[SOLVED] FMAT3888 Project- Portfolio Optimisation with Market Data P0

In this project, you will work in groups to do portfolio optimisation based on real-world market data. You should complete the project in your group, but write up your report individually. Your group will also give an in-class presentation on your results (and everyone in your group must speak).
All submission is on Canvas.Assignment Project Exam Help
The spreadsheet (available with this project on Canvas) contains monthly returns data for 8 different assets :
1. Australian Equities (AEQ)
2. Developed Market Equities (DEQ)
3. Emerging Market Equities (EMEQ)
4. Australian Listed Property (ALP)
5. Hedge Funds (HF)
6. Australian Fixed Interest (AFI)
7. Global Fixed Interest (GOV)
8. Cash (CASH)
Let Si = (Si)t∈N be the price of asset i (for i = 1,…,8) at the end of month t. The spreadsheet contains the monthly returns
,
Asset Dynamics We will assume that each asset follows a lognormal returns distribution. That is, given the log-returns (for asset i in month t)
,
we assume that the joint returns Xt := (Xt1,…,Xt8)T ∈R8 for each month t are i.i.d. multivariate normally distributed with mean a := (a1,…,a8)T ∈R8 and covariance B = (bij)i,j=1,…,8 ∈R8×8. The log-returns are related to the (simple) returns in the spreadsheet via
Rti = eXti − 1 ⇐⇒ Xti = log(1 + Rti).
Note: since we assume Xti is normally distributed, this means that we expect Rti > −1 to always hold
(why is this useful?). If we just took Rti to be normally distributed, then we could get Rti ≤ −1 with positive probability.
Parameter Estimation
Question 2.Assignment Project Exam HelpFor n ∈N, the n-month return for asset i (from month t to t + n) is given by
.
, (1)
Question 3. Let the random vector R (respectively, R
model the joint annual (respectively, two-year) returns for the eight assets. For k = 1,2 denote
= Cov .
Use the results in Q1 and Q2 to compute/estimate for all i,j and k = 1,2 for the two time intervals (A) and (B) from Q1.
In general, if we have a normally distributed random variable X which tends to take small values
(i.e. X ≈ 0 with high probability), then the random variable Y := eX − 1 satisfies Y ≈ X (since ex − 1 ≈ x for x ≈ 0).
So even though Y is actually lognormally distributed, for the purposes of computation it would be reasonable to simplify and assume Y is normally distributed instead. We could assume the mean/variance (or covariance for multivariate distributions) of Y is the same as X, since Y ≈ X, but it would be better to directly compute E[Y ] = E[eX − 1] and Var(Y ) = Var(eX − 1) and use these instead.
Static Portfolio Optimisation
Question 4. Consider an investor who statically invests all their wealth in these eight assets for two years. Answer the following questions using both sets of parameters from Q1, namely for periods (A) and (B).
(a) Solve the utility maximisation problem:
max EhU(wT R(2))i,
s.t. ,
where w = (w1,…,w8)T is the vector of weights, and U(x) = −e−γx with γ = 1.
(b) Comment on the differences of your results corresponding to the two datasets (A) and (B).
Question 5. Under the same setup of Q4, answer the following questions for both datasets (A) and (B) from Q1.
(a) Find the efficient frontier for the market consisting of these eight assets, using the estimated
parametersAssignment Project Exam Helpµi := µ(2)i , cij := c(2)ij , ρij = ρ(2)ij for all i,j = 1,…,8.
(b) Find the portfolio with minimum variance which yields at least 10% expected return. That is, solve:
T
where w = (w1,…,w8)T is the vector of weights and C = [cij] is the covariance matrix for R(2).
(c) Comment on the differences between your results for the two datasets.
(e) Compare your results to those from Q4.
Dynamic Portfolio Optimisation
Question 6. Consider an investor who invests all their wealth in these eight assets for two years, during which they will adjust their portfolio weights at the beginning of the second year. For k = 1,2, denote
V the returns of the eight asset classes for the k-th year. Note that V 1 and V 2 are i.i.d. copies of R(1). Let w = (w1,…,w8)T (respectively u = (u1,…,u8)T ) be the portfolio weights at the beginning of the first year (respectively second year). Then the return of the profolio over the two-year investment period is given by
G(w,u) = (1 + wT V 1)(1 + uT V 2) − 1. (why?)
Suppose the investor believes that parameters estimated using the dataset (B) are valid. Answer the following questions.
