Assume an individual that has a logarithmic utility

function and a level of wealth of $$10,000.$ This

individual is faced with two gambles. The first gamble

is an $80\%$ chance of losing $$1,000$ and a $20\%$ chance of

losing $$5,000.$ The second gamble is a $\frac{50}{50}$ chance of

gaining$/$losing $$100.$ First, for each gamble calculate

the Markowitz risk premium. Second, for each gamble

calculate the Pratt$-$Arrow measure of risk premium.

Finally, explain what these numbers mean and why

they might differ.

An investor with a logarithmic utility function of wealth

and initial wealth of $$200,000.$ Assume a two state

world where the pure security prices are $$0\mathrm{.}60($for

security $1)$ and $$0\mathrm{.}70($for security $2)$ and the

corresponding state probabilities are $\mathrm{.}4$ and $\mathrm{.}6.$ What is

the optimal portfolio decision in terms of C $($current

consumption$),$ Q$1($quantity of pure security $1),$ and Q$2$

$($quantity of pure security $2).$ Is your answer feasible

for this investor? Finally, what important implication

can we make from our portfolio allocation decision?

Security $1$ has an expected return of $7\%$ and a standard

deviation of $8\%.$ Security $2$ has an expected return of $10\%$

and a standard deviation of $9\%.$ The securities have a

correlation of $-0\mathrm{.}167.$ First, for the minimum variance

portfolio, solve for the optimal weight in securities $1$ and $2.$

Second, based on these numbers, calculate the expected

portfolio return, the portfolio variance, and portfolio

standard deviation for the minimum variance portfolio.