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Subject of Economics

Degree of MA(SocSci)/BAcc/BSc/MA

Degree Exam

Economic Analysis with MATLAB, ECON4101

Tuesday, 26 April 2022 9:15-11:15

Exam duration: 2 hours

How to complete this exam:

Answer two questions out of four. Submit the whole work, including the MATLAB code. Youcan

submit MATLAB code as additional files (the best), or you can copy-paste the code into the doc file that you will be submitting. The MATLAB code or the corresponding doc document should have brief comments on what is done by the code.

Materials Supplied:

Question 4 Data: PortfolioChoiceExam.xlsx

Instructions to students:

Read the exam student guidance onMoodle.

For this exam, the required number of questions is 2.

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Instructions.

Answer two questions out of four. Submit the whole work, including the MATLAB code. You can submit MATLAB code as additional files (the best), or you can copy-paste the code into the doc file that you will be submitting. The MATLAB code or the corresponding doc document should have brief comments on what is done by the code.

Question 1.

The elk and wolves (Non-linear systems).

Grey wolves were eliminated in Northern America in the early 1990s. The elk herd in Yel- lowstone National Park expanded rapidly, and ña§ected by climate and seasonal factors ñgrew to some 20000 animals by 1995. This number greatly exceeded the carrying capacity of vegeta- tion on the Park, that was estimated to be 10000. About 30 grey wolves were reintroduced into the Greater Yellowstone Ecosystem in 1995. We want to investigate the population dynamics of wolves and the elk.

We can write down a mathematical model describing the coexistence of elks and wolves in the same habitat. Suppose the elk is the main resource for wolves. Suppose that, in the absence of wolves, the population of elks can be described by the following dynamic equation

where coe¢ cient r is the reproduction rate of the elk, and coe¢ cient K is carrying capacity of vegetation.

(a) Find stationary points for Et (derive an analytical expression),i.e. points where Et = Et+1 for any time period t: Interpret parameter K. Assume K = 10000 and E0 = 20000 and compute how many years are needed to bring the elk herd to be 5% above the carrying capacity.

The elk is consumed by a wolf at the attack rate a when they meet, so the abundance of the elk will decrease and the above equation becomes

where Wt is the number of wolves.

The number of wolves is described by the following equation:

Wt+1 = Wt - μWt + TaEtWt:

Wolves, in the absence of the elk would go extinct with rate μ: However, the number of prays consumed a§ects the number of new wolves born, and parameter T determines the birth rate of wolves.

Assume the following parameters

r = 0:01

a = 0:0001

μ = 0:13

T = 0:15

and the following initial conditions

E0 = 20000;

W0 = 30:

(b) What is the population of the elk and wolves will be in 2025?

(d) Find stationary points for Et and Wt (derive analytical expressions and get numerical values),i.e. points where Et = Et+1 and Wt = Wt+1 for any time period t: Find numerical values for stationary E and W and comment on its consistency of these values with findings in (c).

Question 2

Zombie Apocalypse (Markov Chains).

In Zombieland people are neither born nor they die. Suppose there is a full outbreak of a zombie epidemic so that, at any one time, individuals can be in one of three states: healthy (H), bitten (B) or Zombie (Z). The probabilities of moving from one state to another in any period can be described by the following graph:

(a) Write the transition matrix for this society (ordering the states as Z, B, H such that the row probabilities sum to one.

(b) If the initial state of the people of Zombieland at time zero is given by a vector of probabilities U0 = [0; 0:1; 0:9], calculate the proportion of zombies in the population at the end of 2 periods.

Suppose that now a team of doctors has found a partial antidote for people bitten: reducing the probability of becoming a zombie, and therefore increasing the probability of a bitten person returning to a healthy state of 0.9. This is illustrated in the following graph:

(d) What is the long run distribution after the cure is discovered? Will the cure reverse the epidemic? Explain your answer.

(e) Finally suppose that the team of doctors then discovers a drug that can return a zombie to Health. If injected, the zombie becomes healthy with a positive probability. Without performing any specific calculation, explain the implications of this for the long run distribution of the people of Zombieland.

Question 3

Flu epidemic (Networks).

The following picture illustrates the standard epidemic SIR model in a networks framework. We distinguish Susceptible, Infected and Recovered individuals. The network is directed, show- ing contacts of individuals and the direction of passing the virus. When contact happens, the probability of passing the virus is one. The illness lasts one period only, after which an individual recovers with probability one. There is no immunity and any individual can be re-infected. We assume that in the initial moment individuals 7 and 8 are infected.

(a) Write down the adjacency matrix for this network.

(b) Using the adjacency matrix and appropriate calculations, find for how many periods at least one individual remains not infected. Identify these individuals. Explain your answer.

(c) Suppos the epidemic has started. If the government has an option to isolate some indi- viduals, who should be isolated first? Explain your reasoning.

Question 4

Portfolio Choice

Please download PortfolioChoiceExam.xlsx file. The file has 10 columns, and 254 rows. The Örst column contains dates, the first row contains names of companies. Columns B contains the data for a S&P500 market index. Columns C-J contain data for 8 individual assets. Each column in B2:K254 contains daily price observations over the span of 253 days. Use these data and answer the following questions. Assume that short-selling is allowed. Assume there are 252 trading days in a year.

(a) Draw the efficient frontier in coordinates standard deviation and expected return, draw position of each individual asset in the same diagram. Comment on your findings.

(b) Introduce a risk free rate, rf = 0:01 per annum. Draw the capital allocation line, compute Sharpe ratio and the standard deviation and return of the market portfolio.

(c) The Capital Asset Pricing Model predicts the relationship we should observe between the risk of an asset and its expected return. It states that the excess return on a risky asset A is related to an excess return on the market portfolio:

E (rA ) - rf = βAM (E (rM ) - rf)

where

and superscript M refers to the market portfolio.

Suppose investor cannot compute the market portfolio weights and uses equally-weighted portfolio W instead. (To build an equally-weighted portfolio you invest 1/8 th of your financial wealth into each of 8 risky assets.) For each risky asset, compute E (rA ) - rf; and E (rW ) - rf. In coordinates (β; E (r) - rf) plot 10 positions (βAW ; E (rA ) - rf), one for each risky asset in your database. Plot a straight line, passing (0,0) and which has slope E (rW ) - rf: Comment on your Öndings.