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CRICOS 00099F Financial Modelling and Analysis
创建日期：2024-05-22 14:41:51
所属分类：数学mathematics
标签： CRICOS 00099F
Financial Modelling and Analysis

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CRICOS 00099F

Financial Modelling and Analysis
Seminar 1
Introduction and Descriptive statistics
2Communication
• Content related questions
o Discussion Board in Canvas
• Administrative questions and consultation hours
o Via email to the Subject Coordinator
o [email protected]
Seminar 1 - Introduction and Descriptive statistics
3Subject contents
• Seminar 1 Descriptive statistics
• Seminar 2 Statistical inference – Confidence intervals
• Seminar 3 Statistical inference – Hypothesis Testing
• Seminar 4 Forecasting
• Seminar 5 Measuring forecasting accuracy
• Seminar 6 Linear regressions – Simple and Multiple
• Seminar 7 Linear regressions – Time series
• Seminar 8 Linear regressions – Model evaluation
• Seminar 9 Linear regressions – Additional topics
• Seminar 10 Excel modelling – Best practices
• Seminar 11 Excel modelling – Applications
• Seminar 12 Review
Seminar 1 - Introduction and Descriptive statistics
4Seminar structure - Wednesdays
In-person
2. In-class activity 3. Q&A + Quiz1. Lecture
12:00 – 12:50 13:00 – 13:50 14:00 – 14:50
Online 18:00 – 18:50 19:00 – 19:50 20:00 – 20:50
Seminar 1 - Introduction and Descriptive statistics
5Assessment
• In-class quizzes 20%
o 50 minutes with MCQs
o Quiz #1: Week 5 | Topics in weeks 1 - 4
o Quiz #2: Week 8 | Topics in weeks 5 - 7
• Case Study 20%
o Individual
o Due on Friday 12 May 2023
• Final exam 60%
o Covers the whole 11 lectures
Seminar 1 - Introduction and Descriptive statistics
6What Do I Need to Do to Succeed?
• 10 – 15 minutes of reading before coming to the seminars
• Resolve any question ASAP, e.g. during the same lecture.
• Following the in-class examples and activities
• Self-assessment: can I explain the concept or method to others?
Seminar 1 - Introduction and Descriptive statistics
Descriptive Statistics
• Population, sample, and probability
• Expectation, variance, covariance, and correlation of
random variables
• Population values when probability distribution is
known
• Sample values when probability distribution is
unknown
8Population versus Sample
• Population contains ALL potential observations.
o Parameters (Greek letters): mean μ, variance σ2, correlation ρ, etc.
• A sample contains a portion of the population.
o Estimates (Roman letters): mean ത, variance s2, correlation r, etc.
• Statistic inference: how close is ത to μ, s2 to σ2, etc.?
Population
Sample 2:
Y1,…,Yn
Sample 1:
X1,…,Xm
Seminar 1 - Introduction and Descriptive statistics
9Random Variables and Probability Distributions
• The value of a random variable is subject to chance (or randomness)
o Future values are unknown
o Degree of randomness: short-term interest rate and wage are “deterministic” compared
to asset prices and volatility.
