Optimization and Financial Mathematics

Due on Sunday 10th November at 11:59pm on TurnitIn
 Submit exactly two files: a pdf with your report and m file with your Matlab
code.
Report should be pleasant to read and include project formulations,
descriptions and outputs
(tables, plots, histograms etc), all answers and discussion should be there.
Marking will be based on: accuracy, programming and presentation.
 Please do not write your name on any sheet.
 The deadline is a hard deadline in the sense that in case of a late submission
(maximum
up to 10 days), you will be deducted 5% of the total marks for each day of
delay. This
is non-negotiable, so make sure you submit in time; a submission on Monday
the 11th at
12:01am will be an automatic deduction of 5%. It is your responsibility to
check that your
submission was successful.
MATH2070: Do all questions except Question 6.
MATH2970: Do all questions.

 

 

In this computer project you will be analysing real stock market data downloaded from Yahoo!Finance.
The file Data_2013_2019.csv which you can download from Ed, contains the daily closing prices of
the 30 stocks which make up the Dow Jones Industrial Average Index and closing prices of two indexes,
Dow Jones Industrial Average Index and S&P 500 Index. Prices are recorded on a (business)-daily
basis between 2/01/2013 and 30/09/2019.
There is one particularity with this time series: On 31st of Ausgust 2017 Dow and DuPont merged and
were traded as a new entity DowDuPont, then in 2019 Dow spun off of DowDuPont and was added to
the Dow Jones Industrial Average. Therefore only consider the 29 stocks without Dow (due to a short
trading history).
All prices are in US dollars.

Copyright
c 2019 The University of Sydney 1

Correlations and the covariance matrix
1. Export the data into Matlab using csvread and/or readtable. This question investigates the
correlations of the return rates of the 29 stocks. When analysing return rate data one has
several choices. A commonly used variable is the logarithmic change of price or the so called log
return rate: Let Skt be the price of k-th stock at time t, then consider Ykt = log Skt − log Sk(t−1)
(wrt the natural base).
(i) Calculate the maximal correlation between the Yk, name and plot the two stock prices
associated with the highest correlation as a function of time. On the graph present
normalised time series so that they start from the same value 100 on 2/01/2013.
(ii) Calculate the minimal correlation between the Yk, name and plot the two stock prices
associated with the smallest correlation as a function of time. On the graph present
normalised time series so that they start from the same value 100 on 2/01/2013.
(iii) Visualise the correlation matrices for two subperiods: 1/12/2014–1/09/2016 and 1/09/2016–
1/02/2018 (you may use Matlab’s command imagesc). Can you spot differences?
Plot the price of Dow Jones Industrial Average in the whole period. Can you relate it to
your observations about the correlation matrices?
(iv) Plot the histogram of the correlation coefficients ρij for the two periods from the previous
point. Comment on your result.
(v) Find the maximal risk (expressed via standard deviation) of the 29 stocks and the most
risky stock corresponding to this value. Calculate the maximal correlation between the
most risky stock and one of the remaining stocks, name and plot these two stock prices as
a function of time. On the graph present normalised time series so that they start from the
same value 100 on 2/01/2013. (Here work with the entire time series.)
(vi) Find the minimal risk (expressed via standard deviation) of the 29 stocks and the least
risky stock corresponding to this value. Calculate the maximal correlation between the
least risky stock and one of the remaining stocks, name and plot these two stock prices as
a function of time. On the graph present normalised time series so that they start from the
same value 100 on 2/01/2013. (Here work with the entire time series.)

Portfolio Theory
2. In this section we consider simple return rates, that is Rkt =

Skt−Sk(t−1)
Sk(t−1)
, where Skt is the price
of the k-th stock at time t. Carry out the following computational tasks for an unrestricted
optimal portfolio P

consisting of the 29 stocks included in the Dow Jones for an agent who

wants to invest $200,000 and has a risk aversion parameter t = 0.2.
(a) Compute the dollar investment in each of the stocks and the corresponding expected return
and risk of P

.

(b) Illustrate the problem graphically and plot on the same graph in the μσ-plane :
(i) The 29 stocks of the Dow Jones.
(ii) The minimum variance and efficient frontiers. Use a t-range |t| ≤ 0.35 for your display.
(iii) A plot of 1000 random feasible portfolios satisfying |xi

| ≤ 20 (for each of the 29 stocks)

and σi ≤ 0.05 for i = 1, . . . , 1000.
You might notice that the random points occupy some region well-separated from the
minimum variance frontier (MVF) – comment on this and explain why (This is a/the
major part of the question).
(iv) The indifference curve of an investor with t = 0.2 and their optimal portfolio P

.
3. Determine which investors shortsell in the market consisting of the 29 stocks, and which stocks
they shortsell. Are there any stocks which no-one will shortsell or which everyone will shortsell?

