Question

In: Economics

Consider an investor with expected utility function who invests her initial wealth between a risky asset...

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?

Solutions

Expert Solution

, 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


Related Solutions

Consider an investor with expected utility function who invests her initial wealth between a risky asset...
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...
An investor invests 65% of her wealth in a risky asset with an expected rate of return of 9.53% and a variance of 3.72%, and she puts 30% in a Treasury bill that pays 2.47%.
An investor invests 65% of her wealth in a risky asset with an expected rate of return of 9.53% and a variance of 3.72%, and she puts 30% in a Treasury bill that pays 2.47%. Her complete portfolio's expected rate of return and standard deviation are ________ and ________ respectively.
Suppose you are allocating your wealth between a risky asset, which has expected return of 0.075...
Suppose you are allocating your wealth between a risky asset, which has expected return of 0.075 and standard deviation of 0.1 and a risk-free asset, which has expected return of 0.045. You want to equally divide your wealth between the risky and the risk-free asset. What would be the Sharpe Ratio on your complete portfolio? 0.2372 0.3000 0.7500 0.0949
Investor B maximizes his utility function by investing 40% of his wealth in the risk-free asset. What fraction of his wealth does investor B invest in X?
Consider an economy consisting of two stocks (X and Y) and a risk-free asset. Investor A maximizes his utility function by investing 10% of his wealth in the risk-free asset, 75% in X, and 15% in Y. Investor B maximizes his utility function by investing 40% of his wealth in the risk-free asset. What fraction of his wealth does investor B invest in X?
Consider the decision problem of investing an amount of wealth WW into a risky asset with...
Consider the decision problem of investing an amount of wealth WW into a risky asset with return R={0.1with probabilityp−0.05with probability1−pR={0.1with probabilityp−0.05with probability1−p and into a risk-less asset with risk-free interest rate r=2%r=2%. You are a risk averse investor with utility function U(WT)=10+ln(WT)U(WT)=10+ln⁡(WT) where WTWT is the amount of wealth at the end of the investment. a) Find the optimal allocation in risky and risk-less assets as a function of the probability pp and the initial amount of wealth WW invested....
You invest $98 in a risky asset and the T-bill. The risky asset has an expected...
You invest $98 in a risky asset and the T-bill. The risky asset has an expected rate of return of 18% and a standard deviation of 0.24, and a T-bill with a rate of return of 5%. A portfolio that has an expected outcome of $111 is formed by investing what dollar amount in the risky asset? Round your answer to the nearest cent (2 decimal places).
Assume that as an investor, you decide to invest part of your wealth in a risky...
Assume that as an investor, you decide to invest part of your wealth in a risky asset that has an expected return of 11%, and a standard deviation of 15%. You invest the rest of your capital in the risk-free rate, which offers a return of 3%. You want the resulting portfolio to have an expected return of 5%. What percentage of your capital should you invest in the risky asset?
What are the similarities and differences between the utility function (i.e., u(x)) in the expected utility...
What are the similarities and differences between the utility function (i.e., u(x)) in the expected utility theory and the value function (i.e., v(x)) in the reference-dependent theory (i.e., the prospect theory)?
Consider a set of risky assets that has the following expected return and standard deviation: Asset...
Consider a set of risky assets that has the following expected return and standard deviation: Asset Expected Return E(r) Standard Deviation 1 0.12 0.3 2 0.15 0.5 3 0.21 0.16 4 0.24 0.21 If your utility function is as described in the book/lecture with a coefficient of risk aversion of 4.0  , then what is the second-lowest utility you can obtain from an investment in one (and only one) of these assets? Please calculate utility using returns expressed in decimal form...
A person with initial wealth w0 > 0 and utility function U(W) = ln(W) has two...
A person with initial wealth w0 > 0 and utility function U(W) = ln(W) has two investment alternatives: A risk-free asset, which pays no interest (e.g. money), and a risky asset yielding a net return equal to r1 < 0 with probability p and equal to r2 > 0 with probability 1 (>,<,=) p in the next period. Denote the fraction of initial wealth to be invested in the risky asset by x. Find the fraction x which maximizes the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT