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ECMT3150: Assignment

创建日期：2024-08-09 17:51:38

所属分类：计算机computer science

标签： ECMT3150

Assignment

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ECMT3150: Assignment
1. [Total: 24 marks]
Note: Please append your R codes (as a separate .R le) for part (g) while you submit the
assignment.
Let Xi denote the log-price of Nvidia (code: NVDA), by the end of trading day i, and let
Xi := Xi Xi1; thus Xi is the log-return on trading day i (i.e., over period (i 1; i]).
Assume fXig_i0 follows the AR(1) model:
Xi = 0 + 1Xi1 + ui: (1)
where ui iid normal with mean 0 and variance 2.
Let fFigi0 be the natural ltration generated by fuigi0.
(a) [2 marks] Express Xi in terms of Xi1 and ui.
(b) [2 marks] Compute E(XijFi1).
(c) [2 marks] Compute V ar(XijFi1).
(d) [2 marks] What is the condition on 0 and 1 such that fXigi1 is a martingale
di¤erence sequence?
A trading strategy is dened by figi0, where i is measurable with respect to Fi.
The log-return associated with the trading strategy over period (0; T ] is given by
rT =
TX
i=1
i1Xi.
Specically, i1 represents the number of NVDA shares a trader buys at the start of day i.
(e) [4 marks] Alice invested in a share of NVDA using a buy-and-hold strategy, with i 1
for all i. Compute E(rT ) and V ar(rT ) with 0 = 0 and 1 = 1.
(f) [4 marks] Bob suggested another strategy, with i Xi for i > 0 and 0 = 1. Compute
E(rT ) and V ar(rT ) with 0 = 0 and 1 = 1.
1
(g) [8 marks] Carol suggested yet another strategy, with i 1fXi > 0g and 0 = 1.
We want to evaluate the risk-return tradeo¤ of the proposed strategies using computer
simulation.
Start an R session, and set a random seed equal to the last 4 digits of your student ID.1
Then generate B sample values of rT (name them as r
(1)
T ; r
(2)
T ; : : : ; r
(B)
T ), and compute
the sample mean and standard error of rT as follows:
rT =
1
B
BX
b=1
r
(b)
T ;
se(rT ) =
vuut 1
B 1
BX
b=1
(r
(b)
T rT )2:
For the purpose of your simulations, set T = 63, 2 = 0:1, B = 1000, and X0 = 0.
The Sharpe ratio, dened as SR = rTse(rT ) , is a common measure of the risk-return
tradeo¤. Trading strategies with higher SR are more preferred by investors.
Complete the following table with SR values. Comment on the performance of the
trading strategies under di¤erent scenarios.
0 1 Alice Bob Carol
0 1
0:01 1
0:01 1
0 0:9
0 1:1
2. [Total: 16 marks]
Note: Correct all numerical answers to two decimal places.
Let M denote the mood of Mimi (h: happy; a: angry), and let W denote the weather (s:
sunny; r: rainy). The joint probability distribution of M and W is given in the table below.
The row and column sums are displayed in the last column and in the last row, respectively.
p(m;w) M = h M = a
W = s 0:4 0:1
W = r 0:2 0:3
A good day occurs when Mimi is happy and the weather is sunny; otherwise, a bad day
occurs. Let D denote the type of a day (g: good; b: bad). Then we have fD = gg = fM =
hg \ fW = sg and fD = bg = fM = ag [ fW = rg.
(a) [2 marks] Compute the probability that today is a bad day, i.e., compute P (D = b).
1This is to ensure that your answers are replicable but di¤erent from those of other students.
2
(b) [2 marks] Derive the conditional distribution of W given M = a.
Assume that, given m and w, your test score S follows a normal distribution with mean
(m;w) := E(SjM = m;W = w) and standard deviation 10. The conditional mean function
(m;w) is given in the table below:
(m;w) m = h m = a
w = s 70 50
w = r 60 40
The passing score is 50 or above.
(c) [3 marks] Compute the mean score E(S).
(d) [3 marks] Given that it was a bad day when you took the test, what is the mean score
you would get?
(e) [3 marks] Compute the probability of failing the test.
(f) [3 marks] Given that you failed the test, what is the probability that it was a bad day?
3. [Total: 20 marks]
Note: Please append your R codes (as a separate .R le) while you submit the assignment.
Correct all numerical answers to three decimal places.
Mimi, an amateur economist, proposes the following time series model for ination rate:
yt =
1
10
+
1
2
yt1 1
4
yt2 + "t; (2)
where "t iid N(0; 0:012) (normal distribution with mean 0 and variance 0:012). The time
period is measured in number of quarters.
(a) [3 marks] Show that the time series fytg generated by model (2) is stationary.
(b) [3 marks] There is a stochastic cycle in the time series generated by model (2). Find its
periodity in number of quarters.
(c) [4 marks] Compute the ACF for the rst 3 lags, i.e., (1), (2) and (3).
(d) [2 marks] Write an R program to simulate a sample path of fytg over 30 years. Set the
initial values y0 and y1 to be y0 = 0:1 and y1 = 0:12. While simulating the random
numbers for "t, set the random seed to be your last 4 digits of your student ID.
(e) [2 marks] Plot the sample ACF and record its value for the rst 3 lags (the values can be
retrieved from the acf command output stored as a list). Why are they di¤erent from
your answers in part (c)?
3
(f) [3 marks] Using the simulated sample path in part (d), estimate an AR(2) model using
the R command arima. Write down the estimated model with the parameter estimates
and their standard error. Also record the estimated variance of the innovations.

(g) [3 marks] Using the simulated sample path in part (d) and the R package forecast,
plot the point forecast and the condence interval for each period over the next 5 years.
Describe the short-run and long-run behaviour of the point forecast and the condence
interval.

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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