Homework answers / question archive / INVESTMENTS – GRA 6534 Computer assignment 2 General instructions You are not required to write any report, but you have to fill in and submit the attached form with your results

INVESTMENTS – GRA 6534 Computer assignment 2 General instructions You are not required to write any report, but you have to fill in and submit the attached form with your results

Finance

Share With

---

INVESTMENTS – GRA 6534

Computer assignment 2

General instructions

• You are not required to write any report, but you have to fill in and submit the attached form with your results. Theoretical answers and argumentations in support of quantitative results must be short, concise and to the point. You must use a 11-point font. This form is the only document that will be used to determine your score. Please submit the form as a PDF file. Fill in the tables in the template by writing all your quantitative results in red, and, likewise, write all your theoretical answers in red as well.  No other document is allowed.

 

• You also have to submit one Excel file with your computations. This Excel file will be used only to determine whether there was plagiarism. Its content will not be taken into account when grading.

 

• In this part you might find useful the links below. You are encouraged to watch these videos before starting with the assignment.

 

http://www.youtube.com/watch?v=FZyAXP4syD8 (This video explains how to perform portfolio optimization in Excel)

 

and

 

http://www.youtube.com/watch?v=ZfJW3ol2FbA (This video explains how to generate the variance-covariance matrix).

 

 

• Please note that collaboration among different groups is strictly forbidden. Any plagiarism from other groups will be punished in accordance with BI rules.

 

 

Part 1. Mean-variance portfolio optimization

 

Assume that past return moments (sample average returns, variances and covariances) over a long period of time are a good proxy of the future return moments. You are provided with monthly returns of the stock market indices of 40 developed and emerging markets, in the period that covers January 1995-December 2019. These are the MSCI Investable indices, which include only the stocks that were actually investable (illiquid stocks are removed) for investors. We approach the portfolio allocation exercise from the perspective of a US investor, hence all returns have been computed from prices converted in USD for each country. The risk-free rate is the US one-month Treasury Bill rate. As a proxy for the market portfolio of a US investor who diversifies globally, we take the MSCI All Country World index, which includes the equity markets of all the developed and emerging countries according to the MSCI classification. In the following tasks, assume that short-selling is not allowed, thus constrain portfolio weights to be greater than or equal to zero.

To start with, consider the set of assets that includes all the stock market indices from the different countries, excluding the global market portfolio MSCI AC World, which we use only as a benchmark. 

1. Find the portfolio with the maximum expected return, i.e. the right extreme of the efficient frontier. Report its average return, which we denote by ?_??? , and its standard deviation.
2. Find the global minimum variance portfolio, i.e. the left extreme of the efficient frontier. Report its average return, which we denote by ?_??? , and standard deviation.
3. Divide the interval between ?_???  and ?_??? in four sub-intervals. Report the values of the three intermediate expected returns, which we denote by ?₂,  ?₃, and  ?₄.
4. Find the portfolio with minimum variance subject to the constraint that the expected return is ?₂.

Re-run the optimization model twice, changing the constraint first from  ?₂ to ?₃, and then to  ?₄ . 

5. Find the portfolio with maximum Sharpe ratio.
6. Draw the efficient frontier of risky assets. In the same graph, plot the point with the maximum Sharpe ratio, and the point that represents the global market portfolio.
7. Which point would be chosen by a risk-neutral investor, and why? 
8. Which point would be chosen by a risk-averse investor who believes in the CAPM? Would you agree with this choice? Please explain why you answered “yes”, “no”, or “we cannot tell”.

 

 

 

Part 2. Incorporating sustainability in portfolio optimization

 

We now want to incorporate a measure of economic sustainability into our portfolio optimization model. In Part 1, you approached the portfolio selection problem by maximizing across two dimensions: expected return and risk. In this Part 2, we would like to maximize across three dimensions: expected return, risk and sustainability, taking into account resilience/preparedness to tackle the effects of climate change. 

