3. Suppose in a market of many securities, there are two securities, A and B. A has a mean return of 7.0% and a beta to some index of 1.5 while B has a mean return of 4.0% and a beta to the same index of 0.5.  a. What is the mean return of the index? (2 marks)  b. What is the risk free rate?(2 marks)  Now suppose that, in this market, only two states are possible over the next period. Security A will record a positive return of 40% in the first state but a loss of 20% in the second state.  c. What is the probability of occurrence of each state?(2 marks)  d. What are the prices of the two Arrow-Debreu securities, (paying 1 in one state, and O in the other) associated with each state?(2 marks)  e. What is the price of a new security with payoffs of 9 in the first state and 11 in the second state?(2 marks)  f. What is the beta of the new security?(2 mark):

This question involves the application of the Capital Asset Pricing Model (CAPM) and the concept of Arrow-Debreu securities. Here's how you can solve each part:

a. The mean return of the index can be calculated using the CAPM formula, which is:

Ri = Rf + βi (Rm - Rf)

Where: Ri = Expected return on the security Rf = Risk-free rate βi = Beta of the security Rm = Expected return on the market portfolio

We have two equations here (one for each security) and two unknowns (Rm and Rf). We can solve these equations simultaneously to find the values of Rm and Rf.

b. The risk-free rate can be calculated in the same way as in part a, by solving the two equations simultaneously.

c. The probability of occurrence of each state cannot be determined from the information given in the question. We would need additional information about the expected returns in each state.

d. The prices of the Arrow-Debreu securities can be calculated using the formula:

Price = Probability of the state * Discount factor

The discount factor is usually the risk-free rate, which we calculated in part b.

e. The price of the new security can be calculated by summing up the products of the payoffs in each state and the prices of the Arrow-Debreu securities associated with each state.

f. The beta of the new security can be calculated using the formula:

Beta = Covariance(Return of the security, Return of the market) / Variance(Return of the market)

We would need additional information about the returns of the market and the new security to calculate this.