$5.)$ Assume that only a single risky asset exists $($the Global index$),$ and a single risk$-$free rate $($the $1-$

month US risk$-$free rate$)$ at which investors can borrow and lend. Assume the risk$-$free rate is

equal to its time$-$series average and has zero variance. Also assume the utility function

provided in the previous problem, with a risk$-$aversion coefficient of $3\mathrm{.}0.$ For all parts use your

estimate of annualized arithmetic average returns $($APR$)$ and annualized variance. $(20$ points$)$

a$.$ Plot the capital allocation line $($CAL$)$ in the expected return$-$standard deviation plane.

What does moving up and down the CAL represent about the allocation between the

two assets? What is the intercept of the line? What is its slope? What is the economic

significance of the intercept and slope?

b$.$ Plot the investor$$s utility of the complete portfolio as a function of the allocation to the

risky asset. What does the curve suggest about the investor$$s approximate optimal

allocation to the risky asset?

c$.$ On the same plot as the CAL in $($a$),$ plot the indifference curve in the expected return$-$

standard deviation plane corresponding to a utility level of $0\mathrm{.}03$ for an investor with a

risk aversion coefficient of $3.$ Plot a second indifference curve corresponding to the

utility level suggested by $($b$).$ How does the second indifference curve differ from the

first? What is the relation of the second indifference curve with respect to the CAL?

d$.$ What is the exact allocation to the risky asset in the complete portfolio that maximizes

the utility of the investor?

$6.)$ Imagine that you are an investor on January $1,2004,$ using the historical data up to that date.

I.e$.,$ unless otherwise indicated only use the data prior to $2004.$ Assume that the Global index

cannot be included in your portfolio. I.e$.,$ only use the returns for North America, Japan, Asia

Pacific, and Europe. $(30$ points$)$

a$.$ Plot the minimum variance frontier for the risky assets assuming that short$-$selling is

allowed. What does the minimum variance frontier represent?

b$.$ What are the weights on the maximum Sharpe Ratio $($SR$)$ portfolio assuming short$-$

selling is allowed? If you applied these weights to your portfolio on January $1,2004,$

what would the SR of your portfolio be over the next $10$ years $($i$.$e$.,$ an out$-$of$-$sample

test from $1/1/2004$ through $12/31/2013)?$

c$.$ What are the weights on the maximum SR portfolio assuming short$-$selling is NOT

allowed? If you applied these weights to your portfolio, what would the SR of your

portfolio be over the next $10$ years?

d$.$ Comparing the performance of the two approaches in $($b$)$ and $($c$),$ which one generated

better out$-$of$-$sample performance? Explain.

e$.$ What are the weights on the minimum global variance portfolio? If you applied these

weights to your portfolio, what would the SR of your portfolio be over the next $10$

years?