FINC3017: Investments and Portfolio
Management
Guanglian Hu
University of Sydney
[S2 2024]
Administrative Details
▶ Lecturer/Course coordinator:
▶ Guanglian Hu, weeks 1 to 6
▶ Pramod Kumar Yadav, weeks 7 to 12
▶ Lectures, 4 - 6pm on Thursdays, H70.B2010
▶ Tutorials start in week 2
▶ Consultation: 12-1pm on Wednesdays or by appointment
▶ Contact: guanglian.hu@sydney.edu.au, H69 534
Assessment: Assignments
▶ Two individual assignments, each accounting for 30% of the
final grade.
▶ They are due by week 8 (20 Sep) and 12 (25 Oct).
▶ You will be assessed on your technical application to
quantitative questions as well as your critical discussion of key
issues.
Assessment: Final Exam
▶ The final exam is scheduled in the final exam period, 40% of
your final grade. Closed book exam.
▶ It covers entire course, a mix of quantitative and conceptual
questions
Learning Materials
▶ Course slides
▶ 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
Overview: Weeks 1 to 6
▶ We will cover the following topics in weeks 1 to 6:
▶ Overview of Asset Classes and Financial Instruments
▶ Portfolio Theory
▶ CAPM
▶ Asset Pricing Theory
▶ The unit emphasizes quantitative methods
An Overview of Asset Classes and Financial Instruments
▶ Debt securities, equities, and derivatives
▶ marked to market, buying on margin, and short selling
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.
CAPM
▶ 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
Asset Pricing Theory
▶ Consumption-based asset pricing model
▶ Stochastic discount factor (SDF)
▶ State prices
Math Preliminaries
▶ Measuring Returns
▶ Matrix Algebra
▶ Probability and Statistics
▶ Regressions
▶ Risk Preferences
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)
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
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
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%
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?
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.
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).
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.
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
]
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
]
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 ̸= 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
]
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
]
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

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.
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
]
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).
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,
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.
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
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
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
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?
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 i.i.d,
Var (r¯ ) = Var (r )T
Regression
▶ The commonly used linear regression model is
Y = Xβ + ϵ
▶ As an example,
y1
y2
.
.
yT
 =

x11 x12
x21 x22
. .
. .
xT1 xT2

[
β1
β2
]
+

ϵ1
ϵ2
.
.
ϵT

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 = σ2ϵ I,
σ2(βˆ) = (X ′X )−1σ2ϵ
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 )] = π1U(W1) + π2U(W2)
where π1 and π2 are probabilities of respective states.
▶ 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
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.