Estimate expected returns and variance-covariance matrix

Assignment Help Financial Econometrics

Reference no: EM131347767

Topics in Financial Economics: Theories and International Finance Assignment

Question 1: (Mean-Variance Analysis)

Note- For this question you will need to use computer software such as Excel or Matlab.

Suppose you are investor who can invest in the US stock market. In particular you can invest in four risky securities: JPMorgan Chase and Co, Exxon Mobil, Johnson & Johnson, and Microsoft. Historical data of all four risky securities is available on the class website.

a) Suppose your investment period is 3 months. Based on historical data, estimate the expected returns and variance-covariance matrix of four risky securities. Plot the mean-variance (MV) frontier.

b) In addition to investment in four risky securities, you can also lend at 1% per annum, but borrow at 5% per annum

If you wish to obtain the standard deviation of 5%, what is your optimal portfolio?
If you wish to obtain the standard deviation of 10%, what is your optimal portfolio?
If you wish to obtain the expected return of 4%, what is your optimal portfolio?
Draw the optimal frontier and explain the intuition behind. In particular, explain how would different lending and borrowing rates affect your optimal frontier?

c) Now, suppose that there is no risk free asset whatsoever so there is no borrowing or lending at any rate. In addition, suppose that market portfolio is mean-variance efficient with an expected return of 3.5%. What would be the market portfolio and identify a zero beta portfolio which is on the frontier?

d) For any portfolio on mean-variance efficient frontier, does there exists a portfolio on the frontier which has zero correlation with it? Prove your claim.

Question 2: (CAPM)

In this question we look at the properties of the minimum variance portfolio. Suppose you have the opportunity to invest in three stocks that A, B and C where the standard deviation of returns are given by: σ_A = 0.15, σ_B = 0.20, σ_C = 0.25 and σ_A,B = σ_A,C = σ_B,C = 0.

a) What is the minimum variance portfolio and what is its variance and covariance with each of the three assets?

b) Redo part a) for arbitrary standard deviations σ_A, σ_B, σ_C

c) Prove that the covariance between the minimum variance portfolio (MV) and any other portfolio equals the variance of the minimum variance portfolio. While one can rely on the specific expression for the minimum variance portfolio show this in by way of contradiction. That is, assume that there is an asset for which the above is not true and consider a portfolio that consists of x in the minimum variance portfolio and 1-x in the other portfolio.

Question 3: (OPTIONS)

We are interested in pricing a 1-year put option on cisco with a strike price of K=100. Currently CISCO is trading at 80 and the risk free rate is 5% per annum. In addition we assume that (i) CISCO will not pay any dividend and (ii) there are no other derivatives that are trading including other call or put options.

a) Suppose that we follow a Binomial model and CISCO may either appreciate by 50% or depreciate by 25%.

What should be the price of the option based on replication?
What should be the price of the option based on risk neutral pricing?

b) In this part we no longer assume a Binomial structure and future stock price can take arbitrary non-negative values. For each of the following possible prices for the put option determine whether it admits an arbitrage strategy: (i) 100 (ii) 60 (iii) 10.

In case there is an arbitrage, describe explicitly such strategy and calculate arbitrage profit today as well as payoff diagram in 1 year; in case there is not you are not required to prove it.

c) Suppose now that there is also a call option with a strike of k = 90 that costs $10. For each of the following possible prices for the put option determine whether it admits an opportunity strategy: (i) 20 (ii) 30.

Again as in part b), in case there is an arbitrage, describe explicitly such strategy and calculate arbitrage profit today as well as payoff diagram in 1 year; in case there is not you are not required to prove it.

Attachment:- Assignment Data.rar

Reference no: EM131347767

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