FINC3017: Investments and Portfolio Management
Guanglian Hu
University of Sydney
[S1 2022]
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Administrative Details
Lecturer/Course coordinator: Guanglian Hu
Lectures via Zoom, 1pm (Sydney time) on Tuesdays
face-to-face and on-line tutorials scheduled on Tuesdays, Wednesdays,
and Thursdays. Note that tutorials start in week 2
Consultation: 10am - noon on Wednesdays or by appointment
Contact: guanglian.hu@sydney.edu.au, H69 Room 543
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Assessment: Assignments
Two individual assignments, each accounting for 30% of the final
grade
They are due by week 6 and 12.
You will be assessed on your technical application to quantitative
questions as well as your critical discussion of key issues.
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Assessment: Final Exam
The final exam is scheduled in the final exam period, 40% of your
final grade
It covers entire course, a mix of quantitative and conceptual questions
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Textbook
Investments, by Bodie, Z., Kane, A. and Marcus, A.J..
You have the free access to the ebook via library.
Additional readings such as journal articles and other online materials
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Course Objective
The purpose of this course is to provide a comprehensive introduction
to investments.
Topics include modern portfolio theory, the Capital Asset Pricing
Model (CAPM), anomalies, asset pricing theory, performance
evaluation, return predictability, and volatility.
The unit emphasizes quantitative methods
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An Overview of Asset Classes and Financial Instruments
Debt securities, equities, and derivatives
marked to market, buying on margin, and short selling
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Mean-Variance Portfolio Theory
The theory seeks to find an optimal multi asset allocation
Derive and understand portfolio theory
This theory has huge impact on practice and forms the cornerstone of
a large industry that focuses on diversified investments.
Professor Harry Markowitz won the 1990 Nobel Prize in Economics
for developing the mean-variance portfolio theory.
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CAPM and Anomalies
The Capital Asset Pricing Model (CAPM) is an extension of modern
portfolio theory. It is an equilibrium outcome of everybody applying
the portfolio theory.
The CAPM has many deep implications. William Sharpe won the
1990 Nobel Prize in Economics for developing the CAPM.
Empirical tests and performance of the model
Anomalies
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Asset Pricing Theory
Consumption-based asset pricing model
Stochastic discount factor (SDF)
State prices
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Forecasting Asset Returns
Empirical evidence on stock return predictability
Present value relationships
Statistical issues with return forecasting regressions
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Trading Volatility
Volatility estimation and modeling
The development of volatility derivatives markets, with a particular
focus on VIX futures
Understand the pricing of volatility claims
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Math Preliminaries
Measuring Returns
Matrix Algebra
Probability and Statistics
Regressions
Risk Preferences
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Measuring Returns
Denote the price of an asset at date t by Pt . Ignoring the dividend,
the simple net return Rt on the asset between dates t − 1 and t is
defined as:
Rt =
Pt
Pt−1
− 1
The simple gross return on the asset is given by 1+ Rt
The asset’s gross return over the most recent k periods from date
t − k to date t, written 1+ Rt(K ), is
1+ Rt(K ) = (1+ Rt)× (1+ Rt−1)× ...× (1+ Rt−k+1)
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Example
Suppose you invest $100 into stock XYZ. In the first year, you lose
