1. I) In order to maximize diversification benefits, an investor has added a security with a correlation to the existing portfolio. In your view, was the correlation closest to 0.0, -1.0, or 1.0? Why? [4 marks]

2) An analyst gathers the following data about the returns for two stocks.

Stock A Stock B

Expected Return 5% 8%

Variance 0.36% 0.81%

The covariance between returns on stocks A and B is 0.12%. Compute the correlation between the returns of Stock A and Stock B. [4 marks]

3) A stock has a beta of 0.92 and an estimated return of 11%. The risk-free rate is 3%, and the expected return on the market is 12%. According to the CAPM, is this stock overvalued, undervalued, or properly valued? [4 marks]

4) The relevant measure of risk for the capital market line (CML) and security market line (SML) differ in terms of standard deviation and covariance. Please comment on their measure of risk respectively. [3 marks]

2. Assume there is one risky asset and one risk-free asset (with a return of 20%) and two states of the world with returns 10% and 20% for the risky asset which occur with probabilities 0.4 and 0.6.

1) Find the optimal portfolio fbr an investor with the utility function U = —W² 4- 10,000W, where W is the future wealth of the investor. [7 marks]

2) Suppose his initial wealth Wo is $10,000. Compute the optimal portfolio.

[3 marks]

3) Comment on the investor's utility function on its representation of preference by analyzing its marginal utility of wealth and its convexity.

[6 marks]

4) What is his expected utility if he does not hold the risk-free asset? [4 marks]

3. Consider four securities and their mean returns, and standard deviations.

Expected return and standard deviation estimates fbr specific assets are summarized in the table below. These estimates are based on yearly discretely compounded return data over certain time period.

Table 1 financial data

1) Sketch a plot in the return-risk space fbr the efficient frontier curve, and mark the locations of the portfolios and assets mentioned in the above table. Draw the Capital Market Line on this plot. [5 marks]

2) Verify that the tangent portfolio is in the attainable set of portfolios, and comment on the risk of the global min. portfolio. [5 marks]

3) Find the efficient portfolio on the efficient frontier of the risky assets that has a mean return value of 22%, [5 marks]

4) Assuming the return data in Table 1 are based on yearly discrete compounding over certain time period, compute the 5% value-at-risk on the tangent portfolio with initial $45,000 investment fbr two years. [5 marks]

5) Please create an efficient poitfblio with expected return equal to 6%. using the tangency portfolio and another asset. [5 marks]

4. An investor might be interested in the maximum expected return to risk 15 marks

portfolio, which is called the MRR portfolio. This portfolio would be the one that maximizes the log-ratio of twice expected return to risk log (半).

Assume that the portfolio has n assets Pi, P2...., andP_n with returns R/, R2,...»

and R_n which have expectations m=3,卩2,…,“")and variance matrix C=(c〃)

(z=7, 2,..., n; j=l, 2,..., 〃). The portfolio has money-weights w = (w/,可2,...,

Wn) determining its composition, and let u = (Z, 1  /) be a n-element row

vector of ones.

I) Write the appropriate Lagrange function L in consideration of the investor's

risk preference in the set of attainable portfolios. [3 marks]

2) Differentiate the function L and make it equal to zero, and determine the

weights of the MRR portfolio. [5 marks]

3) Show that the obtained MRR portfolio is on the efficient frontier by using

the following notations:

a-MmC' m^T, b =MmC'^,ii^r, and c =MuC''u^T, where M=ac-b . [7 marks]

Consider an investment portfolio consisting of the following

The coefficient of correlation is 0.80.

portfolio is y. Derive a quadratic relationship between the change in the portfolio value and the percentage change in the underlying asset price of the option. Assuming that the change in the portfolio is normally distributed, further derive an approximate formula to compute VaR for this portfolio consisting of only options. [5 marks]

6 1) The probability mass function of Poisson distribution is

e~^AA^x

P(X = x) = ―-— where A > 0.

人•

Given a sample of 〃 measured values k_L tor i = 1, which was drawn from the Poisson population,

1) Derive the Maximum Likelihood Estimator fbr the parameter A. [5 marks]

2) Show that the point with respect to A maximizes the probability function for the Poisson population. [4 marks]

3) Claims in a portfolio have a Poisson distribution with A = 4t, where t denotes time which is expressed in days. The amount X_k paid fbr the k-th claim are independent and identically distributed random variables with mean 1,000 pounds and standard deviation 100. Find approximate probability that there are at least 10,000 pounds claimed in 10 days. [6 marks]