.The distances between the hub and these four nodes are 300m, 400m, 500m, and 700m, respectively. Recall that the CSMNCD protocol is used for this Ethernet. Assume that the signal propagation speed is 2*108m/sec.a. What is the minimum required frame length?b. If all frames are 1500 bits long, find the efficiency of this Ethernet.Solution:a). minimum required frame length is given by2*d prop* BW=2*(500+700)/( 2 108) * 10 * 106=120 bits.There is no maximum required packet length.b). Efficiency is given by1/(1+5* d prop/ d trans) =1/(1+5*120/2/1500 )=0.83P37. In this problem, you will put together much of what you have learned about Internetprotocols. Suppose you walk into a room, connect to Ethernet, and want to download a web page. What are all the protocol steps that take place starting from powering on your PC to getting the web page? Assume there is nothing in our DNS or browser caches when you power on your Pc. (Hint: the steps include the use of Ethernet, OHCP, ARP, ONS, TCP, and HTTP protocols.) Explicitly indicate in your steps how you obtain the IP and MAC addresses of a gateway router.Solution:(The following description is short, but contains all major key steps and key protocols involved.)Your computer first uses DHCP to obtain an IP address. You computer first creates a special IP datagram destined to 255.255.255.255 in the DHCP server discovery step, and puts it in a Ethernet frame and broadcast it in the Ethernet. Then following the steps in the DHCP protocol, you computer is able to get an IP address with a given lease time.A DHCP server on the Ethernet also gives your computer a list of IP addresses of first-hop routers, the subnet mask of the subnet where your computer resides, and the addresses of local DNS servers (if they exist).Since your computer’s ARP cache is initially empty, your computer will use ARP protocol to get the MAC addresses of the first-hop router and the local DNS server.Your computer first will get the IP address of the Web page you would like to download. If the local DNS server does not have the IP address, then your computer will use DNS protocol to find the IP address of the Web page.Once your computer has the IP address of the Web page, then it will send out the HTTP request via the first-hop router if the Web page does not reside in a local Web server. The HTTP request message will be segmented and encapsulated into TCP packets, and then further encapsulated into IP packets, and finally encapsulated into Ethernet frames. Your computer sends the Ethernet frames destined to the first-hop router. Once the router receives the frames, it passes them up into IP layer, checks its routing table, and then sends the packets to the right interface out of all of its interfaces.Then your IP packets will be routed through the Internet until they reach the Web server.The server hosting the Web page will send back the Web page to your computer via HTTP response messages. Those messages will be encapsulated into TCP packets and then further into IP packets. Those IP packets follow IP routes and finally reach your first-hop router, and then the router will forward those IP packets to your computer by encapsulating them into Ethernet frames.Additional 1: Please answer the following question after reading Chapter 6 and referring tothe ftp:///chapt6.pdf . What are MAC methods for WiFi, Bluetooth, WiMax and Cellular network respectively? What are RDT approaches for WiFi, Bluetooth, WiMax and Cellular network respectively? error detection + retransmisstion or error correction?Solution:The following table summarizes the wireless MAC and RDT technologies. Each needs to be explained in detail.Additional 2:Why WiFi can’t CSMA/CD?Solution:Because Collisions Detection is difficult for WiFi:∙hidden terminal problem !∙difficult to sense because of fading∙Most radios are half duplexAdditional 3: Please explain how public key cryptography RSA is used in email security protocol: PGP (Pretty Good Privacy). Please refer to Chapter 8Solution:Step 1: Calculate hash (MD5) of message H(m). => you’ve got the digest of the messageStep 2: Encrypt H(m) with Alice’s private key => you’ve got Alice’s signature.Step 3: Append signature to text, then encrypt it with shared key Ks.Step 4: Encrypt shared key Ks with Bob’s public key, then append to the cipher text of step 2 => Bob can get the Ks.Step 5: The result message of step 4 is converted to base64 and be put into an RFC 822 body and be expected to arrive unmodified.D2 (don’t submit) Many of the functions of an adapter can be performed in software that runs onthe node's CPU. What are the advantages and disadvantages of moving this functionality from the adapter to the node?。