You are provided with country-level data for the Notre Dame - Gain index (ND-Gain), which summarizes a country's economic readiness to leverage investments and convert them to adaptation actions towards climate related challenges. A higher value of ND-Gain implies that the country is more economically resilient to climate change and better prepared to embrace the climate change challenges. A lower value of ND-Gain implies that a country is less economically prepared for climate change challenges. More information can be found here: https://gain.nd.edu/our-work/country-index/. 

By incorporating a measure of economic sustainability into our portfolio selection process we aim to reward countries that are better prepared to combat climate change challenges, since investors will add them to their portfolios and hence invest in the country. In this portfolio selection approach, we aim to create a portfolio that yields a higher risk-adjusted performance, as well as invest in countries that are economically resilient to climate change (i.e. has a higher value of economic sustainability and resiliency). 

In this assignment we implement a simple way to achieve the goals of this investor, which begins with the maximum Sharpe ratio portfolio computed in Part 1. We then incorporate the sustainability rating into the portfolio selection process. 

1. Report the average sustainability rating of the portfolio that maximizes the Sharpe ratio.
2. Re-run the optimization model that maximizes the Sharpe ratio four times, but adding the constraint that the average sustainability score is, respectively, 2.5, 5, 7.5 and 10 points above the average sustainability score of the unconstrained maximum Sharpe ratio portfolio. If the model gives no solution, just write “No feasible solution” in the form with the results.
3. Re-run the optimization model that maximizes the Sharpe ratio four times, but adding the constraint that the average sustainability score is, respectively, 2.5, 5, 7.5 and 10 points below the average sustainability score of the unconstrained maximum Sharpe ratio portfolio. If the model gives no solution, just write “No feasible solution” in the form with the results.
4. Represent all these maximum Sharpe ratio points, together with the unconstrained maximum Sharpe ratio, in a graph with the Sharpe ratio on the y axis, and the sustainability scores on the x axis. Connect these points, thus creating a frontier for investors who care not only about expected returns and risk, but also about sustainability.
5. Which point will be chosen by an investor who is unaware of climate change risk, and neglects sustainability issues? Why?
6. Is the entire frontier that you depicted in point d) above the efficient frontier of risky assets for investors who care about optimizing across these three dimensions? Why?
7. Focus on the dynamics of the maximum Sharpe ratio when the investor progressively increases the sustainability score of her portfolio. Is this surprising or expected, from a portfolio optimization perspective? Reflect on one reason that should happen in real life in order for us to observe the opposite pattern in the risk-adjusted performance of the portfolio some months or years after its construction.

 

[Aside from the assignment: An ideal tool to maximize across three dimensions is to implement multiobjective portfolio optimization. This is outside the scope of this course, but if you are interested to learn more about this please do not hesitate to get in touch with me for further explanations.

Part 3. Theory

 

A friend of yours tells you that she has her money invested in the risk-free asset, but she wants to increase the riskiness of her portfolio to gain a higher expected return. Her financial advisor recommends that she invests her money in either of the following two funds. The first is an aggressive fund that invests 90% in riskier stocks and 10% in less risky stocks. The second is more conservative and invests 40% in the same risky stocks and 60% in the same less risky stocks. You run your own analysis and find that both funds are on the mean-variance efficient frontier of risky assets, with the aggressive fund being much more to the right on the frontier compared to the more conservative fund.

1. Her financial advisor recommends that your friend invests either in the conservative fund if she is more risk averse, or in the aggressive fund if she is less risk averse. Would you agree? Why? 
2. Your friend is willing to bear more risk than the one of the conservative fund, and the advisor suggests that, since both funds are on the efficient frontier of risky assets, she should create a new portfolio mixing with 50% weights the conservative and the aggressive fund to find an intermediate solution that best fits her risk profile. Would you agree? Why?
3. Suppose that your friend, like most investors, cannot borrow money to invest. With the presence of this funding constraint, what would happen to the price and expected return of the very risky fund?
4. The CAPM theory is derived under several strict model assumptions. List two of the CAPM model assumptions, and for each assumption that you list briefly outline why it may not be true in reality.