• A probability distribution specifies all the possible outcomes along with the
probability of occurrence of each outcome
Seminar 1 - Introduction and Descriptive statistics
10
Measures of Central Tendency
Expectation
• Weighted average:
μ = E X =෍
i=1
M
XiP(Xi)
where
• Xi = the i
th outcome of the (discrete) random variable X
• P(Xi)= probability of occurrence of the i
th outcome of X
• M = number of possible outcomes/values
• σi=1
M () is the sum over i = 1, …, M
Seminar 1 - Introduction and Descriptive statistics
11
Measures of Dispersion
Variance and Standard Deviation
• Variance of discrete random variable X
σ2 = σi=1
M Xi − E X
2 P(Xi)
• Standard deviation of discrete random variable X
σ = σi=1
M Xi − E X 2P(Xi)
Seminar 1 - Introduction and Descriptive statistics
12
IPO Distribution Example
• Consider the following table on the number of IPOs per month on the ASX since 2010:
IPOs per month Probability
0 0.35
1 0.25
2 0.2
3 0.1
4 0.05
5 0.05
Seminar 1 - Introduction and Descriptive statistics
13
IPO Distribution Example
• The expected number of IPOs per month is
E X = σi=1
M XiP(X = Xi)
= 0 × 0.35 + 1 × 0.25 + 2× 0.20 + 3 × 0.10 + 4 × 0.05 + 5 × 0.05
= 1.40
Seminar 1 - Introduction and Descriptive statistics
14
IPO Distribution Example
• Since the mean = 1.4, the Variance is
σ2 =෍
i=1
M
Xi − E X
2 P(Xi)
σ2 = 0 − 1.4 2 × 0.35 + 1 − 1.4 2 × 0.25 + 2 − 1.4 2 × 0.20 +
+ 3 − 1.4 2 × 0.10 + 4 − 1.4 2 × 0.05 + 5 − 1.4 2 × 0.05
σ2 = 2.04
• Standard Deviation is
σ = σ2 = 2.04 = 1.428
Seminar 1 - Introduction and Descriptive statistics
15
Two Funds Exercise
• Consider two alternative investments. The first is an aggressive fund with high returns
during economic boom. The second is a defensive fund that performs better during
economic downturn.
• The following table summarizes your estimate of the annual returns under three
economic conditions, with each with probability occurrence:
Seminar 1 - Introduction and Descriptive statistics
16
Two Funds Exercise
Probability
Economic
Condition
Aggressive Fund
(X)
Defensive Fund
(Y)
0.2 Recession -30% 20%
0.5 Stable Economy 10% 5%
0.3
Expanding
Economy
25% -10%
Seminar 1 - Introduction and Descriptive statistics
17
Two Funds Exercise
X = Aggressive Fund
• E X = −0.3 0.2 + 0.1 0.5 + 0.25 0.3
= 0.065 = 6.5%
Y = Defensive Fund
• E Y = 0.2 0.2 + 0.05 0.5 + −0.1 0.3
= 0.035 = 3.5%
Seminar 1 - Introduction and Descriptive statistics
18
Two Funds Exercise
X = Aggressive Fund
• σX
2 = Var(X) = 0.2(−0.3−0.065)2+ 0.5(0.1−0.065)2+
0.3(0.25−0.065)2
= 0.0375
• σX = 0.194
Seminar 1 - Introduction and Descriptive statistics
Y = Defensive Fund
• σY
2 = Var(Y) = 0.2(0.2−0.035)2+ 0.5(0.05−0.035)2+
0.3(−0.10−0.035)2
= 0.011
• σY = 0.105
19
Measures of Association
Covariance and Correlation
• The covariance and correlation measure the strength of the linear relationship between
two random variables X and Y.
• A positive covariance/correlation implies that X and Y often move in the same direction.
• A negative covariance/correlation implies that X and Y often move in the opposite
direction.
Seminar 1 - Introduction and Descriptive statistics
20
Measures of Association
Covariance
• Covariance:
Cov X, Y = E{ Xi − E X Yi − E Y }
= σi=1
M Xi − E X Yi − E Y P(XiYi)
where P(XiYi) = prob. of occurrence of i
th outcome of X and ith outcome of Y.
o The sign indicates the direction of the relationship
o The size/magnitude depends on the size/magnitude of X and Y.
Seminar 1 - Introduction and Descriptive statistics
Cov(X,Y) > Cov(V,W) does not necessarily imply greater co-movement between X and Y
3.5%
Defensive Mean
Return
6.5%
Aggressive Mean
Return
21
Measures of Association
Correlation Coefficient
• Correlation coefficient:
ρXY = Cor(X, Y) =
Cov(X, Y)
σXσY
o The sign indicates the direction of the relationship
o The size/magnitude is -1 < ρXY < 1 and does not depend on the size/magnitude of X
and Y.
