Let us revisit the portfolio problem in Topic 2 (and in the problem above), but now let us take into account your human capital. As before, you can invest your wealth into either the riskless asset (1% per year) or in three stock market indices with the following characteristics:

Asset	E(r)
US	8.0%	19%
Japan	3.7%	25%
Mexico	6.0%	15%

Correlations
Assets	US	J	M
US	1.0	0.5	0.8
J	0.5	1.0	0.2
M	0.8	0.2	1.0

Now, let us add a fifth asset, your human capital. Assume that human capital accounts for 50% of your total wealth. This is an asset that you are endowed with, but that you cannot rebalance (you cannot buy more human capital and you cannot sell your human capital).

(i) First assume that your human capital is riskless. Solve your portfolio choice problem assuming you are a mean variance investor with = 4. That is, what are the optimal weights (of financial wealth) in each stock market index and the riskless asset.

(ii) Redo part (i) assuming that your human capital is only 20% of your total wealth.

(iii) Now assume that human capital is risky. To keep things simple (at first), assume that your human capital has the same return as the US stock market.

(a) If human capital is 20% of your wealth, can you achieve the same Sharpe ratio as in part (ii). If so, how? If not, why not?

(b) If human capital is 50% of your wealth, can you achieve the same Sharpe ratio as in part (i). If so, how? If not, why not?

(iv) Optional "Challenge" Question: you are not required to answer this question to get full credit on the problem set. But, it might be fun to try :) A more realistic assumption is that your human capital is only positively correlated with the US stock market.

Suppose that the mean return to your human wealth is 9% per year, with a standard deviation of 25%. The correlation of your human capital returns with the return on US, Japan, and Mexico stock market indices is 0.3, 0.1, and 0.1, respectively. Assume that human capital accounts for 50% of your total wealth. Solve the portfolio choice problem assuming you are a mean-variance investor with = 4. That is, what are the optimal weights (of financial wealth) in each stock market index and the riskless asset.

(Hint: Use the Alternative Approach on Slide 47 of the MPT Lecture Notes. You will be optimizing over the weights in US, Japan and Mexico having fixed the weight in human capital at 0.5. Write down the first-order conditions for each of the three risky asset weights. This is a system with three linear equations and three unknowns. Solve this system and use the solution answer the question.)