10% and in the second year you make 10%. What is the value of your
investment at the end of the second year?
A = 100
B = 99
C = 101
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Measuring Returns
The continuously compounded return or log return rt is defined as the
natural log of its gross return 1+ Rt :
rt = log(1+ Rt) = log(
Pt
Pt−1
) = pt − pt−1
where pt = log(Pt).
Continuously compounded multiperiod return is the sum of continuously
compounded single period returns
rt(K ) = log(1+ Rt(K ))
= log((1+ Rt)× (1+ Rt−1)× ...× (1+ Rt−k+1))
= log(1+ Rt) + log(1+ Rt−1) + .....+ log(1+ Rt−k+1)
= rt + rt−1 + ...+ rt−k+1
It is much easier to derive the statistical properties of additive process than
of multiplicative process
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Measuring Average Returns
Assume returns for stock XYZ over the past 4 years are 10%, 25%, -20%,
20% respectively. What is the average return of the stock?
Arithmetic Average: sum of returns in each period divide by the total
number of periods
0.1+ 0.25− 0.2+ 0.2
4 = 8.75%
Geometric Average: single per-period return that gives the same
cumulative performance as the sequence of actual returns
(1+ rG)4 = (1+ 0.1)× (1+ 0.25)× (1− 0.2)× (1+ 0.2)
⇒ rG = 7.19%
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Compounding
Suppose James graduated from college at 25 and he invested $10,000
into the S&P 500. Assuming that the S&P 500 would return 10% per
year going forward, what would this investment worth when James
retired at 65?
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Compounding
The answer is: $10,000*(1+10%)40 =$452,593
If the investment horizon is 50 years, he will get
$10,000*(1+10%)50 =$1,173,909
Each dollar you spend now is expensive in terms of the
opportunity cost.
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Matrix Algebra
A matrix is a set of elements (e.g., real numbers), organized into rows
and columns.
Information is described by data. Matrix is a nice tool to organize the
data.
Matrices are like plain numbers in many ways: they can be added,
subtracted, and, in some cases, multiplied and inverted (divided).
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Matrix Algebra
Examples
A =
[
a11 a12
a21 a22
]
b =
 b1b2
b3
 C = [ c11 c12 c13c21 c22 c23
]
Dimensions of a matrix: the number of rows by the number of
columns. A is a 2x2 matrix, b is a 3x1 matrix, and C is a 2x3 matrix.
A matrix with only 1 column or only 1 row is called a vector. b is a
column vector.
If a matrix has an equal number of rows and columns, it is called a
square matrix. Matrix A, above, is a square matrix.
In matrix A, a11 and a22 are diagonal elements and a12 and a21 are
off-diagonal elements.
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Matrix Addition and Subtraction
Matrix addition and subtraction is only defined for the matrices that
are of the same order, or, in other words, share the same
dimensionality.
Matrix addition[
a b
c d
]
+
[
e f
g h
]
=
[
a+ e b + f
c + g d + h
]
Matrix subtraction[
a b
c d
]
−
[
e f
g h
]
=
[
a− e b − f
c − g d − h
]
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Matrix Addition and Subtraction: Examples
matrix addition [
2 1
7 9
]
+
[
3 1
0 2
]
=
[
5 2
7 11
]
matrix subtraction[
2 1
7 9
]
−
[
1 0
2 3
]
=
[
1 1
5 6
]
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Matrix Multiplication
Multiplication of matrices requires a conformability condition: the column
dimension of the lead matrix A (NxT) must be equal to the row dimension
of the lag matrix B (TxK). The product of AB is a NxK matrix
When the matrices do conform, we multiply rows of the first matrix (pre
multiplier) with columns of the second matrix (post multiplier)[
a b
c d
]
×
[
e f
g h
]
=
[
ae + bg af + bh
ce + dg cf + dh
]
For matrices, AB 6= BA. For example, suppose A is 2x3 and B is 3x2, then
AB is a 2x2 matrix and BA is a 3x3 matrix.
Scalar multiplication [
a b
c d
]
×m =
[
am bm
cm dm
]
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Matrix Multiplication: Examples
Matrix multiplication 2 13 6
7 9