2

4. Three funds with different risk profiles: In this question you will divide 29 stocks with
respect to their risk profile into 3 funds. Sort stocks from highest to lowest risk (expressed via
variance or standard deviation). Assuming that each stock has the same contribution to a given
fund, form high-risk fund from the 9 most risky stocks, low-risk fund from the 10 least risky
stocks and mid-risk fund from the rest.
(a) Compute expected returns and covariance matrix of the 3 funds.
(b) Let Pˆ be an unrestricted optimal portfolio consisting of the 3 funds for an agent who wants
to invest $200,000 and has a risk aversion parameter t = 0.2.
(i) Compute the dollar investment in each of the stocks and the corresponding expected
return and risk of Pˆ.
(ii) Plot on the second μσ-plane graph :
• The 3 funds.
• The minimum variance and efficient frontiers. Use a t-range |t| ≤ 0.35 for your
display.
• The indifference curve of an investor with t = 0.2 and their optimal portfolio Pˆ.
• The minimum variance frontier and optimal portfolio P

from Question 2. Compare

solutions P

∗ and Pˆ to the two problems based on computations and graphs.

Capital Asset Pricing Model
5. Assume that the daily risk free rate in the studied period was 0.002906%. Suppose that Standard
& Poor’s 500 Index is the market portfolio (S&P 500 Index prices are included in the data file).
Make a new μσ-plane graph showing the risk free asset, market portfolio, and the Security
Market Line. Compute the β’s of all relevant assets in this project (29 stocks, 3 funds, and
two optimal portfolio P

∗ and Pˆ from Questions 2 and 4). Plot these assets on the same graph.
Identify assets with β’s greater than 1 and lower than 1. Comment on the result and describe
what Portfolio Theory would recommend an investor to do.
Log Optimal Portfolio
In this section we consider logarithmic utility maximisation problem. As in Markowitz Portfolio Theory,
there is a random vector of returns on n stocks R = [R1, R2, …, Rn]. The objective is to maximise the
expected logarithmic utility of the final wealth W1, i.e.,

E

log(W1)

= E

log(W0x
T
(1 + R))
= E

log(x
T
(1 + R))
+ log(W0)

where W0 is initial wealth. To ensure well-posedness of the problem there is no-shortselling constraint
imposed on feasible portfolio vector x, that is x = [x1, x2, …, xn] ∈ R
n
satisfies xi ≥ 0 for i = 1, …, n

and Pn
i=1 xi = 1.
Let us introduce R ̄ := 1 + R = [1 + R1, 1 + R2, …, 1 + Rn], and denote the cumulative distribution
function of R ̄ by F, i.e., F(y) := P(R ̄ ≤ y). We assume that R ̄ is a discrete random variable, and
therefore F is of the form F(y) = P
k:yk≤y
pk with pk = P(R ̄ = yk) for each yk in a countable set

{y1, y2, …}.
The problem of logarithmic utility maximisation can be written as

maximise v(x, F) := E
h
log
x
T R ̄

i

subject to x ≥ 0,
Xn
i=1
xi = 1

Note that the above expectation is computed with respect to the distribution of R ̄ which is uniquely
determined by F.

3

For a fixed F, a feasible portfolio x

that achieves the maximum of v(x, F) is called log optimal

portfolio, i.e.,

v(x

, F) = v

(F) := max
x≥0,
Pn
i=1 xi=1
v(x, F).

6. (a) Show that
(i) v(x, F) is concave in x and linear in F,
(ii) v

(F) is convex,
(iii) the set of log optimal portfolios with respect to a fixed F is convex.
(b) Prove the following theorem:
Theorem: The log optimal portfolio x

for a fixed distribution F satisfies the following

necessary and sufficient conditions:
E

R ̄
i
(x∗)
T R ̄

= 1 if x

i > 0,

E

R ̄
i
(x∗)
T R ̄

≤ 1 if x

i = 0.

Hint: Using 6(a)(i) argue that log optimal portfolio can be characterised locally by an
appropriate condition on the directional derivative of v in the direction from x

to any

other feasible portfolio x.