Consider the case of three assets (A,B, and C). The corresponding expected returns are
1 0.5 0.4
0.5
1 0.5
0.4 0.5 1
μ=
0.07
0.10
0.06
and the correlation matrix is II
The return variances of the asset returns are σ = 0.04, o = 0.09, o2 = 0.04. The risk-free
rate is r = 0.02.
a) Determine the covariance matrix.
b) For a required expected return = 0.09, determine the portfolio return volatility as
a square root of a polynomial function of Ac. (For this subquestion, do not use matrix
notation)
c) For a required expected return = 0.09, determine the portfolio weights and the volatility
of the portfolio with the lowest return volatility. (For this subquestion, do not use matrix
notation)
1
d) Generalize your finding in c) for a general required expected return and use it obtain
the volatility of the overall minimum variance portfolio. (For this subquestion, do not use
matrix notation)
e) Agent 1 has the preference #pp -0.5.opp. Calculate the optimal investment in the risky
assets.
f) Determine agent I's optimal investment in the risk-free asset and determine the portfolio
weights of the tangency portfolio.
g) Provide the formula for the capital market line.
h) Determine the risk-aversion coefficient b₂ of the agent 2, who holds the tangency portfolio
exactly.
i) The price vector of the assets is p = (50, 100, 200). How many units of each asset does
investor 1 hold, if his/her total wealth is 1000 EUR.
k) In total there are 4 investors in the economy, each having 1000 EUR wealth. The net
investment in the risk-free asset is 1000 EUR. Calculate the market capitalization of each of
the three risky assets.