MATH2070/2970: Optimisation and Financial Mathematics Semester 2, 2023
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Assignment
MATH2070/2970: Optimisation and Financial Mathematics
Semester 2, 2023
Data provided
The le project_data.csv is a comma-separated text le containing the stock prices from 1 July 2022 to 30 June 2023 (inclusive) for 5 large companies listed on the Australian Stock Exchange (ASX):
❼ BHP BHP Group Ltd, a mining company (mostly coal, copper, iron ore and oil)
❼ CSL CSL Ltd, a biotechnology company (manufactures vaccines, antivenoms, etc)
❼ NAB National Australia Bank, a retail bank (manages savings, o ers business loans, mort- gages, credit cards, etc)
❼ TCL Transurban Group, a road operator (manages toll roads, mostly in Sydney and Mel- bourne)
❼ TLS Telstra Group Ltd, a telecommunications company (sells mobile phone plans, home/busi- ness internet, etc)
The stock prices are the end-of-day prices, adjusted to exclude the impact of corporate events (like payments of dividends) which a ect the stated share price but do not re ect trading activity.
Questions
1. For each of the 5 stocks, compute their simple returns Rt = 
, where St is the stock price on day t. Calculate the mean and covariance matrix for the returns to 5 decimal places, and con rm they are
' 0.00(l0.00)0(0)2(9)3(9)
r = ' 0.00013 ' and C = ' 0.00003 0.00003 0.00015 0.00004 0.00002 ' .
[0.(.)00058l [ −0(.).00001 0.(.)00003 0.(.)00002 0.0(.)0003 0.(.)00007 l
Use the above rounded values of r and C for the remainder of the project.
2. Find the optimal portfolio x(t) for investors with risk-aversion parameter t (i.e. your answer should be a function of t). For what value(s) of t ≥ 0 does the optimal portfolio have no short selling of any asset?
3. Consider a speci c investor with risk aversion parameter t = 0.04 who wants to invest $10,000 in these assets.
(a) What is their optimal investment allocation, mean return and risk (standard deviation) of their investment? Give all answers in $.
(b) Suppose now the investor does not want to short sell any asset. What is their new optimal portfolio, mean return and risk (standard deviation)? Again, give all answers in $.
4. Illustrate this problem graphically: in the (σ,µ) plane, show (on the same graph):
(i) The 5 assets: BHP, CSL, NAB, TCL, TLS
(ii) The optimal (unconstrained) portfolio for the investor from Question 3(a) (iii) The minimum risk portfolio
(iv) The e cient frontier and minimum variance frontier
(v) 5000 randomly generated feasible portfolios
(vi) MATH2970 only: 5000 randomly generated feasible portfolios with no short-selling
You should format the plot with appropriate labels, etc. I recommend using the axis ranges σ ∈ [0.005, 0.02] and µ ∈ [−0.001, 0.002], but you may choose other values if you prefer.
Note: MATH2070 students should show (i) (v), MATH2970 students should show all (i) (vi).
5. Suppose now that the market also has a risk-free cash asset with return 0.0001.
(a) What is the new optimal portfolio for the investor from Question 3 (with short selling allowed)? Is this a borrowing or lending portfolio?
(b) In a new plot, show in the (σ,µ) plane:
(i) The 5 assets
(ii) The optimal unconstrained portfolios for the investor with only risky assets (Ques- tion 3a) and with a risk-free asset (Question 5a)
(iii) The e cient frontier and minimum variance frontier
(iv) The capital market line
(v) The market portfolio
I recommend you use axis range σ ∈ [0, 0.02] for this plot, but you may choose other values if you prefer.
6. Take the market portfolio from Question 5b(v) as the market portfolio for this question.
(a) Compute the betas of the 5 stocks in the market.
(b) With reference to the underlying businesses, explain why the betas of BHP and TCL are so di erent. Your answer should be brief, at most 1 paragraph.
7. (MATH2970 only) There are many di erent ways to calculate the `risk' of an investment. In lectures we focused on the standard deviation of returns, σ . In practice, σ is not widely used for risk management, because it can be made larger by having an investment with the potential for very large pro ts (which most people would not call a `risk'). More common risk measures
focus only on the size/likelihood of losses.
Two widely used risk measures are:
❼ Value-at-Risk (VaR): the (100 − X)% VaR is the amount L such that such that P[Rt ≤ L] = X%. Common values are X = 1, 5 (i.e. 99% or 95% VaR).
❼ Expected shortfall (ES): the (100−X)% ES is the average size of returns that are worse than the (100 − X)% VaR. That is, the (100 − X)% ES is equal to the conditional expectation E[Rt|Rt ≤ (100 − X)%-VaR]. ✶
Generate 10,000 simulated values of Rt for the portfolio in Question 3a, and use these simulated values to estimate the 99% VaR and 95% ES of this investment. Give your VaR and ES values in $. Both values should be negative (since they represent losses).
Hint: in our model, the returns Rt of any portfolio are normally distributed.
2023-10-18