Seminar 1 - Introduction and Descriptive statistics
Cov(X,Y) = 25 > Cov(V,W) = 0.1
σX= σy= 8 σV= σW= 0.4
Cor(X,Y) = 0.391 < Cor(V,W) = 0.625
22
Two Funds Exercise
• Cov X, Y = −0.30 − 0.065 0.20 − 0.035 0.2
+ 0.10 − 0.065 0.05 − 0.035 0.5
+ 0.25 − 0.065 −0.10 − 0.035 0.3
= −0.0193
• ρXY =
σXY
σXσY
=
−0.019
0.194×0.105
= −0.9476
When the return of the aggressive fund is high, the return of the defensive fund is low;
and vice versa.
Seminar 1 - Introduction and Descriptive statistics
23
Measures of Association
Autocovariance
Cov(Xt, Xt+k) = E{ Xt − E Xt Xt+k − E Xt+k }
• The value depends on k, not t:
Cov(Xt−k, Xt) = Cov(Xt−k+1, Xt+1)
= Cov(Xt−k+2, Xt+2) = …= Cov Xt, Xt+k
• The sign indicates the direction of the relation
• The size/magnitude depends on the size/magnitude of X.
Seminar 1 - Introduction and Descriptive statistics
24
Measures of Association
Autocorrelation
ρX(k) = Cor Xt, Xt+k =
Cov(Xt,Xt+k)
σtσt+k
σt = var(Xt) and σt+k = var(Xt+k)
• The sign indicates the direction of the relationship
• -1 < ρX(k) < +1
Seminar 1 - Introduction and Descriptive statistics
Cov(Xt,Xt+1) = 25 > Cov(Yt,Yt+1) = 0.1
σX,t= σx,t+1= 8 σY,t= σY,t+1= 0.4
rX(1) = 0.391 < rY(1) = 0.625
Sample and Sample Statistics
26
Sample and Sample Statistics
• Probability is not directly observable
o Often we assume a population distribution
• Alternatively we can take a sample from the population and estimate sample statistics
o Central Tendency: the way data clusters around some central value
o Dispersion: the way observations tend to spread out around some central value
o Association: between two variables or two time points for the same variable
Seminar 1 - Introduction and Descriptive statistics
27
Measures of Central Tendency
Sample Mean
• The mean, or average, is the most commonly used measure of central tendency
ഥX =
1
N
σi=1
N Xi
o ഥX = sample mean
o Xi is the i
th observed value of X
o N = number of observations (sample size)
• Mean values are affected by the minimum or the maximum values in the sample
Seminar 1 - Introduction and Descriptive statistics
28
The IPO Example
• We don’t observe the population distribution of the monthly IPOs.
• We observe a 2-year sample.
• The average monthly IPOs:
o ഥX2015 = 1.33
o ഥX2016 = 1.83
o ഥX2015−16 = 1.58
Seminar 1 - Introduction and Descriptive statistics
Month
IPOs per
month
Jan-15 0
Feb-15 1
Mar-15 0
Apr-15 2
May-15 3
Jun-15 2
Jul-15 1
Aug-15 1
Sep-15 3
Oct-15 2
Nov-15 1
Dec-15 0
Jan-16 0
Feb-16 1
Mar-16 1
Apr-16 3
May-16 0
Jun-16 3
Jul-16 2
Aug-16 4
Sep-16 2
Oct-16 5
Nov-16 1
Dec-16 0
29
Sample vs Population
• In the IPO example, the population mean μ = 1.4 (slide 13) under the assumed
distribution
o Number of possible values M = 6
o Each value/outcome has an assumed probability.
o μ = 1.4 is a description of the unobserved population.
• The (full) sample mean ഥX = 1.58.
o Number of observations N = 24.
o Many observations have the same value.
o ഥX = 1.58 is a description of the sample from 2015-16.
Seminar 1 - Introduction and Descriptive statistics
30
Measures of Central Tendency
Sample Median
• The median is the middle value in an ordered array of numbers.
o Half of the observations will be smaller and half will be larger than the median.
• Median is the middle value not affected by the minimum or the maximum values.