3x2
×
[
1 0 2
2 3 1
]
2x3
=
 4 3 515 18 12
25 27 23

3x3
Scalar multiplication [
2 4
6 1
]
× 2 =
[
4 8
12 2
]
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Transpose Matrix
The transpose of a matrix A is another matrix AT (also written as A′)
created by swapping rows and columns
Formally, the (i,j) element of AT is the (j,i) element of A. In other
words, if A is a m x n matrix, then AT is a n x m matrix
A =
 a b cd e f
g h i
 A′ =
 a d gb e h
c f i

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Transpose Matrix: Examples
Example 1
A =
[
3 8 −9
1 0 4
]
A′ =
 3 18 0
−9 4

Example 2
A =
[
2 1
1 2
]
A′ =
[
2 1
1 2
]
If A′=A, then A is called a symmetric matrix. Note that only square
matrices can be symmetric.
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Inverse of a Matrix
The inverse of a matrix A is also a matrix, written as A−1, where
AA−1 = A−1A = I. I is the identity matrix: a square matrix with all
diagonal elements equal to one and off-diagonal elements equal to zero. For
example, I(3)
A =
 1 0 00 1 0
0 0 1

In scalar algebra, a number times its inverse equals one
The inverse of a matrix is usually very difficult to compute by hand, but can
be calculated easily with computer
For a 2x2 matrix, it works as follows[
a b
c d
]−1
=
1
ad − bc
[
d −b
−c a
]
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Probability and Statistics: Random Variables
A random variable can take on values randomly. Two types of random
variables:
A discrete random variable has a countable number of possible values
A continuous random variable takes an infinite number of values
We model stock returns as random variables. For example, the gross
return on a stock might be one of the following four values:
R =
Value Probability
1.10 1/5
1.05 1/5
1.00 2/5
0.00 1/5
probabilities must sum up to one
often times we don’t know the true probabilities. We have a prior
(guess).
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Probability and Statistics: Probability Distribution
A listing of the values a random variable can take on and their
associated probabilities is a probability distribution. For example, the
distribution of the returns in the above example,
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Probability and Statistics: Normal Distribution
Of course stock returns can take on a much wider range of values. It
is common in finance to assume that stock returns are normally
distributed.
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Probability and Statistics: Normal Distribution
However, this assumption is inappropriate for financial data
Strong evidence of excess kurtosis (fat tails) for stock returns: a
higher probability of extreme observations than a normal distribution
would suggest
Negative skewness at the index level and positive skewness at
individual stock level
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Probability and Statistics: Moments
The behavior of a random variable can be characterized by its
moments
Mean: measures the central tendency
Median: middle observation, also measures the central tendency
Variance: dispersion around mean
Standard deviation: the square-root of variance
Skewness: symmetry of the distribution
Covariance and correlation: comovements between two random
variables
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Probability and Statistics: Moments of the Combinations
of Random Variables
Suppose x and y are two random variables and a and b are constants,
then the moments of linear combinations of x and y are given by:
E (ax + by) = aE (x) + bE (y);
Var (ax + by) = a2Var (x) + b2Var (y) + 2abCov(x , y);
Lastly, covariances work linearly:
Cov(ax , by) = abCov(x , y);
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Probability and Statistics: Population and Sample
Population v.s. sample: We don’t know the true probability
distribution of stock return (population); we only observe the
realizations (sample)
It is common to use sample statistics (sample mean, sample standard
deviation, etc.) to proxy for population values. But this approach can
be problematic
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Understanding the Difference Between Population Mean
and Sample Mean
Consider a coin toss game ($1 for heads, $-1 for tails). The expected
value of your payoff in the population is 0.
The sample mean can be different and varies across samples. For
example, a realized sequence of coin tosses might be H,T,T,H,H. In
that sample, the sample mean is $0.2.
How much does the sample mean vary from sample to sample?
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Understanding the Difference Between Population Mean
and Sample Mean
Suppose you observe a sample of returns for a stock (r1, r2...rt ...rT ). By
definition, the sample mean is
r¯ = 1T
T∑
1
rt
The variance of the sample mean is
Var ( 1T
T∑
1
rt) =
1
T 2Var (
T∑
1
rt) =
1
T 2
T∑
1
Var (rt) + covariance terms;
Assuming homoskedasticity (Var (r1) = Var (r2) = .... = Var (rT )) and
Cov(rt , rt+1) = 0 (is this a good assumption?),
Var (r¯ ) = Var (r )T
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Regression
The commonly used linear regression model is
Y = Xβ +
As an example,
y1
y2
.
.
yT
 =

x11 x12
x21 x22
. .
. .
xT1 xT2

[
β1
β2
]
+

1
2
.
.
T

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Regression
It is quite common in finance to use OLS estimates, but consider
more sophisticated estimates of standard error:
βˆ = (X ′X )−1X ′Y
σ2(βˆ) = (X ′X )−1X ′ΩX (X ′X )−1 (1)
where Ω takes into account various forms of autocorrelation and
heteroskedasticity in residuals.
OLS standard error is a special case of (1) with Ω = σ2 I,
σ2(βˆ) = (X ′X )−1σ2
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Risk Preferences
We capture risk preferences with the expected utility framework
The expected utility framework assumes that U(W ) (W denotes
wealth) is increasing and twice differentiable and that an investor
maximizes E [U(W )] when considering risky investments
In a two-state example,
E [U(W )] = pi1U(W1) + pi2U(W2)
where pi1 and pi2 are probabilities of respective states.
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Consider an investor with initial wealth of W0, offered a gamble that
pays +h or −h with probability 1/2. It is a fair gamble because the
expected payoff is zero.
An investor is said to be risk averse if she rejects a fair gamble
W0 or W0 + h
W0
W0 or W0 − h
1/2
1/2
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Risk Preferences
Risk aversion puts some discipline on the utility function
Investor rejects the fair gamble on the previous slide if
U(W0) >
1
2U(W0 + h) +
1
2U(W0 − h)
which implies
U(W0)−U(W0 − h) > U(W0 + h)−U(W0)
For investors to be risk averse, the utility function must satisfy the
above inequality
Equivalently, this means that U”(W ) < 0, or U(W ) is a concave
function. The marginal utility falls as wealth increases.
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