代写ECMT3150、R语言编程代写

日期：2023-03-13 08:13

ECMT3150: Assignment 1 (Semester 1, 2023)

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 a stock, Cherry Inc. (code: CRRY), 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 de?ned by figi0, where i is measurable with respect to Fi. Speci?-

cally, i represents the number of CRRY shares a trader buys at the start of day i.

The log-return due to the trading strategy over period (0; T ] is given by

rT =

TX

i=1

i1Xi.

(e) [4 marks] Alice invested in a share of CRRY 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 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 3 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 variance of rT as follows:

rT =

1

B

BX

b=1

r

(b)

T ;

se(rT ) =

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.

The Sharpe ratio, de?ned 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] LetM 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) [2 marks] Compute P (M = a).

(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 5. The conditional mean function

(m;w) is given in the table below:

1This is to ensure that your answers are replicable but di¤erent from those of other students.

2

(m;w) m = h m = a

w = s 80 50

w = r 70 40

The passing score is 50 or above.

(c) [3 marks] Compute the mean score E(S).

(d) [3 marks] Given that Mimi was angry, 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 Mimi was angry?

3. [Total: 20 marks]

Note: Please append your R codes (as a separate .R ?le) while you submit the assignment.

Carol, an amateur economist, proposes the following time series model for unemployment

rate:

yt =

1

20

+

p

3

2

yt1 1

4

yt2 + "t; (2)

where "t iid N(0; 0:022) (normal distribution with mean 0 and variance 0:022). The time

period is measured in number of quarters.

(a) [3 marks] Show that the time series fytg generated by model (1) is stationary.

(b) [3 marks] There is a stochastic cycle in the time series generated by model (1). 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 3 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)?

(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.

[Important note: the ?intercept?estimate in the arima output is in fact the unconditional

mean; see Rob Hyndman?s page for details: https://robjhyndman.com/hyndsight/

arimaconstants/.]

(g) [3 marks] Using the simulated sample path in part (d) and the R package forecast,

plot the point forecast and the con?dence interval for each period over the next 5 years.

Describe the short-run and long-run behaviour of the point forecast and the con?dence

interval.