Seminar 1 - Introduction and Descriptive statistics
31
Measures of Central Tendency
Calculation of the median
• To find the median first order the N observations from smallest to largest:
X(1) ≤ X(2) ≤ ··· ≤ X(N)
o if N is odd = +1
2
o if N is even =
1
2

2
+
2
+1
Seminar 1 - Introduction and Descriptive statistics
32
Median Example
• The price-earnings ratios of six listed companies are:
10.3 , 4.9 , 8.9 , 11.7 , 6.3 , 7.7
• The ordered sequence is:
4.9 , 6.3 , 7.7 , 8.9 , 10.3 , 11.7
• Positioning point =
6+1
2
= 3.5 (N = 6)
• The median is the average of the third and fourth ordered observations:
=
7.7 + 8.9
2
= 8.3
Seminar 1 - Introduction and Descriptive statistics
33
Median Exercise
• What is the median of the following data?
10 , 2 , 3 , 2 , 4 , 2 , 5
• Ordered array:
X(1) … X(7)
2 , 2 , 2 , 3 , 4 , 5 , 10
• Location of the median = (7+1)/2 = 4
• Therefore, the median equals 3.
• What is the median of 30 , 2 , 3 , 2 , 4 , 2 , 5 ?
o Is the mean the same?
Seminar 1 - Introduction and Descriptive statistics
34
Measures of Dispersion
Sample Standard Deviation
• The standard deviation measures the dispersion around the mean and is calculated as:
sX =
σi=1
N xi − തx 2
N − 1
o There is a greater contribution to s as x moves away from the mean
• Variance is the square of the standard deviation.
o Neither measure can be negative
Seminar 1 - Introduction and Descriptive statistics
35
Measures of Association
Sample Covariance and Correlation
• Sample covariance
Cov X, Y =
σi=1
N (Xi − ഥX)(Yi − ഥY)
N − 1
• Sample Correlation
Cor , =
Cov(X, Y)
sXsY
Seminar 1 - Introduction and Descriptive statistics
36
Linear Correlation
• The above definitions of covariance and correlation measure the strength of linear
relationship between two random variables.
o They do not measure non-linear relationships.
• For example:
o Let X = -5,…,0,..,5.
o Therefore X2 = 25,…,0,…25
o Cor(X,X2) = 0
o No linear correlation
Seminar 1 - Introduction and Descriptive statistics
37
Measures of Association
Sample Autocorrelation
• For a time series of X1,…,Xt,…,XT, the sample autocorrelation (AC) of order k is given by
=
1
T − 1
σt=1
T−k(Xt − ഥX)(Xt+k − ഥX)
1
T − 1
σt=1
T (Xt − ഥX)2
• The X’s are assumed to be taken from the same population, and have the same mean
and variance.
=
σt=1
T−k(Xt − ഥX)(Xt+k − ഥX)
σt=1
T (Xt − ഥX)2
Seminar 1 - Introduction and Descriptive statistics
38
Two Funds Sample
• In the fund return example (slide 16), if the observed time-series returns are:
Seminar 1 - Introduction and Descriptive statistics
Observation
Aggressive Fund
(X)
Defensive Fund
(Y)
1 -30% 20%
2 10% 5%
3 25% -10%
• What are the measures of central tendency, dispersion and association for this sample?
39
Two Funds Sample
Sample Mean and Standard Deviation
• Mean
ഥX =
−0.30 + 0.10 + 0.25
3
= 0.0167 = 1.67%
ഥY =
0.20 + 0. 05 − 0.10
3
= 0.05 = 5.00%
• Standard Deviation
=
−0.30 − 0.0167 2 + 0.10 − 0.0167 2 + 0.25 − 0.0167 2
3 − 1
= 28.43%
=
0.20 − 0.05 2 + 0.05 − 0.05 2 + −0.10 − 0.05 2
3 − 1
= 15.00%
Seminar 1 - Introduction and Descriptive statistics
40
Two Funds Sample
Sample Covariance and Correlation
• Sample covariance
Cov X, Y =
(−0.3 − 0.0167)(0.2 − 0.05) + (0.1 − 0.0167)(0.05 − 0.05) + (0.25 − 0.0167)(−0.1 − 0.05)
3 − 1
= −0.0413 < 0
• Sample correlation
Cor(X, Y) =
−0.0413
0.2843 × 0.1500
= −0.9672
• It is a reasonable estimate of the population correlation ρXY = -0.9476 (slide 22)
Seminar 1 - Introduction and Descriptive statistics
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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 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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