(a) Solve the utility maximisation problem:
max E[U(G(w,u))],
s.t. ,
Question 7. Under the setup of Q6, answer the following questions.
(a) Solve the portfolio optimisation problem:
min Var(G(w,u)),
s.t. E[G(w,u)] ≥ 0.10,
.
(b) Compare your result with that for Q5(b).
(c) Compare your results to those from Q6.Assignment Project Exam Help
Possible Extensions
• No short selling allowed, wi ≥ 0 for all i.
• Limits on exposure to any given asset, −L ≤ wi ≤ L for some constant L.
• Alternative risk measures (other than portfolio variance) that only consider the risk of losses, such as 95% Value-at-Risk (i.e. the 5th percentile of the distribution). There are many other measures, such as Conditional Value-at-Risk and lower partial moments (e.g. semivariance).
• Change in market dynamics: suppose you are told by your CIO that the correlation between two assets is now to be different to the estimated value from Q3 because of some recent market disruption. Find a change which makes C no longer positive semidefinite. Then calculate—it’s up to you exactly how to do this—a positive semidefinite approximation to your new C, and see how this affects your portfolio optimisation in Q5.
This could be for either the static or dynamic case. I recommend that you consider at most one or two of these, but you should consider this a chance to explore questions that you find interesting.
Submission Guidelines
Report
1. Executive Summary
– A short (< 1 page) overview of the key work and conclusions in your report. This must be non-technical, and should be understood by a finance professional with limited mathematical knowledge. This means you should avoid technical mathematical terms such as “vector” and “dynamic programming”.
2. Introduction
– Background of the task, e.g. popularity of portfolio optimisaton in investing; theory of portfolio optimisation (modern portfolio theory, utility theory, stochastic processes theory).
– Summary of the work to be discussed in the report, including an explanation of the structure of the (rest of the) report.
Assignment Project Exam Help
4. Theoretical Results
5. Computational Results
– Explain your parameter estimation procedure.
– Static & dynamic portfolio optimisation: interpret your results and compare static vs. dynamic.
– Separate discussion of any extensions considered.
6. Conclusions
7. References
– Any reasonable bibliography/citation formatting is acceptable.
8. Appendix
– Provide brief pseudocode for your computational results (i.e. the main steps in your code), approx. 1–2 pages.
The report must be written as a self-contained document that tells a coherent story. You should not say things like “the answer to Q4 is…”.
Presentation
• You will have a strict 9 minutes for your presentation, with 1 minute for questions afterwards (while the next group gets ready).
• Every group member must speak during the presentation for roughly the same amount of time (i.e. you will each speak for approx. 1–2 minutes).
• Your slides must be in pdf form (e.g. by using LaTeX’s beamer package or exporting a Microsoft Powerpoint presentation to pdf using “File → Export → Create PDF”).
There is not enough time to discuss everything you did: you should pick some of the most interesting ideas and results from your project to share. Remember that everyone in the audience has been working
on this project, so there is no need to spend lots of time introducing the problem, data, etc.Assignment Project Exam Help
Grading
Your report is worth 40% of your FMAT3888 grade, and will be assessed individually. It will be graded based on:
• Mathematical correctness: mathematical reasoning is correct, efficient/elegant methods are used
• Writing: report is well-organised and easy to understand, graphs are properly formatted (axes labelled, font size not too small, etc.), spelling & grammar are correct, proper use of citations The relative weight of each part of the report is as follows:
Report Part
10%
Parameter Estimation 6 15%
Static Portfolio Optimisation 14 35%
Dynamic Portfolio Optimisation 10 25%
Quality of presentation & writing 6 15%
Total 40 100%
Presentation
Your presentation is worth 10% of your FMAT3888 grade. You will be assessed as a group (i.e. everyone in your group receives the same grade). It will be graded based on similar criteria to the report:
• Completeness: there isn’t time to show everything you did, but did you pick interesting/important results? Is your presentation detailed enough for everyone to understand what you did?
• Mathematical correctness: is your reasoning accurate, have you shown a good amount of detail (not every step in a calculation, but enough for us to understand), are your computational results relevant?
• Clarity of presentation: both slides (easy to understand, not too cluttered, figures/data well presented) and the live presentation (clear explanations, not rushed, keep to time) The relative weight of each part of the presentation is as follows:
50%
In-class presentation 5 50%
Total 10 100%