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MTH312

Assignment Two

Q 1.

(a) List the assumptions underlying mean variance portfolio theory. [5]

Assume that you are considering selecting assets from among the following four candidates:

The unit of Return is %. Assume that there is no relationship between the amount of rainfall and the condition of the stock market.

(b) Calculate the expected return and the standard deviation of return for each asset. [4] (c) Calculate the correlation coefficient and the covariance between each pair of assets. [4]

(d) For Assets 1 and 2, find the composition, standard deviation, and expected return of the portfolio P that has minimum risk, assuming short selling is not allowed. [8]

(e) Comment on the above result. [3]

(f) Assuming that the riskless lending and borrowing rate is 5%, and short sales are     allowed, describe the influence on portfolio P. Repeat for a rate of 8%. [3]

(g) Assume that the average variance of return for an individual security is 50 and that the average covariance is 10. What is the expected variance of an equally weighted     portfolio of 5, 10, 20, 50, and 100 securities, and what is your conclusion from it? [3]

[Total 30]

Q 2.

CAPM suggests that the higher beta is for any security, the higher must be its            equilibrium return. A group of students is first exposed to the CAPM, one or more    students will find a high-beta stock that last year produced a smaller return than low- beta stocks. How are you explaining this within CAPM framework.

[Total 25]

Q 3.

Suppose X1 , X2 , … are independent identically distributed random variables. Let   m(t) = E(etXi),  S0  = 0, and  Sn  = X1  + X2  + ⋯ + Xn . For a fixed value of t, and m(t)<∞, please prove

{Mn  = [m(t)]−ne tSn}

is a martingale related to  X1 , X2 , … , Xn .

[Total 25]

Q 4.

Find the solution of dSt  = μSt dt + σSt dBt .

[Total 20]