What is the Gamma of a portfolio of 100 long puts and how can we Delta-Gamma hedge with the following parameterization? ୮ 1 1 = e Soo√T √√2π (So = 100, K = 110, T = 2,σ = 0.10, r = 0.05)

Mrs Z's utility function is given by the following equation: Uz(R, o²) = 0.3R – 0.50? Find the overall optimal portfolio for Mrs Z and compute its expected return and the standard deviation of its returns.

There are two assets 1 and 2, with returns X and Y correspondingly. Return X is a random variable with mean 1 and variance 1; return Y is a random variable with mean 1 and variance 3. We know that the expectation of X*Y is equal to 0. Find the share of asset 1 in the risk-minimizing portfolio.

There is one period. Assume a representative agent with utility function U(ct) = αc_t − βc^2_tassume the following: α = 100, β = 1, and δ = 0.97. Consumption at t = 0 is C0 = 24. At t = 1 one of two states θ1 and θ2 eventuate with probability π1 = 0.5, and π2 = 0.5,respectively. There are two complex securities s^1 and s^2.s^1 has a payoff of 23 in θ1 and 27 in θ2.s^2 has a payoff of 20 in θ1 and 32 in θ2.What is the stochastic discount factor mt+1? hint: Recall mt+1 =δU′(ct+1)/U′(ct)

Please solve c part and get a thumbs up

4. Suppose that there are 2 assets with ri 012 = 0.005. = 0.20, 01 = = 0.40, 2 = 0.10, 02 = 0.25 and (a) If ro = 0.02, what are the market portfolio return and variance? What are the corre- sponding weights (i.e. how much to invest in asset 1, asset 2, and the risk-free asset to get the market portfolio)? Answer. (b) If ro 0.05, what are the market portfolio return and variance? What are the corre- sponding weights? Answer.

2. Suppose that you have a riskfree asset and N risky assets for investment. The rate of return on the riskfree asset is r,, while the (Nx1) vector of the rate of return on the N risky assets is r, which is multivariate normal, i.e., r N(u, E). Your utility function for a portfolio that consists of the riskfree asset and the N risky asset is u(r,)=r,-=o, 2 Suppose that the sum of investment proportions on the riskfree and risky assets is one. Answer the following question. A. What is your optimal investment proportion in the risky assets? How is your investment on the riskfree asset affected by different values of 2? B. Suppose that there is only one risky asset i. Show the effects of the Sharpe ratio (4,/0, ) on the investment proportion in the risky asset.

3.1 Show that for any portfolio w = equal to (wi,..., wn) the Beta of the portfolio is Bw = wiB1 + + wn Bn, !! ... where B; for i = 1,...,n are Betas of individual assets.

Use the following CAPM equation for a portfolio to answer the questions that follow:E(RP) = RF + βP (RM – RF) = 1 + 0.8 (5 – 1) = 4.2% a) Is the portfolio defensive or aggressive. Why? b) If the actual portfolio return is 6%, what is the portfolio’s alpha?

The beta of the risk-free asset is: 0.5 1.0 -1.0 2.0