Question

Consider an investor with expected utility function who invests her initial wealth between a risky asset with state-dependent rate of return r and a risk-free asset with rate of return r0. The investor is strictly risk averse. Suppose that E(r) < r0, that is, negative risk premium. Show - using either a diagram or calculus - that the optimal investment in the risky asset is to short sell the asset, that is, hold a portfolio with negative fraction of wealth in risky asset and more than 100% in risk-free. Suppose that the investor is risk neutral instead of strictly risk averse. What would her optimal investment be in this case?

Answer 1

, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation.

Return

Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formulas of expected return for n assets is:

                           (X.1)

where:

	=	1.0;
n	=	the number of securities;
	=	the proportion of the funds invested in security i;
	=	the return on ith security and portfolio p; and
	=	the expectation of the variable in the parentheses.

The return computation is nothing more than finding the weighted average return of the securities included in the portfolio.

Risk

The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio